arXiv subjects

Vicente Cortés

Publications and source records attributed to Vicente Cortés.

At least 19 recordsLinked to original sources

arXiv2026

Deformations of generalized complex structures on $\mathcal{G}$-flat transitive Courant algebroids

We extend Gualtieri's deformation theorem for generalized complex structures on exact Courant algebroids to the more general class of $\mathcal{G}$-flat transitive Courant algebroids. Our approach relies on the framework of tame Fréchet spaces and Hamilton's Nash-Moser (implicit function) theorem. As an important step of the proof of our deformation theorem, we endow the group of autoequivalences of a $\mathcal{G}$-flat transitive Courant algebroid with a tame Fréchet Lie group structure and show that there exists a tame subgroup of exact autoequivalences, whose infinitesimal action is by inner derivations.

math.DG↗

arXiv2026

Darboux theorem for generalized complex structures on transitive Courant algebroids

Under natural assumptions we find local normal forms for generalized complex structures on transitive Courant algebroids, which extend Gualtieri's Darboux theorem for generalized complex structures on manifolds. When the base of the Courant algebroid is a point, they reduce to Wang's description of invariant complex structures on compact semisimple Lie groups.

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arXiv2026

Deformation theory of complex structures via Hamilton-Nash-Moser theory

We revisit the classical proof by Kuranishi on the existence of a locally complete family of complex deformations on a compact complex manifold using Hamilton's Nash-Moser implicit function theorem. This is relevant to existing and future extensions of this theorem to generalized complex structures on Courant algebroids.

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arXiv2026

Affine hypersurfaces and superintegrable systems

It was recently shown that under mild assumptions second-order conformally superintegrable systems can be encoded in a $(0,3)$-tensor, called structure tensor. For abundant systems, this approach led to algebraic integrability conditions that essentially allow one to restore a system from the knowledge of its structure tensor in a point on the manifold. Here we study the geometric structure formalising such systems, which we call an abundant manifold. The underlying Riemannian manifold is necessarily conformally flat. We establish a correspondence between these superintegrable systems and the geometry of affine hypersurfaces. More precisely, we show that abundant manifolds correspond to certain non-degenerate relative affine hypersurfaces normalisations in $\mathbb R^{n+1}$ ($n\ge 2$). We also formulate the necessary and sufficient conditions non-degenerate relative affine hypersurface normalisations in $\mathbb R^{n+1}$ need to satisfy, if they arise from abundant manifolds. These relative affine hypersurface normalisations are called abundant hypersurface normalisations. Both for abundant manifolds and for relative affine hypersurface normalisations a natural concept of conformal equivalence can be defined. We prove that they are compatible, permitting us to identify conformal classes of abundant manifolds with abundant hypersurface immersions (without specified normalisation).

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arXiv2026

Integrability of generalized structures on odd exact Courant algebroids using generalized connections

Odd exact Courant algebroids constitute a simple class of transitive Courant algebroids. Their underlying vector bundle is of odd rank and differs from a generalized tangent bundle by the addition of a line bundle. In this article we study natural analogues of almost complex and almost pseudo-Hermitian structures on such Courant algebroids, which are called B_n-generalized almost complex/pseudo-Hermitian structures. The corresponding integrable structures are known as B_n-generalized complex structures and B_n-generalized pseudo-Kähler structures, respectively. We characterize the integrability of B_n-generalized almost complex/pseudo-Hermitian structures on odd exact Courant algebroids in terms of existence of adapted generalized connections. We describe the affine spaces of adapted generalized connections for such integrable generalized structures.

