Non-compact Calabi-Yau Manifolds Research Articles

On the L2-Hodge theory of Landau-Ginzburg models

Let X be a non-compact Calabi-Yau manifold and f be a holomorphic function on X with compact critical locus. We introduce the notion of f-twisted Sobolev spaces for the pair (X,f) and prove the corresponding Hodge-to-de Rham degeneration property via L2-Hodge theoretical methods when f satisfies an asymptotic condition of strongly ellipticity. This leads to a Frobenius manifold via the Barannikov-Kontsevich construction, unifying the Landau-Ginzburg and Calabi-Yau geometry. Our construction can be viewed as a generalization of K. Saito's higher residue and primitive form theory for isolated singularities. As an application, we construct Frobenius manifolds for orbifold Landau-Ginzburg B-models which admit crepant resolutions.

Long way to Ricci flatness

We study two-dimensional weighted mathcal{N} = (2) supersymmetric ℂℙ models with the goal of exploring their infrared (IR) limit. \U0001d54eℂℙ(N, tilde{N} ) are simplified versions of world-sheet theories on non-Abelian strings in four-dimensional mathcal{N} = 2 QCD. In the gauged linear sigma model (GLSM) formulation, \U0001d54eℂℙ(N, tilde{N} ) has N charges +1 and tilde{N} charges −1 fields. As well-known, at tilde{N} = N this GLSM is conformal. Its target space is believed to be a non-compact Calabi-Yau manifold. We mostly focus on the N = 2 case, then the Calabi-Yau space is a conifold.On the other hand, in the non-linear sigma model (NLSM) formulation the model has ultra-violet logarithms and does not look conformal. Moreover, its metric is not Ricci-flat. We address this puzzle by studying the renormalization group (RG) flow of the model. We show that the metric of NLSM becomes Ricci-flat in the IR. Moreover, it tends to the known metric of the resolved conifold. We also study a close relative of the \U0001d54eℂℙ model — the so called zn model — which in actuality represents the world sheet theory on a non-Abelian semilocal string and show that this zn model has similar RG properties.

On B-type Open–Closed Landau–Ginzburg Theories Defined on Calabi–Yau Stein Manifolds

We consider the bulk algebra and topological D-brane category arising from the differential model of the open-closed B-type topological Landau-Ginzburg theory defined by a pair $(X,W)$, where $X$ is a non-compact Calabi-Yau manifold and $W$ has compact critical set. When $X$ is a Stein manifold (but not restricted to be a domain of holomorphy), we extract equivalent descriptions of the bulk algebra and of the category of topological D-branes which are constructed using only the analytic space associated to $X$. In particular, we show that the D-brane category is described by projective matrix factorizations defined over the ring of holomorphic functions of $X$. We also discuss simplifications of the analytic models which arise when $X$ is holomorphically parallelizable and illustrate these analytic models in a few classes of examples.

Differential Models for B-Type Open–Closed Topological Landau–Ginzburg Theories

We propose a family of differential models for B-type open-closed topological Landau-Ginzburg theories defined by a pair $(X,W)$, where $X$ is any non-compact Calabi-Yau manifold and $W$ is any holomorphic complex-valued function defined on $X$ whose critical set is compact. The models are constructed at cochain level using smooth data, including the twisted Dolbeault algebra of polyvector valued forms and a twisted Dolbeault category of holomorphic factorizations of $W$. We give explicit proposals for cochain level versions of the bulk and boundary traces and for the bulk-boundary and boundary-bulk maps of the Landau-Ginzburg theory. We prove that most of the axioms of an open-closed topological field theory are satisfied on cohomology and conjecture that the remaining axioms are also satisfied.

