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<p>Quantum <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(L_{\infty }\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>∞</mi> </msub> </math></EquationSource> </InlineEquation> algebras are higher loop generalizations of cyclic <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(L_{\infty }\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>∞</mi> </msub> </math></EquationSource> </InlineEquation> algebras. Motivated by the problem of defining morphisms between such algebras, we construct a linear category of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\((-1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-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 <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(L_{\infty }\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>∞</mi> </msub> </math></EquationSource> </InlineEquation> algebras. Finally, using this category, we propose a new notion of a relation between quantum <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(L_{\infty }\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>∞</mi> </msub> </math></EquationSource> </InlineEquation> algebras.</p>

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Lagrangian Relations and Quantum \(L_{\infty }\) Algebras

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年5月28日

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  • Branislav Jurčo,
  • Ján Pulmann,
  • Martin Zika

摘要

Quantum \(L_{\infty }\) L ∞ algebras are higher loop generalizations of cyclic \(L_{\infty }\) L ∞ algebras. Motivated by the problem of defining morphisms between such algebras, we construct a linear category of \((-1)\) ( - 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 \(L_{\infty }\) L ∞ algebras. Finally, using this category, we propose a new notion of a relation between quantum \(L_{\infty }\) L ∞ algebras.