Open superstring field theory on the restricted Hilbert space

Journal of High Energy Physics, Apr 2016

It appears that the formulation of an action for the Ramond sector of open superstring field theory requires to either restrict the Hilbert space for the Ramond sector or to introduce auxiliary fields with picture −3/2. The purpose of this note is to clarify the relation of the restricted Hilbert space with other approaches and to formulate open superstring field theory entirely in the small Hilbert space.

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Open superstring field theory on the restricted Hilbert space

HJE eld theory on the restricted Hilbert space Sebastian Konopka 0 1 Ivo Sachs 0 1 0 Theresienstra e 37 , D-80333 Munchen , Germany 1 Arnold Sommerfeld Center for Theoretical Physics , Ludwig-Maximilians Universitat It appears that the formulation of an action for the Ramond sector of open superstring eld theory requires to either restrict the Hilbert space for the Ramond sector or to introduce auxiliary elds with picture 3=2. The purpose of this note is to clarify the relation of the restricted Hilbert space with other approaches and to formulate open superstring eld theory entirely in the small Hilbert space. String Field Theory; Superstrings and Heterotic Strings; BRST Quantization - Open superstring 1 Introduction 2 3 1 eld theory in the restricted Hilbert space Introduction can be remedied by smearing out the picture changing operator [2] (see also [3] for earlier kinetic term in the Ramond (R) sector was not addressed in [10]. On the other hand, in [11] and [12] the degeneracy of the Ramond kinetic term was avoided with the help of a suitable restriction of the Ramond Hilbert space. Indeed, it was noticed [13] in the early days of string eld theory that Witten's theory propagates only a subset of constrained string elds [14]{[19]. This was subsequently related to the presence of an extra gauge symmetry (not generated by the BRST charge) that can be xed to remove all elds that do not satisfy the constraint [20] (see also [21]). A gauge invariant action for the interacting theory was recently proposed in [12] (see also [22]) with smeared picture changing operators and Ramond elds in the restricted Hilbert space. The above problem with cyclicity of the vertices was avoided by taking { 1 { the the NS eld to live in the large Hilbert space akin to the Berkovits formulation. On another front, in [11] a geometric approach, based on the decomposition of the supermoduli space was outlined, which is formulated in the small Hilbert space with a constrained Ramond sector. Furthermore, in [23] another geometric construction was proposed where the restriction on the Ramond elds is substituted by the introduction of auxiliary elds.1 The purpose of this note is twofold. First we clarify the relation between the restricted and unrestricted Ramond Hilbert spaces. In particular, we show explicitly that the restrictions used in [12] and [11] are the same and furthermore that the cohomology of the restricted Hilbert space is the same as that of the unrestricted space. The latter result was previously obtained in [24].2 In the second part we propose a modi cation of the construction [10] for the R-NS vertices which is cyclic in the small, restricted Hilbert space. Provided the picture changing operators used in [11, 12] can be de ned in a way that is compatible with the interaction vertices, our construction immediately gives a classical action for the open superstring in the small, restricted Hilbert space. More generally, the vertices can be regarded as an algebraic construction of the interaction vertices of the auxiliary eld construction of [23]. Then, invoking the results of [9] one concludes that the resulting action reproduces the correct tree-level S-matrix. 