Showing papers on "Mirror symmetry published in 2011"

From real affine geometry to complex geometry

Abstract: We construct from a real ane 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 that 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.

Mirror symmetry for log Calabi-Yau surfaces I

Abstract: We give a canonical synthetic construction of the mirror family to a pair (Y,D) of a smooth projective surface with an anti-canonical cycle of rational curves, as the spectrum of an explicit algebra defined in terms of counts of rational curves on Y meeting D in a single point. In the case D is contractible, the family gives a smoothing of the dual cusp, and thus a proof of Looijenga's 1981 cusp conjecture.

Torus knots and mirror symmetry

Abstract: We propose a spectral curve describing torus knots and links in the B-model. In particular, the application of the topological recursion to this curve generates all their colored HOMFLY invariants. The curve is obtained by exploiting the full Sl(2, Z) symmetry of the spectral curve of the resolved conifold, and should be regarded as the mirror of the topological D-brane associated to torus knots in the large N Gopakumar-Vafa duality. Moreover, we derive the curve as the large N limit of the matrix model computing torus knot invariants.

Superconformal indices of three-dimensional theories related by mirror symmetry

Abstract: Recently, Kim, and Imamura and Yokoyama derived an exact formula for superconformal indices in three-dimensional field theories. Using their results, we prove analytically the equality of superconformal indices in some U(1)-gauge group theories related by mirror symmetry. The proofs are based on well-known identities in the theory of q-special functions. We also suggest a general index formula taking into account the U(1) J global symmetry present for abelian theories.

Tropical Geometry And Mirror Symmetry

Abstract: The three worlds: The tropics The A- and B-models Log geometry Example: $\mathbb{P}^2$: Mikhalkin's curve counting formula Period integrals The Gross-Siebert program: The program and two-dimensional results Bibliography Index of symbols General index

Gapless interface states between topological insulators with opposite Dirac velocities.

Abstract: The Dirac cone on a surface of a topological insulator shows linear dispersion analogous to optics and its velocity depends on materials. We consider a junction of two topological insulators with different velocities, and calculate the reflectance and transmittance. We find that they reflect the backscattering-free nature of the helical surface states. When the two velocities have opposite signs, both transmission and reflection are prohibited for normal incidence, when a mirror symmetry normal to the junction is preserved. In this case we show that there necessarily exist gapless states at the interface between the two topological insulators. Their existence is protected by mirror symmetry, and they have characteristic dispersions depending on the symmetry of the system.

to complex geometry

Abstract: We construct from a real ane 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 that 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.

Collapsing of abelian fibred Calabi-Yau manifolds

Abstract: We study the collapsing behaviour of Ricci-flat Kahler metrics on a projective Calabi-Yau manifold which admits an abelian fibration, when the volume of the fibers approaches zero. We show that away from the critical locus of the fibration the metrics collapse with locally bounded curvature, and along the fibers the rescaled metrics become flat in the limit. The limit metric on the base minus the critical locus is locally isometric to an open dense subset of any Gromov-Hausdorff limit space of the Ricci-flat metrics. We then apply these results to study metric degenerations of families of polarized hyperkahler manifolds in the large complex structure limit. In this setting we prove an analog of a result of Gross-Wilson for K3 surfaces, which is motivated by the Strominger-Yau-Zaslow picture of mirror symmetry.

SYZ Mirror Symmetry for Toric Calabi-Yau Manifolds

Abstract: We investigate mirror symmetry for toric Calabi-Yau manifolds from the perspective of the SYZ conjecture. Starting with a non-toric special Lagrangian torus fibration on a toric Calabi-Yau manifold $X$, we construct a complex manifold $\check{X}$ using T-duality modified by quantum corrections. These corrections are encoded by Fourier transforms of generating functions of certain open Gromov-Witten invariants. We conjecture that this complex manifold $\check{X}$, which belongs to the Hori-Iqbal-Vafa mirror family, is inherently written in canonical flat coordinates. In particular, we obtain an enumerative meaning for the (inverse) mirror maps, and this gives a geometric reason for why their Taylor series expansions in terms of the Kahler parameters of $X$ have integral coefficients. Applying the results in \cite{Chan10} and \cite{LLW10}, we compute the open Gromov-Witten invariants in terms of local BPS invariants and give evidences of our conjecture for several 3-dimensional examples including $K_{\proj^2}$ and $K_{\proj^1\times\proj^1}$.

