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The cyclic open-closed map, u-connections and R-matrices

Kai Hugtenburg

Abstract

This paper considers the (negative) cyclic open-closed map $\mathcal{OC}^{-}$, which maps the cyclic homology of the Fukaya category of a symplectic manifold to its $S^1$-equivariant quantum cohomology. We prove (under simplifying technical hypotheses) that this map respects the respective natural connections in the direction of the equivariant parameter. In the monotone setting this allows us to conclude that $\mathcal{OC}^{-}$ intertwines the decomposition of the Fukaya category by eigenvalues of quantum cup product with the first Chern class, with the Hukuhara-Levelt-Turrittin decomposition of the quantum cohomology. We also explain how our results relate to the Givental-Teleman classification of semisimple cohomological field theories: in particular, how the R-matrix is related to $\mathcal{OC}^{-}$ in the semisimple case; we also consider the non-semisimple case.

The cyclic open-closed map, u-connections and R-matrices

Abstract

This paper considers the (negative) cyclic open-closed map , which maps the cyclic homology of the Fukaya category of a symplectic manifold to its -equivariant quantum cohomology. We prove (under simplifying technical hypotheses) that this map respects the respective natural connections in the direction of the equivariant parameter. In the monotone setting this allows us to conclude that intertwines the decomposition of the Fukaya category by eigenvalues of quantum cup product with the first Chern class, with the Hukuhara-Levelt-Turrittin decomposition of the quantum cohomology. We also explain how our results relate to the Givental-Teleman classification of semisimple cohomological field theories: in particular, how the R-matrix is related to in the semisimple case; we also consider the non-semisimple case.
Paper Structure (44 sections, 102 theorems, 345 equations, 1 figure)

This paper contains 44 sections, 102 theorems, 345 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Formal TEP-structures
  3. Cyclic open-closed map
  4. Image of the cyclic open-closed map for monotone symplectic manifolds
  5. Semi-simple quantum cohomology
  6. Speculations on the general case
  7. Outline of the paper
  8. Acknowledgements
  9. Formal TEP-structures
  10. E-structures
  11. Semi-simple TEP-structures
  12. TE-structure on the cyclic homology of an $A_\infty$-algebra
  13. Hochschild invariants
  14. The u-connection
  15. Cyclic open-closed map respects connections
  16. ...and 29 more sections

Key Result

Theorem 1.7

Let $L\subset X$ be an oriented, relatively-spin Lagrangian submanifold equipped with a $U(\Lambda)$-local system. Suppose there exists a complex structure $J$ such that $(L,J)$ satisfy Assumptions assumptions. Then there exists a bulk-deformed Fukaya $A_\infty$-algebra $CF^*(L,L)$. This is an $R=\L which is a morphism of pre-TE-structures over $R$.

Figures (1)

  • Figure 1: the coloured regions show the areas swept out by $w_2$.

Theorems & Definitions (226)

  • Definition 1.1: see Definition \ref{['formal TEP structures defi']}
  • Remark 1.2
  • Definition 1.3
  • Definition 1.4
  • Remark 1.5
  • Conjecture 1.6
  • Theorem 1.7: see Theorem \ref{['cyclic open-closed morphism of u-VSHS']}
  • Corollary 1.8
  • Remark 1.9
  • Theorem 1.10: RS
  • ...and 216 more