arXiv subjects
Publications and source records attributed to Mikhail Khovanov.
This is a brief introduction to link homology theories that categorify Reshetikhin--Turaev $\mathsf{SL}(N)$-quantum link invariants. A recently discovered surprising connection between finite state automata and Boolean TQFTs in dimension one is explained as a warm-up.
Unoriented SL(3) foams are two-dimensional CW complexes with generic singularities embedded in 3- and 4-manifolds. They naturally come up in the Kronheimer-Mrowka SO(3) gauge theory for 3-orbifolds and, in the oriented case, in a categorification of the Kuperberg bracket quantum invariant. The present paper studies the more technically complicated case of SL(4) foams. Combinatorial evaluation of unoriented SL(4) foams is defined and state spaces for it are studied. In particular, over a suitably localized ground ring, the state space of any web is free of the rank given by the number of its 4-colorings.
The paper studies the Karoubi envelope of a one-dimensional topological theory with defects and inner endpoints, defined over a field. It turns out that the Karoubi envelope is determined by a symmetric Frobenius algebra K associated to the theory. The Karoubi envelope is then equivalent to the quotient of the Frobenius-Brauer category of K modulo the ideal of negligible morphisms. Symmetric Frobenius algebras, such as K, describe two-dimensional TQFTs for the category of thin flat surfaces, and elements of the algebra can be turned into defects on the side boundaries of these surfaces. We also explain how to couple K to the universal construction restricted to closed surfaces to define a topological theory of open-closed two-dimensional cobordisms which is usually not an open-closed 2D TQFT.
This paper clarifies basic definitions in the universal construction of topological theories and monoidal categories. The definition of the universal construction is given for various types of monoidal categories, including rigid and symmetric. It is also explained how to set up the universal construction for non-monoidal categories. The second part of the paper explains how to associate a rigid symmetric monoidal category to a small category, a sort of the Brauer envelope of a category. The universal construction for the Brauer envelopes generalizes some earlier work of the first two authors on automata, power series and topological theories. Finally, the theory of pseudocharacters (or pseudo-representations), which is an essential tool in modern number theory, is interpreted via one-dimensional topological theories and TQFTs with defects. The notion of a pseudocharacter is studied for Brauer categories and the lifting property to characters of semisimple representations is established in characteristic 0 for Brauer categories with at most countably many objects. The paper contains a brief discussion of pseudo-holonomies, which are functions from loops in a manifold to real numbers similar to traces of the holonomies along loops of a connection on a vector bundle on the manifold. It concludes with a classification of pseudocharacters (pseudo-TQFTs) and their generating functions for the category of oriented two-dimensional cobordisms in the characteristic 0 case.
This paper explains how any nondeterministic automaton for a regular language $L$ gives rise to a one-dimensional oriented Topological Quantum Field Theory (TQFT) with inner endpoints and zero-dimensional defects labelled by letters of the alphabet for $L$. The TQFT is defined over the Boolean semiring $\mathbb{B}$. Different automata for a fixed language $L$ produce TQFTs that differ by their values on decorated circles, while the values on decorated intervals are described by the language $L$. The language $L$ and the TQFT associated to an automaton can be given a path integral interpretation. In this TQFT the state space of a one-point 0-manifold is a free module over $\mathbb{B}$ with the basis of states of the automaton. Replacing a free module by a finite projective $\mathbb{B}$-module $P$ allows to generalize automata and this type of TQFT to a structure where defects act on open subsets of a finite topological space. Intersection of open subsets induces a multiplication on $P$ allowing to extend the TQFT to a TQFT for one-dimensional foams (oriented graphs with defects modulo a suitable equivalence relation). A linear version of these constructions is also explained, with the Boolean semiring replaced by a commutative ring.
We consider the odd analogue of the category of Soergel bimodules. In the odd case and already for two variables, the transposition bimodule cannot be merged into the generating Soergel bimodule, forcing one into a monoidal category with a larger Grothendieck ring compared to the even case. We establish biadjointness of suitable functors and develop graphical calculi in the 2-variable case for the odd Soergel category and the related singular Soergel 2-category. We describe the odd analogue of the Rouquier complexes and establish their invertibility in the homotopy category. For three variables, the absence of a direct sum decomposition of the tensor product of generating Soergel bimodules presents an obstacle for the Reidemeister III relation to hold in the homotopy category.
This is a brief review of the categorification of the Jones polynomial and its significance and ramifications in geometry, algebra, and low-dimensional topology.
The paper explains the connection between topological theories for one-manifolds with defects and values in the Boolean semiring and automata and their generalizations. Finite state automata are closely related to regular languages. To each pair of a regular language and a circular regular language we associate a topological theory for one-dimensional manifolds with zero-dimensional defects labelled by letters of the language. This theory takes values in the Boolean semiring. Universal construction of topological theories gives rise in this case to a monoidal category of Boolean semilinear combinations of one-dimensional cobordisms with defects modulo skein relations. The latter category can be interpreted as a semilinear rigid monoidal closure of standard structures associated to a regular language, including minimal deterministic and nondeterministic finite state automata for the language and the syntactic monoid. The circular language plays the role of a regularizer, allowing to define the rigid closure of these structures. When the state space of a single point for a regular language describes a distributive lattice, there is a unique associated circular language such that the resulting theory is a Boolean TQFT.