2 Restricted Hilbert space Let us start with the restricted Ramond Hilbert space spanned by vectors of the form [12]{[21] = 1j #i + 0 2j #i ( 1)j 1jG0 2j "i ; where j #i = b0j "i, j j denotes the Grassman parity of , 0 is the zero mode of the commuting superconformal ghost and G0 the (matter plus ghost) supercharge with the 0b0 contribution subtracted. More concretely, we decompose the BRST charge Q as Q = c0L0 + b0M + 0G0 + 0K 0 b0 + Q~ 2 where L0; M; G0; K; Q~ have no dependence on the ghost zero modes (see e.g. [21] for details). Then, using that fQ~; G0g = 0 and G20 = L0 it is not hard to see that Q = M (G0 2) + K( 2) + Q~( 1) j #i + 0 G0( 1) + Q~( 2) j #i + ( 1)j 1jG0 G0( 1) + Q~( 2) j "i : According to [21], 2 can be gauged away completely.3 The closedness condition reduces to with a residual gauge freedom Q~ 1 = G0 1 = 0 ; 1 = Q~ ; G0 = 0 : 1In fact, the proposals [11] and [23] were worked out for the closed type II superstring but the idea is easily adapted to the open string. 2We would like to thank Y. Okawa for pointing out this reference to us. 3Notice however, that there are some subtleties when G0 2 = 0. { 2 { (2.1) (2.2) (2.3) (2.4) (2.5) Because the cohomology of Q is known to be isomorphic to the relative cohomology Hrel(Q) calculated on on the subspace de ned by b0 A generic vector in this subspace is given by = = 0 Then, Q = 0 reduces to = 0 [24, 25] we consider this case. j #i with independent of 0 and c0. ~ Q = G0 a cokernel.4 This leads to the condition on picture ( 12 ) states , This operator acts on picture ( 32 ) states and existence of Y0 implies that X0 cannot have (2.6) (2.7) (2.8) (2.9) (2.10) (2.11) (2.12) (2.13) (2.14) with general solution, 2 0 (1) = 0 (0) = 1 j #i + 0 1 j #i + 2 j "i + 0 2 j "i (0) (1) where i(j) are independent of 0 and c0. Now requiring that the condition (2.8) is preserved by Q implies that (21) = 0 and 2 (0) = ( 1)j (11)jG0 1 (1) and thus (2.8) and (2.1) de ne the same invariant subspace. Finally we note that X0 is indeed no cokernel, i.e. every vector in this subspace can be written as = X0 ~, where ~ is an arbitrary string eld with picture 3 2 . This follows from the identities [ 26 ] ( 0) = j0; ( 0) = j0; 0( 0) = j0; 3 2 ih0; 1 2 ih0; 1 2 ih1; 3 2 1 2 j j ; 1 2 j + j1; 1 2 ih0; 1 2 j ~ = 1j #i + 2j "i with i = P n (n) ( 0) we nd 0 i 1 n=0 where the index 12 resp. 32 denotes the picture and jn; 1 2 i = 0nj0; 12 i. Then, for X0 ~ = G0( (10)) ( 1)j 2j (21) (0) j #i ( 1)j 2j 0 2 j #i + G0 2 j "i (0) where we have used that ( 0) ( 0) = j0; 32 ih0; 1 . We then see that X0 ~ is indeed of 2 j the form (2.1) with 1 = G0( (10)) ( 1)j 2j (21) and 2 = ( 1)j 2j (20) : 0 and natural numbers nk and ml. This is not a problem for free string eld theory but becomes an issue in the presence of interaction vertices which generically do not preserve this de nition. nk ml k j i = l j i = 0 for l > 0 and The vertices of open superstring eld theory can be written as Cn( 1 ; ; n) = !( 1; Mn 1( 2 ; ; n 1)); (3.1) where denotes a combined string eld in the R- and NS-sector and Mn are string nproducts. These products were constructed through a gauge transformation of the free theory de ned by a hierarchy of gauge products on the large Hilbert space with each gauge product obtained from lower order products by means of a contracting homotopy for the nilpotent operator 0, that enters in the bosonization of the superconformal ghost (z). More precisely, we require the existence of an operator such that [ 0; ] = 1. Upon changing , the construction of [2] produces actions that are related by eld rede nitions, so that any choice for is equally good. One additional condition on is that the resulting vertices should be non-singular. In [2] a class of such good homotopies built out of = I dz 2 i f (z) (z) was proposed, where f (z) is required to be holomorphic in some annulus that contains the unit circle and (z) enters in the bosonization of superconfomal ghost, (z) = @ (z)e In [10] the homotopy for [ 0; ] was taken to be the same irrespective of whether the string products de ning the string products have zero or one Ramond input. To illustrate this we consider the string product 1 3 M2 = fX; m2gP2<0> + Xm2P2<1> + m2P2<2> where P2<n> is the projector on n Ramond inputs among the two inputs of m2 and m2 = is Witten's string product. The picture changing operator, X is related to through the graded commutator, X = [Q; ]. Finally, fX; m2g is the graded anti-commutator of X and m2. For zero Ramond inputs M2 is cyclic with respect to the standard symplectic form by construction since the combination fX; m2g sums over all possible insertions of a picture changing operator (see [2] for details and notation). For vertices involving two Ramond elds we have (3.2) (z). (3.3) !