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Title: An infrared bootstrap of the Schur index with surface defects

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 Córdova, Gaiotto and Shao, includes contributions by many 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 A1 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.

Authors:
 [1]; ORCiD logo [2]
  1. California Inst. of Technology (CalTech), Pasadena, CA (United States); Univ. of Tokyo (Japan)
  2. ETH Zurich (Switzerland)
Publication Date:
Research Org.:
California Institute of Technology (CalTech), Pasadena, CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC), High Energy Physics (HEP)
OSTI Identifier:
1596782
Grant/Contract Number:  
SC0011632
Resource Type:
Accepted Manuscript
Journal Name:
Journal of High Energy Physics (Online)
Additional Journal Information:
Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2019; Journal Issue: 9; Journal ID: ISSN 1029-8479
Publisher:
Springer Berlin
Country of Publication:
United States
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS

Citation Formats

Fluder, Martin, and Longhi, Pietro. An infrared bootstrap of the Schur index with surface defects. United States: N. p., 2019. Web. doi:10.1007/JHEP09(2019)062.
Fluder, Martin, & Longhi, Pietro. An infrared bootstrap of the Schur index with surface defects. United States. https://doi.org/10.1007/JHEP09(2019)062
Fluder, Martin, and Longhi, Pietro. Tue . "An infrared bootstrap of the Schur index with surface defects". United States. https://doi.org/10.1007/JHEP09(2019)062. https://www.osti.gov/servlets/purl/1596782.
@article{osti_1596782,
title = {An infrared bootstrap of the Schur index with surface defects},
author = {Fluder, Martin and Longhi, Pietro},
abstractNote = {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 Córdova, Gaiotto and Shao, includes contributions by many 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 A1 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.},
doi = {10.1007/JHEP09(2019)062},
journal = {Journal of High Energy Physics (Online)},
number = 9,
volume = 2019,
place = {United States},
year = {2019},
month = {9}
}

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Works referenced in this record:

Phases of N = 2 theories in two dimensions
journal, August 1993


N = 2 dualities
journal, August 2012


Spectral Networks
journal, March 2013

  • Gaiotto, Davide; Moore, Gregory W.; Neitzke, Andrew
  • Annales Henri Poincaré, Vol. 14, Issue 7
  • DOI: 10.1007/s00023-013-0239-7

On classification ofN=2 supersymmetric theories
journal, December 1993

  • Cecotti, Sergio; Vafa, Cumrun
  • Communications in Mathematical Physics, Vol. 158, Issue 3
  • DOI: 10.1007/BF02096804

2d TQFT structure of the superconformal indices with outer-automorphism twists
journal, March 2013

  • Mekareeya, Noppadol; Song, Jaewon; Tachikawa, Yuji
  • Journal of High Energy Physics, Vol. 2013, Issue 3
  • DOI: 10.1007/JHEP03(2013)171

Classification of complete $N = 2$ supersymmetric theories in 4 dimensions
journal, January 2013


Self-dual strings and N = 2 supersymmetric field theory
journal, October 1996


Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD
journal, December 1994


Seiberg-Witten Prepotential from Instanton Counting
journal, January 2003


A new supersymmetric index
journal, November 1992


Jenkins–Strebel differentials with poles
journal, January 2008


An Index for 4 Dimensional Super Conformal Theories
journal, June 2007

  • Kinney, Justin; Maldacena, Juan; Minwalla, Shiraz
  • Communications in Mathematical Physics, Vol. 275, Issue 1
  • DOI: 10.1007/s00220-007-0258-7

Integrable systems and supersymmetric gauge theory
journal, January 1996


New phenomena in SU(3) supersymmetric gauge theory
journal, August 1995


Gauge Theories and Macdonald Polynomials
journal, November 2012

  • Gadde, Abhijit; Rastelli, Leonardo; Razamat, Shlomo S.
  • Communications in Mathematical Physics, Vol. 319, Issue 1
  • DOI: 10.1007/s00220-012-1607-8

