Supersymmetric double-well matrix model as two-dimensional type IIA superstring on RR background

In the previous paper, the authors pointed out correspondence of a supersymmetric double-well matrix model with two-dimensional type IIA superstring theory on a nontrivial Ramond-Ramond background from the viewpoint of symmetries and spectrum. In this paper we further investigate the correspondence from dynamical aspects by comparing scattering amplitudes in the matrix model and those in the type IIA theory. In the latter, cocycle factors are introduced to vertex operators in order to reproduce correct transformation laws and target-space statistics. By a perturbative treatment of the Ramond-Ramond background as insertions of the corresponding vertex operators, various IIA amplitudes are explicitly computed including quantitatively precise numerical factors. We show that several kinds of amplitudes in both sides indeed have exactly the same dependence on parameters of the theory. Moreover, we have a number of relations among coefficients which connect quantities in the type IIA theory and those in the matrix model. Consistency of the relations convinces us of the validity of the correspondence.


Introduction
Although matrix models [1,2,3] have been proposed as nonperturbative formulations of superstring/M theory, it is still difficult to compute perturbative string amplitudes from these models. Regarding two of them [1,3], since the models are formulated relying on nonperturbative objects (D-branes), it is not straightforward to see perturbative aspects of fundamental strings. The remaining one [2] is based on the Schild gauge formulation of the type IIB superstring theory. It is a fully interacting theory, and its analytical treatment to carry out the computation of a perturbative S-matrix has not been found yet.
In this situation, it will be an interesting direction to make correspondence between a supersymmetric matrix model and simpler noncritical superstring theory, in both of which perturbative scattering amplitudes are computable and the correspondence is explicitly confirmed. In fact, we pointed out in the previous paper [4] correspondence of a supersymmetric double-well matrix model to two-dimensional type IIA superstring theory on a nontrivial Ramond-Ramond (RR) background from the viewpoint of symmetries and spectrum. In this paper, we further investigate dynamical aspects of the correspondence. We compute various amplitudes in the type IIA theory, and compare with the calculation of matrix-model correlators obtained in [4]. We carefully introduce cocycle factors to vertex operators in the IIA theory in order to realize correct transformation laws and target-space statistics. In the calculation of amplitudes, the RR background is treated in a perturbative manner by insertions of the corresponding RR vertex operators. Amplitudes evaluated at the on-shell momenta are often indefinite or divergent, for which we find a reasonable regularization scheme preserving mutual locality of physical vertex operators. We thus obtain several kinds of regularized amplitudes in the type IIA theory including precise numerical factors, which allows direct comparison with the corresponding correlators in the matrix model at the quantitative level. As a result, we find that they indeed have exactly the same dependence on parameters of the theory. Furthermore, we obtain a number of relations among coefficients that connect quantities in the type IIA theory to those in the matrix model. Remarkably, all of them are consistent with each other, which provides strong evidence for the validity of the correspondence.
The rest of this paper is organized as follows. In the next section, we present some results for amplitudes of the supersymmetric double-well matrix model computed in [4]. In section 3, a brief review of two-dimensional type IIA superstring theory is given, together with discussion of cocycle factors in detail that is important for precise evaluation of amplitudes. We also explain how to take into account an RR background of the IIA theory. In section 4, we compute basic amplitudes among vertex operators in the type IIA theory on the trivial background. In section 5, the results of the basic amplitudes are transcribed to amplitudes in the type IIA theory on the RR background, that are compared with the matrix-model results presented in section 2. As a result of the comparison, they agree with each other as functions of parameters in the theory, and various relations are obtained among coefficients which link quantities in the type IIA theory with those in the matrix model. The (R−, R+) vertex operators representing the background contain nonlocal vertex operators which violate the Seiberg bound [5]. They do not satisfy the Dirac equation constraint. It would be acceptable in the sense that they do not describe on-shell particles but represent the background. However, since the nonlocal operators turn out not to be BRST-closed, we discuss consistency of amplitudes in the presence of them in section 6. The results obtained so far are summarized and some future directions are discussed in section 7. An identity concerning matrix-model amplitudes is proved in appendix A. We give a brief summary of the worldsheet superconformal symmetry in the type IIA theory in appendix B, and discuss cocycle factors for 0-picture NS vertex operators in appendix C. Integral formulas needed to evaluate IIA string amplitudes in the text are presented in appendix D. Appendix E is devoted to a note on the picture changing manipulation of Friedan-Martinec-Shenker [6] in a certain amplitude in the presence of the nonlocal operator.
(2.11) Some genus-one amplitudes are presented in appendix A in [4]. Among them, 1 N tr B 1 = 0 (2.12) 1 The explicit form of the "mixing" for the first few Φ 2k+1 is presented in eqs. (4.40) in [4]. 2 In general, matrix models can be regarded as a sort of lattice models for string theory. Then, entire functions are analogous to lattice artifacts that are irrelevant to continuum physics.
3 Ψ 1 andΨ 1 have no "mixing" term. The "mixing" terms for Ψ 3 andΨ 3 are given in eqs. (6.29), (6.30) in [4]. means that the torus free energy is a constant independent of µ 2 . It is reasonable to expect that the constant vanishes. Actually, from the result of eq. (3.42) in [7], the partition function in the sector of the filling fraction (ν + , ν − ) becomes (2.13) It is valid in all order in 1/N expansion, indicating that the free energy defined by − ln |Z (ν + ,ν − ) | is zero at each topology, and that the expectation is correct. (The sign factor (−1) ν − N could not be seen from the conventional string perturbation theory.) 3 Two-dimensional type IIA superstring In this section, we explain the two-dimensional type IIA superstring theory, which is discussed in [8,9,10,11,12]. Then, we mention correspondence of physical vertex operators in the type IIA theory with operators in the matrix model [4]. The target space is (ϕ, x) ∈ (Liouville direction) × (S 1 with self-dual radius), and the holomorphic energy-momentum tensor on the string worldsheet is given by with Q = 2. Here, ψ x and ψ ℓ are superpartners of x and ϕ, respectively and b, c (β, γ) represent conformal (superconformal) ghosts. OPEs for the fields are and the others are regular. In order to treat the Ramond sector in the RNS formalism, it is convenient to bosonize ψ x , ψ ℓ , β and γ as Some properties of worldsheet superconformal generators are summarized in appendix B.
Vertex operators are constructed by combining the NS "tachyon" vertex operator (in the (−1) picture): and the R vertex operator (in the (− 1 2 ) picture): Here the local scale invariance on the worldsheet imposes p ℓ = 1 ± k for T k , V k, ǫ and p ℓ = 1 ±k forTk,Vk ,ǭ . We consider a branch of p ℓ = 1 − |k|, p ℓ = 1 − |k| satisfying the locality bound (p ℓ ≤ Q/2) [5] for a while. Target-space supercurrents are As discussed in [8,9,10,11,12], physical vertex operators should satisfy locality with the supercurrents, mutual locality, superconformal invariance (including the Dirac equation constraint) and the level matching condition. Two consistent sets of physical vertex operators are found in ref. [11], which are called as "momentum background" and "winding background". As considered in [4], we focus on the "winding background". It is given by before imposing the Dirac equation constraint. It requires the states corresponding to the Ramond vertex operators to be annihilated by the zero-mode of worldsheet superconformal generator T m,F : Then, imposing the Dirac equation constraint amounts to (NS, NS) : (R+, NS) : , · · · ). (3.12) Note that the spectrum (3.12) is invariant under not only the N = 1 but also the N = 2 superconformal symmetry. In particular, each of G + 0 and G − 0 annihilates the Ramond states in (3.12).

