Instanton partition functions in N=2 SU(N) gauge theories with a general surface operator, and their W-algebra duals

TL;DR

The paper addresses computing $Z_{ m inst}$ for 4d $\mathcal{N}=2$ $\mathrm{SU}(N)$ gauge theories with a general surface operator labeled by a partition $N=p_1+\cdots+p_n$, proposing an explicit combinatorial character for the bifundamental in $\mathrm{SU}(N)\times\mathrm{SU}(N)$ that yields the instanton partition function as a fixed-point sum. It shows consistency with Nekrasov's results in the no-surface-operator limit ($N=N$) and with Feigin et al. for the full surface operator ($N=1+\cdots+1$), and extends to the $1+ (N-1)$ and $1+\cdots+1+2$ partitions where $\,\mathcal{W}$-algebra or quasi-superconformal algebra considerations provide cross-checks. The work also analyzes the $N=1+(N-1)$ case via a dual restricted $SU(N)\times SU(N)$ quiver and a 2d defect description, finding agreement between distinct constructions and supporting the interconnected web of ramified instantons, $\mathcal{W}$-algebras, and integrable systems. Together with discussion of potential LMNS-like contour formulations and differential equations, these results push toward a unified framework for surface operators in $\mathcal{N}=2$ gauge theories and their algebraic duals.

Abstract

We write down an explicit conjecture for the instanton partition functions in 4d N=2 SU(N) gauge theories in the presence of a certain type of surface operator. These surface operators are classified by partitions of N, and for each partition there is an associated partition function. For the partition N=N we recover the Nekrasov formalism, and when N=1+...+1 we reproduce the result of Feigin et. al. For the case N=1+(N-1) our expression is consistent with an alternative formulation in terms of a restricted SU(N)xSU(N) instanton partition function. When N=1+...+1+2 the partition functions can also be obtained perturbatively from certain W-algebras known as quasi-superconformal algebras, in agreement with a recent general proposal.