Showing papers on "Mirror symmetry published in 2006"
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TL;DR: In this paper, a universal construction of Calabi-Yau algebras in terms of a noncommutative symplectic DG algebra resolution is given, in dimension 3, where the resolution is determined by a non commutative potential representation variety.
Abstract: We introduce some new algebraic structures arising naturally in the geometry of Calabi-Yau manifolds and mirror symmetry We give a universal construction of Calabi-Yau algebras in terms of a noncommutative symplectic DG algebra resolution
In dimension 3, the resolution is determined by a noncommutative potential Representation varieties of the Calabi-Yau algebra are intimately related to the set of critical points, and to the sheaf of vanishing cycles of the potential Numerical invariants, like ranks of cyclic homology groups, are expected to be given by `matrix integrals' over representation varieties
We discuss examples of Calabi-Yau algebras involving quivers, 3-dimensional McKay correspondence, crepant resolutions, Sklyanin algebras, hyperbolic 3-manifolds and Chern-Simons Examples related to quantum Del Pezzo surfaces will be discussed in [EtGi]
563 citations
TL;DR: In this paper, the authors studied the mirror symmetry of Calabi-Yau manifolds with singularities and showed an isomorphism between log complex moduli of a log CY space and the mirror of its mirror.
Abstract: This paper lays the foundations of a program to study mirror symmetry by studying the log structures of Illusie-Fontaine and Kato on degenerations of Calabi-Yau manifolds. The basic idea is that one can associate to certain sorts of degenerations of Calabi-Yau manifolds a log Calabi-Yau space, which is a log structure on the degenerate fibre. The log CY space captures essentially all the information of the degeneration, and hence all mirror statements for the "large complex structure limit" given by the degeneration can already be derived from the log CY space. In this paper we begin by discussing affine manifolds with singularities. Given such an affine manifold along with a polyhedral decomposition, we show how to construct a scheme consisting of a union of toric varieties. In certain non-degenerate cases, we can also construct log structures on these schemes. Conversely, given certain sorts of degenerations, one can build an affine manifold with singularities structure on the dual intersection complex of the degeneration. Mirror symmetry is then obtained as a discrete Legendre transform on these affine manifolds, thus providing an algebro-geometrization of the Strominger-Yau-Zaslow conjecture. The deepest result of this paper shows an isomorphism between log complex moduli of a log CY space and log Kahler moduli of its mirror.
245 citations
TL;DR: In this paper, the authors studied homological mirror symmetry for Del Pezzo surfaces and their mirror Landau-Ginzburg models and showed that the derived category of coherent sheaves obtained by blowing up ℂℙ2 at k points is equivalent to the derived categories of vanishing cycles of a certain elliptic fibration Wk:Mk→ℂ with k+3 singular fibers, equipped with a suitable symplectic form.
Abstract: We study homological mirror symmetry for Del Pezzo surfaces and their mirror Landau-Ginzburg models. In particular, we show that the derived category of coherent sheaves on a Del Pezzo surface Xk obtained by blowing up ℂℙ2 at k points is equivalent to the derived category of vanishing cycles of a certain elliptic fibration Wk:Mk→ℂ with k+3 singular fibers, equipped with a suitable symplectic form. Moreover, we also show that this mirror correspondence between derived categories can be extended to noncommutative deformations of Xk, and give an explicit correspondence between the deformation parameters for Xk and the cohomology class [B+iω]∈H2(Mk,ℂ).
233 citations
Posted Content•
TL;DR: The geometric Langlands program can be described in a natural way by compactifying on a Riemann surface C a twisted version of N=4 super Yang-Mills theory in four dimensions as mentioned in this paper.
Abstract: The geometric Langlands program can be described in a natural way by compactifying on a Riemann surface C a twisted version of N=4 super Yang-Mills theory in four dimensions. The key ingredients are electric-magnetic duality of gauge theory, mirror symmetry of sigma-models, branes, Wilson and 't Hooft operators, and topological field theory. Seemingly esoteric notions of the geometric Langlands program, such as Hecke eigensheaves and D-modules, arise naturally from the physics.
145 citations
TL;DR: In this paper, the authors describe the physics of the gauged linear sigma models that formally coincide with the mathematical description of toric stacks and check that physical predictions of those gauged lasso models exactly match the corresponding stacks.
