Tropical mirror

In this talk I will discuss tropical mirror developed by me and S. Lysov in three papers: arXiv:2204.06896, arXiv:2301.01687, arXiv:2305.00423 based on the concept of Higher Topological Quantum Mechanics (HTQM) on graphs (arXiv:2112.12756). HTQM is a collection of data in the linear algebra and I will start with explaining it. Then I will briefly explain how HTQM may be used in proving Kadeishvili theorems for $A$-infinity and $L$-infinity case (see also arXiv:2112.12756). I will proceed with explanation how problem of counting of tropical curves passing through tropical cycles may be understood as HTQM with circle action — we will call it $A$-model. Then sum over graphs in $A$-model would be transformed into sum over graphs in another HTQM $B$-model (of BCOV type). In this process differential in $B$-model will happen to be differential in LG Quantum mechanics, and evaluation operators would be transformed into mirror states. These states would form a good section in terms of K. Saito theory with exponential superpotential.