Surface defect indices and 2d-4d BPS states

Córdova, Clay; Gaiotto, Davide; Shao, Shu-Heng

doi:10.1007/jhep12(2017)078

Title: Surface defect indices and 2d-4d BPS states

Abstract

We conjecture a formula for the Schur index of four-dimensional N=2 theories coupled to (2, 2) surface defects in terms of the 2d-4d BPS spectrum in the Coulomb phase of the theory. The key ingredient in our conjecture is a refined 2d-4d wall-crossing invariant, which we also formulate. Our result intertwines recent conjectures expressing the four-dimensional Schur index in terms of infrared BPS particles, with the Cecotti-Vafa formula for limits of the elliptic genus in terms of two-dimensional BPS solitons. We extend our discussion to framed 2d-4d BPS states, and use this to demonstrate a general relationship between surface defect indices and line defect indices. We illustrate our results in the example of su(2) super Yang-Mills coupled to the CP¹ sigma model defect.

Authors:

Córdova, Clay ^[1]; Gaiotto, Davide ^[2]; Shao, Shu-Heng ^[1]

Inst. for Advanced Study, Princeton, NJ (United States). School of Natural Sciences
Perimeter Inst. for Theoretical Physics, Waterloo, ON (Canada)

Publication Date:: Thu Dec 14 00:00:00 EST 2017

Research Org.:: Institute for Advanced Study, Princeton, NJ (United States)

Sponsoring Org.:: USDOE Office of Science (SC)

OSTI Identifier:: 1502495

Grant/Contract Number:: SC0009988

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: 2017; Journal Issue: 12; Journal ID: ISSN 1029-8479

Publisher:: Springer Berlin

Country of Publication:: United States

Language:: English

Subject:: 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Conformal and W Symmetry; Supersymmetric Gauge Theory

Citation Formats


                    Córdova, Clay, Gaiotto, Davide, and Shao, Shu-Heng. Surface defect indices and 2d-4d BPS states.  United States: N. p., 2017. 
Web.  doi:10.1007/jhep12(2017)078.

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                    Córdova, Clay, Gaiotto, Davide, & Shao, Shu-Heng. Surface defect indices and 2d-4d BPS states.  United States.  https://doi.org/10.1007/jhep12(2017)078

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                    Córdova, Clay, Gaiotto, Davide, and Shao, Shu-Heng. Thu .  
"Surface defect indices and 2d-4d BPS states".  United States.  https://doi.org/10.1007/jhep12(2017)078.  https://www.osti.gov/servlets/purl/1502495.

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@article{osti_1502495,

  title        = {Surface defect indices and 2d-4d BPS states},

  author       = {Córdova, Clay and Gaiotto, Davide and Shao, Shu-Heng},

  abstractNote = {We conjecture a formula for the Schur index of four-dimensional N=2 theories coupled to (2, 2) surface defects in terms of the 2d-4d BPS spectrum in the Coulomb phase of the theory. The key ingredient in our conjecture is a refined 2d-4d wall-crossing invariant, which we also formulate. Our result intertwines recent conjectures expressing the four-dimensional Schur index in terms of infrared BPS particles, with the Cecotti-Vafa formula for limits of the elliptic genus in terms of two-dimensional BPS solitons. We extend our discussion to framed 2d-4d BPS states, and use this to demonstrate a general relationship between surface defect indices and line defect indices. We illustrate our results in the example of su(2) super Yang-Mills coupled to the CP1 sigma model defect.},

  doi          = {10.1007/jhep12(2017)078},

  journal      = {Journal of High Energy Physics (Online)},

  number       = 12,

  volume       = 2017,

  place        = {United States},

  year         = {Thu Dec 14 00:00:00 EST 2017},

  month        = {Thu Dec 14 00:00:00 EST 2017}

}

Copy to clipboard

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https://doi.org/10.1007/jhep12(2017)078

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Cited by: 26 works

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Figures / Tables:

Figure 1: The unwrapping process of a surface defect to a sum of lines. Starting in the upper left we have a surface defect $\mathbb{S}$ (shown in blue) wrapping an equator in $S$³ as well as the $S$¹ which is not shown. The defect is cut open using a resolutionmore »

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Journal of High Energy Physics, Vol. 2019, Issue 2
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Journal of High Energy Physics, Vol. 2020, Issue 2
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Logarithmic W-algebras and Argyres-Douglas theories at higher rank
journal, November 2018

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Journal of High Energy Physics, Vol. 2018, Issue 11
DOI: 10.1007/jhep11(2018)188

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text, January 2018

Hayling, Joseph; Niarchos, Vasilis; Papageorgakis, Constantinos
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DOI: 10.48550/arxiv.1809.10485

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Title: Surface defect indices and 2d-4d BPS states

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

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

2d index and surface operators
journal, March 2014

Conformal manifolds in four dimensions and chiral algebras
journal, October 2016

Infrared computations of defect Schur indices
journal, November 2016

Refined, Motivic, and Quantum
journal, November 2009

On the Landau-Ginzburg Description of n=2 Minimal Models
journal, October 1994

Spectral Networks with Spin
journal, August 2015

Quantum Holonomies from Spectral Networks and Framed BPS States
journal, August 2016

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

Chiral algebras for trinion theories
journal, February 2015

Bubbling surface operators and S-duality
journal, June 2007

On 4d rank-one N = 3 $$ \mathcal{N}=3 $$ superconformal field theories
journal, September 2016

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

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

Semiclassical framed BPS states
journal, July 2016

Elliptic Genera of Two-Dimensional $${\mathcal{N} = 2}$$ N = 2 Gauge Theories with Rank-One Gauge Groups
journal, November 2013

ADE spectral networks
journal, August 2016

Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes
journal, October 1994

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

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

Chiral algebras of class S $$ \mathcal{S} $$
journal, May 2015

Framed BPS states, moduli dynamics, and wall-crossing
journal, April 2011

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

Framed BPS states
journal, January 2013

Y -systems, Q -systems, and 4D $\mathcal {N}=2$ supersymmetric QFT
journal, November 2014

Rigid Surface Operators
journal, January 2010

Bootstrapping the superconformal index with surface defects
journal, January 2013

Bubbling Surface Operators And S-Duality
text, January 2007

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

Framed BPS States
preprint, January 2010

Spectral networks
text, January 2012

Bootstrapping the superconformal index with surface defects
text, January 2012

BPS Degeneracies and Superconformal Index in Diverse Dimensions
text, January 2012

2d Index and Surface operators
text, January 2013

Infinite Chiral Symmetry in Four Dimensions
text, January 2013

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

Semiclassical framed BPS states
text, January 2015

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

A New Supersymmetric Index
text, January 1992

Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes
text, January 1993

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

Spectral Networks with Spin
journal, August 2015

A New Supersymmetric Index
text, January 1992

Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes
text, January 1993

Deconstructing defects
journal, February 2019

Testing Macdonald index as a refined character of chiral algebra
journal, February 2020

Logarithmic W-algebras and Argyres-Douglas theories at higher rank
journal, November 2018

Deconstructing Defects
text, January 2018