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Generalised 4d Partition Functions and Modular Differential Equations

A. Ramesh Chandra, Sunil Mukhi, Palash Singh

TL;DR

This work establishes a robust link between generalized Schur partition functions in 4d N=2 SCFTs and contour-integral representations of vector-valued modular forms that solve modular linear differential equations. Focusing on USp(2N) theories with 2N+2 fundamentals, it proves that the one-parameter generalized Schur partition function Z_USp(2N)(q; α) satisfies a fixed-order MLDE with vanishing Wronskian, with α controlling the MLDE data, and it extends to a two-parameter version. The authors connect these q-series to RCFT characters via contour integrals, provide explicit analyses for the N=1 (SU(2)) and N=2 (USp(4)) cases, and discuss interpretations in terms of unitary and non-unitary theories as well as monodromy traces Tr M^k. They also outline a natural conjecture linking higher powers of monodromy to MLDEs and discuss generalisations to other gauge groups, offering a broad framework for 4d/2d interplay through MLDEs and contour integrals.

Abstract

We prove the equivalence of a class of generalised Schur partition functions $\mathcal Z_G(q;α)$ of 4d $\mathcal N=2$ superconformal gauge theories to contour integral representations of vector-valued modular forms of the type that arise in 2d rational conformal field theories (RCFT). Concretely, we consider the $USp(2N)$ theory with $2N+2$ fundamental hypermultiplets and analytically prove that $\mathcal Z_{USp(2N)}(q;α)$ satisfies an order-$(N+1)$ modular linear differential equation (MLDE) with vanishing Wronskian index, explaining how the parameter $α$ of the former determines the parameters of the latter. Several connections are made to characters of RCFTs including unitary ones. We then propose a two-parameter extension $\mathcal Z_{USp(2N)}(q;α,β)$ of the generalised Schur partition function. Finally, we relate the $α=-k$ specialisation to quantum monodromy traces ${\rm Tr}\,M^k$ and formulate a conjecture linking their $k$-dependence to MLDEs.

Generalised 4d Partition Functions and Modular Differential Equations

TL;DR

This work establishes a robust link between generalized Schur partition functions in 4d N=2 SCFTs and contour-integral representations of vector-valued modular forms that solve modular linear differential equations. Focusing on USp(2N) theories with 2N+2 fundamentals, it proves that the one-parameter generalized Schur partition function Z_USp(2N)(q; α) satisfies a fixed-order MLDE with vanishing Wronskian, with α controlling the MLDE data, and it extends to a two-parameter version. The authors connect these q-series to RCFT characters via contour integrals, provide explicit analyses for the N=1 (SU(2)) and N=2 (USp(4)) cases, and discuss interpretations in terms of unitary and non-unitary theories as well as monodromy traces Tr M^k. They also outline a natural conjecture linking higher powers of monodromy to MLDEs and discuss generalisations to other gauge groups, offering a broad framework for 4d/2d interplay through MLDEs and contour integrals.

Abstract

We prove the equivalence of a class of generalised Schur partition functions of 4d superconformal gauge theories to contour integral representations of vector-valued modular forms of the type that arise in 2d rational conformal field theories (RCFT). Concretely, we consider the theory with fundamental hypermultiplets and analytically prove that satisfies an order- modular linear differential equation (MLDE) with vanishing Wronskian index, explaining how the parameter of the former determines the parameters of the latter. Several connections are made to characters of RCFTs including unitary ones. We then propose a two-parameter extension of the generalised Schur partition function. Finally, we relate the specialisation to quantum monodromy traces and formulate a conjecture linking their -dependence to MLDEs.

Paper Structure

This paper contains 22 sections, 3 theorems, 179 equations, 1 table.

Table of Contents

  1. Introduction and Summary
  2. Background review
  3. Superconformal Index
  4. Schur index
  5. Relation to the 4d/2d correspondence and MLDE
  6. Schur indices for gauge theories
  7. Generalised Schur partition function
  8. Modular differential equations and the indicial equation
  9. Contour integral representation for RCFT characters
  10. $SU(2)$ SCFT with $4$ fundamental hypers
  11. $USp(2N)$ SCFT with $2N+2$ fundamental hypers
  12. First non-trivial case: $N=2$
  13. General $N$
  14. Two-parameter generalised Schur partition function
  15. Discussion and open questions
  16. ...and 7 more sections

Key Result

Proposition 1

The generalised Schur partition function ${\cal Z}_{SU(2)}(q;\alpha)$, with analytically continued $\alpha\in\mathbb C$, is a solution to the following second order $\ell=0$ modular linear differential equation, the MMS equation:

Theorems & Definitions (4)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Conjecture 1