A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Model for assembly map of bordism-invariant functors.
A modular functor is constructed from non-semisimple 3d TFTs.
We consider a category whose morphisms are bordisms of -dimensional pseudomanifolds equipped with a certain additional structure (coloring). On the other hand, we consider the product of copies of infinite symmetric group. We show that unitary representations of produce functors from the category of …
The book develops a new bordism-theoretic approach to understanding orientations of moduli spaces.
We show that once-extended anomalous 3-dimensional topological quantum field theories valued in the 2-category of k-linear categories are in canonical bijection with modular tensor categories equipped with a square root of the global dimension in each factor.
Paper establishes equivalence between algebraic and functorial QFTs.
The goal of this paper is to study the Pontrjagin dual of (reduced) 4-dimensional Spin bordism. That is to say, we consider the functor from the category of topological spaces to the category of compact abelian groups that associates to each space X the compact group of homomorphisms from the reduced 4-dimensional Spin…
We give a simple sufficient condition for Quinn's "bordism-type spectra" to be weakly equivalent to strictly associative ring spectra. We also show that Poincare bordism and symmetric L-theory are naturally weakly equivalent to monoidal functors. Part of the proof of these statements involves showing that Quinn's funct…
The paper contains a survey of train constructions for infinite symmetric groups and related groups. For certain pairs (a group , a subgroup ), we construct categories, whose morphisms are two-dimensional surfaces tiled by polygons and colored in a certain way. A product of morphisms is a gluing of combinatorial …
Quantum field theory uses Lorentzian bordisms to describe time evolution.
Logarithmic representations of the bordism category are considered as a framework for capturing a class of additive invariants characterising Reidemeister torsions.
We investigate the relationship between the algebra of tensor categories and the topology of framed 3-manifolds. On the one hand, tensor categories with certain algebraic properties determine topological invariants. We prove that fusion categories of nonzero global dimension are 3-dualizable, and therefore provide 3-di…
Study homotopy sheaves on categories and their presheaves, proving descent properties.
Open 2D TFTs extend to closed theories with circle value as Hochschild homology.
In terms of category theory, the Gromov homotopy principle for a set valued functor asserts that the functor can be induced from a homotopy functor. Similarly, we say that the bordism principle for an abelian group valued functor holds if the functor can be induced from a (co)homology functor. We examin…
Study of embedding calculus using infinite operads.
For a fixed closed manifold , we construct a cobordism category of embedded manifolds with a single Baas-Sullivan singularity of type . Our main theorem identifies the homotopy type of the classifying space of this cobordism category with that of the infinite loop-space of a certain spectrum related to the spectr…
Ricci-positive manifolds span the kernel of the -genus in rational Spin bordism.
Smooth FFT from B-fields and D-branes.
For simple and simply-connected complex algebraic group G, we conjecture the existence of a functor eta_G from the category of 2-bordisms to the category of holomorphic symplectic varieties with Hamiltonian action, such that gluing of boundaries corresponds to the holomorphic symplectic quotient with respect to the dia…
A modular tensor category gives rise to a Reshetikhin-Turaev type topological quantum field theory which is defined on 3-dimensional bordisms with embedded -coloured ribbon graphs. We extend this construction to include bordisms with surface defects which in turn can meet along line defects. …
Floer field theory is a construction principle for e.g. 3-manifold invariants via decomposition in a bordism category and a functor to the symplectic category, and is conjectured to have natural 4-dimensional extensions. This survey provides an introduction to the categorical language for the construction and extension…
Solves linearity problem for acyclic groups, bounds Cheeger-Gromov ρ-invariants.
Paper proves orientability of gauge theory moduli spaces using bordism theory.
We prove that smooth 1-dimensional topological field theories over a manifold are equivalent to vector bundles with connection. The main novelty is our definition of the smooth 1-dimensional bordism category, which encodes cutting laws rather than gluing laws. We make this idea precise through a smooth version of Rezk'…
New algebraic framework for studying surfaces in 3-manifolds.
