Characterization of 2D rational local conformal nets and its boundary conditions: the maximal case

Let $\mathcal{A}$ be a completely rational local M\"obius covariant net on $S^1$, which describes a set of chiral observables. We show that local M\"obius covariant nets $\mathcal{B}_2$ on 2D Minkowski space which contains $\mathcal{A}$ as chiral left-right symmetry are in one-to-one correspondence with Morita equivalence classes of Q-systems in the unitary modular tensor category $\mathrm{DHR}(\mathcal{A})$. The M\"obius covariant boundary conditions with symmetry $\mathcal{A}$ of such a net $\mathcal{B}_2$ are given by the Q-systems in the Morita equivalence class or by simple objects in the module category modulo automorphisms of the dual category. We generalize to reducible boundary conditions. To establish this result we define the notion of Morita equivalence for Q-systems (special symmetric $\ast$-Frobenius algebra objects) and non-degenerately braided subfactors. We prove a conjecture by Kong and Runkel, namely that Rehren's construction (generalized Longo-Rehren construction, $\alpha$-induction construction) coincides with the categorical full center. This gives a new view and new results for the study of braided subfactors.Comment: 44 pages, many tikz figures. Some improvements. Some typos fixe

Modular invariant Frobenius algebras from ribbon Hopf algebra automorphisms

For any finite-dimensional factorizable ribbon Hopf algebra H and any ribbon automorphism omega of H, we establish the existence of the following structure: an H-bimodule F_omega and a bimodule morphism Z_omega from Lyubashenko's Hopf algebra object K for the bimodule category to F_omega. This morphism is invariant under the natural action of the mapping class group of the one-punctured torus on the space of bimodule morphisms from K to F_omega. We further show that the bimodule F_omega can be endowed with a natural structure of a commutative symmetric Frobenius algebra in the monoidal category of H-bimodules, and that it is a special Frobenius algebra iff H is semisimple. The bimodules K and F_omega can both be characterized as coends of suitable bifunctors. The morphism Z_omega is obtained by applying a monodromy operation to the coproduct of F_omega; a similar construction for the product of F_omega exists as well. Our results are motivated by the quest to understand the bulk state space and the bulk partition function in two-dimensional conformal field theories with chiral algebras that are not necessarily semisimple.Comment: 44 pages, some figures. v2: Several changes and clarifications, conclusions unchanged. v3: typos correcte

Torsion-free, divisible, and Mittag-Leffler modules

We study (relative) K-Mittag-Leffler modules, with emphasis on the class K of absolutely pure modules. A final goal is to describe the K-Mittag-Leffler abelian groups as those that are, modulo their torsion part, aleph_1-free, Cor.6.12. Several more general results of independent interest are derived on the way. In particular, every flat K-Mittag-Leffler module (for K as before) is Mittag-Leffler, Thm.3.9. A question about the definable subcategories generated by the divisible modules and the torsion-free modules, resp., has been left open, Quest.4.6

Boundary Conditions for Topological Quantum Field Theories, Anomalies and Projective Modular Functors

We study boundary conditions for extended topological quantum field theories (TQFTs) and their relation to topological anomalies. We introduce the notion of TQFTs with moduli level $m$, and describe extended anomalous theories as natural transformations of invertible field theories of this type. We show how in such a framework anomalous theories give rise naturally to homotopy fixed points for $n$-characters on $\infty$-groups. By using dimensional reduction on manifolds with boundaries, we show how boundary conditions for $n+1$-dimensional TQFTs produce $n$-dimensional anomalous field theories. Finally, we analyse the case of fully extended TQFTs, and show that any fully extended anomalous theory produces a suitable boundary condition for the anomaly field theory.Comment: 26 pages, 6 figures. Exposition improved, bibliography updated. Final version, to appear in Comm. Math. Phy

Schemes over \F_1 and zeta functions

We determine the {\em real} counting function $N(q)$ ($q\in [1,\infty)$) for the hypothetical "curve" C=\overline {\Sp \Z} over \F_1, whose corresponding zeta function is the complete Riemann zeta function. Then, we develop a theory of functorial \F_1-schemes which reconciles the previous attempts by C. Soul\'e and A. Deitmar. Our construction fits with the geometry of monoids of K. Kato, is no longer limited to toric varieties and it covers the case of schemes associated to Chevalley groups. Finally we show, using the monoid of ad\`ele classes over an arbitrary global field, how to apply our functorial theory of \Mo-schemes to interpret conceptually the spectral realization of zeros of $L$-functions.Comment: 1 figure, 32 page

Induced quadratic modules in ∗*-algebras

Positivity in $\ast$-algebras can be defined either algebraically, by quadratic modules, or analytically, by $\ast$-representations. By the induction procedure for $\ast$-representations we can lift the analytical notion of positivity from a $\ast$-subalgebra to the entire $\ast$-algebra. The aim of this paper is to define and study the induction procedure for quadratic modules. The main question is when a given quadratic module on the $\ast$-algebra is induced from its intersection with the $\ast$-subalgebra. This question is very hard even for the smallest quadratic module (i.e. the set of all sums of hermitian squares) and will be answered only in very special cases.Comment: 30 pages, to appear in Comm. Algebr

ON FINITE LENGTH SMOOTH REPRESENTATIONS OF pp-ADIC GENERAL LINEAR GROUPS (Automorphic form, automorphic LL-functions and related topics)

We survey qualitative aspects of the study of the decomposition of finite-length smooth representations of the groups GLn(F), for a p-adic field F, with emphasis on techniques that have been developing in the recent years. We state general goals and questions on decomposition of parabolic induction of irreducible representations, and review applications for branching laws such as the local Gan-Gross-Prasad program. We give a flavor of the categorical links, and their possible applications, between the p-adic setting and the representation theory of quiver Hecke algebras of type A. Finally, we review, as a case study, the recent RSK classification of irreducible representations, introduced by the author with Lapid

A local global question in automorphic forms

In this paper, we consider the \SL(2) analogue of two well-known theorems about period integrals of automorphic forms on \GL(2): one due to Harder-Langlands-Rapoport, and the other due to Waldspurger.Comment: 28 page