Lectures on Instanton Counting

Abstract

These notes have two parts. The first is a study of Nekrasov's deformed partition functions $Z(\ve_1,\ve_2,\vec{a};\q,\vecτ)$ of N=2 SUSY Yang-Mills theories, which are generating functions of the integration in the equivariant cohomology over the moduli spaces of instantons on ${\mathbb{R}}^4$. The second is review of geometry of the Seiberg-Witten curves and the geometric engineering of the gauge theory, which are physical backgrounds of Nekrasov's partition functions. The first part is continuation of math.AG/0306198, where we identified the Seiberg-Witten prepotential with $Z(0,0,\vec{a};\q,0)$. We put higher Casimir operators to the partition function and clarify their relation to the Seiberg-Witten $u$-plane. We also determine the coefficients of $\ve_1\ve_2$ and $(\ve_1^2+\ve_2^2)/3$ (the genus 1 part) of the partition function, which coincide with two measure factors $A$, $B$ appeared in the $u$-plane integral. The proof is based on the blowup equation which we derived in the previous paper.

60 pages, to appear in Proceedings of "Workshop on algebraic structures and moduli spaces", July 14 - 20, 2003, Centre de recherches mathematiques, Universite de Montreal

References in corpus (7)

• Monopole Condensation, And Confinement In N=2 Supersymmetric Yang-Mills Theory
• Instantons and recursion relations in N=2 Susy gauge theory
• Vanishing theorems and character formulas for the Hilbert scheme of points in the plane
• Small Instantons, Little Strings and Free Fermions
• On blowup formulae for the S-duality conjecture of Vafa and Witten
• The uses of Whitham hierarchies
• Hilbert schemes, integrable hierarchies, and Gromov-Witten theory

Cited by in corpus (8)

• The Quiver Matrix Model and 2d-4d Conformal Connection
• Non-Perturbative Quantum Geometry III
• Direct integration for general Omega backgrounds
• Beta-ensembles for toric orbifold partition function
• Instantons and the 5D U(1) gauge theory with extra adjoint
• Combinatorial realizations of crystals via torus actions on quiver varieties
• Super Topological Recursion and Gaiotto Vectors For Superconformal Blocks
• Blowup relations on ${\mathbb{C}}^2/{\mathbb{Z}}_2$ from Nakajima-Yoshioka blowup relations