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Domenico Fiorenza

Publications and source records attributed to Domenico Fiorenza.

At least 19 recordsLinked to original sources
arXiv2026

A categorification of Kauffman states for planar graphs

Given a decorated planar graph $(G,\omega)$, where $G$ is a planar graph and $\omega\in H^1(|\mathcal{Q}G|,\mathbb{Z})$ with $\mathcal{Q}G$ the directed medial graph of $G$, we call some angular functions $\omega$-compatible and study two distinct but related directed graphs: $\mathcal{L}(G,\omega)$, which is the directed graph of such functions, and $BMS(G,\omega)$, the directed graph of BMS states which are some pairs of $\omega$-compatible functions plus additional data. We give sufficient conditions for $\mathcal{L}(G,\omega)$ to be a graded distributive lattice, recovering Kauffman's Clock Theorem when $G$ is a knot diagram. We also define a potential on $\mathcal{Q} G$ and associate a representation of the corresponding quiver with potential to every BMS state. Under suitable assumptions, this construction yields an isomorphism between $\mathcal{L}(G,\omega)$ and the lattice of subrepresentations of a maximal representation, generalizing a result of Bazier-Matte--Schiffler.

math.RT↗
arXiv2026

$L_\infty$-morphisms between twisted Courant $r$-Lie algebras and untwisted Courant $(r{+}1)$-Lie algebroids

In "Lie infinity algebras and higher analogues of Dirac structures and Courant algebroids" [arXiv:1003.1004], Marco Zambon constructs an $L_\infty$-algebra associated with any higher standard or twisted Courant algebroid (also known as a Vinogradov algebroid), and exhibits an explicit $L_\infty$-morphism from the Lie algebra associated with a standard Lie algebroid twisted by a closed 2-form to the Lie-2 algebra of the standard Courant algebroid. He poses the question of whether analogous canonical $L_\infty$-morphisms exist in higher degrees -- namely, for any standard higher Courant algebroid twisted by a closed $(r+1)$-form. We adfirmatively answer this question, presenting a general framework that naturally yields such canonical $L_\infty$-morphisms for arbitrary $r$, while at the same time clarifying the geometrical and homotopical structures underlying the construction. We also show how this framework accommodates the canonical morphism between Roger's observable $L_\infty$-algebra of a pre-$r$-plectic manifold and the higher Courant algebra described by Zambon and one of the authors in "Observables on multisymplectic manifolds and higher Courant algebroids" [arXiv:2209.05836].

math.SG↗
arXiv2025

Invertible projective 2-representations from invertible 2d TQFTs with defects

We investigate invertible projective representations and their 2-categorical analogues using the language of TQFTs with defects. The main result is a freeness property for invertible projective representatios. While trivial in the 1-categorical setting, this result becomes interesting for 2-representations: as an application, only relying only on invertibility of Clifford algebras and Fock bimodules in the Morita 2-category of super vector spaces we recover Ludewig--Roos' result that the Clifford/Fock construction is a projective 2-representation of the category of Lagrangian correspondences.

math.CT↗
arXiv2025

Projective and anomalous representations of categories and their linearizations

We invesigate the relation between projective and anomalous representations of categories, and show how to any anomaly $J\colon \mathcal{C}\to 2\mathrm{Vect}$ one can associate an extension $\mathcal{C}^J$ of $\mathcal{C}$ and a subcategory $\mathcal{C}^J_{\mathrm{ST}}$ of $\mathcal{C}^J$ with the property that: (i) anomalous representations of $\mathcal{C}$ with anomaly $J$ are equivalent to $\mathrm{Vect}$-linear functors $E\colon \mathcal{C}^J\to \mathrm{Vect}$, and (ii) these are in turn equivalent to linear representations of $\mathcal{C}^J_{\mathrm{ST}}$ where "$J$ acts as scalars". This construction, inspired by and generalizing the technique used to linearize anomalous functorial field theories in the physics literature, can be seen as a multi-object version of the classical relation between projective representations of a group $G$, with given $2$-cocycle $\alpha$, and linear representations of the central extension $G^\alpha$ of $G$ associated with $\alpha$.

math.CT↗
arXiv2024

Computing the Khovanov homology of 2 strand braid links via generators and relations

In "Homfly polynomial via an invariant of colored plane graphs", Murakami, Ohtsuki, and Yamada provide a state-sum description of the level $n$ Jones polynomial of an oriented link in terms of a suitable braided monoidal category whose morphisms are $\mathbb{Q}[q,q^{-1}]$-linear combinations of oriented trivalent planar graphs, and give a corresponding description for the HOMFLY-PT polynomial. We extend this construction and express the Khovanov-Rozansky homology of an oriented link in terms of a combinatorially defined category whose morphisms are equivalence classes of formal complexes of (formal direct sums of shifted) oriented trivalent plane graphs. By working combinatorially, one avoids many of the computational difficulties involved in the matrix factorization computations of the original Khovanov-Rozansky formulation: one systematically uses combinatorial relations satisfied by these matrix factorizations to simplify the computation at a level that is easily handled. By using this technique, we are able to provide a computation of the level $n$ Khovanov-Rozansky invariant of the 2-strand braid link with $k$ crossings, for arbitrary $n$ and $k$, confirming and extending previous results and conjectural predictions by Anokhina-Morozov, Beliakova-Putyra-Wehrli, Carqueville-Murfet, Dolotin-Morozov, Gukov-Iqbal-Kozcaz-Vafa, Nizami-Munir-Sohail-Usman, and Rasmussen.

