Super-spectral curve of irregular conformal blocks

We use super-spectral curve to investigate irregular conformal states of integer and half-odd integer rank. The spectral curve is the loop equation of supersymmetrized irregular matrix model. The case of integer rank corresponds to the colliding limit of supersymmetric vertex operators of NS sector and half-odd integer to the Ramond sectors. The spectral curve is simply integrable at Nekrasov-Shatashvili limit and the partition function (inner product of irregular conformal state) is obtained from the superconformal structure manifest in the spectral curve. We present some explicit forms of the partition function of integer (NS sector) and of half-odd ranks (Ramond sector).


Introduction
Irregular conformal state is a conformal state, but is not a primary or descendent state. Rather it is similar to a coherent state since it is a simultaneous eigenstate of some of positive mode of conformal generator. The simplest irregular state is the eigenstate of Virasoro L +1 mode which is called Gaiotto state [1] or Whittaker state [2]. More irregular states have been systematically investigated for Virasoro and W-irregular state [3,4,5,6,7].
The irregular state is termed as rank n if it is the simultaneous eigenstate of Virasoro generators L k with n ≤ k ≤ 2n or of W (q) generators W (q) k with (q − 1)n ≤ k ≤ qn with spin q. However, the construction of the irregular state is not easy to find because the eigenvalues are not enough to define the state of rank n ≥ 2. One needs additional information how the irregular state behaves when all other positive generators applied on the irregular state such as W (q) k with 0 ≤ k < (q − 1)n. The progress is achieved according to AGT [8] and the idea of colliding limit [9,10]. AGT connects Nekrasov partition function of N=2 super Yang-Mills theory in 4 dimension with the Liouville conformal block in 2 dimension. Colliding limit of Liouville conformal block describes the irregular state and in turn closely related with Argyres-Douglas theory of N=2 super Yang-Mills theory. Colliding limit of the Liouville conformal block is easily investigated in terms of irregular matrix model. Originally, Penner-type matrix model is suggested from the Liouville conformal block [11,9,12] and colliding limit of the Penner-type matrix model results in the irregular matrix model. The irregular matrix model is successful to describe irregular states and their inner product. The partition function is related with the inner product between primary state and irregular state or between two irregular states depending on the potential of the irregular matrix model. However, the success is limited to the case of irregular states of integer rank. Virasoro irregular state of integer rank n has eigenvalue of the highest Virasoro generator L 2n . Question arises. Can we find irregular state with half-odd rank, that is, irregular state of highest Virasoro generator L 2n−1 ? The state of rank 1/2 is easily found from the rank 1 if one limits the eigenvalue of L 2 vanish. However, this trick does not work for rank greater than 1 since this limit does not exist since other eigenvalues diverges unless special limit is achieved so that the state is a simultaneous eigenstate of L 1 and L 2n−1 only [4].
In this paper, we will present the irregular matrix model of half-odd integer rank using supersymmetrizing the theory. The irregular vertex operator is constructed similar to the regular vertex operator [13,14,15] and is supersymmetrized in [16]. It is noted that the irregular vertex operator with half-odd rank appears naturally with Ramond sector in the super-symmetrized version. This operator is useful for the free field formalism. If one includes the screening operators, then one can investigate the interacting system of irregular states. This paper is organized as following. In section 2, we present irregular super-matrix model and its loop equation. The matrix model is related with the N=1 super Liouville conformal block and its colliding limit. The loop equation is simply integrable at Nekrasov-Shatashvili limit (NS limit), which is called super-spectral curve. In section 3, we consider irregular states with integer rank. This state is obtained from the NS sector. Using the super-spectral curve we obtain partition function and present the explicit form of rank 1. In section 4, irregular states with half-odd rank are considered. Partition functions of rank 1/2 and 3/2 are presented. In section 5, we present an idea on RG flow equation corresponding to the operator algebra of the irregular vertices from the string field theory. Section 6 is the conclusion and discussion. Super-spectral curve of the irregular matrix model is presented in the appendix.
2 Irregular super-matrix model and its spectral curve Super-vertex operator V α (ζ) in the NS sector is considered in the super-field formalism V α (ζ) = e αΦ (ζ) (2.1) where ζ = (z, θ) is the holographic super-coordinate, Φ is the super-field and α is the Liouville momentum. Two point correlation of the vertex operator is normalized as in [17] To find the multi-point correlation in the super Liouville formalism one may use screening operator V b (ζ) in the presence of background charge Q = b + 1/b. Primary operator has the conformal dimension ∆ α = α(Q − α)/2 and the superconformal system has central charge c = 3/2(1 + 2Q 2 ). Explicitly, (n + 2)-point holomorphic correlation can be calculated in the presence of Nscreening operators V b (ζ) and be put into Selberg integrals where neutrality condition I α I + Nb = Q is assumed.
