K-theoretical boundary rings in [formula omitted] coset models

This paper explores further the connection between Langlands duality and T-duality for compact simple Lie groups, which appeared in work of Daenzer-Van Erp and Bunke-Nikolaus. We show that Langlands duality gives rise to isomorphisms of twisted K-groups, but that these K-groups are trivial except in the simplest case of SU(2) and SO(3). Along the way we compute explicitly the map on $H^3$ induced by a covering of compact simple Lie groups, which is either 1 or 2 depending in a complicated way on the type of the groups involved. We also give a new method for computing twisted K-theory using the Segal spectral sequence, giving simpler computations of certain twisted K-theory groups of compact Lie groups relevant for D-brane charges in WZW theories and rank-level dualities. Finally we study a duality for orientifolds based on complex Lie groups with an involution.

Non-hermitian symmetric N = 2 coset models, Poincaré polynomials, and string compactification

We construct string topology operations in twisted K-theory. We study the examples given by symplectic Grassmannians, computing K*τ(LℍPℓ) in detail. Via the work of Freed–Hopkins–Teleman, these computations are related to completions of the Verlinde algebras of Sp(n). We compute these completions, and other relevant information about the Verlinde algebras. We also identify the completions with the twisted K-theory of the Gruher–Salvatore pro-spectra. Further comments on the field-theoretic nature of these constructions are made.

We study conformal field theory with the symmetry algebra $ \mathcal{A}\left( {2,\ p} \right)={{{\widehat{\mathfrak{gl}}{(n)_2}}} \left/ {{\widehat{\mathfrak{gl}}{{{\left( {n-p} \right)}}_2}}} \right.} $ . In order to support the conjecture that this algebra acts on the moduli space of instantons on ℂ2/ℤ p , we calculate the characters of its representations and check their coincidence with the generating functions of the fixed points of the moduli space of instantons. We show that the algebra $ \mathcal{A}\left( {2,\ p} \right) $ can be realized in two ways. The first realization is connected with the cross-product of p Virasoro and p Heisenberg algebras: $ {{\mathcal{H}}^p} $ × Vir p . The second realization is connected with: $ {{\mathcal{H}}^p} \times \widehat{\mathfrak{sl}}{(p)_2}\times \left( {\widehat{\mathfrak{sl}}{(2)_p}\times {{{\widehat{\mathfrak{sl}}{(2)_{n-p }}}} \left/ {{\widehat{\mathfrak{sl}}{(2)_n}}} \right.}} \right) $ . The equivalence of these two realizations provides the non-trivial identity for the characters of $ \mathcal{A}\left( {2,\ p} \right) $ . The moduli space of instantons on ℂ2/ℤ p admits two different compactifications. This leads to two different bases for the representations of $ \mathcal{A}\left( {2,\ p} \right) $ . We use this fact to explain the existence of two forms of the instanton pure partition functions.

Massive integrable soliton theories

Some mathematical questions relating to coset conformal field theories (CFT) are considered in the framework of algebraic quantum field theory as developed previously by us. We consider the issue of fix point resolution in the diagonal cosets of type A. We show how to decompose certain reducible representations into irreducibles, and prove that the coset CFT gives rise to a unitary modular category and therefore may be used to construct 3-manifold invariants. We prove that if the coset inclusion satisfies certain conditions which can be checked in examples, the Kac–Wakimoto Hypothesis (KWH) is equivalent to the Kac–Wakimoto Conjecture (KWC), a result which seems to be hard to prove by purely representation considerations. Examples are also presented.

We analyze various perspectives on the elliptic genus of non-compact supersymmetric coset conformal field theories with central charge larger than three. We calculate the holomorphic part of the elliptic genus via a free field description of the model, and show that it agrees with algebraic expectations. The holomorphic part of the elliptic genus is directly related to an Appell-Lerch sum and behaves anomalously under modular transformation properties. We analyze the origin of the anomaly by calculating the elliptic genus through a path integral in a coset conformal field theory. The path integral codes both the holomorphic part of the elliptic genus, and a non-holomorphic remainder that finds its origin in the continuous spectrum of the non-compact model. The remainder term can be shown to agree with a function that mathematicians introduced to parameterize the difference between mock theta functions and Jacobi forms. The holomorphic part of the elliptic genus thus has a path integral completion which renders it non-holomorphic and modular.

Level-Rank Duality of WZW Theories and Isomorphisms of N = 2 Coset Models

The fusion rules in [Formula: see text] for a finite group [Formula: see text] can be computed in terms of character inner products. Using an explicit formula (Theorem 3.2) for these fusion rules, we show that [Formula: see text] is multiplicity free for two infinite families of finite groups: the Dihedral groups and the Dicyclic groups. In fact, we will compute all fusion rules in these categories. Multiplicity freeness is a desired property for modular tensor categories, since it greatly simplifies the computation of [Formula: see text]-matrices. Furthermore, we observe that the fusion rules for Dihedral groups [Formula: see text] with [Formula: see text] odd are extremely similar to the fusion rules of type-B level-[Formula: see text] fusion algebras of Wess–Zumino–Witten conformal field theories. Moreover, we give a proof of the fusion rule formula by using Mackey theory.