Surface defects and instanton–vortex interaction
journal, July 2017


Schur indices, BPS particles, and Argyres-Douglas theories
journal, January 2016


The Self-Duality Equations on a Riemann Surface
journal, July 1987


The structure of BPS spectra
text, January 2015


On the phase transition towards permanent quark confinement
journal, June 1978


Refined, Motivic, and Quantum
journal, November 2009


Solutions of four-dimensional field theories via M-theory
journal, September 1997


Spectral Networks with Spin
journal, August 2015

  • Galakhov, Dmitry; Longhi, Pietro; Moore, Gregory W.
  • Communications in Mathematical Physics, Vol. 340, Issue 1
  • DOI: 10.1007/s00220-015-2455-0

Geometry and integrability of topological-antitopological fusion
journal, March 1993

  • Dubrovin, B.
  • Communications in Mathematical Physics, Vol. 152, Issue 3
  • DOI: 10.1007/BF02096618

Branes and N = 2 theories in two dimensions
journal, March 1998


Tinkertoys for Gaiotto duality
journal, November 2010

  • Chacaltana, Oscar; Distler, Jacques
  • Journal of High Energy Physics, Vol. 2010, Issue 11
  • DOI: 10.1007/JHEP11(2010)099

BPS graphs: from spectral networks to BPS quivers
journal, July 2017

  • Gabella, Maxime; Longhi, Pietro; Park, Chan Y.
  • Journal of High Energy Physics, Vol. 2017, Issue 7
  • DOI: 10.1007/JHEP07(2017)032

S-duality and 2d topological QFT
journal, March 2010

  • Gadde, Abhijit; Pomoni, Elli; Rastelli, Leonardo
  • Journal of High Energy Physics, Vol. 2010, Issue 3
  • DOI: 10.1007/JHEP03(2010)032

ADE spectral networks
journal, August 2016


Supersymmetric Yang-Mills theory and integrable systems
journal, February 1996


Wall-crossing in coupled 2d-4d systems
journal, December 2012

  • Gaiotto, Davide; Moore, Gregory W.; Neitzke, Andrew
  • Journal of High Energy Physics, Vol. 2012, Issue 12
  • DOI: 10.1007/JHEP12(2012)082

New = 2 superconformal field theories in four dimensions
journal, February 1996


Ising model andN=2 supersymmetric theories
journal, October 1993

  • Cecotti, Sergio; Vafa, Cumrun
  • Communications in Mathematical Physics, Vol. 157, Issue 1
  • DOI: 10.1007/BF02098023

5D SYM on 3D sphere and 2D YM
journal, October 2012


Topological—anti-topological fusion
journal, December 1991


Framed BPS states
journal, January 2013

  • Gaiotto, Davide; Moore, Gregory W.; Neitzke, Andrew
  • Advances in Theoretical and Mathematical Physics, Vol. 17, Issue 2
  • DOI: 10.4310/ATMP.2013.v17.n2.a1

Rigid Surface Operators
journal, January 2010


General Argyres-Douglas theory
journal, January 2013


Electric-magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory
journal, September 1994


Liouville Correlation Functions from Four-Dimensional Gauge Theories
journal, January 2010

  • Alday, Luis F.; Gaiotto, Davide; Tachikawa, Yuji
  • Letters in Mathematical Physics, Vol. 91, Issue 2
  • DOI: 10.1007/s11005-010-0369-5

Localization of Gauge Theory on a Four-Sphere and Supersymmetric Wilson Loops
journal, May 2012


Loop operators and S-duality from curves on Riemann surfaces
journal, September 2009


Loop and surface operators in $ \mathcal{N} = 2 $ gauge theory and Liouville modular geometry
journal, January 2010

  • Alday, Luis F.; Gaiotto, Davide; Gukov, Sergei
  • Journal of High Energy Physics, Vol. 2010, Issue 1
  • DOI: 10.1007/JHEP01(2010)113

Gauge theory loop operators and Liouville theory
journal, February 2010

  • Drukker, Nadav; Gomis, Jaume; Okuda, Takuya
  • Journal of High Energy Physics, Vol. 2010, Issue 2
  • DOI: 10.1007/JHEP02(2010)057

