Universal construction of topological theories in two dimensions

Abstract

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. Contents 1. Universal construction in n dimensions 1 2. Universal construction in two dimensions 6 2.1. State space of a circle and the generating function 6 2.2. State space and the Hankel matrix 9 2.3. Rational theories over a field. 12 2.4. Semi-universal rational theories α M,N 13 2.5. One-circle state space for a polynomial generating function. 19 2.6. State spaces for unions of circles 22 2.7. Example of the constant generating function 26 2.8. Free theory 31 2.9. More examples 32 2.10. Recursive sequences 38 3. Overlapping theta-foams and the Sergeev-Pragacz formula 40 References 53 1. Universal construction in n dimensionsConsider the tensor category Cob n of oriented n-dimensional cobordisms. Its objects are oriented closed (n − 1)-manifolds N . Morphisms from N 0 to N 1 are equivalence classes of oriented compact n-manifolds M with ∂M = (−N 0 ) N 1 , and the equivalence relation is diffeomorphism rel boundary. This is a symmetric tensor category, and by a tensor category in this note we will mean a symmetric tensor category.

Two-dimensional topological theories, rational functions and their tensor envelopes

“…also denoted φ. This isomorphism intertwines nondegenerate R-valued bilinear forms (, ) α and (, ) α on these spaces and shows that α and α define equivalent topological theories as defined in [Kh1].…”

“…Object A is unknown to us, but we can observe values of closed cobordisms, which are α n for a connected genus n cobordism. Then the universal pairing construction of [Kh1,KS] in dimension two (and its counterpart [BHMV] in three dimensions) consists of recovering a minimal model for X and C from the closed cobordism data. This toy example in two dimensions can be compared to more complicated reconstructions in control theory.…”

“…Here C and C are k-linear symmetric monoidal categories. Such algebras and categories exists in view of [KS,Kh1].…”

“…As observed by Comes [C], there is a functor from the category Cob 2 of oriented twodimensional cobordisms between one-manifolds to the Deligne category. Modifications of this functor, coupled with the universal construction of two-dimensional topological theories [BHMV,Kh1], lead to generalizations of the Deligne category Rep (S t ) and of its quotient Rep (S t ) by the negligible ideal [KS].…”

“…It is shown in [KS] that the hom spaces in Cob α are finite-dimensional over k iff the generating function Z α (T ) is rational, see formula (2), although the category Cob α for any sequence α. Notation (8) A α (n) := Hom Cobα (0, n) is used in [Kh1] to denote the state space of n circles in the theory associated to α. Vector space A α (n) can be described as the quotient of the k-vector space with a basis {[S]} S of viewable cobordisms S with n boundary circles modulo skein relations that hold in the evaluation α for any closure of these n circles by a cobordism T on the other size, so that T S are closed surfaces and evaluations α(T S) make sense as elements of k. These state spaces are finite-dimensional for rational sequences α.…”