Generalized Complex Geometry Research Articles

On a B-field transform of generalized complex structures over complex tori

Let (Xn,Xˇn) be a mirror pair of an n-dimensional complex torus Xn and its mirror partner Xˇn. Then, by SYZ transform, we can construct a holomorphic line bundle with an integrable connection from each pair of a Lagrangian section of Xˇn→Rn/Zn and a unitary local system along it, and those holomorphic line bundles with integrable connections form a dg-category DGXn. In this paper, we focus on a certain B-field transform of the generalized complex structure induced from the complex structure on Xn, and interpret it as the deformation XGn of Xn by a flat gerbe G. Moreover, we construct the deformation of DGXn associated to the deformation from Xn to XGn, and also discuss the homological mirror symmetry between XGn and its mirror partner on the object level.

Integrable sigma model with generalized [formula omitted] structure, Yang-Baxter sigma model with generalized complex structure and multi-Yang-Baxter sigma model

We construct an integrable sigma model with a generalized F structure, which involves a generalized Nijenhuis structure J satisfying J3=−J. Utilizing the expression of the generalized complex structure on the metric Lie group manifold G in terms of operator relations on its Lie algebra g, we formulate a Yang-Baxter sigma model with a generalized complex structure. Additionally, we present multi-Yang-Baxter sigma models featuring two and three compatible Nijenhuis structures. Examples for each of these models are provided.

D-terms in generalised complex geometry

Generalised Complex Geometry provides a natural interpretation of the N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 supersymmetry conditions for warped solutions of type II supergravity as differential equations on polyforms on the internal manifold. Written in this language the supersymmetry conditions correspond to calibration conditions for probe D-branes: D-string, domain-wall or space-filling branes, depending on the directions they span in the non-compact four-dimensional space. The BPS condition corresponding to the calibration of space-filling D-branes has been reformulated by Tomasiello, eliminating the explicit dependence on the metric. We generalise this derivation to the case of non-supersymmetric backgrounds violating the domain-wall and D-string calibration conditions. We use this reformulation to derive constraints that the ten-dimensional solutions with BPS space-filling sources must respect in order to dimensionally reduce to solutions of four-dimensional N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 supergravity with non-vanishing F-terms and potentially non-vanishing D-terms. We give the equations of motion for the class of type II vacua satisfying these constraints in the language of pure spinors. We investigate how restrictive these constraints are for the class of type IIB SU(3) backgrounds with BPS space-filling O5-planes.

Generalized Luttinger surgery and other cut-and-paste constructions in generalized complex geometry

Exploiting the affinity between stable generalized complex structures and symplectic structures, we explain how certain constructions coming from symplectic geometry can be performed in the generalized complex setting. We introduce generalized Luttinger surgery and generalized Gluck twist along J-symplectic submanifolds. We also study branched coverings in the generalized complex setting. As an application, we produce stable generalized complex structures on a variety of high-dimensional manifolds. Remarkably, some of them have non-homotopy-equivalent path-connected components of their type change locus.

Integration of generalized complex structures

We solve the integration problem for generalized complex manifolds, obtaining as the natural integrating object a holomorphic stack with a shifted symplectic structure; in other words, a real symplectic groupoid with a compatible complex structure is defined only up to Morita equivalence. We explain how such objects differentiate to give generalized complex manifolds, and we show that a generalized complex manifold is integrable in this sense if and only if its underlying real Poisson structure is integrable. We describe several concrete examples of these integrations. Crucial to our solution are new technical tools, which are of independent interest, namely, a reduction procedure for Lie groupoid actions on Courant algebroids, as well as certain local-to-global extension results for multiplicative forms on local Lie groupoids.

A remark about mirror symmetry of elliptic curves and generalized complex geometry

In this short note we describe the isomorphism of generalized complex structure between T-dual manifolds introduced by Cavalcanti-Gualtieri, in the case of elliptic curves. We also compare this isomorphism with the mirror map for elliptic curves described by Polishchuk and Zaslow.

Smearing and unsmearing KKLT AdS vacua

Gaugino condensation on D-branes wrapping internal cycles gives a mechanism to stabilize the associated moduli. According to the effective field theory, this gives rise, when combined with fluxes, to supersymmetric AdS4 solutions. In this paper we provide a ten-dimensional description of these vacua. We first find the supersymmetry equations for type II AdS4 vacua with gaugino condensates on D-branes, in the framework of generalized complex geometry. We then solve them for type IIB compactifications with gaugino condensates on smeared D7-branes. We show that supersymmetry requires a (conformal) Calabi-Yau manifold and imaginary self-dual three-form fluxes with an additional (0,3) component. The latter is proportional to the cosmological constant, whose magnitude is determined by the expectation value of the gaugino condensate and the stabilized volume of the cycle wrapped by the branes. This confirms, qualitatively and quantitatively, the results obtained using effective field theory. We find that exponential separation between the AdS and the KK scales seems possible as long as the three-form fluxes are such that their (0,3) component is exponentially suppressed. As for the localized solution, it requires going beyond SU(3)-structure internal manifolds. Nevertheless, we show that the action can be evaluated on-shell without relying on the details of such complicated configuration. We find that no “perfect square” structure occurs, and the result is divergent. We compute the four-fermion contributions, including a counterterm, needed to cancel these divergences.

