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D-brane categories

C. I. Lazaroiu

Abstract

This is an exposition of recent progress in the categorical approach to D-brane physics. I discuss the physical underpinnings of the appearance of homotopy categories and triangulated categories of D-branes from a string field theoretic perspective, and with a focus on applications to homological mirror symmetry.

D-brane categories

Abstract

This is an exposition of recent progress in the categorical approach to D-brane physics. I discuss the physical underpinnings of the appearance of homotopy categories and triangulated categories of D-branes from a string field theoretic perspective, and with a focus on applications to homological mirror symmetry.

Paper Structure

This paper contains 20 sections, 4 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Generalized D-branes and open string field theory
  3. Open string field theory with D-branes
  4. Vacua and D-brane composites
  5. Closure under composite formation and the quasiunitary cover
  6. Topological A/B strings and graded topological D-branes
  7. Graded open string field theory
  8. The $A_\infty$ structure induced by gauge-fixing and the deformation potential
  9. Topological B-type branes and the derived category
  10. Topological A branes and the Fukaya category
  11. A physical overview
  12. Graded A-type branes
  13. Relative spin structure
  14. A (very) brief review of Fukaya's results
  15. Graded Chern-Simons field theory
  16. ...and 5 more sections

Key Result

Theorem 1

Fix $t\in H^2(Y,{\bf Z}_2)$ and a cover ${\tilde{\@fontswitch{}{\mathcal{}} L}}(Y)$ of ${\@fontswitch{}{\mathcal{}} L}(Y)$. Also fix an appropriate countable collection of graded Lagrangian submanifolds $L$ of $X$, each endowed with a relative spin structure (with respect to $t$) and with a complex

Theorems & Definitions (5)

  • Theorem 1
  • Proposition 2
  • Definition 3
  • Theorem 4
  • Theorem 5