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

TL;DR

This work develops a cohomological localization framework for supersymmetric Yang-Mills-Chern-Simons theories on Seifert manifolds, reducing the partition function and Wilson-loop observables to finite-dimensional integrals and discrete flux sums. By exploiting equivariant cohomology and a BRST-like charge with Q^2 generating isometries, the authors derive explicit 1-loop determinants and fixed-point equations, obtaining exact results for the YMCS partition function and the supersymmetric index on $S^1\times\Sigma$. They extend the formalism to include matter, notably ABJM-type sectors, and demonstrate that the index matches brane predictions and field-theory expectations, including no quantum level shift in the supersymmetric case. The results connect 3D localization to 2D BF-type theories, yield q-deformed YM expressions, and illuminate vacuum structures and dualities, with broader implications for higher-dimensional generalizations and large-N dynamics.

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.

Figures (3)

• Figure 1: An example of the correspondence between $\vec{\tilde{n}}$ and the restrictied Young diagram (partition) for $k=10$ and $N=4$. The right-down and right-up edge of the boxes correspond to the black and white circle, respectively. (a) The empty box is for $\vec{\tilde{n}} = (3,2,1,0)$. (b) The partition $\vec{\tilde{\nu}} = (6,3,2)$ is for $\vec{\tilde{n}} = (9,5,3,0)$. (c) The full box is for $\vec{\tilde{n}} = (9,8,7,6)$. The total number of the partitions is ${}_{10}C_4=210$, which is the supersymmetric index of the $U(4)_{10}$ CS theory.
• Figure 2: The supersymmetric YMCS theory can be realized by a configuration with NS5-D3-$(k,1)$5 branes in Type IIB string theory. The brane configuration is lifted up to a M5-M2-M5 system in M-theory on torus. The positions of M2 branes, which are interpreted by the vev of the Wilson loop, are quantized along the fiber direction. We depict the brane configuration by a diagram (b), which corresponds to the restricted Young diagram in Figure \ref{['Young diagrams']}.
• Figure 3: A vacuum configuration of the ABJ theory with the gauge group $U(N+l)_k \times U(N)_{-k}$. The branes are placed on the compact circle along the $x^6$-direction. So left and right M5(NS5)-branes are identified on the circle. If positions of fractional M2-branes coincide with each other, they become a single M2-brane wrapping around the $x^6$-direction and are decoupled from the M5-brane system (or can be freely removed from the tip of the $\mathbb{C}^4/\mathbb{Z}_k$ orbifold in the dual M-theory picture). Thus the partition function of the ABJ theory is factorized into the fractional M2-brane sector (CS theory) and ordinary (non-fractional) M2-brane sector (ABJM theory).