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OPERATOR METHODS IN THE SUPERSTRING THEORY By MEIFANG CHU 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 1990 To My Parents Digitized by the Internet Archive in 2010 with funding from University of Florida, George A. Smathers Libraries with support from Lyrasis and the Sloan Foundation http://www.archive.org/details/operatormethodsiOOchum ACKNOWLEDGEMENTS First of all, I would like to thank Professor Charles B. Thorn, my thesis advisor, for his guidance, patience and constant encouragement. Secondly, it is my pleasure to thank the committee members for their supervision on the dissertation. I am also grateful to the professors and the colleagues in the high energy theory group for the wonderful time I had in discussing physics with them. A special thank you is given to Professor C. Hooper and Professor J. Fry for the assistance they gave me with my visit to Princeton University in the 1986 fiscal year. Finally, I wish to thank my family and my close friends in Gainesville for the sharing and helping through all the highs and lows. This research was supported in part by the United States Department of Energy under contracts No. DEFG0586ER40272 and also in part by the graduate fellowship of the Institute for Fundamental Theory from the Division of Sponsored Research at the University of Florida. TABLE OF CONTENTS page ACKNOWLEDGEMENTS ..................................................iii ABSTRACT ........................................................... v INTRODUCTION ............................................... .... ...... 1 BRST INVARIANT VERTICES FOR OPENCLOSED TRANSITION OF THE NSR SPINNING STRINGS ...................................5 OpenClosed Transition As An Overlap ..................................5 BRST Invariant Vertices................................................20 Picture Changing And Insertion ........................................ 36 OPERATOR QUANTIZATION OF GREENSCHWARZ HETEROTIC STRINGS .............................................. 44 Covariant Quantization.................................................44 Canonical Quantization ............................................... 51 Quantum Anomaly ......................................................61 CONCLUSION ......................................................... .68 APPENDIX A PATH INTEGRALS FOR OVERLAP VERTICES........71 APPENDIX B USEFUL FORMULAE ................................75 REFERENCES ....................................................... 77 BIOGRAPHICAL SKETCH..................................... ........80 Abtract 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 OPERATOR METHODS IN THE SUPERSTRING THEORY By Meifing Chu May 1990 Chairman: Charles B. Thorn Major Department: Physics Two aspects of the superstring theory are st died in the operator lan guage. The interaction between the open string and the closed string is dis cussed in the NeveuSchwarzRamond formalism. Transition amplitudes are given by the overlap vertex of the string wave functions. To ensure the BRST invariance, a supercurrentlike operator insertion is required. By shifting the picture of the open string vacuum, this insertion is identified as the picture changing operator. Its role is to keep both the open string vacuum and the closed string vacuum in the same picture. This is a consequence of the nonzero superghost background charge in the RNS formalism. Also, a canonical quantization of the GreenSchwarz superstring is pur sued here. A new method of operator quantization developed by Batalin and Fradkin is applied to the heterotic string in a semilight cone gauge. Using a normal ordering regularization, it is shown that quantum anomalies cancel at the critical dimension, D = 10. INTRODUCTION String theory is considered a promising candidate for unifying all the fundamental forces of the universe. It describes the propagation and the inter actions of a one dimensional objectstring. Its low energy limit is equivalent to an effective field theory of point particles. For example, the low energy limit of the open string gives the YangMills theory at the tree level 1 and that of the closed string gives the perturbative gravity of Einstein.2 Among all the theories for strings, the E8 x Eg heterotic string is the most promising one to unify the four fundamental interactions. There are two different formalisms for the first quantized superstring; the RamondNeveuSchwarz (RNS) and the GreenSchwarz (GS). The RNS formalism is manifestly Lorentz covariant, and it contains the ten dimensional space time coordinates X' and ten world sheet spinors ?k. The GS formalism is manifestly supersymmetric and contains the coordinates XP and the MajoranaWeyl spinor 0 in ten dimensions. It has been shown that these two formulations are equivalent in the light cone gauge.3 They are related by the SO(8) triality. One important content of the string field theory is the description of string interactions. Similar to the Feynman rules for the particles, string am plitudes are described with propagators and vertices. It is natural from the geometrical point of view that strings can split, join and interchangea combi nation of the first two. If g is the coupling constant for the splitting or joining of strings, from a crossing symmetry argument,4 the interchange of the strings 2 should be of the order of g2. Including both open strings and closed strings, there are at least five fundamental interactions, S= 1 (go 4o o + g2o o o* o0 + g c* 4c c (1.1) + go c + g24c o o) where 4o,c correspond to the open, closed string wave functions. Different string field theories give different descriptions for the products among the strings. But on shell, they should all reproduce the Dual Resonance ampli tudes. For example, in the light cone parametrization, Mandelstam set up the path integrals for the scattering amplitudes of the RNS spinning strings. He gave explicit Neumann functions needed to construct the npoint functions for the open (or closed) strings. Green and Schwarz constructed the overlap vertices for the first four types of interactions in the light cone GS formal ism. In Witten's open string field theory,5 Cremmer, Schwimmer, Thorn6 and Gross, Jevicki7 constructed the threepoint vertex for the open bosonic strings. Zwiebach and Sonoda studied the closed string threepoint vertex.8'9'10 By showing that no threestring vertex can generate a BRST invariant offshell fourpoint amplitude, they concluded that covariant closed string field theory could not be cubic only. As to the interaction between open strings and closed strings, Shapiro and Thorn11 constructed a covariant transition vertex between one open string state and one closed string state. However, they discussed the transition for bosonic strings only. Among all string vertices being studied,12 the coupling between the open string and the closed string has received less attention. This is partly because none of the existing covariant string field theories consistently contains fields associated with both open and closed strings. So far, there is a self consistent open string field theory by Witten, but there is no satisfactory covariant string 3 field theory for the closed string alone. The openclosed transition vertex operator can provide a tool for us to construct a closed string theory from the existing open string theory. In the first part of the dissertation, I will extend the openclosed transition of the bosonic string by Shapiro and Thorn" to the RNS spinning string. These transition vertices will be constructed from the overlaps of string wave functions. A supercurrent like operator will be inserted to the vertices in order to maintain the BRST invariance. This insertion will be later identified as the picture changing operator in order to keep the superghost vacua for both open and closed strings in the same picture. Because of the manifest Lorentz covariance, string field theories are usu ally studied in the RNS formalism. However, from the point of view that supersymmetry is very important in explaining finiteness and renormalizabil ity, it is more "reasonable" to study the string field theory in the GS formalism. Unfortunately, the gauge structures of the GS model make the covariant quan tization very cumbersome. The second part of this dissertation will focus on the quantization problems of GS. In particular, a new method proposed by Batalin and Fradkin will be applied to the operator quantization of the Green Schwarz heterotic string in the semilight cone gauge. This is meant to be an intermediate step between the lightcone quantization and the covariant quantization. The idea of the BF method is to introduce new variables to convert the second class constraints into first class. One can then construct a nilpotent BRST charge to generate the constraint algebra. The hamiltonian and other physical observables can be constructed in the extended phase space by impos ing BRST invariance. However, in the BF formalism, quantum corrections to the observables are ambiguous. They are usually fixed by hand to retain the 4 symmetries of the theory. For the GS heterotic string in the semilight cone, I find that there exist unique quantum corrections such that, at D = 10, the nor mal ordered BRST charge is nilpotent and the super Poincare algebra closes. Thus, there is no quantum anomaly in this gauge. This conclusion agrees with the path integral result by Carlip13 and by Kallosh and Morozov.14 The reason for studying the superstring theory in the operator language is that the operator method is more efficient and unanibigous in comparison with the path integral formalism. From the second quantization point of view, it is desirable to do the first quantization canonically. As to the string interac tion, the difference between the two languages can be seen as follows. In the path integral language, string amplitudes are calculated by integrating over the string world sheets with vertices inserted at the punctures on the surface. These vertices correspond to the emissions and the absorptions of the string states. There are infinitely many string excitation states and thus infinitely many corresponding vertices. However, in the operator language, it is sufficient to construct the string propagators and the overlap vertices for the fundamental couplings in (1.1). All string amplitudes can be obtained from these building blocks. Therefore, operator methods are chosen to study the superstrings in this dissertation. BRST INVARIANT VERTEX FOR OPENCLOSED TRANSITION OF THE RNS SPINNING STRINGS OpenClosed Transition As An Overlap The covariant openclosed transition for the bosonic string