Cocycle factors
Here, we introduce cocycle factors to realize correct transformation laws and target-space statistics for vertex operators 4 .
Vertex operators without cocycle factors have the following two problems. First, the OPE between q + and T k (k ∈ Z + 1 2 ): implies that the radial ordering should be defined as (3.14) The factor (−1) − 1 2 −k ensures continuity at |z| = |w|. Then, the target-space supercharge Q + = dz 2πi q + (z) acts on T k in a manner (3.15) so that the transformation law can be given by the contour integral. The k-dependent sign factor is due to the fact that the supercurrents (3.9) carry x,x-momenta, which is peculiar to noncritical superstring theory [8,9,10]. On the other hand, target-space statistics suggests that Q + should act on an (NS, NS) field T kT−k in the form of a commutator. It will become consistent with (3.15), if we makeT −k and q + noncommuting such as by introducing cocycle factors. If this condition is met, it automatically follows that Q + transformation of V k,+1Tk is given by an anticommutator in accordance with the targetspace statistics. (See the first formula in (3.17).) For other vertex operators, we have the radial orderings: Second, the OPE between an (NS, NS) field T kT−k and an (NS, R−) field T kVk, −1 (k, k ′ ∈ Z + 1 2 ): with z − w = r e iθ ,z −w = r e −iθ leads to the radial ordering It is consistent with the target-space statistics when − 1 2 + 2kk ′ is even, but not otherwise. Similar is the situation for the radial ordering of other fields.
Let us introduce cocycle factors to resolve these problems. We put the hat (ˆ) on vertex operators with cocycle factors. For the target-space supercurrents, (3.20) where β is a constant to be determined. p φ , p h and p x (pφ, ph and px) are momentum modes of holomorphic part (anti-holomorphic part) of free bosons: with · · · representing oscillator modes. From the commutation relations we see noncommuting properties which lead toq The target-space statistics requires Hereafter we assume that β satisfies this condition. Modified target-space supercharges are given byQ For vertex operators, we introduce cocycle factors aŝ It turns out to be a choice resolving the above problems.
In the first problem, sincē hold, we haveTk (w)q + (z) = e i2πβ(− 1 2 +k)q + (z)Tk(w) (3.30) that realizes (3.16) for (3.26). Then,Q + acts on a hatted (NS, NS) field in the form of a commutator: In (3.32), the cocycle factors do not include any modes in the holomorphic sector and can be treated as constants in the radial ordering. For other cases, owing to the noncommuting propertieŝ the modified supercharges consistently act on hatted fields as In the second problem, due to the cocycle factors, we have the following noncommuting relations:Tk Let us see the OPE considered in (3.18) in the presence of the cocycle factors. By using is obtained when |z| > |w|. In the normal ordering of holomorphic fields :T k (z)T k ′ (w) : , the cocycle factors can be treated as constants. It is similar for : . On the other hand, when |z| < |w|, we use (3.37) to havê Note that (3.39) has the same form as (3.40) since holds for (3.26) and k, k ′ ∈ Z + 1 2 . Thus, the radial ordering of the hatted fields becomes consistent with the target-space statistics: Similarly, we can show that the radial ordering of all other hatted vertex operators is consistent. Cocycle factors for 0-picture NS fields are discussed in appendix C.

Correspondence to matrix model operators
In order to make correspondence to the matrix model, we first note that the vertex operatorsV form a quartet underQ + andQ − : which are isomorphic to (2.2) and (2.3) in the matrix model under identification between supercharges in both sides. It leads to the correspondence of single-trace operators in the matrix model to integrated vertex operators in the type IIA theory: where the bare string coupling g s is put in the r.h.s. to count the number of external lines of amplitudes in the IIA theory. Furthermore, it can be naturally extended as for higher k(= 1, 2, · · · ). As discussed in [4], if we regard ψ (ψ) as a target space fermion in the (NS, R) sector (the (R, NS) sector) in the corresponding type IIA theory, φ and B are interpreted as an operator in the (R, R) sector and that in the (NS, NS) sector, respectively. Then, (ν + − ν − ) represents the RR charge (up to a proportional constant). The correspondence (3.46) and (3.47) are consistent with this interpretation. The (R−, R+) vertex operators behave as singlets underQ + andQ − . Actually, for k = 0, 1, 2, · · · is shown by taking the OPEs. The vertex operators can be expressed asQ + -andQ − -exact forms: where the homotopy operatorŝ give the inverses of the supercurrents in the sense that However, : (∂ kq−1 + )V −k, −1 (z) : and : (∂ kq−1 − )V k, +1 (z) : appearing in the r.h.s. of (3.49) are not physical operators, because they belong to the NS sector but have integer x-and x-momenta 5 . Thus, we conclude that the (R−, R+) vertex operators are singlets under the target-space supersymmetry.
In the correspondence (3.46) and (3.47), the (R−, R+) vertex operators seem to have no counterparts in the matrix model. The result of the amplitudes (2.6) and (2.10) implies that correlators of operators with nonzero Ramond charges do not vanish from the viewpoint of the correspondence. Hence it is anticipated that the matrix model represents the type IIA theory on a nontrivial background of (R−, R+) operators.

