Mirror symmetry,D-brane superpotentials and Ooguri-Vafa invariants of Calabi-Yau manifolds

The D-brane superpotential is very important in the low energy effective theory.As the generating function of all disk instantons from the worldsheet point of view,it plays a crucial role in deriving some important properties of the compact Calabi-Yau manifolds.By using the generalized GKZ hypergeometric system,we will calculate the D-brane superpotentials of two non-Fermat type compact Calabi-Yau hypersurfaces in toric varieties,respectively.Then according to the mirror symmetry,we obtain the A-model superpotentials and the Ooguri-Vafa invariants for the mirror Calabi-Yau manifolds.     

Open Topological Strings and Integrable Hierarchies: Remodeling the A-Model

We set up, purely in A-model terms, a novel formalism for the global solution of the open and closed topological A-model on toric Calabi-Yau threefolds. The starting point is to build on recent progress in the mathematical theory of open Gromov-Witten invariants of orbifolds; we interpret the localization formulae as relating D-brane amplitudes to closed string amplitudes perturbed with twisted masses through an analogue of the “loop insertion operator” of matrix models. We first generalize this form of open/closed string duality to general toric backgrounds in all chambers of the stringy Kähler moduli space; secondly, we display a neat connection of the (gauged) closed string side to tau functions of 1+1 Hamiltonian integrable hierarchies, and exploit it to provide an effective computation of open string amplitudes. In doing so, we also provide a systematic treatment of the change of flat open moduli induced by a phase transition in the closed moduli space. We test our proposal in detail by providing an extensive number of checks. We also use our formalism to give a localization-based derivation of the Hori-Vafa spectral curves as coming from a resummation of A-model disc instantons.     

The Topological Vertex

We construct a cubic field theory which provides all genus amplitudes of the topological A-model for all non-compact toric Calabi-Yau threefolds. The topology of a given Feynman diagram encodes the topology of a fixed Calabi-Yau, with Schwinger parameters playing the role of Kähler classes of the threefold. We interpret this result as an operatorial computation of the amplitudes in the B-model mirror which is the quantum Kodaira-Spencer theory. The only degree of freedom of this theory is an unconventional chiral scalar on a Riemann surface. In this setup we identify the B-branes on the mirror Riemann surface as fermions related to the chiral boson by bosonization.Acknowledgement We would like to thank D.-E.Diaconescu, R. Dijkgraaf, J. Gomis, A. Grassi, A. Iqbal, A. Kapustin, S. Katz, V. Kazakov, I. Kostov, C-C. Liu, H. Ooguri, J. Schwarz, S. Shenker and E. Zaslow for valuable discussions (and the cap!). The research of MA and CV was supported in part by NSF grants PHY-9802709 and DMS-0074329. In addition, CV thanks the hospitality of the theory group at Caltech, where he is a Gordon Moore Distinguished Scholar. M.A. is grateful to the Caltech theory group for hospitality during part of this work. A.K. is supported in part by the DFG grant KL-1070/2-1.     

Opening Mirror Symmetry on the Quintic

Aided by mirror symmetry, we determine the number of holomorphic disks ending on the real Lagrangian in the quintic threefold. We hypothesize that the tension of the domainwall between the two vacua on the brane, which is the generating function for the open Gromov-Witten invariants, satisfies a certain extension of the Picard-Fuchs differential equation governing periods of the mirror quintic. We verify consistency of the monodromies under analytic continuation of the superpotential over the entire moduli space. We further check the conjecture by reproducing the first few instanton numbers by a localization computation directly in the A-model, and verifying Ooguri-Vafa integrality. This is the first exact result on open string mirror symmetry for a compact Calabi-Yau manifold.     

Effective Superpotentials for Compact D5-Brane Calabi-Yau Geometries

For compact Calabi-Yau geometries with D5-branes we study N = 1 effective superpotentials depending on both open- and closed-string fields. We develop methods to derive the open/closed Picard-Fuchs differential equations, which control D5-brane deformations as well as complex structure deformations of the compact Calabi-Yau space. Their solutions encode the flat open/closed coordinates and the effective superpotential. For two explicit examples of compact D5-brane Calabi-Yau hypersurface geometries we apply our techniques and express the calculated superpotentials in terms of flat open/closed coordinates. By evaluating these superpotentials at their critical points we reproduce the domain wall tensions that have recently appeared in the literature. Finally we extract orbifold disk invariants from the superpotentials, which, up to overall numerical normalizations, correspond to orbifold disk Gromov-Witten invariants in the mirror geometry.     

