Generalized Kähler Geometry

Generalized Kähler geometry is the natural analogue of Kähler geometry, in the context of generalized complex geometry. Just as we may require a complex structure to be compatible with a Riemannian metric in a way which gives rise to a symplectic form, we may require a generalized complex structure to be compatible with a metric so that it defines a second generalized complex structure. We prove that generalized Kähler geometry is equivalent to the bi-Hermitian geometry on the target of a 2-dimensional sigma model with (2, 2) supersymmetry. We also prove the existence of natural holomorphic Courant algebroids for each of the underlying complex structures, and that these split into a sum of transverse holomorphic Dirac structures. Finally, we explore the analogy between pre-quantum line bundles and gerbes in the context of generalized Kähler geometry.     

Generalized Pseudo-Kähler Structures

In this paper we consider pseudo-bihermitian structures – pairs of complex structures compatible with a pseudo-Riemannian metric. We establish relations of these structures with generalized (pseudo-) Kähler geometry and holomorphic Poisson structures similar to that in the positive definite case. We provide a list of compact complex surfaces which could admit pseudo-bihermitian structures and give examples of such structures on some of them. We also consider a naturally defined null plane distribution on a generalized pseudo-Kähler 4-manifold and show that under a mild restriction it determines an Engel structure.     

Special geometry in hypermultiplets

We give a detailed analysis of pairs of vector and hypermultiplet theories with N = 2 supersymmetry in four space-time dimensions that are related by the (classical) mirror map. The symplectic reparametrizations of the special Kähler space associated with the vector multiplets induce corresponding transformations on the hypermultiplets. We construct the Sp(1) × Sp(n) one-forms in terms of which the hypermultiplet couplings are encoded and exhibit their behaviour under symplectic reparametrizations. Both vector and hypermultiplet theories allow vectorial central charges in the supersymmetry algebra associated with integrals over the Kähler and hyper-Kähler forms, respectively. We show how these charges and the holomorphic BPS mass are related by the mirror map.     

Generalized Kähler Geometry, Gerbes, and all that

We review recent advances in generalized Kähler geometry while stressing the use of Poisson and symplectic geometry. The derivation of a generalized Kähler potential is sketched and relevant global issues are discussed.     

Complex/Symplectic Mirrors

We construct a class of symplectic non-Kähler and complex non-Kähler string theory vacua, extending and providing evidence for an earlier suggestion by Polchinski and Strominger. The class admits a mirror pairing by construction. Comparing hints from a variety of sources, including ten-dimensional supergravity and KK reduction on SU(3)-structure manifolds, suggests a picture in which string theory extends Reid’s fantasy to connect classes of both complex non-Kähler and symplectic non-Kähler manifolds.     

Anti-holomorphic involutive isometry of hyper-Kähler manifolds and branes

We study complex Lagrangian submanifolds of a compact hyper-Kähler manifold and prove two results: (a) that an involution of a hyper-Kähler manifold which is antiholomorphic with respect to one complex structure and which acts non-trivially on the corresponding symplectic form always has a fixed point locus which is complex Lagrangian with respect to one of the other complex structures, and (b) there exist Lagrangian submanifolds which are complex with respect to one complex structure and are not the fixed point locus of any involution which is anti-holomorphic with respect to one of the other complex structures.     

Equivariant Volumes of Non-Compact Quotients and Instanton Counting

Motivated by Nekrasov’s instanton counting, we discuss a method for calculating equivariant volumes of non-compact quotients in symplectic and hyper-Kähler geometry by means of the Jeffrey-Kirwan residue formula of non-abelian localization. In order to overcome the non-compactness, we use varying symplectic cuts to reduce the problem to a compact setting, and study what happens in the limit that recovers the original problem. We implement this method for the ADHM construction of the moduli spaces of framed Yang-Mills instantons on \({\mathbb{R}^{4}}\) and rederive the formulas for the equivariant volumes obtained earlier by Nekrasov-Shadchin, expressing these volumes as iterated residues of a single rational function.     