Abstract: In this paper, we will discuss gauged linear sigma model descriptions of toric stacks. Toric stacks have a simple description in terms of (symplectic, GIT) C × quotients of homogeneous coordinates, in exactly the same form as toric varieties. We describe the physics of the gauged linear sigma models that formally coincide with the mathematical description of toric stacks and check that physical predictions of those gauged linear sigma models exactly match the corresponding stacks. We also see in examples that when a given toric stack has multiple presentations in a form accessible as a gauged linear sigma model, that the IR physics of those different presentations matches, so that the IR physics is presentation-independent, making it reasonable to associate CFTs to stacks, not just presentations of stacks. We discuss mirror symmetry for stacks, using Morrison– Plesser–Hori–Vafa techniques to compute mirrors explicitly, and also find a natural generalization of Batyrev’s mirror conjecture. In the process of studying mirror symmetry, we find some new abstract CFTs, involving fields valued in roots of unity.
130 citations
TL;DR: In this paper, the authors describe new autoequivalences of derived categories of coherent sheaves arising from what they call $\mathbb P^n$-objects of the category.
Abstract: We describe new autoequivalences of derived categories of coherent sheaves arising from what we call $\mathbb P^n$-objects of the category. Standard examples arise from holomorphic symplectic manifolds. Under mirror symmetry these autoequivalences should be mirror to Seidel's Dehn twists about lagrangian $\mathbb P^n$ submanifolds.
We give various connections to spherical objects and spherical twists, and include a simple description of Atiyah and Kodaira-Spencer classes in an appendix.
110 citations
TL;DR: In this article, an exact real-space renormalization method is developed to address the electronic transport in mirror Fibonacci chains at a macroscopic scale by means of the Kubo-Greenwood formula.
Abstract: An exact real-space renormalization method is developed to address the electronic transport in mirror Fibonacci chains at a macroscopic scale by means of the Kubo-Greenwood formula. The results show that the mirror symmetry induces a large number of transparent states in the dc conductivity spectra, contrary to the simple Fibonacci case. A length scaling analysis over ten orders of magnitude reveals the existence of critically localized states and their ac conduction spectra show a highly oscillating behaviour. For multidimensional quasiperiodic systems, a novel renormalization plus convolution method is proposed. This combined renormalization + convolution method has shown an extremely elevated computing efficiency, being able to calculate electrical conductance of a three-dimensional non-crystalline solid with 10 30 atoms. Finally, the dc and ac conductances of mirror Fibonacci nanowires are also investigated, where a quantized dc-conductance variation with the Fermi energy is found, as observed in gold nanowires.
106 citations
TL;DR: In this paper, the authors studied the space of stability conditions on the non-compact Calabi-Yau threefold, which is the total space of the canonical bundle of the quantum cohomology.
Abstract: We study the space of stability conditions Stab(X) on the non-compact Calabi-Yau threefold X which is the total space of the canonical bundle of \(\mathbb{P}^2\). We give a combinatorial description of an open subset of Stab(X) and state a conjecture relating Stab(X) to the Frobenius manifold obtained from the quantum cohomology of \(\mathbb{P}^2\). We give some evidence from mirror symmetry for this conjecture.
98 citations
TL;DR: In this article, Kontsevich's homological mirror symmetry conjecture was shown to hold in the context of toric varieties, where the origin point of a toric variety is a convex hull of the primitive vertices of the 1-cones of a simplicial rational polyhedral fan.
Abstract: In this paper we give some evidence for M Kontsevich’s homological mirror symmetry conjecture [13] in the context of toric varieties. Recall that a smooth complete toric variety is given by a simplicial rational polyhedral fan such that jj D R and all maximal cones are non-singular (Fulton [10, Section 2.1]). The convex hull of the primitive vertices of the 1–cones of is a convex polytope which we denote by P , containing the origin as an interior point, and may be thought of as the Newton polytope of a Laurent polynomial W W .C/ ! C. This Laurent polynomial is the Landau–Ginzburg mirror of X .
90 citations
TL;DR: What is believed to be a new class of solutions of the three-flat problem for circular flats is described in terms of functions that are symmetric or antisymmetric with respect to reflections at a single line passing through the center of the flat surfaces.
