arXiv research

Study non-semisimple TQFT for Burau representation density and unitarity.

New skein categories for non-semisimple settings, extending existing theory.

Constructs maps on skein modules using non-semisimple quantum invariants.

New pseudo-Hermitian models from non-semisimple TQFTs.

Proves Witten-Reshetikhin-Turaev 3-TQFT as a boundary condition of Crane-Yetter 4-TQFT.

ETQFTs created from non-semisimple modular categories.

Introduces admissible skein modules for non-semisimple categories.

New non-semisimple Ising anyons enable robust universal quantum computation.

A Hermitian TQFT from non-semisimple quantum sl(2) modules.

We use modified traces to renormalize Lyubashenko's closed 3-manifold invariants coming from twist non-degenerate finite unimodular ribbon categories. Our construction produces new topological invariants which we upgrade to 2+1-TQFTs under the additional assumption of factorizability. The resulting functors provide mon…

Restricts quantum representations of mapping class groups to integral coefficients.

Study projective representations from non-semisimple TQFTs on surfaces.

Develops Hermitian TQFTs from quantum groups, defining new topological phases.

The paper explores how the unit inclusion affects topological quantum field theories in non-semisimple categories.

New relation between quantum groups and BPS series.

New invariant calculates 4-manifolds using trisection diagrams and combings.

Recent work extends Turaev's modular categories to non-semisimple settings.

The paper defines new TQFTs from non-semisimple categories and proves spherical categories are chromatic.

New proof and formula linking fusion trees to quantum knot invariants.

Defines a new 3D TQFT from non-semisimple categories.

We develop the general theory for the construction of Extended Topological Quantum Field Theories (ETQFTs) associated with the Costantino-Geer-Patureau quantum invariants of closed 3-manifolds. In order to do so, we introduce relative modular categories, a class of ribbon categories which are modeled on representations…

A modular functor is constructed from non-semisimple 3d TFTs.

Study para-Sasakian φ-symmetric spaces using Boothby-Wang fibration.

Quantum invariants for fibered links determined by genus and Hopf invariant.

We show that unrolled quantum groups at odd roots of unity give rise to relative modular categories. These are the main building blocks for the construction of 1+1+1-TQFTs extending CGP invariants, which are non-semisimple quantum invariants of closed 3-manifolds decorated with ribbon graphs and cohomology classes. Whe…

Constructs TQFTs for cobordisms with cohomology class decorations.

New invariants derived from Kauffman bracket for 3-manifolds.

New (3+1) TQFTs created from non-semisimple categories.

Study on special coordinates for Dubrovin-Frobenius manifolds in low dimensions.

The paper solves a problem in constructing a bicategory of algebra bundles.

Generalizes string-net modular functors to non-spherical categories.

Constructs 3D topological field theories from a specific quantum group, linking to physics invariants.

This survey covers some of the results contained in the papers by Costantino, Geer and Patureau (https://arxiv.org/abs/1202.3553) and by Blanchet, Costantino, Geer and Patureau (https://arxiv.org/abs/1404.7289). In the first one the authors construct two families of Reshetikhin-Turaev-type invariants of 3-manifolds, $\…

We construct some explicit quasihomogeneous algebraic solutions to the associativity (WDVV) equations by using analytical methods of the finite gap integration theory. These solutions are expanded in the uniform way to non-semisimple Frobenius manifolds.

New invariants from symplectic fermions categorify link and manifold structures.

We study cohomogeneity one Riemannian manifolds and we establish some simple criterium to test when a singular orbit is totally geodesic. As an application, we classify compact, positively curved Riemannian manifolds which are acted on isometrically by a non semisimple Lie group with an hypersurface orbit.

We prove a 20-year-old conjecture concerning two quantum invariants of three manifolds that are constructed from finite dimensional Hopf algebras, namely, the Kuperberg invariant and the Hennings-Kauffman-Radford invariant. The two invariants can be viewed as a non-semisimple generalization of the Turaev-Viro-Barrett-W…

We find and propose an explanation for a large variety of modularity-related symmetries in problems of 3-manifold topology and physics of 3d $\mathcal{N}=2$ theories where such structures a priori are not manifest. These modular structures include: mock modular forms, $SL(2,\mathbb{Z})$ Weil representations, quantum mo…

In this paper we study $F$-manifolds equipped with multiple flat connections (and multiple $F$-products), that are required to be compatible in a suitable sense. In the semisimple case we show that a necessary condition for the existence of such multiple flat connections can be expressed in terms of the integrability o…

New theory captures framing anomaly in gauge theory.

Coloured Alexander polynomials form a sequence of non-semisimple quantum invariants coming from the representation theory of the quantum group $U_q(sl(2))$ at roots of unity. This sequence recovers the original Alexander polynomial as the first term. We give a topological model for this invariants, showing that they ca…

Study of 4D symmetric spaces with (2,2) signature.

The long-standing Alekseevskii conjecture states that a connected homogeneous Einstein space G/K of negative scalar curvature must be diffeomorphic to R^n. This was known to be true only in dimensions up to 5, and in dimension 6 for non-semisimple G. In this work we prove that this is also the case in dimensions up to …

Riemannian and pseudo-Riemannian symmetric spaces with semisimple transvection group are known and classified for a long time. Contrary to that the description of pseudo-Riemannian symmetric spaces with non-semisimple transvection group is an open problem. In the last years some progress on this problem was achieved. I…

Direct formula found for ADO invariants from homological representations.

We construct non-semisimple $2+1$-TQFTs yielding mapping class group representations in Lyubashenko's spaces. In order to do this, we first generalize Beliakova, Blanchet and Geer's logarithmic Hennings invariants based on quantum $\mathfrak{sl}_2$ to the setting of finite-dimensional non-degenerate unimodular ribbon H…

A formula connects two algebraic structures derived from a category.

New invariant from Viro's gl(1|1) polynomial distinguishes lens spaces.