Quaternionic Geometry Research Articles

Membrane and fivebrane instantons from quaternionic geometry

We determine the one-instanton corrections to the universal hypermultiplet moduli space coming both from Euclidean membranes and NS-fivebranes wrapping the cycles of a (rigid) Calabi-Yau threefold. These corrections are completely encoded by a single function characterizing a generic four-dimensional quaternion-Kahler metric without isometries. We give explicit solutions for this function describing all one-instanton corrections, including the fluctuations around the instanton to all orders in the string coupling constant. In the semi-classical limit these results are in perfect agreement with previous supergravity calculations.

Generalized planar curves and quaternionic geometry

Motivated by the analogies between the projective and the almost quaternionic geometries, we first study the generalized planar curves and mappings. We follow, recover, and extend the classical approach, see e.g., (Sov. Math. 27(1) 63–70 (1983), Rediconti del circolo matematico di Palermo, Serie II, Suppl. 54 75–81) (1998), Then we exploit the impact of the general results in the almost quaternionic geometry. In particular we show, that the natural class of ℍ-planar curves coincides with the class of all geodesics of the so called Weyl connections and preserving this class turns out to be the necessary and sufficient condition on diffeomorphisms to become morphisms of almost quaternionic geometries.

Reduction of Vaisman structures in complex and quaternionic geometry

We consider locally conformal Kähler geometry as an equivariant (homothetic) Kähler geometry: a locally conformal Kähler manifold is, up to equivalence, a pair ( K , Γ ) , where K is a Kähler manifold and Γ is a discrete Lie group of biholomorphic homotheties acting freely and properly discontinuously. We define a new invariant of a locally conformal Kähler manifold ( K , Γ ) as the rank of a natural quotient of Γ , and prove its invariance under reduction. This equivariant point of view leads to a proof that locally conformal Kähler reduction of compact Vaisman manifolds produces Vaisman manifolds and is equivalent to a Sasakian reduction. Moreover, we define locally conformal hyperKähler reduction as an equivariant version of hyperKähler reduction and in the compact case we show its equivalence with 3-Sasakian reduction. Finally, we show that locally conformal hyperKähler reduction induces hyperKähler with torsion (HKT) reduction of the associated HKT structure and the two reductions are compatible, even though not every HKT reduction comes from a locally conformal hyperKähler reduction.

Membrane Instantons and de Sitter Vacua

We investigate membrane instanton effects in type-IIA strings compactified on rigid Calabi-Yau manifolds. These effects contribute to the low-energy effective action of the universal hypermultiplet. In the absence of additional fivebrane instantons, the quaternionic geometry of this hypermultiplet is determined by solutions of the three-dimensional Toda equation. We construct solutions describing membrane instantons, and find perfect agreement with the string theory prediction. In the context of flux compactifications we discuss how membrane instantons contribute to the scalar potential and the stabilization of moduli. Finally, we demonstrate the existence of meta-stable de Sitter vacua.

Quaternionic geometry of matroids

Abstract Building on a recent paper [8], here we argue that the combinatorics of matroids are intimately related to the geometry and topology of toric hyperkähler varieties. We show that just like toric varieties occupy a central role in Stanley’s proof for the necessity of McMullen’s conjecture (or g-inequalities) about the classification of face vectors of simplicial polytopes, the topology of toric hyperkähler varieties leads to new restrictions on face vectors of matroid complexes. Namely in this paper we will give two proofs that the injectivity part of the Hard Lefschetz theorem survives for toric hyperkähler varieties. We explain how this implies the g-inequalities for rationally representable matroids. We show how the geometrical intuition in the first proof, coupled with results of Chari [3], leads to a proof of the g-inequalities for general matroid complexes, which is a recent result of Swartz [20]. The geometrical idea in the second proof will show that a pure O-sequence should satisfy the g-inequalities, thus showing that our result is in fact a consequence of a long-standing conjecture of Stanley.

Harmonicity and minimality of oriented distributions

We consider an oriented distribution as a section of the corresponding Grassmann bundle and, by computing the tension of this map for conveniently chosen metrics, we obtain the conditions which the distribution must satisfy in order to be critical for the functionals related to the volume or the energy of the map. We show that the three-dimensional distribution ofS4m+3 tangent to the quaternionic Hopf fibration defines a harmonic map and a minimal immersion and we extend these results to more general situations coming from 3-Sasakian and quaternionic geometry.

