K-decompositions and 3d gauge theories

Dimofte, Tudor; Gabella, Maxime; Goncharov, Alexander B.

doi:10.1007/JHEP11(2016)151

Title: K-decompositions and 3d gauge theories

Abstract

This paper combines several new constructions in mathematics and physics. Mathematically, we study framed flat PGL(K, C)-connections on a large class of 3-manifolds M with boundary. We introduce a moduli space $$\mathcal{L}$$_K(M) of framed flat connections on the boundary ∂M that extend to M. Our goal is to understand an open part of $$\mathcal{L}$$_K(M) as a Lagrangian subvariety in the symplectic moduli space X^un_K(∂M) of framed flat connections on the boundary — and more so, as a “K₂-Lagrangian,” meaning that the K₂-avatar of the symplectic form restricts to zero. We construct an open part of $$\mathcal{L}$$_K(M) from elementary data associated with the hypersimplicial K-decomposition of an ideal triangulation of M, in a way that generalizes (and combines) both Thurston’s gluing equations in 3d hyperbolic geometry and the cluster coordinates for framed flat PGL(K, C)-connections on surfaces. By using a canonical map from the complex of configurations of decorated flags to the Bloch complex, we prove that any generic component of $$\mathcal{L}$$_K(M) is K₂-isotropic as long as ∂M satisfies certain topological constraints (theorem 4.2). In some cases this easily implies that $$\mathcal{L}$$_K(M) is K₂-Lagrangian. For general M, we extend a classic result of Neumann and Zagier on symplectic properties of PGL(2) gluing equations to reduce the K₂-Lagrangian property to a combinatorial statement. Physically, we translate the K-decomposition of an ideal triangulation of M and its symplectic properties to produce an explicit construction of 3d N = 2 superconformal field theories T_K [M] resulting (conjecturally) from the compactification of K M5-branes on M. This extends known constructions for K = 2. Just as for K = 2, the theories T_K [M] are described as IR fixed points of abelian Chern-Simons-matter theories. Changes of triangulation (2-3 moves) lead to abelian mirror symmetries that are all generated by the elementary duality between N_f = 1 SQED and the XYZ model. In the large K limit, we find evidence that the degrees of freedom of T_K [M] grow cubically in K.

Authors:

Dimofte, Tudor ^[1]; Gabella, Maxime ^[2]; Goncharov, Alexander B. ^[3]

Inst. for Advanced Study, Princeton, NJ (United States); Univ. of California, Davis, CA (United States). Dept. of Mathematics. Center for Quantum Mathematics and Physics
Inst. for Advanced Study, Princeton, NJ (United States); Alternative Energies and Atomic Energy Commission (CEA), Gif-sur-Yvette (France). Inst. of Theoretical Physics
Yale Univ., New Haven, CT (United States). Mathematics Dept.

Publication Date:: Thu Nov 24 00:00:00 EST 2016

Research Org.:: Institute for Advanced Study, Princeton, NJ (United States); Yale Univ., New Haven, CT (United States); Alternative Energies and Atomic Energy Commission (CEA), Gif-sur-Yvette (France)

Sponsoring Org.:: USDOE Office of Science (SC), High Energy Physics (HEP); National Science Foundation (NSF); European Research Council (ERC)

Contributing Org.:: Univ. of California, Davis, CA (United States)

OSTI Identifier:: 1368083

Grant/Contract Number:: FG02-90ER40542; DMS-1059129; DMS-1301776; 259133

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: 2016; Journal Issue: 11; Journal ID: ISSN 1029-8479

Publisher:: Springer Berlin

Country of Publication:: United States

Language:: English

Subject:: 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Differential and Algebraic Geometry; Supersymmetric gauge theory; Supersymmetry and Duality

Citation Formats


                    Dimofte, Tudor, Gabella, Maxime, and Goncharov, Alexander B. K-decompositions and 3d gauge theories.  United States: N. p., 2016. 
Web.  doi:10.1007/JHEP11(2016)151.

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                    Dimofte, Tudor, Gabella, Maxime, & Goncharov, Alexander B. K-decompositions and 3d gauge theories.  United States.  https://doi.org/10.1007/JHEP11(2016)151

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                    Dimofte, Tudor, Gabella, Maxime, and Goncharov, Alexander B. Thu .  
"K-decompositions and 3d gauge theories".  United States.  https://doi.org/10.1007/JHEP11(2016)151.  https://www.osti.gov/servlets/purl/1368083.

