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Showing papers on "Mirror symmetry published in 2007"


Posted Content
TL;DR: In this article, the geometry of complexified moduli spaces of special Lagrangian submanifolds in the complement of an anticanonical divisor in a compact Kahler manifold is studied.
Abstract: We study the geometry of complexified moduli spaces of special Lagrangian submanifolds in the complement of an anticanonical divisor in a compact Kahler manifold. In particular, we explore the connections between T-duality and mirror symmetry in concrete examples, and show how quantum corrections arise in this context.

343 citations


Journal ArticleDOI
TL;DR: In this article, the authors review and extend the progress made over the past few years in understanding the structure of toric quiver gauge theories; those which are induced on the world-volume of a stack of D3-branes placed at the tip of a toric Calabi-Yau cone, at an ''orbifold point'' in Kaehler moduli space.
Abstract: We review and extend the progress made over the past few years in understanding the structure of toric quiver gauge theories; those which are induced on the world-volume of a stack of D3-branes placed at the tip of a toric Calabi-Yau cone, at an ``orbifold point'' in Kaehler moduli space These provide an infinite class of four-dimensional N=1 superconformal field theories which may be studied in the context of the AdS/CFT correspondence It is now understood that these gauge theories are completely specified by certain two-dimensional torus graphs, called brane tilings, and the combinatorics of the dimer models on these graphs In particular, knowledge of the dual Sasaki-Einstein metric is not required to determine the gauge theory, only topological and symplectic properties of the toric Calabi-Yau cone By analyzing the symmetries of the toric quiver theories we derive the dimer models and use them to construct the moduli space of the theory both classically and semiclassically Using mirror symmetry the brane tilings are shown to arise in string theory on the world-volumes of the fractional D6-branes that are mirror to the stack of D3-branes at the tip of the cone

189 citations


Posted Content
TL;DR: In this paper, a new, conjectural recursion solution for Hurwitz numbers at all genera was proposed, based on recent progress in solving type B topological string theory on the mirrors of toric Calabi-Yau manifolds.
Abstract: We propose a new, conjectural recursion solution for Hurwitz numbers at all genera. This conjecture is based on recent progress in solving type B topological string theory on the mirrors of toric Calabi-Yau manifolds, which we briefly review to provide some background for our conjecture. We show in particular how this B-model solution, combined with mirror symmetry for the one-leg, framed topological vertex, leads to a recursion relation for Hodge integrals with three Hodge class insertions. Our conjecture in Hurwitz theory follows from this recursion for the framed vertex in the limit of infinite framing.

172 citations


Journal ArticleDOI
TL;DR: In this paper, the authors study the topological sector of N = 2 sigma-models with HH-flux and show that the results are naturally expressed in the language of twisted generalized complex geometry.
Abstract: We study the topological sector of N = 2 sigma-models with HH-flux. It has been known for a long time that the target-space geometry of these theories is not Kahler and can be described in terms of a pair of complex structures, which do not commute, in general, and are parallel with respect to two different connections with torsion. Recently an alternative description of this geometry was found, which involves a pair of commuting twisted generalized complex structures on the target space. In this paper, we define and study the analogs of A and B-models for N = 2 sigma-models with HH-flux and show that the results are naturally expressed in the language of twisted generalized complex geometry. For example, the space of topological observables is given by the cohomology of a Lie algebroid associated to one of the two twisted generalized complex structures. We determine the topological scalar product, which endows the algebra of observables with the structure of a Frobenius algebra. We also discuss mirror symmetry for twisted generalized Calabi-Yau manifolds.

164 citations


Journal ArticleDOI
TL;DR: In this paper, the authors show that the tension of the domainwall between the two vacua on the brane satisfies a certain extension of the Picard-Fuchs differential equation governing periods of the mirror quintic.
Abstract: 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.