(N; M2(R; R)) = !(N; m2(R; R)) = !(R; m2(R; N )) (3.4) where N and R denote NS- and R- string elds respectively. At rst sight it looks as if M2 were not cyclic since there is an X missing in front of m2 on the right hand side of (3.4). However, we will see in the end that this is exactly what we need, because of subtleties in de ning a symplectic form on the R-string elds. Next, let us consider the 4-vertex. First, we have from (3.3) [M2; M2](R; R; R) = 2Xm2 m2(R; R; R) = 0 (3.5) due to associativity of the star product (m2 m2 = 0). Thus, to this order the A1 consistency condition (or equivalently the BV-equation) allows us to set M3(R; R; R) = 0. { 4 { For two Ramond inputs we have where 1 2 1 2 Then, [M2; M2](R; N; R) = m2 Xm2(R; N; R) = [Q; [m2; 2]](R; N; R) ; 2 = m2P2<1> + f ; m2gP2<0> : 1 3 M3(R; N; R) = m3(R; N; R) ; sistency condition, 12 [M2; M2] + [Q; M3] = 0, then xes M3 completely as Since the gauge products n never have more than one Ramond input [10], the A1 conHJEP04(216) where m3 = [m2; 2] and we have used associativity of m2. Associativity then also implies that M3(R; N; R) = [m2; 2](R; N; R) = 0 and thus M3 is in the small Hilbert space. Similarly, for one Ramond input 1 2 3 = 1 4 f ; m3gP3<0> + m3P3<1> : [M2; M2](N; R; N ) = Xm2 Xm2(N; R; N ) = [Q; [Xm2P2<1>; 2P2<1>]](N; R; N ) To continue we choose the homoptopy for de ning the gauge product 3 as 1 2 1 2 = [Q; [M2; 2P2<1>](N; R; N ) = [Q; [M2; 2]](N; R; N ) : (3.8) 3(N; R; N ) = m3(N; R; N ) = m2 m2(N; R; N ) : Using, associativity of m2 again we then nd 1 2 M3(N; R; N ) = ([M2; 2] + [Q; 3]) (N; R; N ) = M2<1> 2(N; R; N ) = Xm2<1> 2(N; R; N ) = Xm3P3<1>(N; R; N ) which is in the small Hilbert space. More generally, for a generic permutation of the Rand N inputs M3P3<1> = Xm2<1> 2P3<1> = Xm3P3<1> holds. Thus, modulo the factor X that will be dealt with below, proving cyclicity of M3 is reduced to show cyclicity of m3. Explicitly, we have !(N1; M3(R1; N2; R2)) = !(N1; m3(R1; N2; R2)) = !L(N1; 0m2( m2(R1; N2); R2)) + !L(N1; 0m2(R1; m2(N2; R2))) ; { 5 { (3.6) (3.7) (3.9) (3.10) (3.11) (3.12) (3.13) where !L is the symplectic form evaluated in the large Hilbert space and which reproduces the symplectic form, !, on the small Hilbert space upon insertion of the zero mode 0 [2]. Now, commuting 0 through to R1 and using cyclicity of m2 we get Since is BPZ-even we then have Thus, m3 is cyclic with respect to the symplectic form !( ; ). In order to prove cyclicity to arbitrary order we rst recall the recursion relations de ning the higher order products [10]. For zero or one Ramond input we have Mn<+02=1> = X[Mk+1; n k+2]Pn<+02=1> ; M1 = Q Similarly, for two adjacent Ramond inputs, !(N1; M3(R1; R2; N2)) = !(N1; m2(R1; 2(R2; N2))) !(N1; 2(m2(R1; R2); N2)) = !L(N1; m2( 0R1; 2(N2; R2))) !L(N1; 2(m2( 0R1; R2); N2)) : Now, for the rst term we use cyclicity of m2 while for the second we use cyclicity of 2 for two R-inputs which gives !(N1; M3(R1; R2; N2)) = !L( 0R1; m2( 2(R2; N2); N1)) + !L(m2( 0R1; R2); 2(N2; N1)) 1 n + 1 n k=0 mn+3 = 1 (3.15) (3.16) (3.17) (3.18) (3.19) (3.20) (3.21) and for two Ramond inputs where with m2 = . Finally, the gauge products n are given by Mn<+23> = mn+3Pn<+23> = X[mk+2; n k+2]Pn<+23> 1 n + 1 n k=0 1 n + 1 n k=0 X[mk+2; n k+2] n+2 = n + 3 f ; mn+2gPn<+02> + mn+2Pn<+12> : { 6 { ) for all n. Indeed, upon inspection of (3.20), subject to the homotopy (3.21), it is apparent that such a term would have to be of the form P mn k+2mk+2 which vanishes due to the A1 condition [m; m] = 0. Furthermore, it (n 1)Mn<+11> = X mn<1> 2 + mn<1>1 3 + = (n 1)Xmn+1Pn<+11> : (3.22) To show this identity we proceed by induction. We have from (3.18) HJEP04(216) nMn<+11> = [Mn<1>; 2<1>] + [Mn<11>; 3<1>] + + [Q; n<+1>1] +Mn<1> <0> + Mn<11> 3<0> + 2 [Q; mp<1>] together with the identity, [m; M ] = 0, that is, 1 2 1 3 1 2 S = !( ; Q ) !( ~; XQ ~) + !( ~; Q ) + !( ; M2( ; )) + !