Type IIA theory on RR background
The worldsheet action of the type IIA theory consists of the free CFT part and the Liouville-like interaction part: where d 2 z = d(Re z) d(Im z), each boson with the suffix "tot" represents the sum of its holomorphic and anti-holomorphic parts, and the 0-picture NS fieldsT (z) do not have the ǫ = −1 andǭ = +1 parts in (C.6): Here and in what follows, superscripts indicating the picture numbers are put on vertex operators except that they have the natural pictures ((−1) for NS fields and (− 1 2 ) for R fields). The form of S int corresponds to the term Ntr(−iµ 2 B) in the matrix model action via (3.46) (up to a choice of the picture under identification of 1/N and g s ). The Liouville coupling µ 1 is related to µ 2 in the matrix model, which is clarified in section 5.
In the trivial background, the genus-zero amplitude with insertion of integrated vertex Dividing by the conformal Killing group of the sphere is equivalent to fixing the positions of three vertex operators with cc inserted at each of the fixed positions: We take a usual choice of (z 1 , z 2 , z 3 ) = (∞, 1, 0). As the amplitude on a nontrivial (R−, R+) background, we consider 6 where the background is incorporated as a linear combination of vertex operators in the (R−, R+) sector with numerical coefficients a k : (3.12). On the other hand, for V RR k (k = 1, 2, · · · ), we choose (R−, R+) vertex operators of the nonlocal branch (p ℓ = 1 + |k|). Since the nonlocal vertex operators are invariant underQ + andQ − as well as the local ones, W RR consists of the maximal set of (R−, R+) vertex operators preserving the target-space supersymmetry 7 . Notice that we do not regard the (R−, R+) operators as particles in asymptotic states, but as a background. From this point of view it will be natural to include nonlocal operators in addition to the local ones in W RR 8 . Actually, as we will see later, the inclusion of nonlocal operators is crucial to match the matrix-model amplitudes with the type IIA ones. It should be pointed out that the nonlocal operators do not satisfy the Dirac equation constraint. More precisely, they are invariant under a half of the worldsheet supersymmetry transformations ( . This point will be discussed in some detail in section 6. It seems somewhat similar to a boundary operator, and tempts us to interpret it as a certain brane-like object, which however preserves linear combinations of holomorphic and anti-holomorphic generators (for example, G + 0 + iḠ − 0 and G − 0 + iḠ + 0 for an A-brane configuration [16]). Such a brane could exist without breaking the target-space supersymmetry in our case, since the supersymmetry does not induce translations in the target space. It would be interesting to proceed analysis from the viewpoint of this interpretation.
The treatment of the background (3.58) is a perturbation from the trivial background, and valid for small |q RR |. We exactly compute the path-integral with respect to the constant mode of the Liouville coordinate ϕ tot as performed in refs. [17,18] denotes the Euler number of the manifold M), V L is the volume of the Liouville direction, and the suffix "CFT" means the correlator computed under the free CFT action S CFT in (3.54). Calculation of the amplitude (3.61) 7 Ref. [11] considers different nonlocal RR vertex operators with ǫ,ǭ flipped in addition to p ℓ = 1 + |k|. They satisfy the Dirac equation constraint, but do not preserve the target-space supersymmetry. 8 According to ref. [14], we find some similarity in the c = 1 Liouville theory. Nonlocal W ∞ operators (forbidden by the Seiberg bound) there deform the linear dilaton background to a two-dimensional blackhole one [15] with an infinitesimally small mass.
can be explicitly carried out only when s is a nonnegative integer. Then, according to [19], the divergent factor Γ(−s) is regularized as As usual in computations in the RNS formalism, the total picture in each of holomorphic and anti-holomorphic sectors should be adjusted to 2h − 2 for a handle-h Riemann surface. Although cocycle factors in (3.54) and (3.60) might seem to induce nonlocal interactions, we will see in the following that they merely give phase factors to amplitudes reflecting target-space statistics.

Basic amplitudes
As a preparation to obtain IIA amplitudes on the RR background, we compute some basic CFT amplitudes in the form (3.61). We will consider various amplitudes which contain RR fields and are relevant to comparison with the matrix model results. The nonlocal branch (p ℓ = 1 + |k|) as well as the local one (p ℓ = 1 − |k|) are considered for (R−, R+) vertex operators. Amplitudes which consist only of "tachyons" are briefly mentioned at the end of this section.
Before computation, we notice that in the spectrum (3.12) the target-space bosons coming from the (NS, NS) and (R±, R∓) sectors are "winding-like"k = −k, while the target-space fermions from the (NS, R−) and (R+, NS) sectors are "momentumlike"k = k. Then it immediately follows that momentum/winding in the x-direction is conserved separately in the bosons and fermions. As a corollary, we conclude that if we have a fermion in the (NS, R−) sector, there must be a one in the (R+, NS) sector and vice versa to obtain a nontrivial amplitude.

(NS, NS)-(R+, R−)-(R−, R+)
We first compute the three-point amplitude among the fields of (NS, NS), (R+, R−) and (R−, R+):V which is compared to the matrix model amplitude (2.6) in section 5.1. From the conservation of H andH charges (or equivalently, integrals over the zero-modes h 0 andh 0 in (3.61)), only the s = 0 case is possibly nonvanishing. The s = 0 amplitude reads Here, the bra vacuum 0| has the background charge (+2, +2) for the bosonized superconformal ghost (φ,φ) and the background charge − Q 2 χ(S 2 ) = −2 for the Liouville field, while the ket vacuum |0 is neutral for both of these charges. By using (3.36) and (3.38), the last line in (4.2) becomes We move the three cocycle factors in the last line to act on |0 . Since |0 is annihilated by p φ , p h and p x , the cocycle factors do not work anymore: Similarly, by moving the three cocycle factors in the first line to act on 0|, the phase factor e i4πβ arises picking up the background charge forφ. However, because of β ∈ Z + 1 2 the phase is trivial. Thus, Now, the amplitude is factorized into the holomorphic and the anti-holomorphic part as The last line in (4.6) is computed by the Wick contraction. We end up with where the factor 1 V L in (4.6) is canceled with V L from the delta-function of the conservation of the Liouville momentum: 8) and the phase factor in (4.6) was recast by using the x-winding conservation i k i = 0. We fix three positions as (z 1 , z 2 , z 3 ) = (∞, 1, 0) with inserting cc at each of them. The result is (4.9) The kinematical constraints ( i k i = 0 and i p ℓ i = 2) are met by for the local branch ofV 3 , and by with k = 1, 2, · · · for the nonlocal branch. Notice that if we did not allow the nonlocal branch, k 2 could not take values in N + 1 2 .