To formulate this integral in terms matrix model, we put (n + 2) external operator contribution (z IA − θ I θ A ) −bα I into an exponential of a super-potential V (ζ I ) = Aα I ln(z IA − θ I θ A ) withα = α; This will be called deformed super Penner-type matrix model. Here β = −b 2 is used instead of b. In addition, g = i is introduced for later convenience. In terms of the new notations, . If one applies the colliding limit by fusing n operators to the one at origin and let the rest to the infinity after accordingly normalizing the partition function at infinity, one obtains a new super potential of the form V (ζ I ) = V B (z I ) + θ I V F (z I ): V B (z) and V F (z) are bosonic and fermionic part of super-potential.
The loop equation provides the super-spectral curve with the deformed parameter ǫ. (Its derivation is found in Appendix A).
Explicitly, the potential obtained from the colliding limit of (n + 2) number of NS sector of N=1 super Liouville vertex operators is of the form (2.8) where V B (z I ) is the bosonic part and V F (z I ) the fermionic part. c k is a commuting variable defined as c k = Aα A z k A and ξ k is an anti-commuting variable defined as ξ k = Aα A z k A θ A . The partition function with the new super potential will be called irregular super-matrix model of integer rank n.
It is noted that the matrix model is closely related with irregular vertex operator was investigated in [16] W where D θ = θ∂ z + ∂ θ . γ k is commuting (anti-commuting) when k is even (odd). The same potentials V B (z I ) and V F (z I ) in (2.7) and (2.8) are obtained if one contracts W n with N screening operators V b (ζ). There is a slightly different form of the super-matrix model due to Ramond sector. If one uses the vertex operator of Ramond sector and contracts W n− 1 2 with N screening operators V b (ζ), one obtains the irregular matrix model of Ramond sector. The resulting irregular potential is the one similar to (2.7) and (2.8): (2.12) The difference from the model of rank n (NS sector) is that the commuting variable c k has unusual constraints. c k contains the product of two anti-commuting variables so that c 2 n = 0 = c n ξ n−1 . This model is called irregular super-matrix model of half-odd rank (n−1/2).

Partition function of integer rank
For the integer rank n, the potential is given in (2.7) and (2.8): Here c k (ξ k ) is a commuting (anti-commuting ) variable. We are using the super-spectral curve (2.5) and (2.6) using the explicit form of F F (F B ) using the potential (3.1).
Ω r is anti-commuting number and is defined as This expectation value is the basic tool to find the partition function from the superspectral curves (2.5) and (2.6) as noted in bosonic cases [18,19,20]. The super-flow equation (A.18) and (A.22) is essential to find the moments d m and η r . In the following we provide an explicit calculation for the simplest case (rank 1).
For the rank 1, there are 3 flow equations: 2 bosonic and one fermionic: To solve the flow equations, we need to find the moments d 0 , η 1/2 and d 1 in the left hand side of the flow equations as the functional dependence of variables c 1 , ξ 1 and ξ 2 from the super-spectral curve. The moment d 0 is easily identified if one considers the dominant contribution of the bosonic spectral curve (2.6) at large z limit.
Therefore, the bosonic flow equation (3.4) requires the partition function to be of the form Here, ξ 0 ξ 1 /c 2 1 is the homogeneous solution and C is a constant independent of c 1 , ξ 1 and ξ 2 , which can be normalized to be 0.
