Dirac geometry, deformation theory and shifted symplectic geometry

Abstract

We develop a Dirac deformation theory that interpolates between twisted Dirac geometry and Poisson geometry, and prove that this deformation is compatible with the principal structural operations of Dirac geometry: reduction along strong Dirac maps, integration to quasi-symplectic groupoids, and Morita equivalence of Lie algebroids. The fundamental example is the deformation of the Cartan--Dirac structure $L_G$ on a compact Lie group~$G$ to the Kirillov--Kostant--Souriau Poisson structure on $\mathfrak{g}^*$, and its lift to a deformation of the quasi-symplectic groupoid $D(G)\rightrightarrows G$ to the symplectic groupoid $T^*G\rightrightarrows\mathfrak{g}^*$. As applications, we obtain a uniform deformation theory recovering, as special cases, the deformation of quasi-Hamiltonian to Hamiltonian reduction along conjugacy classes, the Steinberg and Sevostyanov slices to their additive (Kostant, Slodowy) counterparts, the multiplicative parabolic and unipotent reductions, the quasi-Hamiltonian implosion to symplectic implosion, and the multiplicative Moore--Tachikawa varieties to their additive analogues. In the language of quasi-symplectic groupoid presentations of $1$-shifted symplectic stacks, our main reduction theorem yields a smooth deformation of the corresponding reduced spaces.