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The Quantum $A_{\infty}$-Relations on the Elliptic Curve

Michael Slawinski

Abstract

We define and prove the existence of the Quantum $A_{\infty}$-relations on the Fukaya category of the elliptic curve, using the notion of the Feynman transform of a modular operad, as defined by Getzler and Kapranov. Following Barannikov, these relations may be viewed as defining a solution to the quantum master equation of Batalin-Vilkovisky geometry.

The Quantum $A_{\infty}$-Relations on the Elliptic Curve

Abstract

We define and prove the existence of the Quantum -relations on the Fukaya category of the elliptic curve, using the notion of the Feynman transform of a modular operad, as defined by Getzler and Kapranov. Following Barannikov, these relations may be viewed as defining a solution to the quantum master equation of Batalin-Vilkovisky geometry.

Paper Structure

This paper contains 30 sections, 14 theorems, 117 equations, 9 figures.

Table of Contents

  1. Acknowledgements
  2. Modular Operads
  3. The Feynman Transform
  4. The Free Modular Operad $\mathbb{M}P$
  5. The Twisted Modular Operad $\mathcal{E}_V$
  6. Tropical Morse Graphs
  7. Holomorphic Polygons and Annuli
  8. Labeling of Tropical Morse Graphs
  9. Construction of Riemann Surfaces with Boundary
  10. The Lagrangian Condition
  11. The Dimension of The Moduli Space $G_{d,b,\sigma}^{\mathrm{trop}}$
  12. Moduli Space Degeneration
  13. Topological Change Through a Vertex
  14. The genus zero case
  15. The Genus One Case
  16. ...and 15 more sections

Key Result

Proposition 1.4

If $G$ is a stable graph, then Note that when $b(v)=0$ for all $v\in\mathrm{Vert}(G)$ this formula reduces to Euler's formula for planar graphs, i.e., $|\mathrm{Edge}(G)|=\dim\mathrm{H}_1(G)-1+|\mathrm{Vert}(G)|$, where the number of "faces" of a planar graph $G$ is given by $\dim\mathrm{H}_1(G)+1$.

Figures (9)

  • Figure 1: $G$ has three vertices, four edges, and one external leg.
  • Figure 2: The left-hand figure is the domain of the tropical Morse graph $u:G\longrightarrow B$ and the right-hand figure is its fatgraph. We see that as the boundary of the fatgraph is traversed in the counterclockwise direction, the interior of the fatgraph lies on the left, and the Lagrangians are encountered in the order $(L_{n_j}$, $L_{n_i})$, where the corresponding edge on the left is labeled $n_j-n_i$. If, or example, the upper right-hand edge were oriented outward, the labeling of the edge would change to $n_1-n_2$. The same holds for the remaining two edges.
  • Figure 3: Pinching the strip yields two triangles meeting at a point. The labeling procedure for legs described above applied to the left half-edge gives $n_j-n_i$, as this half-edge is directed toward the corner (pinch). Applying the procedure to the right half-edge also gives $n_j-n_i$, as this half-edge is directed away from the corner.
  • Figure 4: The tail of the vector is given by $R_e(s,0)=\sigma_{n_i}(\phi(s))\in L_{n_i}$, and the head is given by $R_e(s,1)=\sigma_{n_j}(\phi(s))\in\L_{n_j}$. See Section \ref{['LConditionSection']} for justification.
  • Figure 5: The contributions, counterclockwise from upper left, are: $+1$, $+1$, $-1$, and $-1$. Note that even if the outgoing edge $e$ of a vertex $v$ is such that $n_e>0$, the signs $(-1)^{s(u)}$ remain the same. The only difference is the polygon may not close once $e$ terminates. The arrow in the segment below each triangle indicates the direction of motion of $\phi(s)\in\mathbb{R}$. The corner points $p_{i,j}$ are, as always, ordered according to a boundary traversal defined by the interior lying to the left, which, in this case, results in a counterclockwise ordering of the corners.
  • ...and 4 more figures

Theorems & Definitions (61)

  • Definition 1.1
  • Example 1.2
  • Definition 1.3
  • Proposition 1.4
  • proof
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Definition 1.8
  • Definition 1.9
  • ...and 51 more