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Brane structures in microlocal sheaf theory

Xin Jin, David Treumann

Abstract

Let $L$ be an exact Lagrangian submanifold of a cotangent bundle $T^* M$, asymptotic to a Legendrian submanifold $Λ\subset T^{\infty} M$. We study a locally constant sheaf of $\infty$-categories on $L$, called the sheaf of brane structures or $\mathrm{Brane}_L$. Its fiber is the $\infty$-category of spectra, and we construct a Hamiltonian invariant, fully faithful functor from $Γ(L,\mathrm{Brane}_L)$ to the $\infty$-category of sheaves of spectra on $M$ with singular support in $Λ$.

Brane structures in microlocal sheaf theory

Abstract

Let be an exact Lagrangian submanifold of a cotangent bundle , asymptotic to a Legendrian submanifold . We study a locally constant sheaf of -categories on , called the sheaf of brane structures or . Its fiber is the -category of spectra, and we construct a Hamiltonian invariant, fully faithful functor from to the -category of sheaves of spectra on with singular support in .

Paper Structure

This paper contains 87 sections, 34 theorems, 180 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Exact Lagrangians and wavefronts
  3. Isotopies
  4. Compact and noncompact Lagrangians
  5. Coefficients
  6. Singular support and front projections
  7. Smooth part
  8. Double and multiple points
  9. Singular points of a front
  10. Microlocal monodromy and brane obstruction
  11. The Nadler-Zaslow functor
  12. The $J$-homomorphism
  13. More general microlocal categories
  14. Precedents
  15. Questions
  16. ...and 72 more sections

Key Result

Theorem 1

Let $L$ be an exact Lagrangian submanifold of a cotangent bundle $T^* M$, asymptotic to a Legendrian submanifold $\Lambda \subset T^{\infty} M$. If the brane obstructions on $L$ vanish, there is a fully faithful functor from local systems on $L$, to constructible sheaves on $M$ with singular support over $\Lambda$.

Figures (4)

  • Figure 1: The filled cone is $C_{(x,a,\epsilon)}$. The outer edges emitting from $(x,a)$ form the boundary of $C_{x,a,\epsilon'}$ for $0<\epsilon'<\epsilon$. The portion of gray curve above $(x,a)$ is part of $\partial H_t^{\epsilon'}$ as a smoothing of $\partial C_{x,a,\epsilon'}$.
  • Figure 2: The entire triangle illustrates $R$ in case (3). The upper triangle illustrates $R^+$.
  • Figure 3: An illustration of the assumptions in Corollary \ref{['cor: F,G,a1b1,eta1eta2']}.
  • Figure 4: An illustration of the assumptions in Corollary \ref{['lemma: disjoint support']}.

Theorems & Definitions (73)

  • Theorem : Nadler-Zaslow
  • Example
  • Proposition
  • Remark
  • Proposition
  • proof
  • Lemma
  • proof
  • Remark
  • Definition
  • ...and 63 more