Abstract
In three dimensions, a free, periodic scalar field is related by duality to an abelian gauge field. Here I explore aspects of this duality when both theories are quantized on a Riemann surface of genus g. At higher genus, duality involves an identification of winding with momentum on the Jacobian variety of the Riemann surface. I also consider duality for monopole and loop operators on the surface and exhibit the operator algebra, a refinement of the Wilson-’t Hooft algebra.
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References
C. Beasley, Localization for Wilson loops in Chern-Simons theory, arXiv:0911.2687 [INSPIRE].
C. Beasley, Global aspects of abelian duality in dimension three, arXiv:1405.2123 [INSPIRE].
V. Borokhov, A. Kapustin and X.-k. Wu, Topological disorder operators in three-dimensional conformal field theory, JHEP 11 (2002) 049 [hep-th/0206054] [INSPIRE].
V. Borokhov, A. Kapustin and X.-k. Wu, Monopole operators and mirror symmetry in three-dimensions, JHEP 12 (2002) 044 [hep-th/0207074] [INSPIRE].
T.H. Buscher, Quantum corrections and extended supersymmetry in new σ models, Phys. Lett. B 159 (1985) 127 [INSPIRE].
T.H. Buscher, A symmetry of the string background field equations, Phys. Lett. B 194 (1987) 59 [INSPIRE].
T.H. Buscher, Path integral derivation of quantum duality in nonlinear σ-models, Phys. Lett. B 201 (1988) 466 [INSPIRE].
N. Drukker, T. Okuda and F. Passerini, Exact results for vortex loop operators in 3d supersymmetric theories, arXiv:1211.3409 [INSPIRE].
D.S. Freed, G.W. Moore and G. Segal, The uncertainty of fluxes, Commun. Math. Phys. 271 (2007) 247 [hep-th/0605198] [INSPIRE].
D.S. Freed, G.W. Moore and G. Segal, Heisenberg groups and noncommutative fluxes, Annals Phys. 322 (2007) 236 [hep-th/0605200] [INSPIRE].
R. Friedman, Algebraic surfaces and holomorphic vector bundles, Springer-Verlag, New York U.S.A. (1998).
A. Giveon, M. Porrati and E. Rabinovici, Target space duality in string theory, Phys. Rept. 244 (1994) 77 [hep-th/9401139] [INSPIRE].
P. Griffiths and J. Harris, Principles of algebraic geometry, John Wiley and Sons, Inc., New York U.S.A. (1978).
S. Gukov and E. Witten, Gauge theory, ramification, and the geometric langlands program, in Current developments in mathematics 2006, D. Jerison et al. eds., International Press, Somerville U.S.A. (2008), hep-th/0612073 [INSPIRE].
G. ’t Hooft, On the phase transition towards permanent quark confinement, Nucl. Phys. B 138 (1978) 1 [INSPIRE].
K. Intriligator and N. Seiberg, Aspects of 3d N = 2 Chern-Simons-matter theories, JHEP 07 (2013) 079 [arXiv:1305.1633] [INSPIRE].
A. Kapustin and E. Witten, Electric-magnetic duality and the geometric Langlands program, Commun. Num. Theor. Phys. 1 (2007) 1 [hep-th/0604151] [INSPIRE].
A. Kapustin, B. Willett and I. Yaakov, Exact results for supersymmetric abelian vortex loops in 2 + 1 dimensions, JHEP 06 (2013) 099 [arXiv:1211.2861] [INSPIRE].
M. Roček and E.P. Verlinde, Duality, quotients and currents, Nucl. Phys. B 373 (1992) 630 [hep-th/9110053] [INSPIRE].
E.P. Verlinde, Fusion rules and modular transformations in 2D conformal field theory, Nucl. Phys. B 300 (1988) 360 [INSPIRE].
E. Witten, Dynamics of quantum field theory, in Quantum fields and strings: a course for mathematicians, volume 2, P. Deligne et al. eds., American Mathematical Society, Providence, Rhode Island U.S.A. (1999).
E. Witten, Geometric langlands and the equations of Nahm and Bogomolny, arXiv:0905.4795 [INSPIRE].
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Beasley, C. Abelian duality at higher genus. J. High Energ. Phys. 2014, 157 (2014). https://doi.org/10.1007/JHEP07(2014)157
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DOI: https://doi.org/10.1007/JHEP07(2014)157