ADE string chains and mirror symmetry

6d superconformal field theories (SCFTs) are the SCFTs in the highest possible dimension. They can be geometrically engineered in F-theory by compactifying on non-compact elliptic Calabi-Yau manifolds. In this paper we focus on the class of SCFTs whose base geometry is determined by −2 curves intersecting according to ADE Dynkin diagrams and derive the corresponding mirror Calabi-Yau manifold. The mirror geometry is uniquely determined in terms of the mirror curve which has also an interpretation in terms of the Seiberg-Witten curve of the four-dimensional theory arising from torus compactification. Adding the affine node of the ADE quiver to the base geometry, we connect to recent results on SYZ mirror symmetry for the A case and provide a physical interpretation in terms of little string theory. Our results, however, go beyond this case as our construction naturally covers the D and E cases as well.

Calabi–Yau Monopoles for the Stenzel Metric

We construct the first nontrivial examples of Calabi-Yau monopoles. Our main interest on these, comes from Donaldson and Segal's suggestion \cite{Donaldson2009} that it may be possible to define an invariant of certain noncompact Calabi-Yau manifolds from these gauge theoretical equations. We focus on the Stenzel metric on the cotangent bundle of the $3$-sphere $T^* \mathbb{S}^3$ and study monopoles under a symmetry assumption. Our main result constructs the moduli of these symmetric monopoles and shows that these are parametrized by a positive real number known as the mass of the monopole. In other words, for each fixed mass we show that there is a unique monopole which is invariant in a precise sense. Moreover, we also study the large mass limit under which we give precise results on the bubbling behavior of our monopoles. Towards the end an irreducible $SU(2)$ Hermitian-Yang-Mills connection on the Stenzel metric is constructed explicitly.

Refined stable pair invariants for E-, M- and [p, q]-strings

We use mirror symmetry, the refined holomorphic anomaly equation and modularity properties of elliptic singularities to calculate the refined BPS invariants of stable pairs on non-compact Calabi-Yau manifolds, based on del Pezzo surfaces and elliptic surfaces, in particular the half K3. The BPS numbers contribute naturally to the five-dimensional N=1 supersymmetric index of M-theory, but they can be also interpreted in terms of the superconformal index in six dimensions and upon dimensional reduction the generating functions count N=2 Seiberg-Witten gauge theory instantons in four dimensions. Using the M/F-theory uplift the additional information encoded in the spin content can be used in an essential way to obtain information about BPS states in physical systems associated to small instantons, tensionless strings, gauge symmetry enhancement in F-theory by [p,q]-strings as well as M-strings.

Refined black hole ensembles and topological strings

We formulate a refined version of the Ooguri-Strominger-Vafa (OSV) conjecture. The OSV conjecture that Z_{BH} = |Z_{top}|^2 relates the BPS black hole partition function to the topological string partition function Z_{top}. In the refined conjecture, Z_{BH} is the partition function of BPS black holes counted with spin, or more precisely the protected spin character. Z_{top} becomes the partition function of the refined topological string, which is itself an index. Both the original and the refined conjecture are examples of large N duality in the 't Hooft sense. The refined conjecture applies to non-compact Calabi-Yau manifolds only, so the black holes are really BPS particles with large entropy, of order N^2. The refined OSV conjecture states that the refined BPS partition function has a large N dual which is captured by the refined topological string. We provide evidence that the conjecture holds by studying local Calabi-Yau threefolds consisting of line bundles over a genus g Riemann surface. We show that the refined topological string partition function on these geometries is computed by a two-dimensional TQFT. We also study the refined black hole partition function arising from N D4 branes on the Calabi-Yau, and argue that it reduces to a (q,t)-deformed version of two-dimensional SU(N) Yang-Mills. Finally, we show that in the large N limit this theory factorizes to the square of the refined topological string in accordance with the refined OSV conjecture.

Calabi-Yau manifolds from pairs of non-compact Calabi-Yau manifolds

Most of Calabi-Yau manifolds that have been considered by physicists are complete intersection Calabi-Yau manifolds of toric varieties or some quotients of product types. Purpose of this paper is to introduce a different and rather new kind of construction method of Calabi-Yau manifolds by pasting two non-compact Calabi-Yau manifolds. We will also in some details explain a curious and mysterious similarity with construction of some $G_2$-manifolds (also called Joyce manifolds), which are base spaces for M-theory.