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.     

Comments on worldsheet description of the Omega background

Nekrasov?s partition function is defined on a flat bundle of R⁴ over S¹ called the Omega background. When the fibration is self-dual, the partition function is known to be equal to the topological string partition function, which computes scattering amplitudes of self-dual gravitons and graviphotons in type II superstring compactified on a Calabi-Yau manifold. We propose a generalization of this correspondence when the fibration is not necessarily self-dual.     

Conifold transitions and mirror symmetry for Calabi-Yau complete intersections in Grassmannians

In this paper we show that conifold transitions between Calabi-Yau 3-folds can be used for the construction of mirror manifolds and for the computation of the instanton numbers of rational curves on complete intersection Calabi-Yau 3-folds in Grassmannians. Using a natural degeneration of Grassmannians G(k, n) to some Gorenstein toric Fano varieties P(k, n) with conifolds singularities which was recently described by Sturmfels, we suggest an explicit mirror construction for Calabi-Yau complete intersections X ⊂ G(k, n) of arbitrary dimension. Our mirror construction is consistent with the formula for the Lax operator conjectured by Eguchi, Hori and Xiong for gravitational quantum cohomology of Grassmannians.     

The elliptic genus of Calabi-Yau 3- and 4-folds,product formulae and generalized Kac-Moody algebras

In this paper the elliptic genus for a general Calabi-Yau 4-fold is derived. The recent work of Kawai calculating N = 2 heterotic string one-loop threshold corrections with a Wilson line turned on is extended to a similar computation where K3 is replaced by a general Calabi-Yau 3- or 4-fold. In all cases there seems to be a generalized Kac-Moody algebra involved, whose denominator formula appears in the result.     

Counting BPS States on the Enriques Calabi-Yau

We study topological string amplitudes for the FHSV model using various techniques. This model has a type II realization involving a Calabi-Yau threefold with Enriques fibres, which we call the Enriques Calabi-Yau. By applying heterotic/type IIA duality, we compute the topological amplitudes in the fibre to all genera. It turns out that there are two different ways to do the computation that lead to topological couplings with different BPS content. One of them gives the standard D0-D2 counting amplitudes, and from the other one we obtain information about bound states of D0-D4-D2 branes on the Enriques fibre. We also study the model using mirror symmetry and the holomorphic anomaly equations. We verify in this way the heterotic results for the D0-D2 generating functional for low genera and find closed expressions for the topological amplitudes on the total space in terms of modular forms, and up to genus three. This model turns out to be much simpler than the generic B-model and might be exactly solvable.     

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.     

Möbius and super-Möbius gauge fixing for the closed string amplitudes on the desk

We systematically study the gauge fixing of the Möbius and the super-Möbius transformations for the N-point closed-string amplitudes on the disk. Using the Faddeev-Popov method, we obtain the explicit formulas for the Koba-Nielsen factors for these amplitudes.     

Cosmological solutions with Calabi-Yau compactification

Assuming a Calabi-Yau compactification, cosmological solutions are presented in ten-dimensional, N=1 Yang-Mills supergravity theory with the curvature squared term (R²_μνϱσ −4R_μν² + R²). In a vacuum state, Kasner-type soluti ons exist as well as (four-dimensional Minkoswki space-time)×(a Calabi-Yau space). In the later stage of the universe the (four-dimensional Friedmann universe)×(a constant Calabi-Yau space) is realized asymptotically like an attractor. This solution is asymptotically stable against small perturbations.     

