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 Title:
 Maximally supersymmetric theories in lightcone superspace
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 Ananth, Sudarshan
 Publication Date:
 2005
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 English
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 vi, 65 leaves : ill. ; 29 cm.
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 Algebra ( jstor )
Coordinate systems ( jstor ) Light cones ( jstor ) Physics ( jstor ) Supergravity ( jstor ) Superspaces ( jstor ) Supersymmetry ( jstor ) Symmetry ( jstor ) Vertices ( jstor ) Yang Mills theory ( jstor ) Dissertations, Academic  Physics  UF Physics thesis, Ph.D
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theses ( marcgt ) nonfiction ( marcgt )
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 Thesis (Ph. D.)University of Florida, 2005.
 Bibliography:
 Includes bibliographical references.
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 Printout.
 General Note:
 Vita.
 Statement of Responsibility:
 by Sudarshan Ananth.
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Full Text 
MAXIMALLY SUPERSYMMETRIC THEORIES IN
LIGHTCONE SUPERSPACE
By
SUDARSHAN ANANTH
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA 2005
To my parents and my sister, for all their love and support.
ACKNOWLEDGMENTS
I am deeply indebted to Prof. Pierre Ramond for five wonderful years of graduate school. I am grateful to him for the academic freedom that he has always given me, his constant interest and support, his kindness as a human being and above all for his delightful sense of humour. I will greatly miss the sense of excitement from working with him, the unpredictability of our exchanges and most of all, being across the hall from him.
I am grateful to Prof. Lars Brink for very interesting discussions on a variety of issues and for sharing with me his insight into topics presented in this thesis. I have greatly enjoyed our joint work/lunch sessions and look forward to many more.
I owe a special debt of gratitude to Prof. Guruswamy Rajasekaran for his invaluable guidance and wise counsel.
I have benefited greatly from conversations (and innumerable sets of tennis) with Professors Zongan Qiu and Sergei Shabanov.
I thank Prof. Richard Woodard for all the help he has given me during my stint at UF. I also thank the other members of my supervisory committee, Professors Pierre Sikivie and David Groisser, for their support.
I thank Prof. Jim Dufty for his interest and advice, Professors Ranga and Vasudha Narayanan for their generous hospitality, Ms. Yvonne Dixon and Ms. Darlene Latimer for being so helpful with everything administrative, and Marc Soussa for being so much fun to share an office with.
Finally, I thank Ali Nayeri, Chao leong, Amas Khan, Kyoungchul Kong, Lisa Everett, Suhas Gangadhariah and Tuan Tran for the fun food jaunts.
iii
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ....................... iii
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
CHAPTER
1 INTRODUCTION ............... ........... 1
1.1 Divergences and the LightCone ..................... . 3
1.2 Overview ... .. ... . ... . ..... . ... . .. . . . ........ 5
1.3 Dimensional Oxidation .................... ..... 6
2 THE SUPERPOINCARE ALGEBRA ... . . . ... .. . . . . . . . 8
2.1 Supersymmetry ...... ............................ 8
2.2 Supersymmetry and the LightCone ..................... 10
2.3 The Complete d = 4 SuperPoincard Algebra ............... 13
3 THE YANGMILLS SYSTEM ............. ........ 16
3.1 TenDimensional Supersymmetric YangMills ............... 16
3.2 (N = 4, d = 4) SuperYangMills in LightCone Superspace ....... 18 3.3 Oxidizing (N = 4, d = 4) SuperYangMills to (NA = 1, d = 10) YangMills 21
3.3.1 Ten Dimensions .... ............ ... ............ 21
3.3.2 The SuperPoincard Algebra in 10 Dimensions .......... 22
3.3.3 The Generalized Derivatives ................... . 26
3.3.4 Invariance of the Action .... ................. 28
4 THE SUPERGRAVITY SYSTEM .. . . . . . . . . . . . . . . . . 32
4.1 ElevenDimensional Supergravity ... ................ . 32
4.2 (KI = 8, d = 4) Supergravity in LightCone Superspace ......... 38
4.2.1 Field Content ................... .......... 38
4.2.2 A Simpler N = 8 ThreePoint Vertex . . . . . . . . . . . . . . . 41
4.3 Oxidizing (A = 8, d = 4) Supergravity to (n = 1,d = 11) Supergravity 42
4.3.1 Eleven Dimensions ........................... 42
4.3.2 The SuperPoincard Algebra in 11 Dimensions ... . ... . . 43
4.3.3 The Generalized Derivatives .................... . . . . 45
iv
4.3.4 Invariance of the Action . ... ........ ... ... .... . 47
5 DISCUSSION AND FUTURE DIRECTIONS ... . .. . . . . . . 50
5.1 Extending Gravity, to Order 2 ........ ................ . . . . 50
5.1.1 Conjectured Quartic Vertex . .................. . 50
5.1.2 Supersymmetry Variations . .......... .......... 51
5.1.3 Chiralization ................... ......... 53
5.2 Dual Descriptions ........... ......... .......... 54
APPENDIX
A YANGMILLS ...... ... ........... ..... 55
A.1 Duality Theorems .......... ....... .. ........... 55
A.2 Component Check ......... ......... ......... 56
A.3 Useful Results ...................... ........ . 58
B SUPERGRAVITY .... .. .. ................ 59
B.1 Duality Theorem ...... ...................... ..59
B.2 Useful Results ....................... ......... 60
REFERENCES . . . . . . . . . . . . . . . . . . . . . . 62
BIOGRAPHICAL SKETCH .. . . . . . .. . . . . . . . . . . . 65
v
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy MAXIMALLY SUPERSYMMETRIC THEORIES IN LIGHTCONE SUPERSPACE By
Sudarshan Ananth
August 2005
Chairman: Pierre Ramond
Major Department: Physics
Reduced supersymmetric field theories retain a great deal of information regarding their higherdimensional progenitors. In this dissertation, we describe how the (Kn = 4, d = 4) SuperYangMills theory may be "oxidized" into its parent theory, the fully tendimensional n = 1 YangMills. Remarkably, this is achieved by adding a single term to the fourdimensional transverse space derivatives. We work in lightcone superspace which is entirely free of auxiliary fields.
Very similar in structure to (n = 4, d = 4) YangMills is the maximally supersymmetric (n = 8, d = 4) Supergravity. This theory is obtained by reduction from elevendimensional A( = 1 Supergravity. We show that this fourdimensional Supergravity theory may be restored to eleven dimensions in a very similar fashion.
vi
CHAPTER 1
INTRODUCTION
The Standard Model of particle physics remains the towering achievement of the last half century. It describes all known interactions (probed by experiments), from the Hubble radius down to scales of order 1016 cm. The SU(3) x SU(2) x U(1) Standard Model describes three of the four fundamental interactions with a remarkable degree of precision. However the theory is not without its shortcomings. It fails to explain why such widely varying scales seem to occur in nature. Gravity, for example, operates at the Planck scale (1033cm) while the length scales involved in the Standard Model are around 1020 times larger. A natural question to ask is: why is the Standard Model consistent at a scale so much greater than the Planck scale?
Experimental evidence is also raising serious doubts regarding the completeness of the Standard Model in its present form. Flavor oscillations of neutrinos produced by cosmic rays offer evidence for neutrino masses (as do measurements of neutrino fluxes from the sun)the Standard Model, however, has only massless neutrinos accompanying the charged leptons in 3 decays. The Standard Model also seems arbitrary in the sense that it involves 19 parameters (plus 10 for neutrino masses) such as gauge and Yukawa couplings of varying magnitudes. Observations also indicate the presence of a dominant amount of nonluminous matter in the universe, something the Standard Model does not explain. Thus, there are several indications that we must look beyond the Standard Model [1] for an accurate description of our universe.
Gravity along with the Standard Model describes all four interactions, one of which is mediated by the spin 2 graviton and the remaining three by SU(3) x SU(2) x U(1) spin 1 gauge bosons. In addition, the theory involves the spin 0 Higgs, quarks, leptons and fifteen multiplets of spin 1 fermions (in 3 generations of 5). The union of
1
gravity with quantum mechanics leads to a nonrenormalizable quantum field theory, an indication that new physics appears at very high energy. Thus, our current understanding of Nature and its interactions (as governed by the Standard Model+gravity) cannot be viewed as a fundamental picture but merely as fragments of a bigger theory. The aim is to determine the single theoretical structure that will explain all fundamental interactions and hopefully have few (if any) undetermined parameters.
Without affecting the consistency of a theory of gravity, the best method to smooth out its divergences, involves spreading out the gravitational interaction. This is string theory: In this theory, the graviton (and other elementary particles) are onedimensional objects rather than point particles (as in quantum field theory). The benefits String Theory offers are enormous: to start with, every consistent string theory must contain a massless spin 2 particle whose interactions (at low energy) are general relativityit also leads to a consistent theory of quantum gravity (perturbatively). In addition, String Theories lead to gauge groups that are large enough to include the Standard Model in them. String theories however suffer from too many symmetries and supersymmetries, and require ten spacetime dimensions for consistency. Comparison with data requires an understanding, perhaps dynamical, of the breaking of these symmetries, for which we have no hint. The answer may lie in the elevendimensional Mtheory [2]. Although very little is known about this theory, we do know that it subsumes all the known superstring theories and may contain guideposts for supersymmetrybreaking and dimensional reduction. Its infrared limit is the much studied N = 1 Supergravity in eleven dimensions [3], believed to be ultraviolet divergent.
Although Mtheory casts welldefined shadows on lowerdimensional manifolds, its actual structure remains a mystery. Our hope therefore lies in unraveling its structure, starting from its known low energy limit, elevendimensional Supergravity. This .N = 1 theory in eleven dimensions is the largest supersymmetric local field theory
with maximal helicity two (on reduction to four dimensions). It is ultraviolet divergent and its divergences are presumably tamed by MTheory. A clearer understanding of how this occurs will offer us a window into the workings of MTheory.
1.1 Divergences and the LightCone
A technically difficult but conceptually simple framework for discussing divergences is the lightcone [4] frame formulation. Working in lightcone gauge is advantageous for two reasons. Primarily, all spurious degrees of freedom are eliminated and one deals exclusively with the physical degrees of freedom. Secondly, the role of the transverse little group [5] is made apparent. This opens up the possibility of relating the spacetime group structure to the divergences that occur in a given theory.
Such a relation (between spacetime spin and divergences) was noticed by Richard Hughes [6] who pointed out that the coefficients of the oneloop 3 function in QCD were proportional to (1)2s(1  12s2)/3 where s is the helicity of the circulating particle. This result was extended by Curtright [7] to dimensionally reduced theories. Curtright considered bosonic and fermionic contributions to the one loop vacuum polarization graph. He conjectured that these contributions (for a theory in d dimensions, obtained by reduction from D dimensions) was
1 2f.cfb 1 [ I(0) 1(2) 2(D) 1
(1) (16r2) (Ega q[ 1  qq,)  r  qqv) (1.1)
where f is 0 for bosons and 1 for fermions and the indices are those appropriate to either the boson or spinor representations of SO(D  2). r is the rank of SO(D 2) and q", the usual fourmomenta. This result is obtained, using supersymmetric dimensional regularization [8]. The p'th Dynkin index is defined by
I(P) [R]= (w.w) (1.2) w in R
4
where w are the weight vectors in representation R. It follows that I(O) is simply the dimension of the representation. In 10 dimensions for example, a gauge boson has 8 components which means I(o)boson = 8.
A perfect check of this conjecture is the KN = 4 SuperYangMills theory. This theory is obtained by reducing tendimensional YangMills where the relevant little group is SO(8). Thanks to triality,
() vector (spinor = 8
I(2vector = (2)spinor = 8,
and since fermions enter with an additional sign, finiteness of the N = 4 theory at one loop follows. In fact triality also ensures that these indices match to all orders, offering an alternate proof of finiteness for the N = 4 theory [9, 10].
Higher loop processes would then involve generalized higher order representation indices. In ten dimensions, the lack of divergences in string theories may be attributed to the triality of SO(8), the lightcone little group. Applying a similar reasoning to eleven dimensions, we see that the incomplete cancelation of the Dynkin indices of the SO(9) representations that describe N = 1 Supergravity is responsible for the divergences in that theory. Indeed, the mismatch between the eighthorder bosonic and fermionic Dynkin indices seems to support this conjecture (an eighthorder mismatch implies divergences for the KN = 8 Supergravity theory, at three loops or beyond).
The role of the little group is key to this analysis (and hence the need to work on the lightcone). Since MTheory resides in d = 11, we expect special features of elevendimensional spacetime to be reflected in its physical little group, SO(9). The first step is therefore to understand the divergent behavior of N = 1 Supergravity. This requires a lightcone gauge description of the theory and will be one of the topics presented in this thesis.
Lightcone Supergravity actions (in component form) are simply too bulky to be useful. The sheer volume of terms at order n and K2 makes it unsuitable for reading off Feynman rules (or for further calculations).
Superspace offers a compact alternative to this component approach. Superspace is an extension of ordinary spacetime, to include extra anticommuting coordinates in the form of N twocomponent Weyl spinors 0. We then define functions over this modified space, called superfields. These superfields may be expanded in a Taylor series, with respect to the anticommuting coordinates (the series is finite since the square of an anticommuting variable is zero). The coefficients obtained in this process are the usual component fields. Supersymmetry is manifest in superspace and its algebra is represented by translations and rotations, involving both commuting and anticommuting coordinates.
Lightcone superspace is the perfect venue to study supersymmetric theories. There is a complete lack of auxiliary fields, due to working on the lightcone and we will find that the actions are tractable even at higher orders in the coupling constant. In the next few chapters, we will describe the formulation of the ten and elevendimensional theories (discussed above), in lightcone superspace.
1.2 Overview
The fourdimensional N= 4 SuperYangMills was formulated in lightcone superspace by Brink, Bengtsson and Bengtsson (and independently by Mandelstam) [12]. In reference [12] the authors introduced a single superfield that captured all the degrees of freedom of the N = 4 theory. Due to the maximal supersymmetry in the theory, the superfield is extremely constrained. The action for the A = 4 theory (in terms of this superfield) was obtained by requiring SuperPoincard invariance in four dimensions.
6
This thesis assumes as a starting point their paper. We show that the above introduced KN = 4 superfield is sufficiently rich in structure to completely describe the fully tendimensional A = 1 YangMills. We achieve this by introducing a new "generalized" derivative. This involves extending the d = 4 transverse space derivatives into superspace. We prove that the derivative transforms appropriately under the SuperPoincar6 generators and then show that simply generalizing the derivatives that appear in the fourdimensional action leads to the fully tendimensional theory.
(An = 8, d = 4) Supergravity was also formulated in lightcone superspace (up to first order in the gravitational coupling) by the same authors [11, 12]. Again, since the theory is maximally supersymmetric, its superfield contains sufficient information, to describe its higherdimensional progenitor. Along similar lines to the YangMills case, we introduce the missing coordinates and derivatives and show that this "d=4" superfield may be used to completely describe the fully elevendimensional N = 1 Supergravity.
We call this process dimensional Oxidation [13,14] and explain in detail the various checks one may perform to verify its consistency. While the aim is to build lightcone actions for field theories with a view to analyzing their divergences, this oxidation process by itself is fascinating and warrants some discussion.
1.3 Dimensional Oxidation
Dimensional reduction [15] offers a mechanism to introduce new physics into a lowerdimensional universe [16]. It is, however, far more difficult (and often impossible) to start with a lowerdimensional theory and uniquely reconstruct its higherdimensional parent. Reduced supersymmetric field theories are fairly unique in this regard. They tend to retain a great deal of information regarding their once higherdimensional existence. This "memory" coded into their spectra allows for the unique recovery of their higherdimensional progenitors.
7
A perfect candidate that displays such "memory" is the reduced maximally supersymmetric (M = 4, d = 4) YangMills theory. This theory is obtained by reduction from (A = 1, d = 10) YangMills. The lightcone little group decomposition relevant to this reduction is SO(8) D SO(2) x SO(6)  SO(2) x SU(4). Signatures of ten dimensions are obvious in the d = 4 theory: the six scalar fields in the N = 4 spectrum serve as reminders of a lost (compactified) SO(6) while the SU(4) spinors assemble into a single eightspinor: 41/2 + 41/2 = 8s (of SO(8)). Also, the spectrum of the parent theory remains totally intact in the d = 4 version: for example, the 8, (in d = 10) is simply reinterpreted (in four dimensions) as, 60 + 11 + 11. Thus, the spectrum of the KN = 4 theory hints at a tendimensional formulation (governed by the little group SO(8)). In principle, one simply needs to reintroduce the six missing coordinates (and their derivatives) to recover the SO(8)invariant theory.
