David N. Yetter

ON ALGEBRAIC STRUCTURES IMPLICIT IN TOPOLOGICAL QUANTUM FIELD THEORIES

We show that any 3D topological quantum field theory satisfying physically reasonable factorizability conditions has associated to it in a natural way a Hopf algebra object in a suitable tensor category. We also show that all objects in the tensor category have the structure of left-left crossed bimodules over the Hopf algebra object. For 4D factorizable topological quantum filed theories, we provide by analogous methods a construction of a Hopf algebra category.

Measurable Categories and 2-Groups

Using the theory of measurable categories developed in [10], we provide a notion of representations of 2-groups better suited to physically and geometrically interesting examples than that using 2-VECT (cf. [8]). Using this theory we sketch a 2-categorical approach to the state-sum model for Lorentzian quantum gravity proposed in [6], and suggest state-integral constructions for 4-manifold invariants.

TOPOLOGICAL QUANTUM FIELD THEORIES ASSOCIATED TO FINITE GROUPS AND CROSSED G-SETS

Using methods suggested by the work of Turaev and Viro [11, 12], we provide a detailed construction of topological quantum field theories associated to finite crossed G-sets. Our construction of theories associated to finite groups fills in some details implicit in Dijkgraaf and Wittens [3] discussion of topological gauge theories with finite gauge group, while the theories associated to finite crossed G-sets simultaneously extend Dijkgraaf and Wittens [3] results to 3-manifolds equipped with links and Freyd and Yetters [5] construction of link invariants from crossed G-sets from links in the 3-sphere to links in arbitrary 3-manifolds. Topological interpretations of the manifold and link invariants associated to these TQFTs are provided. We conclude discussion of our results as a toy model for QFT and of their relation to quantum groups.

TQFT's from homotopy 2 types

Using techniques from [14] we construct topological quantum field theories using an algebraic model of a homotopy 2-type as initial data.

GENERALIZED BARRETT–CRANE VERTICES AND INVARIANTS OF EMBEDDED GRAPHS

In [1] Barrett and Crane introduce a modification of the generalized CraneYetter state-sum (cf. [2]) based on the category of representations of Spin(4) ∼= SU(2)×SU(2), which provides a four-dimensional analogue of Regge and Ponzano’s [9] spin-network formulation of three-dimensional gravity. The key to the modification of the Crane-Yetter state-sum is the use of a different intertwiner between the “inbound” and “outbound” tensor products of objects assigned to faces of the tetrahedron, thereby imposing a “quantum analogue” of the condition that the sum of the simple bivectors represented by two faces with a common edge is itself simple. The purposes of this paper are

State-sum invariants of 3-manifolds associated to artinian semisimple tortile categories

Abstract The method of Turaev and Viro is generalized to construct state-sum invariants of 3-manifolds using an artinian semisimple tortile category as initial data. In the first two sections of this paper we lay the topological and algebraic groundwork for the construction of a large class of (2+1)-dimensional TQFTs and their associated 3-manifold invariants. Then, following Turaev and Viro [21], we will first construct the 3-manifold invariants, and then handle the “relative case” to obtain TQFTs. In doing so, we isolate the analogues of the (classical or quantum) 6j-symbols in any artinian semisimple tortile category over an arbitrary field, in the correct normalization to repeat the construction of Turaev and Viro [21] with some modifications. Throughout, we assume all manifolds and homeomorphisms are piecewise linear and write compositions in diagrammatic order.

Quandles and Monodromy

We show that a variety of monodromy phenomena arising in geometric topology and algebraic geometry are most conveniently described in terms of homomorphisms from a(n augmented) knot quandle associated with the base to a suitable (augmented) quandle associated with the fiber. We consider the cases of the monodromy of a branched covering, braid monodromy and the monodromy of a Lefschetz fibration. The present paper is an expanded and corrected version of [1].

Abelian categories of modules over a (lax) monoidal functor

Abstract Crane and Yetter (Deformations of (bi)tensor categories, Cahier de Topologie et Geometrie Differentielle Categorique, 1998) introduced a deformation theory for monoidal categories. The related deformation theory for monoidal functors introduced by Yetter (in: E. Getzler, M. Kapranov (Eds.), Higher Category Theory, American Mathematical Society Contemporary Mathematics, Vol. 230, American Mathematical Society, Providence, RI, 1998, pp. 117–134.) is a proper generalization of Gerstenhabers deformation theory for associative algebras (Ann. Math. 78(2) (1963) 267; 79(1) (1964) 59; in: M. Hazewinkel, M. Gerstenhaber (Eds.), Deformation Theory of Algebras and Structure and Applications, Kluwer, Dordrecht, 1988, pp. 11–264). In the present paper we solidify the analogy between lax monoidal functors and associative algebras by showing that under suitable conditions, categories of functors with an action of a lax monoidal functor are abelian categories. The deformation complex of a monoidal functor is generalized to an analogue of the Hochschild complex with coefficients in a bimodule, and the deformation complex of a monoidal natural transformation is shown to be a special case. It is shown further that the cohomology of a monoidal functor F with coefficients in an F,F-bimodule is given by right derived functors.

The Area of the Medial Parallelogram of a Tetrahedron

The midpoints of any four edges of a Euclidean tetrahedron that form a cycle are coplanar, and are the vertices of a parallelogram. The purpose of this note is to derive a simple formula for the area of this medialparallelogram of a tetrahedron in terms of the lengths of the six edges. It would appear that this result is either new or long-forgotten. Despite the very classical nature of the problem our formula solves, there is some serious contemporary interest arising from recently proposed simplicial models for quantum gravity, in which such a formula is needed to approach the problem of length operators; see [1], [2]. Consider a tetrahedron with edge-lengths as in Figure 1. Fix a pair of nonincident edges, say those of lengths e and f. It is then easy to see that the midpoints of the remaining four edges lie in a plane parallel to both of the chosen edges, and equidistant from the planes containing each chosen edge and parallel to both, and that they form the vertices of a parallelogram in this plane.

On 2-dimensional Dijkgraaf-Witten theory with defects

In this paper, we provide a construction of a state-sum model for finite gauge-group Dijkgraaf-Witten theory on surfaces with codimension 1 defects. The construction requires not only that the triangulation be subordinate to the filtration, but flag-like: each simplex of the triangulation is either disjoint from the defect curve, or intersects it in a closed face. The construction allows internal degrees of freedom in the defect curves by introducing a second gauge-group from which edges of the curve are labeled in the state-sum construction. Edges incident with the defect, but not lying in it, have states lying in a set with commuting actions of the two gauge-groups. We determine the appropriate generalizations of the 2-cocycles specifying twistings of defect-free 2D Dijkgraaf-Witten theory. Examples arising by restriction of group 2-cocycles, and constructed from characters of the 2-dimensional guage group are presented. This research was carried out at Summer Undergraduate Mathematics Research (SUMaR) math REU at Kansas State University, funded by NSF under DMS award #1262877.