Lagrangian Relations and Quantum $L_\infty$ Algebras
Mathematical Institute, Faculty of Mathematics and Physics, Charles University（查尔斯大学数学物理学院数学研究所）；School of Mathematics, University of Edinburgh（爱丁堡大学数学学院）
Branislav Jurčo, Ján Pulmann, Martin Zika
备注：
43 pages; v4 - version accepted for publication; added some missing relevant references, changed the convention for degree shift, added minor clarifications including Remarks 4.8, 4.19
Quantum $L_\infty$ algebras are higher loop generalizations of cyclic $L_\infty$ algebras. Motivated by the problem of defining morphisms between such algebras, we construct a linear category of $(-1)$-shifted symplectic vector spaces and distributional half-densities, originally proposed by Ševera. Morphisms in this category can be given both by formal half-densities and Lagrangian relations; we prove that the composition of such morphisms recovers the construction of homotopy transfer of quantum $L_\infty$ algebras. Finally, using this category, we propose a new notion of a relation between quantum $L_\infty$ algebras.
We explore the relationship between the classical constructions of cumulants and Koszul brackets, showing that the former are an expontial version of the latter. Moreover, under some additional technical assumptions, we prove that both constructions are compatible with standard homological perturbation theory in an appropriate sense. As an application of these results, we provide new proofs for the homotopy transfer Theorem for $L_\infty$ and $IBL_\infty$ algebras based on the symmetrized tensor trick and the standard perturbation Lemma, as in the usual approach for $A_\infty$ algebras. Moreov…
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A version of mirror symmetry predicts a ring isomorphism between quantum cohomology of a symplectic manifold and Jacobian algebra of the Landau-Ginzburg mirror, and for toric manifolds Fukaya-Oh-Ohta-Ono constructed such a map called Kodaira-Spencer map using Lagrangian Floer theory. We discuss a general construction of Kodaira-Spencer ring homomorphism when LG mirror potential $W$ is given by $J$-holomorphic discs with boundary on a Lagrangian $L$: we find an $A_{\infty}$-algebra $\mathcal{B}$ whose $m_1$-complex is a Koszul complex for $W$ under mild assumptions on $L$. Closed-open map gives…
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In this paper, we first construct the controlling algebras of embedding tensors and Lie-Leibniz triples, which turn out to be a graded Lie algebra and an $L_\infty$-algebra respectively. Then we introduce representations and cohomologies of embedding tensors and Lie-Leibniz triples, and show that there is a long exact sequence connecting various cohomologies. As applications, we classify infinitesimal deformations and central extensions using the second cohomology groups. Finally, we introduce the notion of a homotopy embedding tensor which will induce a Leibniz$_\infty$-algebra. We realize Ko…
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It is shown that the closure of the infinitesimal symmetry transformations underlying classical ${\cal W}$ algebras give rise to L$_\infty$ algebras with in general field dependent gauge parameters. Therefore, the class of well understood ${\cal W}$ algebras provides highly non-trivial examples of such strong homotopy Lie-algebras. We develop the general formalism for this correspondence and apply it explicitly to the classical ${\cal W}_3$ algebra.
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