Search arXiv⌕ Search

arXiv subjects

Luca Scala

Publications and source records attributed to Luca Scala.

10 recordsLinked to original sources
arXiv2026

$\alpha'$-Bootstrap

Due to the exponential growth in the number of terms, computing $\alpha'$-corrections to string theory's low-energy effective actions is a challenging matter. In order to fix all the couplings, one has usually to deal with a large number of string scattering amplitudes. This difficulty can be overcome by exploiting T-duality, which severely constrains the allowed structure of the effective action. It is then convenient to work in a formulation where T-duality is a manifest symmetry. Building on a series of previous works by some of the authors, we present a refined version of an elegant and effective procedure that allows to obtain all the higher-derivative corrections of the NS-NS sector of sting theories at order $\alpha'$ and $\alpha'^2$, up to an overall coefficient. We dub this approach $\alpha'$-bootstrap, since it is based only on consistency conditions and avoids the direct computation of scattering amplitudes. The procedure relies on an infinite dimensional algebraic structure that we present in full detail, and it is conjectured to work at all orders. Although, at the moment, it still misses the $\zeta$-like corrections starting at order $\alpha'^3$, the ease with which it can be generalized is promising for future developments in this direction.

hep-th↗
arXiv2024

$\varrho$-Poincar\'e: bicrossproduct structure, $\star$-products and quantum Lie algebra

We discuss the bicrossproduct structure of the quantum group $\varrho$-Poincar\'e and of the dual quantum universal enveloping algebra, expanding the construction to general Lie algebra-type deformations of Poincar\'e coming from classical $r$-matrices. We review the relation between different bases of the quantum universal enveloping algebra of $\varrho$-Poincar\'e and noncommutative $\star$-products defined on the $\varrho$-Minkowski spacetime, analysing some of their relevant features. Furthermore, we comment on the role of physical bases and introduce the $\varrho$-Poincar\'e quantum Lie algebra.

hep-th↗
arXiv2023

Generalized Dualities for Heterotic and Type I Strings

We define generalized dualities for heterotic and type I strings based on consistent truncations to half-maximal gauged supergravities in more than three dimensions. The latter are constructed from a generalized Scherk-Schwarz ansatz in heterotic double field theory that satisfies the strong constraint. Necessary and sufficient conditions on the resulting embedding tensor are discussed, showing that only certain gaugings, called geometric, can arise from this procedure. For all of them, we explicitly construct the internal geometry and gauge potentials. In general, this construction is not unique and permits different uplifts which are used to define generalized T-duality. Two examples are worked out underlying the utility of our approach to explore new dualities and uplifts of half-maximal gauged supergravities.

hep-th↗
arXiv2022

Localization and observers in $\varrho$-Minkowski spacetime

We consider the $\varrho$-Minkowski spacetime, a model with linear noncommutativity involving the time and the azimuthal angle. We study its quantum symmetries, the $\varrho$-Poincar\'e quantum group, and analyse the concepts of localizability and quantum observers.

hep-th↗
arXiv2015

Higher symmetric powers of tautological bundles on Hilbert schemes of points on a surface

We study general symmetric powers $S^k L^{[n]}$ of a tautological bundle $L^{[n]}$ on the Hilbert scheme $X^{[n]}$ of $n$ points over a smooth quasi-projective surface $X$, associated to a line bundle $L$ on $X$. Let $V_L$ be the $\mathfrak{S}_n$-vector bundle on $X^n$ defined as the exterior direct sum $L \boxplus \cdots \boxplus L$. We prove that the Bridgeland-King-Reid transform $\mathbfΦ(S^k L^{[n]})$ of symmetric powers $S^k L^{[n]}$ is quasi isomorphic to the last term of a finite decreasing filtration on the natural vector bundle $S^k V_L$, defined by kernels of operators $D^l_L$, which operate locally as higher order restrictions to pairwise diagonals. We use this description and the natural filtration on $(S^k V_L)^{\mathfrak{S}_n}$ induced by the decomposition in direct sum, to obtain, for $n =2$ or $k \leq 4$, a finite decreasing filtration $\mathcal{W}^\bullet$ on the direct image $μ_*(S^k L^{[n]})$ for the Hilbert-Chow morphism whose graded sheaves we control completely. As a consequence of this structural result, we obtain a chain of cohomological consequences, like a spectral sequence abutting to the cohomology of symmetric powers $S^k L^{[n]}$, an effective vanishing theorem for the cohomology of symmetric powers $S^k L^{[n]} \otimes \mathcal{D}_A$ twisted by the determinant, in presence of adequate positivity hypothesis on $L$ and $A$, as well as universal formulas for their Euler-Poincaré characteristic.

