[Paper Review] Deformation of Dirac structures via $L_\infty$ algebras

Marco Gualtieri, Mykola Matviichuk+1 more|arXiv (Cornell University)|Feb 28, 2017

Homotopy and Cohomology in Algebraic Topology1 references3 citations

TL;DR

This paper establishes a canonical $L_∞$-isomorphism between $L_\infty$ algebras controlling deformations of a Dirac structure in an exact Courant algebroid, regardless of the choice of transversal Dirac or almost Dirac structure. Using a simplified $BV_\infty$-based formalism and pure spinor methods, it proves that these algebras are canonically equivalent, leading to formality maps and explicit formulas for Maurer-Cartan elements under change of transversal, with applications to (quasi)-Poisson geometry, Lie bialgebras, and complex manifold deformations including Kodaira-Spencer theory.

ABSTRACT

The deformation theory of a Dirac structure is controlled by a differential graded Lie algebra which depends on the choice of an auxiliary transversal Dirac structure; if the transversal is not involutive, one obtains an $L_\infty$ algebra instead. We develop a simplified method for describing this $L_\infty$ algebra and use it to prove that the $L_\infty$ algebras corresponding to different transversals are canonically $L_\infty$-isomorphic. In some cases, this isomorphism provides a formality map, as we show in several examples including (quasi)-Poisson geometry, Dirac structures on Lie groups, and Lie bialgebras. Finally, we apply our result to a classical problem in the deformation theory of complex manifolds: we provide explicit formulas for the Kodaira-Spencer deformation complex of a fixed small deformation of a complex manifold, in terms of the deformation complex of the original manifold.

Motivation & Objective

• To resolve the non-canonical dependence of deformation theory on the choice of transversal Dirac structure in Courant algebroids.
• To construct a canonical $L_\infty$-isomorphism between deformation $L_\infty$ algebras arising from different transversal structures.
• To generalize the formalism to non-integrable Dirac structures and curved $L_\infty$ algebras.
• To apply the isomorphism to derive explicit formulas for the Kodaira-Spencer complex of deformed complex manifolds.
• To establish a Tian-Todorov-type formality result for the deformation complex of complex structures under small deformations.

Proposed method

• Using pure spinors and the Clifford algebra structure of the Courant algebroid, the authors reformulate the deformation $L_\infty$ algebra in a simplified way.
• Applying a $BV_\infty$ formalism as a quantization of Roytenberg's derived bracket construction.
• Defining the $L_\infty$-isomorphism via a family of multilinear maps $f_k$ constructed from the operator $R_\varepsilon$ associated to the deformation parameter $\varepsilon$.
• Explicitly computing the higher brackets and morphism components using the bilinear operation $R_\varepsilon(\alpha_1,\alpha_2) = i_\varepsilon(\alpha_1 \wedge \alpha_2) - (i_\varepsilon \alpha_1) \wedge \alpha_2 - \alpha_1 \wedge i_\varepsilon \alpha_2$.
• Verifying that the resulting maps satisfy the $L_\infty$ morphism axioms, particularly the generalized Jacobi identity.
• Applying the isomorphism to Maurer-Cartan elements, deriving the transformation formula $B = (1 + \rho\varepsilon)^{-1}\rho$ for deformed solutions.

Experimental results

Research questions

• RQ1 How does the deformation $L_\infty$ algebra of a Dirac structure depend on the choice of transversal Dirac structure?
• RQ2 Can one construct a canonical $L_\infty$-isomorphism between deformation algebras arising from different transversals, even when the transversal is not involutive?
• RQ3 Does this isomorphism induce a formality map in geometric contexts such as Poisson or generalized complex geometry?
• RQ4 Can the Kodaira-Spencer DGLA of a deformed complex manifold be expressed explicitly in terms of the original DGLA using this isomorphism?
• RQ5 What is the explicit transformation law for Maurer-Cartan elements under change of transversal structure?

Key findings

• The $L_\infty$ algebras associated to different transversal Dirac structures on a fixed Dirac structure are canonically $L_\infty$-isomorphic, resolving the ambiguity in deformation theory.
• The isomorphism is constructed explicitly via a $BV_\infty$-based formalism, yielding a family of multilinear maps $f_k$ that satisfy the $L_\infty$ morphism axioms.
• For Maurer-Cartan elements $\rho$, the deformed solution is given by $B = (1 + \rho\varepsilon)^{-1}\rho$, which converges when $1 + \rho\varepsilon$ is invertible.
• The isomorphism induces a Tian-Todorov-type formula relating the Schouten-Nijenhuis bracket of the original and deformed complex structures.
• The cohomology brackets induced by the original and deformed brackets are isomorphic, implying identical obstruction maps in cohomology.
• The method applies to curved $L_\infty$ algebras and provides explicit formulas for the Kodaira-Spencer complex of a deformed complex manifold in terms of the original complex structure's deformation complex.