Local Calabi-Yau Manifolds Research Articles

Exact multi-instantons in topological string theory

Topological string theory has multi-instanton sectors which lead to non-perturbative effects in the string coupling constant and control the large order behavior of the perturbative genus expansion. As proposed by Couso, Edelstein, Schiappa and Vonk, these sectors can be described by a trans-series extension of the BCOV holomorphic anomaly equations. In this paper we find exact, closed form solutions for these multi-instanton trans-series in the case of local Calabi-Yau manifolds. The resulting multi-instanton amplitudes turn out to be very similar to the eigenvalue tunneling amplitudes of matrix models. Their form suggests that the flat coordinates of the Calabi-Yau manifold are naturally quantized in units of the string coupling constant, as postulated in large NN dualities. Based on our results we obtain a general picture for the resurgent structure of the topological string in the local case, which we illustrate with explicit calculations for local \mathbb{P}^2ℙ2.

Mathematical Structures of Non-perturbative Topological String Theory: From GW to DT Invariants

We study the Borel summation of the Gromov–Witten potential for the resolved conifold. The Stokes phenomena associated to this Borel summation are shown to encode the Donaldson–Thomas (DT) invariants of the resolved conifold, having a direct relation to the Riemann–Hilbert problem formulated by Bridgeland (Invent Math 216(1), 69–124, 2019). There exist distinguished integration contours for which the Borel summation reproduces previous proposals for the non-perturbative topological string partition functions of the resolved conifold. These partition functions are shown to have another asymptotic expansion at strong topological string coupling. We demonstrate that the Stokes phenomena of the strong-coupling expansion encode the DT invariants of the resolved conifold in a second way. Mathematically, one finds a relation to Riemann–Hilbert problems associated to DT invariants which is different from the one found at weak coupling. The Stokes phenomena of the strong-coupling expansion turn out to be closely related to the wall-crossing phenomena in the spectrum of BPS states on the resolved conifold studied in the context of supergravity by Jafferis and Moore (Wall crossing in local Calabi Yau manifolds, arXiv:0810.4909, 2008).

Topological string amplitudes and Seiberg-Witten prepotentials from the counting of dimers in transverse flux

Important illustration to the principle “partition functions in string theory are τ-functions of integrable equations” is the fact that the (dual) partition functions of 4d mathcal{N} = 2 gauge theories solve Painlevé equations. In this paper we show a road to self-consistent proof of the recently suggested generalization of this correspondence: partition functions of topological string on local Calabi-Yau manifolds solve q-difference equations of non-autonomous dynamics of the “cluster-algebraic”integrable systems.We explain in details the “solutions” side of the proposal. In the simplest non-trivial example we show how 3d box-counting of topological string partition function appears from the counting of dimers on bipartite graph with the discrete gauge field of “flux” q. This is a new form of topological string/spectral theory type correspondence, since the partition function of dimers can be computed as determinant of the linear q-difference Kasteleyn operator. Using WKB method in the “melting” q → 1 limit we get a closed integral formula for Seiberg-Witten prepotential of the corresponding 5d gauge theory. The “equations” side of the correspondence remains the intriguing topic for the further studies.

Quantum K-theory of Calabi-Yau manifolds

The disk partition function of certain 3d N = 2 supersymmetric gauge theories computes a quantum K-theoretic ring for Kähler manifolds X. We study the 3d gauge theory/quantum K-theory correspondence for global and local Calabi-Yau manifolds with several K¨ahler moduli. We propose a multi-cover formula that relates the 3d BPS world- volume degeneracies computed by quantum K-theory to Gopakumar-Vafa invariants.

Stringy dyonic solutions and clifford structures

Using the toroidal compactification of string theory on [Formula: see text]-dimensional tori, [Formula: see text], we investigate dyonic objects in arbitrary dimensions. First, we present a class of dyonic black solutions formed by two different D-branes using a correspondence between toroidal cycles and objects possessing both magnetic and electric charges, belonging to [Formula: see text] dyonic gauge symmetry. This symmetry could be associated with electrically charged magnetic monopole solutions in stringy model buildings of the standard model (SM) extensions. Then, we consider in some detail such black hole classes obtained from even-dimensional toroidal compactifications, and we find that they are linked to [Formula: see text] Clifford algebras using the vee product. It is believed that this analysis could be extended to dyonic objects which can be obtained from local Calabi–Yau manifold compactifications.

Local Calabi–Yau manifolds of type [formula omitted] via SYZ mirror symmetry

We carry out the SYZ program for the local Calabi–Yau manifolds of type A˜ by developing an equivariant SYZ theory for the toric Calabi–Yau manifolds of infinite-type. Mirror geometry is shown to be expressed in terms of the Riemann theta functions and generating functions of open Gromov–Witten invariants, whose modular properties are found and studied in this article. Our work also provides a mathematical justification for a mirror symmetry assertion of the physicists Hollowood–Iqbal–Vafa (Hollowood et al., 2008).

