Chiral algebra, localization, modularity, surface defects, and all that

Dedushenko, Mykola; Fluder, Martin

doi:10.1063/5.0002661

Title: Chiral algebra, localization, modularity, surface defects, and all that

Abstract

We study the 2D vertex operator algebra (VOA) construction in 4D N = 2 superconformal field theories on S³ × S¹, focusing on both old puzzles and new observations. The VOA lives on a two-torus $T^{2} \subset S^{3} \times S^{1}$ , it is $\frac{1}{2} Z$ -graded, and this torus is equipped with the natural choice of spin structure (1,0) for the $Z + \frac{1}{2}$ -graded operators, corresponding to the NS sector vacuum character. By analyzing the possible refinements of the Schur index that preserves the VOA, we find that it admits discrete deformations, which allows access to the remaining spin structures (1,1), (0,1), and (0,0), of which the latter two involve the inclusion of a particular surface defect. For Lagrangian theories, we perform the detailed analysis: we describe the natural supersymmetric background, perform localization, and derive the gauged symplectic boson action on a torus in any spin structure. In the absence of flavor fugacities, the 2D and 4D path integrals precisely match, including the Casimir factors. We further analyze the 2D theory: we identify its integration cycle and the two-point functions and interpret flavor holonomies as screening charges in the VOA. Next, we make some observations about modularity; the T-transformation acts on our four partition functions and lifts to a large diffeomorphism on S³ × S¹. More interestingly, we generalize the four partition functions on the torus to an infinite family labeled by both the spin structure and the integration cycle inside the complexified maximal torus of the gauge group. Members of this family transform into one another under the full modular group, and we confirm the recent observation that the S-transform of the Schur index in Lagrangian theories exhibits logarithmic behavior. Finally, we comment on how locally our background reproduces the Ω-background.

Authors:

^[1]; Fluder, Martin ^[2]

California Institute of Technology (CalTech), Pasadena, CA (United States)
California Institute of Technology (CalTech), Pasadena, CA (United States); Univ. of Tokyo (Japan). Kavli Institute for the Physics and Mathematics of the Universe (WPI)

Publication Date:: Fri Sep 11 00:00:00 EDT 2020

Research Org.:: California Institute of Technology (CalTech), Pasadena, CA (United States)

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

OSTI Identifier:: 1802219

Alternate Identifier(s):: OSTI ID: 1659533

Grant/Contract Number:: SC0011632; de-sc0011632

Resource Type:: Accepted Manuscript

Journal Name:: Journal of Mathematical Physics

Additional Journal Information:: Journal Volume: 61; Journal Issue: 9; Journal ID: ISSN 0022-2488

Publisher:: American Institute of Physics (AIP)

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; Physics; Gauge group; Quantum field theory; Differential geometry; Algebraic structures; Conformal algebra; Superconformal algebras; Lagrangian field theories; Particle symmetry; Supersymmetric gauge theory; Operator product expansion

Citation Formats


                    Dedushenko, Mykola, and Fluder, Martin. Chiral algebra, localization, modularity, surface defects, and all that.  United States: N. p., 2020. 
Web.  doi:10.1063/5.0002661.

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                    Dedushenko, Mykola, & Fluder, Martin. Chiral algebra, localization, modularity, surface defects, and all that.  United States.  https://doi.org/10.1063/5.0002661

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                    Dedushenko, Mykola, and Fluder, Martin. Fri .  
"Chiral algebra, localization, modularity, surface defects, and all that".  United States.  https://doi.org/10.1063/5.0002661.  https://www.osti.gov/servlets/purl/1802219.