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@article{osti_1368083,

  title        = {K-decompositions and 3d gauge theories},

  author       = {Dimofte, Tudor and Gabella, Maxime and Goncharov, Alexander B.},

  abstractNote = {This paper combines several new constructions in mathematics and physics. Mathematically, we study framed flat PGL(K, C)-connections on a large class of 3-manifolds M with boundary. We introduce a moduli space $\mathcal{L}$K(M) of framed flat connections on the boundary ∂M that extend to M. Our goal is to understand an open part of $\mathcal{L}$K(M) as a Lagrangian subvariety in the symplectic moduli space XunK(∂M) of framed flat connections on the boundary — and more so, as a “K2-Lagrangian,” meaning that the K2-avatar of the symplectic form restricts to zero. We construct an open part of $\mathcal{L}$K(M) from elementary data associated with the hypersimplicial K-decomposition of an ideal triangulation of M, in a way that generalizes (and combines) both Thurston’s gluing equations in 3d hyperbolic geometry and the cluster coordinates for framed flat PGL(K, C)-connections on surfaces. By using a canonical map from the complex of configurations of decorated flags to the Bloch complex, we prove that any generic component of $\mathcal{L}$K(M) is K2-isotropic as long as ∂M satisfies certain topological constraints (theorem 4.2). In some cases this easily implies that $\mathcal{L}$K(M) is K2-Lagrangian. For general M, we extend a classic result of Neumann and Zagier on symplectic properties of PGL(2) gluing equations to reduce the K2-Lagrangian property to a combinatorial statement. Physically, we translate the K-decomposition of an ideal triangulation of M and its symplectic properties to produce an explicit construction of 3d N = 2 superconformal field theories TK [M] resulting (conjecturally) from the compactification of K M5-branes on M. This extends known constructions for K = 2. Just as for K = 2, the theories TK [M] are described as IR fixed points of abelian Chern-Simons-matter theories. Changes of triangulation (2-3 moves) lead to abelian mirror symmetries that are all generated by the elementary duality between Nf = 1 SQED and the XYZ model. In the large K limit, we find evidence that the degrees of freedom of TK [M] grow cubically in K.},

  doi          = {10.1007/JHEP11(2016)151},

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

  number       = 11,

  volume       = 2016,

  place        = {United States},

  year         = {Thu Nov 24 00:00:00 EST 2016},

  month        = {Thu Nov 24 00:00:00 EST 2016}

}

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DOI: 10.48550/arxiv.1204.4824

Contact Terms, Unitarity, and F-Maximization in Three-Dimensional Superconformal Theories
text, January 2012

Closset, Cyril; Dumitrescu, Thomas T.; Festuccia, Guido
arXiv
DOI: 10.48550/arxiv.1205.4142

Comments on Chern-Simons Contact Terms in Three Dimensions
text, January 2012

Closset, Cyril; Dumitrescu, Thomas T.; Festuccia, Guido
arXiv
DOI: 10.48550/arxiv.1206.5218

3d analogs of Argyres-Douglas theories and knot homologies
text, January 2012

Fuji, Hiroyuki; Gukov, Sergei; Stosic, Marko
arXiv
DOI: 10.48550/arxiv.1209.1416

Holomorphic Blocks in Three Dimensions
text, January 2012

Beem, Christopher; Dimofte, Tudor; Pasquetti, Sara
arXiv
DOI: 10.48550/arxiv.1211.1986

Tangles, Generalized Reidemeister Moves, and Three-Dimensional Mirror Symmetry
preprint, January 2012

Cordova, Clay; Espahbodi, Sam; Haghighat, Babak
arXiv
DOI: 10.48550/arxiv.1211.3730

RG Domain Walls and Hybrid Triangulations
preprint, January 2013

Dimofte, Tudor; Gaiotto, Davide; van der Veen, Roland
arXiv
DOI: 10.48550/arxiv.1304.6721

3d Chern-Simons Theory from M5-branes
text, January 2013

Lee, Sungjay; Yamazaki, Masahito
arXiv
DOI: 10.48550/arxiv.1305.2429

On the 6d origin of discrete additional data of 4d gauge theories
text, January 2013

Tachikawa, Yuji
arXiv
DOI: 10.48550/arxiv.1309.0697

3d-3d Correspondence Revisited
text, January 2014

Chung, Hee-Joong; Dimofte, Tudor; Gukov, Sergei
arXiv
DOI: 10.48550/arxiv.1405.3663

Complex Chern-Simons theory at level k via the 3d-3d correspondence
preprint, January 2014

Dimofte, Tudor
arXiv
DOI: 10.48550/arxiv.1409.0857

Aspects of Defects in 3d-3d Correspondence
text, January 2015

Gang, Dongmin; Kim, Nakwoo; Romo, Mauricio
arXiv
DOI: 10.48550/arxiv.1510.05011

A Schlafli-type formula for convex cores of hyperbolic 3-manifolds
preprint, January 1997