Invariant Generalized Almost Complex Structures on Real Flag Manifolds

We characterize those real flag manifolds that can be endowed with invariant generalized almost complex structures. We show that no $GM_2$-maximal real flag manifolds admit integrable invariant generalized almost complex structures. We give a concrete description of the generalized complex geometry on the maximal real flags of type $B_2$, $G_2$, $A_3$, and $D_l$ with $l\geq 5$, where we prove that the space of invariant generalized almost complex structures under invariant $B$-transformations is homotopy equivalent to a torus and we classify all invariant generalized almost Hermitian structures on them.

Complex Dirac Structures: Invariants and Local Structure

We study complex Dirac structures, that is, Dirac structures in the complexified generalized tangent bundle. These include presymplectic foliations, transverse holomorphic structures, CR-related geometries and generalized complex structures. We introduce two invariants, the order and the (normalized) type. We show that, together with the real index, they allow us to obtain a pointwise classification of complex Dirac structures. For constant order, we prove the existence of an underlying real Dirac structure, which generalizes the Poisson structure associated to a generalized complex structure. For constant real index and order, we prove a splitting theorem, which gives a local description in terms of a presymplectic leaf and a small transversal.

Fibrations in semitoric and generalized complex geometry

AbstractThis paper studies a class of singular fibrations, called self-crossing boundary fibrations, which play an important role in semitoric and generalized complex geometry. These singular fibrations can be conveniently described using the language of Lie algebroids. We will show how these fibrations arise from (nonfree) torus actions, and how to use them to construct and better understand self-crossing stable generalized complex four-manifolds. We moreover show that these fibrations are compatible with taking connected sums, and use this to prove a singularity trade result between two types of singularities occurring in these types of fibrations (a so-called nodal trade).

Brane Quantization of Toric Poisson Varieties

In this paper we propose a noncommutative generalization of the relationship between compact K\"ahler manifolds and complex projective algebraic varieties. Beginning with a prequantized K\"ahler structure, we use a holomorphic Poisson tensor to deform the underlying complex structure into a generalized complex structure, such that the prequantum line bundle and its tensor powers deform to a sequence of generalized complex branes. Taking homomorphisms between the resulting branes, we obtain a noncommutative deformation of the homogeneous coordinate ring. As a proof of concept, this is implemented for all compact toric K\"ahler manifolds equipped with an R-matrix holomorphic Poisson structure, resulting in what could be called noncommutative toric varieties. To define the homomorphisms between generalized complex branes, we propose a method which involves lifting each pair of generalized complex branes to a single coisotropic A-brane in the real symplectic groupoid of the underlying Poisson structure, and compute morphisms in the A-model between the Lagrangian identity bisection and the lifted coisotropic brane. This is done with the use of a multiplicative holomorphic Lagrangian polarization of the groupoid.

Toric generalized Kähler structures. I

This is a sequel of [20], which provides a general formalism for this paper. We mainly investigate thoroughly a subclass of toric generalized Kähler manifolds of symplectic type introduced by Boulanger [3]. We find that torus actions on such manifolds are all strong Hamiltonian in the sense of [20]. For each such a manifold, we prove that besides the ordinary complex structures J± underlying the biHermitian description, there is a third canonical complex structure J0, which makes the manifold toric Kähler. The other generalized complex structure besides the symplectic one is always a B-transform of a generalized complex structure induced from a J0-holomorphic Poisson structure β characterized by an anti-symmetric constant matrix. Stimulated by the above results, we introduce a generalized Delzant construction which starts from a Delzant polytope with d faces of codimension 1, the standard Kähler structure of Cd and an anti-symmetric d×d matrix. This construction is used to produce non-abelian examples of strong Hamiltonian actions.

Self-crossing stable generalized complex structures

We extend the notion of (smooth) stable generalized complex structures to allow for an anticanonical section with normal self-crossing singularities. This weakening not only allows for a number of natural examples in higher dimensions but also sheds some light into the smooth case in dimension four. We show that in four dimensions there is a natural connected sum operation for these structures as well as a smoothing operation which changes a self-crossing stable generalized complex structure into a smooth stable generalized complex structure on the same manifold. This allows us to construct large families of stable generalized complex manifolds.

