Lagrangian Correspondences Research Articles

Abelian TQFTS and Schrödinger local systems

In this paper, we construct an action of 3-cobordisms on the finite dimensional Schrödinger representations of the Heisenberg group by Lagrangian correspondences. In addition, we review the construction of the abelian Topological Quantum Field Theory (TQFT) associated with a q -deformation of U(1) for any root of unity q . We prove that, for 3-cobordisms compatible with Lagrangian correspondences, there is a normalization of the associated Schrödinger bimodule action that reproduces the abelian TQFT.The full abelian TQFT provides a projective representation of the mapping class group \mathrm{Mod}(\Sigma) on the Schrödinger representation, which is linearizable at odd root of 1. Motivated by homology of surface configurations with Schrödinger representation as local coefficients, we define another projective action of \mathrm{Mod}(\Sigma) on Schrödinger representations. We show that the latter is not linearizable by identifying the associated 2-cocycle.

Functoriality in categorical symplectic geometry

Categorical symplectic geometry is the study of a rich collection of invariants of symplectic manifolds, including the Fukaya A ∞ A_\infty -category, Floer cohomology, and symplectic cohomology. Beginning with work of Wehrheim and Woodward in the late 2000s, several authors have developed techniques for functorial manipulation of these invariants. We survey these functorial structures, including Wehrheim and Woodward’s quilted Floer cohomology and functors associated to Lagrangian correspondences, Fukaya’s alternate approach to defining functors between Fukaya A ∞ A_\infty -categories, and the second author’s ongoing construction of the symplectic ( A ∞ , 2 ) (A_\infty ,2) -category. In the last section, we describe a number of direct and indirect applications of this circle of ideas, and propose a conjectural version of the Barr–Beck monadicity criterion in the context of the Fukaya A ∞ A_\infty -category.

Categories of Lagrangian Correspondences in Super Hilbert Spaces and Fermionic Functorial Field Theory

In this paper, we study Lagrangian correspondences between Hilbert spaces. A main focus is the question when the composition of two Lagrangian correspondences is again Lagrangian. Our answer leads in particular to a well-defined composition law in a category of Lagrangian correspondences respecting given polarizations of the Hilbert spaces involved. As an application, we construct a functorial field theory on geometric spin manifolds with values in this category of Lagrangian correspondences, which can be viewed as a formal Wick rotation of the theory associated to a free fermionic particle in a curved spacetime.

Topological defects as Lagrangian correspondences

Topological defects attract much recent interest in high-energy and condensed matter physics, because they encode (noninvertible) symmetries and dualities. We study codimension-1 topological defects from a Hamiltonian point of view, with the defect location playing the role of ``time.'' We show that the Weinstein symplectic category governs topological defects and their fusion: Each defect is a Lagrangian correspondence, and defect fusion is their geometric composition. We illustrate the utility of these ideas by constructing S- and T-duality defects in string theory, including a novel topology-changing non-Abelian T-duality defect.

Explicit constructions of quilts with seam condition coming from symplectic reduction

Associated to a symplectic quotient $M/\!/G$ is a Lagrangian correspondence $\Lambda_G$ from $M/\!/G$ to $M$. In this note, we construct in two examples quilts with seam condition on such a correspondence, in the case of $S^1$ acting on $\mathbb{CP}^2$ with symplectic quotient $\mathbb{CP}^2/\!/ S^1 = \mathbb{CP}^1$. First, we study the quilted strips that would, if not for figure eight bubbling, identify the Floer chain groups $CF(\gamma,S_{\text{Cl}}^1)$ and $CF(\mathbb{RP}^2,T_{\text{Cl}}^2)$, where $\gamma$ is the connected double-cover of $\mathbb{RP}^1$. Second, we answer a question due to Akveld-Cannas da Silva-Wehrheim by explicitly producing a figure eight bubble which obstructs an isomorphism between two Floer chain groups. The figure eight bubbles we construct in this paper are the first concrete examples of this phenomenon.

Aspects of functoriality in homological mirror symmetry for toric varieties

We study homological mirror symmetry for toric varieties, exploring the relationship between various Fukaya-Seidel categories which have been employed for constructing the mirror to a toric variety. In particular, we realize tropical Lagrangian sections as objects of a partially wrapped category and construct a Lagrangian correspondence mirror to the inclusion of a toric divisor. As a corollary, we prove that tropical sections generate the Fukaya-Seidel category, completing a Floer-theoretic proof of homological mirror symmetry for projective toric varieties. In the course of the proof, we develop techniques for constructing Lagrangian cobordisms and Lagrangian correspondences in Liouville domains, which may be of independent interest.

