Abstract
We consider coupling an ordinary quantum field theory with an infinite number of degrees of freedom to a topological field theory. On \( \mathbb{R} \) d the new theory differs from the original one by the spectrum of operators. Sometimes the local operators are the same but there are different line operators, surface operators, etc. The effects of the added topological degrees of freedom are more dramatic when we compactify \( \mathbb{R} \) d, and they are crucial in the context of electric-magnetic duality. We explore several examples including DijkgraafWitten theories and their generalizations both in the continuum and on the lattice. When we couple them to ordinary quantum field theories the topological degrees of freedom allow us to express certain characteristic classes of gauge fields as integrals of local densities, thus simplifying the analysis of their physical consequences.
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References
N. Seiberg, Modifying the Sum Over Topological Sectors and Constraints on Supergravity, JHEP 07 (2010) 070 [arXiv:1005.0002] [INSPIRE].
T. Banks and N. Seiberg, Symmetries and Strings in Field Theory and Gravity, Phys. Rev. D 83 (2011) 084019 [arXiv:1011.5120] [INSPIRE].
O. Aharony, N. Seiberg and Y. Tachikawa, Reading between the lines of four-dimensional gauge theories, JHEP 08 (2013) 115 [arXiv:1305.0318] [INSPIRE].
O. Aharony, S.S. Razamat, N. Seiberg and B. Willett, 3d dualities from 4d dualities for orthogonal groups, JHEP 08 (2013) 099 [arXiv:1307.0511] [INSPIRE].
E. Witten, Five-brane effective action in M-theory, J. Geom. Phys. 22 (1997) 103 [hep-th/9610234] [INSPIRE].
G.W. Moore, Anomalies, Gauss laws and Page charges in M-theory, Comptes Rendus Physique 6 (2005) 251 [hep-th/0409158] [INSPIRE].
D. Belov and G.W. Moore, Holographic Action for the Self-Dual Field, hep-th/0605038 [INSPIRE].
E. Witten, Conformal Field Theory In Four And Six Dimensions, arXiv:0712.0157 [INSPIRE].
E. Witten, Geometric Langlands From Six Dimensions, arXiv:0905.2720 [INSPIRE].
D.S. Freed and C. Teleman, Relative quantum field theory, Commun. Math. Phys. 326 (2014) 459 [arXiv:1212.1692] [INSPIRE].
S. Gukov and A. Kapustin, Topological Quantum Field Theory, Nonlocal Operators and Gapped Phases of Gauge Theories, arXiv:1307.4793 [INSPIRE].
A. Kapustin and R. Thorngren, Higher symmetry and gapped phases of gauge theories, arXiv:1309.4721 [INSPIRE].
R. Dijkgraaf and E. Witten, Topological Gauge Theories and Group Cohomology, Commun. Math. Phys. 129 (1990) 393 [INSPIRE].
S. Gukov and E. Witten, Gauge Theory, Ramification, And The Geometric Langlands Program, hep-th/0612073 [INSPIRE].
S. Gukov and E. Witten, Rigid Surface Operators, Adv. Theor. Math. Phys. 14 (2010) [arXiv:0804.1561] [INSPIRE].
G. ’t Hooft, On the Phase Transition Towards Permanent Quark Confinement, Nucl. Phys. B 138 (1978) 1 [INSPIRE].
P. Goddard, J. Nuyts and D.I. Olive, Gauge Theories and Magnetic Charge, Nucl. Phys. B 125 (1977) 1 [INSPIRE].
E. Witten, Lecture II-9: Wilson loops, ’t Hooft loops, and ’t Hooft’s model of confinement, in P. Deligne et al., Quantum fields and strings: A course for mathematicians, vol. 1, 2, AMS, Providence, U.S.A.(1999).
D. Gaiotto, G.W. Moore and A. Neitzke, Framed BPS States, arXiv:1006.0146 [INSPIRE].
G.T. Horowitz, Exactly Soluble Diffeomorphism Invariant Theories, Commun. Math. Phys. 125 (1989) 417 [INSPIRE].
J.M. Maldacena, G.W. Moore and N. Seiberg, D-brane charges in five-brane backgrounds, JHEP 10 (2001) 005 [hep-th/0108152] [INSPIRE].
T.H. Hansson, V. Oganesyan and S.L. Sondhi, Superconductors are topologically ordered, Ann. Phys. 313 (2004) 497 [cond-mat/0404327].
