Physmatics

Public lecture given at The Fields Institute, June 2, 2005

Seidel's Mirror Map for the Torus

Using only the Fukaya category and the monodromy around large complex structure, we reconstruct the mirror map in the case of a symplectic torus. This realizes an idea described by Paul Seidel.Comment: 6 pages, some typos and references fixe

Local Mirror Symmetry at Higher Genus

We discuss local mirror symmetry for higher-genus curves. Specifically, we consider the topological string partition function of higher-genus curves contained in a Fano surface within a Calabi-Yau. Our main example is the local P^2 case. The Kodaira-Spencer theory of gravity, tailored to this local geometry, can be solved to compute this partition function. Then, using the results of Gopakumar and Vafa and the local mirror map, the partition function can be rewritten in terms of expansion coefficients, which are found to be integers. We verify, through localization calculations in the A-model, many of these Gromov-Witten predictions. The integrality is a mystery, mathematically speaking. The asymptotic growth (with degree) of the invariants is analyzed. Some suggestions are made towards an enumerative interpretation, following the BPS-state description of Gopakumar and Vafa.Comment: 29 pages, 4 figures, references and some genus 5 results adde

Categorical Mirror Symmetry: The Elliptic Curve

We describe an isomorphism of categories conjectured by Kontsevich. If $M$ and $\widetilde{M}$ are mirror pairs then the conjectural equivalence is between the derived category of coherent sheaves on $M$ and a suitable version of Fukaya's category of Lagrangian submanifolds on $\widetilde{M}.$ We prove this equivalence when $M$ is an elliptic curve and $\widetilde{M}$ is its dual curve, exhibiting the dictionary in detail.Comment: harvmac, 29 pages (big); updated with correction

Open-String Gromov-Witten Invariants: Calculations and a Mirror "Theorem"

We propose localization techniques for computing Gromov-Witten invariants of maps from Riemann surfaces with boundaries into a Calabi-Yau, with the boundaries mapped to a Lagrangian submanifold. The computations can be expressed in terms of Gromov-Witten invariants of one-pointed maps. In genus zero, an equivariant version of the mirror theorem allows us to write down a hypergeometric series, which together with a mirror map allows one to compute the invariants to all orders, similar to the closed string model or the physics approach via mirror symmetry. In the noncompact example where the Calabi-Yau is K_{\PP^2}, our results agree with physics predictions at genus zero obtained using mirror symmetry for open strings. At higher genera, our results satisfy strong integrality checks conjectured from physics.Comment: 20 page

Seidel's Mirror Map for Abelian Varieties

We compute Seidel's mirror map for abelian varieties by constructing the homogeneous coordinate rings from the Fukaya category of the symplectic mirrors. The computations are feasible as only linear holomorphic disks contribute to the Fukaya composition in the case of the planar Lagrangians used. The map depends on a symplectomorphism $\rho$ representing the large complex structure monodromy. For the example of the two-torus, different families of elliptic curves are obtained by using different $\rho$ which are linear in the universal cover. In the case where $\rho$ is merely affine linear in the universal cover, the commutative elliptic curve mirror is embedded in noncommutative projective space. The case of Kummer surfaces is also considered.Comment: 12 pages, final versio

Constructible Sheaves and the Fukaya Category

Let $X$ be a compact real analytic manifold, and let $T^*X$ be its cotangent bundle. Let $Sh(X)$ be the triangulated dg category of bounded, constructible complexes of sheaves on $X$. In this paper, we develop a Fukaya $A_\infty$-category $Fuk(T^*X)$ whose objects are exact, not necessarily compact Lagrangian branes in the cotangent bundle. We write $Tw Fuk(T^*X)$ for the $A_\infty$-triangulated envelope of $Fuk(T^*X)$ consisting of twisted complexes of Lagrangian branes. Our main result is that $Sh(X)$ quasi-embeds into $Tw Fuk(T^*X)$ as an $A_\infty$-category. Taking cohomology gives an embedding of the corresponding derived categories.Comment: 56 pages; to appear in JAM

Ribbon Graphs and Mirror Symmetry I

Given a ribbon graph $\Gamma$ with some extra structure, we define, using constructible sheaves, a dg category $CPM(\Gamma)$ meant to model the Fukaya category of a Riemann surface in the cell of Teichm\"uller space described by $\Gamma.$ When $\Gamma$ is appropriately decorated and admits a combinatorial "torus fibration with section," we construct from $\Gamma$ a one-dimensional algebraic stack $\widetilde{X}_\Gamma$ with toric components. We prove that our model is equivalent to $Perf(\widetilde{X}_\Gamma)$, the dg category of perfect complexes on $\widetilde{X}_\Gamma$.Comment: 28 page

Legendrian knots and constructible sheaves

We study the unwrapped Fukaya category of Lagrangian branes ending on a Legendrian knot. Our knots live at contact infinity in the cotangent bundle of a surface, the Fukaya category of which is equivalent to the category of constructible sheaves on the surface itself. Consequently, our category can be described as constructible sheaves with singular support controlled by the front projection of the knot. We use a theorem of Guillermou-Kashiwara-Schapira to show that the resulting category is invariant under Legendrian isotopies, and conjecture it is equivalent to the representation category of the Chekanov-Eliashberg differential graded algebra. We also find two connections to topological knot theory. First, drawing a positive braid closure on the annulus, the moduli space of rank-n objects maps to the space of local systems on a circle. The second page of the spectral sequence associated to the weight filtration on the pushforward of the constant sheaf is the (colored-by-n) triply-graded Khovanov-Rozansky homology. Second, drawing a positive braid closure in the plane, the number of points of our moduli spaces over a finite field with q elements recovers the lowest coefficient in 'a' of the HOMFLY polynomial of the braid closure.Comment: 92 pages, final journal version, Inventiones Mathematicae (2016

Wavefunctions for a Class of Branes in Three-space

Wavefunctions are proposed for a class of Lagrangian branes in three complex-dimensional space. The branes are asymptotic to Legendrian surfaces of genus g. The expansion of these wavefunctions in appropriate coordinates conjecturally encodes all-genus open Gromov-Witten invariants, i.e. the free energy of the topological open string. This paper is written in physics language, but tries to welcome mathematicians. Most results stem from joint mathematical works with Linhui Shen and David Treumann.Comment: 10 page