Quantum Spectrum of BPS Instanton States in $ 5d $ Guage Theories

TL;DR

This paper constructs and classifies 1/2 BPS instanton states in 5d gauge theories, revealing their spectrum, representation structure, and correspondence with string webs in brane web realizations.

Contribution

It provides a detailed construction and classification of BPS instanton states in 5d gauge theories, including their supermultiplets and relation to string webs.

Findings

Infinite BPS instanton spectrum with specific group representations

Correspondence between string webs and instanton states in Coulomb branch

Complete supermultiplet structure for 5d $SU(2)$ with $N_f \,\leq 4$

Abstract

We construct $ 1/2 $ BPS 1-instanton states in the Hilbert space of $ 5d $ gauge theories. The states are local with the scale size $ \rho=0 $. With the degeneracy from the gauge orientation taken into account, the BPS spectrum consists of an infinite number of states in the definite representations of the gauge group and the $ (J,0) $ representation of the $ SO(4)\cong SU(2)_{L} \times SU(2)_{R} $ rotation group. In the brane webs realization of the $ 5d $ $ \mathcal{N}=1 $ gauge theory, BPS states are string webs inside the 5-brane webs. For a theory with the gauge group $ SU(N) $ and the Chern-Simons level $ \kappa $, we give a classification of string webs in the Coulomb branch and show that for each web, one can always find an instanton state carrying the same spin and the same electric charge. The action of supercharges on instanton states gives the instanton supermultiplets. For the $ \mathcal{N} =1$ $ SU(2) $ gauge theory with $ N_{f} \leq 4$, we construct the instanton vector and hyper multiplets, which, together with the original multiplets of the theory, furnish the complete representation of the $ E_{N_{f}+1} $ group at UV.