Another equally impressive illustration of such "memory" is N = 8 Supergravity in four dimensions. Starting from elevendimensional Supergravity, the littlegroup decomposition reads
SO(9) D SO(2) x SO(7). (1.3) This theory is reminded of its origins because
* The SO(9) structure is apparent in 81/2 + 81/2 = 16s
* The 35 + 5 scalars point to the, 7 4 of SO(7).
Again, obtaining the covariant (K = 1, d = 11) Supergravity action should involve the introduction of the missing directions. Both the A = 4 and K = 8 theories are maximally supersymmetric  this produces the mirrorsymmetry in their superfields (which one may argue, is the source of their "memory"). This "memory" is the key to our oxidation process. We will show that these theories are lifted perfectly into their once higherdimensional avatars and the entire process is consistent.
CHAPTER 2
THE SUPERPOINCARE ALGEBRA
2.1 Supersymmetry
This chapter is a quick review of some relevant results from existing literature, pertaining to supersymmetry and specifically to the lightcone gauge. We will then use these results in subsequent chapters.
The known symmetries of the Smatrix in particle physics are Poincard invariance, internal global symmetries (whose generators are Lorentz scalars) related to conserved quantum numbers like electric charge, and the discrete symmetries, C, P and T. Subject to certain assumptions, Coleman and Mandula [17] showed that these are the only possible symmetries. However, their statement assumed that the symmetry algebra of the Smatrix involved only commutators. Introducing anticommutators into the algebra leads to the possibility of supersymmetry. Supersymmetry [1820], involves the introduction of anticommuting symmetry generators which transform in the (, 0) and (0, ) representations of the Lorentz group. Since the genrators are not scalars it is not an internal symmetry. The bosonic generators are, therefore, the fourmomenta p,, the six Lorentz generators, M,,, and a certain number of Hermitian internal symmetry generators, Br. The algebra is that of the Poincar6 group (in four dimensions, we set r" = (, +, +, +) with p, v 1 ... 4)
[p, p,] = 0
[PY,, M,] = i ( 77,pp,,  71? pp) (2.1)
IMpI Mp0] = i (7 rp MpU  r/pPMp 7 r7,pMl P + 7r1p M")
8
9
together with that of the internal symmetry group(with indices r, s...) [Br, B,] = if,t Bt (2.2) The Casimir operators are
2 = (2.3) W2 = W W"
where W 2 E~~"" p, M,,. These operators commute not only with the members of the Poincar6 group, but also with the internal symmetry generators, [Br, p2] = [Br, W2] = 0 (2.4) This implies two things: First, all members of the same irreducible multiplet of the internal group have the same mass [21] and secondly that they are bound to have the same spin. In fourdimensional Weyl spinor notation, the K supersymmetry generators are denoted by Qi, i = 1 ... N. The fermionic supersymmetry generators satisfy
1
[ Qai , M,] = (ap,), Q3i
2 (2.5) [Q, My] %
We also have the key anticommutator, {Q,, Q3) = (y"),3 p, (2.6)
10
with the " defined by
(y=  (Y')pa , (2.7)
and
Tr (y,"y) = 4rf". (2.8)
2.2 Supersymmetry and the LightCone
We continue working with the metric (, +, +,..., +). As a first step to formulating our theory in lightcone gauge, we introduce the following lightcone coordinates and their derivatives.
1 1
z = ( zx0 X3) 3 (0003) ; (2.9)
1 1
x = (x1 + ix2); Z2 (01  i(2) ; (2.10)
1 1
= (xl  ix2); = (01 + i02) , (2.11)
such that
+ x = Oz = 1; Ox = 0: =+1. (2.12) This then yields the following 4product,
A"B, = AiBi  A+B_  AB+ (2.13)
In the lightcone frame, p =  i '9 is interpreted as the Hamiltonian. We will consider the generators of the SuperPoincard algebra at the lightcone time x+ = 0. Generators that are capable of shoftong away from a given x+ surface are referred to as dynamical. The rest are kinematical. We use m, n,p, q... vector indices to denote
transverse spacetime indices. These run over SO(d  2) values in ddimensions. In order to be able to translate between spinor and vector notation, we define gammamatrices by
(ym)ap = (bm)/a, (2.14) such that
Tr (ym n ) (d  2) 6mn. (2.15) This then allows us to write, Cap  Cm (yn)o . (2.16) We also define
7mn m, y] , m / n. (2.17) We now project out equation (2.6), using 2 7' 7 to obtain {Q+., Q,} = (7+) p+ (2.18)
which represents the "restricted" supersymmetry algebra. We simply pick a specific representation for the 7matrices to readoff the lightcone supersymmetry algebra, {q+ , q+01 = , v/2OP+,
f{q!, q,}  V,2 p  (2.19) {q', q0}= v62 p .
12
The q+ part of the algebra is linearly realized in superspace, characterized by a Grassmann parameter 0a and its complex conjugate 0,. These variables are defined by { 0a, } = {0, 0} = {0. ,08 = 0, (2.20) The corresponding Grassmann derivatives are written as
  ;  0 (2.21) with canonical anticommutation relations {Oa, } = ; {&o, 0 } = 6 . (2.22) Under conjugation, upper and lower spinor indices are interchanged, so that V = O, while
(Oa) = 0o; (6 ) =  0g. (2.23) Also, the order of the operators is interchanged; that is 0a0P = Op o and a 0 = 0 0,. The kinematical supersymmetry is then realized by choosing
q2 O + 0 + ; = 0p  8 +, (2.24) so that they satisfy
{q, Q+ } = iV' ~ p+ , (2.25) The dynamical supersymmetries are obtained in the next section (these are obtained using the lightcone superspace Poincar6 algebra which we need to define first). These
13
fermionic variables are also used to define chiral derivatives in superspace,
da = 0a I 00 + a= o+ , a+, (2.26) which satisfy
{ da, dp} = i v p + . (2.27)
2.3 The Complete d = 4 SuperPoincard Algebra
The entire (superspace)lightcone SuperPoincar6 algebra, splits up into kinematical and dynamical pieces. Apart from a few minor alterations, this algebra is presented in reference [12]. The kinematical generators are
* the three momenta,
p =  iO , p = iO, p = i8, (2.28)
* the transverse space rotation, j = xc8  ; + S12 , (2.29) where
S12 ( Oa 0  (d d  do d) . (2.30)
2 40 + The last term is included to ensure that these generators commute with the chiral derivatives (this is related to chiralitypreservation, when acting on superfields. We will discuss this when we introduce the superfields), [j,d"] = [j,d] = 0. (2.31)
14
* and the "plusrotations"
j+ = ixO+, j+ = i0+9. (2.32) j+ = ix 0+ 1 (a +O a) , (2.33)
2
which satisfy
[j+, d ] = d, [j+, dp] = d , (2.34) and thus preserve the chiral derivatives. Note that it is only for the choice x+= 0 that the generator j+ is kinematical, since the dynamical part is multiplied by x+ = 0. The dynamical generators are
* the lightcone Hamiltonian, S=  (2.35)
* and the dynamical boosts,
j = i  ix + i 00 + (dd d ) ,
=  ix8 + i( O8i+ (d dPdd') ) (2.36) The helicity counter added to the expressions above implies that
[j,d] d j, ] = 2 (2.37)
2 j+ d, 2 .,+
15
These generators also satisfy
[J,j+] = ij+j, [j ,j ] = ij. (2.38) In a similar fashion, the supersymmetries split into
* the already introduced kinematical supersymmetries,
q = an + 0 +; +0 = o , (2.39) satisfying
{q+ , q+} = iV 6P,3 " + , (2.40) and anticommuting with the chiral derivatives {q', d,3}= {d, q+} = 0. (2.41)
* dynamical supersymmetries, which may be obtained by boosting the kinematical ones
q0 ilq  + ,  = i[j , q ] = + . (2.42) They obey
S, i a a + , (2.43) and
{q _, Op } = i v"26 0 . (2.44)
CHAPTER 3
THE YANGMILLS SYSTEM
Having set up the relevant tools in superspace we now focus on formulating our theories in that space. Both the (KN = 1, d = 10) and (N = 4, d = 4) SuperYangMills theories are known in component form (in the lightcone gauge). However, while the fourdimensional version also has a superspace description, the d = 10 variety does not. This is the problem we resolve in this section. After a brief review of the fourdimensional theory (in superspace), we oxidize it up to ten dimensions thus obtaining a complete lightcone superspace description of (Af = 1, d = 10) YangMills.
3.1 TenDimensional Supersymmetric YangMills
A YangMills theory based on a local symmetry G is a field theory with symmetry currents coupled minimally to vectorboson fields. The space integrals of the time components of the currents formally define the generators T of the symmetry group G. The generators of the group and consequently the currents from which they are constructed transform as the adjoint representation. For the current vector boson coupling to be invariant under G the bosons must also transform as the adjoint.
Leptons and quarks which are spin 1 fermions are usually assumed to be fundamental fields in the Lagrangian. The generators Ta of the Lie group G satisfy
[Ta, Tb] = i fabc TC (3.1)
where the f* are structure constants. Mathematically, gauge fields are introduced through the covariant derivative
D, = 0, + igA,T (3.2)
16
17
from which we construct the field strengths Fm, = F,aT = i[D,, D,] (3.3) The d = 10 Yang Mills action reads S= Fo a 2 ,wa 2' DaIDA } (3.4) where the vector and spinor fields transform according to the adjoint representation of some semisimple Lie group. The spinor satisfies both the Majorana and Weyl constraints.
This action is invariant under the supersymmetry transformations
6 A, = i2 , Aa
S1 (3.5) Light cone gauge is fixed by choosing A_ = 0. The spinor breaks up as A = 2 (y+ 7 + y +)A = A+ + A (3.6) As an illustration of the LC2 approach, we focus purely on the bosonic portion of the above action. The bosonic equations of motion corresponding are ,F',Va + g fabc F'"bA,,c = 0 (3.7) LC2 indicates that all spurious degrees of freedom have been eliminated using the equations of motion. For example, the p = + component of this equation implies
18
that
+a (OA Aia) g fab {(_ Akb) Akc} (3.8)
When substituted back into the bosonic part of the action we obtain
1 1 A
+ g fb [ Aia (0+ Ayb Aj)  [ Ab Ah A (3.9)
1 A ) +(0+ A ) +g2f [fade[_ AibACAidAe _ Ab c A" Aed
This procedure is easily extended to include the fermions. The .+ may be thought of as a Green's function that satsifies
0+ G(x) = 6(x)
This determines G upto a factor h where 0+ h = 0. i.e. up to a zero mode h = h(x+) of the operator 0+ (for more details on this prescription, we refer the reader to reference [22].
3.2 (nA = 4, d = 4) SuperYangMills in LightCone Superspace
When tendimensional M = 1 YangMills is dimensionally reduced to four dimensions it yields one complex bosonic field (the gauge field), four complex Grassmann fields and six scalars. In four dimensions any massless particle can be described by a complex field and its complex conjugate of opposite helicity, the SO(2) coming from the little group decomposition
SO(8) D SO(2) x SO(6). (3.10)
19
Particles with no helicity are described by real fields. The eight vectors fields in ten dimension reduce to
8, = 60 + 11 + 11 , (3.11)
and the eight spinors to
8, = 41/2 + 41/2 . (3.12) The representations on the righthand side belong to SO(6)  SU(4) with subscripts denoting the helicity: there are six scalar fields, two vector fields, four spinor fields and their conjugates. We introduce the anticommuting Grassmann variables 0' and
{On,90 } = {9,,0a} = {0, 4} = 0, (3.13) which transform as the spinor representations of SO(6)  SU(4) O  41/2; p ~ 41/2 (3.14) where a, /, , , ... 1, 2, 3, 4, denote SU(4) spinor indices. Their derivatives are written as
0 0
S ; 0 =  (3.15) with canonical anticommutation relations {oa,p} = 60; {o,O0} = . (3.16) Having introduced these variables, we may now capture it all the physical degrees of freedom of the (N = 4, d = 4) theory in one complex superfield [12]
1 i 1 S(y) 0+ A (y) + 0' 01 Zc3 (y)+ 0 Y ,E,36 0+ A (Y)
+ Oa )C(Y) + 26a O00Y EO X (y). (3.17)
20
In this notation, the eight original gauge fields Ai , i = 1,..., 8 appear as
1 1 A = (A, +iA2) , A = (AiA2), (3.18) while the six scalar fields are written as antisymmetric SU(4) bispinors
Ca4 = (A+3 + i Aa+6) 4 (A+3  iAa+6) , (3.19) for a / 4; complex conjugation is akin to duality
1
Ca = 2 E,0 C~r . (3.20) The fermion fields are denoted by Xa and Ra. All have adjoint indices (not shown here) and are local fields in the modified lightcone coordinates y = (, , , y e) . (3.21) In this LC2 lightcone formulation all the unphysical degrees of freedom have been integrated out leaving only the physical ones. One verifies that 4 and its complex conjugate satisfy the chiral constraints d" = 0 ; d = 0 , (3.22) as well as the "insideout" constraints
1
do d3 = 2 e~my d' d , (3.23)
1
d d = 21~e'dd . (3.24)
21
The (A = 4, d = 4) YangMills action is then simply
/ d4x d40 d40 I (3.25)
where
2 4ï¿½ak 1 ld 1 C= W2 + fa c
g2 fab fade ( 1 ( 1 b  d c (3.26) (ï¿½ +) ( + ) + 0+ 0' j7,6
Grassmann integration is normalized so that fd4 01020304 = 1, and fab are the structure functions of the Lie algebra.
3.3 Oxidizing (n = 4, d = 4) SuperYangMills to (A = 1, d = 10) YangMills
Apart from some minor changes most of the material presented up to this point appears in references [11] and [12]. We now embark on the program outlined in our overview: we will show that the compact d = 4 formalism discussed so far can be generalized to entirely describe the d = 10 theory.
3.3.1 Ten Dimensions
The very compact formalism of the previous section is specific to the N = 4 theory in four dimensions. We now generalize this formalism to restore the theory to ten dimensions without altering the superfield (except for an added dependence on new coordinates). This is achieved simply by introducing generalized derivative operators.
First of, the transverse lightcone variables need to be generalized to eight. We stick to the previous notation and introduce the six extra coordinates and their
22
derivatives as antisymmetric bispinors
x4 (1 (x,+3 + iXa+6) , 04 _ 1 (0+3 + i%+6) , (3.27)
'
for a / 4, and their complex conjugates
1 3 .21
6 y zP ; d s a . (3.28) Their derivatives satisfy
O, x'' = (, a6 0, 6c 60 Y 00&a Z .2 = ( s ( t,6 6P,) , (3.29) and
0af Yb = bp21 o , = ep a . (3.30) There are no modifications to be made to the chiral superfield except for the dependence on the extra coordinates A(y) = A(x, 2, xz , ~,, y) , etc... . (3.31) These extra variables will be acted on by new operators that generate the higherdimensional symmetries.
