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Title: Fivebranes and 3-manifold homology

Abstract

Motivated by physical constructions of homological knot invariants, we study their analogs for closed 3-manifolds. We show that vebrane compacti cations provide a universal description of various old and new homological invariants of 3-manifolds. In terms of 3d/3d correspondence, such invariants are given by the Q-cohomology of the Hilbert space of partially topologically twisted 3d N = 2 theory T[M 3] on a Riemann surface with defects. We demonstrate this by concrete and explicit calculations in the case of monopole/Heegaard Floer homology and a 3-manifold analog of Khovanov-Rozansky link homology. The latter gives a categori cation of Chern-Simons partition function. Finally, some of the new key elements include the explicit form of the S-transform and a novel connection between categori cation and a previously mysterious role of Eichler integrals in Chern-Simons theory.

Authors:
 [1];  [2];  [3]
  1. California Inst. of Technology (CalTech), Pasadena, CA (United States). Walter Burke Inst. for Theoretical Physics
  2. School of Natural Sciences, Inst.for Advanced Study, Princeton, NJ (United States)
  3. Harvard Univ., Cambridge, MA (United States). Jefferson Physical Lab.
Publication Date:
Research Org.:
California Inst. of Technology (CalTech), Pasadena, CA (United States)
Sponsoring Org.:
USDOE; National Science Foundation (NSF)
OSTI Identifier:
1393398
Grant/Contract Number:  
SC0011632; PHY-1067976
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: 7; Journal ID: ISSN 1029-8479
Publisher:
Springer Berlin
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Chern-Simons Theories; Topological Field Theories; M-Theory; Topological Strings

Citation Formats

Gukov, Sergei, Putrov, Pavel, and Vafa, Cumrun. Fivebranes and 3-manifold homology. United States: N. p., 2017. Web. doi:10.1007/JHEP07(2017)071.
Gukov, Sergei, Putrov, Pavel, & Vafa, Cumrun. Fivebranes and 3-manifold homology. United States. doi:10.1007/JHEP07(2017)071.
Gukov, Sergei, Putrov, Pavel, and Vafa, Cumrun. Fri . "Fivebranes and 3-manifold homology". United States. doi:10.1007/JHEP07(2017)071. https://www.osti.gov/servlets/purl/1393398.
@article{osti_1393398,
title = {Fivebranes and 3-manifold homology},
author = {Gukov, Sergei and Putrov, Pavel and Vafa, Cumrun},
abstractNote = {Motivated by physical constructions of homological knot invariants, we study their analogs for closed 3-manifolds. We show that vebrane compacti cations provide a universal description of various old and new homological invariants of 3-manifolds. In terms of 3d/3d correspondence, such invariants are given by the Q-cohomology of the Hilbert space of partially topologically twisted 3d N = 2 theory T[M3] on a Riemann surface with defects. We demonstrate this by concrete and explicit calculations in the case of monopole/Heegaard Floer homology and a 3-manifold analog of Khovanov-Rozansky link homology. The latter gives a categori cation of Chern-Simons partition function. Finally, some of the new key elements include the explicit form of the S-transform and a novel connection between categori cation and a previously mysterious role of Eichler integrals in Chern-Simons theory.},
doi = {10.1007/JHEP07(2017)071},
journal = {Journal of High Energy Physics (Online)},
number = 7,
volume = 2017,
place = {United States},
year = {2017},
month = {7}
}

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Cited by: 10 works
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Works referenced in this record:

Matrix model as a mirror of Chern-Simons theory
journal, February 2004


Notes on the Seiberg-Witten Equations, the Weinstein Conjecture and Embedded Contact Homology
journal, January 2007


Monopoles and lens space surgeries
journal, January 2007


Dualities for 3d theories with tensor matter
journal, December 2011

  • Kapustin, Anton; Kim, Hyungchul; Park, Jaemo
  • Journal of High Energy Physics, Vol. 2011, Issue 12
  • DOI: 10.1007/JHEP12(2011)087

Patterns in Knot Cohomology, I
journal, January 2003


Walls, lines, and spectral dualities in 3d gauge theories
journal, May 2014

  • Gadde, Abhijit; Gukov, Sergei; Putrov, Pavel
  • Journal of High Energy Physics, Vol. 2014, Issue 5
  • DOI: 10.1007/JHEP05(2014)047

