Three-Dimensional Superconformal Index on ${M}^2 \times S^1_\beta$

Abstract

In this chapter, we review recent development on the three-dimensional superconformal index (SCI) $I_{\mathrm{Theory}}^{M^{2}}(x,\alpha _{a})={\mathrm{Tr}}_{H(M^{2})}\left( (-1)^{\hat{F}}x^{\prime \{Q,Q^{\dagger }\}}{{\Pi }_{a}}\alpha _{a}^{\hat{f}_{a}}\right)$ based on supersymmetric localization principle. In Sect. 3.1, we give the physical meaning for the SCI, and represent it in the path integral formalism. In Sect. 3.2, we turn to define supersymmetric actions on $M^2\times S_\beta ^1$ where $\beta$ corresponds to the inverse temperature. In Sect. 3.3, we explain the supersymmetric localization principle. We will perform the exact calculations in later chapters based on this technique.