Bonahon, Francis
arXiv
DOI: 10.48550/arxiv.dg-ga/9704017

Electric-Magnetic Duality And The Geometric Langlands Program
preprint, January 2006

Kapustin, Anton; Witten, Edward
arXiv
DOI: 10.48550/arxiv.hep-th/0604151

Mirror Symmetry in Three Dimensional Gauge Theories
text, January 1996

Intriligator, K.; Seiberg, N.
arXiv
DOI: 10.48550/arxiv.hep-th/9607207

Aspects of N=2 Supersymmetric Gauge Theories in Three Dimensions
text, January 1997

Aharony, O.; Hanany, A.; Intriligator, K.
arXiv
DOI: 10.48550/arxiv.hep-th/9703110

Solutions Of Four-Dimensional Field Theories Via M Theory
text, January 1997

Witten, Edward
arXiv
DOI: 10.48550/arxiv.hep-th/9703166

The Holographic Weyl anomaly
text, January 1998

Henningson, Mans; Skenderis, Kostas
arXiv
DOI: 10.48550/arxiv.hep-th/9806087

Moduli spaces of local systems and higher Teichmuller theory
preprint, January 2003

Fock, V. V.; Goncharov, A. B.
arXiv
DOI: 10.48550/arxiv.math/0311149

The quantum dilogarithm and representations quantum cluster varieties
text, January 2007

Fock, V. V.; Goncharov, A. B.
arXiv
DOI: 10.48550/arxiv.math/0702397

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preprint, January 1998

Frohman, Charles; Gelca, Razvan; Lofaro, Walter
arXiv
DOI: 10.48550/arxiv.math/9812048

The colored Jones polynomials and the simplicial volume of a knot
preprint, January 1999

Murakami, Hitoshi; Murakami, Jun
arXiv
DOI: 10.48550/arxiv.math/9905075

Works referencing / citing this record:

All-Order Volume Conjecture for Closed 3-Manifolds from Complex Chern–Simons Theory
journal, March 2018

Gang, Dongmin; Romo, Mauricio; Yamazaki, Masahito
Communications in Mathematical Physics, Vol. 359, Issue 3
DOI: 10.1007/s00220-018-3115-y

Complex Chern-Simons from M5-branes on the Squashed Three-Sphere
preprint, January 2013

Cordova, Clay; Jafferis, Daniel L.
arXiv
DOI: 10.48550/arxiv.1305.2891

All-Order Volume Conjecture for Closed 3-Manifolds from Complex Chern-Simons Theory
text, January 2017

Gang, Dongmin; Romo, Mauricio; Yamazaki, Masahito
arXiv
DOI: 10.48550/arxiv.1704.00918

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

Gabella, Maxime; Longhi, Pietro; Park, Chan Y.
arXiv
DOI: 10.48550/arxiv.1704.04204

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Title: K-decompositions and 3d gauge theories

Abstract

Citation Formats

Spectral Networks journal, March 2013

N = 2 dualities journal, August 2012

The A-polynomial from the noncommutative viewpoint journal, October 2001

The exact superconformal R-symmetry extremizes Z journal, May 2012

Aspects of N = 2 supersymmetric gauge theories in three dimensions journal, August 1997

A-polynomial, B-model, and quantization journal, February 2012

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The colored Jones function is q-holonomic journal, January 2005

Volumes of hyperbolic three-manifolds journal, January 1985

Spontaneous compactification of seven-dimensional supergravity theories journal, September 1985

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

Exact results for Wilson loops in superconformal Chern-Simons theories with matter journal, March 2010

$S$-duality of boundary conditions in ${\mathcal N}=4$ super Yang-Mills theory journal, January 2009

A 3d-3d appetizer journal, November 2016

Quantum Riemann surfaces in Chern-Simons theory journal, January 2013

The quantum dilogarithm and representations of quantum cluster varieties journal, September 2008

The colored Jones polynomials and the simplicial volume of a knot journal, January 2001

Electric-magnetic duality and the geometric Langlands program journal, January 2007

Index for three dimensional superconformal field theories with general R-charge assignments journal, April 2011

A quantum Teichmüller space journal, September 1999

The ?-invariant of hyperbolic 3-manifolds journal, October 1985

Gauge Theories Labelled by Three-Manifolds journal, December 2013

Combinatorial computation of characteristic classes journal, April 1975

Fivebranes and knots journal, January 2012

On the characteristic and deformation varieties of a knot conference, September 2004

