arXiv subjects

Andrés Viña

Publications and source records attributed to Andrés Viña.

At least 19 recordsLinked to original sources

arXiv2025

Holomorphic Yang-Mills fields on $B$-branes

Considering $B$-branes over a complex manifold $X$ as objects of the bounded derived category of coherent sheaves over $X$, we define holomorphic gauge fields on $B$-branes and introduce the Yang-Mills functional for these fields. These definitions extend well-known concepts in the context of vector bundles to the setting of $B$-branes. For a given $B$-brane, we show that its Atiyah class is the obstruction to the existence of gauge fields. When $X$ is the variety of complete flags in a $3$-dimensional complex vector space, we prove that any $B$-brane over $X$ admits at most one holomorphic gauge field. Furthermore, we establish that the set of Yang-Mills fields on a given $B$-brane, if nonempty, is in bijective correspondence with the points of an algebraic set defined by $m$ complex polynomials of degree less than four in $m$ indeterminates, where $m$ is the dimension of the space of morphisms from the brane to its tensor product with the sheaf of holomorphic one-forms.

math.AG↗

arXiv2025

Holomorphic Gauge Fields on $B$-Branes

Considering the $B$-branes over a complex manifold as the objects of the bounded derived category of coherent sheaves on that manifold, we extend the definition of holomorphic gauge fields on vector bundles to $B$-branes. We construct a family of coherent sheaves on the complex projective space, which generates the corresponding bounded derived category and such that the supports of the elements of this family are two by two disjoint. Using that family, we prove that the cardinal of the set of holomorphic gauge fields on any $B$-brane over the projective space is less than $2.$

math.AG↗

arXiv2023

Yang-Mills fields on $B$-branes

Considering the $B$-branes over a complex manifold $Y$ as objects of the bounded derived category $D^b(Y)$, we define holomorphic gauge fields on $B$-branes and the Yang-Mills functional for these fields.These definitions are a generalization to $B$-branes of concepts that are well known in the context of vector bundles. Given ${\mathscr F}^{\bullet}\in D^b(Y)$, we show that the Atiyah class $a({\mathscr F}^{\bullet})\in{\rm Ext}^1({\mathscr F}^{\bullet},\,Ω^1({\mathscr F}^{\bullet}))$ is the obstruction to the existence of gauge fields on ${\mathscr F}^{\bullet}$. We determine the $B$-branes over $\mathbb{ CP}^n$ that admit holomorphic gauge fields. We prove that the set of Yang-Mills fields on the $B$-brane ${\mathscr F}^{\bullet} $, if it is nonempty, is in bijective correspondence with the points of an algebraic subset of ${\mathbb C}^m$ defined by $m\cdot s$ polynomial equations of degree $\leq 3$, where $m={\rm dim}\,{\rm Hom}({\mathscr F}^{\bullet},\,Ω^1({\mathscr F}^{\bullet}))$ and $s$ is the number of non-zero cohomology sheaves ${\mathscr H}^i({\mathscr F}^{\bullet})$. We show sufficient conditions under them any Yang-Mills field on a reflexive sheaf of rank $1$ is flat.

math.AG↗

arXiv2022

Gauge fields on $B$-branes over $\mathbb{CP}^n$

Considering the $B$-branes over the complex projective space ${\mathbb P}^n$ as the objects of the bounded derived category $D^b({\mathbb P}^n)$, we prove that the cardinal of the set of holomorphic gauge fields on a given $B$-brane ${\mathscr G}^{\bullet}$ is $\leq 1$. Moreover, the cardinal is $1$ iff each ${\mathscr G}^p$ is isomorphic to a direct sum of copies of ${\mathscr O}_{{\mathbb P}^n}$.

math.AG↗

arXiv2021

Gauge fields on coherent sheaves

Given a flat gauge field $\nabla$ on a vector bundle $F$ over a manifold $M$ we deduce a necessary and sufficient condition for the field $\nabla+ E$, with $E$ an ${\rm End}(F)$-valued $1$-form, to be a Yang-Mills field. For each curve of Yang-Mills fields on $F$ starting at $\nabla$, we define a cohomology class of $H^2(M,\,{\mathscr P})$, with ${\mathscr P}$ the sheaf of $\nabla$-parallel sections of $F$. This cohomology class vanishes when the curve consists of flat fields. We prove the existence of a curve of Yang-Mills fields on a bundle over the torus $T^2$ connecting two vacuum states. We define holomorphic and meromorphic gauge fields on a coherent sheaf and the corresponding Yang-Mills functional. In this setting, we analyze the Aharonov-Bohm effect and the Wong equation.

