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arxiv: 2512.02107 · v3 · submitted 2025-12-01 · ✦ hep-th · math-ph· math.MP

Generalised 4d Partition Functions and Modular Differential Equations

A. Ramesh Chandra , Sunil Mukhi , Palash Singh This is my paper

Pith reviewed 2026-05-17 02:39 UTC · model grok-4.3

classification ✦ hep-th math-phmath.MP
keywords generalised Schur partition functionsmodular linear differential equationsUSp(2N) gauge theoriesvector-valued modular forms4d N=2 superconformal theoriesrational conformal field theoryquantum monodromy traces
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The pith

Generalised Schur partition functions of 4d N=2 theories equal contour integrals of vector-valued modular forms from 2d RCFTs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that generalised Schur partition functions arising in four-dimensional N=2 superconformal gauge theories admit contour integral representations that match vector-valued modular forms of the kind appearing in two-dimensional rational conformal field theories. For the concrete case of the USp(2N) theory with 2N+2 fundamental hypermultiplets, the function Z_USp(2N)(q;α) is shown to satisfy a modular linear differential equation of exact order N+1 whose Wronskian index vanishes. The single parameter α fixes the coefficients of this differential equation. A reader would care because the result supplies an explicit bridge that lets modular properties of four-dimensional observables be read off from two-dimensional conformal field theory structures.

Core claim

We prove the equivalence of a class of generalised Schur partition functions Z_G(q;α) of 4d 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 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 Z_USp(2

What carries the argument

The contour integral representation of the generalised Schur partition function Z_USp(2N)(q;α), whose analytic continuation yields an order-(N+1) MLDE with vanishing Wronskian index whose coefficients are controlled by α.

If this is right

  • The parameter α in the 4d partition function fixes the concrete coefficients of the associated order-(N+1) MLDE.
  • The same functions are connected to characters of both unitary and non-unitary rational conformal field theories.
  • A two-parameter extension Z_USp(2N)(q;α,β) of the generalised Schur partition function is defined.
  • The specialisation α=-k corresponds to quantum monodromy traces Tr M^k, with a conjecture that their k-dependence is governed by the same family of MLDEs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same contour-integral technique may produce MLDEs for generalised Schur functions of other classical gauge groups.
  • Once the MLDE is known, its solutions can be used to generate the q-expansion of the 4d partition function without summing over instanton configurations.
  • Numerical checks of the monodromy-trace conjecture for small k and small N would test whether the k-dependence is indeed captured by the differential equation.

Load-bearing premise

The generalised Schur partition function admits a contour integral representation whose analytic continuation directly produces an MLDE of exact order N+1 with vanishing Wronskian index.

What would settle it

Expand Z_USp(2N)(q;α) as a q-series for N=1 (where the predicted order is 2) and check whether the coefficients satisfy the explicit second-order MLDE whose parameters are fixed by the chosen value of α.

read the original abstract

We prove the equivalence of a class of generalised Schur partition functions $\mathcal Z_G(q;\alpha)$ 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 $\mathcal Z_{USp(2N)}(q;\alpha)$ satisfies an order-$(N+1)$ modular linear differential equation (MLDE) with vanishing Wronskian index, explaining how the parameter $\alpha$ 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 $\mathcal Z_{USp(2N)}(q;\alpha,\beta)$ of the generalised Schur partition function. Finally, we relate the $\alpha=-k$ specialisation to quantum monodromy traces ${\rm Tr}\,M^k$ and formulate a conjecture linking their $k$-dependence to MLDEs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript claims to analytically prove that the generalised Schur partition function Z_USp(2N)(q; α) of the USp(2N) N=2 superconformal gauge theory with 2N+2 fundamental hypermultiplets admits a contour integral representation equivalent to vector-valued modular forms arising in 2d RCFTs. Specifically, it shows that this partition function satisfies an order-(N+1) modular linear differential equation (MLDE) with vanishing Wronskian index, maps the parameter α to the MLDE parameters, makes connections to RCFT characters, proposes a two-parameter extension Z_USp(2N)(q; α, β), and relates the α = -k case to quantum monodromy traces with a conjecture on their k-dependence to MLDEs.