Surface operators in $ \mathcal{N} $ = 2 4d gauge theories
journal, November 2012


NILPOTENT ORBITS AND CODIMENSION-2 DEFECTS OF 6d${\mathcal N} = (2, 0)$ THEORIES
journal, February 2013

  • Chacaltana, Oscar; Distler, Jacques; Tachikawa, Yuji
  • International Journal of Modern Physics A, Vol. 28, Issue 03n04
  • DOI: 10.1142/S0217751X1340006X

Four Dimensional Superconformal Index from q -Deformed Two Dimensional Yang-Mills Theory
journal, June 2011


Counting chiral primaries in , superconformal field theories
journal, July 2006


Wall Crossing Invariants from Spectral Networks
journal, December 2017


BPS degeneracies and superconformal index in diverse dimensions
journal, November 2014


Four-Dimensional Wall-Crossing via Three-Dimensional Field Theory
journal, July 2010

  • Gaiotto, Davide; Moore, Gregory W.; Neitzke, Andrew
  • Communications in Mathematical Physics, Vol. 299, Issue 1
  • DOI: 10.1007/s00220-010-1071-2

Quantum Wall Crossing in $${{\mathcal N}=2}$$ Gauge Theories
journal, October 2010

  • Dimofte, Tudor; Gukov, Sergei; Soibelman, Yan
  • Letters in Mathematical Physics, Vol. 95, Issue 1
  • DOI: 10.1007/s11005-010-0437-x

Bootstrapping the superconformal index with surface defects
journal, January 2013

  • Gaiotto, Davide; Rastelli, Leonardo; Razamat, Shlomo S.
  • Journal of High Energy Physics, Vol. 2013, Issue 1
  • DOI: 10.1007/JHEP01(2013)022

Surface defects, the superconformal index and q-deformed Yang-Mills
journal, October 2013

  • Alday, Luis F.; Bullimore, Mathew; Fluder, Martin
  • Journal of High Energy Physics, Vol. 2013, Issue 10
  • DOI: 10.1007/JHEP10(2013)018

Infrared computations of defect Schur indices
journal, November 2016

  • Córdova, Clay; Gaiotto, Davide; Shao, Shu-Heng
  • Journal of High Energy Physics, Vol. 2016, Issue 11
  • DOI: 10.1007/JHEP11(2016)106

Surface defect indices and 2d-4d BPS states
journal, December 2017

  • Córdova, Clay; Gaiotto, Davide; Shao, Shu-Heng
  • Journal of High Energy Physics, Vol. 2017, Issue 12
  • DOI: 10.1007/JHEP12(2017)078

Surface defects and chiral algebras
journal, May 2017

  • Córdova, Clay; Gaiotto, Davide; Shao, Shu-Heng
  • Journal of High Energy Physics, Vol. 2017, Issue 5
  • DOI: 10.1007/JHEP05(2017)140

Line defect Schur indices, Verlinde algebras and U(1)r fixed points
journal, November 2017


The BPS spectrum generator in 2D-4D systems
journal, November 2012


Spectral Networks and Fenchel–Nielsen Coordinates
journal, May 2016


2d SCFTs from M2-branes
journal, November 2013

  • Hori, Kentaro; Park, Chan Y.; Tachikawa, Yuji
  • Journal of High Energy Physics, Vol. 2013, Issue 11
  • DOI: 10.1007/JHEP11(2013)147

ADE spectral networks and decoupling limits of surface defects
journal, February 2017


Orbifold Schur index and IR formula
journal, April 2018

  • Imamura, Yosuke
  • Progress of Theoretical and Experimental Physics, Vol. 2018, Issue 4
  • DOI: 10.1093/ptep/pty025

Line defects, tropicalization, and multi-centered quiver quantum mechanics
journal, September 2014


Geometric engineering of (framed) BPS states
journal, January 2014

  • Chuang, Wu-Yen; Diaconescu, Duiliu-Emanuel; Manschot, Jan
  • Advances in Theoretical and Mathematical Physics, Vol. 18, Issue 5
  • DOI: 10.4310/ATMP.2014.v18.n5.a3