Mahler’s Measure and the Dilogarithm (I) journal, June 2002

Combinatorics of Triangulations and the Chern-Simons Invariant for Hyperbolic 3-Manifolds book, December 1992

Tangles, generalized Reidemeister moves, and three-dimensional mirror symmetry journal, May 2014

A TQFT from Quantum Teichmüller Theory journal, June 2014

Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial journal, March 2005

Contact terms, unitarity, and F -maximization in three-dimensional superconformal theories journal, October 2012

Comments on Chern-Simons contact terms in three dimensions journal, September 2012

Plane curves associated to character varieties of 3-manifolds journal, December 1994

Moduli spaces of local systems and higher Teichmüller theory journal, June 2006

Mirror symmetry in three-dimensional gauge theories, and D-brane moduli spaces journal, May 1997

The quantum content of the gluing equations journal, January 2013

Gauge Noninvariance and Parity Nonconservation of Three-Dimensional Fermions journal, January 1984

Framed BPS states journal, January 2013

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

Invariants of algebraic curves and topological expansion journal, January 2007

Spectral Networks and Snakes journal, March 2013

The gravity dual of supersymmetric gauge theories on a squashed three-sphere journal, November 2012

Dynamics of N = 2 supersymmetric gauge theories in three dimensions journal, September 1997

The volume and Chern-Simons invariant of a representation journal, December 2009

On the 6d origin of discrete additional data of 4d gauge theories journal, May 2014

Non-abelian tensor-multiplet anomalies journal, September 1998

Towards the F-theorem: $ \mathcal{N} = 2 $ field theories on the three-sphere journal, June 2011

A Schläfli-type formula for convex cores of hyperbolic 3-manifolds journal, January 1998

SUSY gauge theories on squashed three-spheres journal, May 2011

Knot invariants and topological strings journal, June 2000

Supersymmetric Vacua and Bethe Ansatz journal, July 2009

Mirror symmetry in three dimensional gauge theories journal, October 1996

:{unav) journal, January 1997

Aspects of defects in 3d-3d correspondence journal, October 2016

Exactly marginal deformations and global symmetries journal, June 2010

3d-3d correspondence revisited journal, April 2016

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

$ {\text{SL}}\left( {2,\mathbb{R}} \right) $ Chern-Simons, Liouville, and gauge theory on duality walls journal, August 2011

Exact results for perturbative Chern–Simons theory with complex gauge group journal, January 2009

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

Super-A-polynomial for knots and BPS states journal, February 2013

Holographic Uniformization journal, February 2013

Analytic continuation of Chern-Simons theory book, January 2010

Vortex Counting and Lagrangian 3-Manifolds journal, October 2011

S 3/Z n partition function and dualities journal, November 2012

2 + 1 dimensional gravity as an exactly soluble system journal, December 1988

Geometry of Configurations, Polylogarithms, and Motivic Cohomology journal, September 1995

The holographic Weyl anomaly journal, July 1998

3d Chern-Simons theory from M5-branes journal, December 2013

The gravity duals of $ \mathcal{N}=2 $ superconformal field theories journal, October 2012

RG domain walls and hybrid triangulations journal, January 2015

Volumes of hyperbolic manifolds and mixed Tate motives journal, January 1999

The Classical Polylogarithms, Algebraic K-Theory And ζ F (n) book, January 1993

The volume and Chern-Simons invariant of a representation text, January 2007

Supersymmetric vacua and Bethe ansatz text, January 2009

Exact Results for Perturbative Chern-Simons Theory with Complex Gauge Group text, January 2009

Liouville Correlation Functions from Four-dimensional Gauge Theories text, January 2009

Spectral Networks
journal, March 2013

N = 2 dualities
journal, August 2012

The A-polynomial from the noncommutative viewpoint
journal, October 2001

The exact superconformal R-symmetry extremizes Z
journal, May 2012

Aspects of N = 2 supersymmetric gauge theories in three dimensions
journal, August 1997

A-polynomial, B-model, and quantization
journal, February 2012

Complex Chern–Simons Theory at Level k via the 3d–3d Correspondence
journal, June 2015

The colored Jones function is q-holonomic
journal, January 2005

Volumes of hyperbolic three-manifolds
journal, January 1985

Spontaneous compactification of seven-dimensional supergravity theories
journal, September 1985

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

Exact results for Wilson loops in superconformal Chern-Simons theories with matter
journal, March 2010

$S$-duality of boundary conditions in ${\mathcal N}=4$ super Yang-Mills theory
journal, January 2009

A 3d-3d appetizer
journal, November 2016

Quantum Riemann surfaces in Chern-Simons theory
journal, January 2013

The quantum dilogarithm and representations of quantum cluster varieties
journal, September 2008

