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.
We present a novel derivation of the duality between the two-dimensional Euclidean black hole and supersymmetric Liouville theory. We realise these (1+1)-dimensional conformal field theories on the worldvolume of domain walls in a (2+1)-dimensional gauge theory. We show that there exist two complementary descriptions of the domain wall dynamics, resulting in the two mirror conformal field theories. In particular, effects which are usually attributed to worldsheet instantons are captured by the classical scattering of domain walls.
We prove the equivalence of the SL(2,R)/U(1) Kazama-Suzuki model, which is a fermionic generalization of the 2d Black Hole, and N=2 Liouville theory. We show that this duality is an example of mirror symmetry. The essential part of the derivation is to realize the fermionic 2d Black Hole as the low energy limit of a gauged linear sigma-model. Liouville theory is obtained by dualizing the charged scalar fields and taking into account the vortex-instanton effects, as proposed recently in non-dilatonic models. The gauged linear sigma-model we study has many useful generalizations which we briefly…
We study D-branes in the mirror pair N=2 Liouville / supersymmetric SL(2,R)/U(1) coset superconformal field theories. After revisiting the duality between the two models, we build D0, D1 and D2 branes, on the basis of the boundary state construction for the Euclidean AdS(3) conformal field theory. We also construct D0-branes in an orbifold that rotates the angular direction of the cigar. We show how the poles of correlators associated to localized states and bulk interactions naturally decouple in the one-point functions of localized and extended branes. We stress the role played in the analys…
We consider unoriented strings in the supersymmetric SL(2,R)/U(1) coset, which describes the two-dimensional Euclidean black hole, and its mirror dual N=2 Liouville theory. We analyze the orientifolds of these theories from several complementary points of view: the parity symmetries of the worldsheet actions, descent from known AdS_3 parities, and the modular bootstrap method (in some cases we can also check our results against known constraints coming from the conformal bootstrap method). Our analysis extends previous work on orientifolds in Liouville theory, the AdS_3 and SU(2) WZW models an…
We define the notion of mirror of a Calabi-Yau manifold with a stable bundle in the context of type II strings in terms of supersymmetric cycles on the mirror. This allows us to relate the variation of Hodge structure for cohomologies arising from the bundle to the counting of holomorphic maps of Riemann surfaces with boundary on the mirror side. Moreover it opens up the possibility of studying bundles on Calabi-Yau manifolds in terms of supersymmetric cycles on the mirror.
We consider F-theory compactifications on a mirror pair of elliptic Calabi-Yau threefolds. This yields two different six-dimensional theories, each of them being nonperturbatively equivalent to some compactification of heterotic strings on a K3 surface S with certain bundle data E --> S. We find evidence for a transformation of S together with the bundle that takes one heterotic model to the other.
In this technical note we describe a new (to the physics literature) construction of bundles on Calabi-Yaus. We primarily study this construction in the special case of K3 surfaces, for which interesting results can be obtained. For example, we use this construction to give plausibility arguments for a relationship between spaces of solutions of Hitchin's equations and moduli spaces of bundles on K3s. Also, in a recent paper it was proposed by C. Vafa that the mirror to a bundle on a Calabi-Yau n-fold is, in a particular sense, a supersymmetric n-cycle on the mirror Calabi-Yau. We use this new…
We construct a class of exactly solved (0,2) heterotic compactifications, similar to the (2,2) models constructed by Gepner. We identify these as special points in moduli spaces containing geometric limits described by non-linear sigma models on complete intersection Calabi-Yau spaces in toric varieties, equipped with a bundle whose rank is strictly greater than that of the tangent bundle. These moduli spaces do not in general contain a locus exhibiting (2,2) supersymmetry. A quotient procedure at the exactly solved point realizes the mirror isomorphism, as was the case for Gepner models. We f…
We consider tensor products of N=2 minimal models and non-compact conformal field theories with N=2 superconformal symmetry, and their orbifolds. The elliptic genera of these models give rise to a large and interesting class of real Jacobi forms. The tensor product of conformal field theories leads to a natural product on the space of completed mock modular forms. We exhibit families of non-compact mirror pairs of orbifold models with c=9 and show explicitly the equality of elliptic genera, including contributions from the long multiplet sector. The Liouville and cigar deformed elliptic genera…
In this paper we study the moduli space of standard holomorphic structures on a noncommutative complex two torus. It will be shown that the moduli space is naturally identified with the moduli space of stable bundles on an elliptic curve. We also propose that the mirror reflection of the noncommutative complex torus is the mirror reflection of the elliptic curve together with a linear foliation. From this we identify the moduli space of super cycles on the mirror reflection with the moduli space of standard holomorphic bundles on a noncommutative complex torus.