Abstract: In interferometric surface and wavefront metrology, three-flat tests are the archetypes of measurement procedures to separate errors in the interferometer reference wavefront from errors due to the test part surface, so-called absolute tests. What is believed to be a new class of solutions of the three-flat problem for circular flats is described in terms of functions that are symmetric or antisymmetric with respect to reflections at a single line passing through the center of the flat surfaces. The new solutions are simpler and easier to calculate than the known solutions based on twofold mirror symmetry or rotation symmetry. Strategies for effective azimuthal averaging and a method for determining the averaging error are also discussed.
87 citations
01 Jan 2006
TL;DR: The concept of motivic integration was introduced by Kontsevich and Batyrev as mentioned in this paper, who constructed a measure on the arc space of an algebraic variety, the motivic measure, with the subtle and crucial property that it takes values not in $\mathbb{R}$, but in the Grothendieck ring of algebraic varieties.
Abstract: The concept of motivic integration was invented by Kontsevich to show that birationally equivalent Calabi-Yau manifolds have the same Hodge numbers. He constructed a certain measure on the arc space of an algebraic variety, the motivic measure, with the subtle and crucial property that it takes values not in $\mathbb{R}$, but in the Grothendieck ring of algebraic varieties. A whole theory on this subject was then developed by Denef and Loeser in various papers, with several applications. Batyrev introduced with motivic integration techniques new singularity invariants, the stringy invariants, for algebraic varieties with mild singularities, more precisely log terminal singularities. He used them for instance to formulate a topological Mirror Symmetry test for pairs of singular Calabi-Yau varieties. We generalized these invariants to almost arbitrary singular varieties, assuming Mori's Minimal Model Program. The aim of these notes is to provide a gentle introduction to these concepts. There exist already good surveys by Denef-Loeser [DL8] and Looijenga [Loo], and a nice elementary introduction by Craw [Cr]. Here we merely want to explain the basic concepts and first results, including the $p$-adic number theoretic pre-history of the theory, and to provide concrete examples. The text is a slightly adapted version of the 'extended abstract' of the author's talks at the 12th MSJ-IRI "Singularity Theory and Its Applications" (2003) in Sapporo. At the end we included a list of various recent results.
TL;DR: In this paper, a relative version of T -duality in generalized complex geometry is proposed as a manifestation of mirror symmetry, and a bijective correspondence between ∇ -semi-flat generalized almost complex structures on the total space of a real manifold and ∇ ∨ semi-flat GAs on the manifold is established.
TL;DR: In this article, the authors give the first examples of stability conditions on the A-model side of mirror symmetry, where the triangulated category is not naturally the derived category of an abelian category.
Abstract: We find stability conditions [6, 3] on some derived categories of differential graded modules over a graded algebra studied in [12, 10]. This category arises in both derived Fukaya categories and derived categories of coherent sheaves. This gives the first examples of stability conditions on the A-model side of mirror symmetry, where the triangulated category is not naturally the derived category of an abelian category. The existence of stability conditions, however, gives many such abelian categories, as predicted by mirror symmetry. In our examples in 2 dimensions, we completely describe a connected component of the space of stability conditions. It is the universal cover of the configuration space C k+1 of k + 1 points in C with centre of mass zero, with deck transformations the braid group action of [10, 15]. This gives a geometric origin for these braid group actions and their faithfulness, and axiomatises the proposal for stability of Lagrangians in [18] and the example proved by mean curvature flow in [19].
TL;DR: In this paper, the homological mirror conjecture for toric del Pezzo surfaces was shown to coincide with the derived Fukaya category of coherent sheaves on the original manifold, where the mirror object is a regular function on an algebraic torus.
Abstract: We prove the homological mirror conjecture for toric del Pezzo surfaces. In this case, the mirror object is a regular function on an algebraic torus Open image in new window We show that the derived Fukaya category of this mirror coincides with the derived category of coherent sheaves on the original manifold.
TL;DR: In this paper, the authors revisited open string mirror symmetry for the elliptic curve, using matrix factorizations for describing D-branes on the B-model side, and showed how flat coordinates can be intrinsically defined in the Landau-Ginzburg model, and derived the A-model partition function counting disk instantons that stretch between three Dbranes.
Abstract: We revisit open string mirror symmetry for the elliptic curve, using matrix factorizations for describing D-branes on the B-model side. We show how flat coordinates can be intrinsically defined in the Landau-Ginzburg model, and derive the A-model partition function counting disk instantons that stretch between three D-branes. In mathematical terms, this amounts to computing the simplest Fukaya product m2 from the LG mirror theory. In physics terms, this gives a systematic method for determining non-perturbative Yukawa couplings for intersecting brane configurations.
