An infrared bootstrap of the Schur index with surface defects

Abstract

The infrared formula relates the Schur index of a 4d N = 2 theory to its wallcrossing invariant, a.k.a. BPS monodromy. A further extension of this formula, proposed by Córdova, Gaiotto and Shao, includes contributions by various types of line and surface defects. We study BPS monodromies in the presence of vortex surface defects of arbitrary vorticity for general class S theories of type A 1 engineered by UV curves with at least one regular puncture. The trace of these defect BPS monodromies is shown to coincide with the action of certain q-difference operators acting on the trace of the (pure) 4d BPS monodromy. We use these operators to develop a "bootstrap" (of traces) of BPS monodromies, relying only on their infrared properties, thereby reproducing the very general ultraviolet characterization of the Schur index.A relation between Schur indices and BPS monodromies was conjectured in [23], following 6 See also [45], for some results of an IR formula for the lens index. 9 The "no-exotics" property asserts that BPS states have R = 0 in 4d N = 2 gauge theories [33,54,55]. 10 In practice, BPS spectra at arbitrary moduli can be rather involved, and by moving onto a "simple" sector of the theory, in which one can explicitly compute the protected spin character, the wall-crossing formulae allow to extract the BPS degeneracies in other sectors. the latter reference are precisely the "vortex defects" considered in [31] via the Higgsing procedure (see also Appendix A.3), and in the present paper via the IR formalism.

Superconformal surfaces in four dimensions

“…For defects admiting a Lagrangian description in terms of a 2d-4d coupled system the index was computed in [39,43,108]. In [42,43,109] the index was computed via a conjectural formula in terms of the 2d-4d BPS spectrum in the Coulomb branch of the theory, being thus applicable to non-Lagrangian theories as well. Finally, for vortex defects a prescrisption to compute index was given in [37] and used in a variety of different theories [42,48,[110][111][112].…”

Testing Macdonald index as a refined character of chiral algebra

“…where the Haar measure ∆(x i ) is given by 19) and we see that the integral above extracts out the term χ su(3) ∅ (z 1 , z 2 ) in the HL polynomial P HL λ (x 1 , x 2 ). Using the expressions of the HL polynomials expanded in terms of SU(3) characters, (B.21)-(B.28), we obtain for |λ| ≤ 4, 22) and zero for other configurations. It is not difficult to see that the wavefunction vanishes unless the representation of λ contains the Cartan part, i.e.…”

“…In particular, the Schur index with surface operator with label {s i } gives the character of the module labeled by the same set of parameters {s i } in a clean way in Argyres-Douglas theories [21]. The bootstrap of surface operators in the Schur limit has also been studied in [22]. We would like to try to compute the Macdonald index with surface operator for (A n−1 , A m−1 ) AD theories with gcd(n, m) = 1 in this article, and compare our results with the corresponding refined character.…”

A slow review of the AGT correspondence

“…studied in [69,[563][564][565][566][567][568][569][570][571][572][573][574][575][576]. The same operators are important in the 5d version and S 1 × S 3 b version of the correspondence [64,65,190,570,571,[577][578][579][580]; see also [537,539,[581][582][583][584][585][586][587][588][589][590] for other considerations on this class of surface operators.…”