Significance. If the central result holds, this provides a rigorous analytic bridge between 4d N=2 SCFT partition functions and the modular properties of 2d RCFTs via contour integrals and MLDEs. The explicit construction for general N, the vanishing Wronskian index, and the parameter mapping are strengths that could facilitate further studies of modular invariance in higher-dimensional theories. The proposal of the two-parameter extension and the conjecture linking to monodromy traces add potential for new research directions in connecting 4d and 2d structures.

major comments (1)
  1. [Proof of the MLDE (contour integral section)] The central claim that the contour integral representation of Z_USp(2N)(q;α) yields an MLDE of exact order N+1 with vanishing Wronskian index after analytic continuation is load-bearing. The manuscript does not provide explicit verification that contour deformation for |q|<1 versus the fundamental domain introduces no extra poles or requires no additional regularization that would change the differential operator order or Wronskian index (see the discussion of the contour integral and modular continuation in the proof of the main theorem).
minor comments (2)
  1. The abstract would benefit from a one-sentence outline of the key analytic steps used to establish the MLDE order and Wronskian vanishing.
  2. [Introduction] Notation for the generalised partition function Z_G(q;α) should be introduced with a brief comparison to the ordinary Schur index to avoid potential confusion for readers.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We appreciate the recognition of the potential significance of connecting 4d N=2 SCFT partition functions to 2d RCFT modular forms via MLDEs. We address the major comment below, providing a point-by-point response while maintaining the integrity of our analytic proofs.

read point-by-point responses

  1. Referee: [Proof of the MLDE (contour integral section)] The central claim that the contour integral representation of Z_USp(2N)(q;α) yields an MLDE of exact order N+1 with vanishing Wronskian index after analytic continuation is load-bearing. The manuscript does not provide explicit verification that contour deformation for |q|<1 versus the fundamental domain introduces no extra poles or requires no additional regularization that would change the differential operator order or Wronskian index (see the discussion of the contour integral and modular continuation in the proof of the main theorem).

    Authors: We acknowledge the referee's point that the contour deformation step in the proof of the main theorem (Section 3) would benefit from more explicit verification to confirm that no additional poles are introduced and that the MLDE order and vanishing Wronskian index are preserved. In the manuscript, the argument proceeds by analyzing the analytic properties of the integrand, which is a ratio of theta functions and q-Pochhammer symbols whose poles are located at specific positions determined by the parameter α; for the chosen contour and |q|<1 region, deformation to the fundamental domain avoids these poles without residue contributions or regularization needs that would alter the order. However, to strengthen the exposition as suggested, we will add a dedicated paragraph in the revised proof detailing the pole locations, confirming the absence of crossings during deformation, and explicitly stating that the resulting operator remains of order N+1 with Wronskian index zero. This clarification does not change the results but addresses the concern directly. revision: yes

Circularity Check

0 steps flagged

No significant circularity; analytic derivation from contour integrals is self-contained

full rationale

The paper analytically derives the equivalence of the generalised Schur partition function Z_USp(2N)(q;α) to a contour integral representation of a vector-valued modular form, then shows it satisfies an order-(N+1) MLDE with vanishing Wronskian index by direct computation of the modular properties and analytic continuation. This chain begins from the explicit definition of Z_G(q;α) and proceeds via standard properties of contour integrals and modular forms without reducing any prediction or central claim to a fitted input, self-definition, or load-bearing self-citation. Connections to RCFT characters are presented as consequences rather than foundational assumptions, and the two-parameter extension and monodromy conjecture are proposed separately. The derivation remains independent of external benchmarks and does not invoke uniqueness theorems or ansatze from prior self-work in a circular manner.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the definition of the generalized Schur partition function and standard properties of contour integrals and modular forms; no new free parameters beyond the theory parameters N and alpha, no invented entities, and only standard mathematical axioms are used.

axioms (1)
  • standard math Contour integrals of the given form admit analytic continuation to vector-valued modular forms satisfying MLDEs
    Invoked to equate the partition function representation to the stated differential equation properties.

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Cited by 1 Pith paper

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    hep-th 2026-04 unverdicted novelty 6.0

    Generalized Schur indices of N=2 class S theories are expressed using eigenfunctions of non-relativistic elliptic Calogero-Moser models, with extensions claimed for N=1 SCFTs via limits of models like Inozemtsev.

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