has been stud ied by Shapiro and Thorn.11 They constructed the BRST invariant vertex of one closed string state transiting into one open string state, and vice versa. I would like to extend their construction to the RNS spinning string. Although Green and Schwarz 15 had constructed, in the light cone GreenSchwarz su perstring formalism, a supersymmetric transition vertex between the open and closed strings, their vertex was not covariant. Therefore, it is interesting to construct the openclosed transition vertex that is both Lorentz covariant and BRST invariant. Generally speaking, string interactions can be described by the overlap among the strings. Here, the overlap means that the string world sheets are patched smoothly together so that string coordinates are continuous over the boundaries where the strings are connected. Certain singularities may appear at the interaction points where the strings join or split. Each string field theory gives its prescriptions on how strings overlap. Such prescriptions must respect symmetries like duality and unitarity of superstring and also reproduce the correct onshell tree amplitudes. I will use the light cone striplike world sheet picture to study the openclosed transition for the RNS spinning strings. Let < rl be the transition vertex which acts on one open string state and one closed string state. In order to give reasonable amplit udes, < rF must 6 meet the following requirements. First, string coordinates and physical opera tors of the open string should be identified with those of the closed string at the boundary. The transition vertex should therefore obey the corresponding overlap conditions. Secondly, unphysical states must decouple from the phys ical amplitudes. This means that < FI should be annihilated by the BRST charge Q. Knowing these properties of the transition vertex, we can divide the vertex as follows. < fol =< VBI < VF < (cb)l < (,/3)(insertion) The first four parts are the overlap vertices for the space time coordinates, the spin variables, the reparametrization ghosts and the superghosts. Shapiro and Thorn constructed the overlap vertices for the bosonic coordinates and the reparametrization ghosts (cb). For completeness, I will quote their result here. Fr Xcr) = q + p + EnO (Ane2in(rT) + A 2in(r+)) XY(r, o) = q + pr + En 0 2A ncos(na)enr" For { (r, a) = q + p + Zn iO 2a~eos()i B(r, a) = En(Bne2in(r) + Bne2in(r+o)) : (Bn n) BO(r, a) = 2 E bi cos(n)einr and a similar expansion for the C ghost, < TI =< VBI < (cb)l < B I =< 0 exp(W) 00 _\k+1 1 1 00o k+l 1 \ 1 00 k,l=0 + 4(k + + 1) k I a2k+1 a21+1 E 2k(Ak + Ak) a2k k,l=0 k=1 00 /k+1 1 1 +_ 2 2 1( k ) ( )a2k+l (Al il) =(2.1) (2.1) 7 And, the overlap vertex for the reparametrization ghosts, < (cb) H( (l)ml+n 1f / I < exp, I 21c~b E CA60b 01ep 2(7 + n) n m n  _l+k+1 1 131 + 2(1+ k + 1) 2 c k+1l+1 k c2b i,k>O m>1 i ln 1 3 + ^ g ^S 2 V2. '2 fiS ( b2> + c o(bi boll) 21 1 cb+ m>l m>1,1>0 2 cm(bm o b ; 2k +1 22n E 1 k+lb m> k,n0> (2.2) These vertices are constructed from the overlap equations of X/^ and the bc ghosts. In a similar fashion, we can construct the fermionic part from the overlap equations of 'P. Of course, one can also construct the overlap vertex using functional integrals according to Mandelstam.16 The Neumann functions for each openclosed coupling are given in the appendix. Both methods give the same transition vertices. We can therefore be very confident about constructing the transition vertices from the overlaps of strings. Also, < (fly) will be constructed from the overlaps of the superghosts. Besides the overlap vertices, there is an insertion operator which was not present for the pure bosonic string case. In the path integral of the light cone string amplitudes, Mandelstam inserted the supercurrent at the interac tion points to recover Lorentz invariance. Similarly, in our covariant formula tion, the supercurrent insertion is needed for BRST invariance. This ensures that spurious states decouple. There will be ghost contributions also to the insertion which are determined by imposing BRST invariance of the vertices. This modified supercurrent insertion can be interpreted as the picture changing operator to keep all superghost vacua in the same picture. 8 Before we begin the construction, recall that for the RNS strings, there are two different sectors in the open string; Ramond sector and Neveu Schwarz sector, corresponding respectively to spacetime fermions and bosons. In closed strings, there are four different sectors; NSNS, RR, NSR, and RNS for the left moving and the right moving modes. The first two sectors describe space time bosons and the other two sectors spacetime fermions. Therefore, the couplings between open and closed strings are restricted to three types: (1) NS open strings couple to NSNS closed strings; (2) NS open strings couple to RR closed strings; (3) Ramond open strings couple to NSR (or RNS) closed strings. The idea of constructing the overlap vertex < VFI for '# will follow those for the overlap vertex < VB I for XP in Ref.[11]. Lorentz covariance will be maintained throughout the construction. Since the overlap of two strings on a strip is identifying the point a on one string to the point 7r a on the other string, an operator of conformal weight J acting on the transition vertex should have the following identities:7 < VoclAJpen( a) ()J < VoclAoseda The phase factor comes from a conformal transformation which takes a of the closed string to ir a of the open string on a strip. Let 0o,c denote the right (left) moving fermion field of the open (closed) string. The standard mode expansions of the i field are, 0,." A= Foo bl/'iror V r=iRoo r fr is a half integer for NS sector ' ei2rra I an integer for Ramond sector 2r=oo Bei2r7 9 Since 4 is a conformal operator of weight 1/2, there is an extra phase i in its overlap equation. < VFIl' (7r a) = i< VFI+'c () (2.3) Let me outline the idea of constructing the transition vertex from the overlap equations following Ref.[11]. First, one should mode expand the overlap equations of 0 as follows. < VFI B~Nr B NB = 0 \ s>0 / These equations determine the fermionic vertex directly. where Nrs = Nr s Nsr That is, we should expand these overlap equations as one creation oscillator acting on the vertex and giving infinite sums of annihilation oscillators. This particular mode structure is projected out by the socalled moment functions. They are functions which have the required asymptotic limits of the open string and the closed string and they isolate only one creation mode. The guideline of finding these moment functions is to take a chosen polyno mial, multiply it with the conformal mapping function to some power that corresponds to the boundary conditions of the strings. The polynomial will be determined by the asymptotic limits. It is the conformal mapping function that controls which interaction prescriptions we use, lightcone, Witten's, etc. Since the construction of mapping functions varies case by case, I will come back to this later when explicit calculations are carried out. 10 In general, given a moment function, we can expand it into an infinite series, En h en(rFio), as r oo, h:(, T) = En hCe2n(r:ia), as r oo The overlap equation Eq. (2.3) is then multiplied by the moment function and integrated over a at r = 0 where the two strings are connected. j doa < ri i+(o)h () + i c(a)hI(u)} = 0 Then, one can read off the overlap vertex directly from the overlap equations according to Eq.(2.4). At last, to maintain the BRST invariance of the overlap vertex, we should insert the supercurrent at the interaction point. Note that the conformal map ping function from a strip to the upper half plane is /e2r+2i 1, the super 3 current insertion diverges as 4 near the interaction point. < rol < VB < VFIIO =< VBI < VFI(4+O+X)(O,O + e) (2.6) The decoupling of spurious states will give a good consistency check for the overlap vertex. Let us now go to each coupling and construct its vertex explic itly. NS Open String Coupled To NSNS Closed String For the NS sector, I choose the vacuum of the fermionic strings to be the one annihilated by all the positive modes of the spin variables. Bi'O >= 0 for all i and for r >1 r 2 This is purely a choice of convenience. Once the vacuum is chosen, the creation and annihilation operators are well defined. We can now construct the moment 11 functions h(x) according to the outline that we gave in the beginning of this chapter. Denote x = eP where p = r + ia. (a) Single out B3r + iBr, r > : Let h+(x) = P(x2)(x2 1)/4 21 + o(x). (2.7) x  0 With the power 1/4, the asymptotic limit goes to a negative odd integer power of x for the NS closed string. P(x2) is the polynomial that serves to isolate the creation mode. It can be found by truncating the infinite expansion of the inverse mapping function. Only the negative modes are kept. P(2) = negative modes of [x2m1(x2 1)1], as x + 0 = ei x(2m1 (1 (7)lX21 1=0 Substitute this polynomial into (2.7) and expand h+ in the limits of the open string and the closed string respectively. Two sets of moment functions are obtained this way. as x 0, h = ei 2m1 ( )1x21 ei ( ()kr2k 1=0 k=0 oo 1 \+1 5 3 = x2m1+ ( q)2+1 1+m+ l f1m 1=0 as x + oo, =i =2nl )/x2 k 2k 1=0 k=0 S 7m+1 +13 = e 1+1 K2 1=0 I n + where I have used the identities in Appendix B. Choose h = i[h+(x)]*, and substitute them into Eq.(2.5), the following overlap equation is obtained: S0 I m+1 5 3 < VFI{Bm i+Bm + E 1 + (m + )1 2 1=01+M (1 + i) b21+ } = 0. 1=0 4 (2.8) (b) Single out Br iBr, r > ": Let h+ = P(x2)(x2 1)1/4 z2m1 + o(x) x  0 P(x) can be derived using the same trick in (a). P(^= ei2m1 x21 1=0 as x  0, h' = e i 2m1 4 \(_lx2 1=0 00 /n+ /m+ 5\ = .2m1  1 4( l  1=0 m+1 m l= 0 k=O 41 x21+1 *^^If as x 4 00, =n ho = ei4x2m1 1=0 00 1/ n+1 / 5 = ei! 1 =0 + m  1=0 '4 )() 1 kz2k 4) (7 ' S()i21 E k=0 3( 4)T 21+. I2 Again, setting h = i[h+(x)]*, we have 00 l(m+l 5 + (1 i)( 1 b2+_ }= 0. 1=04+m1 m 1 (2.9) 4 iB1+ ) (c) Single out b_2m,n, m > 0: 2 Let h+ = P(x)(x2 1)4 zx2m + o() x  00 For the open string, the polynomial can be obtained similarly as in (a) and (b), but only the positive modes are kept. in P(x) = x2m+1 E(_) 1=0 (121 as x  00, in 1 1 S= + () 1=0 /\\l 00 21 1 I k=0 5(/ 7 00 1 \l+m =x2m + 1 + + I=0 3 \ )4 X212 in2 m +/ h' = x2m+ 1 l=0 00 1 +m n 5 q1 ] 1=0 After setting h = [h+(x)]*, we have 00 1\ x1: k _) k 2k k=0 X21+1 km/ S77 21 ) b21+3 2 o 11+ l  1 i=0 Q +( m ( (43) in7 (d) Single out b_2m _;, m > 0: 2 Let h+ = P(x2)(x2 1) X 4 00 as x  0, (1B + iBl+ ))=0. (2.10) k1)( (^) x2k 00 / \l+m / 1+1=+m + I= 0 < VFl(b2m 2mn + o(4 ). 