Scherk–Schwarz reduction of D = 5 special and quaternionic geometry

We give the N = 2 gauged supergravity interpretation of a generic D = 4, N = 2 theory as it comes from the generalized Scherk–Schwarz reduction of D = 5, N = 2 (ungauged) supergravity. We focus on the geometric aspects of the D = 4 data such as the general form of the scalar potential and masses in terms of the gauging of a ‘flat group’. Higgs and super-Higgs mechanisms are discussed in some detail.

Almost Hermitian structures and quaternionic geometries

Gray and Hervella gave a classification of almost Hermitian structures ( g, I) into 16 classes. We systematically study the interaction between these classes when one has an almost hyper-Hermitian structure ( g, I, J, K). In general dimension we find at most 167 different almost hyper-Hermitian structures. In particular, we obtain a number of relations that give hyper-Kähler or locally conformal hyper-Kähler structures, thus generalising a result of Hitchin. We also study the types of almost quaternion-Hermitian geometries that arise and tabulate the results.

C-map, very special quaternionic geometry and dual Kähler spaces

We show that for all very special quaternionic manifolds a different N=1 reduction exists, defining a Kähler geometry which is “dual” to the original very special Kähler geometry with metric Gab̄=−∂a∂blnV (V=(1/6)dabcλaλbλc). The dual metric gab=V−2(G−1)ab is Kähler and it also defines a flat potential as the original metric. Such geometries and some of their extensions find applications in type IIB compactifications on Calabi–Yau orientifolds.

4-D gauged supergravity analysis of type-IIB vacua onK3×T2/Bbb Z2

We analyze $N=2,1,0$ vacua of type IIB string theory on $K3\times T^2/Z_2$ in presence of three-form fluxes from a four dimensional supergravity viewpoint. The quaternionic geometry of the $K3$ moduli space together with the special geometry of the NS and R-R dilatons and of the $T^2$-complex structure moduli play a crucial role in the analysis. The introduction of fluxes corresponds to a particular gauging of N=2, D=4 supergravity. Our results agree with a recent work of Tripathy and Trivedi. The present formulation shows the power of supergravity in the study of effective theories with broken supersymmetry.

N= 2 Super-Higgs,N= 1 Poincaré Vacua and Quaternionic Geometry

In the context of N=2 supergravity we explain the occurrence of partial super-Higgs with vanishing vacuum energy and moduli stabilization in a model suggested by superstring compactifications on type IIB orientifolds with 3-form fluxes. The gauging of axion symmetries of the quaternionic manifold, together with the use of degenerate symplectic sections for special geometry, are the essential ingredients of the construction.

Non-standard singular invariant differential operators for quaternionic parabolic geometries — Beyond the An-series of Lie algebras

R.J. Baston initiated various techniques in the study of invariant differential operators, fitting them into a. scheme now usually called parabolic invariant theory (parabolic geometry). We shall discuss in particular the roles of generalized Penrose transform and techniques of spectral sequences in generating non-standard singular invariant differential operators. In particular we shall derive, in the case of Lie algebraB 3, sequences of invariant differential operators (containing the non-standard ones), known in the cases ofA n-series of Lie algebras as quaternion complexes, [1].

On fermion masses, gradient flows and potential in supersymmetric theories

In any low energy effective supergravity theory general formulae exist which allow one to discuss fermion masses, the scalar potential and breaking of symmetries in a model independent set up. A particular role in this discussion is played by Killing vectors and Killing prepotentials. We outline these relations in general and specify then in the context of N=1 and N=2 supergravities in four dimensions. Useful relations of gauged quaternionic geometry underlying hypermultiplets dynamics are discussed.

Invariant operators on manifolds with almost Hermitian symmetric structures, III. Standard operators

This paper demonstrates the power of the calculus developed in the two previous parts of the series for all real forms of the almost Hermitian symmetric structures on smooth manifolds, including, e.g., conformal Riemannian and almost quaternionic geometries. Exploiting some finite-dimensional representation theory of simple Lie algebras, we give explicit formulae for distinguished invariant curved analogues of the standard operators in terms of the linear connections belonging to the structures in question, so in particular we prove their existence. Moreover, we prove that these formulae for kth order standard operators, k=1,2,…, are universal for all geometries in question

Instanton action for Type II hypermultiplets

We analyze the hypermultiplet moduli space describing the universal sector of type IIA theory compactified on a Calabi-Yau threefold. The classical moduli space is described in terms of the coset SU(2,1) U(2) . The flux quantization condition of the antisymmetric tensor field of M-theory implies discrete identifications for the scalar fields describing this four-dimensional quaternionic geometry. Non-perturbative corrections of the classical theory are described in terms of a membrane-fivebrane instanton action which we construct herein.