The fermionic moment η 1/2 is obtained if we use the large z expansion of (2.5): where N F = I θ I . To get the information on N F , we use the fact that d m and η r should obey the consistency condition due to commutation relations (A.24) between generators. Note that [l m , g r ] = (r − m/2)g r+m . This requires Therefore, d 0 in (3.7) and η 1/2 in (3.9) has the relation: This shows that η 1/2 behaves as the primary of conformal dimension 1/2. There are two anti-commuting variables ξ 0 and ξ 1 /c 1 of dimension 1/2. This shows that N F should be proportional to either ξ 0 or ξ 1 /c 1 . Fermionic filling fraction is anti-commuting and is concentrated at ξ 0 or ξ 1 . Putting N F = N 1 ξ 0 + N 2 ξ 1 /c 1 with commuting numbers N 1 and N 2 , one has The flow equation (3.5) together with (3.8) and (3.11) is rewritten as Therefore, the flow equation reduces to algebraic identities: which fixes N 1 and A as a function of c 0 , N and N 2 : As a result, the partition function is given as Finally, the bosonic flow equation (3.6) provides additional information on the system. The right hand side of the flow equation vanishes if one uses the partition function of the form (3.15). Therefore, d 1 should vanish. On the other hand one can obtain d 1 using the spectral curve (2.6). It is noted in [21,22] that the resolvent at NS limit is of the form with a monic polynomial of degree N with p 0 = 1. Then, (2.6) results in Note that p 1 is the sum of all the poles p 1 = α z α and is same as the expectation value I z I of the matrix model. Since d 1 vanishes, one concludes that In general, there are two variables c 1 and ξ 0 ξ 1 /c 1 of conformal dimension 1 for the rank 1 case. However, the term proportional to ξ 0 ξ 1 /c 1 turns out to vanish and the only term proportional to c 1 survives. The pole structure of the bosonic resolvent also shares with that of the fermionic one. This can be seen from (2.5). First, note that if one uses (3.16), one may put where τ F (z) = e −V B (z)/ǫ z dyF F (y) P N (y)/ǫ. Since τ F (z α ) is not zero in general (except 0 accidentally), the obvious conclusion is that the pole position z α is also the pole position of x F (z).

Partition function of half-odd rank
The partition function of integer rank n is interpreted as the inner-product between a primary state and an irregular state of rank n. The spectral curve shows that the irregular state is the simultaneous eigenstate of super-current G r with r = n + 1/2, · · · , 2n − 1/2 and Virasoro current L m with m = n, · · · , 2n if d n = 0 (otherwise, m = n + 1, · · · , 2n). One may wonder if the eigenvalue of the highest Virasoro mode L 2n vanishes. Note that the eigenvalue of the highest Virasoro mode is given as Λ 2n = c 2 n . Therefore, unless c n = 0 the case is not achieved in the NS sector. Instead, if one includes the Ramond sector also, one may have the potential of half-odd rank as in (2.11) and (2.12). (4.1) The difference from the NS sector is that the commuting variable vanishes when squared c 2 n = 0. This is because c n is commuting but is the product of two anti-commuting variables. Therefore, the eigenvalue Λ 2n vanishes so that the non-vanishing highest mode becomes L 2n−1 .
The super-flow equations are simply given as The partition function is formally given as as given in (3.7). Non-trivial case starts with rank 3/2. In this case there are three commuting parameters c 0 , c 1 , c 2 and two anti-commuting parameters ξ 0 , ξ 1 . The parameters have the relation with the original γ k in (2.10) as follows: c 0 = γ 0 , c 1 = (γ 1 θ + γ 2 ), c 2 = γ 3 θ, ξ 0 = γ 1 , and ξ 1 = (γ 2 θ + γ 3 ). This shows that c 2 2 = c 2 ξ 1 = 0. Then we have 4 flow equations The bosonic spectral curve (2.6) shows that d 0 is the same as in (3.7) and the solution of (4.5) is given as where we use the fact t = c 2 /c 2 1 and ξ 0 ξ 1 /c 2 1 are homogeneous solutions. The fermionic spectral curve (2.5) shows that η 1/2 has the same form (3.11). However, the right hand side of the fermionic flow equation (4.6) has a different result.
This fermionic flow equation reduces to another algebraic identity whose solves A(t) and B(t): so that the partition function is given as N − 1)))/2 and B 1 = ǫ(N − 1 + N 1 (c 0 + ǫ(N − 1))/2. Therefore, the partition function (4.9) is given in terms of potential variables together with N, N 1 and N 2 . Two more flow equations provide additional information on the system. d 1 is given as (3.18) and corresponding flow equation (4.7) shows that This gives the information on p 1 = I z I ; (4.14) Finally, the fermionic flow equation (4.8) shows that the right side is given as On the other hand, fermionic spectral curve (2.5) shows that the left hand side is LHS = ǫ(c 0 + ǫ(N − 2))q 1 + ǫ(ξ 0 p 1 + ξ 1 N + (c 1 + ǫp 1 )N F ) (4. 16) where q 1 = I z I θ I , fermionic partner of p 1 . Therefore, the flow equation determines q 1 .