150 citations


Posted Content
TL;DR: In this paper, the degrees of the Noether-Lefschetz divisors in 1-parameter families of K3 surfaces were shown to be Fourier coefficients of an explicitly computed modular form of weight 21/2 and level 8.
Abstract: Noether-Lefschetz divisors in the moduli of K3 surfaces are the loci corresponding to Picard rank at least 2. We relate the degrees of the Noether-Lefschetz divisors in 1-parameter families of K3 surfaces to the Gromov-Witten theory of the 3-fold total space. The reduced K3 theory and the Yau-Zaslow formula play an important role. We use results of Borcherds and Kudla-Millson for O(2,19) lattices to determine the Noether-Lefschetz degrees in classical families of K3 surfaces of degrees 2, 4, 6 and 8. For the quartic K3 surfaces, the Noether-Lefschetz degrees are proven to be the Fourier coefficients of an explicitly computed modular form of weight 21/2 and level 8. The interplay with mirror symmetry is discussed. We close with a conjecture on the Picard ranks of moduli spaces of K3 surfaces.

142 citations


Journal ArticleDOI
TL;DR: In this paper, free fermionic construction of four dimensional string vacua, related to the Z2XZ2 orbifolds at special points in the moduli space, yielded the most realistic three family string models to date.

113 citations


Posted Content
TL;DR: In this paper, the notion of a Gorenstein polytope of index r was introduced as a natural combinatorial duality for d-dimensional polytopes and the Borisov duality between two nef-partitions was considered as a duality.
Abstract: The purpose of this paper is to review some combinatorial ideas behind the mirror symmetry for Calabi-Yau hypersurfaces and complete intersections in Gorenstein toric Fano varieties. We suggest as a basic combinatorial object the notion of a Gorenstein polytope of index r. A natural combinatorial duality for d-dimensional Gorenstein polytopes of index r extends the well-known polar duality for reflexive polytopes (case r=1). We consider the Borisov duality between two nef-partitions as a duality between two Gorenstein polytopes P and P^* of index r together with selected special (r-1)-dimensional simplices S in P and S' in P^*. Different choices of these simplices suggest an interesting relation to Homological Mirror Symmetry.

108 citations


Posted Content
TL;DR: The main focus of as discussed by the authors is the calculation of the cohomology of a Calabi-Yau variety associated to a given affine manifold with singularities B, as expected.
Abstract: This paper continues the authors' program of studying mirror symmetry via log geometry and toric degenerations, relating affine manifolds with singularities, log Calabi-Yau spaces, and toric degenerations of Calabi-Yaus. The main focus of this paper is the calculation of the cohomology of a Calabi-Yau variety associated to a given affine manifold with singularities B. We show that the Dolbeault cohomology groups of the Calabi-Yau associated to B are described in terms of some cohomology groups of sheaves on B, as expected. This is proved first by calculating the log de Rham and log Dolbeault cohomology groups on the log Calabi-Yau space associated to B, and then proving a base-change theorem for cohomology in our logarithmic setting. As applications, this shows that our mirror symmetry construction via Legendre duality of affine manifolds results in the usual interchange of Hodge numbers expected in mirror symmetry, and gives an explicit description of the monodromy of a smoothing.

94 citations


Journal ArticleDOI
TL;DR: In this article, it is conjectured that one can define invariants J α (Z, P) e Q for stable stable sheaves on a Calabi-Yau 3-fold manifold, and combine these invariants into a family of holomorphic generating functions F α : Stab(T) → C for a ∈ K(T).
Abstract: Let X be a Calabi-Yau 3-fold, T = D b (coh(X)) the derived category of coherent sheaves on X, and Stab(T) the complex manifold of Bridgeland stability conditions on T. It is conjectured that one can define invariants J α (Z, P) e Q for (Z, P) ∈ Stab(T) and a ∈ K(T) generalizing Donaldson-Thomas invariants, which "count" (Z, P)-semistable (complexes of) coherent sheaves on X, and whose transformation law under change of (Z, P) is known. This paper explains how to combine such invariants J α (Z, P), if they exist, into a family of holomorphic generating functions F α : Stab(T) → C for a ∈ K(T). Surprisingly, requiring the F α to be continuous and holomorphic determines them essentially uniquely, and implies they satisfy a p.d.e., which can be interpreted as the flatness of a connection over Stab(T) with values in an infinite-dimensional Lie algebra L. The author believes that underlying this mathematics there should be some new physics, in String Theory and Mirror Symmetry. String Theorists are invited to work out and explain this new physics.

72 citations


Journal ArticleDOI
TL;DR: In this paper, the authors derive the complete supergravity description of the N = 2 scalar potential which realizes a generic flux-compactification on a Calabi-Yau manifold.