About the Absence of Exotics and the Coulomb Branch Formula
journal, November 2017


BPS Quivers and Spectra of Complete $${\mathcal{N} = 2}$$ N = 2 Quantum Field Theories
journal, August 2013

  • Alim, Murad; Cecotti, Sergio; Córdova, Clay
  • Communications in Mathematical Physics, Vol. 323, Issue 3
  • DOI: 10.1007/s00220-013-1789-8

$\mathcal{N} = 2$ quantum field theories and their BPS quivers
journal, January 2014

  • Alim, Murad; Cecotti, Sergio; Córdova, Clay
  • Advances in Theoretical and Mathematical Physics, Vol. 18, Issue 1
  • DOI: 10.4310/ATMP.2014.v18.n1.a2

Wilson-’t Hooft operators in four-dimensional gauge theories and S -duality
journal, July 2006


Reading between the lines of four-dimensional gauge theories
journal, August 2013

  • Aharony, Ofer; Seiberg, Nathan; Tachikawa, Yuji
  • Journal of High Energy Physics, Vol. 2013, Issue 8
  • DOI: 10.1007/JHEP08(2013)115

Surface operators and separation of variables
journal, January 2016

  • Frenkel, Edward; Gukov, Sergei; Teschner, Jörg
  • Journal of High Energy Physics, Vol. 2016, Issue 1
  • DOI: 10.1007/JHEP01(2016)179

Exact WKB analysis and cluster algebras
journal, November 2014


2d index and surface operators
journal, March 2014


Surface defects and resolvents
journal, September 2013

  • Gaiotto, Davide; Gukov, Sergei; Seiberg, Nathan
  • Journal of High Energy Physics, Vol. 2013, Issue 9
  • DOI: 10.1007/JHEP09(2013)070

The superconformal index and an elliptic algebra of surface defects
journal, October 2014

  • Bullimore, Mathew; Fluder, Martin; Hollands, Lotte
  • Journal of High Energy Physics, Vol. 2014, Issue 10
  • DOI: 10.1007/JHEP10(2014)062

The BPS spectra of two-dimensional supersymmetric gauge theories with twisted mass terms
journal, November 1998


Wild quiver gauge theories
journal, February 2012

  • Bonelli, Giulio; Maruyoshi, Kazunobu; Tanzini, Alessandro
  • Journal of High Energy Physics, Vol. 2012, Issue 2
  • DOI: 10.1007/JHEP02(2012)031

Classification of Argyres-Douglas theories from M5 branes
journal, September 2016


Central charges and RG flow of strongly-coupled $ \mathcal{N}=2 $ theory
journal, March 2013


Infinite Chiral Symmetry in Four Dimensions
journal, January 2015

  • Beem, Christopher; Lemos, Madalena; Liendo, Pedro
  • Communications in Mathematical Physics, Vol. 336, Issue 3
  • DOI: 10.1007/s00220-014-2272-x

Nekrasov-Shatashvili limit of the 5D superconformal index
journal, August 2016

  • Papageorgakis, Constantinos; Pini, Alessandro; Rodríguez-Gómez, Diego
  • Physical Review D, Vol. 94, Issue 4
  • DOI: 10.1103/PhysRevD.94.045007

Exponential networks and representations of quivers
journal, August 2017

  • Eager, Richard; Selmani, Sam Alexandre; Walcher, Johannes
  • Journal of High Energy Physics, Vol. 2017, Issue 8
  • DOI: 10.1007/JHEP08(2017)063

S-duality and the $ \mathcal{N}=2 $ lens space index
journal, October 2013

  • Razamat, Shlomo S.; Yamazaki, Masahito
  • Journal of High Energy Physics, Vol. 2013, Issue 10
  • DOI: 10.1007/JHEP10(2013)048

Down the rabbit hole with theories of class S $$ \mathcal{S} $$
journal, October 2014

