Mathematics

There is exactly one bosonic (3+1)-dimensional topological order whose only nontrivial particle is an emergent boson: pure Z 2 gauge theory. There are exactly two (3+1)-dimensional topological orders whose only nontrivial particle is an emergent fermion: pure "spin- Z 2 " gauge theory, in which the dynamical field is a spin structure; and an anomalous version thereof. I give three proofs of this classification, varying from hands-on to abstract. Along the way, I provide a detailed study of the braided fusion 2 -category Z (1) (ΣSVec) of string and particle operators in pure spin- Z 2 gauge theory.

We generalize the notion of ends and coends in category theory to the realm of module categories over finite tensor categories. We call this new concept "module (co)end". This tool allows us to give different proofs to several known results in the theory of representations of finite tensor categories. As a new application, we present a description of the relative Serre functor for module categories in terms of a module coend, in a analogous way as a Morita invariant description of the Nakayama functor of abelian categories presented in [J. Fuchs, G. Schaumann and C. Schweigert, Eilenberg-Watts calculus for finite categories and a bimodule Radford S^4 theorem, Trans. Amer. Math. Soc. 373 (2020), 1-40]

In this paper, we study (G, χ ϕ ) -equivariant ϕ -coordinated quasi modules for nonlocal vertex algebras. Among the main results, we establish several conceptual results, including a generalized commutator formula and a general construction of weak quantum vertex algebras and their (G, χ ϕ ) -equivariant ϕ -coordinated quasi modules. As an application, we also construct (equivariant) ϕ -coordinated quasi modules for lattice vertex algebras by using Lepowsky's work on twisted vertex operators.

Let L l =L( sl 2l+1 ,−l− 1 2 ) be the simple vertex operator algebra based on the affine Lie algebra sl ˆ 2l+1 at boundary admissible level −l− 1 2 . We consider a lift ν of the Dynkin diagram involution of A 2l = sl 2l+1 to an involution of L l . The ν -twisted L l -modules are A (2) 2l -modules of level −l− 1 2 with an anti-homogeneous realization. We classify simple ν -twisted highest-weight (weak) L l -modules using twisted Zhu algebras and singular vectors for sl ˆ 2l+1 at level −l− 1 2 obtained by Perše. We find that there are finitely many such modules up to isomorphism, and the ν -twisted (weak) L l -modules that are in category O for A (2) 2l are semi-simple.

Two descriptions of the dual −1 Hahn algebra are presented and shown to be related under Howe duality. The dual pair involved is formed by the Lie algebra o(4) and the Lie superalgebra osp(1|2) .

Murakami and Ohtsuki have shown that the Alexander polynomial is an R -matrix invariant associated with representations V(t) of unrolled restricted quantum sl 2 at a fourth root of unity. In this context, the highest weight t∈ C × of the representation determines the polynomial variable. For any semisimple Lie algebra g of rank n , we extend their construction to a link invariant Δ g , which takes values in n -variable Laurent polynomials. We prove general properties of these invariants, but the focus of this paper is the case g= sl 3 . For any knot K , evaluating Δ sl 3 at t 1 =1 , t 2 =1 , or t 2 =i t −1 1 recovers the Alexander polynomial of K . We emphasize that this is not obvious from an examination of the R -matrix and that our proof requires several representation-theoretic results. We tabulate Δ sl 3 for all knots up to seven crossings along with various other examples. In particular, it distinguishes the Kinoshita-Terasaka knot and Conway knot mutant pair and is nontrivial on the Whitehead double of the trefoil.

A new proof of the fusion rules formula in the context of vertex operator algebra is given. Some more general relations between the space of intertwining operators and A(V) bimodules are obtained.

The moduli spaces of compact and connected Riemann surfaces has been a central topic in modern mathematics in recent years. Thus their homological dimensions become important invariants. Motivated by the emergence mathematical counterparts of open-closed string theory, we give an estimate of rational homological dimension of Riemann suraces with possible boundary and marked points(can lie on both interior and boundary). We hope it will have applications in open-closed theory, for example, open-closed Gromov-Witten theory in the future.

We construct a braid group action on a homotopy category of p -DG modules of a deformed Webster algebra.

We give a proof of the Homotopy Transfer Theorem following Kadeishvili's original strategy. Although Kadeishvili originally restricted himself to transferring a dg algebra structure to an A ∞ -structure on homology, we will see that a small modification of his argument proves the general case of transferring any kind of ∞ -algebra structure along a quasi-isomorphism, under weaker hypotheses than existing proofs of this result.