Topological Strings and (Almost) Modular Forms
journal, November 2007

  • Aganagic, Mina; Bouchard, Vincent; Klemm, Albrecht
  • Communications in Mathematical Physics, Vol. 277, Issue 3
  • DOI: 10.1007/s00220-007-0383-3

Khovanov-Rozansky Homology and Topological Strings
journal, October 2005

  • Gukov, Sergei; Schwarz, Albert; Vafa, Cumrun
  • Letters in Mathematical Physics, Vol. 74, Issue 1
  • DOI: 10.1007/s11005-005-0008-8

Seiberg-Witten Prepotential from Instanton Counting
journal, January 2003


Holomorphic disks and knot invariants
journal, August 2004


Reidemeister torsion, the Alexander polynomial and U (1,1) Chern-Simons theory
journal, February 1994


On the gauge theory/geometry correspondence
journal, January 1999

  • Gopakumar, Rajesh; Vafa, Cumrun
  • Advances in Theoretical and Mathematical Physics, Vol. 3, Issue 5
  • DOI: 10.4310/ATMP.1999.v3.n5.a5

Modular forms and quantum invariants of 3-manifolds
journal, January 1999


Euclidean SYM theories by time reduction and special holonomy manifolds
journal, December 1997


sl(3) link homology
journal, January 2004


On a universal perturbative invariant of 3-manifolds
journal, May 1998


A 3d-3d appetizer
journal, November 2016


Refined, Motivic, and Quantum
journal, November 2009


On the Floer homology of plumbed three-manifolds
journal, January 2003


Chern-Simons Theory, Matrix Integrals, and Perturbative Three-Manifold Invariants
journal, November 2004


On the relationship between the Rozansky–Witten and the 3-dimensional Seiberg–Witten invariants
journal, January 2001

  • Blau, Matthias; Thompson, George
  • Advances in Theoretical and Mathematical Physics, Vol. 5, Issue 3
  • DOI: 10.4310/ATMP.2001.v5.n3.a3

Supersymmetric gauge theories, intersecting branes and free fermions
journal, February 2008


Fukaya-Seidel category and gauge theory
journal, January 2015


Gauge Theories Labelled by Three-Manifolds
journal, December 2013

  • Dimofte, Tudor; Gaiotto, Davide; Gukov, Sergei
  • Communications in Mathematical Physics, Vol. 325, Issue 2
  • DOI: 10.1007/s00220-013-1863-2

Black hole condensation and duality in string theory
journal, March 1996


Chern–Simons theory on L ( p , q ) lens spaces and Gopakumar–Vafa duality
journal, March 2010


R ‐matrix approach to quantum superalgebras su q ( mn )
journal, November 1992

  • Chang, D.; Phillips, I.; Rozansky, L.
  • Journal of Mathematical Physics, Vol. 33, Issue 11
  • DOI: 10.1063/1.529866

tt * geometry in 3 and 4 dimensions
journal, May 2014

  • Cecotti, Sergio; Gaiotto, Davide; Vafa, Cumrun
  • Journal of High Energy Physics, Vol. 2014, Issue 5
  • DOI: 10.1007/JHEP05(2014)055

On the Ozsvath-Szabo invariant of negative definite plumbed 3-manifolds
journal, January 2005


Large N Duality, Lagrangian Cycles, and Algebraic Knots
journal, October 2012

  • Diaconescu, D. -E.; Shende, V.; Vafa, C.
  • Communications in Mathematical Physics, Vol. 319, Issue 3
  • DOI: 10.1007/s00220-012-1563-3

The spectral curve of the lens space matrix model
journal, November 2009


AdS spacetimes from wrapped M5 branes
journal, November 2006

  • Gauntlett, Jerome P.; Conamhna, Oisín A. P. Mac; Mateos, Toni
  • Journal of High Energy Physics, Vol. 2006, Issue 11
  • DOI: 10.1088/1126-6708/2006/11/053

Superconformal index and 3d-3d correspondence for mapping cylinder/torus
journal, January 2014

  • Gang, Dongmin; Koh, Eunkyung; Lee, Sangmin
  • Journal of High Energy Physics, Vol. 2014, Issue 1
  • DOI: 10.1007/JHEP01(2014)063

A polynomial invariant of rational homology 3-spheres
journal, February 1996


On the Quantum Invariants for the Spherical Seifert Manifolds
journal, September 2006


Topological gauge theories and group cohomology
journal, April 1990

  • Dijkgraaf, Robbert; Witten, Edward
  • Communications in Mathematical Physics, Vol. 129, Issue 2
  • DOI: 10.1007/BF02096988

