arXiv subjects
Publications and source records attributed to Matthew B. Young.
We construct relative modular categories from the weighted representation theory of the unrolled quantum group of the orthosymplectic Lie superalgebra $\mathfrak{osp}(2 \vert 2n)$.
This is an introduction to Hall algebras from the perspective of $2$-Segal spaces or decomposition spaces, as introduced by Dyckerhoff and Kapranov and Gálvez-Carrillo, Kock and Tonks, respectively. We explain how linearizations of the $2$-Segal space arising as the Waldhausen $\mathcal{S}_{\bullet}$-construction of a proto-exact category recover various previously known Hall algebras. We use the $2$-Segal perspective to study functoriality of the Hall algebra construction and explain how relative variants of $2$-Segal spaces lead naturally to representations of Hall algebras.
We construct three-dimensional non-semisimple topological field theories from the unrolled quantum group of the Lie superalgebra $\mathfrak{osp}(1 \vert 2)$. More precisely, the quantum group depends on a root of unity $q=e^{\frac{2 \pi \sqrt{-1}}{r}}$, where $r$ is a positive integer greater than $2$, and the construction applies when $r$ is not congruent to $4$ modulo $8$. The algebraic result which underlies the construction is the existence of a relative modular structure on the non-finite, non-semisimple category of weight modules for the quantum group. We prove a Verlinde formula which allows for the computation of dimensions and Euler characteristics of topological field theory state spaces of unmarked surfaces. When $r$ is congruent to $\pm 1$ or $\pm 2$ modulo $8$, we relate the resulting $3$-manifold invariants with physicists' $\widehat{Z}$-invariants associated to $\mathfrak{osp}(1 \vert 2)$. Finally, we establish a relation between $\widehat{Z}$-invariants associated to $\mathfrak{sl}(2)$ and $\mathfrak{osp}(1 \vert 2)$ which was conjectured in the physics literature.
We develop representation theoretic techniques to construct three dimensional non-semisimple topological quantum field theories which model homologically truncated topological B-twists of abelian Gaiotto-Witten theory with linear matter. Our constructions are based on relative modular structures on the category of weight modules over an unrolled quantization of a Lie superalgebra. The Lie superalgebra, originally defined by Gaiotto and Witten, is associated to a complex symplectic representation of a metric abelian Lie algebra. The physical theories we model admit alternative realizations as Chern-Simons-Rozansky-Witten theories and supergroup Chern-Simons theories and include as particular examples global forms of $\mathfrak{gl}(1 \vert 1)$-Chern-Simons theory and toral Chern-Simons theory. Fundamental to our approach is the systematic incorporation of non-genuine line operators which source flat connections for the topological flavour symmetry of the theory.
We construct a two dimensional unoriented open/closed topological field theory from a finite graded group $π:\hat{G} \twoheadrightarrow \{1,-1\}$, a $π$-twisted $2$-cocycle $\hatθ$ on $B \hat{G}$ and a character $λ: \hat{G} \rightarrow U(1)$. The underlying oriented theory is a twisted Dijkgraaf-Witten theory. The construction is based in the $(\hat{G}, \hatθ,λ)$-twisted Real representation theory of $\ker π$. In particular, twisted Real representations are boundary conditions and the generalized Frobenius-Schur element is its crosscap state.
This paper is a self-contained introduction to the theory of renormalized Reshetikhin-Turaev invariants of links defined by Geer, Patureau-Mirand and Turaev. Whereas the standard Reshetikhin-Turaev theory of a $\mathbb{C}$-linear ribbon category assigns the trivial invariant to any link with a component colored by a simple object of vanishing quantum dimension, the renormalized theory does not. We give a streamlined development of the renormalized Reshetikhin-Turaev theory of links for the category of weight modules over the restricted unrolled quantum group of $\mathfrak{sl}_2(\mathbb{C})$ at an even root of unity.