Supersymmetric Quantum Theory and Non-Commutative Geometry

Classical differential geometry can be encoded in spectral data, such as Connes' spectral triples, involving supersymmetry algebras. In this paper, we formulate non-commutative geometry in terms of supersymmetric spectral data. This leads to generalizations of Connes' non-commutative spin geometry encompassing non-commutative Riemannian, symplectic, complex-Hermitian and (Hyper-) K?hler geometry. A general framework for non-commutative geometry is developed from the point of view of supersymmetry and illustrated in terms of examples. In particular, the non-commutative torus and the non-commutative 3-sphere are studied in some detail. Received: 1 April 1997 / Accepted: 24 November 1998     

The Foundations of Quantum Mechanics and the Evolution of the Cartan-Kähler Calculus

In 1960–1962, E. Kähler enriched É. Cartan’s exterior calculus, making it suitable for quantum mechanics (QM) and not only classical physics. His “Kähler-Dirac” (KD) equation reproduces the fine structure of the hydrogen atom. Its positron solutions correspond to the same sign of the energy as electrons. The Cartan-Kähler view of some basic concepts of differential geometry is presented, as it explains why the components of Kähler’s tensor-valued differential forms have three series of indices. We demonstrate the power of his calculus by developing for the electron’s and positron’s large components their standard Hamiltonian beyond the Pauli approximation, but without resort to Foldy-Wouthuysen transformations or ad hoc alternatives (positrons are not identified with small components in K ähler’s work). The emergence of negative energies for positrons in the Dirac theory is interpreted from the perspective of the KD equation. Hamiltonians in closed form (i.e. exact through a finite number of terms) are obtained for both large and small components when the potential is time-independent. A new but as yet modest new interpretation of QM starts to emerge from that calculus’ peculiarities, which are present even when the input differential form in the Kähler equation is scalar-valued. Examples are the presence of an extra spin term, the greater number of components of “wave functions” and the non-association of small components with antiparticles. Contact with geometry is made through a Kähler type equation pertaining to Clifford-valued differential forms.     

Special Symplectic Connections and Poisson Geometry

By a special symplectic connection we mean a torsion free connection which is either the Levi-Civita connection of a Bochner-Kähler metric of arbitrary signature, a Bochner-bi-Lagrangian connection, a connection of Ricci type or a connection with special symplectic holonomy. A manifold or orbifold with such a connection is called special symplectic. We show that any special symplectic connection can be constructed using symplectic realizations of quadratic deformations of a certain linear Poisson structure. Moreover, we show that these Poisson structures cannot be symplectically integrated by a Hausdorff groupoid. As a consequence, we obtain a canonical principal line bundle over any special symplectic manifold or orbifold, and we deduce numerous global consequences.     

Holonomy groups,complex structures andD=4 topological Yang-Mills theory

We study conditions for the existence of extended supersymmetry in topological Yang-Mills theory. These conditions are most conveniently formulated in terms of the holonomy group of the underlying manifold, on which the topological Yang-Mills theory is defined. For irreducible manifolds we find that extended supersymmetries are in 1–1 correspondence with covariantly constant complex structures. Therefore, the topological Yang-Mills theory on any Kähler manifold possesses one additional supersymmetry and on any hyper Kähler manifold there are three additional supersymmetries. The Donaldson map, which plays a crucial role in the construction of the topological invariants, is generalized for Kähler manifolds, thus providing candidates for new invariants of complex manifolds.     

Generalized Kähler Geometry from Supersymmetric Sigma Models

We give a physical derivation of generalized Kähler geometry. Starting from a supersymmetric nonlinear sigma model, we rederive and explain the results of Gualtieri (Generalized complex geometry, DPhil thesis, Oxford University, 2004) regarding the equivalence between generalized Kähler geometry and the bi-hermitean geometry of Gates et al. (Nucl Phys B248:157, 1984). When cast in the language of supersymmetric sigma models, this relation maps precisely to that between the Lagrangian and the Hamiltonian formalisms. We also discuss topological twist in this context.     

