Lectures on instanton counting

By H. Nakajima, K. Yoshioka. 2003

206 citations recorded by the source corpus.

These notes have two parts. The first is a study of Nekrasov's deformed partition functions $Z(\ve_1,\ve_2,\vec{a}:\q,\vec{\tau})$ 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.

Subfields: Geometry and Topology, Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/e2335b29af857802d602f2bb203318bb58058da1

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Citations in the corpus

Affine SL(2) Conformal Blocks from 4d Gauge Theories (2010, unknown): “[14,21].”
Correlation functions of Coulomb branch operators (2016, background): “(While it is available for U(N) theories [56–58] it is an open problem to compute them for SU(N).”
Instanton counting on blowup. I. 4-dimensional pure gauge theory (2003, methodology): “In this sense, this paper does not rely on [ 49 ].”
Instanton counting with a surface operator and the chain-saw quiver (2011, background): “It is known that the resolution of singularity of the ADHM moduli space is given by the following affine algebro-geometric quotient [52, 53];”
Localization of Gauge Theory on a Four-Sphere and Supersymmetric Wilson Loops (2007, unknown): “Instanton corrections to the operator exp(βΦ) in the N = 2 equivariant theory on R(4) for a given asymptotic of Φ at infinity were computed in [5, 79, 82, 83].”
Vortex Counting and Lagrangian 3-Manifolds (2010, background): “the resulting partition function1 is a power series expansion in N: Zinst( 1; 2) = X1 k=0 2NkZinst k ( 1; 2); (2.7) with coecients given by equivariant integrals on instanton moduli spaces. (See [15] for an excellent set of lectures on this subject.) In the presence of a half-BPS surface operator in”
Instanton counting, Macdonald function and the moduli space of D-branes (2005, background): “There is a rather long history of attempts to recover the instanton expansion of the Seiberg-Witten theory from the calculus of multi-instantons (see [22] and references thererin).”
Refined curve counting on complex surfaces (2012, background): “We thank Paolo Aluffi, Florian Block, Barbara Fantechi, Ilia Itenberg, Davesh Maulik, Luca Migliorini, Grigory Mikhalkin, Hiraku Nakajima, Andrei Okounkov, Rahul Pandharipande, Sam Payne, Duco van Straten, Richard Thomas, Kota Yoshioka, and Don Zagier for helpful discussions.”