Irregular Vertex Operators and RG Flow Equations
In this section, we provide RG flow equations to the operator algebra of the irregular vertices from the string field theory. The main idea is that, in the formalism of irregular vertex operators, we may have conformal β-function equations on the wavefunctions of these operators, generalized to the off-shell case. For simplicity, we shall limit ourselves to the non-supersymmetric case and to the rank one, however, the discussion is straightforward to generalize to higher ranks and the supersymmetry. The most general form of the rank 1 vertex operator is given by where ξ(α, β) is the wavefunction for the irregular state. In case if U(α, β) were a regular vertex operator, its leading order contribution to the string sigma-model partition function would be given by Z σ ∼ e S(ξ) where S(ξ) is the low-energy effective action, defined by the vanishing β-function condition where Λ is the worldsheet cutoff and C are the structure constants defined by 3-point worldsheet correlators. The above condition ensures that the conformal invariance is preserved by inserting the on-shell operators on the worldsheet. The irregular vertex operators are, however, the off-shell objects, therefore they do not have any associate β-function in a naive literal sense. Nevertheless, the relation of the type (5.2) still retains some important meaning off-shell, in particular, in the context of background-independent string field theory -and can be related to the flow equations derived above. That is, in the on-shell case, the equations of motion (5.2) define the perturbative background deformations preserving the worldsheet conformal symmetry, ensured by the Weyl invariance combined along with the condition of absence of logarithmic singularities in the partition function due to collisions between vertex operators. It is furthermore important that, in the on-shell case, all the vertex operators have conformal dimension 1, and the only OPE terms contributing to the β-functions as a result of collision of two such vertices, are those involving operators of dimension one. In the off-shell case, such as ours, all these conditions have to be modified. First of all, U(α, β) becomes a string field which wavefunction, ξ(α, β) now describes a nonperturbative background deformation from the original to the one defined by the appropriate analytic solution in string field theory. The "β-function"-like constraint of the type (5.2) is now precisely the condition that the string field U is that analytic solution, producing the nonperturbative background change. Moreover, contrary to the perturbative on-shell-case, the "effective action" S(ξ) is typically nonlocal. For the irregular vertices, we can no longer require the absence of the OPE singularities for the colliding operators, as this constraint has an essentially on-shell origin in string perturbation theory. However, we still have to retain the Weyl invariance constraints on the operators, since a) these constraints are imposed off-shell even in standard string perturbation theory b) Weyl invariance is essential to fix the (super)conformal gauge which we are using here. In order to elucidate the constraints due to the scale invariance, one has to calculate the OPE of the irregular vertex sitting on the disc boundary, with the trace of the stress-energy tensor, integrated over the bulk of the disc, and to extract the logarithmic divergence stemming from the OPE integration. This shall lead to the first set of the constraints, analogous to the flow equations. Straightforward calculation gives: lim z,z→τ : T zz : (z,z) : e αφ+β∂φ : (τ ) = lim Integrating over z the contributions proportional to ∼ d 2 z 1 |z−τ | 2 ∼ ln Λ leads to logarithmic singularities defining the variations of the operators under Weyl transformations. Cancellation condition for these variations defines the flow equations we are looking for. In what follows we shall ignore the OPE terms with higher derivatives of φ. That is, the terms proportional to ∂ 2 φ and higher derivatives, are only relevant for the RG flows for the higher rank operators, related to variational derivatives with respect momenta, conjugate to higher derivatives in the irregular vertices (e.g. : ∂ 2 φe αφ+β∂φ+γ∂ 2 φ :∼ ∂ ∂γ e αφ+β∂φ+γ∂ 2 φ ) Then the flow equation describing the Weyl deformations of the irregular operators is This extended β-function relation is related to the Legendre transformed bosonic part of the flow equation (3.4) for the free energy ln Z, expressed in terms of the wavefunction ξ(α, β), related to the partition function according to where V (α, β, τ ) =: e αφ+β∂φ : (τ ). The relation to the bosonic part of the flow equation (3.4) is not straightforward because the generalized RG flow (5.4) is expressed in terms of very different variables. To obtain this relation, one has to insert the differential operator on the right hand side of (5.4) inside the generating functional < e dτ dαdβξ(τ,α)V (α,β,β) >. The relation will then follow as the 2d Ward identity inside the worldsheet correlators.
It is straightforward to generalize this calculation to the supersymmetric case. In this case, the irregular vertices are not eigenvalues of positive Virasoro generators, but the Jordan blocks. In the simplest rank 1 2 case such a block has a multiplicity 2 with components: Applying the Weyl transformation now leads to separate equations on the wavefunctions η 1 and η 2 : Note that, unlike (4.3) in the pair of the flow equations, one of the equations for the rank 1 2 is algebraic.

