Heterotic non-Kähler geometries via polystable bundles on Calabi-Yau threefolds

In arXiv:1008.1018 it is shown that a given stable vector bundle V on a Calabi-Yau threefold X which satisfies c₂(X)=c₂(V) can be deformed to a solution of the Strominger system and the equations of motion of heterotic string theory. In this note we extend this result to the polystable case and construct explicit examples of polystable bundles on elliptically fibered Calabi-Yau threefolds where it applies. The polystable bundle is given by a spectral cover bundle, for the visible sector, and a suitably chosen bundle, for the hidden sector. This provides a new class of heterotic flux compactifications via non-Kähler deformation of Calabi-Yau geometries with polystable bundles. As an application, we obtain examples of non-Kähler deformations of some three generation GUT models.     

On the existence of stable bundles with prescribed Chern classes on Calabi–Yau threefolds

We prove a case of the conjecture of Douglas, Reinbacher and Yau about the existence of stable vector bundles with prescribed Chern classes on a Calabi–Yau threefold. For this purpose we prove the existence of certain stable vector bundle extensions over elliptically fibered Calabi–Yau threefolds.     

Spectral bundles and the DRY-Conjecture

Supersymmetric heterotic string models, built from a Calabi-Yau threefold X endowed with a stable vector bundle V, usually start from a phenomenologically motivated choice of a bundle V_v in the visible sector, the spectral cover construction on an elliptically fibered X being a prominent example. The ensuing anomaly mismatch between c₂(V_v) and c₂(X), or rather the corresponding differential forms, is often ‘solved’, on the cohomological level, by including a fivebrane. This leads to the question whether the difference can be alternatively realized by a further stable bundle. The ‘DRY’-conjecture of Douglas, Reinbacher and Yau in math.AG/0604597 gives a sufficient condition on cohomology classes on X to be realized as the Chern classes of a stable sheaf. In 1010.1644 [hep-th], we showed that infinitely many classes on X exist for which the conjecture is true. In this note, we give the sufficient condition for the mentioned fivebrane classes to be realized by a further stable bundle in the hidden sector. Using a result obtained in 1011.6246 [hep-th], we show that corresponding bundles exist, thereby confirming this version of the DRY-Conjecture.     

A Vertex Formalism for Local Ruled Surfaces

We develop a vertex formalism for topological string amplitudes on ruled surfaces with an arbitrary number of reducible fibers embedded in a Calabi-Yau threefold. Our construction is based on large N duality and localization with respect to a degenerate torus action. We also discuss potential generalizations of our formalism to a broader class of Calabi-Yau threefolds using the same underlying principles.     

Energy bands: Chern numbers and symmetry

Energy bands formed by rotation–vibrational states of molecules in the presence of symmetry and their qualitative modifications under variation of some control parameters are studied within the semi-quantum model. Rotational variables are treated as classical whereas a finite set of vibrational states is considered as quantum. In the two-state approximation the system is described in terms of a fiber bundle with the base space being a two-dimensional sphere, the classical phase space for rotational variables. Generically this rank 2 complex vector bundle can be decomposed into two complex line bundles characterized by a topological invariant, the first Chern class. A general method of explicit calculation of Chern classes and of their possible modifications under variation of control parameters in the presence of symmetry is suggested. The construction of iso-Chern diagrams which split the space of control parameters into connected domains with fixed Chern numbers is suggested. A detailed analysis of the rovibrational model Hamiltonian for a D₃ invariant molecule possessing two vibrational states transforming according to the two-dimensional irreducible representation is done to illustrate non-trivial restrictions imposed by symmetry on possible values of Chern classes.     

Geometric singularities and spectra of Landau-Ginzburg models

Some mathematical and physical aspects of superconformal string compactification in weighted projective space are discussed. In particular, we recast the path integral argument establishing the connection between Landau-Ginzburg conformal theories and Calabi-Yau string compactification in a geometric framework. We then prove that the naive expression for the vanishing of the first Chern class for a complete intersection (adopted from the smooth case) is sufficient to ensure that the resulting variety, which is generically singular, can be resolved to a smooth Calabi-Yau space. This justifies much analysis which has recently been expended on the study of Landau-Ginzburg models. Furthermore, we derive some simple formulae for the determination of the Witten index in these theories which are complimentary to those derived using semiclassical reasoning by Vafa. Finally, we also comment on the possible geometrical significance ofunorbifolded Landau-Ginzburg theories.     