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


3D-partition functions on the sphere: exact evaluation and mirror symmetry
journal, May 2012

  • Benvenuti, Sergio; Pasquetti, Sara
  • Journal of High Energy Physics, Vol. 2012, Issue 5
  • DOI: 10.1007/JHEP05(2012)099

On the Heegaard Floer homology of a surface times a circle
journal, June 2008


Holomorphic triangles and invariants for smooth four-manifolds
journal, June 2006


Kac-Moody algebras
journal, November 1994


Seiberg–Witten invariants and surface singularities
journal, January 2002

  • Némethi, András; Nicolaescu, Liviu I.
  • Geometry & Topology, Vol. 6, Issue 1
  • DOI: 10.2140/gt.2002.6.269

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

Knot invariants and topological strings
journal, June 2000


Branes and Supergroups
journal, September 2015


Mirrors of 3d Sicilian theories
journal, September 2010

  • Benini, Francesco; Tachikawa, Yuji; Xie, Dan
  • Journal of High Energy Physics, Vol. 2010, Issue 9
  • DOI: 10.1007/JHEP09(2010)063

Holomorphic disks and topological invariants for closed three-manifolds
journal, January 2004


A topologically twisted index for three-dimensional supersymmetric theories
journal, July 2015

  • Benini, Francesco; Zaffaroni, Alberto
  • Journal of High Energy Physics, Vol. 2015, Issue 7
  • DOI: 10.1007/JHEP07(2015)127

A categorification of the Jones polynomial
journal, February 2000


3d analogs of Argyres-Douglas theories and knot homologies
journal, January 2013

  • Fuji, Hiroyuki; Gukov, Sergei; Stošić, Marko
  • Journal of High Energy Physics, Vol. 2013, Issue 1
  • DOI: 10.1007/JHEP01(2013)175

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


$\underline{SW}$ = Milnor Torsion
journal, January 1996


A strong coupling test of S-duality
journal, December 1994


Chern-Simons theory and S-duality
journal, May 2013


Classical and quantum conformal field theory
journal, June 1989

  • Moore, Gregory; Seiberg, Nathan
  • Communications in Mathematical Physics, Vol. 123, Issue 2
  • DOI: 10.1007/BF01238857

On the Quantum Invariant for the Brieskorn Homology Spheres
journal, July 2005


An endomorphism of the Khovanov invariant
journal, November 2005


Vortex Counting and Lagrangian 3-Manifolds
journal, October 2011

  • Dimofte, Tudor; Gukov, Sergei; Hollands, Lotte
  • Letters in Mathematical Physics, Vol. 98, Issue 3
  • DOI: 10.1007/s11005-011-0531-8

Brane tilings and their applications
journal, June 2008


Framed sheaves on root stacks and supersymmetric gauge theories on ALE spaces
journal, January 2016


Hyper-Kähler geometry and invariants of three-manifolds
journal, September 1997


Gauge theory and knot homologies
journal, May 2007


Holomorphic disks and three-manifolds invariants: Properties and applications
journal, January 2004


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


Knot Homology and Refined Chern–Simons Index
journal, November 2014


Holomorphic anomalies in topological field theories
journal, September 1993


Matrix factorizations and link homology
journal, January 2008

  • Khovanov, Mikhail; Rozansky, Lev
  • Fundamenta Mathematicae, Vol. 199, Issue 1
  • DOI: 10.4064/fm199-1-1

From Weak to Strong Coupling in ABJM Theory
journal, May 2011

  • Drukker, Nadav; Mariño, Marcos; Putrov, Pavel
  • Communications in Mathematical Physics, Vol. 306, Issue 2
  • DOI: 10.1007/s00220-011-1253-6

Jones Polynomials of Torus Knots via DAHA
journal, September 2012

  • Cherednik, Ivan
  • International Mathematics Research Notices, Vol. 2013, Issue 23
  • DOI: 10.1093/imrn/rns202

    Works referencing / citing this record:

    Twisted indices of 3d N $$ \mathcal{N} $$ = 4 gauge theories and enumerative geometry of quasi-maps
    journal, July 2019

    • Bullimore, Mathew; Ferrari, Andrea; Kim, Heeyeon
    • Journal of High Energy Physics, Vol. 2019, Issue 7
    • DOI: 10.1007/jhep07(2019)014