We find an agreement of equivariant indices of semi-classical homomorphisms between pairwise mirror branes in the GL(2) Higgs moduli space on a Riemann surface. On one side we have the components of the Lagrangian brane of U(1,1) Higgs bundles whose mirror was proposed by Nigel Hitchin to be certain even exterior powers of the hyperholomorphic Dirac bundle on the SL(2) Higgs moduli space. The agreement arises from a mysterious functional equation. This gives strong computational evidence for Hitchin's proposal.
This article discusses the relativistic kinetic theory for a simple collisionless gas from a geometric perspective. We start by reviewing the rich geometrical structure of the tangent bundle TM of a given spacetime manifold, including the splitting of the tangent spaces of TM into horizontal and vertical subspaces and the natural metric and symplectic structure it induces on TM. Based on these structures we introduce the Liouville vector field L and a suitable Hamiltonian function H on TM. The Liouville vector field turns out to be the Hamiltonian vector field associated to H. On the other han…
We study a supersymmetry breaking deformation of the 2d N=(2,2) cigar=Liouville mirror pair, first introduced by Hori and Kapustin. We show that mirror symmetry flows in the infra-red to 2d bosonization, with the theories reducing to massive Thirring and Sine-Gordon respectively. The exact bosonization map emerges at one-loop. We further compactify non-supersymmetric 3d bosonization dualities on a circle and argue that these too flow to 2d bosonization at long distances.
The Fateev-Zamolodchikov-Zamolodchikov (FZZ) duality relates Witten's cigar model to sine-Liouville theory. This duality was proven in the path integral formulation and extended to the case of higher genus closed Riemann surfaces by Schomerus and one of the authors. In this note we further extend the duality to the case with boundary. Specifically, we relate D1-branes in the cigar model to D2-branes in the sine-Liouville theory. In particular, the boundary action for D2-branes in the sine-Liouville theory is constructed. We also consider the fermionic version of the FZZ duality. This duality w…
Mirror symmetry suggests that on a Calabi-Yau 3-fold moduli spaces of stable bundles, especially those with degree zero and indivisible Chern class, might be smooth (i.e. unobstructed, though perhaps of too high a dimension). This is because smoothly embedded special lagrangian cycles in the mirror have unobstructed deformations. As there does not seem to be a counterexample in the literature we provide one here, showing that such a Tian-Todorov-McLean-type result cannot hold.
We study certain compactifications of the type I string on K3. The three topologically distinct choices of gauge bundle for the type I theory are shown to be equivalent to type IIB orientifolds with different choices of background anti-symmetric tensor field flux. Using a mirror transformation, we relate these models to orientifolds with fixed seven planes, and without any antisymmetric tensor field flux. This map allows us to relate these type I vacua to particular six-dimensional F theory and heterotic string compactifications.