10 Sep 2007
TL;DR: In this paper, a new, conjectural recursion solution for Hurwitz numbers at all genera was proposed, based on recent progress in solving type B topological string theory on the mirrors of toric Calabi-Yau manifolds.
Abstract: We propose a new, conjectural recursion solution for Hurwitz numbers at all genera. This conjecture is based on recent progress in solving type B topological string theory on the mirrors of toric Calabi-Yau manifolds, which we briefly review to provide some background for our conjecture. We show in particular how this B-model solution, combined with mirror symmetry for the one-leg, framed topological vertex, leads to a recursion relation for Hodge integrals with three Hodge class insertions. Our conjecture in Hurwitz theory follows from this recursion for the framed vertex in the limit of infinite framing.

Journal ArticleDOI
TL;DR: In this paper, the difference between the standard and reduced genus-one Gromov-Witten invariants of complete intersections has been studied and an explicit formula for the difference has been given.
Abstract: We give an explicit formula for the difference between the standard and reduced genus-one Gromov-Witten invariants. Combined with previous work on geometric properties of the latter, this paper makes it possible to compute the standard genus-one GW-invariants of complete intersections. In particular, we obtain a closed formula for the genus-one GW-invariants of a Calabi-Yau projective hypersurface and verify a recent mirror symmetry prediction for a sextic fourfold as a special case.

Journal ArticleDOI
TL;DR: In this article, the authors apply mirror symmetry to the problem of counting holomorphic rational curves in a Calabi-Yau threefold X with Z3 � Z3 Wilson lines.
Abstract: We apply mirror symmetry to the problem of counting holomorphic rational curves in a Calabi-Yau threefold X with Z3 � Z3 Wilson lines. As we found in Part A [1], the integral homology group H2(X;Z) = Z 3 � Z3 � Z3 contains torsion curves. Using the B-model on the mirror of X as well as its covering spaces, we compute the instanton numbers. We observe that X is self-mirror even at the quantum level. Using the selfmirror property, we derive the complete prepotential on X, going beyond the results of Part A. In particular, this yields the first example where the instanton number depends on the torsion part of its homology class. Another consequence is that the threefold X provides a non-toric example for the conjectured exchange of torsion subgroups in mirror manifolds.

Posted Content
TL;DR: In this article, the authors studied real and integral structures in the space of solutions to the quantum differential equations and gave a natural explanation why the quantum parameter should specialize to a root of unity in Ruan's crepant resolution conjecture.
Abstract: We study real and integral structures in the space of solutions to the quantum differential equations. First we show that, under mild conditions, any real structure in orbifold quantum cohomology yields a pure and polarized tt^*-geometry near the large radius limit. Secondly, we use mirror symmetry to calculate the "most natural" integral structure in quantum cohomology of toric orbifolds. We show that the integral structure pulled back from the singularity B-model is described only in terms of topological data in the A-model; K-group and a characteristic class. Using integral structures, we give a natural explanation why the quantum parameter should specialize to a root of unity in Ruan's crepant resolution conjecture.

Posted Content
TL;DR: In this paper, mirror symmetry is used to determine and sum up a class of membrane instanton corrections to the hypermultiplet moduli space metric arising in Calabi-Yau threefold compactifications of type IIA strings.
Abstract: We use mirror symmetry to determine and sum up a class of membrane instanton corrections to the hypermultiplet moduli space metric arising in Calabi-Yau threefold compactifications of type IIA strings. These corrections are mirror to the D1 and D(-1)-brane instantons on the IIB side and are given explicitly in terms of a single function in projective superspace. The corresponding four-dimensional effective action is completely fixed by the Euler number and the genus zero Gopakumar-Vafa invariants of the mirror Calabi-Yau.

Journal ArticleDOI
Abstract: We use mirror symmetry to determine and sum up a class of membrane instanton corrections to the hypermultiplet moduli space metric arising in Calabi-Yau threefold compactifications of type IIA strings. These corrections are mirror to the D1 and D(-1)-brane instantons on the IIB side and are given explicitly in terms of a single function in projective superspace. The corresponding four-dimensional effective action is completely fixed by the Euler number and the genus zero Gopakumar-Vafa invariants of the mirror Calabi-Yau.

