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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 $$a({\mathscr F}^{\bullet})\in{\rm Ext}^1({\mathscr F}^{\bullet},\,\Omega^1({\mathscr F}^{\bullet}))$$ is the obstruction to the existence of gauge fields on ${\mathscr F}^{\bullet}.$ When $Y$ be either the projective space ${\mathbb P}^n$ or the variety of complete flags in ${\mathbb C}^3,$ we prove that the cardinal of the set of holomorphic gauge fields on any $B$-brane over $Y$ is $\leq 1.$ 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 $m$ polynomial equations of degree $\leq 3$, where $m={\rm dim}\,{\rm Ext}^0({\mathscr F}^{\bullet},\,\Omega^1({\mathscr F}^{\bullet})).$, Comment: 36 pages. The corrected content of preprints arXiv:2206.10238 and arXiv:2210.07635 has been incorporated into this new preprint

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 ${\mathscr O}_{{\mathbb P}^n}$., Comment: 6 pages. arXiv admin note: text overlap with arXiv:2206.10238

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 $a({\mathscr F}^{\bullet})\in{\rm Ext}^1({\mathscr F}^{\bullet},\,\Omega^1({\mathscr F}^{\bullet}))$ 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 $m={\rm dim}\,{\rm Hom}({\mathscr F}^{\bullet},\,\Omega^1({\mathscr F}^{\bullet}))$ 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., Comment: 30 pages. The manuscript has been rewritten, the results presented in arXiv:2210.07635 have been incorporated and an appendix has been added

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., Comment: 29 pages

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., Comment: Extended version to be published in Reports on mathematical Physics

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., Comment: 15 pages

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., Comment: 19 pages

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 $G$., Comment: This paper contains also revised versions of results included in arXiv:1502.01869 [math.AG]. To be published in Journal of Geometry and Symmetry in Physics

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 $k-1$., Comment: Version which will appear in Journal of Advances in Applied Mathematics

Let $\varphi$ and $\varphi'$ be two homotopic actions of the topological group $G$ on the topological space $X$. To an object $A$ in the $G$-equivariant derived category $D_{\varphi}(X)$ of $X$ relative to the action $\varphi$ we associate an object $A'$ of category $D_{\varphi'}(X)$, 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.

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., Comment: 2 new references added

Let $G$ be a Lie group, $H$ a closed subgroup and $M$ the homogeneous space $G/H$. Each representation $\Psi$ 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 $#\,\pi_1({\mathcal G})\geq #\,\Psi(Z(G))$. This relation is applicable to subgroups ${\mathcal G}$ of the Hamiltonian groups of the flag varieties of a semisimple group $G$. Let $M_{\Delta}$ be the toric manifold determined by the Delzant polytope $\Delta$. We put $\varphi_{\bf b}$ for the the loop in the Hamiltonian group of $M_{\Delta}$ defined by the lattice vector ${\bf b}$. We give a sufficient condition, in terms of the mass center of $\Delta$, for the loops $\varphi_{\bf b}$ and $\varphi_{\bf\tilde b}$ to be homotopically inequivalent., Comment: 23 pages, 1 figure. To be published in Int. J. Geom. Methods Mod. Physics

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 $\pi$ 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 $\pi(g)$, 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 $\pi$., Comment: 18 pages, 1 figure

Let $\Delta$ 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 $(\Delta,\,{\mathbf b})$. Under certain hypotheses, we prove the equivalence between the fact that $(\Delta,\,{\mathbf b})$ 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 ${\rm Ham}(M_{\Delta})$ the Hamiltonian group of the symplectic manifold defined by $\Delta$, we determine loops in ${\rm Ham}(M_{\Delta})$ that define infinite cyclic subgroups in $\pi_1({\rm Ham}(M_{\Delta}))$, when $\Delta$ satisfies any of the following conditions: (i) it is the trapezium associated with a Hirzebruch surface, (ii) it is a $\Delta_p$ bundle over $\Delta_1$, (iii) $\Delta$ is the truncated simplex associated with the one point blow up of ${\mathbb C}P^n$., Comment: Revised version which will appear in ISRN Geometry

Let $\Delta$ 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 $(\Delta, {\bf b})$. We prove the equivalence between the fact that $(\Delta, {\bf b})$ 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 $\Delta$ is a $\Delta_{n-1}$ bundle over $\Delta_1$; when $\Delta$ is the polytope associated with the one point blow up of ${\mathbb C}P^n$; and when $\Delta$ is the polytope associated with a Hirzebruch surface., Comment: 17 pages. Section 3 and Introduction have been rewritten

Let $G$ be a reductive Lie group and ${\mathcal O}$ the coadjoint orbit of a hyperbolic element of ${\frak g}^*$. By $\pi$ 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 $\pi(g)$, for $g\in Z(G)$. We also offer a description of the invariant $\pi(g)$ 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 $\pi$., Comment: Completely revised version to appear in Letters in Mathematical Physics