( ; M3( ; ; )) + [Q; n<+1>1] = Xmn<+1>1 + [mn<1>; M2<1>] + [mn<1>1; M3<1>] + + M2<1>mn<0> + M3<1>mn<0>1 + + mn<1>M2<0> + mn<1>1M3<0> + Upon substitution of (3.24) into (3.23) and using (3.21) as well as [m; m] = 0 the result follows. Thanks to (3.19) and (3.22) the problem of proving cyclicity of Mn is again reduced to show cyclicity of mn. To prove cyclicity of mn+3, n 1, one then proceeds exactly as in (3.13){(3.17) expressing mn+3 in terms of [mk+2; n k+2] and then using cyclicity of mq, q n + 2 as well as cyclicity of p, p n + 2 for p NS-inputs. Let us now explain how these vertices lead to a gauge-invariant action for the open superstring in the small Hilbert space. Following [23] we write 1 4 { 7 { where, = + string products Mn are given by and ~ is an auxiliary Ramond string eld with picture ( 32 ). The higher Mn = MnP <0> + mn(P <1> + P <2>) which di ers from (3.3) by the ubiquitous factor X. To prove gauge invariance we use that Mn is cyclic w.r.t. !. The standard proof of gauge-invariance has to be modi ed as M is not an A1-algebra. However, M is an A1-algebra and di ers from M in that it contains an additional X-insertion on Ramond outputs and contains no BRST operator Q. There are three di erent types of gauge-transformations with odd parameters , and ~ having picture 1, 12 and 3 2 . (3.23) (3.24) (3.25) (3.26) Using antisymmetry of ! and cyclicity of Mn one arrives at the identities (n; k 2), !( ; Mn Mk) = !( ; Mn Mk) !( ; QMk) = !(Q ; Mk) = !(Q ; Mk); !( ; Mn Q) = !(Mn ; Q): = !(Mn ; P1<0> Mk + XP1<1> Mk) = !(Mn ; Mk); where denotes the coderivation built from as its 0-string map and we suppressed the string eld . Explicitly, (3:29) reads as !( ; Mn(Q ; : : : ; ) + Mn( ; Q ; : : : ; ) + ) = !(Mn( ; ; : : : ; ) + Mn( ; ; : : : ; ) + : : : ; Q ): HJEP04(216) Summing over (3.27){(3.29) we obtain zero on the left-hand side due to the A1 relations, while on the right-hand side we nd, 0 = !( ; Q ) + !( ~; Q ) + X !(( + ); Mk( ; ; : : : ; )) = = S; 1 2 !( ; Q ) + !( ~; Q ) + X !( ; Mk( ; ; : : : ; )) ! !( ~; Q ) De ne the transformation , (3.27) (3.28) (3.29) (3.30) (3.31) (3.32) (3.33) (3.34) (3.35) (3.36) where we used !( ~; Q ) = 12 !( ~; QX ~) in the last step. Consequently, the transformations (3.30) and (3.31) are a bosonic gauge symmetry of the action. By replacing with ~ in (3.27){(3.29) one veri es that the following transformation is a fermionic gauge symmetry, where X~ denotes the coderivation with 0-string product X . In order to derive the gauge transformations corresponding to the parameter , let us recall that Mn and mn(P <0> + P <1>) give two commuting A1 structures [10]. Together with cyclicity of mn(P <0> + P <1>) w.r.t. ! one can then deduce that the following transformations are a gauge symmetry of S, by imitating the previous derivation, , ~ as + ~ = Q = X ~ { 8 { Notice that all gauge transformations preserve the constraint form Q with not expressible in the form = X for some picture = X ~ up to states of the 32 state . Let us now comment on the applicability of our formalism to writing the proposal for the superstring action [12] in the small Hilbert space. Assuming the constraint (2.8), we can rewrite (3.25) without the need for the auxiliary eld ~ as 1 2 1 2 1 3 S = !( ; Q ) + !( ; Y Q ) + !( ; M2( ; )) + !( ; M3( ; ; )) + (3.37) 1 4 where Y = c0 0( 0) is the inverse picture changing operator in the restricted Hilbert space. The gauge transformation of this action agrees with that of (3.25) up to the contribution coming from the kinetic term that is S / !((X X0)(m2( ; ) + m2( ; ) + m3( ; ; + )); Y Q ) (3.38) Formally this term can be removed by replacing X by X0 (as well as by ( 0)) in the de nition of the higher string products Mn and the gauge products n when applied to states containing one or two Ramond states, e.g. instead of (3.3) we take and instead of (3.6) we take 1 3 M2 = fX; m2gP2<0> + X0m2P2<1> + m2P2<2> 2 = ( 0)m2P2<1> + f ; m2gP2<0> : 1 3 (3.39) (3.40) However, for this choice of homotopy to be well de ned, one needs that the mns are compatible with the particular realisation of the picture ( 12 ) states in terms of the zero modes 0 and 0 described in section 2. Acknowledgments We would like to thank Ted Erler and Barton Zwiebach for helpful discussions. 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Sebastian Konopka, Ivo Sachs. Open superstring field theory on the restricted Hilbert space, Journal of High Energy Physics, 2016, 164, DOI: 10.1007/JHEP04(2016)164