Non-compact mirror bundles and (0,2) Liouville theories

We study (0,2) deformations of N=2 Liouville field theory and its mirror duality. A gauged linear sigma model construction of the ultraviolet theory connects (0,2) deformations of Liouville field theory and (0,2) deformations of N=2 SL(2,R)/U(1) coset model as a mirror duality. Our duality proposal from the gauged linear sigma model completely agrees with the exact CFT analysis. In the context of heterotic string compactifications, the deformation corresponds to the introduction of a non-trivial gauge bundle. This non-compact Landau-Ginzburg construction yields a novel way to study the gauge bundle moduli for non-compact Calabi-Yau manifolds.

Integrability of the holomorphic anomaly equations

We show that modularity and the gap condition make the holomorphic anomaly equation completely integrable for non-compact Calabi-Yau manifolds. This leads to a very efficient formalism to solve the topological string on these geometries in terms of almost holomorphic modular forms. The formalism provides in particular holomorphic expansions everywhere in moduli space including large radius points, the conifold loci, Seiberg-Witten points and the orbifold points. It can be also viewed as a very efficient method to solve higher genus closed string amplitudes in the $\frac{1}{N^2}$ expansion of matrix models with more then one cut.

Supersymmetric gauge theories, intersecting branes and free fermions

We show that various holomorphic quantities in supersymmetric gauge theories can be conveniently computed by configurations of D4-branes and D6-branes. These D-branes intersect along a Riemann surface that is described by a holomorphic curve in a complex surface. The resulting I-brane carries two-dimensional chiral fermions on its world-volume. This system can be mapped directly to the topological string on a large class of non-compact Calabi-Yau manifolds. Inclusion of the string coupling constant corresponds to turning on a constant B-field on the complex surface, which makes this space non-commutative. Including all string loop corrections the free fermion theory is elegantly formulated in terms of holonomic D-modules that replace the classical holomorphic curve in the quantum case.

Gauge symmetry and supersymmetry breaking from intersecting branes

We study a system of intersecting NS- and D-branes in type IIA string theory in R 9 , 1 . We show that the ( 3 + 1 ) -dimensional non-supersymmetric theory at the intersection has unstable vacua which are long-lived in some regions of the parameter space of brane configurations, and disappear in others. We also comment on the relation of our construction to systems of D- and D ¯ -branes wrapped around cycles of non-compact Calabi–Yau manifolds and to other related systems.

A quantum McKay correspondence for fractional 2p-branes on LG orbifolds

We study fractional 2p-branes and their intersection numbers in non-compact orbifolds as well the continuation of these objects in Kahler moduli space to coherent sheaves in the corresponding smooth non-compact Calabi-Yau manifolds. We show that the restriction of these objects to compact Calabi-Yau hypersurfaces gives the new fractional branes in LG orbifolds constructed by Ashok et. al. in hep-th/0401135. We thus demonstrate the equivalence of the B-type branes corresponding to linear boundary conditions in LG orbifolds, originally constructed in hep-th/9907131, to a subset of those constructed in LG orbifolds using boundary fermions and matrix factorization of the world-sheet superpotential. The relationship between the coherent sheaves corresponding to the fractional two-branes leads to a generalization of the McKay correspondence that we call the quantum McKay correspondence due to a close parallel with the construction of branes on non-supersymmetric orbifolds. We also provide evidence that the boundary states associated to these branes in a conformal field theory description corresponds to a sub-class of the boundary states associated to the permutation branes in the Gepner model associated with the LG orbifold.

On the boundary coupling of topological Landau-Ginzburg models

I propose a general form for the boundary coupling of B-type topological Landau-Ginzburg models. In particular, I show that the relevant background in the open string sector is a (generally non-Abelian) superconnection of type (0,1) living in a complex superbundle defined on the target space, which I allow to be a non-compact Calabi-Yau manifold. This extends and clarifies previous proposals. Generalizing an argument due to Witten, I show that BRST invariance of the partition function on the worldsheet amounts to the condition that the (0,<= 2) part of the superconnection's curvature equals a constant endomorphism plus the Landau-Ginzburg potential times the identity section of the underlying superbundle. This provides the target space equations of motion for the open topological model.