2m '+3 /1\ Where P(r) = x~'"+ i 1 21 1=0 as x  oo m1 1=0 / lx I '~ 1 / 00 (_)'.2'1 k=0 2m+ )[+11 1=0 as x  0 , S=2m 4 ()1x21 1=0 00 1 5 S)'+ 1 A +,,, 1=0 + With h = [h+(x)]* we have VI, 1 )+,m 5 < 2m 1 + 1+m m ) 1=0 l+i 0 )1+0 2 1= + 7l Thus, together with (2.8),(2.9),(2.10), these overlap equations determine the fermionic part of the transition vertex completely. The result is listed in the following. < VF= < Ojexp 1++ 1rn+ Bm+l'+ 1 + 1+iB 1=0 1 + l_1 +77 15 3 + 1 b2 [ m+ 1 iBm+ i I=0 4 m +"m( 2 2 2 i0 + m I m b [ +B 00 1(./ 1+m + .1 + 1 + n 1=0 (2.12) 00 / 0k k=0 ( 3 I 14 2 5 3 (q) iB ,)) =20 )2 2+f (2.11) I (_ks2k+2 ki (J)(y_ 5 3 M4 4z2 _)kx2k S I ) h21+ 2mn+2 15 Next, we insert the supercurrent to the vertex near the interaction point (a = 0 + e). < VBFIo =(2e) V < VB,FI ( )+m ( m,1=0 (2.13) a2m+l i(Arn Amn)] [b21+1 ((l iB,] [a 2 2 2 It is straight forward to check that the spurious states decouple from this vertex, which assures the BRST invariance of the vertex. I also find that (2.13) agrees with the one from the path integral calculation (see Appendix A). Before we go to the other two couplings, let me pause for a moment to comment about the uniqueness of the fermionic vertex. It is independent of the moment functions selected, as long as the conformal mapping function stays the same. For example, let us replace the power 1/4 for the moment functions in (a) and (c) with 3/4 instead: h+ = P(x2)(x2 1) z2m+ + o(x ) X + 00 h+ = P(x2(2 1) 2m1 + o(x) x + 0 These will give us two new sets of overlap equations: o0 3/ l+m 1 7 + ()(117 1)b2 < VF Ib_2m + m 1 1=0 S0 l+m 1 7 + (B11 + iB+ = 0 for 7 > 1 2 l=4 +m i 1 ( 2 2_ < VF ((Brn + iBm, ) + +1 ( (( B + iB+) 1>0 0' 3( (+1+n 1 ) 7 +(1 + i) E l 3 1 m1 2 16 At first glance, these identities may be different from what we obtained earlier. However, we can replace b_1 by 2 < V i_ =< VFIl(( 2 ) (B+ + iB,++) (2.14) (4 + 41) ) 2 1>o and the new identities return exactly to the ones we had in Eq.(2.8) and Eq.(2.10). Therefore, as long as we keep the same conformal mapping factor in the moment function, any other variation to the moment function should not affect the final results of the transition vertex. NS Open String Coupled To RR Closed String This coupling involves the Ramond zero modes in the closed string. Its vacuum has two helicities defined by D0 iDo. To compare the result with the path integral calculation, we choose the ground state to be I >, which satisfies (Db iDo)l >= 0. To expand the overlap equations, we use the following moment functions to single out one creation operator for the closed or open string. Notice that in the closed string limit, h should go to a negative even power (Ramond sector) asymptotically. (a) Dm + iDm : h+ = P(x2)(2 1) 2m + o(l) z 4 0 00oo 1/ \l+m 5 3 he = z21n + l  + ) 4 21+2 =+m+1 1 m 1=0 00 l+ 3 5 h=o 4 +1 (~m m') ( (b) Dm iDm : h+ = P(x2)(x2 1 2m+ cost. x  0 00 I m /3\ 5 h' = x2m 1 1( ) (+7 2) + 1= + m + r m 1 1=0 (c) g2m : 2 S00 / 1 I+t+l / 3\ ho iW/4 H I+m+1 1=0  1 ) 2( 2 4 ,2m+! 1 2 4 x 00 S 4 +4 (j ) x21 h+3 +m m 1 1=0 I 00 I+ 1l+m) 5\/ 32 + = 32,n+ E 4 ;Z f4 121 1=0 + (d) 92m : h+ = P(x24(x2li 2 2m+f + o(xl) x + 00 00 1 l+, / 5 3 / x \1 3\ hc = ei/4 E ( x21 1 l +m\ m 1=0 I o 2m+l 0 4 (4 21 12 ho = 1 f + 2 + Y+m+l1 11 M 1=0 Substituting these functions into (2.5), one obtains < VFI according to (2.4). 100 1 1+m 5 \ 3 < VFI =< Iexp( ) (1) () [l + iD,] [Din iDm] + 1+22 + l,r=0 1+1+m77 2 1 by b2m+ 1+ i 4  + I'1 + l,m=O1 I 100 i l+m _5_( 3 , )( [D61 + iDt] b2m+2) =,=o +1 m 2 m (2.15) (_ 21 I ./ f ( 4[ iDl] b2m+i 18 Again, applying the supercurrent insertion, the BRST invariant vertex for NS open string coupled to RR closed string is < B,FI1o 31 =(2E)4 < VB,FI ( ,( ) (ml [a, I i(Am Am)] (2.16) l,m=O S1+(l+i b213 + 2 )(D1 iDj) Ramond Open String Coupled To NSR (RNS) Closed String This vertex describes the openclosed transition of spacetime fermions. It has rather different features than the vertices we discussed earlier. First, there are two ways to define the helicity of the vacuum: either 1(DO2k1 + iDo2k)[ T>= 1 (d2k1 id k) T>= 0; or D+l t >= 1(d + iD)Ifl >= 0. The path integral result corresponds to the former basis. However, we find that it is easier to construct the whole vertex with the latter. Second, because of the asymmetry of the leftmoving and rightmoving sectors in the closed string, the moment functions will be generated with a slight modifica tion. For the right moving modes, we have h+ = P(x)(1 x 42l(v V, +) xk + o(l) p : 00 For the left moving modes, we use h_ = P(x)(1 x2fV(/ T ^/+ )  x^ + o(l) p 00oo where is the complex conjugate of x. Similar to the other two couplings, these moment functions are expanded in modes and substituted into the overlap 19 equations. To avoid redundency, I will not list the expansions of the overlap equations, but list the result of the fermionic vertex in the I T> basis. < VFI = Wh =Z 2 2 (21+1 2m C21 2m1 S4(1 + I + m) ( 212,r 21/+12n2m+ ( 21 + 2m + 3 (21 22m  211l2m 21+ 2m + 1 (11 2m+  1 2m+l 2 p$2m S((2nm 1 2n p 2m Ip2m I 1 2m+1+ ) (lICm + (1 lnCm1 rn ) [Di + D+ + B .B ] 1+ 1 Di+1 1 B+ D,,+ 1+ m  DC+2m 1+ )1 1+1 n+ l) (1 + i()') d D 1) ( i()') dl+ Bm+ +1 m+ S1 i()) d din+1 iem+l(d4m+2 Dm+l) + ien(c2 D;) i2(d2m+l Bm+n) i2+1 (d +l 1 2+) 22 S1 (D2k1 iD2k) AB = (A42k'1 i2A 2)(B2l T iB2) k=1 (2.18) I and m are summed over from 0 to oo. The coefficients (n, en are given in Appendix B. This vertex agrees with the path integral calculation in Appendix 1i 4 1+i 4 1i 4 1i 4 *^ ^ 20 A, as expected. However, if we choose the ground state to be 10 > instead, the vertex becomes < V I =< liexp(1 ( 16 P21i2mn+I P2m~l21+ 1 I1 m+2 DI+ Dm++1 l+m+21 'Dr n+l (P2mrn121 P2112 B B ., iTm 3  ^+4 B,,,+ + fP2,nm21 ~ P211C2,n+1 4 E ( 21+ 2m + 3 1+' i + i2 l#2m+1 P21nm PlI (2m+1 S2m I + 1 (1 i()) dl+1 Dm+ + 2 +l)2,+ + Fn,+2 (2m,+)d,)m+2 Dmn+1 1i +8 p#2m P2nm1 1 Pl 12m 2m nip + ('2,, E +)d,,, + Bm1+  (1 i()m))) () l + 11m2 P+ ) 1  + '+ (i?'2m+2Dn+l 72rnm+IBSm+ m=0 + 2 (1 i()m) lm+ljdm+1)) (2.19) where pm and rm are given in Appendix B. For convenience, we shall use 0 > as the ground state to construct the complete vertex. The insertion corresponding to this basis is < rol 00 F ()( m+1 i(Am An)]  1,771=0 ,m=0O i 1 (D+ (P21 2(21+1)D1+1 + (P211 +i i()(1 2C)d+) +2". (2.20) 1()d+1 Bmi(+1) + Bi dl+1 dm+1 =< VB,Fp(2E)~  2(21)B1+1 21 BRST Invariant Vertices In this section, the construction of the openclosed transition vertices will be completed by including the ghost contributions. The idea is to keep the BRST invariance of the vertices. < FIQBRST = 0 (2.21) To simplify the formula, I will use the following notations: Lm = da 7 JW 7r fd7 Lmo =Im Gro = g , bno = bn Cno = c l SPr = Or {7ro = 7r e"i(4Y(O OAX, + +^) a"+ 11) 2 + and Lim = j (m Ln) and G,.* ( ,, iGr) 1 ~ and b?:, (Bn Bn) and cn = (CnCn) and ,. = (Br+ iBr) and Y ( T irr) V2_J; (2.22) The onshell conditions, (Lo ao) Istates >= 0, are 1 = { in the NS sector ao+ = ^ ao0 = 0, Q00 2 0 in the Ramond sector Also, it is useful to define the following constants 2 for (ijk) =(+ + +),( +),( + ),(+ ) dijk = 1 for (ijk) = (ooo) 0 otherwise for (ijk) = ( ),(+ +),(+ +),( ++) for (ijk) = (ooo) otherwise 0 22 With these new notations, the BRST charge can be written as QBRST =E c n(L u,,00) + 00, r 00 00 m +(m n) : cm A : djk : Uijk 2c c 00 00 + 0(m + r) : ci rT +r : dijk 00 (2.23) To solve Eq.(2.21), let us consider the following ansatz. < Ir =< VB,FI < ghosts (0 + E 11';, 3 + Zsc/i mj' r>0,s>O m>1,s>O / (2.24) where < ghosts =< vacuumI exp ( MnjnCm bn + E Yr'is ir ) m>1,n>0 r>0,s>0 It is not difficult to guess the structure of the ansatz. < ghosts is basi cally the overlap vertex of bc and #y. The cb ghost overlap vertex was given in the beginning of this chapter. The coefficients M are listed in Table 21. The superghost overlap vertex (Y) comes from the overlap equations of the supercurrent. This is because the f superghost has conformal weight 3/2 also. Therefore, we don't have to derive a separate set of overlap equations for the superghost vertex. The new insertion can be understood as an extension of the supercurrent. The first term I0 is the supercurrent insertion that we constructed in the last section. The second term 7 b has the same conformal weight 3/2 as the supercurrent. The cf terms are included in order to satisfy Eq.(2.21). In other words, constructing the complete vertex is to find the coefficients M, Y, W and Z that satisfy Eq.(2.21). Let's substitute the ansatz into Eq.(2.21). 0 /^ 0 IN C4 CD C C,1 o 0 'c tcM I ^ 1.... 0o o 0 s 2 I o 11~ *CO II Si^ ^ *^ C o ~0 micN 0 mI" In +C 45,>, A e eo 0 23 I I I iiII II II I .II  I II 24 The linear terms of C and in (2.21) can be interpreted as a set of Ward identities for the energy momentum tensor and the supercurrent. Based on the consistency between the Ward identities and the Virasoro algebras, the terms of cubic ghosts will cancel among themselves. This argument follows those in the pure bosonic string case."1 In general, these W\ard identities are of the following forms : < VB,FIo(L. + EAI Li K,) = < VBl E rG for m> 1 n>O r>0 (2.25) < VB,FIIo(Gir ;s G) =< VB,FI( E W L + T) for r > 0 s>0 p>O (2.26) Notice that the right hand sides of these identities are singular at the interaction points. This is because operators with conformal weights greater than 1 diverge when acting on the vertex. Moreover, these singularities deter mine the ghost insertion (W, Z). To derive these identities, we first construct the overlap equations of the energy momentum tensor and the supercurrent. < VBFI(L' + AE. L", k) = 0 m> 1 (2.27) n>O < VB,FI(Gir ZE Y;ij) < VB,rFIlIo r > 0 (2.28) s>O In the Ramond sector, the zero mode is related to the insertion in the following way: < VB,FI( Fo + IpFp) = a < VB,FI (2.29) p>O The difference between the Ward identities and the overlap equations is the I0 insertion. Replacing I0 in the Ward identities with the supercurrent modes in (2.28) or (2.29), we recover the overlap equations. Thus, the su perghosts insertions HI, Z, K and T of the ansatz can be determined from 25 M, Y, CD, w, k in the overlap equations. It should not be too difficult to con struct these overlap equations. For the energy momentum tensor, its overlap equations remain the same as for pure bosonic strings, except the different constants k due to the extra fermionic contribution. For the supercurrent conformall weight 3/2), one can use the same set of moment functions for i (weight 1/2). NS Open String And NSNS Closed String At first, let us discuss the ghost vacua for the NS open string and the NSNS closed string. Recall that the reparametrization ghost vacuum is doubly degenerated. I will choose the up" vacuum for all three sectors. bn T>= cn T>= cl T>= 0 for n > 1 and for all i (2.30) This choice will be independent on the sector (NS or Ramond) of the string and will be used for the other two couplings also. On the other hand, the superghost vacua are infinitely degenerate and depend on the sector also. We must first specify the picture of the vacuum for each string. For convenience, I will choose the superghost vacuum for the NS open string and the one for the NSNS closed string to be both in the 1 picture, i.e. they are annihilated by all the positive modes of the superghosts. f^ (1,1,1) >= o 1 for all i and for r> (2.31) l4 (1,1,1) >=0 2 As mentioned before, the overlap equations for the energy momentum tensor are the same as in Ref.