math.GT↗
arXiv2023

Unital $C_\infty$-algebras and the real homotopy type of $(r-1)$-connected compact manifolds of dimension $\le \ell(r-1)+2$

We encode the real homotopy type of an $n$-dimensional $(r-1)$-connected compact manifold $M$, $ r\ge 2$ into a minimal unital $C_\infty$-structure on $H^* (M,\mathbb R)$, obtained via homotopy transfer of the unital DGCA structure of the small quotient algebra associated with a Hodge decomposition of the de Rham algebra $\mathcal A^*(M)$, which has been proposed by Fiorenza-Kawai-L\^e-Schwachh\"ofer in [Ann. Sc. Norm. Super Pisa (5), vol. XXII (2021), 79-107]. We prove that if $n \le \ell (r-1) +2$, with $\ell \geq 4$, the multiplication $\mu_k$ on the minimal unital $C_\infty$-algebra $H^*(M,\mathbb R)$ vanishes for all $k \ge \ell-1$. This extends the results from [loc. cit.], extending the bound on the dimension from $5r-3$ to the general bound $\ell(r-1) +2$. We also prove a variant of this result, conjectured by Zhou, stating that if $n \le \ell(r-1)+4$ and $b_r (M) =1$ then the multiplication $\mu_k$ for all $k \ge \ell-1$ vanishes. This implies two formality results by Cavalcanti [Math. Proc. Cambridge Philos. Soc. 141 (2006), 101-112]. We show that in any dimension $n$ the Harrison cohomology class $[\mu_3]\in \mathrm {HHarr}^{3,-1}(H^* (M, \mathbb R), H^*(M, \mathbb R)) $ is a homotopy invariant of $M$ and the first obstruction to formality, and provide a detailed proof that if $n\leq 4r-1$ this is the only obstruction. Furthermore, we show that in any dimension $n$ the class $[\mu_3]$ and the Bianchi-Massey tensor invented by Crowley-Nordstr\"om in [J. Topol. 13(2020), 539-575] define each other uniquely.

math.AT↗
arXiv2023

The (anti-)holomorphic sector in $\mathbb{C}/Λ$-equivariant cohomology, and the Witten class

Atiyah's classical work on circular symmetry and stationary phase shows how the $\hat{A}$-genus is obtained by formally applying the equivariant cohomology localization formula to the loop space of a simply connected spin manifold. The same technique, applied to a suitable ''antiholomorphic sector'' in the $\mathbb{C}/Λ$-equivariant cohomology of the conformal double loop space $\mathrm{Maps}(\mathbb{C}/Λ,X)$ of a rationally string manifold $X$ produces the Witten genus of $X$. This can be seen as an equivariant localization counterpart to Berwick-Evans supersymmetric localization derivation of the Witten genus.

math.AT↗
arXiv2022

Integrals detecting degree 3 string cobordism classes

The third string bordism group $\mathrm{Bord}_3^{\mathrm{String}}$ is known to be $\mathbb{Z}/24\mathbb{Z}$. Using Waldorf's notion of a geometric string structure on a manifold, Bunke--Naumann and Redden have exhibited integral formulas involving the Chern-Weil form representative of the first Pontryagin class and the canonical 3-form of a geometric string structure that realize the isomorphism $\mathrm{Bord}_3^{\mathrm{String}} \to \mathbb{Z}/24\mathbb{Z}$. We will show how these formulas naturally emerge when one considers certain natural $\mathrm{U}(1)$-valued and $\mathbb{R}$-valued 3d TQFT associated with the classifying stacks of Spin bundles with connection and of String bundles with geometric structure, respectively.

math.AT↗
arXiv2022

Twistorial Cohomotopy implies Green-Schwarz anomaly cancellation

We characterize the integral cohomology and the rational homotopy type of the maximal Borel-equivariantization of the combined Hopf/twistor fibration, and find that subtle relations satisfied by the cohomology generators are just those that govern Horava-Witten's proposal for the extension of the Green-Schwarz mechanism from heterotic string theory to heterotic M-theory. We discuss how this squares with the Hypothesis H that the elusive mathematical foundation of M-theory is based on charge quantization in J-twisted Cohomotopy theory.

