Exact results for $\mathcal{N}=2$ supersymmetric gauge theories on compact toric manifolds and equivariant Donaldson invariants

Abstract

We provide a contour integral formula for the exact partition function of $\mathcal{N}=2$ supersymmetric $U(N)$ gauge theories on compact toric four-manifolds by means of supersymmetric localisation. We perform the explicit evaluation of the contour integral for $U(2)$ $\mathcal{N}=2$ theory on ${\mathbb{P}}^2$ for all instanton numbers. In the zero mass case, corresponding to the $\mathcal{N}=4$ supersymmetric gauge theory, we obtain the generating function of the Euler characteristics of instanton moduli spaces in terms of mock-modular forms. In the decoupling limit of infinite mass we find that the generating function of local and surface observables computes equivariant Donaldson invariants, thus proving in this case a long-standing conjecture by N. Nekrasov. In the case of vanishing first Chern class the resulting equivariant Donaldson polynomials are new.

arXiv comment37 pages + Appendix, 8 figures. v3: accepted for publication on JHEP

References in corpus (40)

• Localization of gauge theory on a four-sphere and supersymmetric Wilson loops
• A Strong Coupling Test of S-Duality
• Supersymmetric Yang-Mills Systems And Integrable Systems
• Instantons on noncommutative R^4, and (2,0) superconformal six dimensional theory
• Integrability and Seiberg-Witten Exact Solution
• Loop and surface operators in N=2 gauge theory and Liouville modular geometry
• Integrable systems and supersymmetric gauge theory
• Multi-Instanton Calculus and Equivariant Cohomology
• On combinatorial expansion of the conformal blocks arising from AGT conjecture
• An Algorithm for the Microscopic Evaluation of the Coefficients of the Seiberg-Witten Prepotential
• Issues in Topological Gauge Theory
• The perturbative partition function of supersymmetric 5D Yang-Mills theory with matter on the five-sphere
• Supersymmetric Yang-Mills Theory On A Four-Manifold
• Seiberg-Witten Theories on Ellipsoids
• Recursion relations in CFT and N=2 SYM theory
• E-Strings and N=4 Topological Yang-Mills Theories
• Hitchin systems, N=2 gauge theories and W-gravity
• Super Liouville conformal blocks from N=2 SU(2) quiver gauge theories
• Instanton moduli spaces and bases in coset conformal field theory
• Gauge Theories on ALE Space and Super Liouville Correlation Functions
• Instantons on ALE spaces and Super Liouville Conformal Field Theories
• Liouville/Toda central charges from M5-branes
• Extended Supersymmetry on Curved Spaces
• Instantons and 2d Superconformal field theory
• Surface Operators and Knot Homologies
• Sheaves on Toric Varieties for Physics
• Six-dimensional supersymmetric gauge theories, quantum cohomology of instanton moduli spaces and gl(N) Quantum Intermediate Long Wave Hydrodynamics
• N=2 gauge theories on toric singularities, blow-up formulae and W-algebrae
• Euler characteristics of moduli spaces of torsion free sheaves on toric surfaces
• M-Theoretic Derivations of 4d-2d Dualities: From a Geometric Langlands Duality for Surfaces, to the AGT Correspondence, to Integrable Systems
• Multi-Instantons, Supersymmetry and Topological Field Theories
• Poincare polynomial of moduli spaces of framed sheaves on (stacky) Hirzebruch surfaces
• Topological Vector Symmetry of BRSTQFT and Construction of Maximal Supersymmetry
• Instanton on toric singularities and black hole countings
• N=2 quiver gauge theories on A-type ALE spaces
• Instanton Calculus, Topological Field Theories and N=2 Super Yang-Mills Theories
• From strings in 6d to strings in 5d
• Coupling of two conformal field theories and Nakajima-Yoshioka blow-up equations
• Super-Liouville - Double Liouville correspondence
• The geometry of the M5-branes and TQFTs

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