We introduce an unrolled quantization $U_q^E(\mathfrak{gl}(1 \vert 1))$ of the complex Lie superalgebra $\mathfrak{gl}(1 \vert 1)$ and use its categories of weight modules to construct and study new three dimensional non-semisimple topological quantum field theories. These theories are defined on categories of cobordisms which are decorated by ribbon graphs and cohomology classes and take values in categories of graded super vector spaces. Computations in these theories are enabled by a detailed study of the representation theory of $U_q^E(\mathfrak{gl}(1 \vert 1))$, both for generic and root of unity $q$. We argue that by restricting to subcategories of integral weight modules we obtain topological quantum field theories which are mathematical models of Chern--Simons theories with gauge supergroups $\mathfrak{psl}(1 \vert 1)$ and $\mathfrak{gl}(1 \vert 1)$ coupled to background flat $\mathbb{C}^{\times}$-connections, as studied in the physics literature by Rozansky--Saleur and Mikhaylov. In particular, we match Verlinde formulae and mapping class group actions on state spaces of non-generic tori with results in the physics literature. We also obtain explicit descriptions of state spaces of generic surfaces, including their graded dimensions, which go beyond results in the physics literature.
In this paper we construct twisted Real quasi-elliptic cohomology as the twisted KR-theory of loop groupoids. The theory systematically incorporates loop rotation and reflection. After establishing basic properties of the theory, we construct twisted elliptic Pontryagin characters and, without twists, Real analogues of the string power operation of quasi-elliptic cohomology. We also explore the relation of the theory to the Tate curve.
Motivated by periodicity theorems for Real $K$-theory and Grothendieck--Witt theory and, separately, work of Hori-Walcher on the physics of Landau-Ginzburg orientifolds, we introduce and study categories of Real matrix factorizations. Our main results are generalizations of Knörrer periodicity to categories of Real matrix factorizations. These generalizations are structurally similar to $(1,1)$-periodicity for $KR$-theory and $4$-periodicity for Grothendieck-Witt theory. We use techniques from Real categorical representation theory which allow us to incorporate into our main results equivariance for a finite group and discrete torsion twists.
We study the algebraic $K$-theory and Grothendieck-Witt theory of proto-exact categories, with a particular focus on classes of examples of $\mathbb{F}_1$-linear nature. Our main results are analogues of theorems of Quillen and Schlichting, relating the $K$-theory or Grothendieck-Witt theories of proto-exact categories defined using the (hermitian) $Q$-construction and group completion.
We study the algebraic $K$-theory and Grothendieck-Witt theory of proto-exact categories of vector bundles over monoid schemes. Our main results are the complete description of the algebraic $K$-theory space of an integral monoid scheme $X$ in terms of its Picard group $\operatorname{Pic}(X)$ and pointed monoid of regular functions $Γ(X, \mathcal{O}_X)$ and a description of the Grothendieck-Witt space of $X$ in terms of an additional involution on $\operatorname{Pic}(X)$. We also prove space-level projective bundle formulae in both settings.
Given a double cover $π: \mathcal{G} \rightarrow \hat{\mathcal{G}}$ of finite groupoids, we explicitly construct twisted loop transgression maps, $τ_π$ and $τ_π^{ref}$, thereby associating to a Jandl $n$-gerbe $\hatλ$ on $\hat{\mathcal{G}}$ a Jandl $(n-1)$-gerbe $τ_π(\hatλ)$ on the quotient loop groupoid of $\mathcal{G}$ and an ordinary $(n-1)$-gerbe $τ^{ref}_π(\hatλ)$ on the unoriented quotient loop groupoid of $\mathcal{G}$. For $n =1,2$, we interpret the character theory (resp. centre) of the category of Real $\hatλ$-twisted $n$-vector bundles over $\hat{\mathcal{G}}$ in terms of flat sections of the $(n-1)$-vector bundle associated to $τ_π^{ref}(\hatλ)$ (resp. the Real $(n-1)$-vector bundle associated to $τ_π(\hatλ)$). We relate our results to Real versions of twisted Drinfeld doubles and pointed fusion categories and to discrete torsion in orientifold string and M-theory.
