Comments on twisted indices in 3d supersymmetric gauge theories

Abstract

We study three-dimensional N = 2 supersymmetric gauge theories on Σ g × S 1 with a topological twist along Σ g , a genus-g Riemann surface. The twisted supersymmetric index at genus g and the correlation functions of half-BPS loop operators on S 1 can be computed exactly by supersymmetric localization. For g = 1, this gives a simple UV computation of the 3d Witten index. Twisted indices provide us with a clean derivation of the quantum algebra of supersymmetric Wilson loops, for any Yang-Mills-Chern-Simonsmatter theory, in terms of the associated Bethe equations for the theory on R 2 × S 1 . This also provides a powerful and simple tool to study 3d N = 2 Seiberg dualities. Finally, we study A-and B-twisted indices for N = 4 supersymmetric gauge theories, which turns out to be very useful for quantitative studies of three-dimensional mirror symmetry. We also briefly comment on a relation between the S 2 × S 1 twisted indices and the Hilbert series of N = 4 moduli spaces.

Cardy formula for 4d SUSY theories and localization

“…(5.3)-(5.4) we see that 20) and similarly forL M 3 . Consider the case in which the theory before the addition of adjoint matter has Tr(R) negative, and a local minimum in zero.…”

“…Next we consider partition function on T 2 × Σ g , where Σ g is Riemann surface with genus g. The localization computation was presented in [19] (see also [14,15,20,55]). The saddle point configuration of localization is somewhat involved on T 2 × Σ g .…”

“…From the index interpretation it follows that they are independent on continuous coupling constants [9], therefore their computation at weak coupling is valid even when the coupling is large. Exact results on several different geometries exist in the literature [10][11][12][13][14][15][16][17][18][19][20][21], derived either via supersymmetric localization [22,23] or by solving the associated counting problem in the Hilbert space.…”

A-twisted correlators and Hori dualities

“…(Naturally, A (F ) µ sits in a background vector multiplet V (F ) .) The correlation functions of Coulomb branch operators on Σ g , with background fluxes (1.5) turned on, are given by [10,11,17,[31][32][33]: 6) with F an index running over the flavor group. The operator H is the handle-gluing operator [31] and Π F are flavor flux operators [34], as we will review.…”

Supersymmetric AdS $$_2\times \Sigma _2$$ 2 × Σ 2 solutions from tri-sasakian trunc

“…In this context, the black hole entropy is computed using topologically twisted index of three-dimensional superconformal field theories (SCFTs) compactified on a Riemann surface 2 [4][5][6][7][8]. In the dual gravity solutions, the black holes interpolate between the asymptotically AdS 4 and the near horizon AdS 2 × 2 geometries.…”