[ll except the constants k. They can be evaluated by applying arbitrary string states to the equations and requiring them to hold exactly in ten dimensions. D = 10. The results are listed in the following. 1 3D < VB,FI(+ L+ L,,, L) 32'(m 1)) = 0 VBFI( + 2(1 + m) 1 m 1 1>0 ( '1 2 Lol+1, =0 2 1 2211 1 1 2 < VB,FI2m L L 2V(Lrm L+) (m + 1)) = 0 < VB,FI L2m1 + 2(1 + + m) ( 2.32 + 2V2 i 2 + 1 21( = 0 1>0 The overlap equations for the supercurrent (weight 3/2) are similar to those for (weight 1/2), except for an opposite sign between the open string and the closed string. < VB,FIJ0(r a) 2 Ti < VB,FIJ() We can mode expand the equations with the same moment functions for V). However, the supercurrent has a, higher weight than 0. It is more singular when acting on the overlap vertex. This appears at the interaction point with an insertion IO. Such singular contribution (,' ) can be determined by requiring overlap equations to hold exactly when acting on arbitrary string states. 1 \I+m / 5 3 < VBFG+m 1 =< VBF ( GI ) 1 2+ 1+l +m 1 / +m mY ++ 1+ gin 2 (2.33) + i8 4f2 9+'21++ () n 11 ( 1) 3m) 2 f \2 n 27 5 3 < V G =< VF) (1)r + < VBF, =2m=< VBFl(( +1+ (n 2 r (2 1) __( )I(flm (2.35) 1 5+m 3 5 3 < VB,Fg2m1 =< V IB,F ) +?) 921+ S1+(1+ i) m 1 5 3 S24 G+ (_)m  f+lm I m I+ m (2.35) (2.36) These identities tell us the coefficients Y, w, and a's, see Table 22. To derive the Ward identities, we insert I0 next to the vertex in the above overlap equations and take all the creation operators to the left. The coefficients for the superghost insertion in the ansatz can be determined. < r = < VB,F, < (cb)l < (70)1(lo + E irji" + ZE zcPi i) Wrn = E( f ws 9yik 2iu) 2k +nikj S r ns r,ns s8 r+n s>O r1 +2 rM s,n+ r+iio. 6n o(21 . ZW = M ( 3n)w L idklj + (S + il n>O s=1 + 3 + _ 1I, + LyG1. 3mom 1 o wk+m(r + )dijk .',owl b 2m+,r + ( +s6 3 1o i 3m 1 + (, + ) WO a 2 4 ms+ 2 m (2.37) *, l o. CD +) 31v '3 + 0 ,II + + + I I + I E II "T I II  f I * II II 1 I> II0 + s. o II I II II I II o+ II I I I A I 4 C  II II I I ^ ^^ ^ and the constants rnf i ., l ikl 1 io io" K = ki + Z(s + )Wso sdikI I ,,0o + "o s=l m k3m 1 + &i312nW o (2.38) 2 ,72 T = w srsUij + (Ulio + uti+),.(r2 ) fk + 02kr+ 2s= NS Open String And RR Closed String Due to the presence of the Ramond zero modes, the superghost vacuum has two different helicities J >. For convenience, let us choose the superghost vacuum to be {f >=7 >= 0 for n, > 1 1 and POLI >= 71+ >= O{I >= 7 >= 0 for r 0 > Also, the onshell condition for the Ranmond sector differs from the NS sector, we must reevaluate the constants k in the overlaps of the L's. < VB,F\Li L L L) 1D (3nm + 1) = 0 W2 2 0)  im _1 3 < VB,F (Lm + E 2( + m 1) L 1>0 i (_)t+m 3 + 2W 2mn 21 ) 1 m1 +1) 0 < VB,F\ L2m LO 2V2(L L) t(3m 1) = 0 VB,FI( L2rn1 21l (1+)I+7+ 1 1 3 =o +l+m 21+ 1 + 2v\2i 27) LT) = 12m l + 1 Am1 /" 1>0 To mode expand the supercurrent overlap equations, we can use the same moment functions for o's. They are + 1+1+ 13 5 (1) (4)921+ ) 1 8 1_)l+m +1 3 5 1 0\+7m+l 3 5\ < VB,FFm =< VBFI( 1 +1 1+) + +E 1 ( ) 921+1 mm < VB,F IF =< VB,FI(E) ( )F++ + +i ()~ I)921+ 1>1 1>0 1+i + 1 + 10 + 2\/2 !>3o 1lC+m /3\ /5 S_ 1+I+m (3 I)5 < VB,FrlI2 =< VB,F\ (+ 4 1 21m 13 5 3 (2.40) These equations give Y, w, and CD listed in Table 23. After I0 is inserted next to the overlap vertex in (2.40), W, Z, K, T in the superghost insertion of the transition vertex can all be determined. Ml' q iu),T I'1 CClp I + IO _. I + I ,I Z+ IIm + + 'i IV + II+ II + AI II a o0 00 II I II t 3 +1 I I i I I AA A6 I II E E 9S aII II 3D '3 ^ * 0 bo V <4.) 0 u x k a 8 m hlr'e II orn = i kr knsklj + li kil 2 kns Irsu k s>0 + 6,0u++J + 2f u+lu + 21u+il" l 2ukijk+r 2u+kjyik  rl n+r rn oo Zmr s= ins,rd" ( + z w> n.And'(r  s>0 n=0 m d+'il j w+mdkil(r + 3m) r d+lj Mi 2+ mr +2r+ 2 mr + 3 ,+w d+i+ 6,. m d+i And Km = k + (p + I)k p diki + 7nd+l +i+ p0 "22? 2" V2 aTr' = 2u+iik +2 w+k nu+il + +i r2 + iowoD(r2 1 22 4 n=1 r, s are positive half integers for the open string and positive integers for the closed string. And, a = .+i Ramond Open String And NSR (RNS) Closed String Because of the asymmetry between the left moving sector and the right moving sector of the closed string, the superghost vacuum is chosen as follows. BnI >= rnl >=0 for n> 1 1 BrlQ >= NolR >= 0 for r > PnIn >= n7 2>=0 for n> 1 (0lo + iBo)l0 >= ( 1o iro)l >= 0 Constants k in the overlap equations of the energy momentum tensor are re evaluated also in this basis. VB,FI(L+ L+ (L L 3) (3m 2)) = 0 ,n 032V2 < VB,FI(Lm 2(1+ m) ( 1>0 ( 3 M 1 on1lLi (a ()+m /1 + 2m 21 1 i I>0 (31) m 1 21+1i= < VBF L12m LO 2(L ( L+) D(3m + 2)) =0 < VB,Fi(", 111 0~L~ +) 16 )1+ (_)l+ +m 1  2(1+1+ m) 1 + 2/i )'+ 2>0 + 1 21 __O )l (n)Ln+l 2 L2 =+1 S1 m I 3 In order to expand the overlap equations for the supercurrent, we need to modify the moment functions by subtracting the polynomial at x = 1. The complication is due to the singularities at the interaction point and the asymmetry in the closed string. h(x) = [P(x) P(1)](x2 1)(v x 1 +x) SXm + o(1). X + 0,00 Expanding h in the open and closed string limits, we obtain the following overlap equations for the supercurrent. ((2pP2mn1 C2mP2p1 G 8(p + m + 1) P m > 1 i (2mP2p (2p+lP2m1 4 2p 2m+3 p+l (pP2m1 (2mPpl (1 2i()P)fp+ + 721m+1 / O 2m p 2 (2  (2m E2m+1n+ )f2m+ +  (2r+1 + 7'2r1 2772r)I) 4 4,F r=0 < VB,FIFm1 =< V ((I ( 2m+lP2p (2p+lP2nm , < F8(p + m + 2) Fp+ 1+i 16 E p=02m+l (2m+lPp1 (1,P2,m (1 i())f + 2jn+2 2m + 1 p 2 2 S1 m+1 (2m+2 e2m+l)f2m+2 2,  r=0 < VB,FIGM1 2 =< VB,F\ 1i 16 (p p 2m  iF) m > 0 i (2pP2,n (2m+1P2p1 4 2p+ 2m +3 2)  iF) f l2r+llo r=0 < VB,FI(L_2mI 1i ( 2 ) =< VBF (1 i()P) p+2m 2n+1Pp3 fp+i i + i C(2p+lP2m C2n+1P2p + 1+i 911 p+1 +4 2p 2m pmn 1 i 2pP22m 2m+1P2p1 2i r+f0 + 4 2p 211 2G+ + 2mn+2(, iF0) 1l+i ++ (E2m+2 f^mn+l+)Fn+l + + i (2C2m+l (2m (2m+2) 10 ( 2) Va vFl8 ( )P) p + 2m + 2 + 1 + i C2p+lP2m1 C2mP2p F 4 2p+ 1 2m 1 Ci(2pP2711 C2mP2p1 1 i fo + 2p 2n + 2 2,+1 + FLo p~m 1i +i (e2n E + 4 1+ (2(2m (2m1 C2m+l) Io for m > 0, along with the following three identities: 1i *" < VB,FI (G_ + E[72q+2Gq+_ + i'l2q+1Fq + "Sq7qfq+l + 2 ) = 0 q>O < VB,FI(fl+ + + Sq+i lIfq 2112q+lFq + 272G+lG 1o1) = 0 < VBF l + iFo + S[ q 1SqIq+f + i?72q+2Fq+1 + 12q+lGq+] q>o + 10) = where Sq (1 i(1)). (2.41) The coefficients Y, w, and w for the overlaps of the superghosts can be read off from these equations. They are.listed in Table 24. 0 0 w o I S . 0 Q  i 0 *0 o 5 + S E0 S o0 O E II * Sf E E  +< + I + I I >LII 11" S ,S  I 's _1 i ^ I w" I " I I 's> *w II II II II + + + p 1M  S O E ffo 1, I I I I O  '3 I I I rrl '3 +  c,. + II II II + I o + 4 ^ir E r c'1 I I I I 3G Similar to the other two couplings, the coefficients of the superghost insertions can be written in terms of the coefficients in Table 24. o.d = " 5 + gl'Sn s>0 6i o( 6,o + 2w + 1r) i+( i'no + iwi.4 + 2iY ) 4 r + 2w +r 6ioAf + 2iAl+j Si+ mk ,ir "" i 3n 3m 3Zm aZr = ( + )i sr j n r ) (r + )W s>O n=O 3m 1 3 1.3 1 w+( 'o o + 3m(Wm + i )wr 32( 2 2 2i 2 ,0120 2)"06r) + ioY + i+o mY+ + ,, + 'irM.i t (2.42) and Km = ki + E(s + )WM s + ( + ) + o io + mi n12 + MO+ + + 4 aTr = biOk + 2i6bi+k+ + A.w + 2 k s>0 (2.43) where m,n stand for integers, and r,s are intergers for the Ramond sector ( i = + ) and half integers for the NS sector ( i = ). Since the coefficients in Table 24 determine the superghost vertex and (2.42), (2.43) determine the superghost insertions, we have completed the construction of the BRST invariant vertex for the Ramond open string coupled to NSR closed string. Picture Changing And Insertion From the construction in the last section, it may seem that the transition vertices have very complicated insertion operators. However, with a more careful study, we can find the similarity between the insertion and the well known picture changing operator. In order to have a better understanding of this insertion operator, I decided to reconstruct the transition vertex for the NS 37 open string and NSNS closed string in a different superghost vacuum. Notice that, in the earlier construction for the transition vertices, the string states (open or closed) are chosen to be in the 1 picture.17,18 Suppose we choose the superghost vacuum for the open string in 0picture and keep those of the closed string in 1 picture. The total ghost number of the vertex will then be 2, which corresponds to the correct background charge for the superghosts. This indicates that (0, 1, 1) is a more natural picture to construct the open closed transition vertex and no insertion will be necessary. Let's denote the vacuum state with the picture numbers I(Po, P+, P) > for the open string, and for the left/right moving sectors of the closed string respectively. For example, the vacuum IQ0 > in (0, 1, 1) > satisfies 3 ysojo >= 0 for s > 2 1 O*O >= 0 for s2 2 Since, only the superghost components will be affected by the choice of the pictures for the ghost vacuum, we will rederive the superghost part of the transition vertex. What we did earlier for the overlap of these superghosts in the coupling between the NS open string and the NSNS closed string remains correct except that the mode expansions require some rearrangement. The reason is that, in this new picture, g1/2 is an annihilation operator. We can substitute I0 in the overlap equations by g1/2 according to (2.35). After some shuffling, a new set of coefficients Y's are obtained in this picture. They are listed in Table 25. i = ,r > 1; = o,r > 3 s j = ,s > j =o,s >  38 0 I 0+ + cI I 0 I n CD D 0 ( C + 0 8 11 c .t 1 0 C   Z c U2 + 0 II  S I aI   o t A 11 1 II II & W C O ' .s^ w ld 39 In this picture, the insertion 10 does not come into the overlap equations at all. This implies that the overlap vertex for the superghosts in this picture is simply < (y3P),(O,1,1) = < Do exp( ir,,i) 3=4 The complete transition vertex is the tensor product of the four overlap vertices < r, (0, 1, 1) =< VB,FI < (bc)I < (7y),(0,1,1)1 (2.44) It is easy to prove that (2.44) is a BRST invariant operator. When the BRST charge acts on this transition vertex, it simply reproduces the overlap equations of the energy momentum tensor and the supercurrent. Now the issue becomes clear to