AI 中文总结
该研究发展狄拉克变形理论，在扭曲狄拉克几何与泊松几何间插值，证明其与狄拉克几何结构运算兼容，以紧致李群相关结构变形为例，还得到统一变形理论及约化空间的光滑变形。
AI 中文摘要
我们发展了一种在扭曲狄拉克几何与泊松几何之间进行插值的狄拉克变形理论，并证明这种变形与狄拉克几何的主要结构运算兼容：沿强狄拉克映射的约化、对准辛群胚的积分以及李代数胚的莫里塔等价。以紧致李群\(G\)上的嘉当 - 狄拉克结构\(L_G\)到\(\mathfrak{g}^*\)上的基里洛夫 - Kostant - 苏里奥泊松结构的变形为例，及其提升为准辛群胚\(D(G)\rightrightarrows G\)到辛群胚\(T^*G\rightrightarrows\mathfrak{g}^*\)的变形。作为应用，得到了统一的变形理论，涵盖了从准哈密顿到哈密顿约化、斯坦伯格和谢沃斯季亚诺夫切片到其加法对应物、乘法抛物和幂幺约化、准哈密顿内爆到辛内爆以及乘法摩尔 - 立川簇到其加法类似物等特殊情况。在\(1\)-移位辛层的准辛群胚表示语言中，主要约化定理给出了相应约化空间的光滑变形。
英文摘要
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.
Comments32 pages. All comments are welcome