Pseudoholomorphic quilts with figure eight singularity

We show that the novel figure eight singularity in a pseudoholomorphic quilt can be continuously removed when composition of Lagrangian correspondences is cleanly immersed. The proof of this result requires a collection of width-independent elliptic estimates that allow for non-standard complex structures on the domain.

$$A_\infty $$ A ∞ functors for Lagrangian correspondences

We construct $$A_\infty $$ functors between Fukaya categories associated to monotone Lagrangian correspondences between compact symplectic manifolds. We then show that the composition of $$A_\infty $$ functors for correspondences is homotopic to the functor for the composition, in the case that the composition is smooth and embedded.

Gromov compactness for squiggly strip shrinking in pseudoholomorphic quilts

We establish a Gromov compactness theorem for strip shrinking in pseudoholomorphic quilts when composition of Lagrangian correspondences is immersed. In particular, we show that figure eight bubbling occurs in the limit, argue that this is a codimension-0 effect, and predict its algebraic consequences—geometric composition extends to a curved $$A_\infty $$ -bifunctor, in particular the associated Floer complexes are isomorphic after a figure eight correction of the bounding cochain. An appendix with Felix Schmaschke provides examples of nontrivial figure eight bubbles.

Iterated spans and classical topological field theories

We construct higher categories of iterated spans, possibly equipped with extra structure in the form of "local systems", and classify their fully dualizable objects. By the Cobordism Hypothesis, these give rise to framed topological quantum field theories, which are the framed versions of the "classical" TQFTs considered in the quantization programme of Freed-Hopkins-Lurie-Teleman. Using this machinery, we also construct an infinity-category of Lagrangian correspondences between symplectic derived algebraic stacks and show that all its objects are fully dualizable.

Categorification of invariants in gauge theory and symplectic geometry

This is a mixture of survey article and research announcement. We discuss instanton Floer homology for 3 manifolds with boundary. We also discuss a categorification of the Lagrangian Floer theory using the unobstructed immersed Lagrangian correspondence as a morphism in the category of symplectic manifolds. During the year 1998–2012, those problems have been studied emphasizing the ideas from analysis such as degeneration and adiabatic limit (instanton Floer homology) and strip shrinking (Lagrangian correspondence). Recently we found that replacing those analytic approach by a combination of cobordism type argument and homological algebra, we can resolve various difficulties in the analytic approach. It thus solves various problems and also simplify many of the proofs.

Pseudoholomorphic quilts

We define relative Floer theoretic invariants arising from 'quilted pseudo-holomorphic surfaces': Collections of pseudoholomorphic maps to various target spaces with 'seam conditions' in Lagrangian correspondences. As application we construct a morphism on quantum homology associated to any monotone Lagrangian correspondence.

Three lectures on derived symplectic geometry and topological field theories

We give an informal introduction to the new field of derived symplectic geometry, and present some applications to topological field theories. We in particular try to explain that derived symplectic geometry provides a suitable framework for the so-called AKSZ construction (after Alexandrov–Kontsevich–Schwartz–Zaboronski). We start with a brief summary of the main features of derived algebraic geometry. We then continue with the definition of n-symplectic and Lagrangian structures, after Pantev–Toën–Vaquié–Vezzosi (PTVV), and provide examples such as •(for shifted symplectic structures) BG, [g∗/G], mapping stacks with symplectic target and “compact oriented” source.•(for Lagrangian structures) moment maps, quasi-Hamiltonian structures, mapping stacks with boundary conditions. We finally explain how this can be used to construct (fully extended) topological field theories with values in (higher) categories of Lagrangian correspondences.

Eulerian and Lagrangian Correspondence of High-Frequency Radar and Surface Drifter Data: Effects of Radar Resolution and Flow Components

AbstractThis study investigated the correspondence between the near-surface drifters from a mass drifter deployment near Martha’s Vineyard, Massachusetts, and the surface current observations from a network of three high-resolution, high-frequency radars to understand the effects of the radar temporal and spatial resolution on the resulting Eulerian current velocities and Lagrangian trajectories and their predictability. The radar-based surface currents were found to be unbiased in direction but biased in magnitude with respect to drifter velocities. The radar systematically underestimated velocities by approximately 2 cm s−1 due to the smoothing effects of spatial and temporal averaging. The radar accuracy, quantified by the domain-averaged rms difference between instantaneous radar and drifter velocities, was found to be about 3.8 cm s−1. A Lagrangian comparison between the real and simulated drifters resulted in the separation distances of roughly 1 km over the course of 10 h, or an equivalent separation speed of approximately 2.8 cm s−1. The effects of the temporal and spatial radar resolution were examined by degrading the radar fields to coarser resolutions, revealing the existence of critical scales (1.5–2 km and 3 h) beyond which the ability of the radar to reproduce drifter trajectories decreased more rapidly. Finally, the importance of the different flow components present during the experiment—mean, tidal, locally wind-driven currents, and the residual velocities—was analyzed, finding that, during the study period, a combination of tidal, locally wind-driven, and mean currents were insufficient to reliably reproduce, with minimal degradation, the trajectories of real drifters. Instead, a minimum combination of the tidal and residual currents was required.

