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Super Liouville conformal blocks from N=2 SU(2) quiver gauge theories

V. Belavin, B. Feigin

TL;DR

This work extends the AGT correspondence to N=1 superconformal blocks by enforcing a Z2 symmetry on the SU(2) instanton moduli space, yielding a modified partition function represented by colored Young diagrams. The authors formulate a concrete conjecture equating the Z2-symmetric instanton sums (on components labeled by cell parity) with the NSR-based N=1 super Liouville conformal blocks in the Whittaker limit, providing explicit expressions for the modified vector-field determinant det' v. They perform detailed low-instanton checks and pole analyses, finding agreement with the four-point N=1 blocks' structure and residues, and uncovering an algebraic framework on cohomologies given by widetilde{gl}_2(2)×NSR. The results generalize the AGT program to a new class of algebras and hint at extensions to higher Z_m actions and four-point blocks, with significant implications for spinor and parafermionic CFTs coupled to gauge theory.

Abstract

The conjecture about the correspondence between instanton partition functions in the N=2 SUSY Yang-Mills theory and conformal blocks of two-dimensional conformal field theories is extended to the case of the N=1 supersymmetric conformal blocks. We find that the necessary modification of the moduli space of instantons requires additional restriction of Z(2)-symmetry. This leads to an explicit form of the N=1 superconformal blocks in terms of Young diagrams with two sorts of cells.

Super Liouville conformal blocks from N=2 SU(2) quiver gauge theories

TL;DR

This work extends the AGT correspondence to N=1 superconformal blocks by enforcing a Z2 symmetry on the SU(2) instanton moduli space, yielding a modified partition function represented by colored Young diagrams. The authors formulate a concrete conjecture equating the Z2-symmetric instanton sums (on components labeled by cell parity) with the NSR-based N=1 super Liouville conformal blocks in the Whittaker limit, providing explicit expressions for the modified vector-field determinant det' v. They perform detailed low-instanton checks and pole analyses, finding agreement with the four-point N=1 blocks' structure and residues, and uncovering an algebraic framework on cohomologies given by widetilde{gl}_2(2)×NSR. The results generalize the AGT program to a new class of algebras and hint at extensions to higher Z_m actions and four-point blocks, with significant implications for spinor and parafermionic CFTs coupled to gauge theory.

Abstract

The conjecture about the correspondence between instanton partition functions in the N=2 SUSY Yang-Mills theory and conformal blocks of two-dimensional conformal field theories is extended to the case of the N=1 supersymmetric conformal blocks. We find that the necessary modification of the moduli space of instantons requires additional restriction of Z(2)-symmetry. This leads to an explicit form of the N=1 superconformal blocks in terms of Young diagrams with two sorts of cells.

Paper Structure

This paper contains 12 sections, 1 theorem, 77 equations.

Table of Contents

  1. Introduction
  2. AGT conjecture
  3. Integer partitions.
  4. Whittaker vector.
  5. Super Liouville field theory
  6. $\mathcal{N}=1$ conformal blocks.
  7. Supersymmetric Whittaker vector.
  8. ADHM construction and the determinants of the vector field
  9. Modified Moduli space and super conformal blocks
  10. Lowest levels calculations and analytic properties
  11. Analysis of the physical poles.
  12. Conclusion

Key Result

Proposition 5.1

The $2N$-instanton contribution to the moduli integral, evaluated on the $\mathbb{Z}_2$ symmetric subspace of the moduli space $\mathcal{M}_{\text{sym}}$, reproduces the $N$th-level conformal block coefficients in the Neveu--Schwarz sector of the $\mathcal{N}=1$ super Liouville theory up to some fac

Theorems & Definitions (1)

  • Proposition 5.1