3.3.2 The SuperPoincar6 Algebra in 10 Dimensions
We start with the construction of the SO(8) little group using the decomposition SO(8) D SO(2) x SO(6). The SO(2) generator is the same as before while the SO(6) ~ SU(4) generators are given by
1 1
JaO (=XaPOp  40CpO")  O"Oa + 013Oa +  (OPOp  9OP P)6a
2 4
23
i i
+ ( d d  dp d ) + ( dp dp dPdP ) " . (3.32)
20 0+ 8 vr 8+
The extra terms with the d and d operators are not necessary for closure of the algebra however they ensure that the generators commute with the chiral derivatives. They satisfy the commutation relations
[J, Jp] = 0, [j& JP] 6 JP3  P JG, . (3.33) The remaining SO(8) generators lie in the coset SO(8)/(SO(2) x SO(6))
JP = x OF  xP" + P  i V , + d d i 1  iJp = 2 i3 0 +O + iV V n a + d d . (3.34) All SO(8) transformations are specially constructed so as not to mix chiral and antichiral superfields
[J,"dp] = 0; [Jp,d] = 0, (3.35) and satisfy the SO(8) commutation relations
[Jja1]  6 Ia [J,3p  lpia
[jaO j]  t  +a, aJ/p  (36053,  OP 6")Ji
24
Rotations between the 1 or 2 and 4 through 9 directions induce on the chiral fields the changes
(1 1
ï¿½=( 1 wz J,(3.36) where complex conjugation is like duality
WpO W= e . (3.37) For example, a rotation in the 1  4 plane through an angle 0 corresponds to taking 0 = W14 = w23 (= w23 = W14 by reality), all other components being zero. Finally, we verify that the kinematical supersymmetries are duly rotated by these generators
[J +, q+] = 6p q'  [p; [q] = , 6)Pq+a , bPf+ . (3.38)
We now use the SO(8) generators to construct the other SuperPoincard generators
J+ = ixO a; J+ = itO+
J+ab = i 4+; J+op = i d+ +. (3.39) The dynamical boosts are now
ï¿½B + I OP, i
J = ix ix a i Oa + 4 (dPd pddp) +1  ipa 0 V I P 0 0, v OP + d d' (3.40)
4 +  0+2 9+ v/20+
25
and its conjugate
8 + I8, 8" i (
J = i  i + i zj ï¿½ (dP  dP1
+ ++
1 0P v+ /2 1
4 + ,  8o'+ j+d dI . (3.41) The others are obtained by using the SO(8)/(SO(2) x SO(6)) rotations
JO = [J, J/] ; i 0 = [J ,Ja ] (3.42) We do not show their explicit forms as they are too cumbersome. The four supersymmetries in four dimensions turn into one supersymmetry in ten dimensions. In our notation, the kinematical supersymmetries qf and 4+a, are assembled into one SO(8) spinor. The dynamical supersymmetries are obtained by boosting
i[J,] i[J,+]  a, (3.43) where
Q 1 epO
S + + 2 + q+. (3.44) They satisfy the supersymmmetry algebra Q', Up} = id * O + 4po ) , (3.45) (eo, , 35+ 4
26
and can be obtained from one another by SO(8) rotations
1
2 [JP , 0 ] Q 4Q/ , (3.46)
while
[Jpo, Q ] = 0. (3.47) Note also that
{Q, q+} _ 6o (3.48) This framework offers a simple method to introduce the central charges germane to the fourdimensional theory: the sixdimensional derivatives aP are simply replaced by cnumbers Z"O thus yielding the massive supersymmetry algebra in four dimensions with six central charges
However, relaxing this duality condition gives us all 12 central charges of the N = 4 YangMills Theory thus indicating that both varieties of central charges have a common origin in the lightcone gauge formulation. Starting with these tools we proceed to build the interacting theory in ten dimensions.
3.3.3 The Generalized Derivatives
The cubic interaction in the (KN = 4, d = 4) Lagrangian contains explicitly the derivative operators c9 and . To achieve covariance in ten dimensions these must be generalized. We propose the following operator
 + d d P" , (3.49)
4 A2 0+
27
which naturally incorporates the rest of the derivatives dP" with a as an arbitrary parameter. After some algebra we find that V is covariant under SO(8) transformations. We define its rotated partner by
Va a ,[ 1 (3.50)
where
V"aP = O  + dp 8d, eef" . (3.51)
4 V2 0+
If we apply to it the inverse transformation, it goes back to the original form
[p_ , Vaj = (6, 62  6,a"V)V , (3.52) and these operators transform under SO(8)/(SO(2) x SO(6)), and SO(2) x SO(6) as the components of an 8vector.
We introduce the conjugate operator V by requiring that
V (V , (3.53) with
V + dP d"c ,. (3.54) Define
 a V , J , (3.55) which is given by
d = d" E, (3.56)
4 v/20+
28
We then verify that
[Jap, ], = ( 6,0P 6,3 ) V (3.57)
The value of the parameter a is fixed by requiring invariance of the cubic interaction.
3.3.4 Invariance of the Action
The kinetic term is trivially made SO(8)invariant by including the six extra transverse derivatives in the d'Alembertian. The quartic interactions are obviously invariant since they do not contain any transverse derivative operators. Hence we need only consider the cubic vertex.
The whole point of this exercise is to show that covariance in ten dimensions is achieved by simply replacing the transverse derivatives 0 and 0 by V and V respectively. We thus propose the new cubic interaction term
4gf dox d4 0 d ( b Vc a b c) (3.58)
Since it is obviously invariant under SO(6) x SO(2) we need only consider the coset variations. Acting on the chiral superfield we have
6j  j, ,30 = i 0 . 0+ Oa 01, (3.59) Si 1 i 1(3.60)
sJï¿½  wpFJ= wp  j p q  iV 5pOq + dp qj (3.60 We list the conjugate relations for completeness
 = w" P = i V w"a'0+ , , (3.61)
 1 i 1 6J = Jo , = op [ 3 j  iv2 a p d' d . (3.62)
4 V2+ v/2 0+
29
The variations of the generalized derivative and its conjugate are given by
6J V = ,p [ J~, V] = 0 Vp , (3.63) 6J7 = w#" [J,, ,] = 2 V /wp dpd . (3.64) Invariance under SO(8) is checked by doing a 6j variation on the cubic vertex, including its complex conjugate.
In terms of
V + V fa fe a cb cL + , (3.65) explicit calculations yield 6J V fae OmsnI( (  1) a0'cb dï¿½ d" Sc ).66) and
6J = 6,
= fabe 0  a a c d + di 2 + v/2 0+ i a
+ 1a b da d3 ,) (3.67) Adding the two yields
6J(V + V) = ( a1)f 0
X b a c b d' d d . (3.68)
30
The cubic vertex is therefore SO(8) invariant if a = 1. This determines the generalized derivative to be
S=2+ dp dd OP". (3.69)
This result has also been checked explicitly by performing the Grassmann integrations and looking at the components (we present some of those checks in Appendix A).
To obtain this result we have used the antisymmetry of the structure functions, the chiral constraints, the "insideout" constraints, and performed integrations by parts on the coordinates and Grassmann variables. In particular, using the relation between the chiral field and its conjugate implied by the "inside out" constraints,
 1 1
2. 4! a"P"dp d d + dp 0, (3.70)
we deduce two magical identities
fbc a2 b8c = 0, (3.71)
f a b d d30 = 0. (3.72) In this lightcone form Lorentz invariance in ten dimensions is automatic once little group invariance has been established. We have therefore shown tendimensional invariance since the quartic term does not need to be changed. Thus we now have a complete lightcone superspace description of tendimensional N/ = 1 YangMills. The action is simply
SdloxJ d 4Od4 , (3.73)
31
where
4 1 1
D
1= 01 + b + ba 1l b 9c g2fabc ade( (~b ~ (d dO+ e) + d!e) . (3.74) 0+ + The O is now the tendimensional d'Alembertian and the superfields have an added dependence on the six new coordinates.
CHAPTER 4
THE SUPERGRAVITY SYSTEM
Having oxidized fourdimensional YangMills into its tendimensional parent, we now focus on the analogous situation in Supergravity. In many ways the YangMills system serves as a perfect toy model for the Supergravity system. Again, our initial focus is on the maximally supersymmetric theory in four dimensions, KN = 8 Supergravity. Starting with its lightcone superspace description we show how to obtain the fully elevendimensional theory in superspace.
A Supergravity theory requires a fermionic companion to the usual bosonic graviton (gravitational field). This fermionic counterpart is the spin 2 gravitino. The discovery of this particle in a laboratory would be a triumph for Supergravity because it is the only consistent field theory for interacting fields involving spin .
The presence of these fermion fields does not alter the classical predictions of general relativity (fermion exchange leads to a shortrange potential). However, at high energies (short distances) Supergravity is much better behaved as a quantum theory than general relativity. Infinities in the Smatrix in the first and second order quantum corrections cancel due to supersymmetry (as discussed in the Introduction). How far these cancelations persist is an open question. But this finiteness implies that Supergravity has predictive power.
4.1 ElevenDimensional Supergravity
Eleven dimensions is interesting because a maximal Supergravity theory in that dimension, on reduction to d = 4 yields particles with spin less than or equal to two. The bosonic field content of elevendimensional Supergravity consists of the elfbein, ea and a completely antisymmetric 3form potential A,, with field strength
32
33
F,vpa = A,,pa]. In terms of the SO(9) little group in eleven dimensions these correspond to a total of 128 bosonic states. The fermionic content consists of a single spin 3 Majorana field, ', which has 128 fermionic states. The N = 1 Supergravity action in eleven dimensions is [3]
1 e w+w
L = 2x eePbeaCRpacb(e, w)  297 DT(2
1 22
e F FIVP + e eA'.11 F F A
48 "P (12)4 /m ...4 ...5 8 A gIO101
+ Ke (, 'u1P,3 + 12"V P Fupoa + upo.a
The spinconnection w is determined by the variation L = 0 to be
Wpab(e) =eP ( , ebP p peb )  eb (0, eap  Opea, ) + eaP eb (a e, cp Op e, ) ec
K2 K2
+ 4 (T b pb Ta +ap b 8 A oppabT
where p, v,... run over all eleven directions, a, b,. .. are locally flat elfbein coordinates (not to be confused with our notation for the transverse directions) and the curvature is defined as
Rwvab = op Wvab  9 Wvab + Wpiac Wvcb  Wac WyCb (4.1)
y71..,n denotes the completely antisymmetric product of n y matrices and the covariant derivative is defined as D, = 0, + [a b] cal . d and P are supercovariant versions of themselves.
The elevendimensional action is invariant under local supersymmetry transformations. These transformations may be expressed in terms of an infinitesimal spacetime dependent Grassman parameter e(x) which transforms as a Majorana spinor.
34
The transformations that leave the action (4.1) invariant are
5 em T P 7
3
6A,,p = 3 '[,v Tp] (4.2)
1
9=D,(c^)e+144 "pX86/f,7(x "F x
Working in the 1.5 order formalism the aim is to formulate the theory in the lightcone gauge. This may be done in component form (as we will show for gravity) but due to sheer size is not a very useful approach. Mindful of our ultimate aim, we formulate the theory entirely in terms of the elfbein variables ea. This provides us a framework in which fermions may be introduced naturally. On the lightfront p= D, E, @ where ( = 1... 8. a, b ... are the locally flat lightcone indices, a = +, , i where i = 1. . 8 (again, with metric ( + + ...)). We parametrize the elfbeins as follows
e + = e
ee = e2 (4.3)
We define the symmetric object gij (in a metricbased approach, this is the transverse metric) by
g = ee1' = e 7Yi (4.4)
where ( is a real field and yij is a real symmetric unimodular matrix that satisfies
yij jk= 6i k
35
We work in the lightcone gauge by choosing [2328] ee+ = 0
e6k 0
For the final gauge condition, we choose 4 = 1 (.
The EinsteinHilbert part of the Supergravity action is
L= 22 eepb ec Rpcb(e,w) (4.5) and has equations of motion, R., = 0. In order to work in perturbation theory we choose
y ij= (eh )ij (4.6) where hij is a symmetric tracefree matrix. We rescale h =1 h to obtain /K2
ij = ij + K hij + 2 him hj + O (3) (4.7) The R__ = 0 equation of motion can be used to eliminate ( in favour of y = 0 + 0(2)
The Rj_ = 0 equation of motion gives
_ om2)
e = ,_m + 0 ( (4.8)
36
while the R+_ = 0 equation of motion determines ee   him + O (2) (4.9) The gravity action, L =  e e pb e"c Rpa,,b may then be written entirely in terms of the physical variables hij and up to order K reads
1 O K a
L = hi h ( hmn hn ) hmk (i0 hm ) hil
+ K hmk ( Ok  him) Oa, hil +  hmk (0 hmk ) O1 hil + (0 him ) (ak hmk ) 1 hil
K K K
+  hij ( i hki ) Oj hkl  hmk (0, hil )k him   hjl (k hij ) hik
4 2 2
The gravitino gets relatively more cumbersome to deal with in components. The gravitinodependant terms in elevendimensional Supergravity are e w+w.
L o, = 2 p y " D p ( 2 ) + D 2T 2
+ 9e (wp. P I3T + 12 I' yP a ) Fup(
Each spinor may be projected into an upper and lower component as defined below
f+y TP = 2 xP(+) ;  +11 = 2 () Lightcone gauge is chosen by setting
_ =0 (4.10)
37
We also make the additional gauge choice [29]
S IF = yi i  y +  = 0 (4.11)
which implies that 'i Ii () = 0 and thus allows us to define a physical "shifted" gravitino field (with 128 components)
i () i i ) gt() (4.12)
For convenience, we suppress the hat over the T. The equations of motion read
K K(4.13) 71tv^ Dv(W) X =  7ylPvUPa Op Fvpoa+  7 pa ga FIpa (4.13)
In principle, it is straightforward to rewrite the Gravitino action in the LC2 form (some field redefinitions are needed, even at order K to ensure that time derivatives do not appear. Instead, they are pushed into higher order terms in the Lagrangian). However, there is little benefit in doing this since the terms that appear are so numerous that they are not very useful for analytic computations. In superspace these terms shrink to an amazing degree and become tremendously tractable.
Similar arguments apply to the 3form field. However, we must indicate how we work in lightcone gauge. The equations of motion governing the 3form read
,FI" = _ L1.."8 VP F, F ...
576 1***#4 95.**#8
K0 (p )5e
96 ( )
38
We choose lightcone gauge by setting
A_ij = A+ij =0
A+k = A+k 0 (4.14)
Again, we will not work this out in LC2 although it is somewhat simpler than the gravitino.
This elevendimensional Supergravity theory on reduction to four dimensions leads to the maximally supersymmetric K = 8 Supergravity. This theory will be the focus of our next section.
4.2 (A = 8, d = 4) Supergravity in LightCone Superspace
Based on the fact that both the (N = 8, d = 4) Supergravity theory and the (n = 4, d = 4) SuperYangMills theory are maximally supersymmetric, the authors of reference [12] introduced a constrained superfield and a superspace action for the n = 8 theory. We start with this superspace action and show that it may be used to describe the (N/  1, d = 11) Supergravity theory in lightcone superspace.
4.2.1 Field Content
The physical degrees of freedom of the elevendimensional N/ = 1 Supergravity theory are classified in terms of the transverse little group SO(9) with the Graviton G(MN) transforming as a symmetric secondrank tensor, the threeform A[MNP] as an antisymmetric thirdrank tensor and the RaritaSchwinger field as a spinorvector IM (M, N,... are SO(9) indices). This theory on reduction to four dimensions leads to the maximally supersymmetric N = 8 theory.
The relevant decomposition is,
SO(9) D SO(2) x SO(7) . (4.15)
39
All told, after decomposition, the N = 8 theory has a spectrum comprised of a metric, twentyeight vector fields, seventy scalar fields, fiftysix spin onehalf fields, eight spin threehalf fields and their conjugates [29]. The S0(7) symmetry is an internal one and can in fact be upgraded to an SU(8) symmetry. However, it is important to remember that it is really the SO(7) which is relevant when we "oxidize" the theory to d = 11. SO(7) indices are represented by m, n, ...
All the physical degrees of freedom of the K = 8 theory are captured by a single complex superfield [11]
1 1 1 ï¿½h (y) i l0 . (y) + i l' + AP (y)
0aI,1 X. 0_ (Y)  O, 6 C,,p Y(y) + o "X '(y) + i 0 (6)a ( o ()  () i ~ A 3 (y) + X (( 7) +2 Xay) , (4.16)
where
1 1
0a  ..n... ,an ) al ...= . (4.17)
The ymatrices allow us to translate bispinor indices into vector indices and vice versa. For example, the complex Ac, represent
1 1
0 g A,, 8 Am  ~97m0 A nl, (4.18)
while the real Cafa may be decomposed as
Y*' O 07POPC(mp)  7YmPOCp 0 , 0 Cfmpq}" (4.19)
 8 OC 
40
These tensor fields make up the two bosonic representations under the decomposition SO(9) C SO(7) x SO(2) of elevendimensional Supergravity. The G(MN) (44 of SO(9)) split up as
44 = 27 + 7 + 7 + 1 + 1 + 1. (4.20) C(mp) represents the 27 while Am, Am represent the 7 + 7. Similarly the threeform A[MNP] (84 of SO(9)) splits into 84 = 35 + 21 + 21 + 7. (4.21) These correspond to C[mpq], A[mn, A[mn] and Cp respectively. All fields are local in the modified lightcone coordinates S= (, , X+,y  a ) . (4.22) The superfield 0 and its complex conjugate # satisfy the chiral constraints
d = 0 ; d = 0 , (4.23) and are related through the "insideout" constraints
1 1
8! 4 (4.24) where d8 is simply the product of all eight chiral derivatives. The A = 8 Supergravity action to order K is then simply d4xf d" d L ï¿½ (4.25)
41
where
= [ + +4 2 4 88#0 8+ + c.c.) (4.26)
Grassmann integration is normalized so that f d89 (0)" = 1.
