# Fermi-gas correlators of ADHM theory and triality symmetry

### Submission summary

 As Contributors: Tadashi Okazaki Preprint link: scipost_202108_00070v2 Date accepted: 2021-10-20 Date submitted: 2021-10-09 02:39 Submitted by: Okazaki, Tadashi Submitted to: SciPost Physics Academic field: Physics Specialties: High-Energy Physics - Theory Approach: Theoretical

### Abstract

We analytically study the Fermi-gas formulation of sphere correlation functions of the Coulomb branch operators for 3d $\mathcal{N}=4$ ADHM theory with a gauge group $U(N)$, an adjoint hypermultiplet and $l$ hypermultiplets which can describe a stack of $N$ M2-branes at $A_{l-1}$ singularities. We find that the leading coefficients of the perturbative grand canonical correlation functions are invariant under a hidden triality symmetry conjectured from the twisted M-theory. The triality symmetry also helps us to fix the next-to-leading corrections analytically.

###### Current status:
Publication decision taken: accept

Editorial decision: For Journal SciPost Physics: Publish
(status: Editorial decision fixed and (if required) accepted by authors)

We would like to thank the referees for carefully reading the draft and pointing out several improvements.

1. In the Fermi-gas analysis we take the real Wigner transform of the Hamiltonian $H_W$ by specifying the imaginary FI and mass parameters. However, we expect that the expression with arbitrary FI and mass parameters can be reached by analytic continuation. We have added comment after eq.(2.10).

2. The monopole operators are labeled by the GNO charge $\pm A_{a}$, $a=1,\cdots, r$ where $A$ is the cocharacter that specifies an embedding of a $U(1)$ monopole singularity into the gauge group $G$ and $r$ is the rank of $G$. For $G=U(N)$ we have $A=(A_1, \cdots, A_r)\in \mathbb{Z}^r$ and we denote the charge by $n_$.   The correlation function of the Coulomb branch operators can be algebraically presented in terms of the twisted traces over the Verma modules of the quantized Coulomb branch algebra. Since the non-trivial twisted traces involving the monopole operators or equivalently shift operators appear only when they simply shift the vector multiplet scalar fields, only some insertion of non-periodic part'' without the shift, i.e. $R_{n_=0}$ in the integrand will lead to distinct Coulomb branch correlation functions with change of residues.   We have added these clarifications.

3. To clarify, $\langle \mathcal{O} \rangle$ depends on $N$ as in (3.2) and $\rho_s$ depends on N as in (3.59). So they cannot be factored out. We have explicitly written their dependence on $N$ in (3.44) and (3.51)-(3.53).

4. We have moved technical details of the calculations to Appendix.

### List of changes

2. clarifications of notations after eq.(3.2)
3. inclusion of dependence on $N$ in eq.(3.44) and (3.51)-(3.53).
4. additional Appendix for technical details of the calculations.

### Submission & Refereeing History

Resubmission scipost_202108_00070v2 on 9 October 2021
Submission scipost_202108_00070v1 on 30 August 2021

## Reports on this Submission

### Report

The authors have made appropriate modifications. I would like to recommend the manuscript for publication.

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