Journal ArticleDOI
TL;DR: In this paper, the authors apply mirror symmetry to the problem of counting holomorphic rational curves in a Calabi-Yau threefold X with Z 3 x Z 3 Wilson lines.
Abstract: We apply mirror symmetry to the problem of counting holomorphic rational curves in a Calabi-Yau threefold X with Z_3 x Z_3 Wilson lines. As we found in Part A [hep-th/0703182], the integral homology group H_2(X,Z)=Z^3 + Z_3 + Z_3 contains torsion curves. Using the B-model on the mirror of X as well as its covering spaces, we compute the instanton numbers. We observe that X is self-mirror even at the quantum level. Using the self-mirror property, we derive the complete prepotential on X, going beyond the results of Part A. In particular, this yields the first example where the instanton number depends on the torsion part of its homology class. Another consequence is that the threefold X provides a non-toric example for the conjectured exchange of torsion subgroups in mirror manifolds.

Journal ArticleDOI
TL;DR: In this paper, a mirror construction for monomial degenerations of Calabi-Yau varieties in toric Fano varieties was proposed using tropical geometry, and the construction reproduces the mirror constructions by Batyrev and Borisov for complete intersections.
Abstract: Using tropical geometry we propose a mirror construction for monomial degenerations of Calabi-Yau varieties in toric Fano varieties. The construction reproduces the mirror constructions by Batyrev for Calabi-Yau hypersurfaces and by Batyrev and Borisov for Calabi-Yau complete intersections. We apply the construction to Pfaffian examples and recover the mirror given by Rodland for the degree 14 Calabi-Yau threefold in PP^6 defined by the Pfaffians of a general linear 7x7 skew-symmetric matrix. We provide the necessary background knowledge entering into the tropical mirror construction such as toric geometry, Groebner bases, tropical geometry, Hilbert schemes and deformations. The tropical approach yields an algorithm which we illustrate in a series of explicit examples.

Journal ArticleDOI
TL;DR: In this article, the authors construct a class of symplectic non-Kaehler and complex non-kaehler string theory vacua, extending and providing evidence for an earlier suggestion by Polchinski and Strominger.
Abstract: We construct a class of symplectic non-Kaehler and complex non-Kaehler string theory vacua, extending and providing evidence for an earlier suggestion by Polchinski and Strominger. The class admits a mirror pairing by construction. Comparing hints from a variety of sources, including ten-dimensional supergravity and KK reduction on SU(3)-structure manifolds, suggests a picture in which string theory extends Reid's fantasy to connect classes of both complex non-Kaehler and symplectic non-Kaehler manifolds.

Posted Content
TL;DR: In this paper, a family of moduli spaces, a virtual cycle, and a corresponding cohomological field theory associated to the singularity are described for any non-degenerate, quasi-homogeneous hypersurface singularity.
Abstract: For any non-degenerate, quasi-homogeneous hypersurface singularity, we describe a family of moduli spaces, a virtual cycle, and a corresponding cohomological field theory associated to the singularity. This theory is analogous to Gromov-Witten theory and generalizes the theory of r-spin curves, which corresponds to the simple singularity A_{r-1}. We also resolve two outstanding conjectures of Witten. The first conjecture is that ADE-singularities are self-dual; and the second conjecture is that the total potential functions of ADE-singularities satisfy corresponding ADE-integrable hierarchies. Other cases of integrable hierarchies are also discussed.

Journal ArticleDOI
TL;DR: In this article, the instanton corrected hypermultiplet moduli space in type IIB compactifications near a Calabi-Yau conifold point where the size of a two-cycle shrinks to zero was determined.
Abstract: We determine the instanton corrected hypermultiplet moduli space in type IIB compactifications near a Calabi-Yau conifold point where the size of a two-cycle shrinks to zero. We show that D1-instantons resolve the conifold singularity caused by worldsheet instantons. Furthermore, by resumming the instanton series, we reproduce exactly the results obtained by Ooguri and Vafa on the type IIA side, where membrane instantons correct the hypermultiplet moduli space. Our calculations therefore establish that mirror symmetry holds non-perturbatively in the string coupling.

