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David N. Yetter

Publications and source records attributed to David N. Yetter.

16 recordsLinked to original sources
arXiv2024

On Profinite Quandles

We undertake the study of profinite quandles. We provide several constructions of profinite quandles from profinite groups, and from other profinite quandle. We characterize which subquandles of profinite quandles are again profinite. Finally, we provide a characterization of algebraically connected profinite quandles in terms of the profinite completion of their inner automorphism groups $\widehat{\Inn(Q)}$. It is anticipated that the results herein will find applications to the étale homotopy theory of number fields. v.2 has been updated to include an example due to Ariel Davis settling in the negative the question of whether all Stone topological quandles are profinite.

math.GT↗
arXiv2019

Tetrahedral Geometry from Areas

We solve a very classical problem motivated by considerations in quantum gravity: providing a description of the geometry of a Euclidean tetrahedron from the initial data of the areas of the faces and the areas of the medial parallelograms of Yetter or equivalently of the pseudofaces of McConnell. In particular, we derive expressions for the dihedral angles, face angles and (an) edge length, the remaining parts being derivable by symmetry or by identities in the classic 1902 compendium of results on tetrahedral geometry of Richardson. We also provide an alternative proof using (bi)vectors of the result of Yetter that four times the sum of the squared areas of the medial parallelograms is equal to the sum of the squared areas of the faces. The updated version corrects several errors and provides better historical context for the result.

gr-qc↗
arXiv2019

The Area of the Medial Parallelogram of a Tetrahedron

When a pair of non-incident edges of a tetrahedron is chosen, the midpoints of the remaining 4 edges are the vertices of a planar parallelogram. A formula is given in terms of the six edge lengths for the area of this parallelogram. It is not claimed that this formula is new, but it is certainly not well-known. The author would be very grateful for a citation to an occurence of the formula in previously existing literature (by e-mail to dyetter@math.ksu.edu). The result is of some current interest due to the work of Barbieri and Barrett/Crane on attempts to formulate simplicial versions of quantum gravity. The new version corrects the coefficient in the main result. A paper of the same title but with a vectorial version of the derivation of the area formula was published by the author in the American Mathematical Monthly 106 (10) December 1999 pp. 956-598.

math.MG↗
arXiv2015

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.

math.QA↗
arXiv2014

Discrete Conduche Fibrations and C*-algebras

The higher rank graphs of Kumjian and Pask are discrete Conduche fibrations over the monoid of k-tuples of natural numbers for some k in which every morphism in the base has a finite preimage under the the fibration. We examine the generalization of this construction to discrete Conduche fibrations with the same finiteness condition and a lifting property for completions of cospans to commutative squares, over any category satisfying a strong version of the right Ore condition, including all categories with pullbacks and right Ore categories in which all morphisms are monic.

math.OA↗
arXiv2014

Moves on Filtered PL Manifolds and Stratified PL Spaces

We extend results of Pachner and Casali to give finite sets of moves relating triangulations of PL manifolds respecting filtrations by locally flat manifolds and stratifications in which a finite family of simple local models exists for neighborhoods of strata.

math.GT↗
arXiv2001

Abelian Categories of Modules over a (Lax) Monoidal Functor

The analogy between Yetter's deformation theory form (lax) monoidal functors and Gerstenahaber's deformation theory for associative algebras is solidified by shown that under reasonable conditions the category of functors with an action of a lax monoidal functor is abelian, that an analogue of the Hochschild cohomology of an algebra with coefficients in a bimodule exists for monoidal functors, and is given by right derived functors. The deformation cohmology of a monoidal natural transformation is shown to be a special case.

math.CT↗
arXiv1998

Generalized Barrett-Crane Vertices and Invariants of Embedded Graphs

We describe q-analogues of the 4-vertices in the Spin(4)-recoupling theory introduced by Barrett and Crane in gr-qc/9709028 using Kauffman-Lins SU(2)-recoupling theory in each factor and generalize them to obtain operators with the symmetry properties of unframed n-vertices. The elementary properties of the resulting invariants of embedded unframed graphs are examined.

math.QA↗
arXiv1997

Braided Deformations of Monoidal Categories and Vassiliev Invariants

Braided deformations of (symmetric) monoidal categories are related to Vassiliev theory by a direct generalization of well-known results relating "quantum" knot invariants to Vassiliev invariants. The deformation theory of braidings is subsumed by the deformation theory of monoidal functors, which proves surprisingly rich: the deformation complex of a monoidal functor has the same structure as the deformation complex of an algebra, including a pre-Lie structure, from which it is see that the problem of deforming monoidal functors (including braidings) is purely cohomological in nature.

q-alg↗
arXiv1996

Examples of categorification

We construct tensor and bitensor categories with given Grothedieck rig (fusion algebra) in simple cases. The results provide examples on which to test the conjectural construction of 4-D TQFT's proposed by Crane and Frenkel and shed light on several other constructions of TQFT's.

q-alg↗
arXiv1994

State-Sum Invariants of 4-Manifolds I

We provide, with proofs, a complete description of the authors' construction of state-sum invariants announced in [CY], and its generalization to an arbitrary (artinian) semisimple tortile category. We also discuss the relationship of these invariants to generalizations of Broda's surgery invariants [Br1,Br2] using techniques developed in the case of the semi-simple sub-quotient of $Rep(U_q(sl_2))$ ($q$ a principal $4r^{th}$ root of unity) by Roberts [Ro1]. We briefly discuss the generalizations to invariants of 4-manifolds equipped with 2-dimensional (co)homology classes introduced by Yetter [Y6] and Roberts [Ro2], which are the subject of the sequel. (citations refer to bibliography in the paper)

hep-th↗
arXiv1993

On the Classicality of Broda's SU(2) Invariant of 4-manifolds

Recent work of Roberts has shown that the first surgical 4-manifold invariant of Broda and (up to an unspecified normalization factor) the state-sum invariant arising from the TQFT of Crane-Yetter are equivalent to the signature of the 4-manifold. Subsequently Broda defined another surgical invariant in which the 1- and 2-handles are treated differently. We use a refinement of Roberts' techniques developped by the authors in hep-th/9309063 to show that the "improved" surgical invariant of Broda also depends only on the signature and Euler character

hep-th↗
arXiv1993

Evaluating the Crane-Yetter Invariant

We provide an explicit formula for the invariant of 4-manifolds introduced by Crane and Yetter (in hep-th 9301062). A consequence of our result is the existence of a combinatorial formula for the signature of a 4-manifold in terms of local data from a triangulation. Potential physical applications of our result exist in light of the fact that the Crane-Yetter invariant is a rigorous version of ideas of Ooguri on B wedge F theory.

hep-th↗
arXiv1993

We Are Not Stuck With Gluing

We show that the construction of Ocneanu, which yields 1 for any 4D manifold, is not identical to our construction, which gives different numbers for different manifolds.

hep-th↗
arXiv1993

A categorical construction of 4D TQFTs

We construct a four dimensional topological Quantum Field Theory from a modular tensor category. We complete the proof in the case of SU(2)q at a root of unity. Our construction may be important in the physical interpretation of the Chern Simons state in the Ashtekar variables.

hep-th↗