4.2.2 A Simpler A = 8 ThreePoint Vertex
The threepoint vertex in this action, seems highly nonlocal and cumbersome (relative to the A = 4 vertex). However, its form can be greatly simplified leading to a single term very similar to that in the YangMills case [13]. We start by partially integrating the first term with respect to 0 to obtain J { 000 +20 +2} . (4.27)
The last 0 in the first term of equation (4.27) may be rewritten as a ï¿½, using the insideout relation. We then partially integrate the (d)8 onto the q and use the insideout relation again. The second term in equation (4.27) is partially integrated with respect to 0+ to yield two terms.
Thus the first term in the 3point vertex is now
60 1 1 1
J 0 a5+2 5+  0 . (4.28) The third term in this equation exactly cancels the second term in the original vertex. Next, we eliminate the middle term in the equation above by recognizing that
I J ï¿½ 0 ='6 J ï¿½ ï¿½0  I, (4.29)
42
(which follows from a partial integration of the single 0+ in the numerator). This allows us to set
I = 2 +2 0ï¿½ . (4.30) We thus obtain a very concise threepoint vertex
 2  f 2 0 p + c.c.. (4.31)
This simple form allows for comparison with the A = 4 SuperYangMills threepoint vertex
Sg fabc1 a 0b c + c.c.. (4.32)
We will exploit this similarity in structure between the N = 8 and N = 4 cases when tackling the fourpoint vertex.
4.3 Oxidizing (N = 8, d = 4) Supergravity to (N = 1, d = 11) Supergravity
4.3.1 Eleven Dimensions
We now proceed to show that the fully interacting (N/ = 8, d = 4) theory can be restored to its elevendimensional progenitor without altering the superfield. This enables us to formulate the (N = 1, d = 11) theory without auxiliary fields in lightcone superspace.
The first step is to generalize the two transverse variables to nine. In the YangMills case the compactified SO(6) was easily described by SU(4) parameters and we made use of the convenient bispinor notation. In the present case the compactified SO(7) has no equivalent unitary group so we simply introduce additional real coordinates xm and their derivatives 0" (where m runs from 4 through 10). The chiral
43
superfield remains unaltered except for the added dependence on the extra coordinates h(y) = h(x, 2, x',y) , etc... . (4.33) These extra variables will be acted on by new operators that will restore the higherdimensional symmetries.
4.3.2 The SuperPoincar6 Algebra in 11 Dimensions
The SuperPoincar6 algebra needs to be generalized from its fourdimensional version. The SO(2) generators stay the same and we propose generators of the coset SO(9)/(SO(2) x SO(7)) of the form
2 i
+ da(7myr) d (4.34)
Jni(  xJ ) + Z V(y ,  .(n)O303
+ d (7n)a do (4.35) which satisfy the SO(9) commutation relations
[ J, ] ijm j, jJ mmn [Jpq , Jm p q _ 6 Jqmp
[Jm, "]= i Jm"+ 6m" J, (4.36)
44
where J is the same as before, J = j and the S0(7) generators read
Jm i(sm8X  "X6,) + , (m)(B fn)#7
+ , (y )7 m (_n),3 ,7 ~ (&o ) (y, d) (4.37) The full SO(9) transverse algebra is generated by J, Jmn, Jm and J". All rotations are specially constructed to preserve chirality. For example [Jm,d] = 0; [Jf", d] = 0. (4.38) The remaining kinematical generators do not get modified J+ = j+ , J+ = j+ , (4.39) while new kinematical generators appear J+m = ixmO+; J+" = in+. (4.40) We generalize the linear part of the dynamical boosts to
88 + 1 am i
J = ix 2  ix+i 0 + (dd ! d)
0+,'q+ 40F 0+
S+ (on,,Yn) 9f3 00 2 Oa (3')a,13 + Id' )1d" d1
(4.41)
The other boosts may be obtained by using the SO(9)/(SO(2) x SO(7)) rotations
SJ [,jm]; Jjn = [, jn]. (4.42)
45
The dynamical supersymmetries are obtained by boosting
0 i 7 "
[J,q+ ] == iaj+ )p Pq+ , [ i am
[J,] g a = iF q+ + (m )a q4+ (4.43)
They satisfy
{Qq+} =(m)4m, (4.44)
and the supersymmetry algebra
{Q ,Uv} = iJz p 8 + ,p Omm. (4.45)
A few central charges fit nicely into this framework. The d = 11 theory is known to have 517 central charges (corresponding to a two form and a five form which yield 462 + 55): 7 of these may be introduced by simply replacing the seven derivatives am by cnumbers Zm. The remaining 510 charges have an elevendimensional origin.
Having constructed the free A = 1 SuperPoincard generators in eleven dimensions we turn to building the interacting theory.
4.3.3 The Generalized Derivatives
The cubic interaction in the NA = 8 Lagrangian explicitly contains the transverse derivative operators 0 and 0. To achieve covariance in eleven dimensions we proceed to generalize these operators as we did for A = 4 YangMills [13]. We propose the generalized derivative
  m a m
7 = 8 +  d, (7m)ap de +, (4.46) 16 a+
46
which naturally incorporates the coset derivatives 0m. Here a is a parameter still to be determined. We use the coset generators to produce its rotated partner V by
[, Jm V m = idm dï¿½ (ma dP (4.47)
It remains to verify that the original derivative operator is reproduced by undoing this rotation; indeed we find the required closure
[ Vm, P] = 6p V
The new derivative ( V, Vm ) thus transforms as a 9vector under the little group in eleven dimensions. We note that a is not determined by these algebraic requirements. Instead, its value will be fixed in the next section by requiring that our generalized vertex satisfy invariance requirements. We define the conjugate derivative V by requiring that
V  (V4 ) . (4.48) This tells us that
V  + 8  ( do ) d,3 (4.49) 16 +
This construction is akin to that for the A = 4 YangMills theory but this time it applies to the "oxidation" of the (N = 8, d = 4) theory to (K = 1, d = 11) Supergravity. This points to remarkable algebraic similarities between the two theories, with possibly profound physical consequences. It remains to show that the simple replacement of the transverse derivatives 4, 8 by V, V in the (N = 8, d = 4) interacting theory yields the fully covariant Lagrangian in eleven dimensions.
47
4.3.4 Invariance of the Action
While the covariant (nA = 8, d = 4) Supergravity Lagrangian is known to all orders in the gravitational coupling, its lightcone superspace expression has only been constructed to first order in n (threepoint coupling). We note that the fourpoint gravity vertex is indeed known in component form [27, 28] but a superfield expression remains elusive. We will work with the theory to cubic order in this section and in the next section propose a quartic interaction term for the theory. The kinetic term is trivially made SO(9)invariant by including the seven extra transverse derivatives in the d'Alembertian.
We start from our simplified version (4.31) of the N = 8 threepoint vertex. We propose that the elevendimensional vertex is of the same form but with the transverse derivatives replaced by the generalized derivatives introduced in the previous section:
S= 2KS dnxJ d"Od86 I VCVï¿½, (4.50)
together with its complex conjugate. To show SO(9) invariance it suffices to consider the variations
i
i v/2 (4.52) + 2 V2 0+ d( m )od}
6jm V =  m Vm , (4.53)
48
where wm are the parameters of the SO(9)/(SO(7) x SO(2)) coset transformations. The SO(2) invariance is clear from the work in d = 4 and the SO(7) invariance is covariantly realized so if we can show the invariance under the SO(9)/(SO(7) x SO(2)) transformations we have shown invariance under the full SO(9). The Variation
We split the calculation into three parts based on which superfield is being varied. Terms that involve 0 0 or 0m 0" cancel trivially. The remaining terms all involve a single SO(2) derivative and a single 0". We list the contributions from the variations below
Contribution from f 6
4
5 3 (,Y) ' (") if X 0 a+ (4.54) +a
1
Contribution from f ~ ( V
I 2 0ï¿½0m (4.55)
Contribution from f 0 6 ) v
ie 4_ 12/~ { 0 m 2 m) + , (4.56) These results are further simplified by use of the magical identity
1 1
0 3 ï¿½ ï¿½ 5+ , (4.57)
49
which follows from the duality constraint and numerous partial integrations. The final form of the variation then reads
5jV cc +i ,+ ï¿½, (4.58)
Elevendimensional Poincard invariance requires that this variation vanish. This determines a and hence the generalized derivative
1 oam
8V =  d ( 7m ) d + (4.59)
This completes the proof of SO(9) invariance for the threepoint function. It is rather remarkable that this simple replacement of the transverse derivatives renders the action covariant in eleven dimensions. In this lightcone form the Lorentz invariance in eleven dimensions is automatic once the little group invariance has been established. We have therefore proven elevendimensional invariance to order K.
CHAPTER 5
DISCUSSION AND FUTURE DIRECTIONS The fact that these reduced fourdimensional theories remember their parent versions is remarkable. It will be interesting to see if such "memory" extends to nonmaximally supersymmetric theories as well. Although mathematically well defined the physical interpretation of the generalized derivative is unclear. The fact that it oxidizes a d = 4 gauge theory into a d = 10 version indicates that it relaxes the boundary conditions imposed on open strings (by thinking of the system as a D3brane to begin with). Whenever there is a covariant derivative there is curvature at play and this may offer an interesting way of looking at our procedure. The immediate goal however revolves around determining the Supergravity action to order K2
5.1 Extending Gravity, to Order ,2
We now have a superspace description of elevendimensional Supergravity to order K. The obvious next step is to extend the oxidation procedure to order K2. However, the (N = 8, d = 4) fourpoint interaction (in lightcone superspace) is still unknown. In this section we outline a general procedure to derive the fourdimensional quartic interaction. Once this vertex is determined we expect the oxidation procedure to follow along very similar lines (thus yielding the ( = 1, d = 11) action to order K2).
5.1.1 Conjectured Quartic Vertex
The similarity between the threepoint functions of the N = 4 YangMills and KN = 8 Supergravity theories is quite suggestive. Based on this comparison we con50
51
jecture that the (N = 8, d = 4) fourpoint vertex is simply
S= 420 + +2(O+2) (5.1)
where 3 remains to be fixed. A direct check of this result can be achieved by comparison with the component form of lightcone gravity. As mentioned in the previous section the fourpoint function for gravity in lightcone gauge is known but there exists no expression for the quartic (or higher) vertices in terms of (LC2)superfields in the literature. In component form it is well known that the lightcone time derivative 0+, makes an appearance at every order and needs to be eliminated via fieldredefinitions. This redefinition needs to be understood from a superspace point of view. We hope to return to these issues in a future publication [30]. Assuming this fourpoint vertex (in four dimensions) its "oxidized" elevendimensional version is conjectured to be
V = K2 ï¿½ï¿½vov + 0o +0 ) (5.2)
It remains to be seen if this simple form reproduces the full quartic vertex of elevendimensional Supergravity.
5.1.2 Supersymmetry Variations
A possible way to determine the fourpoint vertex is to start from the known twoand threepoint vertices and require invariance of the action (to order n2) under the nonlinear dynamical SuperPoincar6 transformations. Simplest among these are the dynamical supersymmetry generators already derived up to first order in the coupling constant for the (N = 8, d = 4) theory in references [12, 31].
_2 { {0+2ï¿½ +  .} (5.3)
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This variation, clearly preserves the chirality of the superfield it acts on. Note that we can replace the  in the above expression by d or q+. This is possible because the additional 0+ piece cancels between the two terms in the variation. To determine the fourpoint function, we need the dynamical supersymmetry variation at order K2. We do not yet know its exact form but have narrowed it down, based on the following requirements:
* Helicity = + 2 (based on the variations at order 0 and s)
1
* Dimensions = [L]2 (again, based on the earlier variations)
* The chirality of the superfield it acts on must not be affected
* Finally, it must preserve the "insideout" superfield constraints.
The requirement that it leave chirality invariant is satisfied through "Chiralization". The first three constraints offer an ansatz for the variation
6_ oc K2 1 +a 0+b 0+c + . (5.4)
with the restriction, a + b + c = 3 + n (the ... signify that more terms need to be added to ensure chirality). Once this expression is known the fourpoint interaction is easy to determine. In a future publication [30] we will derive the exact form of 6,_ " and prove in addition closure of the supersymmetry algebra with the SuperPoincard generators.
The lowest order dynamical supersymmetry variations for the (N/ = 1, d = 11) theory were detailed in equation (4.43). We conjecture that at first order in coupling the variations are obtained by simply oxidizing the fourdimensional result above. That
53
is,
1 a I bg_' = 2/ + V8682  V (5.5)
This result needs to be checked but we believe it to be correct since it serves as a bridge between the two and threepoint vertices.
5.1.3 Chiralization
"Chiralization" is a descent procedure whereby nonchiral expressions are rendered chiral. For any general nonchiral expression of the form A B (where A is any compound chiral function and B a compound antichiral function) we define a "chiral product" through a descent relation in chiral derivatives
C(AB) = A + ... a (A da a B), (5.6) ,= n! (iv/2+)
where d. ... n, = d, ... da, and dda"  ... dc'.
C (AB) is now a chiral function and satisfies dC = 0. Clearly the descent series involves as many terms as there are supersymmetries in the theory. This procedure is equally applicable to other nonchiral forms. For example the product dAB where both A and B are chiral functions is chiralized by the addition of the term
d
 0+ A d B (5.7)
Similarly the addition of the two terms
 0+ dA B + 0+2A B (5.8) + 0+2
54
to the expression d dA B, chiralizes it and so on.
This procedure is invaluable when dealing with variations (with respect to supersymmetry and the boosts) at higher orders. This simple recipe ensures that all variations respect the superfield chirality structure.
5.2 Dual Descriptions
Despite our focus on formulating supersymmetric theories in lightcone superspace, the ultimate goal (as detailed in the introduction) is to understand MTheory. This thesis studies one possible approach to understanding MTheory through its lowenergy limit elevendimensional Supergravity. There are other venues that also look promising.
MTheory casts well defined shadows on lower dimensions. From the AdS/CFT [32] point of view conformally invariant gauge theories are of great interest (the rank of the gauge group is related to flux in the dual description). The KN = 4 SuperYangMills theory is dual to Type IIB String Theory compactified on AdS5 x S5. Similarly MTheory when compactified on AdS4 x S7 is described in terms of a threedimensional superconformal theory (governed by the symmetry group OSp(814)). This theory has not yet been constructed and will be a very interesting thing to tackle in this lightcone superspace framework. The OSp(8 4) is trivially realized on the superfields [33] using our formalism and this is a good indication that we may be able to write down a Lagrangian for this theory. A better understanding of these dual theories will certainly teach us something about the structure of MTheory.
APPENDIX A
YANGMILLS
A.1 Duality Theorems
Consequence #1;
fabc b ~ 1b = 0 (A.1) Proof: f 1 b c 1 d f34 1bb 1b
which is symmetric in a and b. Consequence #2 ;
fabc a d d 0c = 0 (A.2)
Proof: f b$ bdd = f1dc d)1 Ob
= ae fLdpd .a bf c p Oaddd b again symmetric in a amd b.