Fivebrane instantons, topological wave functions and hypermultiplet moduli spaces

Abstract: We investigate quantum corrections to the hypermultiplet moduli space \( \mathcal{M} \) in Calabi-Yau compactifications of type II string theories, with particular emphasis on instanton effects from Euclidean NS5-branes. Based on the consistency of D- and NS5-instanton corrections, we determine the topology of the hypermultiplet moduli space at fixed string coupling, as previewed in [1]. On the type IIB side, we compute corrections from (p, k)-fivebrane instantons to the metric on \( \mathcal{M} \) (specifically, the correction to the complex contact structure on its twistor space \( \mathcal{Z} \)) by applying S-duality to the D-instanton sum. For fixed fivebrane charge k, the corrections can be written as a non-Gaussian theta series, whose summand for k = 1 reduces to the topological A-model amplitude. By mirror symmetry, instanton corrections induced from the chiral type IIA NS5-brane are similarly governed by the wave function of the topological B-model. In the course of this investigation we clarify charge quantization for coherent sheaves and find hitherto unnoticed corrections to the Heisenberg, monodromy and S-duality actions on \( \mathcal{M} \), as well as to the mirror map for Ramond-Ramond fields and D-brane charges.

The effective action of D6-branes in N=1 type IIA orientifolds

Abstract: We use a Kaluza-Klein reduction to compute the low-energy effective action for the massless modes of a spacetime-filling D6-brane wrapped on a special Lagrangian 3-cycle of a type IIA Calabi-Yau orientifold. The modifications to the characteristic data of the N=1 bulk orientifold theory in the presence of a D6-brane are analysed by studying the underlying Type IIA supergravity coupled to the brane worldvolume in the democratic formulation and performing a detailed dualisation procedure. The N=1 chiral coordinates are found to be in agreement with expectations from mirror symmetry. We work out the Kahler potential for the chiral superfields as well as the gauge kinetic functions for the bulk and the brane gauge multiplets including the kinetic mixing between the two. The scalar potential resulting from the dualisation procedure can be formally interpreted in terms of a superpotential. Finally, the gauging of the Peccei-Quinn shift symmetries of the complex structure multiplets reproduces the D-term potential enforcing the calibration condition for special Lagrangian 3-cycles.

The effective action of D6-branes in $ \mathcal{N} = 1 $ type IIA orientifolds

Abstract: We use a Kaluza-Klein reduction to compute the low-energy effective action for the massless modes of a spacetime-filling D6-brane wrapped on a special Lagrangian 3-cycle of a type IIA Calabi-Yau orientifold. The modifications to the characteristic data of the $ \mathcal{N} = 1 $ bulk orientifold theory in the presence of a D6-brane are analysed by studying the underlying Type IIA supergravity coupled to the brane world volume in the democratic formulation and performing a detailed dualisation procedure. The $ \mathcal{N} = 1 $ chiralcoordinates are found to be in agreement with expectations from mirror symmetry. We work out the Kahler potential for the chiral superfields as well as the gauge kinetic functions for the bulk and the brane gauge multiplets including the kinetic mixing between the two. The scalar potential resulting from the dualisation procedure can be formally interpreted in terms of a superpotential. Finally, the gauging of the Peccei-Quinn shift symmetries of the complex structure multiplets reproduces the D-term potential enforcing the calibration condition for special Lagrangian 3-cycles.

Homological mirror symmetry for Brieskorn–Pham singularities

Abstract: We prove that the derived Fukaya category of the Lefschetz fibration defined by a Brieskorn–Pham polynomial is equivalent to the triangulated category of singularities associated with the same polynomial together with a grading by an abelian group of rank one. Symplectic Picard-Lefschetz theory developed by Seidel is an essential ingredient of the proof.

Strange duality of weighted homogeneous polynomials

Abstract: We consider a mirror symmetry between invertible weighted homogeneous polynomials in three variables. We define Dolgachev and Gabrielov numbers for them and show that we get a duality between these polynomials generalizing Arnold’s strange duality between the 14 exceptional unimodal singularities.

Generalized geometry, calibrations and supersymmetry in diverse dimensions

Abstract: We consider type II backgrounds of the form $$ {\mathbb{R}^{1,d - 1}} \times {\mathcal{M}_{10 - d}} $$ for even d, preserving 2 d/2 real supercharges; for d = 4, 6, 8 this is minimal supersymmetry in d dimensions, while for d = 2 it is $$ \mathcal{N} = \left( {2,0} \right) $$ supersymmetry in two dimensions. For d = 6 we prove, by explicitly solving the Killing-spinor equations, that there is a one-to-one correspondence between background supersymmetry equations in pure-spinor form and D-brane generalized calibrations; this correspondence had been known to hold in the d = 4 case. Assuming the correspondence to hold for all d, we list the calibration forms for all admissible D-branes, as well as the background supersymmetry equations in pure-spinor form. We find a number of general features, including the following: The pattern of codimensions at which each calibration form appears exhibits a (mod 4) periodicity. In all cases one of the pure-spinor equations implies that the internal manifold is generalized Calabi-Yau. Our results are manifestly invariant under generalized mirror symmetry.

