Instanton Bound States in ABJM Theory

TL;DR

The paper investigates non-perturbative corrections to the ABJM partition function within the Fermi-gas formalism, focusing on worldsheet, membrane, and bound-state instantons. By enforcing pole cancellations among contributions at each order, the authors determine bound-state coefficients $f_{\ell,m}(k,\mu)$ and show that the full bound-state sum can be absorbed into the worldsheet sector through a redefinition of the chemical potential $\mu_{\rm eff}$. They provide explicit constructions for $(1,n)$, $(2,n)$ and $(3,n)$ bound states, and derive analytic forms for 3- and 4-membrane instanton corrections, supported by multiple non-trivial checks. A key finding is that the non-perturbative corrections organize into a simpler structure when expressed in terms of $\mu_{\rm eff}$, suggesting a deeper underlying relation to topological data and GV invariants. The work opens avenues to a more complete understanding of instanton effects in M-theory by revealing hidden cancellations and structural simplifications in ABJM theory.

Abstract

The partition function of the ABJM theory receives non-perturbative corrections due to instanton effects. We study these non-perturbative corrections, including bound states of worldsheet instantons and membrane instantons, in the Fermi-gas approach. We require that the total non-perturbative correction should be always finite for arbitrary Chern-Simons level. This finiteness is realized quite non-trivially because each bound state contribution naively diverges at some levels. The poles of each contribution should be canceled out in total. We use this pole cancellation mechanism to find unknown bound state corrections from known ones. We conjecture a general expression of the bound state contribution. Summing up all the bound state contributions, we find that the effect of bound states is simply incorporated into the worldsheet instanton correction by a redefinition of the chemical potential in the Fermi-gas system. Analytic expressions of the 3- and 4-membrane instanton corrections are also proposed.