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@article{osti_1802219,

  title        = {Chiral algebra, localization, modularity, surface defects, and all that},

  author       = {Dedushenko, Mykola and Fluder, Martin},

  abstractNote = {We study the 2D vertex operator algebra (VOA) construction in 4D N = 2 superconformal field theories on S3 × S1, focusing on both old puzzles and new observations. The VOA lives on a two-torus T 2 ⊂ S 3 × S 1 , it is 1 2 Z -graded, and this torus is equipped with the natural choice of spin structure (1,0) for the Z + 1 2 -graded operators, corresponding to the NS sector vacuum character. By analyzing the possible refinements of the Schur index that preserves the VOA, we find that it admits discrete deformations, which allows access to the remaining spin structures (1,1), (0,1), and (0,0), of which the latter two involve the inclusion of a particular surface defect. For Lagrangian theories, we perform the detailed analysis: we describe the natural supersymmetric background, perform localization, and derive the gauged symplectic boson action on a torus in any spin structure. In the absence of flavor fugacities, the 2D and 4D path integrals precisely match, including the Casimir factors. We further analyze the 2D theory: we identify its integration cycle and the two-point functions and interpret flavor holonomies as screening charges in the VOA. Next, we make some observations about modularity; the T-transformation acts on our four partition functions and lifts to a large diffeomorphism on S3 × S1. More interestingly, we generalize the four partition functions on the torus to an infinite family labeled by both the spin structure and the integration cycle inside the complexified maximal torus of the gauge group. Members of this family transform into one another under the full modular group, and we confirm the recent observation that the S-transform of the Schur index in Lagrangian theories exhibits logarithmic behavior. Finally, we comment on how locally our background reproduces the Ω-background.},

  doi          = {10.1063/5.0002661},

  journal      = {Journal of Mathematical Physics},

  number       = 9,

  volume       = 61,

  place        = {United States},

  year         = {Fri Sep 11 00:00:00 EDT 2020},

  month        = {Fri Sep 11 00:00:00 EDT 2020}

}

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DOI: 10.1007/JHEP07(2014)137

Gauge theory, ramification, and the geometric Langlands program
journal, January 2006

Gukov, S.; Witten, E.
Current Developments in Mathematics, Vol. 2006, Issue 1
DOI: 10.4310/CDM.2006.v2006.n1.a2

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Title: Chiral algebra, localization, modularity, surface defects, and all that

Abstract

Citation Formats

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

Vertex operator algebras of Argyres-Douglas theories from M5-branes journal, December 2017

Chiral algebras from Ω-deformation journal, August 2019

Gauge Theories and Macdonald Polynomials journal, November 2012

Extended supersymmetry on curved spaces journal, October 2013

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

Four-dimensional Lens Space Index from Two-dimensional Chiral Algebra text, January 2017

The supersymmetric index in four dimensions journal, October 2017

Hidden exceptional symmetry in the pure spinor superstring journal, January 2020

Supersymmetric Casimir energy and the anomaly polynomial journal, September 2015

Surface defects and chiral algebras journal, May 2017

Boundaries, Mirror Symmetry, and Symplectic Duality in 3d $\mathcal{N}=4$ Gauge Theory text, January 2016

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

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

Exact correlators of BPS Operators from the 3d superconformal bootstrap journal, March 2015

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

Vortices and Vermas journal, January 2018

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

Lagrangians of N = 2 supergravity-matter systems journal, January 1985

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

IR duality in 2D N = ( 0 , 2 ) gauge theory with noncompact dynamics journal, March 2019

Infinite Chiral Symmetry in Four Dimensions journal, January 2015

The Coulomb Branch of 3d $${\mathcal{N}= 4}$$ N = 4 Theories journal, June 2017

Chiral algebras, localization and surface defects journal, February 2018

On the superconformal index of $ \mathcal{N} = 1 $ IR fixed points. A holographic check journal, March 2011

Twisted compactifications of 3d N$$ \mathcal{N} $$ = 4 theories and conformal blocks journal, February 2019

N=2 S-duality Revisited text, January 2017

The superconformal index of class $ \mathcal{S} $ theories of type D journal, May 2014

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

Vertex Operator Algebras and 3d N $$ \mathcal{N} $$ = 4 gauge theories journal, May 2019

Weyl versus Conformal Invariance in Quantum Field Theory text, January 2017

Argyres-Douglas theories, chiral algebras and wild Hitchin characters journal, January 2018

Infrared computations of defect Schur indices journal, November 2016

Extensions of vertex algebras. Constructions and applications journal, August 2017

Supersymmetric vortex defects in two dimensions journal, January 2018

W $$ \mathcal{W} $$ symmetry in six dimensions journal, May 2015

Elliptic hypergeometric integrals and ’t Hooft anomaly matching conditions journal, June 2012