Lagrangian correspondences and Donaldson’s TQFT construction of the Seiberg–Witten invariants of 3–manifolds

Using Morse-Bott techniques adapted to the gauge-theoretic setting, we show that the limiting boundary values of the space of finite energy monopoles on a connected 3-manifold with at least two cylindrical ends provides an immersed Lagrangian submanifold of the vortex moduli space at infinity. By studying the signed intersections of such Lagrangians, we supply the analytic details of Donaldson's TQFT construction of the Seiberg-Witten invariants of a closed 3-manifold.

Geometric composition in quilted Floer theory

We prove that Floer cohomology of cyclic Lagrangian correspondences is invariant under transverse and embedded composition under a general set of assumptions.

Quilted Floer trajectories with constant components: Corrigendum to the article “Quilted Floer cohomology”

We fill a gap in the proof of the transversality result for quilted Floer trajectories in [10] by addressing trajectories for which some but not all components are constant. Namely we show that for generic sets of split Hamiltonian perturbations and split almost complex structures, the moduli spaces of parametrized quilted Floer trajectories of a given index are smooth of expected dimension. An additional benefit of the generic split Hamiltonian perturbations is that they perturb the given cyclic Lagrangian correspondence such that any geometric composition of its factors is transverse and hence immersed. 53D40; 57R56

Floer cohomology and geometric composition of Lagrangian correspondences

We prove an isomorphism of Floer cohomologies under geometric composition of Lagrangian correspondences in exact and monotone settings.

Fusion categories and homotopy theory

We apply the yoga of classical homotopy theory to classification problems of G -extensions of fusion and braided fusion categories, where G is a finite group. Namely, we reduce such problems to classification (up to homotopy) of maps from BG to classifying spaces of certain higher groupoids. In particular, to every fusion category \mathcal C we attach the 3-groupoid \underline{\underline{\mathrm{BrPic}}}(\mathcal C) of invertible \mathcal C -bimodule categories, called the Brauer–Picard groupoid of \mathcal C , such that equivalence classes of G -extensions of \mathcal C are in bijection with homotopy classes of maps from BG to the classifying space of \underline{\underline{\mathrm{BrPic}}}(\mathcal C) . This gives rise to an explicit description of both the obstructions to existence of extensions and the data parametrizing them; we work these out both topologically and algebraically. One of the central results of the article is that the 2-truncation of \underline{\underline{\mathrm{BrPic}}}(\mathcal C) is canonically equivalent to the 2-groupoid of braided auto-equivalences of the Drinfeld center \mathcal Z(\mathcal C) of \mathcal C . In particular, this implies that the Brauer–Picard group \mathrm{BrPic}(\mathcal C) (i.e., the group of equivalence classes of invertible \mathcal C -bimodule categories) is naturally isomorphic to the group of braided auto-equivalences of \mathcal Z(\mathcal C) . Thus, if \mathcal C = \mathrm{Vec}_A , where A is a finite abelian group, then \mathrm{BrPic}(\mathcal C) is the orthogonal group \mathrm{O}(A \oplus A^*) . This allows one to obtain a rather explicit classification of extensions in this case; in particular, in the case G = \mathbb Z_2 , we re-derive (without computations) the classical result of Tambara and Yamagami. Moreover, we explicitly describe the category of all (\mathrm{Vec}_{A_1},\mathrm{Vec}_{A_2}) -bimodule categories (not necessarily invertible ones) by showing that it is equivalent to the hyperbolic part of the category of Lagrangian correspondences.

Functoriality for Lagrangian correspondences in Floer theory

We associate to every monotone Lagrangian correspondence a functor between Donaldson–Fukaya categories. The composition of such functors agrees with the functor associated to the geometric composition of the correspondences, if the latter is embedded. That is “categorification commutes with composition” for Lagrangian correspondences. This construction fits into a symplectic 2-category with a categorification 2-functor, in which all correspondences are composable, and embedded geometric composition is isomorphic to the actual composition. As a consequence, any functor from a bordism category to the symplectic category gives rise to a category valued topological field theory.