Torus Knots and the Topological Vertex

We propose a class of toric Lagrangian A-branes on the resolved conifold that is suitable to describe torus knots on S ³. The key role is played by the \({SL(2, \mathbb{Z})}\) transformation, which generates a general torus knot from the unknot. Applying the topological vertex to the proposed A-branes, we rederive the colored HOMFLY polynomials for torus knots, in agreement with the Rosso and Jones formula. We show that our A-model construction is mirror symmetric to the B-model analysis of Brini, Eynard and Mariño. Compared to the recent proposal by Aganagic and Vafa for knots on S ³, we demonstrate that the disk amplitude of the A-brane associated with any knot is sufficient to reconstruct the entire B-model spectral curve. Finally, the construction of toric Lagrangian A-branes is generalized to other local toric Calabi–Yau geometries, which paves the road to study knots in other three-manifolds such as lens spaces.     

Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric varieties

We formulate general conjectures about the relationship between the A-model connection on the cohomology of ad-dimensional Calabi-Yau complete intersectionV ofr hypersurfacesV ₁ ,...,V _r in a toric varietyP and the system of differential operators annihilating the special generalized hypergeometric series ⁰ constructed from the fan . Using this generalized hypergeometric series, we propose conjectural mirrorsV ofV and the canonicalq-coordinates on the moduli spaces of Calabi-Yau manifolds.In the second part of the paper we consider some examples of Calabi-Yau 3-folds having Picard number >1 in products of projective spaces. For conjectural mirrors, using the recurrent relation among coefficients of the restriction of the hypergeometric function ⁰ on a special line in the moduli space, we determine the Picard-Fuchs equation satisfied by periods of this special one-parameter subfamily. This allows to obtain some sequences of integers which can be conjecturally interpreted in terms of Gromov-Witten invariants. Using standard techniques from enumerative geometry, first terms of these sequence of integers are checked to coincide with numbers of rational curves on Calabi-Yau 3-folds.     

Computing parallel/coincident phase D-brane superpotentials and Type-Ⅱ/F-theory duality

In this paper we study the parallel phase and the coincident phase of D-brane systems with the compactification of one closed modulus. D-brane systems with two phases are described by different 4-folds in terms of Type-Ⅱ/F-theory duality, and the phase transitions are related by the blow-up from a 4-fold with singularities to a 4-fold without. In terms of gauge theory, the phase transition corresponds to the enhancement of gauge group U(1)×U(1)→U(2) connecting the Coulomb branch and the Higgs branch. For the sextic and octic with two D-branes,using mirror symmetry and Type-Ⅱ/F theory duality, A-model superpotentials are obtained from the B-model side for the two phases, and the U(1) Ooguri-Vafa invariants for the parallel phase and U(2) Ooguri-Vafa invariants for the coincident phase are extracted from the A-model superpotential. The difference between the invariants of the two phases is evidence of the phase transition between the Coulomb branch and the Higgs branch.     

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.     

All Loop Topological String Amplitudes from Chern-Simons Theory

We demonstrate the equivalence of all loop closed topological string amplitudes on toric local Calabi-Yau threefolds with computations of certain knot invariants for Chern-Simons theory. We use this equivalence to compute the topological string amplitudes in certain cases to very high degree and to all genera. In particular we explicitly compute the topological string amplitudes for ² up to degree 12 and ¹× ¹ up to total degree 10 to all genera. This also leads to certain novel large N dualities in the context of ordinary superstrings, involving duals of type II superstrings on local Calabi-Yau three-folds without any fluxes.This research is supported in part by NSF grants PHY-9802709 and DMS-0074329     

Emergent Spacetime from Modular Motives

The program of constructing spacetime geometry from string theoretic modular forms is extended to Calabi-Yau varieties of dimensions three and four, as well as higher rank motives. Modular forms on the worldsheet can be constructed from the geometry of spacetime by computing the L?Cfunctions associated to omega motives of Calabi-Yau varieties, generated by their holomorphic n?forms via Galois representations. The modular forms that emerge in this way are related to characters of the underlying rational conformal field theory. The converse problem of constructing space from string theory proceeds in the class of diagonal theories by determining the motives associated to modular forms in the category of pure motives with complex multiplication. The emerging picture suggests that the L?Cfunction can be viewed as defining a map between the geometric category of motives and the category of conformal field theories on the worldsheet.