us: The natural overlap vertex tends to mix the superghost vacua of the open and closed string in different pictures. In order to keep all the superghost vacua in the same picture, a. picture changing operator X is inserted to the vertex at the interaction point. In the following, I will argue that this conjecture is indeed correct by showing the equivalence of the two transition vertices for the coupling of one NS open string state and one NSNS closed string state. Let us insert the picture changing operator1" directly to the vertex in (2.44). Since the superghosts in the vertex are unbosonized, it is easier to first unbosonize those in the picture changing operator. X(z) = b(#)[PO + 2by] c&#b(3) + 6'(#3)zb (2.45) Notice that one should normal order this operator before inserting it to the vertex. The creation modes in X act on the overlap vertex and bring down the sums of annihilation modes as follows. < r,(o, 1, 1)I(o,o) S< B,F, (, 1,1)l exp(E A' ,,^c' E r i) m< rs>1 4( b 1) ( b 1 e2,m+ m>O + ()P+ ()4q~ [i2v/bp bO2p+1 ] eij7+q+l + 702q+11] p,q>O (2.46) By checking the transitions of a few low lying states, I find that this lengthy vertex produces the same amplitudes as those from the transition ver tex < r, (1, 1, 1) in (2.37). A similar check can be done in the bosonized language also. I adopt the notations used by Thorn [ 20]: 7 = e07 and P = e01a where { (z) = E nz" satisfy {(n,7m} = 6n,m I (z) = ZT,nz and 4(z) = cnzn satisfy [,i, Om] = mbn,m Also eO = V : eZEn : z0 For example, the first few low lying states are 7_I(1) >= 77V (1) > 2 73 (1) >= (r2 n1l01)V (1) > 2 2 21  9_A 1(1) >= (262 + 21)V (1) > 1 >= (363 + 2(2 1 + 1(2 1 2))V1(1) > 2 The picture changing operator is more frequently used in the bosonized form. X(z) = eOP P c Oz e2((26b 0, i + 9zb t + 2b r Oz 9) (2.47) Inserting it to the new vertex (2.44) at the interaction point, we have < r, (0, 1,1) x(0,0) =< n,0 < VB,Flexp( M',nnc + +Z A,~,nnl m>l1 m>1 n>0 n>0 Smn> 2f k2k+1 k m,nl> kL>1  (),P+' p,q2>O (2q) icp+i + 2V2co2p+l] + [i,q+1 2V o2q+1]) MnCn + Y mnin + z ,nmn b 5*' ~ AT~q~q m>l n=>O  i2 ) VI m,n>l + < oI < VB,Flexp(A Mncm + iiij E Nij n;j m>l m">1 m,n>l n>O n>0 1 V2 + T( 2k+1 + V ) Vri) x 8V k>1 ([i ()P S[i 1()q q q>1 ( b 7 () b2p+l prO 1 T_1  ) q () 2q+1 ,E "q>Oq +[V2 1] [x[2 ( ) 2q + 1 2q+ q>1 q>O + 2[i 1()'( ()q 2( 7 2q+.4 qP1 q >0 x ()"'( d + 02+11]) (2.48) 42 Note that the overlap vertex for the superghosts has been bosonized. It is not difficult to see that the scalar fields O's overlap the same way as the bosonic coordinates X's, see Eq.(2.1), except the linear terms corresponding to the background charge in the particular picture. The rl, C ghosts (weight 1 and 0) overlap similarly to the b, c ghosts except an opposite sign for the openclosed crossing terms, i.e. M~0 = Mo00 MCC = MCC Mc = Mc MOC ~ MC mn mn mn mn mn mn mn mn BRST invariance is manifest here because the picture changing operator is BRST invariant by itself. Once again, this vertex in (2.48) gives the same transition amplitudes as those of the vertex in (2.37). Although only the first few terms of the insertion were checked explicitly with a few low lying states, the results should hold for all higher terms in the insertion because both vertices are BRST invariant. Therefore, the extended supercurrent insertion in (2.37) can be identified as the picture changing operator acting on the vertex in the (0, 1, 1) picture. < F, (0, 1, 1)IX(0,0) = < VB,F(1, 1, 1)1 < ghost I(Il + W wiy + Z c f8y) (2.49) = < r, (1, 1, )1 From the above examination on the transition between a NS open string state and a NSNS closed string state, I believe that the same interpretation of the insertion as the picture changing operator holds in the other two types of openclosed transitions. To summarize, the covariant openclosed transition vertices for the NSR spinning strings were constructed based on the overlap of string wave func tions and the extended supercurrent ansatz inserted at the interaction point. 43 Superghost contributions are determined by the BRST invariance of the tran sition vertices. By changing the picture of the transition vertex for the NS open string coupled to the NSNS closed string, the role of the extended super current insertion is understood as a picture changing operator. One should be able to identify the insertion with the picture changing operator in the other two types of openclosed transition in a similar fashion. OPERATOR QUANTIZATION OF THE GREENSCHWTARZ HETEROTIC STRINGS Covariant Quantization In 1983, Green and Schwarz proposed a covariant string action in which the spacetime supersymmetry was built manifestly. This theory is described by the bosonic coordinates XI' and the Majorana Weyl spacetime spinors 0 in ten dimensions. Both of them are scalars on the world sheet. It has a local fermionic symmetry (K symmetry) which implements a global spacetime supersymmetry in the theory. For the Type II strings, there is an extra bosonic symmetry (A symmetry). In the light cone gauge, this theory has been proved to be equivalent to the usual lightcone superstring in the RNS formalism.3 Because of the manifest spacetime supersymmetry, it is desirable to construct a string field theory in the GreenSchwarz formalism. Many people21,22,13,23,24,25 have been trying to quantize the theory co variantly since then. Yet, very little success has been made. The difficulties for achieving a canonical quantization of this theory come from the fact that the fermionic constraint is a mixture of first class constraints and second class constraints. Moreover, these constraints are linearly dependent on one another (socalled reducible constraints). Hori and Kamimura were able to separate these constraints covariantly, but the separated constraints remain reducible. So far, only the Super Harmonic Space technique26,27 was able to overcome these problems. The drawback is that at least 112 new harmonic variables must 45 be introduced into the theory. The huge number of these auxiliary variables (pure gauge) makes this method less practical and less attractive. Comparatively, the Lagrangian quantization method seems to be easier. The mixture of the first class and second class constraints is not a problem at all. However, the linear dependence of the constraints requires the introduction of an infinite tower of ghosts for ghosts.28 Batalin and Vilkoviski (BV) formu lated a systematic way to quantize theories like this.29 If lucky, one may be able to handle these infinite ghosts effectively. Most attempts of quantizing the GreenSchwarz superstrings have been following this direction. I will review the recent works on the BV quantization of the GreenSchwarz superstring and discuss the frustration of these attempts. In the next section, I will pursue the canonical quantization of GS superstring. For simplicity and consistency, I will discuss only the heterotic strings. The GreenSchwarz heterotic string action can be written covariantly in ten dimensions as21 S= d2a (illa + ie2 alxerqp+ a4 + (3.1) II = aXP iOFk aO and e01 = e10 = 1. 01 are the left moving world sheet Majorana fermions, I = 1,2,..., 32. They are Lorentz singlets with some internal quantum numbers such as SO(32) or Eg x E8. Like the other string theories, this covariant action has a Weyl symmetry and a local reparametrization symmetry. Its right moving sector has a local fermionic symmetry, called K symmetry, which is closely related to the global space time supersymmetry of the theory. In this article, we consider only the right moving sector. The transformation laws under these gauge symmetries are given as follows: Reparametriza t ion 6XI = eC0AX 60 = e0_0 = + Sga = 87 0 fg97f _97eg K supersymmetry `6X = i0FL60 60 = 2ill PrK+ `5 =0 6gap = 16 g=(Pi P =0) where P0 1 (g2 Ea^/fg) = e7 2 e=F Global super Poincare 6XL, = aPlX + b iOF"P7 60 = apvO + rl 6gf = 0 From the degree counting, the MajoranaWeyl spinor 0 in the ten dimen sional heterotic string has sixteen components. The gauge degree of freedom for the K supersymmetry allows us to eliminate eight of them. As pointed out by Kallosh [28], the parameter t+ itself is another sixteen component MajoranaWeyl spinor. Obviously, twice too many gauge parameters have been introduced. This means that not all the components in K+ are linearly independent. In fact, this can be seen when we set K+ to be =+ = Pt_++ 4++ will parametrize another symmetry of the action, because II2 = 0 is one of the equations of motion. The same argument applies to tc++ and K+++, etc. Correspondingly, one must introduce infinite FadeevPopov ghosts c+, c++, etc.28 Before applying the BV quantization, let us fix the world sheet Lorentz parameter and the Weyl parameter such that the metric can be represented by the vielbeins e. According to BV, the quantized action is the solution of the master equa tion. 6,.1S 6iS _,.S 61S (S, S) =  = (3.2) , i Z Zi  In general, this solution is written as an expansion of the antifields 0*: S = SO+E Sick+ SJ. .+Snonmin, Snonmin = E *ra (3.3) i ij a These antifields are defined as the functional derivatives of the gauge fermion Q with respect to the minimal sectors 4 in the theory, i.e. the coordinates, moment, and the FadeevPopov ghosts. are the rest of the field contents including the antighosts and the extra ghosts. The gauge fermion is chosen so that O*s correspond to the gauge fixing conditions in the nonminimal part of the action. Based on the transformation laws of the various gauge symmetries listed above, Gates et al. solved the master equation for the GreenSchwarz heterotic string.30 The quantized action has a finite number of terms with the antifields only up to quadratic. S = SO + S1 + S2 + Snonmin (3.4) S1 = x*Pa + (A[+P ])[+P + aAab + 2_.lrcilP + (et* P_*P+ (*i+)(A+( 4clA+0) + [e* P P ac e(P P+ PI*P + ( C+ *nc"c, e+*e)] A + c [pcn+1Pp + +ncnA. (Aacn)a] 0 + s4 (3.5) S2 = Ax* A[+P* 2_(circl) cFcrn+)] 0 + (e+* P*P+ +) [(X* A[+P )(circi) + 4*(Elc2) + c cncn+2] 0 0x n Snonmin = :IFF + c n m* , 1n=l m=l where C and i are the reparametrization ghosts. co = Cn(+) are the FaddeevPopov ghosts and cm for n > m > 1 are the extra ghosts and antighost at the nth level. c* is defined as 6* oFX* 2P A_]oP. And, Sf are the four ghost coupling terms. Its details are not crucial and will not affect the following analysis. I will not quote them here. The real task now is to find the right gauge conditions such that the gauge fixed action can be linearized, or manageable. From the experience on the superparticles,31 the authors in references [24], [30], and [ 32] tried the derivative gauge, a0+ = 0. At first glance, they seemed to over gauge fix. But actually, eight of these conditions correspond to one of the equations of motion, IIf~la'0+ = 0. Therefore, this gauge choice is consistent within the 49 BV formulation. There are many ways to choose the gauge conditions for the ghosts. For example, Lindstr6m et al chose the gauge fermion to be 00 n S= e+e + n m+1~ 1 m (3.6) n=0 m=0 The pyramid structure of the ghosts is30 \ \ \ Bergshoeff and Kallosh arranged the gauge fermion slightly differently.33 00 00 S= e + E On"'n+(9m,n + +m+l,n+l) (3.7) n=0 n=m where the pyramid of the ghosts is 00,0 \ 01,0 p0,0 \ \ 02,0 ^10 81,1 \ \ \ 03,0 2,0 02,1 #1,1 50 Once the gauge fermion is chosen, the antifields can be determined. Subsituting them into Eq.