Quantum Cohomology and Periods

Abstract: In a previous paper, the author introduced a Z-structure in quantum cohomology defined by the K-theory and the Gamma class and showed that it is compatible with mirror symmetry for toric orbifolds. Applying the quantum Lefschetz principle to the previous results, we find an explicit relationship between solutions to the quantum differential equation for toric complete intersections and the periods (or oscillatory integrals) of their mirrors. We describe in detail the mirror isomorphism of variations of Z-Hodge structure for a mirror pair of Calabi-Yau hypersurfaces (Batyrev's mirror).

Complete Intersection Moduli Spaces in N=4 Gauge Theories in Three Dimensions

Abstract: We study moduli spaces of a class of three dimensional N=4 gauge theories which are in one-to-one correspondence with a certain set of ordered pairs of integer partitions. It was found that these theories can be realised on brane intervals in Type IIB string theory and can therefore be described using linear quiver diagrams. Mirror symmetry was known to act on such a theory by exchanging the partitions in the corresponding ordered pair, and hence the quiver diagram of the mirror theory can be written down in a straightforward way. The infrared Coulomb branch of each theory can be studied using moment map equations for a hyperKahler quotient of the Higgs branch of the mirror theory. We focus on three infinite subclasses of these singular hyperKahler spaces which are complete intersections. The Hilbert series of these spaces are computed in order to count generators and relations, and they turn out to be related to the corresponding partitions of the theories. For each theory, we explicitly discuss the generators of such a space and relations they satisfy in detail. These relations are precisely the defining equations of the corresponding complete intersection space.

Generalized mirror symmetry and trace anomalies

Abstract: We consider the compactification of M-theory on X7 with Betti numbers (b0, b1, b2, b3, b3, b2, b1, b0) and define a generalized mirror symmetry (b0, b1, b2, b3) → (b0, b1, b2 − ρ/2, b3 + ρ/2) under which ρ ≡ 7b0 − 5b1 + 3b2 − b3 changes sign. Generalized self-mirror theories with ρ = 0 have massless sectors with vanishing trace anomaly (before dualization). Examples include pure supergravity with and supergravity plus matter with .

The Berglund-H\"ubsch-Chiodo-Ruan mirror symmetry for K3 surfaces

Abstract: We prove that the mirror symmetry of Berglund-Hubsch-Chiodo-Ruan, applied to K3 surfaces with a non-symplectic involution, coincides with the mirror symmetry described by Dolgachev and Voisin.

Homological Mirror Symmetry for Calabi-Yau hypersurfaces in projective space

Abstract: We prove Homological Mirror Symmetry for a smooth d-dimensional Calabi-Yau hypersurface in projective space, for any d > 2 (for example, d = 3 is the quintic three-fold). The main techniques involved in the proof are: the construction of an immersed Lagrangian sphere in the `d-dimensional pair of pants'; the introduction of the `relative Fukaya category', and an understanding of its grading structure; a description of the behaviour of this category with respect to branched covers (via an `orbifold' Fukaya category); a Morse-Bott model for the relative Fukaya category that allows one to make explicit computations; and the introduction of certain graded categories of matrix factorizations mirror to the relative Fukaya category.

Algebraic $K$-theory of toric hypersurfaces

Abstract: We construct classes in the motivic cohomology of certain 1-parameter families of Calabi–Yau hypersurfaces in toric Fano n-folds, with applications to local mirror symmetry (growth of genus 0 instanton numbers) and inhomogeneous Picard–Fuchs equations. In the case where the family is classically modular the classes are related to Beilinson’s Eisenstein symbol; the Abel–Jacobi map (or rational regulator) is computed in this paper for both kinds of cycles. For the “modular toric” families where the cycles essentially coincide, we obtain a motivic (and computationally effective) explanation of a phenomenon observed by Villegas, Stienstra, and Bertin.

Quantization via mirror symmetry

Abstract: When combined with mirror symmetry, the A-model approach to quantization leads to a fairly simple and tractable problem. The most inter- esting part of the problem then becomes finding the mirror of the coisotropic brane. We illustrate how it can be addressed in a number of interesting ex- amples related to representation theory and gauge theory, in which mirror geometry is naturally associated with the Langlands dual group. Hyperholo- morphic sheaves and (B,B,B) branes play an important role in the B-model approach to quantization.