math.AG↗

arXiv2020

Flat gauge fields and the Riemann-Hilbert correspondence

The geometric phase that appears in the effects of Aharonov-Bohm type is interpreted in the frame of Deligne's version of the Riemann-Hilbert correspondence. We extend also the concept of flat gauge field to $B$-branes which are coherent sheaves, so that such a field on a sheaf turns it into a holonomic regular $D$-module.

math-ph↗

arXiv2018

Charge of $D$-branes on singular varieties

Considering the $D$-branes on a variety $Z$ as the objects of the derived category $D^b(Z)$, we propose a definition for the charge of $D$-branes on not necessarily smooth varieties. We define the charge $Q({\mathcal G})$ of ${\mathcal G}\in D^b(Z)$ as an element of the homology of $Z$, so that the mapping $Q$ is compatible with the pushforward by proper maps between varieties. Given a generic anticanonical hypersurface $Y$ of a toric variety $X$ defined by a reflexive polytope, we express the charge of a line bundle on $Y$ defined by a divisor $D$ of $X$ in terms of intersections of $D$ with cycles determined by the polytope faces.

math.AG↗

arXiv2018

Spanning Class in the Category of Branes

Given a generic anticanonical hypersurface $Y$ of a toric variety determined by a reflexive polytope, we define a line bundle ${\mathcal L}$ on $Y$ that generates a spanning class in the bounded derivative category $D^b(Y)$. From this fact, we deduce properties of some spaces of strings related with the brane ${\mathcal L}$. We prove a vanishing theorem for the vertex operators associated to strings stretching from branes of the form ${\mathcal L}^{\otimes i}$ to nonzero objects in $D^b(Y)$. We also define a gauge field on ${\mathcal L}$ which minimizes the corresponding Yang-Mills functional.

math.AG↗

arXiv2017

Branes on $G$-manifolds

Let $X$ be Calabi-Yau manifold acted by a group $G$. We give a definition of $G$-equivariance for branes on $X$, and assign to each equivariant brane an element of the equivariant cohomology of $X$ that can be considered as a charge of the brane. We prove that the spaces of strings stretching between equivariant branes support representations of $G$. This fact allows us to give formulas for the dimension of some of such spaces, when $X$ is a flag manifold of $G$.

math-ph↗

arXiv2017

Cohomological vertex operators

Given a Calabi-Yau manifold and considering the $B$-branes on it as objects in the derived category of coherent sheaves, we identify the vertex operators for strings between two branes with elements of the cohomology groups of Ext sheaves. We define the correlation functions for these general vertex operators. Strings stretching between two coherent sheaves are studied as homological extensions of the corresponding branes. In this context, we relate strings between different pairs of branes when there are maps between these branes. We also interpret some strings with ghost number $k$ as obstructions for lifts or extensions of strings with ghost number $k-1$.

math.AG↗

arXiv2016

Invariants under deformation of the actions of topological groups

Let $φ$ and $φ'$ be two homotopic actions of the topological group $G$ on the topological space $X$. To an object $A$ in the $G$-equivariant derived category $D_φ(X)$ of $X$ relative to the action $φ$ we associate an object $A'$ of category $D_{φ'}(X)$, such that the corresponding $G$-equivariant compactly supported cohomologies $H_{G,c}(X,\,A)$ and $H_{G,c}(X,\,A')$ are isomorphic. When $G$ is a Lie group and $X$ is a subanalytic space, we prove that the $G$-equivariant cohomologies $H_{G}(X,\,A)$ and $H_{G}(X,\,A')$ are also isomorphic.

math.AT↗

arXiv2015

Equivariant branes

Given a Calabi-Yau manifold $X$ acted by a group $G$ and considering the $B$-branes on $X$ as objects in the derived category of coherent sheaves, we give a definition of equivariant branes, which generalizes the concept of equivariant sheaves. We also propose a definition of equivariant charge of an equivariant brane. The spaces of strings joining the branes ${\mathcal F}$ and ${\mathcal G}$, are the groups $Ext^i({\mathcal F},\,{\mathcal G})$. We prove that the spaces of strings between two $G$-equivariant branes support representations of $G$. Thus, these spaces can be decomposed in direct sum of invariant spaces for the $G$-action. We show some particular decompositions, when $X$ is a toric variety and when $X$ is a flag manifold of a semisimple Lie group.

