Partition functions of superconformal Chern-Simons theories from Fermi gas approach

Abstract

Abstract:We study the partition function of three-dimensional N = 4 superconformal Chern-Simons theories of the circular quiver type, which are natural generalizations of the ABJM theory, the worldvolume theory of M2-branes. In the ABJM case, it was known that the perturbative part of the partition function sums up to the Airy function as Z(N ) = e A C −1/3 Ai[C −1/3 (N − B)] with coefficients C, B and A and that for the non-perturbative part the divergences coming from the coefficients of worldsheet instantons and membrane instantons cancel among themselves. We find that many of the interesting properties in the ABJM theory are extended to the general superconformal Chern-Simons theories. Especially, we find an explicit expression of B for general N = 4 theories, a conjectural form of A for a special class of theories, and cancellation in the non-perturbative coefficients for the simplest theory next to the ABJM theory.

Instanton effects in rank deformed superconformal Chern-Simons theories from topological strings

“…It is already a non-trivial question whether any quiver superconformal Chern-Simons theory corresponds to some Calabi-Yau geometry or not. In the subsequent works [20][21][22][23][24], the investigation started with a special class of N = 3 theories: the circular quiver superconformal Chern-Simons theories with unitary gauge groups. 2 It was shown in [30][31][32][33][34] that, for the circular quiver, the theory enjoys the supersymmetry N = 4 when the Chern-Simons level k a associated to each vertex is expressed as k a = k(s a − s a−1 )/2 with s a = ±1.…”

“…2 It was shown in [30][31][32][33][34] that, for the circular quiver, the theory enjoys the supersymmetry N = 4 when the Chern-Simons level k a associated to each vertex is expressed as k a = k(s a − s a−1 )/2 with s a = ±1. Namely, the full information of the N = 4 theories is encoded in the list of s a = ±1, and we may refer to the theory with {s a } = {+1, +1, · · · , +1 as (p 1 , q 1 , p 2 , q 2 , · · · ) model [21]. For example, the theory with alternating s a is denoted as the (1, 1, · · · , 1) model.…”

“…The physical interpretation of this theory is the orbifold and its grand potential was studied in [20]. Another limiting case {s a } = {(+1) p , (−1) q }, or the (p, q) model [21], which is expected to be a building block of all other N = 4 theories, was also studied carefully in [21][22][23][24].…”

“…Namely, the theories with different sequences of {s a } are expected to be connected and to form a non-trivial moduli space as a whole. In fact, since the ordering of s a = ±1 corresponds to the ordering of the quantum operators [21] in the one-particle density matrix in the Fermi gas formalism, the theories which are different only in the ordering are equivalent in the classical limit. This may suggest that these theories correspond to the topological string theory on the same Calabi-Yau manifold [35].…”

ABJ theory in the higher spin limit

“…Inspired by the seminal work of Drukker, Marino, and Putrov [71,72] and, in good part, with the use of the elegant Fermi gas approach developed by Marino and Putrov [73], a great deal about the ABJ(M) partition function has been uncovered, in particular, at large N , both in perturbative [73,74] and nonperturbative expansions [75][76][77][78][79][80][81][82]. There has also been significant progress in the study of Wilson loops in the ABJ(M) theory [83][84][85][86] as well as the partition functions of more general Chern-Simons-matter theories [87][88][89][90][91]. However, the ABJ partition function in the HS limit (1.1) has not been much investigated in the literature.…”