We show that certain superpotential and Kahler potential couplings of N=1 supersymmetric compactifications with branes or bundles can be computed from Hodge theory and mirror symmetry. This applies to F-theory on a Calabi-Yau four-fold and three-fold compactifications of type II and heterotic strings with branes. The heterotic case includes a class of bundles on elliptic manifolds constructed by Friedmann, Morgan and Witten. Mirror symmetry of the four-fold computes non-perturbative corrections to mirror symmetry on the three-folds, including D-instanton corrections. We also propose a physical…
We discuss the relativistic kinetic theory for a simple, collisionless, charged gas propagating on an arbitrary curved spacetime geometry. Our general relativistic treatment is formulated on the tangent bundle of the spacetime manifold and takes advantage of its rich geometric structure. In particular, we point out the existence of a natural metric on the tangent bundle and illustrate its role for the development of the relativistic kinetic theory. This metric, combined with the electromagnetic field of the spacetime, yields an appropriate symplectic form on the tangent bundle. The Liouville v…
We use local mirror symmetry in type IIA string compactifications on Calabi-Yau n+1 folds $X_{n+1}$ to construct vector bundles on (possibly singular) elliptically fibered Calabi-Yau n-folds Z_n. The interpretation of these data as valid classical solutions of the heterotic string compactified on Z_n proves F-theory/heterotic duality at the classical level. Toric geometry is used to establish a systematic dictionary that assigns to each given toric n+1-fold $X_{n+1}$ a toric n fold Z_n together with a specific family of sheafs on it. This allows for a systematic construction of phenomenologica…
We study one class of linear sigma models and their T-dualized theories for noncompact Calabi-Yau manifolds. In the low energy limit, we find that this system has various massless effective theories with orbifolding symmetries. This phenomenon is new and there are no analogous structures in the models for compact Calabi-Yau manifolds and for line bundles on the simple toric varieties.
In this paper we develop a relative version of T-duality in generalized complex geometry which we propose as a manifestation of mirror symmetry. Let M be an n-dimensional smooth real manifold, V a rank n real vector bundle on M, and nabla a flat connection on V. We define the notion of a nabla-semi-flat generalized complex structure on the total space of V. We show that there is an explicit bijective correspondence between nabla-semi-flat generalized complex structures on the total space of V and nabla(dual)-semi-flat generalized complex structures on the total space of the dual of V. Similarl…
We point out that insertions of matrix fields in (connected amputated) amplitudes of (generalized) Kontsevich models are given by covariant derivatives with respect to the Kontsevich moduli. This implies that correlators are sections of symmetric products of the (holomorphic) tangent bundle on the (complexified) moduli space of FZZT Liouville branes. We discuss the relation of Kontsevich parametrization of moduli space with that provided by either the (p,1) or the (1,p) boundary conformal field theories. It turns out that the Kontsevich connection captures the contribution of contact terms to …
Let $X$ be a holomorphic symplectic variety with a torus $\mathsf{T}$ action and a finite fixed point set of cardinality $k$. We assume that elliptic stable envelope exists for $X$. Let $A_{I,J}= \operatorname{Stab}(J)|_{I}$ be the $k\times k$ matrix of restrictions of the elliptic stable envelopes of $X$ to the fixed points. The entries of this matrix are theta-functions of two groups of variables: the Kähler parameters and equivariant parameters of $X$. We say that two such varieties $X$ and $X'$ are related by the 3d mirror symmetry if the fixed point sets of $X$ and $X'$ have the same card…
In this paper we propose (0,2) mirrors for general Fano toric varieties with special tangent bundle deformations, corresponding to subsets of toric deformations. Our mirrors are of the form of (B/2-twisted) (0,2) Landau-Ginzburg models, matching Hori-Vafa mirrors on the (2,2) locus. We compare our predictions to (0,2) mirrors obtained by Chen et al for certain examples of toric varieties, and find that they match. We also briefly outline conjectures for analogous results for hypersurfaces in Fano toric varieties. Our methods utilize results from supersymmetric localization, which allows us to …