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56
A.2 Component Check
We start with the Taylor expanded version of the bosonic superfield
S(y) 1 A(x) + t 00A(x)  ( )2 A(x) (O)3 +2A(x)
8+ 4 12V0
1+ (6)4 a3 A (x) + 204 A (x) + B PO Co (x) O 2 p (0 )a+ 03 (a(A.3)
96V 2
i 0 0,0 ( )2O +2Cai (X)
The 3point coupling in the tendimensional theory is obtained by starting with the fourdimensional result and simply 'extending' the superspace derivative
S3 = d1ox d40 d40a fa e a1 b c + complex conjugate
(A.4)
= dx a f [Vabc AAA + Vccc + VAAC + VCCA + complex conjugate
where the generalized derivative is assumed to be
V = + Bd d (A.5)
The cubic terms in A simplify to
VAAA =  12 d+A A A
o (A.6) = + 12AA 6A + 12A d+A A q+
(where we have suppressed the gauge indices). To study this in component form we choose a typical term
Typical A3 Term = 3 A2 A, A2 + A2 OA2 A, (A.7)
57
The cubic terms in C simplify to
Vccc = i 2 2 B 0+ C, C p Op + i 8 v ,, Ca' aoy Op6
+ (A.8) +i 2 B  C~4 CP" O + 0 Which in component form reads
Vccc = i 8 vf2 B +A A A  i 16 B A A O A
+ (A.9) +i 8 V2 B +Ai Aj Oi 8+Ai
We work with a typical term again choosing the term involving one 06, one A6 and two A7's. The first two terms then give
i8V B O+AA7  A6  i 16 B A A6 06 A7 (A.10) 0+
while the third term is T = i 8 V2 B +A6 A7 06 O+A7. We partially integrate this to obtain
T =i8V2B60 1 A6A7 A7
(A.11)
=i8B2 B  A60+A7 A7 + i 8 BA606A7 A7  T
This lets us solve for T. (terms of the form A7a A7b (...) vanish due to the antisymmetric structure constant). The final expression then reads
Typical C3 Term= il2v/2B AA606A + A7 A71As6 (A.12) A comparison with the typical A3 term determines B = (A.13)
4 v
58
which is exactly what we obtain when working directly in superspace.
A.3 Useful Results The extended derivative V is defined by
oVï¿½ = v + 4 2d"dd (A.14)
[,] =  da8 = p (A.15)
4 2[ 8+
Oa ,[On', V]  o , [ 3,08  2 dr ï¿½ (A.16)
_ 0, ia 1
wa p [, a p [aO  20 Yp ] (A.17)
2 2(9a+
[ d, 7]+ OaC (A.18)
d3 I a= [, L] = /"P _ o pd
42 & 8+
APPENDIX B SUPERGRAVITY B.1 Duality Theorem
Consequence #1 ;
1 d
+ (B+ .1) +3
1 
 m0+ am(  I
121
 1(B.2)
We work with the second term to obtain
a+ 3 m(B.3)
 3m a+
1 1
+ 54 ama o+ ama+ (B.4)
59
60
B.2 Useful Results
B i i
d =+ 0+   a+ (B.5)
S + + B+ (B.6)
a oa + a, 0_ 1, a '" 0 +2 (B.7)
v' 2
[o, 7] = (r Y ")""dK (B.8) [ddd, 7V] = (7) 0a~+ ia ddO (yn)a " anO (B.9)
4 2V0
([O 16 4 + i + dV2 O3 (,)o"O (B.10)
16 ( + 41
_ 3 0
J 16 ( OnfPa n 4+  _ 9pdQ, y 8)aK L + b (B.11)
61
[ 0 (1/ al +/k 0,3 a (B.12)
8 + 4
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BIOGRAPHICAL SKETCH
The author received a bachelor's in physics from Loyola College and a master's in physics from the Indian Institute of Technology Madras. He is currently a Mclaughlin Fellow in the Department of Physics at the University of Florida.
65
I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.
Pierr Rmond, Chair
Distinguished Professor of Physics
I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.
Pierre Sikivie
Professor of Physics
I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.
Richard Wooard
Professor of Physics
I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.
Associate Professor of Physics
I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.
David Groisser
Associate Professor of Mathematics
This dissertation was submitted to the Graduate Faculty of the Department of
Physics in the College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August 2005
Dean, Graduate School

Full Text 
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MAXIMALLY SUPERSYMMETRIC THEORIES IN LIGHTCONE SUPERSPACE By SUDARSHAN ANANTH A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2005
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my parents and my sister, for all their love and support.
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ACKNOWLEDGMENTS I am deeply indebted to Prof. Pierre Ramond for five wonderful years of graduate school. I am grateful to him for the academic freedom that he has always given me, his constant interest and support, his kindness as a human being and above all for his delightful sense of humour. I will greatly miss the sense of excitement from working with him, the unpredictability of our exchanges and most of all, being across the hall from him. I am grateful to Prof. Lars Brink for very interesting discussions on a variety of issues and for sharing with me his insight into topics presented in this thesis. I have greatly enjoyed our joint work/lunch sessions and look forward to many more. I owe a special debt of gratitude to Prof. Guruswamy Rajasekaran for his invaluable guidance and wise counsel. I have benefited greatly from conversations (and innumerable sets of tennis) with Professors Zongan Qiu and Sergei Shabanov. I thank Prof. Richard Woodard for all the help he has given me during my stint at UF. I also thank the other members of my supervisory committee, Professors Pierre Sikivie and David Groisser, for their support. I thank Prof. Jim Dufty for his interest and advice, Professors Ranga and Vasudha Narayanan for their generous hospitality, Ms. Yvonne Dixon and Ms. Darlene Latimer for being so helpful with everything administrative, and Marc Soussa for being so much fun to share an office with. Finally, I thank Ah Nayeri, Chao Ieong, Amas Khan, Kyoungchul Kong, Lisa Everett, Suhas Gangadhariah and Tuan Tran for the fun food jaunts. iii
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TABLE OF CONTENTS Page ACKNOWLEDGMENTS iii ABSTRACT vi CHAPTER 1 INTRODUCTION 1 1.1 Divergences and the LightCone 3 1.2 Overview 5 1.3 Dimensional Oxidation 6 2 THE SUPERPOINCARE ALGEBRA 8 2.1 Supersymmetry 8 2.2 Supersymmetry and the LightCone 10 2.3 The Complete d = 4 SuperPoincare Algebra 13 3 THE YANGMILLS SYSTEM 16 3.1 TenDimensional Supersymmetric YangMills 16 3.2 (J\f = 4, d = 4) Super YangMills in LightCone Superspace 18 3.3 Oxidizing (J\f = 4, d = 4) Super YangMills to {Jsf = 1, d = 10) YangMills 21 3.3.1 Ten Dimensions 21 3.3.2 The SuperPoincare Algebra in 10 Dimensions 22 3.3.3 The Generalized Derivatives 26 3.3.4 Invariance of the Action 28 4 THE SUPERGRAVITY SYSTEM 32 4.1 ElevenDimensional Supergravity 32 4.2 (Af = 8, d = 4) Supergravity in LightCone Superspace 38 4.2.1 Field Content 38 4.2.2 A Simpler M = 8 ThreePoint Vertex 41 4.3 Oxidizing {M = 8, d = 4) Supergravity to {M = 1, d = 11) Supergravity 42 4.3.1 Eleven Dimensions 42 4.3.2 The SuperPoincare Algebra in 11 Dimensions 43 4.3.3 The Generalized Derivatives 45 iv
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4.3.4 Invariance of the Action 47 5 DISCUSSION AND FUTURE DIRECTIONS 50 5.1 Extending Gravity, to Order k 2 50 5.1.1 Conjectured Quartic Vertex 50 5.1.2 Supersymmetry Variations 51 5.1.3 Chiralization 53 5.2 Dual Descriptions 54 APPENDIX A YANGMILLS 55 A.l Duality Theorems 55 A.2 Component Check 56 A. 3 Useful Results 58 B SUPERGRAVITY 59 B. l Duality Theorem 59 B.2 Useful Results 60 REFERENCES 62 BIOGRAPHICAL SKETCH 65 v
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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy MAXIMALLY SUPERSYMMETRIC THEORIES IN LIGHTCONE SUPERSPACE By Sudarshan Ananth August 2005 Chairman: Pierre Ramond Major Department: Physics Reduced supersymmetric field theories retain a great deal of information regarding their higherdimensional progenitors. In this dissertation, we describe how the (J\f = 4, d = 4) Super YangMills theory may be "oxidized" into its parent theory, the fully tendimensional J\f = 1 YangMills. Remarkably, this is achieved by adding a single term to the fourdimensional transverse space derivatives. We work in lightcone superspace which is entirely free of auxiliary fields. Very similar in structure to (M = 4, d = 4) YangMills is the maximally supersymmetric (M = 8, d = 4) Supergravity. This theory is obtained by reduction from elevendimensional M = 1 Supergravity. We show that this fourdimensional Supergravity theory may be restored to eleven dimensions in a very similar fashion. vi
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CHAPTER 1 INTRODUCTION The Standard Model of particle physics remains the towering achievement of the last half century. It describes all known interactions (probed by experiments), from the Hubble radius down to scales of order 1(T 16 cm. The SU(S) x SU{2) xU(l) Standard Model describes three of the four fundamental interactions with a remarkable degree of precision. However the theory is not without its shortcomings. It fails to explain why such widely varying scales seem to occur in nature. Gravity, for example, operates at the Planck scale (10 _33 cm) while the length scales involved in the Standard Model are around 10 20 times larger. A natural question to ask is: why is the Standard Model consistent at a scale so much greater than the Planck scale? Experimental evidence is also raising serious doubts regarding the completeness of the Standard Model in its present form. Flavor oscillations of neutrinos produced by cosmic rays offer evidence for neutrino masses (as do measurements of neutrino fluxes from the sun)Â— the Standard Model, however, has only massless neutrinos accompanying the charged leptons in j3 decays. The Standard Model also seems arbitrary in the sense that it involves 19 parameters (plus 10 for neutrino masses) such as gauge and Yukawa couplings of varying magnitudes. Observations also indicate the presence of a dominant amount of nonluminous matter in the universe, something the Standard Model does not explain. Thus, there are several indications that we must look beyond the Standard Model [1] for an accurate description of our universe. Gravity along with the Standard Model describes all four interactions, one of which is mediated by the spin 2 graviton and the remaining three by SU(3) x SU(2) x U(l) spin 1 gauge bosons. In addition, the theory involves the spin 0 Higgs, quarks, leptons and fifteen multiplets of spin \ fermions (in 3 generations of 5). The union of 1
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2 gravity with quantum mechanics leads to a nonrenormalizable quantum field theory, an indication that new physics appears at very high energy. Thus, our current understanding of Nature and its interactions (as governed by the Standard Model+gravity) cannot be viewed as a fundamental picture but merely as fragments of a bigger theory. The aim is to determine the single theoretical structure that will explain all fundamental interactions and hopefully have few (if any) undetermined parameters. Without affecting the consistency of a theory of gravity, the best method to smooth out its divergences, involves spreading out the gravitational interaction. This is string theory: In this theory, the graviton (and other elementary particles) are onedimensional objects rather than point particles (as in quantum field theory). The benefits String Theory offers are enormous: to start with, every consistent string theory must contain a massless spin 2 particle whose interactions (at low energy) are general relativityÂ— it also leads to a consistent theory of quantum gravity (perturbatively). In addition, String Theories lead to gauge groups that are large enough to include the Standard Model in them. String theories however suffer from too many symmetries and supersymmetries, and require ten spacetime dimensions for consistency. Comparison with data requires an understanding, perhaps dynamical, of the breaking of these symmetries, for which we have no hint. The answer may lie in the elevendimensional Mtheory [2]. Although very little is known about this theory, we do know that it subsumes all the known superstring theories and may contain guideposts for supersymmetrybreaking and dimensional reduction. Its infrared limit is the much studied TV = 1 Supergravity in eleven dimensions [3], believed to be ultraviolet divergent. Although Mtheory casts welldefined shadows on lowerdimensional manifolds, its actual structure remains a mystery. Our hope therefore lies in unraveling its structure, starting from its known low energy limit, elevendimensional Supergravity. This N = 1 theory in eleven dimensions is the largest supersymmetric local field theory
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with maximal helicity two (on reduction to four dimensions). It is ultraviolet divergent and its divergences are presumably tamed by MTheory. A clearer understanding of how this occurs will offer us a window into the workings of MTheory. 1.1 Divergences and the LightCone A technically difficult but conceptually simple framework for discussing divergences is the lightcone [4] frame formulation. Working in lightcone gauge is advantageous for two reasons. Primarily, all spurious degrees of freedom are eliminated and one deals exclusively with the physical degrees of freedom. Secondly, the role of the transverse little group [5] is made apparent. This opens up the possibility of relating the spacetime group structure to the divergences that occur in a given theory. Such a relation (between spacetime spin and divergences) was noticed by Richard Hughes [6] who pointed out that the coefficients of the oneloop j3 function in QCD were proportional to (l) 2s (l 12s 2 )/3 where s is the helicity of the circulating particle. This result was extended by Curtright [7] to dimensionally reduced theories. Curtright considered bosonic and fermionic contributions to the one loop vacuum polarization graph. He conjectured that these contributions (for a theory in d dimensions, obtained by reduction from D dimensions) was 1 (ISO 9 faCdfbCd 7 [ d~L {q2 ^ id) ~ Q ^ ~ 4 Â— {q2 ^ D) ~ ^ } < L1 ) where / is 0 for bosons and 1 for fermions and the indices are those appropriate to either the boson or spinor representations of SO(D 2). r is the rank of SO(D 2) and the usual fourmomenta. This result is obtained, using supersymmetric dimensional regularization [8]. The p'th Dynkin index is defined by lW[R]= J2 (WW)* w in R (1.2)
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where w are the weight vectors in representation R. It follows that 1^ is simply the dimension of the representation. In 10 dimensions for example, a gauge boson has 8 components which means f^ DOSOn = 8. A perfect check of this conjecture is the J\f = 4 Super YangMills theory. This theory is obtained by reducing tendimensional YangMills where the relevant little group is 50(8). Thanks to triality, ^ vector = ^%pinor = ^ ^ Vector = ^^spinor = ^ Â» and since fermions enter with an additional sign, finiteness of the M = 4 theory at one loop follows. In fact triality also ensures that these indices match to all orders, offering an alternate proof of finiteness for the M = 4 theory [9, 10]. Higher loop processes would then involve generalized higher order representation indices. In ten dimensions, the lack of divergences in string theories may be attributed to the triality of 50(8), the lightcone little group. Applying a similar reasoning to eleven dimensions, we see that the incomplete cancelation of the Dynkin indices of the 50(9) representations that describe N = 1 Supergravity is responsible for the divergences in that theory. Indeed, the mismatch between the eighthorder bosonic and fermionic Dynkin indices seems to support this conjecture (an eighthorder mismatch implies divergences for the J\f = 8 Supergravity theory, at three loops or beyond). The role of the little group is key to this analysis (and hence the need to work on the lightcone). Since MTheory resides in d = 11, we expect special features of elevendimensional spacetime to be reflected in its physical little group, 50(9). The first step is therefore to understand the divergent behavior of N = 1 Supergravity. This requires a lightcone gauge description of the theory and will be one of the topics presented in this thesis.
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5 Lightcone Supergravity actions (in component form) are simply too bulky to be useful. The sheer volume of terms at order k and k 2 makes it unsuitable for reading off Feynman rules (or for further calculations). Superspace offers a compact alternative to this component approach. Superspace is an extension of ordinary spacetime, to include extra anticommuting coordinates in the form of M twocomponent Weyl spinors 9. We then define functions over this modified space, called superfields. These superfields may be expanded in a Taylor series, with respect to the anticommuting coordinates (the series is finite since the square of an anticommuting variable is zero). The coefficients obtained in this process are the usual component fields. Supersymmetry is manifest in superspace and its algebra is represented by translations and rotations, involving both commuting and anticommuting coordinates. Lightcone superspace is the perfect venue to study supersymmetric theories. There is a complete lack of auxiliary fields, due to working on the lightcone and we will find that the actions are tractable even at higher orders in the coupling constant. In the next few chapters, we will describe the formulation of the ten and elevendimensional theories (discussed above), in lightcone superspace. 1.2 Overview The fourdimensional M = 4 Super YangMills was formulated in lightcone superspace by Brink, Bengtsson and Bengtsson (and independently by Mandelstam) [12]. In reference [12] the authors introduced a single superfield that captured all the degrees of freedom of the J\f = 4 theory. Due to the maximal supersymmetry in the theory, the superfield is extremely constrained. The action for the N = 4 theory (in terms of this superfield) was obtained by requiring SuperPoincare invariance in four dimensions.