N = 2 $$ \mathcal{N}=2 $$ central charge bounds from 2d chiral algebras journal, April 2016

Seiberg-Witten theories on ellipsoids journal, September 2012

Nonunitary Lagrangians and Unitary Non-Lagrangian Conformal Field Theories journal, February 2018

Deformation Quantization and Superconformal Symmetry in Three Dimensions journal, February 2017

Extended supersymmetry in the space-time R × S 3 journal, January 1990

4d index to 3d index and 2d topological quantum field theory journal, September 2012

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

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

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

Global Properties of Supersymmetric Theories and the Lens Space journal, July 2014

Geometry of Higgs-branch superconformal primary bundles journal, September 2018

Localization of the four-dimensional $ \mathcal{N}=4 $ SYM to a two-sphere and 1/8 BPS Wilson loops journal, December 2012

Twisting of n=1 susy Gauge Theories and Heterotic Topological Theories journal, December 1995

Transformation rules of N = 2 supergravity multiplets journal, May 1980

Exact results for supersymmetric abelian vortex loops in 2 + 1 dimensions journal, June 2013

Localization on Hopf surfaces journal, August 2014

Chiral algebras for trinion theories journal, February 2015

Elliptic hypergeometric integrals and ’t Hooft anomaly matching conditions journal, June 2012

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

Four-dimensional lens space index from two-dimensional chiral algebra journal, July 2018

The character of the supersymmetric Casimir energy journal, August 2016

Boundaries, mirror symmetry, and symplectic duality in 3d N = 4 $$ \mathcal{N}=4 $$ gauge theory journal, October 2016

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

Macdonald index and chiral algebra journal, August 2017

The omega deformation, branes, integrability and Liouville theory journal, September 2010

Classification of 4d $ \mathcal{N} $ =2 gauge theories journal, December 2013

SCFT/VOA correspondence via Ω-deformation journal, October 2019

${{{\mathcal N}}=2}$ SUSY gauge theories on S 4 journal, October 2017

Topological quantum field theory and N=2 conformal supergravity journal, September 1988

Holonomy groups, complex structures andD=4 topological Yang-Mills theory journal, August 1991

On the chiral algebra of Argyres-Douglas theories and S-duality journal, April 2018

Stress-tensor OPE in N = 2 $$ \mathcal{N}=2 $$ superconformal theories journal, February 2016

Superconformal index, BPS monodromy and chiral algebras journal, November 2017

Bootstrapping the superconformal index with surface defects journal, January 2013

Vertex operator algebras, Higgs branches, and modular differential equations journal, August 2018

Chern-Simons theory from M5-branes and calibrated M2-branes journal, August 2019

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

Deformation Quantization and Superconformal Symmetry in Three Dimensions journal, February 2017

2d index and surface operators journal, March 2014

On S-duality of the superconformal index on lens spaces and 2d TQFT journal, May 2013

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

Vertex operator algebras of Argyres-Douglas theories from M5-branes
journal, December 2017

Chiral algebras from Ω-deformation
journal, August 2019

Gauge Theories and Macdonald Polynomials
journal, November 2012

Extended supersymmetry on curved spaces
journal, October 2013

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

Four-dimensional Lens Space Index from Two-dimensional Chiral Algebra
text, January 2017

The supersymmetric index in four dimensions
journal, October 2017

Hidden exceptional symmetry in the pure spinor superstring
journal, January 2020

Supersymmetric Casimir energy and the anomaly polynomial
journal, September 2015

Surface defects and chiral algebras
journal, May 2017

Boundaries, Mirror Symmetry, and Symplectic Duality in 3d $\mathcal{N}=4$ Gauge Theory
text, January 2016

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

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

Exact correlators of BPS Operators from the 3d superconformal bootstrap
journal, March 2015

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

Vortices and Vermas
journal, January 2018

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

Lagrangians of N = 2 supergravity-matter systems
journal, January 1985

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

IR duality in 2D $N = (0, 2)$ gauge theory with noncompact dynamics
journal, March 2019

Infinite Chiral Symmetry in Four Dimensions
journal, January 2015

The Coulomb Branch of 3d $${\mathcal{N}= 4}$$ N = 4 Theories
journal, June 2017

Chiral algebras, localization and surface defects
journal, February 2018

On the superconformal index of $ \mathcal{N} = 1 $ IR fixed points. A holographic check
journal, March 2011