(3.5) for the solution of the master equation, we obtain the complete gauge fixed action. Not too surprisingly, the action has complicated interaction terms and kinetic terms. At first, these three groups claimed that after some field redefinitions the gauge fixed action becomes quadratic and the theory is essentially free! Sg = (P+P  Xo+X + E S++Oa( So n ) + (3.8) n=1 m=l Moreover, the conformal anomalies of the string variables and the ghosts cancel among themselves with simple regulnrization with fn = lim Efun n n oo n 10 26 16()"+12[6(2m n)(2m n 1) + 1] = 0. n=0 m=0 However, more careful study shows that BRST invariance is lost after the field redefinitions. Fisch and Henneaux34 pointed out that these field redefi nitions are noninvertible mappings. They found infinite residual symmetries in the shifted action. Various modifications on the gauge fermions and mod ifications on the field redefinitions have been investigated.33 So far, there is no satisfactory way of removing the constraints on the new fields in the "lin earized" action, neither of realizing BRST symmetry in the quantized theory. Therefore, it is an open question whether the covariantly quantized action in the derivative gauge corresponds to a free theory. 51 In summary, the frustration of Lagrangian quantization of the Green Schwarz superstrings is that the right gauge fermion which would lead to a linearized (free) action has not yet been found ,or a more sophisticated but unknown treatment is necessary. It seems that in order to study string field theory from this, one must introduce more variables and more symmetries to overcome the difficulties that we face currently. In other words, direct modifications in the original GreenSchwarz action such as Siegel's23 seems unavoidable. For example, LindstrSm et al have tried to enlarge the phase space to replace all second class constraints by "virtual" first class constraints. So far, they have succeeded in the covariant quantization of the superparticle.35 It is unclear whether their generalization will work also for the superstrings. This is out of the scope of this dissertation and I shall not say anymore about this. Let us now move on to the next section and discuss the canonical quantization. Canonical Quantization As I explained earlier, the covariant canonical quantization of the Green Schwarz superstring is much more difficult than the Lagrangian quantization. Unfortunately, even with the Lagrangian formalism, its covariant quantization is still not very satisfactory. On the other hand, the path integral quantization results are sometimes ambiguous due to the dependence on the regularization of the measure. From the second quantization point of view, it is preferable to have the canonical first quantization. In the following, I will introduce the quantization method of Batalin and Fradkin to canonically quantize the GreenSchwarz heterotic string in the semilight cone gauge. In the path integral quantization by Carlip13 and later by Kallosh and Morozov,14 a partially covariant gauge was considered; namely, the conformal 52 gauge for the metric and the light cone gauge for the spacetime spinors. gaQ = r0o0 F+0 = 0 semilight cone gauge (3.9) Kallosh and Morozov's results indicated that the gauge fixed action is free of the conformal anomaly. They even went further and discussed how the one, two, threepoint functions vanished by counting the zero modes of the spinors in the action. It is therefore interesting to understand the machinery behind the anomaly cancellation in the operator language, i.e. Q2 = 0. Hopefully, the canonical quantization in this semilight cone gauge will teach us something useful for the canonical quantization in a covariant gauge. In the operator formulation, the Hamiltonian for the GreenSchwarz het erotic string is H = d2a(PtX + C Lo) (3.10) L0 is the Lagrangian in (3.1). The dot (the prime) denotes the time (spatial) derivative. From the symmetry transformations listed in the last section, the corresponding constraints can be found with the Noether method. After some rearrangements such that there are no secondary constraints,22 the constraints for the reparametrization and the K symmetry are L(a) =(P X')2 2(0' 2 (3.11) Fa(a) =Ca + i(oF")a(P X' + i6ro'), Since we are interested in the rightmoving sector where supersymme try is implemented, let's examine the constraint algebra under the Poisson brackets: { L(), L(a')p B = 2,L(a)6(a a') 4L(a)o)6(a a') {L(a),Fa(a')} PB = 2Fa(a)O,(a a') (3.12) {Fa(o), F(a')}pB = 2i(cr()ab(P X' + 2iBr8ao)6(a a'). 53 It is clear that the fermionic constraint F is a mixture of the first class con straint and the second class constraint. Only eight of them are the first class constraints which generate the K symmetry. The advantage of choosing the semilight cone gauge is that we can separate these constraints easily by the following projections: f r= FF:, + + r = 1i ll = 1, :l = 0. (3.13) After we impose the gauge conditions, the 1+ projection corresponds to the conjugate momentum of F+0 and can be solved explicitly. The 1 projection corresponds to the original second class constraint in the theory. FI : (l+ = iBrF(P X')k Fl : G = (l iM0O = 0 where 3(a) = (P X') (3.14) We are now left with a theory whose first class constraint L and second class constraint G are irreducible. It is traditional to use Dirac brackets for the canonical quantization in the presence of the second class constraints. In general, these Dirac brackets can be very complicated and ordering ambiguities can cause serious problems. Consequently, it is very difficult to implement BRST quantization with Dirac brackets. Batalin and Fradkin proposed a new method to overcome these prob lems. The idea is very simple. By introducing new variables, the second class constraints can be converted into first class. These effective first class con straints then generate virtual symmetries in the extended phase space, which allows us to eliminate these new variables through gauge fixing and recover the original second class constraints. These constraint algebra can be gener ated by the nilpotency condition of the BRST charge. There are two different formulations for this new canonical quantization.36,37 In the following, I will 54 apply these two different formulations separately to the quantization of the GreenSchwarz heterotic strings. Old Formulation Define the equaltime supercommutator for operators A and B: [A, B] = AB ()e(A)(B)BA. (3.15) where e(A) denotes the Grassmann parity of the operator A. Under these supercommutators, the conjugate relations among the string variables and their moment are preserved. [XP(a),P"(a')] = ihgP'(a a'). (3.16) [Ca(), 0b(O')] = itlbb6(U 0'). For example, in the semilight cone gauge, the constraint algebra reduces to [L(), L(a')] = 29aLS(a a') 4L&a6(a a') [L(a), Ga(a')] = 2Ga(a)a,6(a a') (3.17) [Ga(a), Gb(O)] = WabM6(a a') Wab 2i(Cr)ab, M(U) (p x')+ This constraint algebra can be generated from one equation, namely, the nilpo tency condition of the BRST charge QB. [QB, QB = 0 (3.18) Let us introduce the reparametrization ghosts B, C for L and the spinor ghosts 7, fP for G. (C) = e(B) = (L) + 1 = 1 [C(a), B(')] = b(a ') ghl(C) = gh(B) = 1 (3.19) gh2(C) = gh2(B) = 0 55 ( e(7) = ()= e(G) 1 = 0 [ra(a), b(a')] = 6 56(a a') gh2) = gh2(6f) = 1 (3.20) ghl = ghl(#) =0 To convert G into a first class constraint, we introduce new variables VO which have the same Grassmann parity as the constraint itself. And they obey the following commutation relation: [ca(a), b(ar)] = _abb(a a') (3.21) e(I) = 1 and ow = I Then a linear term in 4 is added to the naive BRST charge for G to make it nilpotent. [, Q2] = 0 Q2 = daa(Ga + abM' ) (3.22) It is easy to see that the integrand is now an effective first class constraint which anticommutes with itself. In general, the solution for Q2 will be a series expansion in 7 and j, such that e(Q2) = 1 and ghl(Q2) = 0, gh2(Q2) = 1. To construct the complete BRST charge, we add in addition to Q2 a nilpotent charge Q1, which generates the Virasoro algebra for the first class constraint and at the same time anticommutes with Q2. QB = Q1 + Q2, [Q,Q11 = [Q1, Q21 = 0 Since Q2 is linear in D, we can write Q2 = 02 + a>" Equivalently, Qi obeys [Q1, Q2] = [Q1, a] If we take the commutator of this equation with the conjugate of .a, i.e. (Sa, fb] = 6ab, this condition can be viewed as a first order differential equation with respect to 4. It has the following solution +4 Q1 =QleD 'D with Aia =[, [t2, Qa]] and [Q(, Pa] = [ l, ia] =0 where Qi is independent of <. On the other hand, [Ql,0Ql]=0 Q l0e eDDi =0 Batalin and Fradkin suggested that (Q can be solved with a series ex pansion in powers of ghost s as follows. Q1 = J(CL+ VICCB + UICy +...) (3.23) (Q1) = 1 and ghl(Qi) = 1, gh2(Q1) = 0 However, with the above expansion, there exists no solution! Modification such as allowing second class constraints at the first order (linear in C) is necessary for a possible solution to exist. This situation happens whenever the original first class constraints do not form a closed algebra with the new effective first class constraints. Adding a term quadratic in the second class constraints to L, the nilpotent BRST charge can be solved right away. QB =Q1 + Q2 / da(C(L + MlbGaaGGb) + 2CaCB Ca() + CyfM^aM + a(G. 4bAfb 1 + 7a(G + .abM )) (3.24) It turns out that the linear term is the same as writing the L constraints in the starred variables which are used to substitute the Dirac brackets. In other words, one can define the starred variables as q* q {q, Gi}A'Gj, Ai = {G, Gj} then {q,p}D = {q*,P*} we find L(X*, P*, 0*, *) = L + MAIGa, GbC^ab It is easy to check that L* now forms a closed algebra with the new effective first class constraint. In other words, the BRST charge generates the constraint algebra of two first class constraints, L* and the extended G constraint. Another remark about this BRST charge QB is that, in this old formu lation of Batalin and Fradkin, the rank of the charge can generally be higher. Fortunately, in this semilight cone gauge, it turns out to be first rank only (cubic in the ghosts). This is favourable for the second quantization. Also, analogous to the bosonic superghosts for the covariant NSR superstrings, we have bosonic superghosts y, # for the fermionic constraints G. Like the con straints, these ghosts are space time spinors and world sheet scalars. Their contributions to the conformal anomaly will be discussed in the last section. The BRST charge not only generates the constraint algebra for the the ory, its kernel also defines the physical space. As a conserved charge, QB commutes with the Hamiltonian. From the consistency of the constraints, all physical observables should be BRST invariant operators. For example, the Super Poincare generators should be extended in the enlarged phase space so that they commute with QB. It is important to check the closure of this algebra after the extension and after gauge fixing. To preserve the gauge fixing conditions, the naive Lorentz transformation laws must be compensated by the transformations from the reparametrization and the K symmetry. The corresponding generators can be found with the Noether method. Jo+ = (dP+X PX+ C(0) JO+= Jd(P+X' P'X+) (3.25) (3.25) Joi= d(PXi pi + i(P X')kBFokio) jJ = I da(Pj' PjX' + Ca'iG) QOa = f d(a + i(r)a.M 2i(P X')k(Irl+).) In the enlarged phase space, one should modify these generators in the same way as the unitarizing Hamiltonian discussed in Ref.