2(R+, R−)-2(R−, R+)
Let us compute the four-point amplitude of two (R+, R−) and two (R−, R+) fields: with a = 1, 2 and b = 3, 4. It gives the counterpart of the matrix model result (2.7) or (2.11) as we see in section 5.2. Only the s = 0 case in the amplitude (3.61) can be nontrivial from the conservation of H andH charges. A parallel argument to (4.2)-(4.7) leads to The corresponding string amplitude on the trivial background is obtained from (4.13) by fixing the first three positions as (z 1 , z 2 , z 3 ) = (∞, 1, 0) and integrating the rest (z 4 ). Then, we have where The kinematics restricts k i and p ℓ i as follows. For both ofV b belonging to the local branch, with the corresponding Liouville momenta respectively. For one ofV b (say,V 3 ) local and the rest (V 4 ) nonlocal, with n 1 , n 2 = 0, 1, 2, · · · but (n 1 , n 2 ) = (0, 0). The case of both ofV b nonlocal is not allowed.
If we try to plug these on-shell values into I (1,0) in (D.12) (or (D.14)) directly to get I (1,0) , it becomes indefinite or divergent. Thus we adopt the following prescription as a regularization. All the powers of the integrand in I (1,0) are uniformly shifted, i.e. α,ᾱ, β,β are shifted by the same small quantity ε. Note that the uniform shift preserves the mutual locality of vertex operators and thus the equality between (D.12) and (D.14) (see (D.15)). In the regularized result, we take 1 ε proportional to the volume of the Liouville direction: with c L being a proportional constant. Since the divergence can be interpreted as a resonance in string theory, it seems plausible to regard 1 ε as the Liouville volume. Namely, it essentially has the same origin as in (3.62). Similar treatment is found in c = 1 noncritical bosonic string theory [20].

Case of both ofV b local
For the case of both ofV b local, α =ᾱ = 0 and β =β = −1 at (4.16) and (4.17). Then, Plugging (4.20) into (4.14), we have For the case of one ofV b (say,V 3 ) local and the rest (V 4 ) nonlocal, α =ᾱ = n 1 + n 2 and β =β = −n 1 − 1 at (4.18). As a result of the regularization with (4.19), Then, the amplitude finally becomes As a consistency check, we can see that the case ofV 4 local andV 3 nonlocal gives the identical result. It should be so, sinceV b are target-space bosons.

(NS, R−)-(R+, NS)-(R−, R+)
Next we turn to the amplitude including the target-space fermions. We consider the three-point amplitude of (NS, R−) and (R+, NS) fermions and an (R−, R+) field: which corresponds to the matrix-model amplitude (2.10) with k = ℓ = 0 as is seen in section 5.3. The s = 0 case alone satisfies the conservation of H andH charges in (3.61).
After a similar calculation as in the previous one, we have Here as we mentioned at the beginning of this section, δ k 1 +k 2 , 0 and δ k 3 , 0 represent the conservations of x-momentum and of x-winding, respectively. The string amplitude is obtained as (4.26) The kinematics allows only the possibility with the local branch ofV 3 . Its nonlocal branch is not allowed.
From the Wick contraction, wherek k 1 or theǭ = +1 part ofT k 2 does not contribute to the amplitude, which ensures the correct target-space statistics as discussed in appendix C.
The kinematics restricts k i and p ℓ i as and all ofV a local. It leads to the string amplitude Here, I (1,1) is the integral I (1,1) given by (D.2) with To evaluate I (1,1) at the on-shell momenta (4.30) by (D.31), we use the regularization mentioned above. Namely, all the powers of the integrand in I (1,1) (α, β, α ′ , β ′ ,ᾱ,β,ᾱ ′ , β ′ , 2σ) are uniformly shifted by ε. Then, γ and γ ′ given by (D.27) become Under the shift, we have from the expressions (D.32) and (D.33). Here, the formula The factors s(β), s(β ′ ), s(β + 2σ) and s(β ′ + 2σ) in (D.31) become O(ε) quantities, which absorb the divergent factors from C 12 's and C 23 's. Thus, the final result of I (1,1) is finite: From (4.31) and (4.36), we end up with (4.38) The conservation of H andH charges shows that the amplitude (3.61) for s = 0, −2 can be nontrivial. Although the s = −2 case is not calculable by the standard CFT technique, the kinematical constraint from the Liouville momentum leads to i |k i | = 1, which however is not met because of |k i | ≥ 1 2 for all i. Thus, the s = −2 amplitude should vanish, and the s = 0 case alone remains to be considered 10 . The Wick contraction leads The kinematical constraints allow Then, the corresponding string amplitude is expressed as where I (1,0) is given by We calculate I (1,0) at the on-shell value (4.40) by the regularization method to obtain However, since the factor (p ℓ 1 + k 1 ) 2 vanishes for (4.40), we conclude that the amplitude is trivial: The result (4.44) can be intuitively understood. From the H andH charge conservation, only the (ǫ,ǭ) = (−1, +1) part of is allowed to contribute to the amplitude. But, that part disappears at the on-shell (4.40) from (C.2), and the amplitude must vanish. It supports the validity of the regularization method.

2(NS, NS)-2(R+, R−)
The four-point amplitude of we have the expression of the string amplitude Here, (4.49) I (1,0) evaluated at the on-shell value by the regularization becomes Since the factor (p ℓ 2 + k 2 ) 2 vanishes at the on-shell, we find

4(NS, NS)
For readers who are interested in "tachyon" amplitudes, we present the s = 0 amplitude of four (NS, NS) "tachyons" 11 where From (D.12) or (D.14), we can formally rewrite (4.54) to the form: The factors of the gamma functions are common in "tachyon" amplitudes in two-dimensional (super)string theory, for example eq. (3.14) in [19]. However, at the on-shell momenta our regularization scheme gives Then we obtain the vanishing amplitude 12 The result (4.59) immediately implies that the three-point function of "tachyons" for s = 1 also vanishes from (3.61) and (3.62). Together with the conservation of H,H charges and the Liouville momentum, we conclude that it vanishes for any s.