In this paper, we introduce a bordism category whose objects are bundles of closed -dimensional piecewise linear manifolds and whose morphisms are bundles of -dimensional piecewise linear cobordisms. In the main theorem of this article, we show that the classifying space $B\mathcal{C}_d^{…
Extends string-net theory to 3D TQFT via surface graphs and surgery.
We study bordism groups and bordism homology theories based on pseudomanifolds and stratified pseudomanifolds. The main seam of the paper demonstrates that when we uses classes of spaces determined by local link properties, the stratified and unstratified bordism theories are identical; this includes the known examples…
Goosen extends TQFTs using generators and relations for a simple example.
New perspective on APS indices preserves orientations and gradings through bordisms.
We call a Morse function on a closed manifold -constrained if neither nor has critical points of indefinite Morse index . In this paper we study bordism groups of -constrained Morse functions, and thus interpolate between the case of bordism groups of Morse functions (computed by Ikegami…
This note provides a computation of the bordism groups of K-Witt spaces for fields K with characteristic 2. We provide a complete computation for the unoriented bordism groups. For the oriented bordism groups, a nearly complete computation is provided as well a discussion of the difficulty of resolving a remaining ambi…
The group of bordism classes of unoriented surfaces in 4-space is determined. The bordism classes are characterized by normal Euler numbers,double linking numbers, and triple linking numbers.
Study PL bordism theories with quantitative bounds on filling simplices.
We introduce a framework, twisted parametrized stable homotopy theory, for describing semi-infinite homotopy types. A twisted parametrized spectrum is a section of a bundle whose fibre is the category of spectra. We define these bundles in terms of modules over a stack of parametrized spectra and in terms of diagrams o…
We introduce characteristics into chromatic homotopy theory. This parallels the prime characteristics in number theory as well as in our earlier work on structured ring spectra and unoriented bordism theory. Here, the K(n)-local Hopkins-Miller classes take the places of the prime numbers, and this allows us to di…
Global group laws connect equivariant bordism rings to formal group laws.
This paper shows that, away from 6, the kernel of the Witten genus is precisely the ideal consisting of (bordism classes of) Cayley plane bundles with connected structure group, but only after restricting the Witten genus to string bordism. It does so by showing that the divisibility properties of Cayley plane bundle c…
We call a closed, connected, orientable manifold in one of the categories TOP, PL or DIFF chiral if it does not admit an orientation-reversing automorphism and amphicheiral otherwise. Moreover, we call a manifold strongly chiral if it does not admit a self-map of degree -1. We prove that there are strongly chiral, smoo…
We define new bordism and spin bordism invariants of certain subgroups of the mapping class group of a surface. In particular, they are invariants of the Johnson filtration of the mapping class group. The second and third terms of this filtration are the well-known Torelli group and Johnson subgroup, respectively. We i…
Proves nearby Lagrangian cocores are homotopically rigid in certain dimensions.
The groups of link bordism can be identified with homotopy groups via the Pontryagin-Thom construction. B.J. Sanderson computed the bordism group of 3 component surface-links using the Hilton-Milnor Theorem, and later gave a geometric interpretation of the groups in terms of intersections of Seifert hypersurfaces and t…
Generalised characteristic classes are constructed for bordism cohomologies which allow a natural extension of classical genera to these bordism cohomology rings taking values in singular cohomology.
We discuss an universal bordism invariant obtained from the Atiyah-Patodi-Singer eta-invariant from the analytic and homotopy theoretic point of view. Classical invariants like the Adams e-invariant, -invariants and -bordism invariants are derived as special cases. The main results are a secondary index theo…
Chern-Simons and Reshetikhin-Turaev theories are shown equivalent.
Constructs differential models for twisted Spin^c-bordism and its dual, defining a new anomaly map.
By results of Loeffler and Comezana, the Pontrjagin-Thom map from geometric G-equivariant bordism to homotopy theoretic equivariant bordism is injective for compact abelian G. If G = S^1 x ... x S^1, we prove that the associated fixed point square is a pull back square, thus confirming a recent conjecture of D. Sinha. …