Two Recursions for the Seiberg-Witten Prepotential: Zamolodchikov's Recursion, the AGT Correspondence and Topological Recursion on the Seiberg-Witten Curve
Abstract
The Seiberg-Witten prepotential of four-dimensional 𝒩=2 gauge theory can be reached by two recursions of very different nature, and this thesis compares them explicitly. The first part is a self-contained review of the microscopic side: the ADHM construction as a hyperkähler quotient, the supersymmetric measure on the instanton moduli space, Seiberg-Witten theory, and the one- and two-instanton contributions computed by non-commutative resolution and localisation. The second part introduces the Ω-background and Nekrasov's partition function, and uses the AGT correspondence to turn Zamolodchikov's recursion for Virasoro conformal blocks into a recursion for the instanton partition function. For pure SU(2), this recursion is run up to five instantons, including the first corrections in ε1,ε2. In the last part, the same quantities are obtained from the Eynard-Orantin topological recursion on the Seiberg-Witten curve, which yields the Taylor expansion of the prepotential around an arbitrary point of the Coulomb branch. For pure SU(2) we give the Bergman kernel and the B-periods of ω0,3, ω0,4 and ω1,1 in closed form. They reproduce the instanton series generated by Zamolodchikov's recursion, remain valid at strong coupling, where they detect the massless monopole, and at genus one give the first correction to the self-dual Ω-background. The two recursions are thus complementary: one is exact in the vacuum expectation value and perturbative in the instanton number, the other is exact in the instanton parameter and perturbative around a fixed vacuum. We formulate a conjecture relating the B-periods of ωg,n to the derivatives of the genus-g free energy, and list open problems.
Related
- Eynard-Orantin Topological Recursions for Higher Weil-Petersson Volumes on Moduli Spaces of Super Riemann Surfaces
- Quantum modularity in refined topological recursion
- Eynard-Orantin Topological Recursion, Moduli-Space Virasoro Constraints, and Semisimple CohFTs
- Topological recursion for spin enumeration of hypermaps
- Topological Recursion on Spectral Curves with Infinite Ramification Loci
BibTeX
@misc{tajik2026recursions,
title = {{Two Recursions for the Seiberg-Witten Prepotential: Zamolodchikov's Recursion, the AGT Correspondence and Topological Recursion on the Seiberg-Witten Curve}},
author = {Bastam Tajik},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.12672},
eprint = {2609.12672},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12672},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Two Recursions for the Seiberg-Witten Prepotential: Zamolodchikov's Recursion, the AGT Correspondence and Topological Recursion on the Seiberg-Witten Curve
AU - Bastam Tajik
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2609.12672
UR - https://arxiv.org/abs/2609.12672
AB - The Seiberg-Witten prepotential of four-dimensional 𝒩=2 gauge theory can be reached by two recursions of very different nature, and this thesis compares them explicitly. The first part is a self-contained review of the microscopic side: the ADHM construction as a hyperkähler quotient, the supersymmetric measure on the instanton moduli space, Seiberg-Witten theory, and the one- and two-instanton contributions computed by non-commutative resolution and localisation. The second part introduces the Ω-background and Nekrasov's partition function, and uses the AGT correspondence to turn Zamolodchikov's recursion for Virasoro conformal blocks into a recursion for the instanton partition function. For pure SU(2), this recursion is run up to five instantons, including the first corrections in ε_1,ε_2. In the last part, the same quantities are obtained from the Eynard-Orantin topological recursion on the Seiberg-Witten curve, which yields the Taylor expansion of the prepotential around an arbitrary point of the Coulomb branch. For pure SU(2) we give the Bergman kernel and the B-periods of ω_0,3, ω_0,4 and ω_1,1 in closed form. They reproduce the instanton series generated by Zamolodchikov's recursion, remain valid at strong coupling, where they detect the massless monopole, and at genus one give the first correction to the self-dual Ω-background. The two recursions are thus complementary: one is exact in the vacuum expectation value and perturbative in the instanton number, the other is exact in the instanton parameter and perturbative around a fixed vacuum. We formulate a conjecture relating the B-periods of ω_g,n to the derivatives of the genus-g free energy, and list open problems.
ER -