Check of the correspondence
In this section, we compute various amplitudes in the IIA theory on the RR background based on the result in the previous section, and show that the correspondence (3.46) and (3.47) hold at the level of the amplitudes.
Here we consider leading nontrivial contributions in the perturbation by W RR . Then, the RR charge q RR will be identified with the quantity (ν + − ν − ) in the matrix model: 13 To make (3.46) and (3.47) more precise, we introduce numerical coefficients c k , d k andd k and write As we will see later, these coefficients appear to contain no divergence. It is contrast to the correspondence of two-dimensional bosonic string theory to the c = 1 matrix model or the Penner model, where momentum-dependent divergent factors, the so-called leg factors, should be put to connect quantities in the string theory with those in the matrix model [20,22]. 13 A similar observation is made in a matrix model for noncritical type 0B string theory [21], where eigenvalues asymmetrically filled in two potential wells are interpreted as an RR field.
14 Here we have implicitly assumed that µ 1 > 0. 15 The additional term − 1 4 ∂ ∂µ1 W RR could appear in the r.h.s. of (5.2) due to the µ 1 -dependence of the RR background. It gives small corrections of the order (ν + − ν − ) compared to the leading 1 4 V (0,0) B (0). In this section, we focus on contributions from the leading and neglect these corrections.

N tr(−iB) Φ 2k+1 C,0
The matrix-model amplitude N tr(−iB) Φ 2k+1 C,0 is obtained by differentiating (2.6) with respect to ω: ln ω + (less singular). (5.5) Note that this is equal to the leading contribution to Ntr(−iB) Φ 2k+1 C at large N as seen from (2.4). The corresponding IIA amplitude is N g −2 where we put an overall normalization constant N independent of fields, and the bare string coupling g −2 s due to the sphere topology (a string tree amplitude) 16 . The leading nontrivial contribution in the small (ν + − ν − )-expansion comes from the linear order of W RR in (3.58). Under an appropriate choice of the picture, it reads (5.6) (4.9)-(4.11) were used in the last equality. We find that dependence on ω as well as ν ± completely coincides in (5.5) and (5.6) for any k. In particular, as we have noticed at the end of section 4.1, the existence of the nonlocal branch enables this agreement to hold for any k ∈ N. Furthermore by identifying their coefficients, we have a relation The large-N leading part of the two-point function Φ 2k 1 +1 Φ 2k 2 +1 C in the matrix model reads from (2.11) as + (less singular) . (5.9) 16 In general, g 2h−2 s is put for a string h-loop amplitude.
The corresponding IIA amplitude is N g −2 , whose leading nontrivial contribution comes from the quadratic order of W RR as From the result in section 4.2, we have There appears the square of the Liouville volume (2 ln ω) 2 . One of them is from the integral over the Liouville constant mode as in (3.62) as usual, while the other from the resonance of on-shell particles and the background as mentioned in (4.19). Thus we obtain (5.9) and (5.12) indeed have the same dependence on ν ± and ω for any k 1 and k 2 . Moreover, the dependence on k 1 and k 2 of the coefficient in (5.12) is written in a factorized form as f (k 1 ) f (k 2 ) g(k 1 + k 2 ), where f and g are some functions. It serves as a nontrivial check to see that the matrix model result exhibits the same factorization as well. It is not manifest at all in the original expression (2.7), but (2.11) and therefore (5.9) are indeed so. Identifying (5.9) with (5.12) leads to Notice that the r.h.s. is independent of k 1 and k 2 , and thus that the product of the first two factors in the l.h.s. must give a function of k 1 + k 2 . It determines the k-dependence ofĉ k andâ k aŝ c k =ĉ 0 e γk (2k + 1)!,â k =â 0 e −γk k!(k + 1)! (k = 0, 1, 2, · · · ) (5.14) with γ being a numerical constant and As another nontrivial check, we can see that (5.14) correctly reproduces the k-dependence of (5.7) which is obtained from a separate amplitude. Then, we have (Note that e i2πβ = −1 due to (3.26).)

Ψ 1Ψ1 C,0
The large-N leading part of Ψ 1Ψ1 C , which is given by the k = ℓ = 0 case of the matrix-model amplitude (2.10), . Its leading nontrivial contribution with respect to small (ν + − ν − ) comes from the linear order of W RR in (3.58): In the last line, we used which is relevant to the target-space supersymmetry as is seen later.

Ψ 3Ψ3 C,0
The large-N leading of Ψ 3Ψ3 C reads from the k = ℓ = 1 case of (2.10) as π ω 3 ln ω + (less singular) .  (1) , whose leading nontrivial contribution in the (ν + − ν − )-expansion arises from the cubic order of W RR : By making use of (4.30) and (4.37), the final expression becomes We again find that (5.22) and (5.24) have exactly the same dependence on ν ± and ω. Comparing their coefficients gives As we have seen so far, it is remarkable that the single choice of the RR background (3.59) and (3.60) realizes the agreement between several kinds of IIA amplitudes ((5.6), (5.12), (5.19) and (5.24)) and the corresponding matrix-model correlators, with respect not only to the dependence of ν ± and ω but also to prefactors (depending on x,xmomenta/powers of matrices).

Target-space supersymmetry
Corresponding to (5.4), let us identify the supercharges in the matrix model Q andQ with those in the IIA theory as by putting the coefficients α,ᾱ. From (5.2) and (5.4), each of the quartet (Φ 1 , Ψ 1 ,Ψ 1 , 1 N tr(−iB)) with respect to Q,Q should be precisely mapped to each of ( from which we obtain (5.21) again. Note that the argument here does not refer to any amplitudes. Nevertheless, it reproduces the relation (5.21) which was derived from amplitudes in the previous subsection. This also shows consistency of the correspondence, in particular, the identification of the supercharges in both sides.