Journal ArticleDOI
TL;DR: In this article, the authors apply mirror symmetry to the super Calabi-Yau manifold CP(n|n+1) and show that the mirror can be recast in a form which depends only on the superdimension and which is reminiscent of a generalized conifold.
Abstract: We apply mirror symmetry to the super Calabi-Yau manifold CP(n|n+1) and show that the mirror can be recast in a form which depends only on the superdimension and which is reminiscent of a generalized conifold. We discuss its geometrical properties in comparison to the familiar conifold geometry. In the second part of the paper examples of special-Lagrangian submanifolds are constructed for a class of super Calabi-Yau's. We finally comment on their infinitesimal deformations.

Journal ArticleDOI
TL;DR: In this article, it was shown that the product of a six-manifold M × hat M is doubly fibered by supersymmetric three-tori, with both sets of fibers transverse to M and hat M, and the mirror map is then realized by T-dualizing the fibers.
Abstract: Fibrations of flux backgrounds by supersymmetric cycles are investigated. For an internal six-manifold M with static SU(2) structure and mirror hat M, it is argued that the product M × hat M is doubly fibered by supersymmetric three-tori, with both sets of fibers transverse to M and hat M. The mirror map is then realized by T-dualizing the fibers. Mirror-symmetric properties of the fluxes, both geometric and non-geometric, are shown to agree with previous conjectures based on the requirement of mirror symmetry for Killing prepotentials. The fibers are conjectured to be destabilized by fluxes on generic SU(3) × SU(3) backgrounds, though they may survive at type-jumping points. T-dualizing the surviving fibers ensures the exchange of pure spinors under mirror symmetry.

Journal ArticleDOI
TL;DR: In this paper, the perturbative aspects of a B-twisted two-dimensional heterotic sigma model on a holomorphic gauge bundle over a complex, hermitian manifold were studied.
Abstract: In this paper, we study the perturbative aspects of a "B-twisted" two-dimensional $(0,2)$ heterotic sigma model on a holomorphic gauge bundle $\mathcal E$ over a complex, hermitian manifold $X$. We show that the model can be naturally described in terms of the mathematical theory of ``Chiral Differential Operators". In particular, the physical anomalies of the sigma model can be reinterpreted as an obstruction to a global definition of the associated sheaf of vertex superalgebras derived from the free conformal field theory describing the model locally on $X$. In addition, one can also obtain a novel understanding of the sigma model one-loop beta function solely in terms of holomorphic data. At the $(2,2)$ locus, one can describe the resulting half-twisted variant of the topological B-model in terms of a $\it{mirror}$ "Chiral de Rham complex" (or CDR) defined by Malikov et al. in \cite{GMS1}. Via mirror symmetry, one can also derive various conjectural expressions relating the sheaf cohomology of the mirror CDR to that of the original CDR on pairs of Calabi-Yau mirror manifolds. An analysis of the half-twisted model on a non-K\"ahler group manifold with torsion also allows one to draw conclusions about the corresponding sheaves of CDR (and its mirror) that are consistent with mathematically established results by Ben-Bassat in \cite{ben} on the mirror symmetry of generalised complex manifolds. These conclusions therefore suggest an interesting relevance of the sheaf of CDR in the recent study of generalised mirror symmetry.

Journal ArticleDOI
TL;DR: In this article, the authors provide a straightforward computational scheme for the equivariant local mirror symmetry of curves, i.e. mirror symmetry for for k ≥ 1, and detail related methods for dealing with mirror symmetry in non-nef toric varieties, based on the theorems of Refs. 2 and 13.
Abstract: We provide a straightforward computational scheme for the equivariant local mirror symmetry of curves, i.e. mirror symmetry for for k ≥ 1, and detail related methods for dealing with mirror symmetry of non-nef toric varieties, based on the theorems of Refs. 2 and 13. The basic tools are equivariant I functions and their Birkhoff factorization.

