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We develop a combinatorial theory of vector bundles with connection on locally ordered simplicial complexes. This is a first step towards a discrete exterior calculus for bundle-valued forms. The basic building block is the discrete…

We give criteria for determining the positivity of line bundles coming from vertex operator algebras (VOAs) on the moduli space $\overline{\mathrm{M}}_{0,n}$ of rational curves with $n$ marked points. The criteria use the multiplicative…

The idea of "summing over all intermediate states" that is central for implementing locality in quantum systems can be realized by coend constructions. In the concrete case of systems of conformal blocks for a certain class of conformal…

We survey some recent work on conformal blocks in genus zero, focussing on (1) Chern classes, global generation and morphisms, and (2) the Knizhnik--Zamolodchikov connection on conformal blocks (and invariants), their motivic realizations,…

We study flat vector bundles over complex parallelizable manifolds.

This is a set of lecture notes on the operator algebraic approach to 2-dimensional conformal field theory. Representation theoretic aspects and connections to vertex operator algebras are emphasized. No knowledge on operator algebras or…

The Courant bracket defined originally on the sections of the vector bundle $TM \oplus T^*M \to M$ is extended to the direct sum of the 1-jet vector bundle and its dual. The extended bracket allows to interpret many structures encountered…

The main purpose of this paper is to study the vector groupoids. This is an algebraic structure which combines the concepts of Brandt groupoid and vector space such that these are compatible.

This note is the sequel of "Geometric structures as variational objects, I." It generalizes the main result and perspectives of that work to a class of geometric structures that includes integrable almost-complex structures.

In the first part of this article we revisit the theory of weighted spinors on conformal manifolds. In the second part we introduce the notions of asymptotically flat Weyl structures and of associated mass, and we prove a conformal version…

Let $C$ be a chain-like curve over $\mathbb{C}$. In this paper, we investigate the rationality of moduli spaces of $w$-semistable vector bundles on $C$ of arbitrary rank and fixed determinant by putting some restrictions on the Euler…

This article is devoted to the investigation of structure of wrap groups of connected fiber bundles over the fields of real $\bf R$, complex $\bf C$ numbers, the quaternion skew field $\bf H$ and the octonion algebra $\bf O$. Iterated wrap…

We derive conformal blocks in an inverse spacetime dimension expansion. In this large D limit, the blocks are naturally written in terms of a new combination of conformal cross-ratios. We comment on the implications for the conformal…

We present a braided circuit topology framework for investigating topology and structural phase transitions in aggregates of semiflexible polymers. In the conventional approach to circuit topology, which specifically applies to single…

We give a factorization of the cycle of a bounded complex of vector bundles in terms of certain associated differential forms and residue currents. This is a generalization of previous results in the case when the complex is a locally free…

We study spaces of conformal blocks associated with line bundles over elliptic curves, with coefficients in a vertex algebra. For vertex algebras satisfying suitable finiteness and semisimplicity conditions, which are met by all admissible…

Transformers are ubiquitous in wide tasks. Interpreting their internals is a pivotal goal. Nevertheless, their particular components, feed-forward (FF) blocks, have typically been less analyzed despite their substantial parameter amounts.…

We give a brief survey of recent work on integral forms in vertex operator algebras (VOAs).

The factorizable vectors of a complete Boolean algebra of type I factors, acting on a separable Hilbert space, are shown to be total, resolving a conjecture of Araki and Woods. En route, the spectral theory of noise-type Boolean algebras of…

Block full rank pencils introduced in [Dopico et al., Local linearizations of rational matrices with application to rational approximations of nonlinear eigenvalue problems, Linear Algebra Appl., 2020] allow us to obtain local information…