In this paper we show that there exists a new class of topological field theories, whose correlators are intersection numbers of cohomology classes in a constrained moduli space. Our specific example is a formulation of 2D topological gravity. The constrained moduli-space is the Poincare' dual of the top Chern-class of the bundle $E_\rightarrow {\cal M}_g$, whose sections are the holomorphic differentials. Its complex dimension is $2g-3$, rather then $3g-3$. We derive our model by performing the A-topological twist of N=2 supergravity, that we identify with N=2 Liouville theory, whose rheonomi…
Fibrewise T-duality (Fourier-Mukai transform) for D-branes on an elliptic Calabi-Yau three-fold $X$ is seen to have an expected adiabatic form for its induced cohomology operation only when an appropriately twisted operation resp. twisted charge is defined. Some differences with the case of $K3$ as well as connections with the spectral cover construction for bundles on $X$ are pointed out. In the context of mirror symmetry Kontsevich's association of line bundle twists (resp. a certain 'diagonal' operation) with monodromies (esp. the conifold monodromy) is made explicit and checked for two exa…
One of the open problems in understanding (0,2) mirror symmetry concerns the construction of Toda-like Landau-Ginzburg mirrors to (0,2) theories on Fano spaces. In this paper, we begin to fill this gap by making an ansatz for (0,2) Toda-like theories mirror to (0,2) supersymmetric nonlinear sigma models on products of projective spaces, with deformations of the tangent bundle, generalizing a special case previously worked out for P1xP1. We check this ansatz by matching correlation functions of the B/2-twisted Toda-like theories to correlation functions of corresponding A/2-twisted nonlinear si…
Metrics of constant negative curvature on a compact Riemann surface are critical points of the Liouville action functional, which in recent constructions is rigorously defined as a class in a Cech-de Rham complex with respect to a suitable covering of the surface. We show that this class is the square of the metrized holomorphic tangent bundle in hermitian-holomorphic Deligne cohomology. We achieve this by introducing a different version of the hermitian-holomorphic Deligne complex which is nevertheless quasi-isomorphic to the one introduced by Brylinski in his construction of Quillen line bun…
We study Landau Ginzburg (LG) theories mirror to 2D N=2 gauged linear sigma models on toric Calabi-Yau manifolds. We derive and solve new constraint equations for Landau Ginzburg elliptic Calabi-Yau superpotentials, depending on the physical data of dual linear sigma models. In Calabi-Yau threefolds case, we consider two examples. First, we give the mirror symmetry of the canonical line bundle over the Hirzebruch surfaces $\bf F_n$. Second, we find a special geometry with the affine so(8) Lie algebra toric data extending the geometry of elliptically fibered K3. This geometry leads to a pure N=…
We compute instanton corrections to correlators in the genus-zero topological subsector of a (0,2) supersymmetric gauged linear sigma model with target space P1xP1, whose left-moving fermions couple to a deformation of the tangent bundle. We then deduce the theory's chiral ring from these correlators, which reduces in the limit of zero deformation to the (2,2) ring. Finally, we compare our results with the computations carried out by Adams et al.[ABS04] and Katz and Sharpe[KS06]. We find immediate agreement with the latter and an interesting puzzle in completely matching the chiral ring of the…
Using Polyakov's functional integral approach with the Liouville action functional defined in \cite{ZT2} and \cite{LTT}, we formulate quantum Liouville theory on a compact Riemann surface X of genus g > 1. For the partition function <X> and for the correlation functions with the stress-energy tensor components $<\prod_{i=1}^{n}T(z_{i})\prod_{k=1}^{l}\bar{T}(\w_{k})X>$, we describe Feynman rules in the background field formalism by expanding corresponding functional integrals around a classical solution - the hyperbolic metric on X. Extending analysis in \cite{LT1,LT2,LT-Varenna,LT3}, we define…
