Line defect Schur indices, Verlinde algebras and U(1)r fixed points

Abstract

Given an N = 2 superconformal field theory, we reconsider the Schur index I L (q) in the presence of a half line defect L. Recently Cordova-Gaiotto-Shao found that I L (q) admits an expansion in terms of characters of the chiral algebra A introduced by Beem et al., with simple coefficients v L,β (q). We report a puzzling new feature of this expansion: the q → 1 limit of the coefficients v L,β (q) is linearly related to the vacuum expectation values L in U(1) r -invariant vacua of the theory compactified on S 1 . This relation can be expressed algebraically as a commutative diagram involving three algebras: the algebra generated by line defects, the algebra of functions on U(1) r -invariant vacua, and a Verlindelike algebra associated to A. Our evidence is experimental, by direct computation in the Argyres-Douglas theories of type (A 1 , A 2 ), (A 1 , A 4 ), (A 1 , A 6 ), (A 1 , D 3 ) and (A 1 , D 5 ). In the latter two theories, which have flavor symmetries, the Verlinde-like algebra which appears is a new deformation of algebras previously considered.

An infrared bootstrap of the Schur index with surface defects

“…The advantage of using BPS graphs is two-fold: it allows to compute the quantum spectrum generator, and more crucially it provides a way to compute 2d-4d BPS particles as well thanks to its underlying spectral network. 12 Alternatively, in [35][36][37][38], the authors leveraged the power of "BPS quivers" [26,[57][58][59] (which are related but not equivalent to BPS graphs), which describe the BPS spectrum in a strong-coupling chamber, where only a finite amount of hypermultiplets contribute; the associated charges can be described in terms of quiver data.…”

“…29) up to an overall sign. Here, I is the Schur index for an arbitrary rank-one class S theory (with at least two regular punctures),38 i.e.…”

Logarithmic W-algebras and Argyres-Douglas theories at higher rank

“…Moreover indices of line and surface defects are related to characters of modules of the vertex operator algebra [11,12]. Interestingly also modularity and Verlinde-like formulae have nice interpretations in the gauge theory [14][15][16]. For some further recent progress see [17][18][19][20][21][22][23][24].…”

“…In [15] the operator algebra of 1 2 -BPS line defects was studied and a main claim is that the homomorphism from this operator algebra to the algebra of functions on U(1) r -invariant vacua of the theory compactified on S 1 factors through the Verlinde algebra of the semisimplification of the category of modules of the chiral algebra.…”

Four-dimensional lens space index from two-dimensional chiral algebra

“…Under this map, various observables of the four-dimensional theory, which are invariant under exactly marginal deformations, are mapped to quantities in the two-dimensional chiral algebra [2][3][4]. Aspects of chiral algebras arising from four-dimensional N = 2 superconformal field theories have been discussed in [5][6][7][8][9][10][11][12][13][14][15][16]. In the following we give a list of observables, which are captured by the protected two-dimensional chiral algebra: The purpose of the current paper is to add the last line to this already rather extensive dictionary.…”