Homological Mirror Symmetry for Local Calabi-Yau Manifolds via SYZ

This is a write-up of the author's talk in the conference Algebraic Geometry in East Asia 2016 held at the University of Tokyo in January 2016. We give a survey on the series of papers [16, 23-25] where the author and his collaborators Daniel Pomerleano and Kazushi Ueda show how Strominger-Yau-Zaslow (SYZ) transforms can be applied to understand the geometry of Kontsevich's homological mirror symmetry (HMS) conjecture for certain local Calabi-Yau manifolds.

A conjectural formula for genus one Gromov-Witten invariants of some local Calabi-Yau n-folds

We conjecture a formula for the generating function of genus one Gromov-Witten invariants of the local Calabi-Yau manifolds which are the total spaces of splitting bundles over projective spaces. We prove this conjecture in several special cases, and assuming the validity of our conjecture we check the integrality of genus one Bogomol’nyi-Prasad-Sommerfield (BPS) numbers of local Calabi-Yau 5-folds defined by Klemm and Pandharipande.

The Topological Open String Wavefunction

We show that, in local Calabi–Yau manifolds, the topological open string partition function transforms as a wavefunction under modular transformations. Our derivation is based on the topological recursion for matrix models, and it generalizes in a natural way the known result for the closed topological string sector. As an application, we derive results for vacuum expectation values of 1/2 BPS Wilson loops in ABJM theory at all genera in a strong coupling expansion, for various representations.

Toric geometry and string theory descriptions of qudit systems

In this paper, we propose a new way to approach qudit systems using toric geometry and related topics including the local mirror symmetry used in the string theory compactification. We refer to such systems as ( n , d ) quantum systems where n and d denote the number of the qudits and the basis states respectively. Concretely, we first relate the ( n , d ) quantum systems to the holomorphic sections of line bundles on n dimensional projective spaces CP n with degree n ( d − 1 ) . These sections are in one-to-one correspondence with d n integral points on a n -dimensional simplex. Then, we explore the local mirror map in the toric geometry language to establish a linkage between the ( n , d ) quantum systems and type II D-branes placed at singularities of local Calabi–Yau manifolds. ( 1 , d ) and ( 2 , d ) are analyzed in some details and are found to be related to the mirror of the ALE space with the A d − 1 singularity and a generalized conifold respectively.

Non-perturbative effects and the refined topological string

The partition function of ABJM theory on the three-sphere has non-perturbative corrections due to membrane instantons in the M-theory dual. We show that the full series of membrane instanton corrections is completely determined by the refined topological string on the Calabi-Yau manifold known as local P1xP1, in the Nekrasov-Shatashvili limit. Our result can be interpreted as a first-principles derivation of the full series of non-perturbative effects for the closed topological string on this Calabi-Yau background. Based on this, we make a proposal for the non-perturbative free energy of topological strings on general, local Calabi-Yau manifolds.

Open Gromov-Witten invariants and SYZ under local conifold transitions

For a local Calabi-Yau manifold which is a smoothing of toric Gorenstein singularity, this paper computes the open Gromov-Witten invariants of a generic fiber of the special Lagrangian fibration constructed by Gross and thereby constructs its SYZ mirror. Moreover it derives a relation for the generating function of open Gromov-Witten invariants of the local Calabi-Yau manifold and that of its conifold transition.

Superpotentials, quantum parameter space and phase transitions in $ \mathcal{N} $ = 1 supersymmetric gauge theories

We study the superpotentials, quantum parameter space and phase transitions that arise in the study of large N dualities between $\mathcal{N}=1$ SUSY U(N) gauge theories and string models on local Calabi-Yau manifolds. The main tool of our analysis is a notion of spectral curve characterized by a set of complex partial 't Hooft parameters and cuts given by projections on the spectral curve of minimal supersymmetric cycles of the underlying Calabi-Yau manifold. In particular we show how prepotentials and superpotentials can be associated to spectral curves without relying on any holomorphic matrix model. As an application, we use a combination of analytical and numerical methods to study the cubic model, determine the analytic condition satisfied by critical one-cut spectral curves, and characterize the transition curves between the one-cut and two-cut phases both in the space of spectral curves and in the quantum parameter space.

The Ω Deformed B-model for Rigid N = 2 Theories

We give an interpretation of the Ω deformed B-model that leads naturally to the generalized holomorphic anomaly equations. Direct integration of the latter calculates topological amplitudes of four-dimensional rigid N = 2 theories explicitly in general Ω-backgrounds in terms of modular forms. These amplitudes encode the refined BPS spectrum as well as new gravitational couplings in the effective action of N = 2 supersymmetric theories. The rigid N = 2 field theories we focus on are the conformal rank one N = 2 Seiberg–Witten theories. The failure of holomorphicity is milder in the conformal cases, but fixing the holomorphic ambiguity is only possible upon mass deformation. Our formalism applies irrespectively of whether a Lagrangian formulation exists. In the class of rigid N = 2 theories arising from compactifications on local Calabi–Yau manifolds, we consider the theory of local $${\mathbb{P}^2}$$ . We calculate motivic Donaldson–Thomas invariants for this geometry and make predictions for generalized Gromov–Witten invariants at the orbifold point.

