Non-Abelian Localization for Supersymmetric Yang-Mills-Chern-Simons Theories on Seifert Manifold

Abstract

We derive non-Abelian localization formulae for supersymmetric Yang-Mills-Chern-Simons theory with matters on a Seifert manifold M, which is the three-dimensional space of a circle bundle over a two-dimensional Riemann surface Σ, by using the cohomological approach introduced by Kallen. We find that the partition function and the vev of the supersymmetric Wilson loop reduces to a finite dimensional integral and summation over classical flux configurations labeled by discrete integers. We also find the partition function reduces further to just a discrete sum over integers in some cases, and evaluate the supersymmetric index (Witten index) exactly on S^1xΣ. The index completely agrees with the previous prediction from field theory and branes. We discuss a vacuum structure of the ABJM theory deduced from the localization.

50 pages, 3 figures, typos corrected, references added

References in corpus (10)

• N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals
• Fractional M2-branes
• Notes on SUSY Gauge Theories on Three-Sphere
• The 4d Superconformal Index from q-deformed 2d Yang-Mills
• Summing Up All Genus Free Energy of ABJM Matrix Model
• Supersymmetric Gauge Theories on the Five-Sphere
• Seiberg-Witten Theory and Random Partitions
• Chern-Simons Theory on S^1-Bundles: Abelianisation and q-deformed Yang-Mills Theory
• On relationships among Chern-Simons theory, BF theory and matrix model
• Two-Dimensional Gauge Theory and Matrix Model

Cited by in corpus (32)

• $\mathcal{N}=1$ supersymmetric indices and the four-dimensional A-model
• The perturbative partition function of supersymmetric 5D Yang-Mills theory with matter on the five-sphere
• Comments on twisted indices in 3d supersymmetric gauge theories
• Seifert fibering operators in 3d $\mathcal{N}=2$ theories
• Higgs branch localization of 3d N=2 theories
• Direct derivation of "mirror" ABJ partition function
• 3d analogs of Argyres-Douglas theories and knot homologies
• Localization of 3d $\mathcal{N}=2$ Supersymmetric Theories on $S^1\times D^2$
• Localization on three-dimensional manifolds
• Supersymmetric theories on squashed five-sphere
• Supersymmetric partition functions on Riemann surfaces
• Generalized Indices for $\mathcal{N}=1$ Theories in Four-Dimensions
• On Supersymmetric Gauge Theories on S^4 x S^1
• Chern-Simons Theory on Seifert 3-Manifolds
• Localization of 4d $\mathcal{N}=1$ theories on ${\mathbb{D}}^2\times {\mathbb{T}}^2$
• G/G gauged WZW model and Bethe Ansatz for the phase model
• q-deformations of two-dimensional Yang-Mills theory: Classification, categorification and refinement
• Localization on round sphere revisited
• ABJ Wilson loops and Seiberg Duality
• A review on SUSY gauge theories on $S^3$
• Superconformal index on ${\mathbb{RP}}^2\times {\mathbb{S}}^1$ and mirror symmetry
• Localization and Duality for ABJM Latitude Wilson Loops
• Coulomb Branch Localization in Quiver Quantum Mechanics
• 3d N=2 Chern-Simons-matter theory, Bethe ansatz, and quantum K-theory of Grassmannians
• Localization and Non-Renormalization in Chern-Simons Theory
• Exact Results in Discretized Gauge Theories
• Higgs and Coulomb Branch Descriptions of the Volume of the Vortex Moduli Space
• Five-dimensional cohomological localization and squashed $q$-deformations of two-dimensional Yang-Mills theory
• The 3d $A$-model and generalised symmetries, Part I: bosonic Chern-Simons theories
• Seven-dimensional super Yang-Mills at negative coupling
• Topologically Twisted SUSY Gauge Theory, Gauge-Bethe Correspondence and Quantum Cohomology
• N=1 Chern-Simons-matter theory and localization