M.C. Diamantini, P. Sodano and C.A. Trugenberger, Superconductors with topological order, Eur. Phys. J. B 53 (2006) 19 [hep-th/0511192] [INSPIRE].
G.Y. Cho and J.E. Moore, Topological BF field theory description of topological insulators, Annals Phys. 326 (2011) 1515 [arXiv:1011.3485] [INSPIRE].
M. Bauer, G. Girardi, R. Stora and F. Thuillier, A Class of topological actions, JHEP 08 (2005) 027 [hep-th/0406221] [INSPIRE].
D.S. Freed, G.W. Moore and G. Segal, Heisenberg Groups and Noncommutative Fluxes, Annals Phys. 322 (2007) 236 [hep-th/0605200] [INSPIRE].
D.S. Freed, Dirac charge quantization and generalized differential cohomology, hep-th/0011220 [INSPIRE].
G. Moore, A Minicourse on Generalized Abelian Gauge Theory, Self-Dual Theories, and Differential Cohomology, lectures at the Simons Center for Geometry and Physics, 12-14 January 2011 http://www.physics.rutgers.edu/∼gmoore/SCGP-Minicourse.pdf.
K. Gawedzki, Topological Actions In Two-dimensional Quantum Field Theories, in proceedings of Cargese 1987, Nonperturbative Quantum Field Theory 185 (1988) 101.
A.Y. Kitaev, Fault tolerant quantum computation by anyons, Annals Phys. 303 (2003) 2 [quant-ph/9707021] [INSPIRE].
M.A. Levin and X.-G. Wen, String net condensation: A Physical mechanism for topological phases, Phys. Rev. B 71 (2005) 045110 [cond-mat/0404617] [INSPIRE].
A. Kapustin and N. Saulina, Topological boundary conditions in abelian Chern-Simons theory, Nucl. Phys. B 845 (2011) 393 [arXiv:1008.0654] [INSPIRE].
A. Kapustin and R. Thorngren, Topological Field Theory on a Lattice, Discrete Theta-Angles and Confinement, arXiv:1308.2926 [INSPIRE].
D. Belov and G.W. Moore, Classification of Abelian spin Chern-Simons theories, hep-th/0505235 [INSPIRE].
A. Kapustin, D-branes in a topologically nontrivial B field, Adv. Theor. Math. Phys. 4 (2000) 127 [hep-th/9909089] [INSPIRE].
I.G. Halliday and A. Schwimmer, The Phase Structure of SU(N )/Z(N ) Lattice Gauge Theories, Phys. Lett. B 101 (1981) 327 [INSPIRE].
I.G. Halliday and A. Schwimmer, Z(2) Monopoles in Lattice Gauge Theories, Phys. Lett. B 102 (1981) 337 [INSPIRE].
G. Mack and V.B. Petkova, Z2 Monopoles in the Standard SU(2) Lattice Gauge Theory Model, Z. Phys. C 12 (1982) 177 [INSPIRE].
J. Polchinski, Order parameters in a modified lattice gauge theory, Phys. Rev. D 25 (1982) 3325 [INSPIRE].
F.J. Wegner, Duality in Generalized Ising Models and Phase Transitions Without Local Order Parameters, J. Math. Phys. 12 (1971) 2259 [INSPIRE].
R. Savit, Duality in Field Theory and Statistical Systems, Rev. Mod. Phys. 52 (1980) 453 [INSPIRE].
S. Elitzur, R.B. Pearson and J. Shigemitsu, The Phase Structure of Discrete Abelian Spin and Gauge Systems, Phys. Rev. D 19 (1979) 3698 [INSPIRE].
A. Ukawa, P. Windey and A.H. Guth, Dual Variables for Lattice Gauge Theories and the Phase Structure of Z(N) Systems, Phys. Rev. D 21 (1980) 1013 [INSPIRE].
E.H. Fradkin and S.H. Shenker, Phase Diagrams of Lattice Gauge Theories with Higgs Fields, Phys. Rev. D 19 (1979) 3682 [INSPIRE].
A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, U.K. (2001).
N. Steenrod, Products of cocycles and extensions of mappings, Ann. Math. 48 (1947) 290.
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Kapustin, A., Seiberg, N. Coupling a QFT to a TQFT and duality. J. High Energ. Phys. 2014, 1 (2014). https://doi.org/10.1007/JHEP04(2014)001
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DOI: https://doi.org/10.1007/JHEP04(2014)001