Torus partition function
In this subsection we confirm that the IIA theory on the RR background is a consistent superstring theory by checking that it has a modular invariant torus partition function.
We also see that the result is consistent with the torus free energy of the matrix model. The genus-one amplitude among vertex operators V i is Here, the volume of the conformal Killing group of T 2 is nothing but the area of the torus of the worldsheet. The corresponding c,c-ghost zero-modes are fixed by the insertion cc(0). B,B are b,b-ghost insertions associated with the integration with respect to the torus moduli τ . The integration is over the fundamental region F . In a similar manner as in section 3.3, the torus partition function under the RR background is expressed as with s = −2 i p ℓ i due to χ(T 2 ) = 0. Note that every vertex operator in W RR has nonpositive x-winding from (3.60). For s ≥ 0, the s = 0 case alone possibly give nonvanishing contribution to 1 , and only V RR k=0 remains in W RR . Furthermore, the conservation of the Liouville momentum i p ℓ i = 0 tells us that even the remaining V RR k=0 gives no effect. Thus, just the first term of n = 0 in the sum ∞ n=0 1 n! (W RR ) n can contribute to the partition function. Because this argument relies only on the conservation of the x-winding and the Liouville momentum, it holds irrespective of a way to distribute the picture charges.
Next, let us consider the s < 0 case. Although the computation cannot be carried out by the standard CFT technique, we can argue that the partition function should be nil from the conservation of the picture and H charges. Since the x-winding of V RR k is k ∈ Z, its conservation law means s = −2, −4, −6, · · · . (5.34) The total picture must be (0, 0) to provide a nontrivial amplitude of the torus topology. We consider contribution from (W RR ) n (n = 0, 1, 2, · · · ) to (5.33). Any RR field there has the (− 1 2 , − 1 2 )-picture originally as in (3.8), (3.59) and (3.60). Hence n should be even and there must be insertion of n 2 picture raising operators (Q BRST (2ξ · )) in the holomorphic sector so that the total picture will be zero: Thus the total H charge of (W RR ) n with the n 2 picture raising operators will be at most − 1 2 × n + 1 × n 2 = 0, while that of V (0,0) B (0) s is s. Recalling (5.34), we see that the total H charge of the amplitude cannot be conserved, and the contribution from (W RR ) n to the partition function should be zero. Now we conclude that the torus partition function becomes As expected from the fact that the two-dimensional string has no dynamical degrees of freedom of oscillator modes, contributions from oscillators cancel leaving those from x-winding/momentum [23]: 17 For example, the explicit form of the (+ 1 2 )-picture R vertex operator (p ℓ = 1 − |k|, k = ǫ|k|) reads with (dτ ) ∝ d 2 τ /(Im τ ) 2 modular invariant. On the integrand in the r.h.s., neither the level matching condition 18 nor the Dirac equation constraint is imposed [10,11]. Holomorphic and anti-holomorphic x-momenta (k,k) in the (NS, NS) state sum are restricted as follows 19 . k,k ∈ Z + 1 2 for the corresponding vertex operator (e −φ+ikx+p ℓ ϕ e −φ+ikx+p ℓφ ) to be local with the target-space supercurrents q + andq − . Also, from the locality between two (NS, NS) vertex operators with the momenta (k,k) and (k ′ ,k ′ ), we have kk ′ −kk ′ ∈ Z. These two conditions lead to 2 ∈ Z is satisfied and Z (NS,NS) (τ + 1,τ + 1) = Z (NS,NS) (τ,τ ).) The former contains the "momentum background", while the latter does the "winding background" (3.10) that we are considering. The restriction is an analog of the GSO projection in critical string theory [8]. Thus, This is consistent with the matrix model results (2.12) and (2.13) which mean the zero torus free energy. We obtain the same result even if the integrand is assumed to be slightly generalized to linear combinations as 18 The level matching condition is taken into account by performing the integral of Re τ . 19 A similar argument is presented in [9]. 20 The result of the torus partition sum is different from that on the trivial background computed in [9,10]. The partition sums in the (NS, R−) and (R+, NS) sectors obtained there are the half of our result, and cancellation with the (NS, NS) sector is observed (See eqs. (3.14), (3.15) in [9] and appendix B.1 in [10].). We guess that the Dirac equation constraint is imposed in the (NS, R) and (R, NS) sectors but not in the (R, R) sector in [9,10]. In contrast, we do not consider the Dirac equation constraint for any R sectors in (5.37). Their result is modular invariant as well as ours.
where coefficients a, b and c are fixed so that (5.42) is modular invariant. From (5.39) and (5.40), for which to be modular invariant there is no other possibility than the prefactor 1+a−b−c being null. Then the torus partition function vanishes. Note that our conclusion of the vanishing torus partition function is not usually expected from the supercurrents (3.9) carrying x,x-momenta [8,9,10]. However, in our case where the (R−, R+) vertex operators represent the fixed RR background and does not take part in the torus partition sum, contributions from the remaining three sectors are balanced as seen in the above. Furthermore, the RR background itself does not spoil the supersymmetry, which is consistent with the fact that the (R−, R+) fields are singlets under the supersymmetries, as shown in section 3.2. As a result, we obtain what is naively expected from a supersymmetric theory.

Remarks on nonlocal RR vertex operators
The nonlocal vertex operators appearing in the background W RR do not satisfy the Dirac equation constraint as we have pointed out below (3.60). It would be acceptable from the point of view of representing a background and not on-shell particles. As a consequence, however they are not BRST-closed: Nevertheless, we can formally see that this violation of the BRST invariance does not contribute to amplitudes among BRST-closed physical vertex operators (Q BRST (cV i, phys (z)) = 0). For example, by deforming the contour of the BRST current 21 . Actually in relevant amplitudes in the previous sections, no more than one nonlocal operator is inserted. 21 We might also give somewhat similar but more formal argument by assuming the existence of the CFT state even corresponding to the nonlocal operator. As mentioned below (3.60), = 0, which implies that the breaking of the BRST invariance would have zero norm G + 0 V (nonlocal) k, −1 = 0 in the matter CFT sector and would be decoupled in our case where the matter CFT is unitary. However, that is not always true as we see that the integrand in B of (E.9) is not zero at z 4 = 0.
We emphasize that (6.1) does not immediately mean inconsistency of the theory. Indeed, the nonlocal vertex operators have the conformal weight (1, 1) preserving the worldsheet conformal symmetry. Namely, they are marginal perturbations around the flat background given by S CFT in (3.53) 22 , under which the theory should make sense as a string theory. We have also seen in section 5.7 that the theory is modular invariant. Furthermore, the breaking (6.1) solely comes from the breaking of the global worldsheet supersymmetry G + 0 . Thus we can construct a BRST-like charge with (B.8) and (B.17). This charge is nilpotent and annihilates the physical vertex operators (3.12) as well as the nonlocal vertex operators. This situation is reminiscent of boundary states as mentioned in section 3.3. For amplitudes containing boundary states in that case, physical vertex operators inserted in the bulk are invariant under each of the holomorphic and anti-holomorphic BRST operators, while the boundary states break the invariance under a half of them and preserve the other half (a certain linear combination of them). As discussed in [24,25], it requires intricate examinations to confirm the BRST invariance for such amplitudes due to b-ghost insertions associated with the boundary states and to continuation to off-shell momenta as a regularization. Although it is not easy to prove decoupling of the BRST-exact operators and independence of the way to distribute the picture charges in a generic amplitude, the literature investigates this issue by taking some concrete amplitudes and gets affirmative consequences. Similarly, let us see evidences supporting that (6.1) does not ruin consistency of the theory at least for amplitudes discussed in the previous sections.
6.1 Target-space gauge symmetry (6.1) may imply that BRST-exact operators potentially do not decouple from amplitudes in the presence of the nonlocal operators. In general, decoupling of BRST-exact operators guarantees gauge symmetry in target space, and its breaking would run into serious inconsistency of theory. Note that in the two-dimensional superstring we are considering, only the RR vertex operators concern gauge particles, and they do not couple to its gauge potential, but directly to gauge invariant U(1) RR field strength. Thus there must be no gauge transformation in the target space expressed by BRST-exact operators. Actually, as is shown in the following, we cannot construct any BRST-exact shift to the (−1)-picture NS field cT k (z) = c e −φ+ikx+p ℓ ϕ (z): in a consistent manner. U(z) should have the ghost number zero, the picture (−1) and the weight zero. Moreover, it should have the term e ikx+p ℓ ϕ of the weight 1 2 which is common to cT k . Hence the prefactor of e ikx+p ℓ ϕ appearing in U(z) has the weight (− 1 2 ). As discussed in [25], let us consider the operators consisting of b, c, ξ, η and φ with the ghost number zero and the picture P given by 23 with their weights (6.6) The prefactor of e ikx+p ℓ ϕ in U(z) has the form of A + or A − with P = −1 multiplied by polynomials of derivatives of x, ϕ, H and φ. It may also be multiplied by factors of a form : ∂ k b∂ ℓ c : or : (∂ p+1 ξ)∂ q η : (k, ℓ, p, q ≥ 0) that do not change the ghost number or the picture number. Note that such multiplicative fields increase or keep the weight, but never decrease it. Thus, the weight must be for P = −1. However, any integer n does not satisfy this condition, meaning that such U(z) does not exist. Similar argument is possible for other vertex operators 24 .