Type II/F-theory superpotentials with several deformations and \mathcal{N} = 1 mirror symmetry

Abstract: We present a detailed study of D-brane superpotentials depending on several open and closed-string deformations. The relative cohomology group associated with the brane defines a generalized hypergeometric GKZ system which determines the off-shell superpotential and its analytic properties under deformation. Explicit expressions for the $ \mathcal{N} = 1 $ superpotential for families of type II/F-theory compactifications are obtained for a list of multi-parameter examples. Using the Hodge theoretic approach to open-string mirror symmetry, we obtain new predictions for integral disc invariants in the A model instanton expansion. We study the behavior of the brane vacua under extremal transitions between different Calabi-Yau spaces and observe that the web of Calabi-Yau vacua remains connected for a particular class of branes.

Mirror symmetry and projective geometry of Reye congruences I

Abstract: Studying the mirror symmetry of a Calabi-Yau threefold $X$ of the Reye congruence in $\mP^4$, we conjecture that $X$ has a non-trivial Fourier-Mukai partner $Y$. We construct $Y$ as the double cover of a determinantal quintic in $\mP^4$ branched over a curve. We also calculate BPS numbers of both $X$ and $Y$ (and also a related Calabi-Yau complete intersection $\tilde X_0$) using mirror symmetry.

Noncommutative mirror symmetry for punctured surfaces

Abstract: Recently Abouzaid, Auroux, Efimov, Katzarkov and Orlov showed that the wrapped Fukaya Categories of punctured spheres and finite unbranched covers of punctured spheres are derived equivalent to the categories of singularities of a superpotential on certain crepant resolutions of toric 3 dimensional singularities. We generalize this result to other punctured Riemann surfaces and reformulate it in terms of certain noncommutative algebras coming from dimer models. In particular, given any consistent dimer model we can look at a subcategory of noncommutative matrix factorizations and show that this category is $A_\infty$-isomorphic to a subcategory of the wrapped Fukaya category of a punctured Riemann surface. The connection between the dimer model and the punctured Riemann surface then has a nice interpretation in terms of a duality on dimer models.

Five-brane superpotentials, blow-up geometries and SU(3) structure manifolds

Abstract: We investigate the dynamics of space-time filling five-branes wrapped on curves in heterotic and orientifold Calabi-Yau compactifications. We first study the leading $ \mathcal{N} = 1 $ scalar potential on the infinite deformation space of the brane-curve around a supersymmetric configuration. The higher order potential is also determined by a brane superpotential which we compute for a subset of light deformations. We argue that these deformations map to new complex structure deformations of a non-Calabi-Yau manifold which is obtained by blowing up the brane-curve into a four-cycle and by replacing the brane by background fluxes. This translates the original brane-bulk system into a unifying geometrical formulation. Using this blow-up geometry we compute the complete set of open-closed Picard-Fuchs differential equations and identify the brane superpotential at special points in the field space for five-branes in toric Calabi-Yau hypersurfaces. This has an interpretation in open mirror symmetry and enables us to list compact disk instanton invariants. As a first step towards promoting the blow-up geometry to a supersymmetric heterotic background we propose a non-Kahler SU(3) structure and an identification of the three-form flux.

Gromov--Witten invariants for mirror orbifolds of simple elliptic singularities

Abstract: We consider a mirror symmetry of simple elliptic singularities. In particular, we construct isomorphisms of Frobenius manifolds among the one from the Gromov--Witten theory of a weighted projective line, the one from the theory of primitive forms for a universal unfolding of a simple elliptic singularity and the one from the invariant theory for an elliptic Weyl group. As a consequence, we give a geometric interpretation of the Fourier coefficients of an eta product considered by K. Saito.

The Influence of Perception on the Distribution of Multiple Symmetries in Nature and Art

Abstract: Much is already known about single mirror symmetry, but multiple mirror symmetry is still understood poorly. In particular, perceptually, multiple symmetry does not seem to behave as suggested by the number of symmetry axes alone. Here, theoretical ideas on single symmetry perception and their extensions to multiple symmetry are discussed alongside empirical findings on multiple symmetry perception. The evidence suggests that, apart from the number of axes, also their relative orientation is perceptually relevant. This, in turn, suggests that perception is responsible for the preponderance of 3-fold and 5-fold symmetries in flowers as well as for their absence in decorative art.

Symmetries in peripheral ocular aberrations

Abstract: A mirror symmetry in the aberrations between the left and right eyes has previously been found foveally, but while a similar symmetry for the peripheral visual field is likely, it has not been inve ...