GLSMs for partial flag manifolds

In this paper we outline some aspects of nonabelian gauged linear sigma models. First, we review how partial flag manifolds (generalizing Grassmannians) are described physically by nonabelian gauged linear sigma models, paying attention to realizations of tangent bundles and other aspects pertinent to (0, 2) models. Second, we review constructions of Calabi–Yau complete intersections within such flag manifolds, and properties of the gauged linear sigma models. We discuss a number of examples of nonabelian GLSMs in which the Kähler phases are not birational, and in which at least one phase is realized in some fashion other than as a complete intersection, extending previous work of Hori–Tong. We also review an example of an abelian GLSM exhibiting the same phenomenon. We tentatively identify the mathematical relationship between such non-birational phases, as examples of Kuznetsov’s homological projective duality. Finally, we discuss linear sigma model moduli spaces in these gauged linear sigma models. We argue that the moduli spaces being realized physically by these GLSMs are precisely Quot and hyperquot schemes, as one would expect mathematically.     

The Equivariant Cohomology Theory of Twisted Generalized Complex Manifolds

It has been shown recently by Kapustin and Tomasiello that the mathematical notion of Hamiltonian actions on twisted generalized Kähler manifolds is in perfect agreement with the physical notion of general (2, 2) gauged sigma models with three-form fluxes. In this article, we study the twisted equivariant cohomology theory of Hamiltonian actions on H-twisted generalized complex manifolds. If the manifold satisfies the ${\overline{\partial} \partial}It has been shown recently by Kapustin and Tomasiello that the mathematical notion of Hamiltonian actions on twisted generalized K?hler manifolds is in perfect agreement with the physical notion of general (2, 2) gauged sigma models with three-form fluxes. In this article, we study the twisted equivariant cohomology theory of Hamiltonian actions on H-twisted generalized complex manifolds. If the manifold satisfies the -lemma, we establish the equivariant formality theorem. If in addition, the manifold satisfies the generalized K?hler condition, we prove the Kirwan injectivity in this setting. We then consider the Hamiltonian action of a torus on an H-twisted generalized Calabi-Yau manifold and extend to this case the Duistermaat-Heckman theorem for the push-forward measure. As a side result, we show in this paper that the generalized K?hler quotient of a generalized K?hler vector space can never have a (cohomologically) non-trivial twisting. This gives a negative answer to a question asked by physicists whether one can construct (2, 2) gauged linear sigma models with non-trivial fluxes.     

A gravitino-rich universe

Journal of High Energy Physics - Recently it has been shown that the two-sphere partition function of a gauged linear sigma model of a Calabi-Yau manifold yields the exact quantum Kähler...     

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.     

Gauge-invariant effective lagrangians for goldstone-like particles of supersymmetric technicolor models

We point out that generally the low-energy spectrum in supersymmetric technicolor models contains quasi-Goldstone fermions and quasi-Goldstone bosons in addition to the usual (pseudo)- Goldstone bosons. Using the language of Kähler geometry, we present a step-by-step procedure for constructing gauge-invariant non-linear lagrangians involving the fermionic and bosonic Goldstone particles in situations in which supersymmetry is preserved. Both the cases of fully gauged and partially gauged global symmetries are considered. We discuss the dynamical version of the super-Higgs mechanism, and we illustrate it with the supersymmetric Susskind-Weinberg technicolor model.     

Mirror symmetry,mirror map and applications to complete intersection Calabi-Yau spaces

We extend the discussion of mirror symmetry, Picard-Fuchs equations, instanton corrected Yukawa couplings and the topological one-loop partition function to the case of complete intersections with higher dimensional moduli spaces. We will develop a new method of obtaining the instanton corrected Yukawa couplings through a study of the solutions of the Picard-Fuchs equations. This leads to closed formulas for the prepotential for the Kähler moduli fields induced from the ambient space for all complete intersections in nonsingular weighted projective spaces. As examples we treat part of the moduli space of the phenomenologically interesting three-generation models which are found in this class. We also apply our method to solve the simplest model in which a topology change was observed and discuss examples of complete intersections in singular ambient spaces.     

On the Hyperkähler/Quaternion Kähler Correspondence

A hyperkähler manifold with a circle action fixing just one complex structure admits a natural hyperholomorphic line bundle. This observation forms the basis for the construction of a corresponding quaternionic Kähler manifold in the work of A.Haydys. In this paper the corresponding holomorphic line bundle on twistor space is described and many examples computed, including monopole and Higgs bundle moduli spaces. Finally a twistor version of the hyperkähler/quaternion Kähler correspondence is established.     