14 Dec 2006
TL;DR: In this paper, an interpretation of mirror symmetry for toric varieties via an equivalence of two conformal field theories is proposed, namely the twisted sigma model of a toric variety in the infinite volume limit (the A-model) and the intermediate model, which is called the I-model.
Abstract: We suggest an interpretation of mirror symmetry for toric varieties via an equivalence of two conformal field theories. The first theory is the twisted sigma model of a toric variety in the infinite volume limit (the A–model). The second theory is an intermediate model, which we call the I–model. The equivalence between the A–model and the I–model is achieved by realizing the former as a deformation of a linear sigma model with a complex torus as the target and then applying to it a version of the T–duality. On the other hand, the I–model is closely related to the twisted Landau-Ginzburg model (the B–model) that is mirror dual to the A–model. Thus, the mirror symmetry is realized in two steps, via the I–model. In particular, we obtain a natural interpretation of the superpotential of the Landau-Ginzburg model as the sum of terms corresponding to the components of a divisor in the toric variety. We also relate the cohomology of the supercharges of the I–model to the chiral de Rham complex and the quantum cohomology of the underlying toric variety.
TL;DR: In this paper, it has been shown that quantities involved in stability conditions for topological D-branes, and containing gauge fields in their expressions, are exchanged by mirror symmetry, which can be considered as an open-string version of the mirror symmetry between pure spinors.
TL;DR: In this paper, it was shown that the Kondo effect may be suppressed under certain conditions in triple quantum dots with mirror symmetry at odd electron occupation, and that the indirect exchange has a ferromagnetic sign in the ground state of three quantum dots in a two-terminal cross geometry for electron occupation $N=3.
Abstract: Indirect exchange interactions between itinerant electrons and nanostructures with nontrivial geometrical configurations manifest a plethora of unexpected results. These configurations can be realized either in quantum dots with several potential valleys or in real complex molecules with strong correlations. Here we demonstrate that the Kondo effect may be suppressed under certain conditions in triple quantum dots with mirror symmetry at odd electron occupation. First, we show that the indirect exchange has a ferromagnetic sign in the ground state of triple quantum dots in a two-terminal cross geometry for electron occupation $N=3$. Second, we show that for electron occupation $N=1$ in three-terminal fork geometry the zero-bias anomaly in the tunnel conductance is absent (despite the presence of Kondo screening) due to the special symmetry of the dot wave function.
TL;DR: In this article, the mirror map is constructed explicitly for a special class of target spaces and the topological A and B model are shown to be mirror pairs in the sense that the observables, the instantons and the anomalies are mapped to each other.
Abstract: We consider topological sigma models with generalized Kahler target spaces. The mirror map is constructed explicitly for a special class of target spaces and the topological A and B model are shown to be mirror pairs in the sense that the observables, the instantons and the anomalies are mapped to each other. We also apply the construction to open topological models and show that A branes are mapped to B branes. Furthermore, we demonstrate a relation between the field strength on the brane and a two-vector on the mirror manifold.
Book•
19 Apr 2006TL;DR: Warming up to enumerative geometry Enumerative geometry of lines Excess intersection Rational curves on the quintic threefold Mechanics Introduction to supersymmetry Introduction to string theory Topological quantum field theory Quantum cohomology as discussed by the authors.
Abstract: Warming up to enumerative geometry Enumerative geometry in the projective plane Stable maps and enumerative geometry Crash course in topology and manifolds Crash course in $C^\infty$ manifolds and cohomology Cellular decompositions and line bundles Enumerative geometry of lines Excess intersection Rational curves on the quintic threefold Mechanics Introduction to supersymmetry Introduction to string theory Topological quantum field theory Quantum cohomology and enumerative geometry Bibliography Index.
CERN1
TL;DR: In this paper, a formalism inspired by matrix models was proposed to compute open and closed topological string amplitudes in the B-model on toric Calabi-Yau manifolds.
