Searcharxiv⌕ Search

arXiv subjects

David Nadler

Publications and source records attributed to David Nadler.

At least 37 records · Page 2Linked to original sources
arXiv2022

Geomorphology of Lagrangian ridges

We prove an "h-principle without pre-conditions" for the elimination of tangencies of a Lagrangian submanifold with respect to a Lagrangian distribution. The main result states that such tangencies can always be completely removed at the cost of allowing the Lagrangian to develop certain non-smooth points, called Lagrangian ridges, modeled on the corner $\{p=|q|\} \subset \mathbb{R}^2$ together with its products and stabilizations. This result plays an essential role in the arborealization program.

math.SG↗
arXiv2022

Arboreal models and their stability

This is the first in a series of papers by the authors on the arborealization program. The main goal of the paper is the proof of uniqueness of arboreal models, defined as the closure of the class of smooth germs of Lagrangian submanifolds under the operation of taking iterated transverse Liouville cones. The parametric version of the stability result implies that the space of germs of symplectomorphisms that preserve a canonical model is weakly homotopy equivalent to the space of automorphisms of the corresponding signed rooted tree. Hence the local symplectic topology around a canonical model reduces to combinatorics, even parametrically.

math.SG↗
arXiv2021

A microlocal criterion for commuting nearby cycles

We present a microlocal criterion for the equivalence of iterated nearby cycles along different flags of subspaces in a higher-dimensional base. The result is motivated by its application to Hitchin systems in the context of Betti geometric Langlands. I thank Claude Sabbah for bringing papers of Maisonobe and Kocherperger to my attention. Among many interesting results, they include assertions implying the main assertions of this paper. I will leave this paper on the arXiv as a possibly useful additional reference.

math.RT↗
arXiv2021

Integral transforms for coherent sheaves

The theory of integral, or Fourier-Mukai, transforms between derived categories of sheaves is a well established tool in noncommutative algebraic geometry. General "representation theorems" identify all reasonable linear functors between categories of perfect complexes (or their "large" version, quasi-coherent sheaves) on schemes and stacks over some fixed base with integral kernels in the form of sheaves (of the same nature) on the fiber product. However, for many applications in mirror symmetry and geometric representation theory one is interested instead in the bounded derived category of coherent sheaves (or its "large" version, ind-coherent sheaves), which differs from perfect complexes (and quasi-coherent sheaves) once the underlying variety is singular. In this paper, we give general representation theorems for linear functors between categories of coherent sheaves over a base in terms of integral kernels on the fiber product. Namely, we identify coherent kernels with functors taking perfect complexes to coherent sheaves, and kernels which are coherent relative to the source with functors taking all coherent sheaves to coherent sheaves. The proofs rely on key aspects of the "functional analysis" of derived categories, namely the distinction between small and large categories and its measurement using $t$-structures. These are used in particular to correct the failure of integral transforms on Ind-coherent sheaves to correspond to such sheaves on a fiber product. The results are applied in a companion paper to the representation theory of the affine Hecke category, identifying affine character sheaves with the spectral geometric Langlands category in genus one.

math.AG↗
arXiv2021

Betti Spectral Gluing

Given a complex reductive group G, Borel subgroup B, and topological surface S with boundary dS, we study the "Betti spectral category" DCoh_N(Loc_G(S, dS)) of coherent sheaves with nilpotent singular support on the character stack of G-local systems on S with B-reductions along dS. Modifications along the components of dS endow this category with commuting actions of the affine Hecke category H_G in its realization as coherent sheaves on the Steinberg stack. We prove a "spectral Verlinde formula" identifying the result of gluing two boundary components with the Hochschild homology of the corresponding H_G-bimodule structure. The equivalence is compatible with Wilson line operators (the action of Perf(Loc_G(S)) realized by Hecke modifications at points) as well as Verlinde loop operators (the action of the center of H_G realized by Hecke modifications along closed loops). The result reduces the calculation of such "Betti spectral categories" to the case of disks, cylinders, pairs of pants, and the M"obius band.