Low-energy analysis of M- and F-theories on Calabi-Yau threefolds

We elucidate the interplay between gauge and supersymmetry anomalies in six-dimensional N = 1 supergravity with generalized couplings between tensor and vector multiplets. We derive the structure of the five-dimensional supergravity resulting from the S₁ reduction of these models and give the constraints on Chem-Simons couplings that follow from duality to M-theory compactified on a Calabi-Yau threefold. The duality is supported only on a restricted class of Calabi-Yau threefolds and requires a special type of intersection form. We derive five-dimensional central-charge formulas and briefly discuss the associated phase transitions. Finally, we exhibit connections with F-theory compactifications on Calabi-Yau manifolds that admit elliptic fibrations. This analysis suggests that F-theory unifies type-IIb three-branes and M-theory five-branes.     

Curvature on determinant bundles and first Chern forms

The Quillen–Bismut–Freed construction associates a determinant line bundle with connection to an infinite dimensional super vector bundle with a family of Dirac-type operators. We define the regularized first Chern form of the infinite dimensional bundle, and relate it to the curvature of the Bismut–Freed connection on the determinant bundle. In finite dimensions, these forms agree (up to sign), but in infinite dimensions there is a correction term, which we express in terms of Wodzicki residues.

We illustrate these results with a string theory computation. There is a natural super vector bundle over the manifold of smooth almost complex structures on a Riemannian surface. The Bismut–Freed superconnection is identified with classical Teichmüller theory connections, and its curvature and regularized first Chern form are computed.     

Power-law corrections to the Kubo formula vanish in quantum Hall systems

In first order perturbation theory conductivity is given by the Kubo formula, which in a Quantum Hall System equals the first Chern class of a vector bundle. We apply the adiabatic theorem to show that these topological constraints quantize the averaged conductivity to all orders of perturbation theory. Hence the Kubo formula is valid to all orders.     

Non-self-dual Yang-Mills connections with quadrupole symmetry

We prove the existence of non-self-dual Yang-Mills connections onSU(2) bundles over the four-sphere, specifically on all bundles with second Chern number not equal±1. We study connections equivariant under anSU(2) symmetry group to reduce the effective dimensionality from four to one, and then use variational techniques. The existence of non-self-dualSU(2) YM connections on the trivial bundle (second Chern number equals zero) has already been established by Sibner, Sibner, and Uhlenbeck via different methods.Research partially supported by NSF Grant DMS-8806731Most of this research was done while the author was a Bantrell Fellow at the California Institute of Technology, and was partially supported by NSF Grant DMS-8801918     

The tangent bundle of a Calabi-Yau manifold-deformations and restriction to rational curves

The tangent bundle _X of a Calabi-Yau threefoldX is the only known example of a stable bundle with non-trivial restriction to any rational curve onX. By deforming the direct sum of _X and the trivial line bundle one can try to obtain new examples. We use algebro-geometric techniques to derive results in this direction. The relation to the finiteness of rational curves on Calabi-Yau threefolds is discussed.     

Chern-Weil Construction for Twisted K-Theory

We give a finite-dimensional and geometric construction of a Chern character for twisted K-theory, introducing a notion of connection on a twisted vectorial bundle which can be considered as a finite-dimensional approximation of a twisted family of Fredholm operators. Our construction is applicable to the case of any classes giving the twisting, and agrees with the Chern character of bundle gerbe modules in the case of torsion classes.     

