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.
Quantum trace map connects Teichmüller theory and quantum groups.
New representations of loop braid group from higher gauge theory effects.
We classify higher-SPTs and their anomalies via cobordism theory.
Researchers create projective representations of Hecke groups using TQFT.
Cone structures in quantum field theory linked to information geometry.
Quantum Frobenius map for skein modules constructed and described.
We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show that these invariants coincide with previous invariants defined by Khovanov for sl(2) and sl(3) and by Mazorchuk-Stroppel and Sussan for sl(n). Our technique uses categorifications of the tensor produ…
We prove that either the images of the mapping class groups by quantum representations are not isomorphic to higher rank lattices or else the kernels have a large number of normal generators. Further we show that the images of the mapping class groups have nontrivial 2-cohomology, at least for small levels. For this pu…
Explains model structures for higher orbifolds and applies them to quantum cohomology.
We show that Khovanov homology (and its sl(3) variant) can be understood in the context of higher representation theory. Specifically, we show that the combinatorially defined foam constructions of these theories arise as a family of 2-representations of categorified quantum sl(m) via categorical skew Howe duality. Uti…
This paper is the continuation of Part I, expanding previous results of math.DG/9803051. This paper uses techniques in noncommutative geometry as developed by Alain Connes in order to study the twisted higher index theory of elliptic operators on orbifold covering spaces of compact good orbifolds, which are invariant u…
The Hennings invariant for the small quantum group associated to an arbitrary simple Lie algebra at a root of unity is shown to agree with Jones- Witten-Reshetikhin-Turaev invariant arising from Chern-Simons filed theory for the same Lie algebra and the same root of unity on all integer homol- ogy three-spheres, at roo…
The purpose of this contribution is to point out connections between recent ideas about gerbes and gerbal actions (as higher categorical extension of representation theory) and old discussion in quantum field theory on commutator anomalies, gauge group extensions, and 3-cocycles. The unifying concept is the classical o…
The paper studies properties of stated SL(n)-skein algebras and their centers.
We propose a method for determining the spins of BPS states supported on line defects in 4d theories of class S. Via the 2d-4d correspondence, this translates to the construction of quantum holonomies on a punctured Riemann surface . Our approach combines the technology of spectral networks…
New insights into 4d YM and 5d topological field theories with higher symmetries.
The volume conjecture states that for a hyperbolic knot K in the three-sphere S^3 the asymptotic growth of the colored Jones polynomial of K is governed by the hyperbolic volume of the knot complement S^3\K. The conjecture relates two topological invariants, one combinatorial and one geometric, in a very nonobvious, no…
This paper studies a particular class of higher order conformally invariant dif- ferential operators and related integral operators acting on functions taking values in particular finite dimensional irreducible representations of the Spin group. The differential operators can be seen as a generalization to higher spin …
Unified geometric framework for quantum states using dual number algebras.
Higher gauge theory via differential nonabelian cohomology
The twisted Connes-Moscovici higher index theorem is generalized to the case of good orbifolds. The higher index is shown to be a rational number, and in fact non-integer in specific examples of 2-orbifolds. This results in a non-commutative geometry model that predicts the occurrence of fractional quantum numbers in t…
The quantum field theory of two-dimensional sigma models with bulk and boundary couplings provides a natural framework to realize and unite different species of geometric flows that are of current interest in mathematics. In particular, the bulk renormalization group equation gives rise to the Ricci flow of target spac…
We summarize our axioms for higher categories, and describe the blob complex. Fixing an n-category C, the blob complex associates a chain complex B_*(W;C)$ to any n-manifold W. The 0-th homology of this chain complex recovers the usual topological quantum field theory invariants of W. The higher homology groups should …
We discuss an approach to quantum gerbes over quantum groups in terms of q-deformation of transition functions for a loop group bundle. The case of the quantum group SUq(2) is treated in some detail.
New quantum invariants derived from unrolled quantum groups match existing Hennings invariants.
New quantum states capture more information, enabling advanced processing tasks.
Restricts quantum representations of mapping class groups to integral coefficients.
Our main result is that the image of the quantum representation of a central extension of the mapping class group of the genus closed orientable surface at a prime is a Zariski dense discrete subgroup of some higher rank algebraic semi-simple Lie group defined over $\Q$. As an applicat…
Analyzes quantization of flux observables in gauge theories.
In this article we give examples which show that the TQFT representations of the mapping class groups derived from quantum SU(N) for N>2 are generically decomposable. One general decomposition of the representations is induced by the symmetry which exchanges SU(N) representation labels by their conjugates. The respecti…
Proposes a topological framework to study modular invariants and related concepts.
It is shown that there is a -algebraic quantum group related to any double Lie group. An algebra underlying this quantum group is an algebra of a differential groupoid naturally associated with a double Lie group
Quantum theory constructs a group and skein module for knot complements.
Madelung transform connects quantum and fluid dynamics via symplectic geometry.
Quantum isometry groups exist for certain metric spaces.
Quantum theory of curved tetrahedrons yields quantum group intertwiners.
A quantum framework optimizes collateral allocation for derivatives.
Under several geometric conditions imposed below, the existence of the discrete spectrum below the essential spectrum is shown for the Dirichlet Laplacian on the quantum layer built over a spherically symmetric hypersurface with a pole embedded in the Euclidean space R4. At the end of this paper, we also show the advan…
The bundle approach and n-contextuality reveal quantum model contextuality.
Physical proof of Floer homology and Verlinde formula using N=2 gauge theory.
Study centers of quantum tori and skein algebras for even roots of unity.
Quantum computing uses Bianchi groups to build gates.
Quantum affine bundles are quantum principal bundles with affine quantum structure groups. A general theory of quantum affine bundles is presented. In particular, a detailed analysis of differential calculi over these bundles is performed, including the description of a natural differential calculus over the structure …
If a compact quantum group acts faithfully and smoothly (in the sense of Goswami 2009) on a smooth, compact, oriented, connected Riemannian manifold such that the action induces a natural bimodule morphism on the module of sections of the co-tangent bundle, then it is proved that the quantum group is necessarily commut…
Quantum groups created from disk configuration space homologies.
New model calculates Wilson surfaces in higher gauge theory.
We generalize the asymptotic faithfulness of the skein quantum representations of mapping class groups of orientable closed surfaces to skein . Skein quantum representations of mapping class groups are different from the Reshetikin-Turaev ones from quantum groups or geometric quantization because they ar…
We expose a K-theoretic approach to study group C*-algebras and C*-algebraic compact quantum groups: 1. The conception of multidimensional geometric quantization and the index of group C*-algebras; 2. the entire homology of noncommutative de Rham currents and the noncommutative Chern characters, and their computation f…