The Heisenberg group and conformal field theory

A mathematical construction of the conformal field theory (CFT) associated to a compact torus, also called the "nonlinear Sigma-model" or "lattice-CFT", is given. Underlying this approach to CFT is a unitary modular functor, the construction of which follows from a "Quantization commutes with reduction"- type of theorem for unitary quantizations of the moduli spaces of holomorphic torus-bundles and actions of loop groups. This theorem in turn is a consequence of general constructions in the category of affine symplectic manifolds and their associated generalized Heisenberg groups.Comment: 45 pages, some parts have been rewritten. Version to appear in Quart. J. Mat

Topological modular forms and conformal nets

We describe the role conformal nets, a mathematical model for conformal field theory, could play in a geometric definition of the generalized cohomology theory TMF of topological modular forms. Inspired by work of Segal and Stolz-Teichner, we speculate that bundles of boundary conditions for the net of free fermions will be the basic underlying objects representing TMF-cohomology classes. String structures, which are the fundamental orientations for TMF-cohomology, can be encoded by defects between free fermions, and we construct the bundle of fermionic boundary conditions for the TMF-Euler class of a string vector bundle. We conjecture that the free fermion net exhibits an algebraic periodicity corresponding to the 576-fold cohomological periodicity of TMF; using a homotopy-theoretic invariant of invertible conformal nets, we establish a lower bound of 24 on this periodicity of the free fermions

Klein Topological Field Theories from Group Representations

We show that any complex (respectively real) representation of finite group naturally generates a open-closed (respectively Klein) topological field theory over complex numbers. We relate the 1-point correlator for the projective plane in this theory with the Frobenius-Schur indicator on the representation. We relate any complex simple Klein TFT to a real division ring

What Chern-Simons theory assigns to a point

In this note, we answer the questions "What does Chern-Simons theory assign to a point?" and "What kind of mathematical object does Chern-Simons theory assign to a point?". Our answer to the first question is representations of the based loop group. More precisely, we identify a certain class of projective unitary representations of the based loop group $\Omega G$ that we locally normal representations. We define the fusion product of such representations and we prove that, modulo certain conjectures, the Drinfel'd centre of that representation category of $\Omega G$ is equivalent to the category of positive energy representations of the free loop group $LG$. The above mentioned conjectures are known to hold when the gauge group is abelian or of type $A_1$. Our answer to the second question is bicommutant categories. The latter are higher categorical analogs of von Neumann algebras: they are tensor categories that are equivalent to their bicommutant inside $\mathrm{Bim}(R)$, the category of bimodules over a hyperfinite $\mathit{III}_1$ factor. We prove that, modulo certain conjectures, the category of locally normal representations of the based loop group is a bicommutant category. The relevant conjectures are known to hold when the gauge group is abelian or of type $A_n$. Our work builds on the formalism of coordinate free conformal nets, developed jointly with A. Bartels and C. Douglas.Comment: Minor changes. Added a summary diagram at the end. Replaced the terminology "positive energy representations of the based loop group" by "locally normal representations of the based loop group