Walker 4-manifolds with proper almost complex structures

It is shown that a Walker 4-manifold, endowed with a canonical neutral metric depending on three arbitrary functions, admits a specific almost complex structure (called proper) and an associated opposite almost complex structure. We study when these two almost complex structures are integrable and when the corresponding Kähler forms are symplectic. The conditions for the canonical neutral metric to be Kähler imply that the three arbitrary functions in the metric are all harmonic with respect to two coordinate variables, and we obtain a useful method of constructing indefinite Kähler 4-manifolds. Petean’s example of a nonflat indefinite Kähler–Einstein 4-manifold is a special case of this construction.     

Fedosov’s formal symplectic groupoids and contravariant connections

Using Fedosov’s approach we give a geometric construction of a formal symplectic groupoid over any Poisson manifold endowed with a torsion-free Poisson contravariant connection. In the case of Kähler–Poisson manifolds this construction provides, in particular, the formal symplectic groupoids with separation of variables. We show that the dual of a semisimple Lie algebra does not admit torsion-free Poisson contravariant connections.     

Anti-Kählerian Geometry on Lie Groups

Let G be a Lie group of even dimension and let (g, J) be a left invariant anti-Kähler structure on G. In this article we study anti-Kähler structures considering the distinguished cases where the complex structure J is abelian or bi-invariant. We find that if G admits a left invariant anti-Kähler structure (g, J) where J is abelian then the Lie algebra of G is unimodular and (G, g) is a flat pseudo-Riemannian manifold. For the second case, we see that for any left invariant metric g for which J is an anti-isometry we obtain that the triple (G, g, J) is an anti-Kähler manifold. Besides, given a left invariant anti-Hermitian structure on G we associate a covariant 3-tensor ?? on its Lie algebra and prove that such structure is anti-Kähler if and only if ?? is a skew-symmetric and pure tensor. From this tensor we classify the real 4-dimensional Lie algebras for which the corresponding Lie group has a left invariant anti-Kähler structure and study the moduli spaces of such structures (up to group isomorphisms that preserve the anti-Kähler structures).     

Local index theorem for orbifold Riemann surfaces

We derive a local index theorem in Quillen’s form for families of Cauchy–Riemann operators on orbifold Riemann surfaces (or Riemann orbisurfaces) that are quotients of the hyperbolic plane by the action of cofinite finitely generated Fuchsian groups. Each conical point (or a conjugacy class of primitive elliptic elements in the Fuchsian group) gives rise to an extra term in the local index theorem that is proportional to the symplectic form of a new Kähler metric on the moduli space of Riemann orbisurfaces. We find a simple formula for a local Kähler potential of the elliptic metric and show that when the order of elliptic element becomes large, the elliptic metric converges to the cuspidal one corresponding to a puncture on the orbisurface (or a conjugacy class of primitive parabolic elements). We also give a simple example of a relation between the elliptic metric and special values of Selberg’s zeta function.

     

Analytic torsion and holomorphic determinant bundles

In this paper, we prove that in the case of holomorphic locally Kähler fibrations, the analytic and algebraic geometry constructions of determinant bundles for direct images coincide. We calculate the curvature of the holomorphic Hermitian connection for the Quillen metric on the determinant bundle. We study the behavior of the Quillen metric under change of metrics in the fibers, and also on the twisting vector bundles. We thus generalize the conformal anomaly formula to Kähler manifolds of arbitrary dimension. We also study the Quillen metrics on determinants associated with exact sequences of vector bundles. We prove that the Quillen metric is smooth on the Grothendieck-Knudsen-Mumford determinant for arbitrary holomorphic fibrations.     

Gauge Linear Sigma Model for Berglund—Hübsch-Type Calabi—Yau Manifolds

We briefly present the results of our computation of special Kähler geometry for polynomial deformations of Berglund–Hübsch type Calabi–Yau manifolds. We also build mirror symmetric Gauge Linear Sigma Model and check that its partition function computed by supersymmetric localization coincides with exponent of the Kähler potential of the special metric.