math.AG↗

arXiv2013

Lifting Hamiltonian loops to isotopies in fibrations

Let $G$ be a Lie group, $H$ a closed subgroup and $M$ the homogeneous space $G/H$. Each representation $Ψ$ of $H$ determines a $G$-equivariant principal bundle ${\mathcal P}$ on $M$ endowed with a $G$-invariant connection. We consider subgroups ${\mathcal G}$ of the diffeomorphism group ${\rm Diff}(M)$, such that, each vector field $Z\in{\rm Lie}({\mathcal G})$ admits a lift to a preserving connection vector field on ${\mathcal P}$. We prove that $#\,π_1({\mathcal G})\geq #\,Ψ(Z(G))$. This relation is applicable to subgroups ${\mathcal G}$ of the Hamiltonian groups of the flag varieties of a semisimple group $G$. Let $M_Δ$ be the toric manifold determined by the Delzant polytope $Δ$. We put $φ_{\bf b}$ for the the loop in the Hamiltonian group of $M_Δ$ defined by the lattice vector ${\bf b}$. We give a sufficient condition, in terms of the mass center of $Δ$, for the loops $φ_{\bf b}$ and $φ_{\bf\tilde b}$ to be homotopically inequivalent.

math.SG↗

arXiv2011

Characteristic number associated to mass linear pairs

Let $Δ$ be a Delzant polytope in ${\mathbb R}^n$ and ${\mathbf b}\in{\mathbb Z}^n$. Let $E$ denote the symplectic fibration over $S^2$ determined by the pair $(Δ,\,{\mathbf b})$. Under certain hypotheses, we prove the equivalence between the fact that $(Δ,\,{\mathbf b})$ is a mass linear pair (D. McDuff, S. Tolman, {\em Polytopes with mass linear functions. I.} Int. Math. Res. Not. IMRN 8 (2010) 1506-1574.) and the vanishing of a characteristic number of $E$. Denoting by ${\rm Ham}(M_Δ)$ the Hamiltonian group of the symplectic manifold defined by $Δ$, we determine loops in ${\rm Ham}(M_Δ)$ that define infinite cyclic subgroups in $π_1({\rm Ham}(M_Δ))$, when $Δ$ satisfies any of the following conditions: (i) it is the trapezium associated with a Hirzebruch surface, (ii) it is a $Δ_p$ bundle over $Δ_1$, (iii) $Δ$ is the truncated simplex associated with the one point blow up of ${\mathbb C}P^n$.

math.SG↗

arXiv2011

Action Integrals and discrete series

Let $G$ be a complex semisimple Lie group and ${G}_{\mathbb R}$ a real form that contains a compact Cartan subgroup $T_{\mathbb R}$. Let $π$ be a discrete series representation of $G_{\mathbb R}$. We present geometric interpretations in terms of concepts associated with the manifold $M:=G_{\mathbb R}/T_{\mathbb R}$ of the constant $π(g)$, for $g\in Z(G_{\mathbb R})$. For some relevant particular cases, we prove that this constant is the action integral around a loop of Hamiltonian diffeomorphims of $M$. As a consequence of these interpretations, we deduce lower bounds for the cardinal of the fundamental group of some subgroups of ${\rm Diff}(M)$. We also geometrically interpret the values of the infinitesimal character of the differential representation of $π$.

math.SG↗

arXiv2009

Action integrals and infinitesimal characters

Let $G$ be a reductive Lie group and ${\mathcal O}$ the coadjoint orbit of a hyperbolic element of ${\frak g}^*$. By $π$ is denoted the unitary irreducible representation of $G$ associated with ${\mathcal O}$ by the orbit method. We give geometric interpretations in terms of concepts related to ${\mathcal O}$ of the constant $π(g)$, for $g\in Z(G)$. We also offer a description of the invariant $π(g)$ in terms of action integrals and Berry phases. In the spirit of the orbit method we interpret geometrically the infinitesimal character of the differential representation of $π$.

math.SG↗

arXiv2008

A characteristic number of bundles determined by mass linear pairs

Let $Δ$ be a Delzant polytope in ${\mathbb R}^n$ and ${\bf b}\in{\mathbb Z}^n$. Let $E$ denote the symplectic fibration over $S^2$ determined by the pair $(Δ, {\bf b})$. We prove the equivalence between the fact that $(Δ, {\bf b})$ is a mass linear pair (D. McDuff, S. Tolman, {\em Polytopes with mass linear functions, part I.} {\tt arXiv:0807.0900 [math.SG]}) and the vanishing of a characteristic number of $E$ in the following cases: When $Δ$ is a $Δ_{n-1}$ bundle over $Δ_1$; when $Δ$ is the polytope associated with the one point blow up of ${\mathbb C}P^n$; and when $Δ$ is the polytope associated with a Hirzebruch surface.

math.SG↗

arXiv2008

Continuous families of Hamiltonian torus actions

We determine conditions under which two Hamiltonian torus actions on a symplectic manifold $M$ are homotopic by a family of Hamiltonian torus actions, when $M$ is a toric manifold and when $M$ is a coadjoint orbit.

math.SG↗