  • Razamat, Shlomo S.; Willett, Brian
  • Journal of High Energy Physics, Vol. 2014, Issue 10
  • DOI: 10.1007/JHEP10(2014)099

Heterotic surface defects and dualities from 2d/4d indices
journal, October 2014


N = 1 $$ \mathcal{N}=1 $$ theories of class S k $$ {\mathcal{S}}_k $$
journal, July 2015

  • Gaiotto, Davide; Razamat, Shlomo S.
  • Journal of High Energy Physics, Vol. 2015, Issue 7
  • DOI: 10.1007/JHEP07(2015)073

Surface defects as transfer matrices
journal, November 2016

  • Maruyoshi, Kazunobu; Yagi, Junya
  • Progress of Theoretical and Experimental Physics, Vol. 2016, Issue 11
  • DOI: 10.1093/ptep/ptw151

Surface defects and elliptic quantum groups
journal, June 2017


Surface Defects in E-String Compactifications and the van Diejen Model
journal, April 2018

  • Nazzal, Belal; Razamat, Shlomo S.
  • Symmetry, Integrability and Geometry: Methods and Applications
  • DOI: 10.3842/SIGMA.2018.036

Flavored surface defects in 4d $$\mathcal{N}=1$$ N = 1 SCFTs
journal, December 2018


On the correspondence between surface operators in Argyres-Douglas theories and modules of chiral algebra
journal, March 2019

  • Nishinaka, Takahiro; Sasa, Shinya; Zhu, Rui-Dong
  • Journal of High Energy Physics, Vol. 2019, Issue 3
  • DOI: 10.1007/JHEP03(2019)091

5D SYM and 2D q-deformed YM
journal, April 2013


A brief review of the 2d/4d correspondences
journal, October 2017


On the superconformal index of Argyres–Douglas theories
journal, December 2015


Argyres-Douglas theories, the Macdonald index, and an RG inequality
journal, February 2016

  • Buican, Matthew; Nishinaka, Takahiro
  • Journal of High Energy Physics, Vol. 2016, Issue 2
  • DOI: 10.1007/JHEP02(2016)159

Superconformal indices of generalized Argyres-Douglas theories from 2d TQFT
journal, February 2016


On irregular singularity wave functions and superconformal indices
journal, September 2017

  • Buican, Matthew; Nishinaka, Takahiro
  • Journal of High Energy Physics, Vol. 2017, Issue 9
  • DOI: 10.1007/JHEP09(2017)066

Enhancement of Supersymmetry via Renormalization Group Flow and the Superconformal Index
journal, April 2017


N = 1 $$ \mathcal{N}=1 $$ deformations and RG flows of N = 2 $$ \mathcal{N}=2 $$ SCFTs
journal, February 2017

  • Maruyoshi, Kazunobu; Song, Jaewon
  • Journal of High Energy Physics, Vol. 2017, Issue 2
  • DOI: 10.1007/JHEP02(2017)075

N $$ \mathcal{N} $$ =1 Deformations and RG flows of N $$ \mathcal{N} $$ =2 SCFTs, part II: non-principal deformations
journal, December 2016

  • Agarwal, Prarit; Maruyoshi, Kazunobu; Song, Jaewon
  • Journal of High Energy Physics, Vol. 2016, Issue 12
  • DOI: 10.1007/JHEP12(2016)103

S-duality and 2d Topological QFT
text, January 2009


Framed BPS States
preprint, January 2010


Spectral networks
text, January 2012


Schur Indices, BPS Particles, and Argyres-Douglas Theories
text, January 2015


BPS Graphs: From Spectral Networks to BPS Quivers
text, January 2017


An Index for 4 dimensional Super Conformal Theories
text, January 2005


A New Supersymmetric Index
text, January 1992


Phases of $N=2$ Theories In Two Dimensions
text, January 1993


New N=2 Superconformal Field Theories in Four Dimensions
text, January 1995


Branes and N=2 Theories in Two Dimensions
text, January 1997


Works referencing / citing this record:

Testing Macdonald index as a refined character of chiral algebra
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