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., Comment: 26 pages, 1 figure. To appear in J. Geom. Phys

Let $M$ be a quantizable symplectic manifold acted on by $T=(S^1)^r$ in a Hamiltonian fashion and $J$ a moment map for this action. Suppose that the set $M^{T}$ of fixed points is discrete and denote by ${\alpha}_{pj}\in{\mathbb Z}^r$ the weights of the isotropy representation at $p$. By means of the $\alpha_{pj}$'s we define a partition ${\mathcal Q}_+$, ${\mathcal Q}_-$ of $M^T$. (When $r=1$, ${\mathcal Q}_{\pm}$ will be the set of fixed points such that the half of the Morse index of $J$ at them is even (odd)). We prove the existence of a map $\pi_{\pm}:{\mathcal Q}_{\pm}\to{\mathcal Q}_{\mp}$ such that $J(q)-J(\pi_{\pm}(q))\in I_{\mp}$, for all $q\in {\mathcal Q}_{\pm}$, where $I_{\pm}$ is the lattice generated by the $\alpha_{pj}$'s with $p\in{\mathcal Q}_{\pm}.$ We define partition functions $N_p$ similar to the ones of Kostant \cite{Gui} and we prove that $\sum_{p\in{\mathcal Q}_+}N_p(l)=\sum_{p\in{\mathcal Q}_-}N_p(l)$, for any $l\in{\mathbb Z}^r$ with $|l|$ sufficiently large., Comment: 10 pages, comments are wellcome

Each loop $\psi$ in the group $\text{Ham}(M)$ of Hamiltonian diffeomorphisms of a symplectic manifold $M$ determines a fibration $E$ on $S^2$, whose coupling class \cite{G-L-S} is denoted by $c$. If $VTE$ is the vertical tangent bundle of $E$, we relate the characteristic number $\int_E c_1(VTE)c^n$ with the Maslov index of the linearized flow $\psi_{t*}$ and the Chern class $c_1(TM)$. We give the value of this characteristic number for loops of Hamiltonian symplectomorphisms of Hirzebruch surfaces., Comment: Version to appear in Journal of Geometry and Physics

We prove that $\pi_1(\text{Ham}(M))$ contains an infinite cyclic subgroup, where $\text{Ham}(M)$ is the Hamiltonian group of the one point blow up of ${\Bbb C}P^3$. We give a sufficient condition for the group $\pi_1(\text{Ham}(M))$ to contain an infinite cyclic subgroup, when $M$ is a general toric manifold., Comment: 9 pages

Let ${\cal O}$ be a quantizable coadjoint orbit of a semisimple Lie group $G$. Under certain hypotheses we prove that $#(\pi_1(\text{Ham}({\cal O})))\geq #(Z(G))$, where $\text{Ham}({\cal O})$ is the group of Hamiltonian symplectomorphisms of ${\cal O}$., Comment: 6 pages. Added new reference. Proposition 4 corrected

The rational cohomology of a coadjoint orbit ${\cal O}$ is expressed as tensor product of the cohomology of other coadjoint orbits ${\cal O}_k$, with $ \hbox{dim} {\cal O}_k< \hbox{dim} {\cal O}$., Comment: Minor revision. Version to appear in Archiv der Mathematik

Let $\text{Ham(M)}$ be the group of Hamiltonian symplectomorphisms of a quantizable, compact, symplectic manifold $(M,\omega)$. We prove the existence of an action integral around loops in $\text{Ham(M)}$, and determine the value of this action integral on particular loops when the manifold is a coadjoint orbit., Comment: Expanded version. To appear in Geometriae Dedicata

Let ${\cal O}$ be the orbit of $\eta\in{\frak g}^*$ under the coadjoint action of the compact Lie group $G$. We give two formulae for calculating the action integral along a closed Hamiltonian isotopy on ${\cal O}$. The first one expresses this action in terms of a particular character of the isotropy subgroup of $\eta$. In the second one is involved the character of an irreducible representation of $G$., Comment: 12 pages, Latex

On the space ${\cal L}$, of loops in the group of Hamiltonian symplectomorphisms of a symplectic quantizable manifold, we define a closed ${\bf Z}$-valued 1-form $\Omega$. If $\Omega$ vanishes, the prequantization map can be extended to a group representation. On ${\cal L}$ one can define an action integral as an ${\bf R}/{\bf Z}$-valued function, and the cohomology class $[\Omega]$ is the obstruction to the lifting of that action integral to an ${\bf R}$-valued function. The form $\Omega$ also defines a natural grading on $\pi_1({\cal L})$., Comment: 18 pages