Conclusion and discussion
In this work, we analyzed the loop equation in supersymmetric matrix model in the superspace formalism, in order to derive the spectral curve for the Argyres-Douglas limit of N = 2 super Yang-Mills theory, related to N=1 super Liouville conformal field theory through generalized AGT conjecture. We have been able to derive and to integrate the loop equation in the supersymmetric case and to obtain partition functions associated with irregular blocks of ranks 1 2 , 1 and 3 2 . The loop equations, as well as the associate flow equations on the free energy, can be reproduced in the irregular vertex operator approach, in terms of the scale invariance constrants for the vertex operators. One particularly promising thing about the vertex operator approach is that it is relatively straightforward to extend to higher ranks, as well as to observe the Jordan cell structure of the flow equations in the supersymmetric case. We hope to be able to extend these results to higher/arbitrary ranks in the future works. It will be also interesting to investigate (super)-spectral curve for the special value of Liouville parameter space as observed in [23].
In general, it is natural to understand the AGT conjecture as an isomorphism between the partition functions of the sigma-models with irregular vertex operators in Toda/superstring theories and those of super Yang-Mills theories. The relation between these theories can be thought of as a generalization of the one between standard string-theoretic sigma-models and their low-energy limit, through the off-shell generalization of the conformal β-functions. The background-independent second-quantized string field theory approach appears to be a promising framework for that. The work in this direction is currently in progress and we hope to be able to elaborate on these issues soon.

A Super-spectral curve
One may derive the loop equation of the irregular super-matrix model corresponding to the super-conformal symmetry. Spin 3/2 current contribution is obtained if one use the supercoordinate transform [23] z I → z I + θ I ǫ F /(z − z I ) and θ I → θ I + ǫ F /(z − z I ) where ǫ F is the small anti-commuting number. The metric contribution Super-Vandermonde determinant has the contribution Finally, the potential has the contribution e √ β g Collecting all terms one has where prime denotes the derivative with respect to z. ω B (z) (ω F (z)) is one-point commuting (anti-commuting) resolvent f F is related with the super-potential Virasoro contribution is obtained if one uses the super-coordinate transform z I → z I + ǫ/(z − z I ) and θ I → θ I (1 + ǫ/(2(z − z I ) 2 ) where ǫ is an infinitesimal commuting number. The metric contribution is (A.8) (Here the anti-commuting measure [dθ I ] is required to maintain the integral property dθ I θ I = 1). Super-Vandermonde determinant has the contribution Finally, the potential has the contribution e √ β g Collecting all terms one has where ω BB (z, z) and ω (1,2) F F (z, z) are the connected two-point resolvents (A.14) and ω (1,2) (A. 15) f B is related with the super-potential (A.16) It is useful to find the explicit holomorphic structure of f F (z) and f B (z) for the given potential (2.7) and (2.8). They are given in terms of the inverse powers of z The moment η r is given as an expectation value and η −1/2 vanishes which is evident from 1/z expansion of (A.4). If one uses the explicit form of the potential, one may put the non-vanishing moment into an interesting form as in non-supersymmetric case [18,20] η r = g r (− 2 log Z) + δ r,1/2 g βξ 0 (A. 18) where g r is the differential representation of the super current (corresponding to right action) This is obtained if one notices that √ β g At the NS limit ( → 0 and b → ∞ so that b = ǫ), the loop equations (A.4) and (A.13) can be put in terms of one-point resolvent only, which is called the deformed spectral curve where we use compact notations: . It is interesting to look into the explicit form of F F (z) and F B (z). where Ω r is an anti-commuting number Ω r = k+ℓ=r−1/2 c k ξ ℓ − ǫ(δ r,1/2 − (r + 1/2))ξ r−1/2 and Λ m is a commuting number Λ m = k+l=m c k c l /2 − ǫ(m + 1)c m /2 Non-vanishing η r (r = 1/2, · · · , n−1/2) and d m (m = 0, · · · , n) are given in (A.17) and (A.21). The anti-commuting number Ω r with r = (n + 1/2, n + 3/2, · · · , 2n + 1/2) corresponds to the eigenvalue of supercurrent positive mode G r and the commuting number Λ m with m = (n + 1, n + 2, · · · , 2n) corresponds to the eigenvalue of Virasoro positive mode L m .
The same analysis can be done for the potential (2.12) and (2.11) of the half-odd rank (n − 1/2) similarly if one considers the constraint of the variables, c 2 n = 0 = c n ξ n−1 . This shows that Λ m = 0 if m ≥ 2n and Ω r = 0 if r ≥ 2n − 1/2.