math.AG↗
arXiv2015

Notes on diagonals of the product and symmetric variety of a surface

Let $X$ be a smooth quasi-projective algebraic surface and let $Δ_n$ the big diagonal in the product variety $X^n$. We study cohomological properties of the ideal sheaves $\mathcal{I}^k_{Δ_n}$ and their invariants $(\mathcal{I}^k_{Δ_n})^{\mathfrak{S}_n}$ by the symmetric group, seen as ideal sheaves over the symmetric variety $S^nX$. In particular we obtain resolutions of the sheaves of invariants $(\mathcal{I}_{Δ_n})^{\mathfrak{S}_n}$ for $n = 3,4$ in terms of invariants of sheaves over $X^n$ whose cohomology is easy to calculate. Moreover, we relate, via the Bridgeland-King-Reid equivalence, powers of determinant line bundles over the Hilbert scheme to powers of ideals of the big diagonal $Δ_n$. We deduce applications to the cohomology of double powers of determinant line bundles over the Hilbert scheme with $3$ and $4$ points and we give universal formulas for their Euler-Poincaré characteristic. Finally, we obtain upper bounds for the regularity of the sheaves $\mathcal{I}^k_{Δ_n}$ over $X^n$ with respect to very ample line bundles of the form $L \boxtimes \cdots \boxtimes L$ and of their sheaves of invariants $( \mathcal{I}^k_{Δ_n})^{\mathfrak{S}_n}$ on the symmetric variety $S^nX$ with respect to very ample line bundles of the form $\mathcal{D}_L$.

math.AG↗
arXiv2015

Singularities of the Isospectral Hilbert Scheme

We study the singularities of the isospectral Hilbert scheme $B^n$ of $n$ points over a smooth algebraic surface and we prove that they are canonical if $n \leq 5$, log-canonical if $n \leq 7$ and not log-canonical if $n \geq 9$. We describe as well two explicit log-resolutions of $B^3$, one crepant and the other $\mathfrak{S}_3$-equivariant.

math.AG↗
arXiv2009

Perturbations of the metric in Seiberg-Witten equations

Let $M$ a compact connected orientable 4-manifold. We study the space $Ξ$ of $Spin^c$-structures of fixed fundamental class, as an infinite dimensional principal bundle on the manifold of riemannian metrics on $M$. In order to study perturbations of the metric in Seiberg-Witten equations, we study the transversality of universal equations, parametrized with all $Spin^c$-structures $Ξ$. We prove that, on a complex Kähler surface, for an hermitian metric $h$ sufficiently close to the original Kähler metric, the moduli space of Seiberg-Witten equations relative to the metric $h$ is smooth of the expected dimension.

math.DG↗
arXiv2007

Cohomology of the Hilbert scheme of points on a surface with values in representations of tautological bundles

Let $X$ a smooth quasi-projective algebraic surface, $L$ a line bundle on $X$. Let $X^{[n]}$ the Hilbert scheme of $n$ points on $X$ and $L^{[n]}$ the tautological bundle on $X^{[n]}$ naturally associated to the line bundle $L$ on $X$. We explicitely compute the image $\bkrh(L^{[n]})$ of the tautological bundle $L^{[n]}$ for the Bridgeland-King-Reid equivalence $\bkrh : \B{D}^b(X^{[n]}) \ra \B{D}^b_{\perm_n}(X^n)$ in terms of a complex $\comp{\mc{C}}_L$ of $\perm_n$-equivariant sheaves in $\B{D}^b_{\perm_n}(X^n)$. We give, moreover, a characterization of the image $\bkrh(L^{[n]} \tens ... \tens L^{[n]})$ in terms of of the hyperderived spectral sequence $E^{p,q}_1$ associated to the derived $k$-fold tensor power of the complex $\comp{\mc{C}}_L$. The study of the $\perm_n$-invariants of this spectral sequence allows to get the derived direct images of the double tensor power and of the general $k$-fold exterior power of the tautological bundle for the Hilbert-Chow morphism, providing Danila-Brion-type formulas in these two cases. This yields easily the computation of the cohomology of $X^{[n]}$ with values in $L^{[n]} \tens L^{[n]}$ and $\Lambda^k L^{[n]}$.

math.AG↗