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6 This thesis assumes as a starting point their paper. We show that the above introduced M = 4 superfield is sufficiently rich in structure to completely describe the fully tendimensional M = 1 YangMills. We achieve this by introducing a new "generalized" derivative. This involves extending the d = 4 transverse space derivatives into superspace. We prove that the derivative transforms appropriately under the SuperPoincare generators and then show that simply generalizing the derivatives that appear in the fourdimensional action leads to the fully tendimensional theory. (J\f = 8, d = 4) Supergravity was also formulated in lightcone superspace (up to first order in the gravitational coupling) by the same authors [11, 12]. Again, since the theory is maximally supersymmetric, its superfield contains sufficient information, to describe its higherdimensional progenitor. Along similar lines to the YangMills case, we introduce the missing coordinates and derivatives and show that this "d=4" superfield may be used to completely describe the fully elevendimensional Af = 1 Supergravity. We call this process dimensional Oxidation [13, 14] and explain in detail the various checks one may perform to verify its consistency. While the aim is to build lightcone actions for field theories with a view to analyzing their divergences, this oxidation process by itself is fascinating and warrants some discussion. 1.3 Dimensional Oxidation Dimensional reduction [15] offers a mechanism to introduce new physics into a lowerdimensional universe [16]. It is, however, far more difficult (and often impossible) to start with a lowerdimensional theory and uniquely reconstruct its higherdimensional parent. Reduced supersymmetric field theories are fairly unique in this regard. They tend to retain a great deal of information regarding their once higherdimensional existence. This "memory" coded into their spectra allows for the unique recovery of their higherdimensional progenitors.
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7 A perfect candidate that displays such "memory" is the reduced maximally supersymmetric (J\f Â— 4, d = 4) YangMills theory. This theory is obtained by reduction from (J\f = l,d= 10) YangMills. The lightcone little group decomposition relevant to this reduction is 50(8) D 50(2) x 50(6) w 50(2) x SU(A). Signatures of ten dimensions are obvious in the d = 4 theory: the six scalar fields in the J\f = 4 spectrum serve as reminders of a lost (compactified) 50(6) while the SU(4) spinors assemble into a single eightspinor: 4i/ 2 + 4_ 1/2 = 8 S (of 50(8)). Also, the spectrum of the parent theory remains totally intact in the d = 4 version: for example, the 8Â„ (in d = 10) is simply reinterpreted (in four dimensions) as, 6 0 + li + l_i. Thus, the spectrum of the M = 4 theory hints at a tendimensional formulation (governed by the little group 50(8)). In principle, one simply needs to reintroduce the six missing coordinates (and their derivatives) to recover the 50 (8)invariant theory. Another equally impressive illustration of such "memory" is M = 8 Supergravity in four dimensions. Starting from elevendimensional Supergravity, the littlegroup decomposition reads 50(9) D 50(2) x 50(7) . (1.3) This theory is reminded of its origins because Â• The 50(9) structure is apparent in 8 1/2 + 8_ 1/2 = 16 s Â• The 35 + 35 scalars point to the, fjff of 50(7). Again, obtaining the covariant (jV = l,d = 11) Supergravity action should involve the introduction of the missing directions. Both the = 4 and Jsf = 8 theories are maximally supersymmetric this produces the mirrorsymmetry in their superfields (which one may argue, is the source of their "memory"). This "memory" is the key to our oxidation process. We will show that these theories are lifted perfectly into their once higherdimensional avatars and the entire process is consistent.
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CHAPTER 2 THE SUPERPOINCARE ALGEBRA 2.1 Supersymmetry This chapter is a quick review of some relevant results from existing literature, pertaining to supersymmetry and specifically to the lightcone gauge. We will then use these results in subsequent chapters. The known symmetries of the Smatrix in particle physics are Poincare invariance, internal global symmetries (whose generators are Lorentz scalars) related to conserved quantum numbers like electric charge, and the discrete symmetries, C, P and T. Subject to certain assumptions, Coleman and Mandula [17] showed that these are the only possible symmetries. However, their statement assumed that the symmetry algebra of the Smatrix involved only commutators. Introducing anticommutators into the algebra leads to the possibility of supersymmetry. Supersymmetry [1820], involves the introduction of anticommuting symmetry generators which transform in the (,0) and (0, ) representations of the Lorentz group. Since the genrators are not scalars it is not an internal symmetry. The bosonic generators are, therefore, the fourmomenta the six Lorentz generators, and a certain number of Hermitian internal symmetry generators, B r . The algebra is that of the Poincare group (in four dimensions, we set rf = (, +, +, +) with fx, v = 1 . . . 4) W , Pu] = 0 [Pn , M pa ) =i( r) wP(T 77^ p p ) (2.1) [M pa , M pc ] = iirj^Mpv rj^Mpv rj^M^ + rj^M^) 8
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together with that of the internal symmetry group(with indices r, s . . .) 9 [B r ,B 8 ] = if rs B t (2.2) The Casimir operators are P 2 = P^Pii W 2 = W^W (23) where W 1 = \ p u M pa . These operators commute not only with the members of the Poincare group, but also with the internal symmetry generators, This implies two things: First, all members of the same irreducible multiplet of the internal group have the same mass [21] and secondly that they are bound to have the same spin. In fourdimensional Weyl spinor notation, the N supersymmetry generators are denoted by Q ai , i = 1 ... A/". The fermionic supersymmetry generators satisfy [B r , p 2 } = [B r ,W 2 }=0 (2.4) [Qai, Mfv] = ^MiQpi (2.5) We also have the key anticommutator, {Q a ,Q p } =(rf aP , (2.6)
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10 with the 7 M defined by (fU = (7 M W , (2.7) and 2.2 Supersymmetry and the LightCone We continue working with the metric (,+,+,...,+). As a first step to formulating our theory in lightcone gauge, we introduce the following lightcone coordinates and their derivatives. x Â± " 7l {x0Â±x3)] d* = ^=(doÂ±ds); (2.9) ^=(x 1 + ix 2 ); S=L(ft_i3,) ; (2.io) * = ^(xiix 2 ); d = Â±={d x + id 2 ), (2.11) such that d + x=dx + = l; dx = dx =+l . (2.12) This then yields the following 4product, A^Bp = ABi A + B_ A_B + (2.13) In the lightcone frame, p~ = iÂ£+is interpreted as the Hamiltonian. We will consider the generators of the SuperPoincare algebra at the lightcone time x + = 0. Generators that are capable of shoftong away from a given x + surface are referred to as dynamical. The rest are kinematical. We use m,n,p,q... vector indices to denote
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11 transverse spacetime indices. These run over SO (d Â— 2) values in ddimensions. In order to be able to translate between spinor and vector notation, we define gammamatrices by {j m ) af3 = ( 7 m ) /3a , (2.14) such that Tr( 7 m 7 ") = (d2)6 mn . (2.15) This then allows us to write, C a/3 = C m ( 7 m ) Q/3 Â• (2.16) We also define 7 mn = \h m n n ], n. (2.17) We now project out equation (2.6), using \ 7+ j_ to obtain {Q + Â«,Qi} =(l + f a P + (218) which represents the "restricted" supersymmetry algebra. We simply pick a specific representation for the 7matrices to readoff the lightcone supersymmetry algebra, {<Â£,?+/Â»} = ~^^p + , {q",qp} = ~V28%p, (2.19) {ql,qp} = V28%p.
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12 The q + part of the algebra is linearly realized in superspace, characterized by a Grassmann parameter 9 a and its complex conjugate 9 a . These variables are defined by {9 a ,9^} = {S at Sp} = {9 a , 6 13 } = 0, (2.20) The corresponding Grassmann derivatives are written as * * s A' (2 2i) with canonical anticommutation relations {d a ,8p} = 5%; {d a , 9^} = 6j>. (2.22) Under conjugation, upper and lower spinor indices are interchanged, so that 6" = 9 a , while (d a ) = 0Â° (&>) = d p . (2.23) Also, the order of the operators is interchanged; that is 9 a 9^ = 9 0 8 a and d a d& = d 0 d a . The kinematical supersymmetry is then realized by choosing q a + = d a + ^=6Â°d + ; q +p = 8 0 L9 0 d\ (2.24) so that they satisfy {ql,q +0 } = iV2S a 0 d + , (2.25) The dynamical supersymmetries are obtained in the next section (these are obtained using the lightcone superspace Poincare algebra which we need to define first). These
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13 fermionic variables are also used to define chiral derivatives in superspace, d a = d a ^ a d + ; dp = Bp + ^=M + , (226) which satisfy {d a , d 0 } = iV2 8 a 0 d + . (2.27) 2.3 The Complete d = 4 SuperPoincare Algebra The entire (superspace) lightcone SuperPoincare algebra, splits up into kinematical and dynamical pieces. Apart from a few minor alterations, this algebra is presented in reference [12]. The kinematical generators are Â• the three momenta, p+ = id + , p= id, p = id, (2.28) Â• the transverse space rotation, j = xdxd + S 12 , (2.29) where S 12 = \(9Â°d a 0 Q dn ~ ^^(d a d a d a dÂ°) . (2.30) The last term is included to ensure that these generators commute with the chiral derivatives (this is related to chiralitypreservation, when acting on superfields. We will discuss this when we introduce the superfields), [J'.d*] = [J,d p ] = 0 (2.31)
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14 Â• and the "plusrotations" j + = ixd + , j + = ixd + . (2.32) j + ~ = ix~d + l { 9 a d a + 6 a d a ) , (2.33) which satisfy lj + ,d*} = l d\ [j + ,dp] = l d p , (2.34) and thus preserve the chiral derivatives. Note that it is only for the choice x + = 0 that the generator j+~ is kinematical, since the dynamical part is multiplied by x + = 0. The dynamical generators are Â• the lightcone Hamiltonian, P = ~ % ^ d (2.35) Â• and the dynamical boosts, ix % ~ " 9 + 'K+i^^*^))!: , ix% x~d + iiW + j^tfbb*))^; . (2.36) The helicity counter added to the expressions above implies that ir.r]Â±eÂ£, ir, i s \ = iÂ±Â± . (2.37)
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15 These generators also satisfy ir,j + ] = lr,j + ~] = ij~ (2.38) In a similar fashion, the supersymmetries split into Â• the already introduced kinematical supersymmetries, < = ~d a + ^ a d + ; Q + p= B 0 ^=6,d + , (2.39) satisfying {q%,q +0 } = iV25 a p d + , (2.40) and anticommuting with the chiral derivatives {q+,dp} = {d a ,q +p } = 0. (2.41) Â• dynamical supersymmetries, which may be obtained by boosting the kinematical ones 0 0
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CHAPTER 3 THE YANGMILLS SYSTEM Having set up the relevant tools in superspace we now focus on formulating our theories in that space. Both the (Af = l,d = 10) and (J\f 4, d = 4) Super YangMills theories are known in component form (in the lightcone gauge). However, while the fourdimensional version also has a superspace description, the d = 10 variety does not. This is the problem we resolve in this section. After a brief review of the fourdimensional theory (in superspace), we oxidize it up to ten dimensions thus obtaining a complete lightcone superspace description of (J\f = 1, d = 10) YangMills. 3.1 TenDimensional Supersymmetric YangMills A YangMills theory based on a local symmetry G is a field theory with symmetry currents coupled minimally to vectorboson fields. The space integrals of the time components of the currents formally define the generators T a of the symmetry group G. The generators of the group and consequently the currents from which they are constructed transform as the adjoint representation. For the current vector boson coupling to be invariant under G the bosons must also transform as the adjoint. Leptons and quarks which are spin Â± fermions are usually assumed to be fundamental fields in the Lagrangian. The generators T a of the Lie group G satisfy [T\T b }=if abc T c (3.1) where the are structure constants. Mathematically, gauge fields are introduced through the covariant derivative Â£> M = a M + igAfT* 16 (3.2)
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IT from which we construct the field strengths if = F^ a T a = i[DÂ„ D v \ (3.3) The d = 10 Yang Mills action reads S = J d 10 x {\f^ F^ a + % \ a Y Dp \ a } (3.4) where the vector and spinor fields transform according to the adjoint representation of some semisimple Lie group. The spinor satisfies both the Majorana and Weyl constraints. This action is invariant under the supersymmetry transformations 5 A* = ia ltx \ a 1 (35) S\ a = _y a/flight cone gauge is fixed by choosing Aa = 0. The spinor breaks up as A = \ (7+7+ 77+ ) A = A + + A_ (3.6) As an illustration of the LC 2 approach, we focus purely on the bosonic portion of the above action. The bosonic equations of motion corresponding are d v F tu,a + g f abc F^ h A v c = 0 (3.7) LC 2 indicates that all spurious degrees of freedom have been eliminated using the equations of motion. For example, the fi = + component of this equation implies
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18 that A + a = ^(diAn + g / a6c i{( d_ V) A k c } (3.8) When substituted back into the bosonic part of the action we obtain S = J d l0 x{^A i a DA i a + 9f abc [ di A1( d + Aj b Aj c ) d { Aj a At Af\ + g 2 f 1 * r" i ~ \ At Aj c At Aj e ~\^{d + At At) (3.9) This procedure is easily extended to include the fermions. The may be thought of as a Green's function that satsifies This determines G upto a factor h where d + h = 0. i.e. up to a zero mode h = h(x + ) of the operator d + (for more details on this prescription, we refer the reader to reference [22]. 3.2 (Af 4, d Â— 4) Super YangMills in LightCone Superspace When tendimensional Jsf = 1 YangMills is dimensionally reduced to four dimensions it yields one complex bosonic field (the gauge field), four complex Grassmann fields and six scalars. In four dimensions any massless particle can be described by a complex field and its complex conjugate of opposite helicity, the SO{2) coming from the little group decomposition d + G(x~) = 6{x~) SO(8) D SO{2) x 50(6) . (3.10)
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19 Particles with no helicity are described by real fields. The eight vectors fields in ten dimension reduce to 8Â„ = Bo + li + l! , (3.11) and the eight spinors to 8, = 4 1/2 + 4_ 1/2 . (3.12) The representations on the righthand side belong to 50(6) ~ SU(A) with subscripts denoting the helicity: there are six scalar fields, two vector fields, four spinor fields and their conjugates. We introduce the anticommuting Grassmann variables 6 a and h {p,eP} = {SaJi,} = R,^} = o, (3.13) which transform as the spinor representations of SO(6) ~ SÂ£/(4) 9 a ~ 4 1/2 ; 6 P ~ 4_ 1/2 , (3.14) where a, /?, 7, 5, Â• = 1,2,3,4, denote SU(4) spinor indices. Their derivatives are written as with canonical anticommutation relations {d\h} = ^; {B a , = 8 a 0 . (3.16) Having introduced these variables, we may now capture it all the physical degrees of freedom of the (Jsf = 4, d = 4) theory in one complex superfield [12] + 0Â° UV) + ^0* 0* ff> e a p l5 X s (y) . (3.17)
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20 In this notation, the eight original gauge fields Ai ,i = 1, . . . , 8 appear as A = Â±=(A 1 + iA 2 ) , A = Â±={A x iA 2 ) , (3.18) while the six scalar fields are written as antisymmetric SU (4) bispinors c Â° 4 = 71 (Aa+3 + iAa+e) ' ^ A = 7! (Aa+3 ~ iAa+&) ' (3>19) for a ^ 4; complex conjugation is akin to duality Cap = \e a ^ s Cn s . (3.20) The fermion fields are denoted by x a and \aAll have adjoint indices (not shown here) and are local fields in the modified lightcone coordinates y = (x, x, x + ,y~ =x~ ^=9 a e a ) . (3.21) v2 In this LC 2 fightcone formulation all the unphysical degrees of freedom have been integrated out leaving only the physical ones. One verifies that = 0; J a 0 = O, (3.22) as well as the "insideout" constraints d a dp = ^e a0yS cP d s 4> , (3.23) ml (3.24)
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21 The (A/ = 4, d = 4) YangMills action is then simply j d A x J d A ed A 6C , (3.25) where g 2 fÂ° bc ! adÂ£ ( ^dV)^: b 4> c t d 4> e ) (3.26) Grassmann integration is normalized so that / d 4 66 1 8 2 6 i 6 i = 1, and are the structure functions of the Lie algebra. 3.3 Oxidizing (Af = 4, d = 4) Super YangMills to (Af = l,d = 10) YangMills Apart from some minor changes most of the material presented up to this point appears in references [11] and [12]. We now embark on the program outlined in our overview: we will show that the compact d = 4 formalism discussed so far can be generalized to entirely describe the d = 10 theory. 3.3.1 Ten Dimensions The very compact formalism of the previous section is specific to the N = 4 theory in four dimensions. We now generalize this formalism to restore the theory to ten dimensions without altering the superfield (except for an added dependence on new coordinates). This is achieved simply by introducing generalized derivative operators. First of, the transverse lightcone variables need to be generalized to eight. We stick to the previous notation and introduce the six extra coordinates and their
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22 derivatives as antisymmetric bispinors x ai = ^(x a + 3 + ix a + fi ) , d a4 = 4=(0Â« + 3 + id a + 6 ) , (3.27) for a / 4, and their complex conjugates x^s = ^ap^x^ d lS = ^e a0lS d a/3 . (3.28) Their derivatives satisfy dapx** = W8p & 6 a s 5{?) ; d a0 x yS = (S%5^ s S a s S^) , (3.29) and There are no modifications to be made to the chiral superfield except for the dependence on the extra coordinates A(y) = A(x,x,x a(i ,x a p,y) , etc... . (3.31) These extra variables will be acted on by new operators that generate the higherdimensional symmetries. 3.3.2 The SuperPoincare Algebra in 10 Dimensions We start with the construction of the SO (8) little group using the decomposition 50(8) D 50(2) x 50(6). The 50(2) generator is the same as before while the 50(6) ~ SU(A) generators are given by J% = \(x ap d p0 XpP d a Â») e a dp + + ^(e"B p e p d")8%
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23 (d a dpd p d a ) + 7r ^(d p d p d p d p )6 Q 0 . (3.32) SV2d+ The extra terms with the d and d operators are not necessary for closure of the algebra however they ensure that the generators commute with the chiral derivatives. They satisfy the commutation relations J,J a t 0, ^0 , J P Â° x^d + ^d+eoe* iV2 Â— d p & r + ^Â— d p dÂ° , V2 d+ y/2d+ Jap = xd a0 x a pd+^d + 0 a 9pi\fi^d a dp + ^Â— v2 C + v2d H d a d 0 . (3.34) All 50(8) transformations are specially constructed so as not to mix chiral and antichiral superfields [J a0 ,d P ] = 0 [Jap, d P ] = 0 , (3.35) and satisfy the 50(8) commutation relations J,J ap ] = J a0 , J > J at JaP , J a p i J, fi a
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24 Rotations between the 1 or 2 and 4 through 9 directions induce on the chiral fields the changes H = ( \u a pJ a(} + \ Jap)*, (336) where complex conjugation is like duality i4 = o>23 (= ^ 23 = u u by reality), all other components being zero. Finally, we verify that the kinematical supersymmetries are duly rotated by these generators [J* 0 , Q +P ) = S^ p q a + 6\ql ; [ J a0 , qÂ» + ] = 8f q +a 8 a "q +0 . (3.38) We now use the SO(8) generators to construct the other Super Poincare generators J + = ixd + ; J+ = ixd + J+Â«P = ix ^d + ; J + af3 = ix a() d + . (3.39) The dynamical boosts are now J = ix dd + \d p(T dPÂ° . . d Â— ix iHfr'TF"*^'}' (3 ' 40)