math.AG↗
arXiv2020

Uniformization of semistable bundles on elliptic curves

Let $G$ be a connected reductive complex algebraic group, and $E$ a complex elliptic curve. Let $G_E$ denote the connected component of the trivial bundle in the stack of semistable $G$-bundles on $E$. We introduce a complex analytic uniformization of $G_E$ by adjoint quotients of reductive subgroups of the loop group of $G$. This can be viewed as a nonabelian version of the classical complex analytic uniformization $ E \simeq \mathbb{C}^*/q^{\mathbb{Z}}$. We similarly construct a complex analytic uniformization of $G$ itself via the exponential map, providing a nonabelian version of the standard isomorphism $\mathbb{C}^* \simeq \mathbb{C}/\mathbb{Z}$, and a complex analytic uniformization of $G_E$ generalizing the standard presentation $E = \mathbb{C}/(\mathbb{Z} \oplus \mathbb{Z} τ)$. Finally, we apply these results to the study of sheaves with nilpotent singular support. As an application to Betti geometric Langlands conjecture in genus 1, we define a functor from $Sh_\mathcal{N}(G_E)$ (the semistable part of the automorphic category) to ${IndCoh}_{\check{\mathcal{N}}}({Locsys}_{\check G} (E))$ (the spectral category).

math.RT↗
arXiv2020

Real and symmetric matrices

We construct a family of involutions on the space $\mathfrak{gl}_n'(\mathbb C)$ of $n\times n$ matrices with real eigenvalues interpolating the complex conjugation and the transpose. We deduce from it a stratified homeomorphism between the space of $n\times n$ real matrices with real eigenvalues and the space of $n\times n$ symmetric matrices with real eigenvalues, which restricts to a real analytic isomorphism between individual $\mathrm{GL}_n(\mathbb R)$-adjoint orbits and $\mathrm{O}_n(\mathbb C)$-adjoint orbits. We also establish similar results in more general settings of Lie algebras of classical types and quiver varieties. To this end, we prove a general result about involutions on hyper-Kähler quotients of linear spaces. We provide applications to the (generalized) Kostant-Sekiguchi correspondence, singularities of real and symmetric adjoint orbit closures, and Springer theory for real groups and symmetric spaces.

math.RT↗
arXiv2020

A stable infinity-category of Lagrangian cobordisms

Given an exact symplectic manifold M and a support Lagrangian Λ, we construct an infinity-category Lag, which we conjecture to be equivalent (after specialization of the coefficients) to the partially wrapped Fukaya category of M relative to Λ. Roughly speaking, the objects of Lag are Lagrangian branes inside of M x T*(R^n), for large n, and the morphisms are Lagrangian cobordisms that are non-characteristic with respect to Λ. The main theorem of this paper is that Lag is a stable infinity-category, so that its homotopy category is triangulated, with mapping cones given by an elementary construction. In particular, its shift functor is equivalent to the familiar shift of grading for Lagrangian branes.

math.SG↗
arXiv2019

Beilinson-Bernstein localization over the Harish-Chandra center

We present a simple proof of a strengthening of the derived Beilinson-Bernstein localization theorem using the formalism of descent in derived algebraic geometry. The arguments and results apply to arbitrary modules without the need to fix infinitesimal character. Roughly speaking, we demonstrate that all Ug-modules are the invariants, or equivalently coinvariants, of the action of intertwining functors (a refined form of Weyl group symmetry). This is a quantum version of descent for the Grothendieck-Springer simultaneous resolution. In an appendix we present an alternative perspective, which identifies the descent data in both classical and quantum versions as a categorical action of Demazure operators.

math.RT↗
arXiv2019

Mirror symmetry for honeycombs

We prove a homological mirror symmetry equivalence between an $A$-brane category for the pair of pants, computed as a wrapped microlocal sheaf category, and a $B$-brane category for a mirror LG model, understood as a category of matrix factorizations. The equivalence improves upon prior results in two ways: it intertwines evident affine Weyl group symmetries on both sides, and it exhibits the relation of wrapped microlocal sheaves along different types of Lagrangian skeleta for the same hypersurface. The equivalence proceeds through the construction of a combinatorial realization of the $A$-model via arboreal singularities. The constructions here represent the start of a program to generalize to higher dimensions many of the structures which have appeared in topological approaches to Fukaya categories of surfaces.