Posted Content
TL;DR: In this article, a real affine manifold with singularities (a tropical manifold) is constructed from a Calabi-Yau manifold, and an explicit and canonical order-by-order description of the degeneration via families of tropical trees is given.
Abstract: We construct from a real affine manifold with singularities (a tropical manifold) a degeneration of Calabi-Yau manifolds. This solves a fundamental problem in mirror symmetry. Furthermore, a striking feature of our approach is that it yields an explicit and canonical order-by-order description of the degeneration via families of tropical trees. This gives complete control of the B-model side of mirror symmetry in terms of tropical geometry. For example, we expect our deformation parameter is a canonical coordinate, and expect period calculations to be expressible in terms of tropical curves. We anticipate this will lead to a proof of mirror symmetry via tropical methods. This paper is the key step of the program we initiated in math.AG/0309070.

Journal ArticleDOI
TL;DR: In this paper, the perturbative aspects of a two-dimensional (0, 2) heterotic sigma model on a holomorphic gauge bundle over a complex, hermitian manifold X were studied.
Abstract: In this paper, we study the perturbative aspects of a ``B-twisted" two-dimensional (0,2) heterotic sigma model on a holomorphic gauge bundle over a complex, hermitian manifold X. We show that the model can be naturally described in terms of the mathematical theory of ``Chiral Differential Operators". In particular, the physical anomalies of the sigma model can be reinterpreted as an obstruction to a global definition of the associated sheaf of vertex superalgebras derived from the free conformal field theory describing the model locally on X. In addition, one can also obtain a novel understanding of the sigma model one-loop beta function solely in terms of holomorphic data. At the (2,2) locus, one can describe the resulting half-twisted variant of the topological B-model in terms of a mirror ``Chiral de Rham complex" (or CDR) defined by Malikov et al. in [1]. Via mirror symmetry, one can also derive various conjectural expressions relating the sheaf cohomology of the mirror CDR to that of the original CDR on pairs of Calabi-Yau mirror manifolds. An analysis of the half-twisted model on a non-Kahler group manifold with torsion also allows one to draw conclusions about the corresponding sheaves of CDR (and its mirror) that are consistent with mathematically established results by Ben-Bassat in [2] on the mirror symmetry of generalised complex manifolds. These conclusions therefore suggest an interesting relevance of the sheaf of CDR in the recent study of generalised mirror symmetry.

Journal ArticleDOI
TL;DR: In this article, the authors studied equivariant local mirror symmetry of curves, i.e., mirror symmetry for Xk = O(k) ⊕O (−2 − k) → P 1 with torus action (λ 1,λ 2) on the bundle.
Abstract: We continue our study of equivariant local mirror symmetry of curves, i.e., mirror symmetry for Xk = O(k) ⊕O (−2 − k) → P 1 with torus action (λ1 ,λ 2) on the bundle. For the antidiagonal action λ1 = −λ2, we find closed formulas for the mirror map, a rational B model Yukawa coupling and consequently Picard–Fuchs equations for all k. Moreover, we give a simple closed form for the B model genus 1 Gromov–Witten potential. For the diagonal action λ1 = λ2, we argue that the mirror symmetry computation is equivalent to that of the projective bundle P( O⊕O (k) ⊕O (−2 − k)) → P 1 . Finally, we outline the computation of equivariant Gromov–Witten invariants for An singularities and toric tree examples via mirror symmetry.

Posted Content
TL;DR: In this paper, the geometric Langlands correspondence has been interpreted as the mirror symmetry of the Hitchin fibrations for two dual reductive groups, which are smooth tori.
Abstract: The geometric Langlands correspondence has been interpreted as the mirror symmetry of the Hitchin fibrations for two dual reductive groups. This mirror symmetry, in turn, reduces to T-duality on the generic Hitchin fibers, which are smooth tori. In this paper we study what happens when the Hitchin fibers on the B-model side develop orbifold singularities. These singularities correspond to local systems with finite groups of automorphisms. In the classical Langlands Program local systems of this type are called endoscopic. They play an important role in the theory of automorphic representations, in particular, in the stabilization of the trace formula. Our goal is to use the mirror symmetry of the Hitchin fibrations to expose the special role played by these local systems in the geometric theory. The study of the categories of A-branes on the dual Hitchin fibers allows us to uncover some interesting phenomena associated with the endoscopy in the geometric Langlands correspondence. We then follow our predictions back to the classical theory of automorphic functions. This enables us to test and confirm them. The geometry we use is similar to that which is exploited in recent work by B.-C. Ngo, a fact which could be significant for understanding the trace formula.