Matsushima–Lichnerowicz type theorems of Lie algebra of automorphisms of generalized Kähler manifolds of symplectic type

In K\"ahler geometry, Fujiki--Donaldson show that the scalar curvature arises as the moment map for Hamiltonian diffeomorphisms. In generalized K\"ahler geometry, one does not have suitable notions of Levi-Civita connection and curvature, however there still exists a precise framework for a moment map and the scalar curvature is defined as the moment map. Then a fundamental question is to understand the existence or non-existence of generalized K\"ahler structures with constant scalar curvature. In the paper, we study the Lie algebra of automorphisms of a generalized complex manifold. We assume that $H^{1}(M)=0$. Then we show that the Lie algebra of the automorphisms is a reductive Lie algebra if a generalized complex manifold admits a generalized K\"ahler structure of symplectic type with constant scalar curvature. This is a generalization of Matsushima and Lichnerowicz theorem in K\"ahler geometry. We explicitly calculate the Lie algebra of the automorphisms of a generalized complex structure given by a cubic curve on $\Bbb C P^2$. Cubic curves are classified into nine cases (see Figure.$1 -- 9$). In the three cases as in Figures. 7, 8 and 9, the Lie algebra of the automorphisms is not reductive and there is an obstruction to the existence of generalized K\"ahler structures of symplectic type with constant scalar curvature in the three cases. We also discuss deformations starting from an ordinary K\"ahler manifold $(X,\omega)$ with constant scalar curvature and show that nontrivial generalized K\"ahler structures of symplectic type with constant scalar curvature arise as deformations if the Lie algebra of automorphisms of $X$ is trivial.

The cohomology of the elliptic tangent bundle

In this note we compute the cohomology of the elliptic tangent bundle, a Lie algebroid introduced in Cavalcanti and Gualtieri (2018), Cavalcanti et al. (2020) used to describe singular symplectic forms arising from generalised complex geometry.

Twisted Rota–Baxter operators and Reynolds operators on Lie algebras and NS-Lie algebras

In this paper, we introduce twisted Rota–Baxter operators on Lie algebras as an operator analog of twisted r-matrices. We construct a suitable L∞-algebra whose Maurer–Cartan elements are given by twisted Rota–Baxter operators. This allows us to define the cohomology of a twisted Rota–Baxter operator. This cohomology can be seen as the Chevalley–Eilenberg cohomology of a certain Lie algebra with coefficients in a suitable representation. We study deformations of twisted Rota–Baxter operators from cohomological points of view. Some applications are given to Reynolds operators and twisted r-matrices. Next, we introduce a new algebraic structure, called NS-Lie algebras, that is related to twisted Rota–Baxter operators in the same way pre-Lie algebras are related to Rota–Baxter operators. We end this paper by considering twisted generalized complex structures on modules over Lie algebras.

Left invariant generalized complex and Kähler structures on simply connected four dimensional Lie groups: Classification and invariant cohomologies

We give a complete classification of left invariant generalized complex structures of type 1 on four dimensional simply connected Lie groups and we compute for each class its invariant generalized Dolbeault cohomology, its invariant generalized Bott-Chern cohomology and its invariant generalized Aeppli cohomology. We classify also left invariant generalized Kähler structures on four dimensional simply connected Lie groups.

Twistor space of a generalized quaternionic manifold

We first make a little survey of the twistor theory for hypercomplex, generalized hypercomplex, quaternionic or generalized quaternionic manifolds. This last theory was initiated by Pantilie (Ann. Mat. Pura. Appl. 193 (2014) 633–641), and allows one to extend the Penrose correspondence from the quaternion to the generalized quaternion case. He showed that any generalized almost quaternionic manifold equipped with an appropriate connection admit a twistor space which comes naturally equipped with a tautological almost generalized complex structure. But he has left open the problem of the integrability. The aim of this article is to give an integrability criterion for this generalized almost complex structure and to give some examples especially in the case of generalized hyperkähler manifolds using the generalized Bismut connection, introduced by Gualtieri (Branes on Poisson varieties, The many facets of geometry: a tribute to Nigel Hitchin (2010) (Oxford: Oxford University Press) pp. 368–395).

Invariant generalized complex geometry on maximal flag manifolds and their moduli

We describe moduli spaces of invariant generalized complex structures and moduli spaces of invariant generalized Kähler structures on maximal flag manifolds under B-transformations. We give an alternative description of the moduli space of generalized complex structures using pure spinors, and describe a cell decomposition of these moduli spaces induced by the action of the Weyl group.

Generalized almost even-Clifford manifolds and their twistor spaces

Abstract Motivated by the recent interest in even-Clifford structures and in generalized complex and quaternionic geometries, we introduce the notion of generalized almost even-Clifford structure. We generalize the Arizmendi-Hadfield twistor space construction on even-Clifford manifolds to this setting and show that such a twistor space admits a generalized complex structure under certain conditions.