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25 and its conjugate dd + Â±8 pa d p a 0Â» + ^{d od p d p dÂ»)}1 d p
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2(5 and can be obtained from one another by SO (8) rotations (3.46) while [Jpa, QÂ° ] Â— 0 Â• (3.47) Note also that (3.48) This framework offers a simple method to introduce the central charges germane to the fourdimensional theory: the sixdimensional derivatives d** 13 are simply replaced by cnumbers Z a/3 thus yielding the massive supersymmetry algebra in four dimensions with six central charges However, relaxing this duality condition gives us all 12 central charges of the N = 4 YangMills Theory thus indicating that both varieties of central charges have a common origin in the lightcone gauge formulation. Starting with these tools we proceed to build the interacting theory in ten dimensions. 3.3.3 The Generalized Derivatives The cubic interaction in the (AT = 4, d = 4) Lagrangian contains explicitly the derivative operators d and 5. To achieve covariance in ten dimensions these must be generalized. We propose the following operator pa V = d + 4\/2d+ dpd^d" 0 , (3.49)
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27 which naturally incorporates the rest of the derivatives d pa with a as an arbitrary parameter. After some algebra we find that V is covariant under 50(8) transformations. We define its rotated partner by V o0 v, r 13 (3.50) where __ Qa/3 _ lOL d 0 d a e a ^d . 4 72*9+ 9 (3.51) If we apply to it the inverse transformation, it goes back to the original form (3.52) and these operators transform under 50(8)/(50(2) x 50(6)), and 50(2) x 50(6) as the components of an 8vector. We introduce the conjugate operator V by requiring that = (V0), (3.53) with Define which is given by v = d + v a/J i a d p dÂ°d pa V, J a , ta V a/ 3 = d a/3 4 ^ d+ d p d a e aj3p(r d (3.54) (3.55) (3.56)
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28 We then verify that pa 6*8/ 6 q a 6f)V . (3.57) The value of the parameter a is fixed by requiring invariance of the cubic interaction. 3.3.4 Invariance of the Action The kinetic term is trivially made 50(8)invariant by including the six extra transverse derivatives in the d'Alembertian. The quartic interactions are obviously invariant since they do not contain any transverse derivative operators. Hence we need only consider the cubic vertex. The whole point of this exercise is to show that covariance in ten dimensions is achieved by simply replacing the transverse derivatives d and 5 by V and V respectively. We thus propose the new cubic interaction term Since it is obviously invariant under SO(6) x SO(2) we need only consider the coset variations. Acting on the chiral superfield we have 6j = u} a pJ a ^ = iV2 uJ af3 d + e a 9 0 (t) (3.59) Sj = ^J M 0 = c^^a+M,i^^ + ^^^ 4> . (3.60) We list the conjugate relations for completeness 6j = u^Jprcf) = iV2 u p(T d + e p 9 , (3.61)
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29 The variations of the generalized derivative and its conjugate are given by Sj V = u a0 [ J al5 , V ] = u af) V a/J , (3.63) SjV = uT [JÂ„ , V] = u>oÂ° d p d a ^ . (3.64) Invariance under 50(8) is checked by doing a 5j variation on the cubic vertex, including its complex conjugate. In terms of V + V = fotcj (Â±T cf> b V0 C + ^ a t V/) , (3.65) explicit calculations yield and SjV = 8jV ict 1 ,Â„& _ M d Adding the two yields 6j(V + V) = (a l)U c u x x / W + . (3.68)
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30 The cubic vertex is therefore 50(8) invariant if a = 1. This determines the generalized derivative to be V = 5 + Â— p Â— dodod"* . (3.69) This result has also been checked explicitly by performing the Grassmann integrations and looking at the components (we present some of those checks in Appendix A). To obtain this result we have used the antisymmetry of the structure functions, the chiral constraints, the "insideout" constraints, and performed integrations by parts on the coordinates and Grassmann variables. In particular, using the relation between the chiral field and its conjugate implied by the "inside out" constraints, 4> = dp ^ d a dp^4>, (3.70) we deduce two magical identities fake J ^0V<90 C = 0, (3.71) fabc f ^4> a ^ b d a d' 3 dt c = 0 . (3.72) In this lightcone form Lorentz invariance in ten dimensions is automatic once little group invariance has been established. We have therefore shown tendimensional invariance since the quartic term does not need to be changed. Thus we now have a complete lightcone superspace description of tendimensional Af = 1 YangMills. The action is simply J d 1Q x J d^ed^C , (3.73)
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where 31 S 2 /^ ^ ( ^ a + f )^ 5 + f ) + ^ # ^ (3.74) The is now the tendimensional d'Alembertian and the superfields have an added dependence on the six new coordinates.
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CHAPTER 4 THE SUPERGRAVITY SYSTEM Having oxidized fourdimensional YangMills into its tendimensional parent, we now focus on the analogous situation in Supergravity. In many ways the YangMills system serves as a perfect toy model for the Supergravity system. Again, our initial focus is on the maximally supersymmetric theory in four dimensions, J\f Â— 8 Supergravity. Starting with its lightcone superspace description we show how to obtain the fully elevendimensional theory in superspace. A Supergravity theory requires a fermionic companion to the usual bosonic graviton (gravitational field). This fermionic counterpart is the spin Â§ gravitino. The discovery of this particle in a laboratory would be a triumph for Supergravity because it is the only consistent field theory for interacting fields involving spin . The presence of these fermion fields does not alter the classical predictions of general relativity (fermion exchange leads to a shortrange potential). However, at high energies (short distances) Supergravity is much better behaved as a quantum theory than general relativity. Infinities in the Smatrix in the first and second order quantum corrections cancel due to supersymmetry (as discussed in the Introduction). How far these cancelations persist is an open question. But this finiteness implies that Supergravity has predictive power. 4.1 ElevenDimensional Supergravity Eleven dimensions is interesting because a maximal Supergravity theory in that dimension, on reduction to d = 4 yields particles with spin less than or equal to two. The bosonic field content of elevendimensional Supergravity consists of the elfbein, e/ and a completely antisymmetric 3form potential A^ p with field strength :>>2
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33 F^vpa = d\,iA vpa \. In terms of the 50(9) little group in eleven dimensions these correspond to a total of 128 bosonic states. The fermionic content consists of a single spin  Majorana field, which has 128 fermionic states. The N = 1 Supergravity action in eleven dimensions is [3] The spinconnection u is determined by the variation , 6 L = 0 to be O^nab Up.ab(e) =e a p ( d v e bp d p e bv ) e b p (d v e ap d p e av ) + e/ e b a ( d a e cp d p e ca ) eÂ„Â° where fi,v 7 ... run over all eleven directions, a, b, . . . are locally flat elfbein coordinates (not to be confused with our notation for the transverse directions) and the curvature is defined as y*i/*n d enotes t h e completely antisymmetric product of n 7 matrices and the covariant derivative is defined as D v = d v + \ [7 0 , 7 6 ] w uab . u and F are supercovariant versions of themselves. The elevendimensional action is invariant under local supersymmetry transformations. These transformations may be expressed in terms of an infinitesimal spacetime dependent Grassman parameter e{x) which transforms as a Majorana spinor. (4.1)
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34 The transformations that leave the action (4.1) invariant are 3 SA llvp = Â€7,^^] (4.2) Â», = iD M (fl)e + l( 7 r ,t 8V7 , ' ,[ )^ Working in the 1.5 order formalism the aim is to formulate the theory in the lightcone gauge. This may be done in component form (as we will show for gravity) but due to sheer size is not a very useful approach. Mindful of our ultimate aim, we formulate the theory entirely in terms of the elfbein variables e^. This provides us a framework in which fermions may be introduced naturally. On the lightfront /i = Â©, 0, Â© where (I) = 1 ... 8. a , b ... are the locally flat lightcone indices, a = +, Â— , i where z = 1 ... 8 (again, with metric ( h + ...)). We parametrize the elfbeins as follows e Q = e*+ (4.3) We define the symmetric object (in a metricbased approach, this is the transverse metric) by where Â£ is a real field and 7jj is a real symmetric unimodular matrix that satisfies Hi l jk = S { k
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35 We work in the lightcone gauge by choosing [2328] For the final gauge condition, we choose
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36 while the R + = 0 equation of motion determines eÂ®=^^h im + 0(n 2 ) (4.9) 2 d~ The gravity action, L = Â— ee ph e ac R pac b may then be written entirely in terms of the physical variables h^ and up to order k reads L g K = hij hij + Â— hij ( d h mn d h mn ) + Â— h mk ( di d~~ h mk ) hu + k h mk ( d k d~ h im ) di h it + h mk ( d~ h mk )dihu + K( d~ h im ) ( d k h mk ) d t h a ft, + Kj ( di h ki ) dj h kl h mk ( d, h it ) d k h im h jt ( d k h {j ) d t h ik The gravitino gets relatively more cumbersome to deal with in components. The gravitinodependant terms in elevendimensional Supergravity are Each spinor may be projected into an upper and lower component as defined below 7 + 7~ ^ = 2 ; 7~7 + ^ M = 2 ^( _) Lightcone gauge is chosen by setting =0 (4.10)
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37 We also make the additional gauge choice [29] 7 = 7*^7+^= 0 (4.11) which implies that Y W ^ = 0 and thus allows us to define a physical "shifted" gravitino field (with 128 components) H =(^+Â§VVj ^ (_) ( 4 12 ) For convenience, we suppress the hat over the The equations of motion read Y uX D v {u)^ x = ^ t*""" ^ F.^a + 1 7 p
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38 We choose lightcone gauge by setting Aij =A +i i = 0 A. + k A+ fe 0 (4.14) Again, we will not work this out in LC 2 although it is somewhat simpler than the gravitino. This elevendimensional Supergravity theory on reduction to four dimensions leads to the maximally supersymmetric Af = 8 Supergravity. This theory will be the focus of our next section. 4.2 (Af = 8, d = 4) Supergravity in LightCone Superspace Based on the fact that both the (Af Â— 8, d = 4) Supergravity theory and the (Af Â— 4, d = 4) Super YangMills theory are maximally supersymmetric, the authors of reference [12] introduced a constrained superfield and a superspace action for the Af = 8 theory. We start with this superspace action and show that it may be used to describe the (Af Â— 1 , d = 11) Supergravity theory in lightcone superspace. 4.2.1 Field Content The physical degrees of freedom of the elevendimensional Af = 1 Supergravity theory are classified in terms of the transverse little group SO (9) with the Graviton q{mn) transforming as a symmetric secondrank tensor, the threeform A [MNP ^ as an antisymmetric thirdrank tensor and the RaritaSchwinger field as a spinorvector ^ M (M, AT, ... are 50(9) indices). This theory on reduction to four dimensions leads to the maximally supersymmetric Af = 8 theory. The relevant decomposition is, 50(9) D 50(2) x 50(7) (4.15)
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39 All told, after decomposition, the M = 8 theory has a spectrum comprised of a metric, twentyeight vector fields, seventy scalar fields, fiftysix spin onehalf fields, eight spin threehalf fields and their conjugates [29]. The 50(7) symmetry is an internal one and can in fact be upgraded to an SU(8) symmetry. However, it is important to remember that it is really the 50(7) which is relevant when we "oxidize" the theory to d = 11. 50(7) indices are represented by m,n, All the physical degrees of freedom of the Af = 8 theory are captured by a single complex superfield [11] *(v) = ^h(y) + i6 a ^4> a (y) + ie a(3 Â±A aP (y) o a(, ^xa 0 Ay)o a ^ s c a0yS (y) + i~et ] M x aM {y) + i 0 d+ A*P(y) + Â§
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40 These tensor fields make up the two bosonic representations under the decomposition 50(9) C 50(7) x 50(2) of elevendimensional Supergravity. The G^ MN ^ (44 of 50(9)) split up as 44 = 27 + 7 + 7 + 1 + 1 + 1. (4.20) C(m P ) represents the 27 while A m , A m represent the 7 + 7. Similarly the threeform A [MNP] (g 4 of S q(q^ gplits intQ 84 = 35 + 21 + 21 + 7. (4.21) These correspond to C[ mpq \, A[ mn ], A[ mn \ and C p respectively. All fields are local in the modified lightcone coordinates y= (x, x, x + , y=x~j=&*h) (422) The superfield (j> and its complex conjugate 4> satisfy the chiral constraints d a = 0 ; 4^ = 0, (4.23) and are related through the "insideout" constraints where d s is simply the product of all eight chiral derivatives. The = 8 Supergravity action to order k is then simply J d 4 x Jd 8 ed 8 ec = J c, (4.25)
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41 where C = t^t+i^tddtd+^^tdd+tdd+t + c.c.) (4.26) Grassmann integration is normalized so that / d B 6 (6) 8 Â— 1. 4.2.2 A Simpler N = 8 ThreePoint Vertex The threepoint vertex in this action, seems highly nonlocal and cumbersome (relative to the M = 4 vertex). However, its form can be greatly simplified leading to a single term very similar to that in the YangMills case [13]. We start by partially integrating the first term with respect to 5 to obtain / d r * . ^2 . 1 d d + (j> (j) d(j>dd + (j) }. (4.27) The last (j> in the first term of equation (4.27) may be rewritten as a , using the insideout relation. We then partially integrate the (d) s onto the 0 and use the insideout relation again. The second term in equation (4.27) is partially integrated with respect to d + to yield two terms. Thus the first term in the 3point vertex is now / 5 ^ + ^4> ddd + + ^ dd + dd^A. (4.28) The third term in this equation exactly cancels the second term in the original vertex. Next, we eliminate the middle term in the equation above by recognizing that 1 = f ^dct>dd + ci> = J Jt BtdtI, (4.29)
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42 (which follows from a partial integration of the single 3 + in the numerator). This allows us to set I = [ Â— ^ 6 36 66 J 2d+ 2 (4.30) We thus obtain a very concise threepoint vertex 2k ^6 36 36 + c.c. (4.31) This simple form allows for comparison with the N = 4 Super YangMills threepoint vertex We will exploit this similarity in structure between the J\f = 8 and N = 4 cases when tackling the fourpoint vertex. 4.3 Oxidizing (J\f = 8, d = 4) Supergravity to (ftf = 1, d = 11) Supergravity 4.3.1 Eleven Dimensions We now proceed to show that the fully interacting (Af = 8, d = 4) theory can be restored to its elevendimensional progenitor without altering the superfield. This enables us to formulate the (J\f = 1, d = 11) theory without auxiliary fields in lightcone superspace. The first step is to generalize the two transverse variables to nine. In the YangMills case the compactified 50(6) was easily described by 517(4) parameters and we made use of the convenient bispinor notation. In the present case the compactified 50(7) has no equivalent unitary group so we simply introduce additional real coordinates x m and their derivatives 3 m (where m runs from 4 through 10). The chiral (4.32)
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43 superfield remains unaltered except for the added dependence on the extra coordinates h(y) = h{x,x,x m ,y~) , etc... . (4.33) These extra variables will be acted on by new operators that will restore the higherdimensional symmetries. 4.3.2 The SuperPoincare Algebra in 11 Dimensions The SuperPoincare algebra needs to be generalized from its fourdimensional version. The 50(2) generators stay the same and we propose generators of the coset SO{9)/(SO(2) x 50(7)) of the form J m = i(xd m x m d) + ^=d + O a ( 1 m ) n ^ =ZÂ— d Q ( 7 m ) n d? 1 2y/2d + d a {T) a ^ (4.34) J n = i(xd n x n d) + ^7=d + M7T% 7^d a {l n rd p + 2>/2d + p (4.35) which satisfy the 50(9) commutation relations J m , J m J m , J n = J m , J,J n = $pÂ™ ji _ s qm J p iJ mn + 5 mn J, = r (4.36)
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44 where J is the same as before, J = j and the 50(7) generators read J mn = i(x m d n x n d n ) + 9 a ( 7 m ) a 0 (7Y " da + h ( 7 m ) a " (rf Â° dÂ° d a (rr 0 m 0 Â° . (4.37) The full 50(9) transverse algebra is generated by J , J mn , J m and J n . All rotations are specially constructed to preserve chirality. For example [JÂ™, d a ] = 0 ; [J\
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45 The dynamical supersymmetries are obtained by boosting [J , q+fi] = Qp Â— % d_ q+f3 ~ y^ 7 "^' 9 [J,<Â£] = Q a = i Â— a^ + Â—(^ m ) a0 a a Â— (4.43) They satisfy (7 m ) a/3 d m (4.44) and the supersymmetry algebra (4.45) A few central charges fit nicely into this framework. The d = 11 theory is known to have 517 central charges (corresponding to a two form and a five form which yield 462 + 55): 7 of these may be introduced by simply replacing the seven derivatives d m by cnumbers Z m . The remaining 510 charges have an elevendimensional origin. Having constructed the free J\f = 1 SuperPoincare generators in eleven dimensions we turn to building the interacting theory. 4.3.3 The Generalized Derivatives The cubic interaction in the Af = 8 Lagrangian explicitly contains the transverse derivative operators d and d. To achieve covariance in eleven dimensions we proceed to generalize these operators as we did for = 4 YangMills [13]. We propose the generalized derivative Â— d m (4.46)
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46 which naturally incorporates the coset derivatives dÂ™. Here a is a parameter still to be determined. We use the coset generators to produce its rotated partner V by [V, J m ] = V m = id m + ^d a ( 1 m ) a0 d^. (4.47) It remains to verify that the original derivative operator is reproduced by undoing this rotation; indeed we find the required closure [ V m ,J P ] = 5 mp V The new derivative ( V , V m ) thus transforms as a 9vector under the little group in eleven dimensions. We note that a is not determined by these algebraic requirements. Instead, its value will be fixed in the next section by requiring that our generalized vertex satisfy invariance requirements. We define the conjugate derivative V by requiring that = . (4.48) This tells us that V = d+^d a (j n r 0 d^ (4.49) This construction is akin to that for the Af = 4 YangMills theory but this time it applies to the "oxidation" of the (Af = 8, d = 4) theory to (Af = 1, d = 11) Supergravity. This points to remarkable algebraic similarities between the two theories, with possibly profound physical consequences. It remains to show that the simple replacement of the transverse derivatives d, d by V, V in the (Af = 8, d = 4) interacting theory yields the fully covariant Lagrangian in eleven dimensions.