hep-th↗
arXiv2022

CR-twistor spaces over manifolds with $G_2$- and $Spin(7)$-structures

In 1984 LeBrun constructed a CR-twistor space over an arbitrary conformal Riemannian 3-manifold and proved that the CR-structure is formally integrable. This twistor construction has been generalized by Rossi in 1985 for $m$-dimensional Riemannian manifolds endowed with a $(m-1)$-fold vector cross product (VCP). In 2011 Verbitsky generalized LeBrun's construction of twistor-spaces to $7$-manifolds endowed with a $G_2$-structure. In this paper we unify and generalize LeBrun's, Rossi's and Verbitsky's construction of a CR-twistor space to the case where a Riemannian manifold $(M, g)$ has a VCP structure. We show that the formal integrability of the CR-structure is expressed in terms of a torsion tensor on the twistor space, which is a Grassmanian bundle over $(M, g)$. If the VCP structure on $(M,g)$ is generated by a $G_2$- or $Spin(7)$-structure, then the vertical component of the torsion tensor vanishes if and only if $(M, g)$ has constant curvature, and the horizontal component vanishes if and only if $(M,g)$ is a torsion-free $G_2$ or $Spin(7)$-manifold. Finally we discuss some open problems.

math.DG↗
arXiv2021

Brackets and products from centres in extension categories

Building on Retakh's approach to Ext groups through categories of extensions, Schwede reobtained the well-known Gerstenhaber algebra structure on Ext groups over bimodules of associative algebras both from splicing extensions (leading to the cup product) and from a suitable loop in the categories of extensions (leading to the Lie bracket). We show how Schwede's construction admits a vast generalisation to general monoidal categories with coefficients of the Ext groups taken in (weak) left and right monoidal (or Drinfel'd) centres. In case of the category of left modules over bialgebroids and coefficients given by commuting pairs of braided (co)commutative (co)monoids in these categorical centres, we provide an explicit description of the algebraic structure obtained this way, and a complete proof that this leads to a Gerstenhaber algebra is then obtained from an operadic approach. This, in particular, considerably generalises the classical construction given by Gerstenhaber himself. Conjecturally, the algebraic structure we describe should produce a Gerstenhaber algebra for an arbitrary monoidal category enriched over abelian groups, but even the bilinearity of the cup product and of the Lie-type bracket defined by the abstract construction in terms of extension categories remain elusive in this general setting.

math.CT↗
arXiv2021

A very short note on the (rational) graded Hori map

The graded Hori map has been recently introduced by Han-Mathai in the context of T-duality as a $\mathbb{Z}$-graded transform whose homogeneous components are the Hori-Fourier transforms in twisted cohomology associated with integral multiples of a basic pair of T-dual closed 3-forms. We show how in the rational homotopy theory approximation of T-duality, such a map is naturally realised as a pull-iso-push transform, where the isomorphism part corresponds to the canonical equivalence between the left and the right gerbes associated with a T-duality configuration.

math.AT↗
arXiv2021

Fullness of exceptional collections via stability conditions -- A case study: the quadric threefold

A powerful tool of investigation of Fano varieties is provided by exceptional collections in their derived categories. Proving the fullness of such a collection is generally a nontrvial problem, usually solved on a case-by-case basis, with the aid of a deep understanding of the underlying geometry. Likewise, when an exceptional collection is not full, it is not straightforward to determine whether its "residual" category, i.e., its right orthogonal, is the derived category of a variety. We show how one can use the existence of Bridgeland stability condition these residual categories (when they exist) to address these problems. We examine a simple case in detail: the quadric threefold $Q_3$ in $\mathbb{P}^{4}$. We also give an indication how a variety of other classical results could be justified or re-discovered via this technique., e.g., the commutativity of the Kuznetsov component of the Fano threefold $Y_4$.

math.AG↗
arXiv2020

Formally integrable complex structures on higher dimensional knot spaces

Let $S$ be a compact oriented finite dimensional manifold and $M$ a finite dimensional Riemannian manifold, let ${\rm Imm}_f(S,M)$ the space of all free immersions $φ:S \to M$ and let $B^+_{i,f}(S,M)$ the quotient space ${\rm Imm}_f(S,M)/{\rm Diff}^+(S)$, where ${\rm Diff}^+(S)$ denotes the group of orientation preserving diffeomorphisms of $S$. In this paper we prove that if $M$ admits a parallel $r$-fold vector cross product $φ\in Ω^r(M, TM)$ and $\dim S = r-1$ then $B^+_{i,f}(S,M)$ is a formally Kähler manifold. This generalizes Brylinski's, LeBrun's and Verbitsky's results for the case that $S$ is a codimension 2 submanifold in $M$, and $S = S^1$ or $M$ is a torsion-free $G_2$-manifold respectively.

math.DG↗
arXiv2020

Twisted Cohomotopy implies level quantization of the full 6d Wess-Zumino term of the M5-brane

The full 6d Hopf-Wess-Zumino term in the action functional for the M5-brane is anomalous as traditionally defined. What has been missing is a condition implying the higher analogue of level quantization familiar from the 2d Wess-Zumino term. We prove that the anomaly cancellation condition is implied by the hypothesis that the C-field is charge-quantized in twisted Cohomotopy theory. The proof follows by a twisted/parametrized generalization of the Hopf invariant, after identifying the full 6d Hopf-Wess-Zumino term with a twisted homotopy Whitehead integral formula, which we establish.

hep-th↗