Gauged linear sigma models for noncompact Calabi–Yau varieties

We study gauged linear sigma models for noncompact Calabi–Yau manifolds described as a line bundle on a hypersurface in a projective space. This gauge theory has a unique phase if the Fayet–Iliopoulos parameter is positive, while there exist two distinct phases if the parameter is negative. We find four massless effective theories in the infrared limit, which are related to each other under the Calabi–Yau/Landau–Ginzburg correspondence and the topology change. In the T-dual theory, on the other hand, we obtain two types of exact massless effective theories: one is the sigma model on a newly obtained Calabi–Yau geometry as a mirror dual, while the other is given by a Landau–Ginzburg theory with a negative power term, indicating N = 2 superconformal field theory on SL ( 2 , R ) / U ( 1 ) . We argue that the effective theories in the original gauged linear sigma model are exactly realized as N = 2 Liouville theories coupled to well-defined Landau–Ginzburg minimal models.

Matrix models, 4D black holes and topological strings on non-compact Calabi-Yau manifolds

We study the relation between c = 1 matrix models at self-dual radii and topological strings on non-compact Calabi-Yau manifolds. Particularly the special case of the deformed matrix model is investigated in detail. Using recent results on the equivalence of the partition function of topological strings and that of four dimensional BPS black holes, we are able to calculate the entropy of the black holes, using matrix models. In particular, we show how to deal with the divergences that arise as a result of the non-compactness of the Calabi-Yau. The main result is that the entropy of the black hole at zero temperature coincides with the canonical free energy of the matrix model, up to a proportionality constant given by the self-dual temperature of the matrix model.

Seiberg Duality is an Exceptional Mutation

The low energy gauge theory living on D-branes probing a del Pezzo singularity of a non-compact Calabi-Yau manifold is not unique. In fact there is a large equivalence class of such gauge theories related by Seiberg duality. As a step toward characterizing this class, we show that Seiberg duality can be defined consistently as an admissible mutation of a strongly exceptional collection of coherent sheaves.

Higher-Order Corrections to Non-Compact Calabi-Yau Manifolds in String Theory

At the leading order, the low-energy effective field equations in string theory admit solutions of the form of products of Minkowski spacetime and a Ricci-flat Calabi-Yau space. The equations of motion receive corrections at higher orders in \alpha', which imply that the Ricci-flat Calabi-Yau space is modified. In an appropriate choice of scheme, the Calabi-Yau space remains Kahler, but is no longer Ricci-flat. We discuss the nature of these corrections at order {\alpha'}^3, and consider the deformations of all the known cohomogeneity one non-compact Kahler metrics in six and eight dimensions. We do this by deriving the first-order equations associated with the modified Killing-spinor conditions, and we thereby obtain the modified supersymmetric solutions. We also give a detailed discussion of the boundary terms for the Euler complex in six and eight dimensions, and apply the results to all the cohomogeneity one examples.

SL(2;R)/U(1) Supercoset and Elliptic Genera of Non-compact Calabi-Yau Manifolds

We first discuss the relationship between the SL(2;R)/U(1) supercoset and N=2 Liouville theory and make a precise correspondence between their representations. We shall show that the discrete unitary representations of SL(2;R)/U(1) theory correspond exactly to those massless representations of N=2 Liouville theory which are closed under modular transformations and studied in our previous work hep-th/0311141. It is known that toroidal partition functions of SL(2;R)/U(1) theory (2D Black Hole) contain two parts, continuous and discrete representations. The contribution of continuous representations is proportional to the space-time volume and is divergent in the infinite-volume limit while the part of discrete representations is volume-independent. In order to see clearly the contribution of discrete representations we consider elliptic genus which projects out the contributions of continuous representations: making use of the SL(2;R)/U(1), we compute elliptic genera for various non-compact space-times such as the conifold, ALE spaces, Calabi-Yau 3-folds with A_n singularities etc. We find that these elliptic genera in general have a complex modular property and are not Jacobi forms as opposed to the cases of compact Calabi-Yau manifolds.