Picture changing operation
Since the picture changing operation discussed in [6] relies on the BRST invariance of vertex operators, one may wonder if (6.1) prevents it. Here we concretely demonstrate the picture changing operation in the presence of the nonlocal operators. Relevant amplitudes investigated in this paper are in sections 4.1 and 4.2. Let us consider the holomorphic part of the amplitude in section 4.1: 25 We change the picture assignment of the NS and R+ fields from the (−1) and (− 1 2 ) pictures to the 0 and (− 3 2 ) pictures by use of 10) 23 Note that ξ and η have the picture charges (+1) and (−1), respectively. 24 We have explicitly seen that no BRST-exact shift is allowed for the 0-picture NS field and R fields with the pictures (− 1 2 ), (+ 1 2 ) and (− 3 2 ). 25 A parallel argument can be applied to the anti-holomorphic part.
where T (0) k (z) is given by (C.2). Note that the picture changing operation does not commute with the multiplication of the c ghost. The (− 3 2 )-picture field is obtained by the inverse picture changing operator Y (z) = 2c (∂ξ) e −2φ (z) as We introduce the ξ zero-mode to move to the large Hilbert space: Here, ξ 0 can be replaced with ξ(z 1 ) = n∈Z ξ n z −n 1 because 0| · · · |0 | large = 0 when · · · does not contain ξ 0 . Plugging (6.10) into (6.12) and manipulating the contour of the BRST current leads to (6.12) (6.13) The first and second terms come from the BRST currents encircling the points z 1 and z 3 , respectively. We move ξ(z 2 ) to the bra vacuum in the first term, and go back to the expression on the small Hilbert space: that is the realization of the picture changing operation. Although the second term would seem to remain due to (6.1), a closer look shows that this is not the case. Actually, it reads (6.15) Notice (6.15) does not conserve any of the (ξ, η) fermion number, the c ghost number, the φ charge and the H charge. Thus, we see that (6.15) vanishes, and that the usual result of picture changing is obtained.
A similar argument for the amplitude in section 4.2 again leads to the usual result (6.17) in spite of the presence of the nonlocal operator. (6.16) and the first equality of (6.17) can be regarded as evidence that the correspondence (3.46) and (3.47) holds independently of the choice of the picture in the IIA theory. We should notice that the usual result of picture changing does not hold for every amplitude in the presence of the nonlocal operators. Appendix E presents an amplitude where the picture changing operation induces a nonvanishing term containing the BRST transformation of nonlocal operators. Since we have not identified the matrix-model counterpart to the positive-winding "tachyons" (V 1 ,V 2 with k 1 , k 2 = 1 2 , 3 2 , · · · in (E.1)), it is not clear which amplitude in the matrix model corresponds to the amplitude discussed there. In general, the BRST charge acting on b ghosts in higher-genus amplitudes amounts to picking up contribution from boundaries of the moduli space [26]. It could give a hint to understand the nonvanishing effect from the nonlocal operators in a geometrical manner.
As far as IIA string amplitudes whose correspondence to matrix-model amplitudes is given in this paper, we have seen in the above that the picture changing manipulation works as usual around the natural pictures.

