SL(2;Z) Action On Three-Dimensional Conformal Field Theories With Abelian Symmetry
By E. Witten. 2003
483 citations recorded by the source corpus.
On the space of three-dimensional conformal field theories with U(1) symmetry and a chosen coupling to a background gauge field, there is a natural action of the group $SL(2,{\bf Z})$. The generator $S$ of $SL(2,{\bf Z})$ acts by letting the background gauge field become dynamical, an operation considered recently by Kapustin and Strassler in explaining three-dimensional mirror symmetry. The other generator $T$ acts by shifting the Chern-Simons coupling of the background field. This $SL(2,{\bf Z})$ action in three dimensions is related by the AdS/CFT correspondence to $SL(2,{\bf Z})$ duality of low energy U(1) gauge fields in four dimensions.
Subfields: Astronomy and Astrophysics, Nuclear and High Energy Physics
View Paper: https://www.semanticscholar.org/paper/ae204be8cecb8b5719a9ccdfeffd3c267983c47d
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Citations in the corpus
- Deconfined quantum critical points: symmetries and dualities (2017, methodology): “It is possible that the theories discussed in this paper do not flow to stable CFTs, which in the context of deconfined critical point means that the transition is ultimately first order.”
- Holography of dyonic dilaton black branes (2010, background): “To implement the SL(2, R) in the boundary one needs to gauge this global symmetry [30].”
- Topological boundary conditions in abelian Chern-Simons theory (2010, background): “It was noticed in [10] that certain abelian Chern-Simons theories with a nontrivial K are nevertheless trivial on the quantum level (isomorphic to the trivial theory).”
- Chern–Simons theory with vector fermion matter (2011, methodology): “This however corresponds to simply modifying the AdS boundary condition on the bulk U(M) spin-1 gauge fields, from the purely magnetic boundary condition to an electric-magnetic mixed boundary condition [62].”
- Towards strange metallic holography (2009, background): “[32] studied the 2+1 dimensional relativistic (z = 1) gauge theory and showed that it had two IR stable realizations, the second corresponding to the gauging of the U(1) symmetry.”
- Lectures on holographic methods for condensed matter physics (2009, background): “This nongeneric fact is interpreted in field theory as a self-duality under particle-vortex duality, and can be traced to the self-duality of the Maxwell equations in the bulk [83, 80].”
- Generalized global symmetries (2014, methodology): “In this section we generalize Witten’s operations on 3d theories with global symmetries [18] to 4d theories with one-form global symmetries.”
- Averaging over Narain moduli space (2020, background): “Triviality of this theory is a special case of a statement in [43] and was analyzed in considerable detail in [44].”
- Quantum critical transport, duality, and M-theory (2007, background): “This conclusion is in accord with those of Kapustin and Strassler [41] and Witten [28] on related models.”
- Boundary conditions and dualities: vector fields in AdS/CFT (2006, methodology): “For d = 3, this conclusion was reached previously in [15, 19] using related pathintegral reasoning.”
- Emergent gauge fields in holographic superconductors (2010, background): “[ 7 ], this CFT, which includes a dynamical gauge field, has also a global U(1) with an associated conserved current given by ˜ J µ = ǫ µ�� ∂�a�, and should not be”
- Comments on twisted indices in 3d supersymmetric gauge theories (2016, background): “In three dimensions, the two-point function of conserved currents contains an interesting conformally-invariant contact term, whose corresponding local term is a ChernSimons functional for background gauge fields [90, 28, 44].”
- Hall conductivity from dyonic black holes (2007, background): “A topological θ term for the four dimensional gauge fields in the bulk gives rise to the antisymmetric contribution ǫμνλp λ in two point current-current correlators of the 2+1 dimensional CFT on the boundary [17].”
- The thermoelectric properties of inhomogeneous holographic lattices (2014, background): “This arises from the electromagnetic duality of the D = 4 Einstein-Maxwell theory, with the dual gauge-field being associated with a second CFT arising from an alternative quantisation scheme [42] (for related discussion see also [43–46].”
- Symmetries and Strings in Field Theory and Gravity (2010, result): “We point out that this observation is consistent with the result of [13] that the theory with p = 1 is trivial.”
- Comments on 3d Seiberg-like dualities (2011, background): “The two-point function of conserved currents in three dimension can contain a conformally invariant contact term [24]”