The colored Jones polynomials and the simplicial volume of a knot
journal, January 2001

Electric-magnetic duality and the geometric Langlands program
journal, January 2007

Index for three dimensional superconformal field theories with general R-charge assignments
journal, April 2011

A quantum Teichmüller space
journal, September 1999

The ?-invariant of hyperbolic 3-manifolds
journal, October 1985

Gauge Theories Labelled by Three-Manifolds
journal, December 2013

Combinatorial computation of characteristic classes
journal, April 1975

Fivebranes and knots
journal, January 2012

On the characteristic and deformation varieties of a knot
conference, September 2004

Mahler’s Measure and the Dilogarithm (I)
journal, June 2002

Combinatorics of Triangulations and the Chern-Simons Invariant for Hyperbolic 3-Manifolds
book, December 1992

Tangles, generalized Reidemeister moves, and three-dimensional mirror symmetry
journal, May 2014

A TQFT from Quantum Teichmüller Theory
journal, June 2014

Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial
journal, March 2005

Contact terms, unitarity, and F -maximization in three-dimensional superconformal theories
journal, October 2012

Comments on Chern-Simons contact terms in three dimensions
journal, September 2012

Plane curves associated to character varieties of 3-manifolds
journal, December 1994

Moduli spaces of local systems and higher Teichmüller theory
journal, June 2006

Mirror symmetry in three-dimensional gauge theories, and D-brane moduli spaces
journal, May 1997

The quantum content of the gluing equations
journal, January 2013

Gauge Noninvariance and Parity Nonconservation of Three-Dimensional Fermions
journal, January 1984

Framed BPS states
journal, January 2013

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

Invariants of algebraic curves and topological expansion
journal, January 2007

Spectral Networks and Snakes
journal, March 2013

The gravity dual of supersymmetric gauge theories on a squashed three-sphere
journal, November 2012

Dynamics of N = 2 supersymmetric gauge theories in three dimensions
journal, September 1997

The volume and Chern-Simons invariant of a representation
journal, December 2009

On the 6d origin of discrete additional data of 4d gauge theories
journal, May 2014

Non-abelian tensor-multiplet anomalies
journal, September 1998

Towards the F-theorem: $ \mathcal{N} = 2 $ field theories on the three-sphere
journal, June 2011

A Schläfli-type formula for convex cores of hyperbolic 3-manifolds
journal, January 1998

SUSY gauge theories on squashed three-spheres
journal, May 2011

Knot invariants and topological strings
journal, June 2000

Supersymmetric Vacua and Bethe Ansatz
journal, July 2009

Mirror symmetry in three dimensional gauge theories
journal, October 1996

:{unav)
journal, January 1997

Aspects of defects in 3d-3d correspondence
journal, October 2016

Exactly marginal deformations and global symmetries
journal, June 2010

3d-3d correspondence revisited
journal, April 2016

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

$ {\text{SL}}\left( {2,\mathbb{R}} \right) $ Chern-Simons, Liouville, and gauge theory on duality walls
journal, August 2011

Exact results for perturbative Chern–Simons theory with complex gauge group
journal, January 2009

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

Super-A-polynomial for knots and BPS states
journal, February 2013

Holographic Uniformization
journal, February 2013

Analytic continuation of Chern-Simons theory
book, January 2010

Vortex Counting and Lagrangian 3-Manifolds
journal, October 2011

S 3/Z n partition function and dualities
journal, November 2012

2 + 1 dimensional gravity as an exactly soluble system
journal, December 1988

Geometry of Configurations, Polylogarithms, and Motivic Cohomology
journal, September 1995

The holographic Weyl anomaly
journal, July 1998

3d Chern-Simons theory from M5-branes
journal, December 2013

The gravity duals of $ \mathcal{N}=2 $ superconformal field theories
journal, October 2012

RG domain walls and hybrid triangulations
journal, January 2015

Volumes of hyperbolic manifolds and mixed Tate motives
journal, January 1999

The Classical Polylogarithms, Algebraic K-Theory And ζ F (n)
book, January 1993

The volume and Chern-Simons invariant of a representation
text, January 2007

Supersymmetric vacua and Bethe ansatz
text, January 2009

Exact Results for Perturbative Chern-Simons Theory with Complex Gauge Group
text, January 2009

Liouville Correlation Functions from Four-dimensional Gauge Theories
text, January 2009

Exact Results for Wilson Loops in Superconformal Chern-Simons Theories with Matter
text, January 2009

Framed BPS States
preprint, January 2010

Vortex Counting and Lagrangian 3-manifolds
text, January 2010