We propose and study the following Mirror Principle: certain sequences of multiplicative equivariant characteristic classes on Kontsevich's stable map moduli spaces can be computed in terms of certain hypergeometric type classes. As applications, we compute the equivariant Euler classes of obstruction bundles induced by any concavex bundles -- including any direct sum of line bundles -- on $¶^n$. This includes proving the formula of Candelas-de la Ossa-Green-Parkes hence completing the program of Candelas et al, Kontesevich, Manin, and Givental, to compute rigorously the instanton prepotential…
We describe the $S^1$-action on the Quot-scheme $\Quot({\cal E}^n)$ associated to the trivial bundle ${\cal E}^n=CP^1\times{\smallBbb C}^n$. In particlular, the topology of the $S^1$-fixed-point components in $\Quot({\cal E}^n)$ and the $S^1$-weights of the normal bundle of these components are worked out. Mirror Principle, as developed by three of the current authors in the series of work [L-L-Y1, I, II, III, IV], is a method for studying certain intersection numbers on a stable map moduli space. As an application, in Mirror Principle III, Sec 5.4, an outline was given in the case of genus ze…
The main purpose of this paper is to give a mathematical definition of ``mirror symmetry'' for Calabi-Yau and G_2 manifolds. More specifically, we explain how to assign a G_2 manifold (M,ϕ,Λ), with the calibration 3-form ϕand an oriented 2-plane field Λ, a pair of parametrized tangent bundle valued 2 and 3-forms of M. These forms can then be used to define various different complex and symplectic structures on certain 6-dimensional subbundles of T(M). When these bundles are integrated they give mirror CY manifolds. In a similar way, one can define mirror dual G_2 manifolds inside of a Spin(7) …
Let $(X,\check{X})$ be a mirror pair of a complex torus $X$ and its mirror partner $\check{X}$. This mirror pair is described as the trivial special Lagrangian torus fibrations $X\rightarrow B$ and $\check{X}\rightarrow B$ on the same base space $B$ by SYZ construction. Then, we can associate a holomorphic line bundle $E(s,\mathcal{L})\rightarrow X$ to a pair $(s,\mathcal{L})$ of a Lagrangian section $s$ of $\check{X}\rightarrow B$ and a unitary local system $\mathcal{L}$ along it. In this paper, we first construct the deformation $X_{\mathcal{G}}$ of $X$ by a certain flat gerbe $\mathcal{G}$ …
In this paper we consider the total space of the canonical bundle of P^2 and we use a proposal by Hosono, together with results in Seidel and Auroux-Katzarkov-Orlov, to deduce the right physical mirror equivalence between D^b(K_{P^2}) and the derived Fukaya category of its mirror. By construction, our equivalence is compatible with the mirror map between moduli spaces and with the computation of Gromov--Witten invariants.
Moving mirrors have been known as tractable setups modeling Hawking radiation from black holes. In this paper, motivated by recent developments regarding the black hole information problem, we present extensive studies of moving mirrors in conformal field theories by employing both field theoretic as well as holographic methods. Reviewing first the usual field theoretic formulation of moving mirrors, we construct their gravity dual by resorting to the AdS/BCFT construction. Based on our holographic formulation, we then calculate the time evolution of entanglement entropy in various moving mirr…
We exhibit a transformation taking special Lagrangian submanifolds of a Calabi-Yau together with local systems to vector bundles over the mirror manifold with connections obeying deformed Hermitian-Yang-Mills equations. That is, the transformation relates supersymmetric A- and B-cycles. In this paper, we assume that the mirror pair are dual torus fibrations with flat tori and that the A-cycle is a section. We also show that this transformation preserves the (holomorphic) Chern-Simons functional for all connections. Furthermore, on corresponding moduli spaces of supersymmetric cycles it identif…
We study the moduli spaces of flat SL(r)- and PGL(r)-connections, or equivalently, Higgs bundles, on an algebraic curve. These spaces are noncompact Calabi-Yau orbifolds; we show that they can be regarded as mirror partners in two different senses. First, they satisfy the requirements laid down by Strominger-Yau-Zaslow (SYZ), in a suitably general sense involving a B-field or flat unitary gerbe. To show this, we use their hyperkahler structures and Hitchin's integrable systems. Second, their Hodge numbers, again in a suitably general sense, are equal. These spaces provide significant evidence …