We categorify a tensor product of two Weyl modules for quantum sl(2) at a prime root of unity.
A pair of biadjoint functors between two categories produces a collection of elements in the centers of these categories, one for each isotopy class of nested circles in the plane. If the centers are equipped with a trace map into the ground field, then one assigns an element of that field to a diagram of nested circles. We focus on the self-adjoint functor case of this construction and study the reverse problem of recovering such a functor and a category given values associated to diagrams of nested circles.
We describe equivariant SL(2) and SL(3) homology for links in the solid torus via foam evaluation. The solid torus is replaced by 3-space with a distinguished line in it. Generators of state spaces for annular webs are represented by foams with boundary that may intersect the distinguished line; intersection points, called anchor points, contribute additional terms, reminiscent of square roots of the Hessian, to the foam evaluation. Both oriented and unoriented SL(3) foams are treated in the paper.
We study generalized Deligne categories and related tensor envelopes for the universal two-dimensional cobordism theories described by rational functions, recently defined by Sazdanovic and one of the authors.
In the unoriented SL(3) foam theory, singular vertices are generic singularities of two-dimensional complexes. Singular vertices have neighbourhoods homeomorphic to cones over the one-skeleton of the tetrahedron, viewed as a trivalent graph on the two-sphere. In this paper we consider foams with singular vertices with neighbourhoods homeomorphic to cones over more general planar trivalent graphs. These graphs are subject to suitable conditions on their Kempe equivalence Tait coloring classes and include the dodecahedron graph. In this modification of the original homology theory it is straightforward to show that modules associated to the dodecahedron graph are free of rank 60, which is still an open problem for the original unoriented SL(3) foam theory.
The paper explores the relation between noncommutative power series and topological theories of one-dimensional cobordisms decorated by labelled zero-dimensional submanifolds. These topological theories give rise to several types of tensor envelopes of noncommutative recognizable power series, including the categories built from the syntactic algebra and syntactic ideals of the series and the analogue of the Deligne category.
We consider recognizable evaluations for a suitable category of oriented two-dimensional cobordisms with corners between finite unions of intervals. We call such cobordisms thin flat surfaces. An evaluation is given by a power series in two variables. Recognizable evaluations correspond to series that are ratios of a two-variable polynomial by the product of two one-variable polynomials, one for each variable. They are also in a bijection with isomorphism classes of commutative Frobenius algebras on two generators with a nondegenerate trace fixed. The latter algebras of dimension n correspond to points on the dual tautological bundle on the Hilbert scheme of n points on the affine plane, with a certain divisor removed from the bundle. A recognizable evaluation gives rise to a functor from the above cobordism category of thin flat surfaces to the category of finite-dimensional vector spaces. These functors may be non-monoidal in interesting cases. To a recognizable evaluation we also assign an analogue of the Deligne category and of its quotient by the ideal of negligible morphisms.
The Deligne category of symmetric groups is the additive Karoubi closure of the partition category. It is semisimple for generic values of the parameter t while producing categories of representations of the symmetric group when modded out by the ideal of negligible morphisms when t is a non-negative integer. The partition category may be interpreted, following Comes, via a particular linearization of the category of two-dimensional oriented cobordisms. The Deligne category and its semisimple quotients admit similar interpretations. This viewpoint coupled to the universal construction of two-dimensional topological theories leads to multi-parameter monoidal generalizations of the partition and the Deligne categories, one for each rational function in one variable.
We consider Blanchet, Habegger, Masbaum and Vogel's universal construction of topological theories in dimension two, using it to produce interesting theories that do not satisfy the usual two-dimensional TQFT axioms. Kronecker's characterization of rational functions allows us to classify theories over a field with finite-dimensional state spaces and introduce their extension to theories with the ground ring the product of rings of symmetric functions in N and M variables. We look at several examples of non-multiplicative theories and see Hankel matrices, Schur and supersymmetric Schur polynomials quickly emerge from these structures. The last section explains how an extension of the Robert-Wagner foam evaluation to overlapping foams gives the Sergeev-Pragacz formula for the supersymmetric Schur polynomials and the Day formula for the Toeplitz determinant of rational power series as special cases.
The first two sections of the paper provide a convenient scheme and additional diagrammatics for working with Frobenius extensions responsible for key flavors of equivariant SL(2) link homology theories. The goal is to clarify some basic structures in the theory and propose a setup to work over sufficiently non-degenerate base rings. The third section works out two related SL(2) evaluations for seamed surfaces.