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Conformal twists, Yang-Baxter \(\sigma\)-models & holographic noncommutativity. (English) Zbl 1436.81084

This paper gives an overview of certain integrable deformations of \(\sigma\)-models in the context of the AdS/CFT correspondence, including several new results and examples. The deformations are studied from four points of view (briefly described below) and the relations between these points of view are identified and given explicitly, either in the form of a general proof, or, when the relation is conjectured, in the form of examples. The range of examples is extensive and interesting relations between some of these examples are derived by considering an outer automorphism of the conformal algebra. The paper elaborates and extends an earlier article by the same authors [“Yang-Baxter \(\sigma\)-models, conformal twists, and noncommutative Yang-Mills theory”, Phys. Rev. D 95, No. 10, Article ID 105006, 7 p. (2017; doi:10.1103/PhysRevD.95.105006)]. Complementary results can be find in [S. J. van Tongeren, Nucl. Phys., B 904, 148–175 (2016; Zbl 1332.81197)]. On the whole I found the paper well organised, but I sometimes missed clear indications when equalities are only valid up to first order in the deformation parameter.
The \(\sigma\)-models in question are deformations of the string worldsheet \(\sigma\)-model in AdS\(_5=SO(4,2)/SO(4,1)\), where the deformations are generated by \(r\)-matrices which solve the homogeneous classical Yang-Baxter Equation and a deformation parameter \(\eta\). These deformations can be identified as twists of the conformal algebra \(\mathfrak{so}(4,2)\) (the first point of view). Each deformed \(\sigma\)-model gives rise to a deformation of the AdS\(_5\) metric, an NS-NS two-form and a dilaton, viewed as a “closed string gravity background” (the second point of view). Applying a field redefinition one obtains so-called open string parameters (the third point of view), which describe a system in AdS\(_5\). Finally, invoking the AdS/CFT correspondence one infers that there should be a corresponding dual noncommutative gauge theory at the conformal boundary (the fourth point of view). The explicit comparisons between these points of view are presented systematically in Poincaré coordinates \(x^M=(x^{\mu},z)\).
One of the key points in the paper is the fact that in the open string picture the deformation is entirely encoded in a non-commutative structure \(\Theta^{MN}\). In this picture the metric is undeformed AdS\(_5\) and the open string coupling is constant, which is used to justify the appeal to the AdS/CFT correspondence to obtain a noncommutative gauge theory. Moreover, for unimodular \(r\)-matrices it is shown that \(\Theta^{MN}\) is divergence free, \(\nabla_M\Theta^{MN}\). From this the authors conclude that \(\Theta^{MN}\) is uniquely determined by the values at the conformal boundary \(z=0\), even if \(\Theta^{MN}\) depends on the holographic coordinate \(z\). This property is what they call “holographic noncommutativity”. More general deformations lead to solutions of generalized supergravity with a Killing field \(I^N\) and in these cases the authors find that \(\nabla_M\Theta^{MN}=I^N\) and \(\Theta^{MN}\) is again determined by the values at the conformal boundary. These relations are verified by identifying \(\Theta^{MN}\) and \(I^N\) in a large class of examples.
The noncommutative structure is related to the \(r\)-matrix and the deformation parameter \(\eta\) by \(\Theta^{MN}=-2\eta r^{MN}\), where the \(r\)-matrix is evaluated in coordinate basis vectors on the right-hand side. (It is shown in Appendix B that this relation even extends to \(r\)-matrices that solve the modified Yang-Baxter equation.) Furthermore, at the conformal boundary one finds that \(\Theta^{\mu z}(z=0)=0\) and \(\Theta^{\mu\nu}(z=0)\) coincides with the noncommutative structure of the deformed gauge theory at the conformal boundary.

MSC:

81T10 Model quantum field theories
81T20 Quantum field theory on curved space or space-time backgrounds
81T35 Correspondence, duality, holography (AdS/CFT, gauge/gravity, etc.)
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
16T25 Yang-Baxter equations
14D15 Formal methods and deformations in algebraic geometry
83E50 Supergravity

Citations:

Zbl 1332.81197
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References:

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