Related papers: Gerbes and homotopy quantum field theories

In this paper we study cobordism categories consisting of manifolds which are endowed with geometric structure. Examples of such geometric structures include symplectic structures, flat connections on principal bundles, and complex…

By means of cohomology groups, we study relationships between equivariant gerbes with connection over a manifold with a Lie group action and gerbes with connection over the quotient space.

Let P be a connected smooth p-manifold. We describe the group of all cobordism classes of smooth maps of n-manifolds to P with singularities of a given $cal K$-invariant class in terms of certain stable homotopy groups by applying the…

Topological quantum field theories (TQFTs) are symmetric monoidal functors out of cobordism categories. In dimension two, oriented TQFTs are famously classified by commutative Frobenius algebras. In the unoriented setting, the…

We construct a certain `cobordism category' ${\cal D}$ whose morphisms are suitably decorated cobordism classes between similarly decorated closed oriented 1-manifolds, and show that there is essentially a bijection between…

The stable envelopes of Okounkov et al. realize some representations of quantum algebras associated to quivers, using geometry. We relate these geometric considerations to quantum field theory. The main ingredients are the supersymmetric…

We review properties of closed meromorphic $1$-forms and of the foliations defined by them. We present and explain classical results from foliation theory, like index theorems, the existence of separatrices, and resolution of singularities…

In this paper we study the topology of cobordism categories of manifolds with corners. Specifically, if {Cob}_{d,<k>} is the category whose objets are a fixed dimension d, with corners of codimension less than or equal to k, then we…

In p-form lattice gauge theory, the fluctuating variables live on p-dimensional cells and interact around (p+1)-dimensional cells. It has been argued that the continuum version of this model should be described by (p-1)-gerbes. However,…

To a homology theory one can associate an additive site and a new homological functor with values in the category of additive sheaves on that site. If this category of sheaves can be shown to be equivalent to a category of comodules of a…

We give a geometric interpretation of sheaf cohomology for higher degrees n in terms of torsors on the member of degree d=n-1 in hypercoverings of type r=n-2, endowed with an additional data, the so-called rigidification. This generalizes…

We give a functorial definition of $G$-gerbes over a simplicial complex when the local symmetry group $G$ is non-Abelian. These combinatorial gerbes are naturally endowed with a connective structure and a curving. This allows us to define a…

The goal of this work is to describe a categorical formalism for (Extended) Topological Quantum Field Theories (TQFTs) and present them as functors from a suitable category of cobordisms with corners to a linear category, generalizing 2d…

We construct geometrically a gerbe assigned to a connection on a principal SU(2)-bundle over an oriented closed 1-dimensional manifold. If the connection is given by the restriction of a connection on a bundle over a compact 2-manifold…

We show that the space of chains of smooth maps from spheres into a fixed compact oriented manifold has a natural structure of a transversal $d$-algebra. We construct a structure of transversal 1-category on the space of chains of maps from…

The representations of the observable algebra of a low dimensional quantum field theory form the objects of a braided tensor category. The search for gauge symmetry in the theory amounts to finding an algebra which has the same…

Gerbes are locally connected presheaves of groupoids. They are classified up to local weak equivalence by path components in a 2-cocycle category taking values in all sheaves of groups, their isomorphisms and homotopies. If F is a full…

We present a precise definition of extended homotopy quantum field theories and develop an orbifold construction for these theories when the target space is the classifying space of a finite group $G$, i.e. for $G$-equivariant topological…

We first give a brief exposition of our recent realization of anyonic quantum states on single M5-brane probes in 11D super-gravity backgrounds, by non-perturbative quantization of the topological sector of the self-dual tensor field on the…

We introduce a framework for internal topological symmetries in quantum field theory, including "noninvertible symmetries" and "categorical symmetries". This leads to a calculus of topological defects which takes full advantage of…