Global properties of topological string amplitudes and orbifold invariants

We derive topological string amplitudes on local Calabi-Yau manifolds in terms of polynomials in finitely many generators of special functions. These objects are defined globally in the moduli space and lead to a description of mirror symmetry at any point in the moduli space. Holomorphic ambiguities of the anomaly equations are fixed by global information obtained from boundary conditions at few special divisors in the moduli space. As an illustration we compute higher genus orbifold Gromov-Witten invariants for \( {{{\mathbb{C}^3}} \mathord{\left/{\vphantom {{{\mathbb{C}^3}} {{\mathbb{Z}_3}}}} \right.} {{\mathbb{Z}_3}}} \) and \( {{{\mathbb{C}^3}} \mathord{\left/{\vphantom {{{\mathbb{C}^3}} {{\mathbb{Z}_4}}}} \right.} {{\mathbb{Z}_4}}} \).

Real mirror symmetry for one-parameter hypersurfaces

We study open string mirror symmetry for one-parameter Calabi-Yau hypersurfaces in weighted projective space. We identify mirror pairs of D-brane configurations, derive the corresponding inhomogeneous Picard-Fuchs equations, and solve for the domainwall tensions as analytic functions over moduli space. Our calculations exemplify several features that had not been seen in previous work on the quintic or local Calabi-Yau manifolds. We comment on the calculation of loop amplitudes.

Topological Strings and (Almost) Modular Forms

The B-model topological string theory on a Calabi-Yau threefold X has a symmetry group Γ, generated by monodromies of the periods of X. This acts on the topological string wave function in a natural way, governed by the quantum mechanics of the phase space H3(X). We show that, depending on the choice of polarization, the genus g topological string amplitude is either a holomorphic quasi-modular form or an almost holomorphic modular form of weight 0 under Γ. Moreover, at each genus, certain combinations of genus g amplitudes are both modular and holomorphic. We illustrate this for the local Calabi-Yau manifolds giving rise to Seiberg-Witten gauge theories in four dimensions and local IP2 and IP1 × IP1. As a byproduct, we also obtain a simple way of relating the topological string amplitudes near different points in the moduli space, which we use to give predictions for Gromov-Witten invariants of the orbifold \({{\mathbb {C}^3} / {\mathbb {Z}_3}}\).

Holomorphic anomaly in gauge theories and matrix models

We use the holomorphic anomaly equation to solve the gravitational corrections to Seiberg-Witten theory and a two-cut matrix model, which is related by the Dijkgraaf-Vafa conjecture to the topological B-model on a local Calabi-Yau manifold. In both cases we construct propagators that give a recursive solution in the genus modulo a holomorphic ambiguity. In the case of Seiberg-Witten theory the gravitational corrections can be expressed in closed form as quasimodular functions of Gamma(2). In the matrix model we fix the holomorphic ambiguity up to genus two. The latter result establishes the Dijkgraaf-Vafa conjecture at that genus and yields a new method for solving the matrix model at fixed genus in closed form in terms of generalized hypergeometric functions.

Geometrically induced metastability and holography

We construct metastable configurations of branes and anti-branes wrapping 2-spheres inside local Calabi–Yau manifolds and study their large N duals. These duals are Calabi–Yau manifolds in which the wrapped 2-spheres have been replaced by 3-spheres with flux through them, and supersymmetry is spontaneously broken. The geometry of the non-supersymmetric vacuum is exactly calculable to all orders of the 't Hooft parameter, and to the leading order in 1 / N . The computation utilizes the same matrix model techniques that were used in the supersymmetric context. This provides a novel mechanism for breaking supersymmetry in the context of flux compactifications.

Quantum entanglement of baby universes

We study quantum entanglements of baby universes which appear in non-perturbative corrections to the OSV formula for the entropy of extremal black holes in type IIA string theory compactified on the local Calabi–Yau manifold defined as a rank 2 vector bundle over an arbitrary genus G Riemann surface. This generalizes the result for G = 1 in hep-th/0504221. Non-perturbative terms can be organized into a sum over contributions from baby universes, and the total wave-function is their coherent superposition in the third quantized Hilbert space. We find that half of the universes preserve one set of supercharges while the other half preserve a different set, making the total universe stable but non-BPS. The parent universe generates baby universes by brane/anti-brane pair creation, and baby universes are correlated by conservation of non-normalizable D-brane charges under the process. There are no other source of entanglement of baby universes, and all possible states are superposed with the equal weight.