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arXiv2025

Components of generalised complex structures on transitive Courant algebroids

Generalised almost complex structures $\mathcal J$ on transitive Courant algebroids $E$ are studied in terms of their components with respect to a splitting $E\cong TM \oplus T^*M \oplus \mathcal G$, where $M$ denotes the base of $E$ and $\mathcal G$ its bundle of quadratic Lie algebras. Necessary and sufficient integrability equations for $\mathcal J$ are established in this formalism. As an application, it is shown that the integrability of $\mathcal J$ implies that one of the components defines a Poisson structure on $M$. Then the structure (normal form) of generalised complex structures for which the Poisson structure is non-degenerate is determined. It is shown that it is fully encoded in a pair $(ω, ρ)$ consisting of a symplectic structure $ω$ on $M$ and a representation $ρ: π_1(M) \to \mathrm{Aut}(\mathfrak g, \langle \cdot ,\cdot \rangle_{\mathfrak{g}}, J_{\mathfrak{g}})$ by automorphism of a quadratic Lie algebra $(\mathfrak g, \langle \cdot ,\cdot \rangle_{\mathfrak{g}})$ commuting with an integrable (in the sense of Lie algebras) skew-symmetric complex structure $J_{\mathfrak{g}}$. Examples of such representations and obstructions for the existence of non-degenerate generalised complex structures are discussed. Finally, a construction of generalised complex structures on transitive Courant algebroids over complex manifolds for which the Poisson structure degenerates along a complex analytic hypersurface is presented.

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arXiv2025

Geometry of effective field theory positivity cones

Positivity bounds are theoretical constraints on the Wilson coefficients of an effective field theory. These bounds emerge from the requirement that a given effective field theory must be the low-energy limit of a relativistic quantum theory that satisfies the fundamental principles of unitarity, locality, and causality. The task of deriving these bounds can be reformulated as the geometric problem of finding the extremal representation of a closed convex cone~$\mathcal C_W$. More precisely, in the presence of multiple particle flavors, the forward-limit positivity cone $\mathcal C_W$ consists of all positive semi-definite tensors in $W =\left\{ S \in \mathrm{Sym}^2 (\mathrm{Sym}^2\, V^*)\oplus \mathrm{Sym}^2 \left(Λ^2 V^*\right) : τS = S \right\} \subset \mathrm{Sym}^2(V^*\otimes V^*)$, where $τ$ denotes transposition in the second and fourth tensor factor and $V\cong\mathbb{R}^n$, where $n$ is the number of flavors. In this work, we solve this question up to three flavors, i.e.~$n=3$, proving a full classification of all extremal elements in these cases. We furthermore study the implications of our findings, deriving the full positivity bounds for amplitudes with and without additional symmetries. In the cases with additional symmetries that we consider, we find that the so-called elastic bounds are sufficient to give rise to the full positivity bounds.

math-ph↗

arXiv2025

The canonical generalised Levi-Civita connection and its curvature

Given a (semi-Riemannian) generalised metric $\mathcal G$ and a divergence operator $\mathrm{div}$ on an exact Courant algebroid $E$, we geometrically construct a canonical generalised Levi-Civita connection $D^{\mathcal G, \mathrm{div}}$ for these data. In this way we provide a resolution of the problem of non-uniqueness of generalised Levi-Civita connections. Since the generalised Riemann tensor of $D^{\mathcal G, \mathrm{div}}$ is an invariant of the pair $(\mathcal G, \mathrm{div})$, we no longer need to discard curvature components which depend on the choice of the generalised connection. As a main result we decompose the generalised Riemann curvature tensor of $D^{\mathcal G, \mathrm{div}}$ in terms of classical (non-generalised) geometric data. Based on this set of master formulas we derive a comprehensive curvature tool-kit for applications in generalised geometry. This includes decompositions for the full generalised Ricci tensor, the generalised Ricci tensor, and three generalised scalar-valued curvature invariants, two of which are new.

math.DG↗

arXiv2025

Exterior Generalised Geometry

It is the aim of this paper to transfer to generalised geometry tools employed in the study of semi-Riemannian immersions, specializing at times to semi-Riemannian hypersurfaces. Given an exact Courant algebroid $E \to M$ and an immersion $ι\colon N \hookrightarrow M$, there is a well-known construction of an exact Courant algebroid $ι^! E \to N$, the pullback of $E$. This paper explains the pullback of generalised metrics and divergence operators. Assuming $N$ is a hypersurface, it develops the notion of generalised exterior curvature, introducing the generalised second fundamental form and the generalised mean curvature. Generalised versions of the Gauß-Codazzi equations are obtained. As an application, the constraint equations for the initial value formulation of the generalised Einstein equations are established in the formalism of generalised geometry. Further applications include a generalised geometry version of the fundamental theorem for hypersurfaces and the result that generalised Kähler and hyper-Kähler structures restrict to submanifolds compatible with the generalised almost complex structure. In particular, we characterise exact semi-Riemannian Courant algebroids which are flat with respect to the canonical generalised connection. These play the role of the ambient space in the fundamental theorem mentioned above.