Kähler metrics and gauge kinetic functions for intersecting D6‐branes on toroidal orbifolds – The complete perturbative story

We systematically derive the perturbatively exact holomorphic gauge kinetic function, the open string Kähler metrics and closed string Kähler potential on intersecting D6‐branes by matching open string one‐loop computations of gauge thresholds with field theoretical gauge couplings in 𝒩 = 1 supergravity. We consider all cases of bulk, fractional and rigid D6‐branes on T⁶/Ω ℛ and the orbifolds T⁶/(ℤ_N × Ω ℛ) and T⁶/(ℤ₂ × ℤ_2M × Ω ℛ) without and with discrete torsion, which differ in the number of bulk complex structures and in the bulk Kähler potential. Our analysis includes all supersymmetric configurations of vanishing and non‐vanishing angles among D6‐branes and O6‐planes, and all possible Wilson line and displacement moduli are taken into account. The shape of the Kähler moduli turns out to be orbifold independent but angle dependent, whereas the holomorphic gauge kinetic functions obtain three different kinds of one‐loop corrections: a Kähler moduli dependent one for some vanishing angle independently of the orbifold background, another one depending on complex structure moduli only for fractional and rigid D6‐branes, and finally a constant term from intersections with O6‐planes. These results are of essential importance for the construction of the related effective field theory of phenomenologically appealing D‐brane models. As first examples, we compute the complete perturbative gauge kinetic functions and Kähler metrics for some T⁶/ℤ₂ × ℤ₂ examples with rigid D‐branes of [1]. As a second class of examples, the Kähler metrics and gauge kinetic functions for the fractional QCD and leptonic D6‐brane stacks of the Standard Model on T⁶/ℤ₆^′T⁶/ℤ₆^′ from [2] are given.     

Two-Sphere Partition Functions and Kähler Potentials on CY Moduli Spaces

JETP Letters - We study the relation between exact partition functions of gauged N = (2, 2) linear sigma-models on S2 and Kähler potentials of Calabi–Yau manifolds proposed by Jockers et...     

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.     

A new visualization of the gauged supersymmetric sigma-model by Killing potentials

The differential geometry of Kähler group manifolds will be thoroughly interpreted through Killing potentials. This enables us to reformulate the four dimensional gauged supersymmetric σ-model on Kähler group manifolds by Killing potentials. In the reformulation the Lagrangian will take a simple form in which the isometry of the manifolds is linearly manifest. The scalar curvature of the manifolds will be ascribed to the spontaneous breaking of supersymmetry in the model.     

Heterotic String Compactification and New Vector Bundles

We propose a construction of Kähler and non-Kähler Calabi–Yau manifolds by branched double covers of twistor spaces. In this construction we use the twistor spaces of four-manifolds with self-dual conformal structures, with the examples of connected sum of n\({\mathbb{P}^{2}}\)s. We also construct K3-fibered Calabi–Yau manifolds from the branched double covers of the blow-ups of the twistor spaces. These manifolds can be used in heterotic string compactifications to four dimensions. We also construct stable and polystable vector bundles. Some classes of these vector bundles can give rise to supersymmetric grand unified models with three generations of quarks and leptons in four dimensions.     

Kähler quantization of vortex moduli

Letters in Mathematical Physics - We discuss the Kähler quantization of moduli spaces of vortices in line bundles over compact surfaces $$\Sigma $$. This furnishes a semiclassical framework...     

Poisson Sigma Model on the Sphere

We evaluate the path integral of the Poisson sigma model on the sphere and study the correlators of quantum observables. We argue that for the path integral to be well-defined the corresponding Poisson structure should be unimodular. The construction of the finite dimensional BV theory is presented and we argue that it is responsible for the leading semiclassical contribution. For a (twisted) generalized Kähler manifold we discuss the gauge fixed action for the Poisson sigma model. Using the localization we prove that for the holomorphic Poisson structure the semiclassical result for the correlators is indeed the full quantum result.     

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.     

D-branes, exceptional sheaves and quivers on Calabi–Yau manifolds: from Mukai to McKay

We present a method based on mutations of helices which leads to the construction (in the large-volume limit) of exceptional coherent sheaves associated with the (∑_al_a=0) orbits in Gepner models. This is explicitly verified for a few examples including some cases where the ambient weighted projective space has singularities not inherited by the Calabi–Yau hypersurface. The method is based on two conjectures which lead to the analog, in the general case, of the Beilinson quiver for . We discuss how one recovers the McKay quiver using the gauged linear sigma model (GLSM) near the orbifold or Gepner point in Kähler moduli space.     

N=1 supergravity theories coupled to matter and duality transformations

The most general action for chiral and complex linear superfields coupled to theN=1 old minimal supergravity is given. Scalar potentials for pure complex linear and mixed cases are found. A condition for the breakdown of the duality transformation, which transforms a theory with complex linear superfields to one with chiral scalar superfields, is obtained. When this condition is satisfied, the potentials and couplings cannot be transformed, in general, into a Kähler form; examples are given. Some aspects of vanishing cosmological constant are considered in this context.     

Vortex Equations in Abelian Gauged σ-Models

We consider nonlinear gauged σ-models with Kähler domain and target. For a special choice of potential these models admit Bogomolny (or self-duality) equations — the so-called vortex equations. Here we describe the space of solutions and energy spectrum of the vortex equations when the gauge group is a torus T ⁿ , the domain is compact, and the target is We also obtain a large family of solutions when the target is a compact Kähler toric manifold.