We further develop the asymptotic analytic approach to the study of scattering diagrams. We do so by analyzing the asymptotic behavior of Maurer-Cartan elements of a differential graded Lie algebra constructed from a (not-necessarily tropical) monoid-graded Lie algebra. In this framework, we give alternative differential geometric proofs of the consistent completion of scattering diagrams, originally proved by Kontsevich-Soibelman, Gross-Siebert and Bridgeland. We also give a geometric interpretation of theta functions and their wall-crossing. In the tropical setting, we interpret Maurer-Cartan elements, and therefore consistent scattering diagrams, in terms of the refined counting of tropical disks. We also describe theta functions, in both their tropical and Hall algebraic settings, in terms of flat sections of the Maurer-Cartan-deformed differential. In particular, this allows us to give a combinatorial description of Hall algebra theta functions for acyclic quivers with non-degenerate skew-symmetrized Euler forms.
Green's theorem states that the Hall algebra of the category of representations of a quiver over a finite field is a twisted bialgebra. Considering instead categories of orthogonal or symplectic quiver representations leads to a class of modules over the Hall algebra, called Hall modules, which are also comodules. A module theoretic analogue of Green's theorem, describing the compatibility of the module and comodule structures, is not known. In this paper we prove module theoretic analogues of Green's theorem in the degenerate settings of finitary Hall modules of $\mathsf{Rep}_{\mathbb{F}_1}(Q)$ and constructible Hall modules of $\mathsf{Rep}_{\mathbb{C}}(Q)$. The result is that the module and comodule structures satisfy a compatibility condition reminiscent of that of a Yetter-Drinfeld module.
Given a finite $\mathbb{Z}_2$-graded group $\hat{\mathsf{G}}$ with ungraded subgroup $\mathsf{G}$ and a twisted cocycle $\hatλ \in Z^n(B \hat{\mathsf{G}}; \mathsf{U}(1)_π)$ which restricts to $λ\in Z^n(B \mathsf{G}; \mathsf{U}(1))$, we construct a lift of $λ$-twisted $\mathsf{G}$-Dijkgraaf--Witten theory to an unoriented topological quantum field theory. Our construction uses a new class of homotopy field theories, which we call orientation twisted. We also introduce an orientation twisted variant of the orbifold procedure, which produces an unoriented topological field theory from an orientation twisted $\mathsf{G}$-equivariant topological field theory.
We compute the motive of the classifying stack of an orthogonal group in the Grothendieck ring of stacks over a field of characteristic different from two.
We introduce and develop a categorification of the theory of Real representations of finite groups. In particular, we generalize the categorical character theory of Ganter--Kapranov and Bartlett to the Real setting. Given a Real representation of a group $\mathsf{G}$, or more generally a finite categorical group, on a linear category, we associate a number, the modified secondary trace, to each graded commuting pair $(g, ω) \in \mathsf{G} \times \hat{\mathsf{G}}$, where $\hat{\mathsf{G}}$ is the background Real structure on $\mathsf{G}$. This collection of numbers defines the Real $2$-character of the Real representation. We also define various forms of induction for Real representations of finite categorical groups and compute the result at the level of Real $2$-characters. We interpret results in Real categorical character theory in terms of geometric structures, namely gerbes, vector bundles and functions on iterated unoriented loop groupoids. This perspective naturally leads to connections with the representation theory of unoriented versions of the twisted Drinfeld double of $\mathsf{G}$ and with discrete torsion in $M$-theory with orientifold. We speculate on an interpretation of our results as a generalized Hopkins--Kuhn--Ravenel-type character theory in Real equivariant homotopy theory.
We introduce a relative version of the $2$-Segal simplicial spaces defined by Dyckerhoff and Kapranov and Gálvez-Carrillo, Kock and Tonks. Examples of relative $2$-Segal spaces include the categorified unoriented cyclic nerve, real pseudoholomorphic polygons in almost complex manifolds and the $\mathcal{R}_{\bullet}$-construction from Grothendieck-Witt theory. We show that a relative $2$-Segal space defines a categorical representation of the Hall algebra associated to the base $2$-Segal space. In this way, after decategorification we recover a number of known constructions of Hall algebra representations. We also describe some higher categorical interpretations of relative $2$-Segal spaces.