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# An infrared bootstrap of the Schur index with surface defects

Fluder, Martin and Longhi, Pietro (2019) An infrared bootstrap of the Schur index with surface defects. . (Submitted) http://resolver.caltech.edu/CaltechAUTHORS:20190709-151753573

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## Abstract

The infrared formula relates the Schur index of a 4d $\mathcal{N}=2$ theory to its wall-crossing invariant, a.k.a. BPS monodromy. A further extension of this formula, proposed by C\'ordova, 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 $\mathcal{S}$ 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.

Item Type:Report or Paper (Discussion Paper)
Related URLs:
URLURL TypeDescription
http://arxiv.org/abs/1905.02724arXivDiscussion Paper
Additional Information:We would like to thank Subhojoy Gupta and Shu-Heng Shao for insightful discussions and correspondence. PL is grateful for hospitality from the Simons Center for Geometry and Physics summer workshop, Caltech, and the Kavli IPMU at Tokyo University where part of this work was carried out. The work of MF is supported by the JSPS Grant-In-Aid for Scientific Research Wakate(A) 17H04837, the WPI Initiative, MEXT, Japan at IPMU, the University of Tokyo, the David and Ellen Lee Postdoctoral Scholarship, and the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award Number de-sc0011632. The work of PL is supported by a grant from the Swiss National Science foundation. He also acknowledges the support of the NCCR SwissMAP that is also funded by the Swiss National Science foundation. PL was also supported by the grants âĂĲGeometry and PhysicsâĂİ and âĂĲExact Results in Gauge and String TheoriesâĂİ from the Knut and Alice Wallenberg foundation during part of this work.
Group:Walter Burke Institute for Theoretical Physics
Funders:
Funding AgencyGrant Number
Japan Society for the Promotion of Science (JSPS)17H04837
Ministry of Education, Culture, Sports, Science and Technology (MEXT)UNSPECIFIED
University of TokyoUNSPECIFIED
David and Ellen Lee Postdoctoral ScholarshipUNSPECIFIED
Department of Energy (DOE)DE-SC0011632
Swiss National Science Foundation (SNSF)UNSPECIFIED
Knut and Alice Wallenberg FoundationUNSPECIFIED
Other Numbering System:
Other Numbering System NameOther Numbering System ID
CALT-TH2019-015
Record Number:CaltechAUTHORS:20190709-151753573
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20190709-151753573
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:97012
Collection:CaltechAUTHORS
Deposited By: Joy Painter
Deposited On:09 Jul 2019 22:42