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Register a new userOn the gl(1|1) Wess-Zumino-Witten Model
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Authors
Jan Troost
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We continue the study of the gl(1|1) Wess-Zumino-Witten model. The Knizhnik-Zamolodchikov equations for the one, two, three and four point functions are analyzed, for vertex operators corresponding to typical and projective representations. We illustrate their interplay with the logarithmic global conformal Ward identities. We compute the four point function for one projective and three typical representations. Three coupled first order Knizhnik-Zamolodchikov equations are integrated consecutively in terms of generalized hypergeometric functions, and we assemble the solutions into a local correlator. Moreover, we prove crossing symmetry of the four point function of four typical representations at generic momenta. Throughout, the map between the gl(1|1) Wess-Zumino-Witten model and symplectic fermions is exploited and extended.
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We revisit various topological issues concerning four-dimensional ungauged and gauged Wess-Zumino-Witten (WZW) terms for $SU$ and $SO$ quantum chromodynamics (QCD), from the modern bordism point of view. We explain, for example, why the definition of the $4d$ WZW terms requires the spin structure. We also discuss how the mixed anomaly involving the 1-form symmetry of $SO$ QCD is reproduced in the low-energy sigma model.
Non-anticommutative deformations have been studied in the context of supersymmetry (SUSY) in three and four space-time dimensions, and the general picture is that highly nontrivial to deform supersymmetry in a way that still preserves some of its important properties, both at the formal algebraic level (e.g., preserving the associativity of the deformed theory) as well as at the physical level (e.g., maintaining renormalizability). The Hopf algebra formalism allows the definition of algebraically consistent deformations of SUSY, but this algebraic consistency does not guarantee that physical models build upon these structures will be consistent from the physical point of view. We will investigate a deformation induced by a Drinfeld twist of the ${cal N}=1$ SUSY algebra in three space-time dimensions. The use of the Hopf algebra formalism allows the construction of deformed ${cal N}=1$ SUSY algebras that should still preserve a deformed version of supersymmetry. We will construct the simplest deformed version of the Wess-Zumino model in this context, but we will show that despite the consistent algebraic structure, the model in question is not invariant under SUSY transformation and is not renormalizable. We will comment on the relation of these results with previous ones discussed in the literature regarding similar four-dimensional constructions.
We consider the problem of the decomposition of the Renyi entanglement entropies in theories with a non-abelian symmetry by doing a thorough analysis of Wess-Zumino-Witten (WZW) models. We first consider $SU(2)_k$ as a case study and then generalise to an arbitrary non-abelian Lie group. We find that at leading order in the subsystem size $L$ the entanglement is equally distributed among the different sectors labelled by the irreducible representation of the associated algebra. We also identify the leading term that breaks this equipartition: it does not depend on $L$ but only on the dimension of the representation. Moreover, a $loglog L$ contribution to the Renyi entropies exhibits a universal form related to the underlying symmetry group of the model, i.e. the dimension of the Lie group.
We shall give an axiomatic construction of Wess-Zumino-Witten actions valued in (G=SU(N)), (Ngeq 3). It is realized as a functor ({WZ}) from the category of conformally flat four-dimensional manifolds to the category of line bundles with connection that satisfies, besides the axioms of a topological field theory, the axioms which abstract Wess-Zumino-Witten actions. To each conformally flat four-dimensional manifold (Sigma) with boundary (Gamma=partialSigma), a line bundle (L=WZ(Gamma)) with connection over the space (Gamma G) of mappings from (Gamma) to (G) is associated. The Wess-Zumino-Witten action is a non-vanishing horizontal section (WZ(Sigma)) of the pull back bundle (r^{ast}L) over (Sigma G) by the boundary restriction (r). (WZ(Sigma)) is required to satisfy a generalized Polyakov-Wiegmann formula with respect to the pointwise multiplication of the fields (Sigma G). Associated to the WZW-action there is a geometric descrption of extensions of the Lie group (Omega^3G) due to J. Mickelsson. In fact we shall construct two abelian extensions of (Omega^3G) that are in duality.
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Bogdan G. Dimitrov (Bogoliubov Laboratory of Theoretical Physics
, n Joint Institute for Nuclear Research
, Dubna
2004
The main purpose of the report is to provide some argumentation that three seemingly distinct approaches of 1. Giveon, Kutasov and Seiberg (hep-th/9806194); 2. Hemming, Keski-Vakkuri (hep-th/0110252); Maldacena, Ooguri (hep-th/0001053) and 3. I. Bars (hep-th/9503205) can be investigated by applying the mathematical methods of integral geometry on the Lobachevsky plane, developed previously by Gelfand, Graev and Vilenkin. All these methods can be used for finding the transformations, leaving the Kac-Moody and Virasoro algebras invariant. The near-distance limit of the Conformal Field Theory of the SL(2, R) WZW model of strings on an ADS3 background can also be interpreted in terms of the Lobachevsky Geometry : the non - euclidean distance is conserved and the Lobachevsky formulae for the angle of parallelism is recovered. Some preliminary technique from integral geometry for inverting the modified integral representation for the Kac- Moody algebra has been demonstrated.
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