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47 4.3.4 Invariance of the Action While the covariant (A/ = 8, d = 4) Supergravity Lagrangian is known to all orders in the gravitational coupling, its lightcone superspace expression has only been constructed to first order in n (threepoint coupling). We note that the fourpoint gravity vertex is indeed known in component form [27, 28] but a superfield expression remains elusive. We will work with the theory to cubic order in this section and in the next section propose a quartic interaction term for the theory. The kinetic term is trivially made 50(9)invariant by including the seven extra transverse derivatives in the d'Alembertian. We start from our simplified version (4.31) of the M = 8 threepoint vertex. We propose that the elevendimensional vertex is of the same form but with the transverse derivatives replaced by the generalized derivatives introduced in the previous section: together with its complex conjugate. To show 50(9) invariance it suffices to consider the variations (4.50) W = 4^md + r( 7 m ) Q/ ^0 (4.51) I 9 + o a (r) a ^ i d a (7 m U^ m 2^2 V2d+ (4.52) Sjm V = Â— m (4.53)
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48 where u m are the parameters of the 50(9)/(50(7) x 50(2)) coset transformations. The 50(2) invariance is clear from the work in d = 4 and the 50(7) invariance is covariantly realized so if we can show the invariance under the 50(9)/ (50(7) x 50(2)) transformations we have shown invariance under the full 50(9). The Variation We split the calculation into three parts based on which superfield is being varied. Terms that involve 3d or d m d n cancel trivially. The remaining terms all involve a single 50(2) derivative and a single d m . We list the contributions from the variations below Contribution from J d+ ^dcf>d m d + Â±;d m d + ( i> (4.54) Contribution from J ^ 4> ( V ] 0 V 0: (4.55) Contribution from / ^ : J>dd m L^ePdtiin^W^d^A , (4.56) These results are further simplified by use of the magical identity (4.57)
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which follows from the duality constraint and numerous partial integrations form of the variation then reads SjVoc J (^m)^W<9">, (4.58) Elevendimensional Poincare invariance requires that this variation vanish. This determines a and hence the generalized derivative V = 8 ^=d a (r) aP d 0 Â— . (4.59) This completes the proof of 50(9) invariance for the threepoint function. It is rather remarkable that this simple replacement of the transverse derivatives renders the action covariant in eleven dimensions. In this lightcone form the Lorentz invariance in eleven dimensions is automatic once the little group invariance has been established. We have therefore proven elevendimensional invariance to order n. 49 . The final
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CHAPTER 5 DISCUSSION AND FUTURE DIRECTIONS The fact that these reduced fourdimensional theories remember their parent versions is remarkable. It will be interesting to see if such "memory" extends to nonmaximally supersymmetric theories as well. Although mathematically well defined the physical interpretation of the generalized derivative is unclear. The fact that it oxidizes a d = 4 gauge theory into a d = 10 version indicates that it relaxes the boundary conditions imposed on open strings (by thinking of the system as a D3brane to begin with). Whenever there is a covariant derivative there is curvature at play and this may offer an interesting way of looking at our procedure. The immediate goal however revolves around determining the Supergravity action to order k 2 5.1 Extending Gravity, to Order k 2 We now have a superspace description of elevendimensional Supergravity to order k. The obvious next step is to extend the oxidation procedure to order k 2 . However, the (AT = 8, d = 4) fourpoint interaction (in lightcone superspace) is still unknown. In this section we outline a general procedure to derive the fourdimensional quartic interaction. Once this vertex is determined we expect the oxidation procedure to follow along very similar lines (thus yielding the (N = \,d= 11) action to order 5.1.1 Conjectured Quartic Vertex The similarity between the threepoint functions of the Af = 4 YangMills and Af = 8 Supergravity theories is quite suggestive. Based on this comparison we con50
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51 jecture that the (A/ = 8, d = 4) fourpoint vertex is simply v = k 2 J j^aV0 + /3^(V0d+ 2 0)^(V0c> +2 0) j . (5.2) It remains to be seen if this simple form reproduces the full quartic vertex of elevendimensional Supergravity. 5.1.2 Supersymmetry Variations A possible way to determine the fourpoint vertex is to start from the known twoand threepoint vertices and require invariance of the action (to order k 2 ) under the nonlinear dynamical SuperPoincare transformations. Simplest among these are the dynamical supersymmetry generators already derived up to first order in the coupling constant for the (Af = 8, d = 4) theory in references [12, 31]. (5 3)
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52 This variation, clearly preserves the chirality of the superfield it acts on. Note that we can replace the j# in the above expression by d or q + . This is possible because the additional 9d + piece cancels between the two terms in the variation. To determine the fourpoint function, we need the dynamical supersymmetry variation at order k 2 . We do not yet know its exact form but have narrowed it down, based on the following requirements: Â• Helicity = + Â§ (based on the variations at order 0 and k) Â• Dimensions = [L] 1 (again, based on the earlier variations) Â• The chirality of the superfield it acts on must not be affected Â• Finally, it must preserve the "insideout" superfield constraints. The requirement that it leave chirality invariant is satisfied through "Chiralization" . The first three constraints offer an ansatz for the variation K2 ^* 2 ^{d + %^d +c c^} + ... (5.4) with the restriction, a + 6+ c = 3 + n (the Â• Â• Â• signify that more terms need to be added to ensure chirality). Once this expression is known the fourpoint interaction is easy to determine. In a future publication [30] we will derive the exact form of <^_ k2 and prove in addition closure of the supersymmetry algebra with the SuperPoincare generators. The lowest order dynamical supersymmetry variations for the (J\f = 1, d = 11) theory were detailed in equation (4.43). We conjecture that at first order in coupling the variations are obtained by simply oxidizing the fourdimensional result above. That
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53 is, Sq_ k 4> = 2k^ ( V^<9 +2 0 d + 4n d + V^J (5.5) d+ 1 a# r Y 06 This result needs to be checked but we believe it to be correct since it serves as a bridge between the twoand threepoint vertices. 5.1.3 Chiralization "Chiralization" is a descent procedure whereby nonchiral expressions are rendered chiral. For any general nonchiral expression of the form A B (where A is any compound chiral function and B a compound antichiral function) we define a "chiral product" through a descent relation in chiral derivatives C(AB) = AB + yLil! d ai an ^ Ad a n ...a lB) (5.6) where d ai ...Â«Â„ = d ai . . . d an and d a " ai = d<* n . . . d a \ C(AB) is now a chiral function and satisfies dC = 0. Clearly the descent series involves as many terms as there are supersymmetries in the theory. This procedure is equally applicable to other nonchiral forms. For example the product dAB where both A and B are chiral functions is chiralized by the addition of the term d ~d + AB (5.7) Similarly the addition of the two terms (5.8)
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54 to the expression ddAB, chiralizes it and so on. This procedure is invaluable when dealing with variations (with respect to supersymmetry and the boosts) at higher orders. This simple recipe ensures that all variations respect the superfield chirality structure. 5.2 Dual Descriptions Despite our focus on formulating supersymmetric theories in lightcone superspace, the ultimate goal (as detailed in the introduction) is to understand MTheory. This thesis studies one possible approach to understanding MTheory through its lowenergy limit elevendimensional Supergravity. There are other venues that also look promising. MTheory casts well defined shadows on lower dimensions. From the AdS/CFT [32] point of view conformally invariant gauge theories are of great interest (the rank of the gauge group is related to flux in the dual description). The J\f = 4 Super YangMills theory is dual to Type IIB String Theory compactified on AdS 5 x S 5 . Similarly MTheory when compactified on AdS^ x S 7 is described in terms of a threedimensional superconformal theory (governed by the symmetry group OSp(S\A)). This theory has not yet been constructed and will be a very interesting thing to tackle in this lightcone superspace framework. The OSp(S\4) is trivially realized on the superfields [33] using our formalism and this is a good indication that we may be able to write down a Lagrangian for this theory. A better understanding of these dual theories will certainly teach us something about the structure of MTheory.
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APPENDIX A YANGMILLS A.l Duality Theorems Consequence #1 ; fate J j^t'FtfdP = 0 (A.l) Proof: J i 8 c = 0 (A.2) Proof: /^/^/d^/ = J d* df>? & <0> d? again symmetric in a amd b. 55
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56 A. 2 Component Check We start with the Taylor expanded version of the bosonic superfield {eefd +2 A{x) + ^(60) 4 d +3 A(x) + 29 4 d + A{x) + ^6 a 6 0 C a p{x) \ff* 6 0 ( 90 ) d + C af3 (a0U) 9b V2 I e a 6 f3 (66) 2 d +2 C al3 (x) The 3point coupling in the tendimensional theory is obtained by starting with the fourdimensional result and simply 'extending' the superspace derivative = J d 10 xa (j) a (f) b V
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57 The cubic terms in C simplify to Voce =i2y/2Bd + C a0 Â—C pa + i8y/2B C ai C aS d n Cps d + (A.8) + i 2 V2 B Â— C aP C pa a"" d + C pa Which in component form reads dVoce =i8V2Bd + A i A i ^A j i 16 \/2 B At Aj dj At d+ (A.9) +i8V2BÂ—A i A j d i d + A j We work with a typical term again choosing the term involving one 0$ , one AÂ§ and two A 7 s. The first two terms then give i8V2Bd + A 7 A 7 ^A 6 ildV2BA 7 A 6 d 6 A 7 (A.10) while the third term is T = i 8 y/2 B ^+A 6 A 7 d 6 d + A 7 . We partially integrate this to obtain T =i8V2Bd 6 d + A 7 (ATI) = i8V2B^A 6 d + A 7 A 7 + i8V2BA 6 d 6 A 7 A 7 T This lets us solve for T. (terms of the form A 7 a A 7 b (...) vanish due to the antisymmetric structure constant). The final expression then reads Typical C 3 Term = i 12 V2 B j A 7 A 6 <9 6 A 7 + A 7 d + A 7 ^ A 6 j (A. 12) A comparison with the typical A 3 term determines
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58 which is exactly what we obtain when working directly in superspace. A. 3 Useful Results The extended derivative V is defined by _ _ in d a0 = 00+ Â—^ Â— 0 (A.14) [d\V] = A d p d^ [V,p] = l^d p ^ (A.15) ua^d" d 0 ,V}4> = ^a+uJ Q/3 [3d 00 2e a d 1 d^\(j) (A.16) 4 v 2 ^uafi [d af3 26 a d 1 d^\cl> (A.17) [d a d 0 , V]0 = ^d + d a0 ( j) V2 (A.18) fV L a0 \ = d a(3 Â— e p
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APPENDIX B SUPERGRAVITY B.l Duality Theorem Consequence #1 ; Â±td4>d m d + 1 d 8 d+ 3 d+ 3 d m Â— Â— d + (j>d(j) 4> d + = + Â£Â§ d m d m dd + dd m d + d m (t)dd + 4> (B.4) 59
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60 B.2 Useful Results = ^=6* d + $ ; a a ^0 = \e a ePd +2 (b.7) v 2 2 [^,V] = ^m 0K dÂ«d n ; [V,^] = ^( 7 Y K 4^ (B.8) [rf*^, V] = ^( 1 n f a d n d + ^=d K d' 3 ('y n ) aK d n (B.9) io 4 v 2 [ae", v] = _^( 7 Y Q 5^ + 0 ?e (5 d K (rr K d n d + (B.n)
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61 [v, r*\ = \{ff a % + ^d K (rr^ (B.12)
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BIOGRAPHICAL SKETCH The author received a bachelor's in physics from Loyola College and a master's in physics from the Indian Institute of Technology Madras. He is currently a Mclaughlin Fellow in the Department of Physics at the University of Florida. or,
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Pierre Ramond, Chair Distinguished Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. JU U ^ Pierre Sikivie Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. LIM JJ Richard Woouard Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. 'ZolngaB^Qkr^' Associate Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. David Groisser Associate Professor of Mathematics
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This dissertation was submitted to the Graduate Faculty of the Department of Physics in the College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August 2005 Dean, Graduate School
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