math.DG↗

arXiv2025

Intrinsic characterization of projective special complex manifolds

We define the notion of an $S^1$-bundle of projective special complex base type and construct a conical special complex manifold from it. Consequently the base space of such an $S^{1}$-bundle can be realized as $\mathbb{C}^{\ast}$-quotient of a conical special complex manifold. As a corollary, we give an intrinsic characterization of a projective special complex manifold generalizing Mantegazza's characterization of a projective special Kähler manifold. Our characterization is in the language of c-projective structures. As an application, a non-trivial $S^1$-family of Obata-Ricci-flat hypercomplex structures (given by a generalization of the rigid c-map) on the tangent bundle of the total space of a $\mathbb{C}^*$-bundle over a complex manifold with certain kind of c-projective structure is constructed. Finally, we show that the quaternionic structure underlying any of these hypercomplex structures is in general not flat and that its flatness implies the vanishing of the c-projective Weyl tensor of the base of the $\mathbb{C}^*$-bundle. Conversely, any c-projectively flat complex manifold satisfying a cohomological integrality condition gives rise to a flat quaternionic structure.

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arXiv2025

Quaternionic Kähler manifolds fibered by solvsolitons

This paper is concerned with the geometry of principal orbits in quaternionic Kähler manifolds $M$ of cohomogeneity one. We focus on the complete cohomogeneity one examples obtained from the non-compact quaternionic Kähler symmetric spaces associated with the simple Lie groups of type A by the one-loop deformation. We prove that for zero deformation parameter the principal orbits form a fibration by solvsolitons (nilsolitons if $4n=\dim M=4$). The underlying solvable group is non-unimodular if $n>1$ and is the Heisenberg group if $n=1$. We show that under the deformation, the hypersurfaces remain solvmanifolds but cease to be Ricci solitons.

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arXiv2024

Compact Locally Conformally Pseudo-Kähler Manifolds with Essential Conformal Transformations

A conformal transformation of a semi-Riemannian manifold is essential if there is no conformally equivalent metric for which it is an isometry. For Riemannian manifolds the existence of an essential conformal transformation forces the manifold to be conformally flat. This is false for pseudo-Riemannian manifolds, however compact examples of conformally curved manifolds with essential conformal transformation are scarce. Here we give examples of compact conformal manifolds in signature $(4n+2k,4n+2\ell)$ with essential conformal transformations that are locally conformally pseudo-Kähler and not conformally flat, where $n\ge 1$, $k, \ell \ge 0$. The corresponding local pseudo-Kähler metrics obtained by a local conformal rescaling are Ricci-flat.

math.DG↗

arXiv2024

Generalised Einstein metrics on Lie groups

We continue the systematic study of left-invariant generalised Einstein metrics on Lie groups initiated in arXiv:2206.01157. Our approach is based on a new reformulation of the corresponding algebraic system. For a fixed Lie algebra $\mathfrak{g}$, the unknowns of the system consist of a scalar product $g$ and a $3$-form $H$ on $\mathfrak{g}$ as well as a linear form $δ$ on $\mathfrak{g}\oplus\mathfrak{g}^*$. As in arXiv:2206.01157, the Lie bracket of $\mathfrak{g}$ is considered part of the unknowns. In the Riemannian case, we show that the generalised Einstein condition always reduces to the commutator ideal and we provide a full classification of solvable generalised Einstein Lie groups. In the Lorentzian case, under the additional assumption $δ=0$, we classify -- up to one case -- all almost Abelian generalised Einstein Lie groups. We then particularize to four dimensions and provide a full classification of generalised Einstein Riemannian Lie groups as well as generalised Einstein Lorentzian Lie groups with $δ=0$ and non-degenerate commutator ideal.