Discussions
In this paper, we computed various amplitudes in two-dimensional type IIA superstring theory on a nontrivial (R−, R+) background, where the background is expressed by vertex operators as a small perturbation. By comparing the results with correlators in the matrix model calculated in [4], we checked the correspondence of the type IIA theory to the supersymmetric double-well matrix model, which was previously discussed from the viewpoint of symmetries and spectrum in [4].
To enable the comparison at the quantitative level, we explicitly constructed cocycle factors to vertex operators in such a way that the target-space statistics is respected. The evaluation of IIA string amplitudes at the on-shell momenta often needs to be regularized. We found a certain regularization scheme which seems reasonable from the viewpoint of resonance structure in the amplitudes. As a result of the comparison, there arise various relations among coefficients that connect quantities of the matrix model to those of the type IIA theory. Remarkably, all of such relations obtained so far are consistent with each other, which convinces us of the validity of the correspondence.
We mainly investigated two-point amplitudes of the IIA theory on the nontrivial (R−, R+) background. By taking into account the background in the perturbation theory, however they amount to the computation of three-, four-and five-point functions on the trivial background. Since the analysis of spectrum in the previous paper [4] is based on the vertex operators (3.12) on the trivial background, the computation here is important to see the effect of the RR background. Some amplitudes have the factor of the square of the Liouville volume. Its physical interpretation is as follows. One of the volume factors is from the integral over the Liouville constant mode as usual, while the other factor is due to the resonance of external particles and the background. Although it increases technical complexity, it will be meaningful to examine the correspondence for higher-point or higher-genus amplitudes. Also, we considered leading nontrivial contributions to the amplitudes in the perturbation of the background W RR . It is interesting to analyze subleading contributions. Then, we would have to take into account deformation of the BRST charge as backreaction from the background, and the relation (5.1) would be renormalized as with u n coefficients.
So far we have not yet clarified the matrix-model counterpart to the positive winding "tachyons" g 2 (z) (k = 0, 1, 2 · · · ), while the negative winding ones (k = −1, −2, · · · ) would correspond to { 1 N tr(−iB) k+1 } up to some mixing terms. We expect to make it clear by introducing source terms of an external matrix to the matrix model as discussed in the case of the Penner model in [22,27]. If it succeeds, it will be interesting to investigate the correspondence for amplitudes concerning the "tachyons" (for example, what we presented in section 4.7 and appendix E.1). Then it would become clear how the picture changing issue of the type IIA theory on the (R−, R+) background in appendix E is understood in the matrix model.
The investigation here and in the previous paper [4] focuses on massless degrees of freedom in the type IIA string theory. As discussed in the two-dimensional NSR string [28,29], the type IIA theory also has massive states at fixed momenta called "discrete states". In the matrix model, it seems natural that such massive states correspond to single-trace operators involving several kinds of matrices like 1 N tr (φ k ψ 2ℓ+1 ), 1 N tr (φ k ψ ℓ B mψn ) and so on. It is intriguing to extend the correspondence to include the massive excitations.
The correspondence we have discussed concerns fundamental string degrees of freedom in the type IIA theory. If we push forward this interpretation, each element of the matrix variables in the matrix model could be regarded as a sort of short string or string bit carrying a unit of winding or momentum along the S 1 direction, which is somewhat similar to the matrix string theory [3]. We could also consider the correspondence from another direction based on a relation of solitonic objects (D-branes) in the IIA theory with the matrix model as in [21,30,31,32,33], where matrix elements are interpreted as open string degrees of freedom on a bunch of D-branes, and the matrix models describe closed string dynamics via open-closed string duality. The single-trace operators could be regarded as sources of closed strings rather than the strings themselves. It would be worth considering both of the interpretations in a complementary manner to obtain deeper correspondence between the matrix model and the type IIA superstring theory. and for (A.2) SinceS andĨ satisfy the same initial conditions: for k = 0, 1, · · · , we can conclude thatS(k, m) =Ĩ(k, m).
(NS (0) , R−) sector Similarly, we see that the ǫ = +1 part of the (NS (0) , R−) field has the correct target-space statistics, but the ǫ = −1 part does not. The ǫ = −1 part does not contribute to the amplitudes in this paper, and the targetspace statistics is correct in the amplitudes.
(R+, NS (0) ) sector In the (R+, NS (0) ) field V k, +1 (z)T We see that theǭ = +1 part gives no contribution to the computed amplitudes. The correct statistics is realized in the amplitudes.

D Integral formulas
In this appendix, we present formulas for the two integrals where z = x + iy,z = x − iy, w = u + iv,w = u − iv, d 2 z = dx dy and d 2 w = du dv. The powers appearing in the integrands α,ᾱ, · · · , β ′ ,β ′ , σ are independent. More general integrals including (D.1), (D.2) are computed in [34]. However, in order to make this paper reasonably self-contained and to remark on the conventions for complex phases in integrands that is not mentioned in [34], we give computational details.

D.1 I (1,0)
As discussed in [34,35], we rotate the integration contour of y by almost −90 degrees as and the real axis by −ǫ, so x). Then, in terms of η ≡ x + y, χ ≡ x − y, (D.1) becomes We assumed no contribution from the infinity in the contour deformation (D.3), which is justified when Re (α +ᾱ + β +β) < −1. (D.5) The complex phase in the integrand should be carefully treated. We consider the following two phase conventions. The one (I) is and the other (II) is By noting the phase in the case (II), we see that (D.4) can be divided into the three parts according to integration regions of χ: Since the integrand of the first line (the second line) in (D.9) is regular with respect to the upper (lower) half plane of η, the η-integral vanishes by closing the contour with a large semi-circle there. Contribution from the large semi-circle can be neglected when Re (α + β) < −1. (D.10) The χ-integral in the last line gives B(ᾱ + 1,β + 1), and the η-integral can be computed as where the first equality holds in the region (D.10). Thus, we end up with for the case (II). The derivation of (D.12) is valid for We define I (1,0) for generic α,ᾱ, β,β by analytic continuation from (D.13).
On the other hand, the result for the case (I) is obtained from (D.12) by the replacement χ ↔ η, α ↔ᾱ and β ↔β: which is obtained in ref. [34]. For amplitudes among mutually local vertex operators as we discuss in the text, the parameters satisfy α −ᾱ, β −β ∈ Z, and thus (D.15).

E 2(NS, NS)-2(R−, R+) amplitude and its picture changing
In this appendix, we compute the four-point genus-zero amplitude of two (NS, NS) and two (R−, R+) fields: Although the matrix-model counterpart of the (NS, NS) fieldsV 1 andV 2 has not been found, we consider this amplitude because it exhibits nontrivial behavior in the picture changing operation whenV 3 orV 4 is nonlocal.

E.1 The amplitude
From the conservation of H andH charges, s = 0, 2 cases can give a nontrivial result. We here consider the s = 0 case. Following the same procedure as in section 4 yields The corresponding string amplitude reads  Here, cT (0) k 2 (z 2 ) can be replaced with (cT k 2 ) (0) (z 2 ) defined in (6.9), because the difference of them (the second term in the r.h.s. of (6.9) with changing (k 1 , z 1 ) to (k 2 , z 2 )) does not contribute to the amplitude from the conservation of various charges. After the same procedure as in section 6.2, we have where B remains, becauseV This satisfies charge conservations, and the Wick contraction leads to a nonvanishing result × dz 4 (z 1 − z 4 ) 1/2 (z 2 − z 4 ) 1/2 i<j (z i − z j ) k i k j −p ℓ i p ℓ j 0| : This should be nonzero because (E.5) is not symmetric underV 1 ↔V 2 .