[ 36]. They will be extended to series expansions in powers of the ghosts with zero total ghost num ber. Adjusting the expansions with second class constraints, one can impose BRST invariance of these generators. The results are P11 =Po J+ = Jo+_ J+i = +i Ji = Ji + jd( M (P X')i(apya + 0) + M( X)kCGki9 M2(P Xt)kGaCkiG) Ji = J i + da(M Gaijc + twaiJ) Qa = Qoa f daM'(P X')k(GTr+r)a If the semilight cone gauge fixing is consistent, then the super Poincare algebra should close up to a BRST transformation. At the classical level, these extended generators indeed give the standard super Poincare algebra. [pP, pv] =O. [PI, JVP] = gPVpp g7p1pv [P", Qa] =0 (3.26) [JPV, JPl = gVPJ'Ua + gPjJVP gV JPP gP PJV + [Q, S"fpl] [Jpv.Qa] = QbOa + [Q, Sa'] [Qa, Qb] = 2i(CP)abP + [Q, Sab] Notice that the S's are determined up to BRST only. The nonzero ones are listed as follows. Si+j = da (c' jfl. + GwiJ'M1) Sijk = Jda(M + A M^G,)aklaij Sij = daM3B(( + iMOi)&ai'(( + iMiT) (3.27) i = 1 /doBM 2( + iMI)b(,r+r)ba Sab= 2i(CP+)ab daBM1. At the quantum level, this closure usually provides an important check for the critical dimension after the normal ordering is taken into account. A discussion of the quantum anomaly will be given in the last section. New Formulation Because the BRST charge constructed from the old formulation is in general of higher rank, its practical use will be severely reduced. That is why Batalin, Fradkin and Fradkina37 derived another formulation in which the BRST charge will, by construction, be first rank only. The punch line is to extend the second class constraints G to a series expansion in powers of the new variables ( such that the new constraints G commute(anticommute) 60 among themselves. Similarly, one can extend the original first class constraint L also in a series expansion of 4, such that L commutes with 6. Consequently, in the extended phase space, these "tilded" first class constraints lead to a first rank BRST charge. For example, for the GreenSchwarz heterotic strings in the semilight cone gauge, we can extend the second class constraint Ga to Ga(a) = Ga + wabbM2 such that [da, Gb] = 0 The extended reparametrization constraint is ~ a b L(a) = L 2Gaa(4 a A) wabo so that [L, Ga] = 0. For example, Virasoro algebra is now extended to [L(7), L(u')] = 20aL6(o a') 4La,6(a a') (3.28) 4cabGaa(Af 1a(Gb6(U ao))) Now that all constraints are first class, the BRST charge is simply Q= Q1 + Q2 = / d (CL + 2COCB + 4COCab8UG, baMl + aGa) (3.29) Physical observables, such as the Hamiltonian or super Poincare gener ators, will be extended in a similar fashion. They should commute (anticom mute) with the BRST charge. Or equivalently, they should satisfy [A, Ga] = 0 and [A, Qi] = Ga This can be done by expanding in a power series of This can be done by expanding A in a power series of #: A = A+ Aa+ AabaIn + ... 61 For example, we can extend the super Poincare algebra generators in the en larged phase space as follows. P = JdaPP J+ = doaP+X PX+ I GaaM + C Gabod.aflM 1') J+i= da (P+Xi iX+) J = d (PXi P'X + i(P X')F aOki 21 + i(P X')kMOrr1 r (P XI)kMl waki + (M4 + 0)rirk8a(c3)Ml(P X')L + 2Co ,c fpi rkrFM2(P ,')k) Ji = Jdu(p'Xj P'Xi i0 + tj G+ Mi awoi' + M1,2(CM)Gai ) Qa = du(C+ i6rM 2i(P Xi)krkl+ 2i M(P X')k4b(CFT)ba + 4iM'(P X')kO1(CJb)cbd(crF)da) Quantum Anomaly In this last section, I will examine the quantum consistency of the semi light cone quantization discussed in the previous section. Due to the operator product singularities, an ordering prescription for the operators must be given to regularize the infinities. It is important to check whether the classical BRST charge remains nilpotent and whether the global symmetry, such as super Poincare, are respected or not. The nilpotency of the BRST charge implies that the quantized theory is free of the conformal anomaly. The closure of the super Poincare algebra assures that the noncovariant gauge fixing does not destroy Lorentz invariance and spacetime supersymmetry of the theory. 62 For a free theory, one can use the normal ordering for regularization. Unfortunately, in this semilight cone gauge, 0 does not have the usual kinetic term and it couples to oX+. Without solving the energy spectrum explicitly, which is extremely difficult because of the interaction, we can expand BX+ around a constant momentum, as in the true light cone, and treat the action perturbatively. Therefore, string coordinates and their conjugate moment can be mode expanded according to the periodic boundary conditions for the closed string. More specifically, we are choosing a particular basis for the Fock space. In this basis, the vacuum is annihilated by all the positive modes and by the zero modes of the moment C, b, and /3. Taking into account of this normal ordering, the quantization that was discussed in the earlier section must be reexamined. The failure of retaining the BRST nilpotency and/or closing the super Poincard algebra will indicate a quantum anomaly in the quantized theory. Let us now examine the results obtained in the old formulation. A straight forward calculation shows that the normal ordered BRST charge does not anticommute with itself. Denote : A : as the normal ordering for an operator A. [: Q:,: Q:] =(i)2 CpCp ( 11)p3 +( + 5)p p>0 + (ih)2 daCC 6iM18M) + M^18) (3.30) Naively, one thinks that Q is anomalous in ten dimensions, and there exist also M dependent terms that violate Lorentz covariance explicitly. This is not so surprising if we compare the result with the Liouville action derived from the heat kernel calculation.14 exp(4 ) (( 13 + 4 x 2)I1ln pl2 + 4(21n p + In ju 2)01n lu12) u = _X+ 63 There exist uterms which violate Lorentz covariance explicitly. If the argument in Ref.[36] is correct, such terms can be removed by shifting the X in the action. We can add o(h) corrections into the BRST charge to remove the M dependent anomaly in (3.30). From the power counting, there are four independent terms that produce the types of anomaly that appear in (3.30): Q= Q + dC (aM18 M + b(MA M) + dM1 OaM +ti where a, b, d, t are free parameters. Therefore, the anomalies that violate Lorentz covariance will disappear, provided a + b = 1 [: :,: :] = 2 Cp ( 11 12a)p3 + ( + 5 3t) 6n V 2 2 p>0 (3.31) The consequence of this shifting is that the nilpotency of : Q : does not determine the critical dimension. There is an extra free parameter a appearing in the cubic term. An interesting observation is that in Ref.[ 38] Kraemmer and Rebhan pointed out a corresponding ambiguity in the path integral result. Let us rewrite the Liouville action as d 2z (( 13)1alnp 2 + 4.26(1l1n p12 + In u 12 n p) + 42(1 )lnI u 2aInp + 41n lul1280Inu2) The second term is a total derivative! This allows us to put an arbitrary parameter in front of the total derivative. Therefore, the critical dimension can not be determined from the heat kernel method. We have seen this ambiguity from the operator quantization method where the free parameter a appears in the shift of the BRST charge in (3.31). To determine the critical dimension, let us examine the closure of the super Poincare algebra. Similar to the BRST charge, one expects that its 64 generators will have quantum corrections also. These corrections can be found by imposing BRST invariance to the generators. With the shifted Q, it is not difficult to find that all the super Poincard generators commute with Q except :Jk: [: Jk: Q :] = (i1 2) daC (aM.OM + ( + )(M~M)2 + (d + 2)MlaM + t)(P X')k (3.32) If Jk were BRST invariant, the nilpotency of the BRST charge in (3.30) will imply that the critical dimension is 22 instead of 10. But the GreenSchwarz heterotic string is constructed in ten dimensions, it is more likely that Jk is not a BRST invariant operator at D = 10. Indeed, one can add the following corrections to Jk and shift the anomalies in (3.32). jk = jk + h/ d ((P X'i)kM. + u(P X)kM1). such that the commutator of jk and the BRST charge Q becomes [: j :,: Q :] =ih2 J daC 6 ((a + )M'M + (b + z)(M1M)2 (3.33) + (d + 2u + 2i)1. + t)(P X)k Note that the coefficients z, u in front of the quantum corrections are arbitrary at this moment, since the coefficients for the quantum corrections of the BRST charge, a, b, d, t are free parameters also. Thus, the quantum corrections to Q and J's are still ambiguous. In the following, we will see how they can be determined uniquely from the closure of the super Poincard algebra. Because some of these generators have cubic and quartic operators, one must be careful with the ordering of their commutators. More specifically, I will regularize the supercommutators as follows. [: AB :,: CD:] =: AB :: CD: _()(A+EB)(CC+'D) : CD:: AB : The quantum ordering effects appearing in the algebra (3.26) are the commutators involving jk and the supersymmetry charge Qa. jk I: jl:] = ih Jda(L + MGoaG Msa(yAM 7 ) + ht)M3YaklY + .