Abstract: We propose a formalism inspired by matrix models to compute open and closed topological string amplitudes in the B-model on toric Calabi-Yau manifolds. We find closed expressions for various open string amplitudes beyond the disk, and in particular we write down the annulus amplitude in terms of theta functions on a Riemann surface. We test these ideas on local curves and local surfaces, providing in this way generating functionals for open Gromov-Witten invariants in the spirit of mirror symmetry. In the case of local curves, we study the open string sector near the critical point which leads to 2d gravity, and we show that toric D-branes become FZZT branes in a double-scaling limit. We use this connection to compute non-perturbative instanton effects due to D-branes that control the large order behavior of topological string theory on these backgrounds
TL;DR: In this paper, a class of local Calabi?Yau supermanifolds with bosonic sub-variety Vb having a vanishing first Chern class was studied and explicit results were given for local Ar supergeometries.
Abstract: We use local mirror symmetry to study a class of local Calabi?Yau supermanifolds with bosonic sub-variety Vb having a vanishing first Chern class. Solving the usual super-CY condition, requiring the equality of the total U(1) gauge charges of bosons ?b and the ghost-like fields ?f one ?bqb = ?fQf, as ?bqb = 0 and ?fQf = 0, several examples are studied and explicit results are given for local Ar supergeometries. A comment on purely fermionic super-CY manifolds corresponding to the special case where qb = 0, ?b and ?fQf = 0 is also made.
14 Dec 2006
TL;DR: In this article, an analysis of weakly coupled two-dimensional spatial optical solitons in a large-aperture class A laser with a saturable absorber is developed.
Abstract: An analysis of clusters of weakly coupled two-dimensional spatial optical solitons in a large-aperture class A laser with a saturable absorber is developed. The symmetries that control the transverse motion of the clusters are described. Numerical solutions of the governing generalized complex Ginzburg-Landau equation demonstrate the existence of four types of clusters of weakly coupled cavity solitons that correspond to symmetries of transverse intensity distributions and energy flows: (1) stationary (with two mirror symmetry axes), (2) rotating about a stationary center of mass (invariant under rotation), (3) translating without rotation (with a single mirror symmetry axis), and (4) asymmetric ones rotating about a center of mass that moves around a circle (with equal periods of rotation and circular motion).
TL;DR: In this article, the authors study the inequivalent quantizations of the N = 3 Calogero model by separation of variables, in which the model decomposes into the angular and the radial parts.
Abstract: We study the inequivalent quantizations of the N=3 Calogero model by separation of variables, in which the model decomposes into the angular and the radial parts. Our inequivalent quantizations respect the “mirror-S3” invariance (which realizes the symmetry under the cyclic permutations of the particles) and the scale invariance in the limit of vanishing harmonic potential. We find a two-parameter family of novel quantizations in the angular part and classify the eigenstates in terms of the irreducible representations of the S3 group. The scale invariance restricts the quantization in the radial part uniquely, except for the eigenstates coupled to the lowest two angular levels for which two types of boundary conditions are allowed independently from all upper levels. It is also found that the eigenvalues corresponding to the singlet representations of the S3 are universal (parameter-independent) in the family, whereas those corresponding to the doublets of the S3 are dependent on one of the parameters. Th...
TL;DR: In this paper, the congruence relation for the number of rational points on a quotient mirror pair of Calabi-Yau varieties over finite fields was studied, and the main result was that rational points in a mirror pair can be represented by a fixed number of points.
Abstract: One of the basic problems in arithmetic mirror symmetry is to compare the number of rational points on a mirror pair of Calabi-Yau varieties. At present, no general algebraic geometric definition is known for a mirror pair. But an important class of mirror pairs comes from certain quotient construction. In this paper, we study the congruence relation for the number of rational points on a quotient mirror pair of varieties over finite fields. Our main result is the following theorem:
TL;DR: In this paper, D-branes in the mirror were extended in T2/4 using the mirror description as a tensor product of minimal models and matrix factorizations in the corresponding Landau-Ginzburg model.
Abstract: We study D-branes extended in T2/4 using the mirror description as a tensor product of minimal models. We describe branes in the mirror both as boundary states in minimal models and as matrix factorizations in the corresponding Landau-Ginzburg model. We isolate a minimal set of branes and give a geometric interpretation of these as D1-branes constrained to the orbifold fixed points. This picture is supported both by spacetime arguments and by the explicit construction of the boundary states, adapting the known results for rational boundary states in the minimal models. Similar techniques apply to a larger class of toroidal orbifolds.