math.DG↗

arXiv2024

Curvature of quaternionic skew-Hermitian manifolds and bundle constructions

This articles is devoted to a description of the second-order differential geometry of torsion-free almost quaternionic skew-Hermitian manifolds, that is, of quaternionic skew-Hermitian manifolds $(M, Q, ω)$. We provide a curvature characterization of such integrable geometric structures, based on the holonomy theory of symplectic connections and we study qualitative properties of the induced Ricci tensor. Then we proceed with bundle constructions over such a manifold $(M, Q, ω)$. In particular, we prove the existence of almost hypercomplex skew-Hermitian structures on the Swann bundle over $M$ and investigate their integrability.

math.DG↗

arXiv2024

Symmetries of one-loop deformed q-map spaces

Q-map spaces form an important class of quaternionic Kähler manifolds of negative scalar curvature. Their one-loop deformations are always inhomogeneous and have been used to construct cohomogeneity one quaternionic Kähler manifolds as deformations of homogeneous spaces. Here we study the group of isometries in the deformed case. Our main result is the statement that it always contains a semidirect product of a group of affine transformations of $\mathbb{R}^{n-1}$ with a Heisenberg group of dimension $2n+1$ for a q-map space of dimension $4n$. The affine group and its action on the normal Heisenberg factor in the semidirect product depend on the cubic affine hypersurface which encodes the q-map space.

math.DG↗

arXiv2023

Classification of odd generalized Einstein metrics on 3-dimensional Lie groups

An odd generalized metric E_{-} on a Lie group G of dimension n is a left-invariant generalized metric on a Courant algebroid E_{H, F} of type B_n over G with left-invariant twisting forms H and F. Given an odd generalized metric E_{-} on G we determine the affine space of left invariant Levi-Civita generalized connections of E_ {-}. Given in addition a left-invariant divergence operator δwe show that there is a left-invariant Levi-Civita generalized connection of E_{-} with divergence δand we compute the corresponding Ricci tensor Ricci^δ of the pair (E_{-}, δ). The odd generalized metric E_{-} is called odd generalized Einstein with divergence δif Ricci^δ =0. We describe all odd generalized Einstein metrics of arbitrary left-invariant divergence on all 3-dimensional Lie groups.

math.DG↗

arXiv2023

S-duality and the universal isometries of instanton corrected q-map spaces

Given a conical affine special Kähler (CASK) manifold together with a compatible mutually local variation of BPS structures, one can construct a quaternionic-Kähler (QK) manifold. We call the resulting QK manifold an instanton corrected c-map space. Our main aim is to study the isometries of a subclass of instanton corrected c-map spaces associated to projective special real (PSR) manifolds with a compatible mutually local variation of BPS structures. We call the latter subclass instanton corrected q-map spaces. In the setting of Calabi-Yau compactifications of type IIB string theory, instanton corrected q-map spaces are related to the hypermultiplet moduli space metric with perturbative corrections, together with worldsheet, D(-1) and D1 instanton corrections. In the physics literature, it has been shown that the hypermultiplet metric with such corrections must have an $\mathrm{SL}(2,\mathbb{Z})$ acting by isometries, related to S-duality. We give a mathematical treatment of this result, specifying under which conditions instanton corrected q-map spaces carry an action by isometries by $\mathrm{SL}(2,\mathbb{Z})$ or some of its subgroups. We further study the universal isometries of instanton corrected q-map spaces, and compare them to the universal isometries of tree-level q-map spaces. Finally, we give an explicit example of a non-trivial instanton corrected q-map space with full $\mathrm{SL}(2,\mathbb{Z})$ acting by isometries and admitting a quotient of finite volume by a discrete group of isometries.

math.DG↗

arXiv2023

Classification of generalized Einstein metrics on 3-dimensional Lie groups

We develop the theory of left-invariant generalized pseudo-Riemannian metrics on Lie groups. Such a metric accompanied by a choice of left-invariant divergence operator gives rise to a Ricci curvature tensor and we study the corresponding Einstein equation. We compute the Ricci tensor in terms of the tensors (on the sum of the Lie algebra and its dual) encoding the Courant algebroid structure, the generalized metric and the divergence operator. The resulting expression is polynomial and homogeneous of degree two in the coefficients of the Dorfman bracket and the divergence operator with respect to a left-invariant orthonormal basis for the generalized metric. We determine all generalized Einstein metrics on three-dimensional Lie groups.

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