Twisted compactifications of 3d N$$ \mathcal{N} $$ = 4 theories and conformal blocks
journal, February 2019

N=2 S-duality Revisited
text, January 2017

The superconformal index of class $ \mathcal{S} $ theories of type D
journal, May 2014

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

Vertex Operator Algebras and 3d N $$ \mathcal{N} $$ = 4 gauge theories
journal, May 2019

Weyl versus Conformal Invariance in Quantum Field Theory
text, January 2017

Argyres-Douglas theories, chiral algebras and wild Hitchin characters
journal, January 2018

Infrared computations of defect Schur indices
journal, November 2016

Extensions of vertex algebras. Constructions and applications
journal, August 2017

Supersymmetric vortex defects in two dimensions
journal, January 2018

W $$ \mathcal{W} $$ symmetry in six dimensions
journal, May 2015

Elliptic hypergeometric integrals and ’t Hooft anomaly matching conditions
journal, June 2012

N = 2 $$ \mathcal{N}=2 $$ central charge bounds from 2d chiral algebras
journal, April 2016

Seiberg-Witten theories on ellipsoids
journal, September 2012

Nonunitary Lagrangians and Unitary Non-Lagrangian Conformal Field Theories
journal, February 2018

Deformation Quantization and Superconformal Symmetry in Three Dimensions
journal, February 2017

Extended supersymmetry in the space-time $R \times S^{3}$
journal, January 1990

4d index to 3d index and 2d topological quantum field theory
journal, September 2012

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

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

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

Global Properties of Supersymmetric Theories and the Lens Space
journal, July 2014

Geometry of Higgs-branch superconformal primary bundles
journal, September 2018

Localization of the four-dimensional $ \mathcal{N}=4 $ SYM to a two-sphere and 1/8 BPS Wilson loops
journal, December 2012

Twisting of n=1 susy Gauge Theories and Heterotic Topological Theories
journal, December 1995

Transformation rules of N = 2 supergravity multiplets
journal, May 1980

Exact results for supersymmetric abelian vortex loops in 2 + 1 dimensions
journal, June 2013

Localization on Hopf surfaces
journal, August 2014

Chiral algebras for trinion theories
journal, February 2015

Elliptic hypergeometric integrals and ’t Hooft anomaly matching conditions
journal, June 2012

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

Four-dimensional lens space index from two-dimensional chiral algebra
journal, July 2018

The character of the supersymmetric Casimir energy
journal, August 2016

Boundaries, mirror symmetry, and symplectic duality in 3d N = 4 $$ \mathcal{N}=4 $$ gauge theory
journal, October 2016

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

Macdonald index and chiral algebra
journal, August 2017

The omega deformation, branes, integrability and Liouville theory
journal, September 2010

Classification of 4d $ \mathcal{N} $ =2 gauge theories
journal, December 2013

SCFT/VOA correspondence via Ω-deformation
journal, October 2019

${{{\mathcal N}}=2}$ SUSY gauge theories on S ⁴
journal, October 2017

Topological quantum field theory and N=2 conformal supergravity
journal, September 1988

Holonomy groups, complex structures andD=4 topological Yang-Mills theory
journal, August 1991

On the chiral algebra of Argyres-Douglas theories and S-duality
journal, April 2018

Stress-tensor OPE in N = 2 $$ \mathcal{N}=2 $$ superconformal theories
journal, February 2016

Superconformal index, BPS monodromy and chiral algebras
journal, November 2017

Bootstrapping the superconformal index with surface defects
journal, January 2013

Vertex operator algebras, Higgs branches, and modular differential equations
journal, August 2018

Chern-Simons theory from M5-branes and calibrated M2-branes
journal, August 2019

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

Deformation Quantization and Superconformal Symmetry in Three Dimensions
journal, February 2017

2d index and surface operators
journal, March 2014

On S-duality of the superconformal index on lens spaces and 2d TQFT
journal, May 2013

Vortices and Vermas
journal, January 2018

Exploring curved superspace
journal, August 2012

Structure of N = 2 supergravity
journal, January 1981