math.SG↗
arXiv2019

Nonlinear Traces

We combine the theory of traces in homotopical algebra with sheaf theory in derived algebraic geometry to deduce general fixed point and character formulas. The formalism of dimension (or Hochschild homology) of a dualizable object in the context of higher algebra provides a unifying framework for classical notions such as Euler characteristics, Chern characters, and characters of group representations. Moreover, the simple functoriality properties of dimensions clarify celebrated identities and extend them to new contexts. We observe that it is advantageous to calculate dimensions, traces and their functoriality directly in the nonlinear geometric setting of correspondence categories, where they are directly identified with (derived versions of) loop spaces, fixed point loci and loop maps, respectively. This results in universal nonlinear versions of Grothendieck-Riemann-Roch theorems, Atiyah-Bott-Lefschetz trace formulas, and Frobenius-Weyl character formulas. On the one hand, we can then linearize by applying sheaf theories, such as the theories of coherent sheaves and D-modules, developed by Gaitsgory and Rozenblyum, as functors out of correspondence categories (in the spirit of topological field theory). This recovers the familiar classical identities, in families and without any smoothness or transversality assumptions. On the other hand, the formalism also applies to higher categorical settings not captured within a linear framework, such as characters of group actions on categories.

math.AG↗
arXiv2019

Spectral action in Betti Geometric Langlands

Let $X$ be a smooth projective curve, $G$ a reductive group, and $Bun_G(X)$ the moduli of $G$-bundles on $X$. For each point of $X$, the Satake category acts by Hecke modifications on sheaves on $Bun_G(X)$. We show that, for sheaves with nilpotent singular support, the action is locally constant with respect to the point of $X$. This equips sheaves with nilpotent singular support with a module structure over perfect complexes on the Betti moduli $Loc_{G^\vee}(X)$ of dual group local systems. In particular, we establish the "automorphic to Galois" direction in the Betti Geometric Langlands correspondence -- to each indecomposable automorphic sheaf, we attach a dual group local system -- and define the Betti version of V. Lafforgue's excursion operators.

math.AG↗
arXiv2018

Wall-crossing for toric mutations

This note explains how to deduce the wall-crossing formula for toric mutations established by Pascaleff-Tonkonog from the perverse schober of the corresponding local Landau-Ginzburg model. Along the way, we develop a general framework to extract a wall-crossing formula from a perverse schober on the projective line with a single critical point.

math.SG↗
arXiv2017

The Character Field Theory and Homology of Character Varieties

We construct an extended oriented $(2+ε)$-dimensional topological field theory, the character field theory $X_G$ attached to a affine algebraic group in characteristic zero, which calculates the homology of character varieties of surfaces. It is a model for a dimensional reduction of Kapustin-Witten theory ($N=4$ $d=4$ super-Yang-Mills in the GL twist), and a universal version of the unipotent character field theory introduced in arXiv:0904.1247. Boundary conditions in $X_G$ are given by quantum Hamiltonian $G$-spaces, as captured by de Rham (or strong) $G$-categories, i.e., module categories for the monoidal dg category $D(G)$ of $D$-modules on $G$. We show that the circle integral $X_G(S^1)$ (the center and trace of $D(G)$) is identified with the category $D(G/G)$ of "class $D$-modules", while for an oriented surface $S$ (with arbitrary decorations at punctures) we show that $X_G(S)\simeq{\rm H}_*^{BM}(Loc_G(S))$ is the Borel-Moore homology of the corresponding character stack. We also describe the "Hodge filtration" on the character theory, a one parameter degeneration to a TFT whose boundary conditions are given by classical Hamiltonian $G$-spaces, and which encodes a variant of the Hodge filtration on character varieties.

math.QA↗
arXiv2017

Mirror symmetry for the Landau-Ginzburg A-model $M=\mathbb C^n, W=z_1 \cdots z_n$

We calculate the category of branes in the Landau-Ginzburg A-model with background $M=\mathbb C^n$ and superpotential $W=z_1 \cdots z_n$ in the form of microlocal sheaves along a natural Lagrangian skeleton. Our arguments employ the framework of perverse schobers, and our results confirm expectations from mirror symmetry.

math.SG↗
arXiv2016

Non-characteristic expansions of Legendrian singularities

This paper presents an algorithm to deform any Legendrian singularity to a nearby Legendrian subvariety with singularities of a simple combinatorial nature. Furthermore, the category of microlocal sheaves on the original Legendrian singularity is equivalent to that on the nearby Legendrian subvariety. This yields a concrete combinatorial model for microlocal sheaves, as well as an elementary method for calculating them.

math.SG↗