The aim of this (mostly expository) article is to show a connection between the finiteness conditions arising in twisted K-theory. There are two different conditions arising naturally in two main approaches to the problem of computing the index of the appropriate family of elliptic operators (the approach of Nistor and Troitsky and the approach of Mathai, Melrose, and Singer). These conditions are formulated absolutely differently, but in some sense they should be close to each other. In this paper, we find this connection and prove the corresponding formal statement. Thereby it is shown that these conditions map to each other. This opens a possibility to synthesize these approaches. It is also shown that the finiteness condition arising in the paper of Nistor and Troitsky is a special case of the finiteness condition that appears in the paper of Emerson and Meyer, where the theorem of Nistor and Troitsky is proved not only for the case of a bundle of Lie groups, but also for the case of a general groupoid.

Conformal field theories corresponding to two-dimensional electrically charged black holes and to two-dimensional anti-de Sitter space with a covariantly constant electric field are simply constructed as SL(2,R)/Z Wess-Zumino-Witten coset models. Four-dimensional spacetime solutions are obtained by tensoring these two-dimensional theories with $\frac{\mathrm{SU}(2)}{Z(m)}$ coset models. These describe a family of dyonic black holes and the Bertotti-Robinson universe.

In a recent article [K. H. Hofmann and F. G. Russo, ‘The probability that$x$and$y$commute in a compact group’,Math. Proc. Cambridge Phil Soc., to appear] we calculated for a compact group$G$the probability$d(G)$that two randomly selected elements$x, y\in G$satisfy$xy=yx$, and we discussed the remarkable consequences on the structure of$G$which follow from the assumption that$d(G)$is positive. In this note we consider two natural numbers$m$and$n$and the probability$d_{m,n}(G)$that for two randomly selected elements$x, y\in G$the relation$x^my^n=y^nx^m$holds. The situation is more complicated whenever$n,m\gt 1$. If$G$is a compact Lie group and if its identity component$G_0$is abelian, then it follows readily that$d_{m,n}(G)$is positive. We show here that the following condition suffices for the converse to hold in an arbitrary compact group$G$: for any nonopen closed subgroup$H$of$G$, the sets$\{g\in G: g^k\in H\}$for both$k=m$and$k=n$have Haar measure$0$. Indeed, we show that if a compact group$G$satisfies this condition and if$d_{m,n}(G)\gt 0$, then the identity component of$G$is abelian.

String propagation on a curved background defines an embedding problem of surfaces in differential geometry. Using this, we show that in a wide class of backgrounds the classical dynamics of the physical degrees of freedom of the string involves 2-dim sigma-models corresponding to coset conformal field theories.

Extremal N=2 black holes in four dimensions can be described by an ensemble of D3-branes wrapped on internal supersymmetric three-cycles of Calabi-Yau threefolds on which type IIB superstring theory is compactified. We construct a similar configuration, with extra RR and NS-NS three-form fluxes being turned on. We can avoid the Freed-Witten anomaly on the D3-branes by enforcing the pullback of these extra fluxes to the D3-branes to vanish at the classical level. In the setup the D3-brane charge is not conserved since it is classified as a trivial class in twisted K-theory. Consequently, the D3-branes may disappear by encountering an instantonic D5-brane localized in time. We discuss what happens on the black hole described by such disappearing D3-branes, relying mainly on topological arguments. Especially, we argue that another RR three-form flux will be left as a lump of remnant which is localized in the uncompactified four-dimensional space-time and that it may carry the same amount of degrees of freedom to describe a black hole, in cooperation with the original NS-NS flux, after this transition of the D3-branes.

In this paper we present a model to calculate the stringy product on twisted orbifold K-theory of Adem-Ruan-Zhang for abelian complex orbifolds. In the first part we consider the non-twisted case on an orbifold presented as the quotient of a manifold acted by a compact abelian Lie group. We give an explicit description of the obstruction bundle, we explain the relation with the product defined by Jarvis-Kaufmann-Kimura and, via a Chern character map, with the Chen-Ruan cohomology, we explicitly calculate the stringy product for a weighted projective orbifold. In the second part we consider orbifolds presented as the quotient of a manifold acted by a finite abelian group and twistings coming from the group cohomology. We show a decomposition formula for twisted orbifold K-theory that is suited to calculate the stringy product and we use this formula to calculate two examples when the group is (ℤ/2) 3 .