Algebra of the infrared and secondary polytopes

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• 作者：M. Kapranov^a ; ^{mikhail.kapranov@ipmu.jp" class="auth_mail" title="E-mail the corresponding author} ; M. Kontsevich^b ; ^{maxim@ihes.fr" class="auth_mail" title="E-mail the corresponding author} ; Y. Soibelman^c ; ^{soibel@math.ksu.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Secondary polytopes ; Deformation theory ; Maurer&ndash ; Cartan elements ; Picard&ndash ; Lefschetz theory ; Fukaya&ndash ; Seidel categories
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：10 September 2016
• 年：2016
• 卷：300
• 期：Complete
• 页码：616-671
• 全文大小：993 K

文摘

We study algebraic structures (L_∞$L_{\infty }$ and A_∞$A_{\infty }$-algebras) introduced by Gaiotto, Moore and Witten in their recent work devoted to certain supersymmetric 2-dimensional massive field theories.

We show that such structures can be systematically produced in any number of dimensions by using the geometry of secondary polytopes, esp. their factorization properties. In particular, in 2 dimensions, we produce, out of a polyhedral “coefficient system”, a dg-category R   with a semi-orthogonal decomposition and an L_∞$L_{\infty }$-algebra g$\mathfrak{g}$. We show that g$\mathfrak{g}$ is quasi-isomorphic to the ordered Hochschild complex of R, governing deformations preserving the semi-orthogonal decomposition.

This allows us to give a more precise mathematical formulation of the (conjectural) alternative description of the Fukaya–Seidel category of a Kahler manifold endowed with a holomorphic Morse function.