Uniqueness and Examples of Compact Toric Sasaki-Einstein Metrics

In [11] it was proved that, given a compact toric Sasaki manifold with positive basic first Chern class and trivial first Chern class of the contact bundle, one can find a deformed Sasaki structure on which a Sasaki-Einstein metric exists. In the present paper we first prove the uniqueness of such Einstein metrics on compact toric Sasaki manifolds modulo the action of the identity component of the automorphism group for the transverse holomorphic structure, and secondly remark that the result of [11] implies the existence of compatible Einstein metrics on all compact Sasaki manifolds obtained from the toric diagrams with any height, or equivalently on all compact toric Sasaki manifolds whose cones have flat canonical bundle. We further show that there exists an infinite family of inequivalent toric Sasaki-Einstein metrics on for each positive integer k.     

Warps,grids and curvature in triple vector bundles

A triple vector bundle is a cube of vector bundle structures which commute in the (strict) categorical sense. A grid in a triple vector bundle is a collection of sections of each bundle structure with certain linearity properties. A grid provides two routes around each face of the triple vector bundle, and six routes from the base manifold to the total manifold; the warps measure the lack of commutativity of these routes. In this paper we first prove that the sum of the warps in a triple vector bundle is zero. The proof we give is intrinsic and, we believe, clearer than the proof using decompositions given earlier by one of us. We apply this result to the triple tangent bundle \(T^3M\) of a manifold and deduce (as earlier) the Jacobi identity. We further apply the result to the triple vector bundle \(T^2A\) for a vector bundle A using a connection in A to define a grid in \(T^2A\). In this case the curvature emerges from the warp theorem.

     

Mirror manifolds in higher dimension

We describe mirror manifolds in dimensions different from the familiar case of complex threefolds. We isolate certain simplifying features present only in dimension three, and supply alternative methods that do not rely on these special characteristics and hence can be generalized to other dimensions. Although the moduli spaces for Calabi-Yaud-folds are not special Kähler manifolds whend>3, they still have a restricted geometry, and we indicate the new geometrical structures which arise. We formulate and apply procedures which allow for the construction of mirror maps and the calculation of order-by-order instanton corrections to Yukawa couplings. Mathematically, these corrections are expected to correspond to calculating Chern classes of various parameter spaces (Hilbert schemes) for rational curves on Calabi-Yau manifolds. Our mirror-aided calculations agree with those Chern class calculations in the limited number of cases for which the latter can be carried out with current mathematical tools. Finally, we make explicit some striking relations between instanton corrections for various Yukawa couplings, derived from the associativity of the operator product algebra.     

BPS relations from spectral problems and blowup equations

Recently, an exact duality between topological string and the spectral theory of operators constructed from mirror curves to toric Calabi–Yau threefold has been proposed. At the same time, an exact quantization condition for the cluster integrable systems associated with these geometries has been conjectured. The consistency between the two approaches leads to an infinite set of constraints for the refined BPS invariants of the toric Calabi–Yau threefold. We prove these constraints for the \(Y^{{N},m}\) geometries using the K-theoretic blowup equations for SU(N) SYM with generic Chern–Simons invariant m.

     

Constraining the Kähler Moduli in the Heterotic Standard Model

Phenomenological implications of the volume of the Calabi-Yau threefolds on the hidden and observable M-theory boundaries, together with slope stability of their corresponding vector bundles, constrain the set of Kähler moduli which give rise to realistic compactifications of the strongly coupled heterotic string. When vector bundles are constructed using extensions, we provide simple rules to determine lower and upper bounds to the region of the Kähler moduli space where such compactifications can exist. We show how small these regions can be, working out in full detail the case of the recently proposed Heterotic Standard Model. More explicitly, we exhibit Kähler classes in these regions for which the visible vector bundle is stable. On the other hand, there is no polarization for which the hidden bundle is stable.     

Topological Properties of the Born–Oppenheimer Approximation and Implications for the Exact Spectrum

The Born–Oppenheimer approximation can generally be applied when a quantum system is coupled with another comparatively slower system which is treated classically: for a fixed classical state, one considers a stationary quantum vector of the quantum system. Geometrically, this gives a vector bundle over the classical phase space of the slow motion. The topology of this bundle is characterized by integral Chern classes. In the case where the whole system is isolated with a discrete energy spectrum, we show that these integers have a direct manifestation in the qualitative structure of this spectrum: the spectrum is formed by groups of levels and these integers determine the precise number of levels in each group.