The quaternionic geometry of four-dimensional conformal field theory

We show that four-dimensional conformal field theory is most naturally formulated on Kulkarni 4-folds, i.e. real 4-folds endowed with an integrable quaternionic structure. This leads to a formalism that parallels very closely that of two-dimensional conformal field theory on Riemann surfaces. In this framework, the notion of Fueter analyticity, the quaternionic analog of complex analyticity, plays an essential role. Conformal fields appear as sections of appropriate either harmonic real or Fueter holomorphic quaternionic line bundles. In the free case, the field equations are statements of either harmonicity or Fueter holomorphicity of the relevant conformal fields. We obtain compact quaternionic expressions of such basic objects as the energy-momentum tensor and the gauge currents for some basic models in terms of Kulkarni geometry. We also find a concise expression of the conformal anomaly and a quaternionic four-dimensional analog of the Schwarzian derivative describing the covariance of the quantum energy-momentum tensor. Finally, we analyze the operator product expansions of free fields.

The complete form of N = 2 supergravity and its place in the general framework of D = 4 N-extended supergravities

Relying on the geometrical set up of Special Kähler Geometry and Quaternionic Geometry, which I discussed at length in my Lectures at the 1995 edition of this Spring School, I present here the recently obtained fully general form of N = 2 supergravity with completely arbitrary couplings. This lagrangian has already been used in the literature to obtain various results: notably the partial breaking of supersymmetry and various extremal black-hole solutions. My emphasis, however, is only on providing the reader with a completely explicit and ready to use component expression of the supergravity action. All the details of the derivation are omitted but all the definitions of the items entering the lagrangian and the supersymmetry transformation rules are given.

Locally conformal Kähler structures in quaternionic geometry

We consider compact locally conformal quaternion Kähler manifolds M M . This structure defines on M M a canonical foliation, which we assume to have compact leaves. We prove that the local quaternion Kähler metrics are Ricci-flat and allow us to project M M over a quaternion Kähler orbifold N N with fibers conformally flat 4-dimensional real Hopf manifolds. This fibration was known for the subclass of locally conformal hyperkähler manifolds; in this case we make some observations on the fibers’ structure and obtain restrictions on the Betti numbers. In the homogeneous case N N is shown to be a manifold and this allows a classification. Examples of locally conformal quaternion Kähler manifolds (some with a global complex structure, some locally conformal hyperkähler) are the Hopf manifolds quotients of H n − { 0 } \mathbb H^n-\{0\} by the diagonal action of appropriately chosen discrete subgroups of C O + ( 4 ) CO^+(4) .

Lectures on Special Kähler geometry and electric—magnetic duality rotations

In these lectures I review the general structure of electric--magnetic duality rotations in every even space--time dimension. In four dimensions, which is my main concern, I discuss the general issue of symplectic covariance and how it relates to the typical geometric structures involved by N=2 supersymmetry, namely Special K\"ahler geometry for the vector multiplets and either HyperK\"ahler or Quaternionic geometry for the hypermultiplets. I discuss classical continuous dualities versus non--perturbative discrete dualities. How the moduli space geometry of an auxiliary dynamical Riemann surface (or Calabi--Yau threefold) relates to exact space--time dualities is exemplified in detail for the Seiberg Witten model of an $SU(2)$ gauge theory.

Topological quantum field theory in quaternionic geometry

A topological quantum field theory on a 4 k-dimensional manifold M admitting an almost quaternionic structure is proposed. Expectation values of certain operators on M are proved to be independent of the choice of an almost quaternionic structure used in calculations and thus carry only smooth information about M. These invariants are explicitly expressed as integrals of differential forms over the instanton moduli space associated with a chosen almost quaternionic structure. When M admits a hyperKähler structure the topological field theory has three additional supersymmetries which induce three complex structures on the associated instanton moduli space proving thus that the latter is a hypercomplex manifold. In this case an analogue of the four-dimensional Donaldson map is constructed which provides a number of candidates for new invariants of 4 k-dimensional hypercomplex structures. In the case k = 1 the proposed topological theory on an almost quaternionic manifold reproduces the Witten interpretation of the four-dimensional Donaldson invariants.