Super-spectral curve of irregular conformal blocks

Dmitri Polyakov,Chaiho Rim

doi:10.1007/jhep12(2016)004

Abstract

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).

Highlights

We present some explicit forms of the partition function of integer (NS sector) and of half-odd ranks (Ramond sector)
Can we find irregular state with half-odd rank, that is, irregular state of highest Virasoro generator L2n−1? The state of rank 1/2 is found from the rank 1 if one limits the eigenvalue of L2 vanish
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

Summary

Irregular super-matrix model and its spectral curve

Super-vertex operator Vα(ζ) in the NS sector is considered in the super-field formalism. Where neutrality condition I αI + N b = Q is assumed To formulate this integral in terms matrix model, we put (n+2) external operator contribution (zIA −θI θA)−bαI into an exponential of a super-potential V (ζI ) = A αI ln(zIA − θI θA) with α = α; Zn =. 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. The partition function with the new super potential will be called irregular super-matrix model of integer rank n. Ck contains the product of two anti-commuting variables so that c2n = 0 = cnξn−1 This model is called irregular super-matrix model of half-odd rank (n − 1/2)

Partition function of integer rank

Partition function of half-odd rank

Irregular vertex operators and RG flow equations

Conclusion and discussion

A Super-spectral curve