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Instanton counting via affine Lie algebras II: from Whittaker vectors to the Seiberg-Witten prepotential

Alexander Braverman, Pavel Etingof

TL;DR

The paper develops a unified framework linking instanton partition functions to Seiberg-Witten prepotentials via Schrödinger operators with periodic potentials and affine Toda integrable systems. It first constructs equivariant partition functions ${\mathcal{Z}}$ in the 1D setting and then generalizes to higher dimensions under integrability, using period integrals on spectral curves to define the instanton prepotential ${\mathcal{F}}^{inst}$. A central achievement is proving Nekrasov's conjecture by showing that the leading small-$\varepsilon$ behavior of the affine partition function ${\mathcal{Z}}_{G}^{aff}$ coincides with the instanton prepotential of the affine Toda system, with a robust parabolic generalization. The method hinges on Whittaker vector realizations, non-stationary Toda Hamiltonians, and a period-based description of the prepotential, offering a distinct route from prior proofs for classical groups. The results illuminate deep connections between equivariant geometry, integrable systems, and gauge-theory prepotentials, extending known cases to a broad algebraic framework.

Abstract

We introduce the notion of the (instanton part of the) Seiberg-Witten prepotential for general Schrodinger operators with periodic potential. In the case when the operator in question is integrable we show how to compute the prepotential in terms of period integrals; this implies that in the integrable case our definition of the prepotential coincides with the one that has been extensively studied in both mathematical and physical literature. As an application we give a proof of Nekrasov's conjecture connecting certain "instanton counting" partition function for an arbitrary simple group G with the prepotential of the Toda integrable system associated with the affine Lie algebra whose affine Dynkin diagram is dual to that of the affinization of the Lie algebra of G (for G=SL(n) this conjecture was proved earlier in the works of Nekrasov-Okounkov and Nakajima-Yoshioka). Our proof is totally different and it is based on the results of the paper math.AG/0401409 by the first author.

Instanton counting via affine Lie algebras II: from Whittaker vectors to the Seiberg-Witten prepotential

TL;DR

The paper develops a unified framework linking instanton partition functions to Seiberg-Witten prepotentials via Schrödinger operators with periodic potentials and affine Toda integrable systems. It first constructs equivariant partition functions in the 1D setting and then generalizes to higher dimensions under integrability, using period integrals on spectral curves to define the instanton prepotential . A central achievement is proving Nekrasov's conjecture by showing that the leading small- behavior of the affine partition function coincides with the instanton prepotential of the affine Toda system, with a robust parabolic generalization. The method hinges on Whittaker vector realizations, non-stationary Toda Hamiltonians, and a period-based description of the prepotential, offering a distinct route from prior proofs for classical groups. The results illuminate deep connections between equivariant geometry, integrable systems, and gauge-theory prepotentials, extending known cases to a broad algebraic framework.

Abstract

We introduce the notion of the (instanton part of the) Seiberg-Witten prepotential for general Schrodinger operators with periodic potential. In the case when the operator in question is integrable we show how to compute the prepotential in terms of period integrals; this implies that in the integrable case our definition of the prepotential coincides with the one that has been extensively studied in both mathematical and physical literature. As an application we give a proof of Nekrasov's conjecture connecting certain "instanton counting" partition function for an arbitrary simple group G with the prepotential of the Toda integrable system associated with the affine Lie algebra whose affine Dynkin diagram is dual to that of the affinization of the Lie algebra of G (for G=SL(n) this conjecture was proved earlier in the works of Nekrasov-Okounkov and Nakajima-Yoshioka). Our proof is totally different and it is based on the results of the paper math.AG/0401409 by the first author.

Paper Structure

This paper contains 17 sections, 8 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. The partition function
  3. Algebraic version
  4. Parabolic generalization of the partition function
  5. Plan of the proof
  6. Acknowledgments
  7. Schrödinger operators and the prepotential: the one-dimensional case
  8. Schrödinger operators
  9. Explicit calculation of $\lim\limits_{\hbar\to 0} b$ via periods
  10. Eigenfunctions of non-stationary Schrödinger operators
  11. Schrödinger operators in higher dimensions and integrable systems
  12. The setup
  13. Integrability
  14. Computation of $\lim\limits_{\hbar\to 0} b$ via periods in the integrable case
  15. Some variants
  16. ...and 2 more sections

Key Result

Theorem 1.4

Let $P\subset G$ be a parabolic subgroup as a above.

Theorems & Definitions (12)

  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Proposition 2.2
  • proof
  • Proposition 2.5
  • proof
  • Definition 2.6
  • Theorem 4.1
  • Lemma 4.2
  • ...and 2 more