( ) i)2 daM2 ((P . X,)ka(P X')1 (P X,)l (P X')k + h2 da M30^ ak + lMaOakIaO 7r2 + 2' a 2 M 2 2 + [(4 z)M5(OaM )2 ( 8 ) OMa + (i + 2unx)M40aM](aki& + [( )M3(a9M)2 +(2 + Z)2a2M +( + r)M2aM] "OwOklO 4 8 2 + [( + z)M4(aM)2 + ( 3 )M3^ + ( 2u7r)M~3aM]C(kIO 2 42 2 Y = ~+ i0rM. (3.34) The operators in the integrands are all normal ordered. The commutator of : Jk : with the supersymmtry charge Qa gives [: jk. : ]: O : = ih Qbabk + de(f z 5 Mz(4 M) +( 5)M^8)M + dax( 4(aoM)2 + (z + )M2M1 5i (3.35) + (u7r )MA39MM) c(r+ k)ba + 8 da ((W )M (aQM) + ( + z)M^M r + (iu + M216 4 + (iur + )M2 M)b(rr+rP)b 66 To close the algebra, these anomalies must be BRST trivial at the critical dimension, in which case the integrals can be written as the commutators of Q with SkI and S^k given in (3.27). [: kI :,: :] = iA d (L+ M1GoaG MAa(#fM1))M3YeakY + da1 _M^ k r akl J 1 2 + 32M 1ak02 3M 3oM(akHCO 2 2 5+ ar)42o.M + (i +dr)M4OaM](aktCO + [(5 + br)M5(~aM)2 + (+ + ar)M4OM + (i + dr)M4M](a1k + [(3 + M3 M2 1+ at)2M + (d)M2aM]OwaklO + [(5 b)M4(aOM)2 ( + M M ( + d)kl 2 2 (3.36) and [: Q :,: sa : = 2e 9 ( + )M4(OA)2 + (13 + M2)M3n M +  d [( + i )MA3(aM)2 + (15 + i)M2 +( + i)M2y. A r+rk + t (ih2) : Sak: (3.37) Comparing the nilpotency condition (3.30) with (3.34) and (3.35), I find that all quantum anomalies cancel and the algebra will close, provided that z= u =0 1 2. ++ b=t=0, a = d=ti 27 7T 4+ D=10 67 The anticommutator of the supercharge with itself does not have a quan tum anomaly, but the commutator of Sab and the BRST charge has an anomaly that vanishes only at D = 10. : Qa :,: Qb :] = 2h(cr")abP, 2h(Cr+)ab I dM'(L + M1^G&oG) [: Q :,: Sb :] = 2h(Cr+)ab daM(L + M1GOc G) + 2i(Cr+)ab2 dr(1 + ar)M2^M 2r 2 Therefore, our conclusion is that the critical dimension of the Green Schwarz heterotic string is ten! We have uniquely determined not only the quantum corrections to the BRST charge Q but also those to the super Poincard generators which close the algebra up to a BRST transformation. This com pletes the realization of the conformal anomaly cancellation in the operator language. CONCLUSION The openclosed transition for the RNS spinning string and the first quantization of the GreenSchwarz heterotic string have been studied using the operator methods. For the openclosed transition, three types of couplings between the NS (R) open string and the RR, NSNS (RNS) closed string were discussed in detail. The transition vertices were constructed based on the overlap equations of the string variables, the energy momentum tensor and the supercurrent. Lorentz covariance is manifest in the construction. Ghost contributions and the insertions are determined by requiring the vertices to be BRST invariant. By shifting the picture of the superghost vacuum for the open string, we understand that the insertion operator not only ensures BRST invariance but also plays the role of a picture changing operator in order to keep all superghost vacua in the same pictures. In the operator quantization of the GreenSchwarz heterotic string, in stead of using the Dirac brackets, I applied the new method of Batalin and Fradkin to derive the BRST charge in the semilight cone gauge. It was found that the old BF method failed to solve a general gauge theory whose original first class constraints and the new effective first class constraints did not form a closed algebra. In this dissertation, a slight modification was suggested to the ansatz which solved the generating equations of the constraint algebra. After this work had been completed, Batalin and Fradkin generalized their formula tion by modifying the generating equations rather than modifying the ansatz. 69 The results of these two modifications give the same BRST charge discussed in this dissertation. The classical super Poincare generators are constructed in the extended phase space. The extensions are made according to BRST invariance of these generators. To study the quantum ordering effects, I chose a particular normal ordering prescription and reexamined the nilpotency of the BRST charge and the super Poincare algebra for the quantization performed in the old formu lation. With the proper quantum corrections to the BRST charge and the Lorentz boost Jk, the super Poincare algebra closes up to a BRST trivial transformation at the critical dimension, D = 10, and conformal anomalies also cancel at the critical dimension. I have also applied the new formulation by Batalin, Fradkin and Fradkina to study the semilight cone gauge. In this new operator formulation, the BRST charge is manifestly first rank (cubic in the ghosts) and no modification is needed as in the BF formulation. I did not discuss the quantum anomaly for the quantization in BFF method. Presumably, the anomalies will also cancel once the proper quantum correction terms to the BRST charge and the super Poincare generators are found. The result on the operator quantization in the semilight cone gauge is very encouraging for the GreenSchwarz formalism. One can further study the BRST cohomology of this theory and possibly construct a noncovariant string field theory out of it. But one should remember that the ultimate goal is to formulate a covariant superstring field theory. The success in the semilight cone quantization is only one small step towards the ultimate goal. So far, there has been no success in achieving a reasonable covariant quantization the GreenSchwarz superstring. In the path integral quantization, the covariantly 70 quantized action gives highly nonlinear interactions. In the operator quantiza tion, one hopes to be able to apply BF or BFF method in a covariant gauge. But, these methods only work for theories with irreducible first class and sec ond class constraints. One must generalize the operator quantization method by BF or BFF to theories with reducible gauge constraints. Otherwise, one should look for alternatives such as enlarging the phase space and the sym metries in order to overcome these difficulties. More study is necessary before one can rule out the possibility of achieving a covariant quantization of the GreenSchwarz superstring. APPENDIX A. PATH INTEGRAL FOR OVERLAP VERTICES According to Mandelstam's prescription of light cone string interaction in the RNS formalism, the transition amplitudes between one open string and one closed string can be written as16 T f dtH < o o SoD C( 0)) b>(= Xcopc = J vxD"(So(X,,)+f{F'X+G+++G } bk k k Bk ) with a = 7 a. (A.1) where So is the classical action for the fermionic string in ten dimensions. T v So(X, ) = dr d((VX)2 + i+(r iO9)+ + 0(r + ijo)) Tc 0 The sources F and G specify the external boundary states. For example, the source term G for the spin variables is G(a, r) = +C(o, 0)6(r + Tc) + o(7r a, 0)6(r To). However, this transition amplitude in (A.1) is not Lorentz invariant. Mandelstam suggested that in order to maintain the Lorentz invariance of the amplitudes in the light cone gauge, a supercurrent must be inserted at the interaction point. To evaluate this integral, one can use the Neumann's function technique. We need to construct the Neumann's functions for the following equations. o+g+,b = 27r+b62(p p') aK_,b = 27rb6 62(p p') The right handed modes are related to the left handed through complex conjugation, therefore we also have K,b(p, p') = K+b,(p*, ') Thus, the amplitude becomes J DXD exp (so(X, ) J1/GalKa,bt",Gb' + surface terms) (insertion) The surface terms will vanish provided the Neumann's functions satisfy the proper boundary conditions. In general, these Neumann's functions are very complicated for finite times, Tc, T. However, the leading contributions come from the limit when Tc, T go to oo. One can Fourier expand these functions and evaluate the amplitude. Thus, a transition vertex can be extracted in terms of oscillators. A.1 NS open string and NSNS closed string. The boundary condition for this type of transition is that the Neumann's functions have to be antiperiodic for both the open string region and the closed string region. Such a function is already known in the upper half plane, 1 z z 73 Since the K's have the same conformal weight as the spin variables, the correct Neumann's function in the p plane is simply (1 (p) p' K(p, p') = z z \z ,') After doing the Fourier expansion, the evaluation of the amplitude is straight forward. We found that the result agrees with the one we did in the overlap method. A.2 NS open string and RR closed string. In order to obtain periodicity in the closed string boundary and antiperi odicity in the open string boundary, we modify the Neumann's function in the p plane to be K1 z i z'+i 2 p 2 p' 2 K__(p,p')= __ \ )a z z' z' i z + i) lz z A.3 Ramond open string and RNS closed string. This boundary condition is a little more complicated than the previous ones. It requires antiperiodicity in the open string and opposite periodicities for the right moving modes and left moving modes in the closed string. This can be done with the following function: 1 z 2 8 2 8 2 K(p, p') z z' For the NSR closed string, all we have to do is to exchange the left moving part with the right moving part. Due to the zero modes in the Ramond sector, the Neumann's functions also carry the helicities (upperindices), and the relations among them are 74 Kab++ = ab = 0 Kab+(z, z) = Kab+ (, Z) After substituting the Neumann's function, the path integral in (A.1) gives the same overlap vertex as from the overlap calculation in the basis whose vacuum is chosen to be I T>. APPENDIX B. USEFUL FORMULAE B.1 Combinatorial Relations E 1=0 1=0 'a) (,a mo 1 a =a ^k1 1 1= 0 k b 1) ( a + b b ,i= a+b1a 1 k1b a (a1 (am a+k k+rnm\ B.2 Identities Of The (,q, p's ...etc. The definitions of the coefficients are 00 (nn (1 + x)4(1 X)4 n=O xn ( 1 + i ?77n 4l(l+ ( x)4 n=O Pn  n + (n+2 7n Cn C n m m+l ~ 2T2r r2r+l) c2m r=0 They obey the following recursion relations, 7p C p 2p p (_)k (2) L,=0 1=0 (B.5) En = 7n r7n1 m = (2r 72'r1) r=0 for p > 1 (B.6) (B.7) for p = 2k for p odd (B.1) (B.2) (B.3) (B.4) 1) mn (a1 1 rm m 1=0 a a1 1 km i2 Sk _)k for p = 2k + 1 S2(3)()k for p= 2k + 2 1=0 { fop= 2k+2 () 1 772m21 = (2m C2m1 1=0 () 1 72m+I21 = (2m+I1 C2m 1=0 Note that (n, 7n, ,n vanish for negative index n. B.3 Gamma matrices. (B.8) (B.9) (B.10) Following the notation by Peskin [ 39], r's are the 32 x32 gamma matrices in ten dimensions. ro=f ') r9=(0 ) rk = ( y (i 0 i 00  where y7 are the 16x16 gamma matrices for SO(8). 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She has wanted to be a theorist in particle physics since high school. She expects to continue research in the high energy theory after her degree is completed. 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. Charles B. Thorn, Chairman 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 Rmond, Cochairman 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. ItichardField 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. Paul Robinson Assistant Professor of Mathematics This thesis 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. May 1990 Dean, Graduate School 83 UNIVERSITY OF FLORIDA 3II 12208l553III I III8IlllI111111 3 1262 08553 8113 