Homotopy Loday Algebras and Symplectic $2$-Manifolds

李代数上的dg Loday-Pirashvili模

Dg Loday-Pirashvili modules over Lie algebras

本文研究李代数上dg Loday-Pirashvili模的推广与等价刻画，证明其可消解任意经典Loday-Pirashvili模，并通过dg导子与Kapranov函子构造Leibniz∞[1]代数结构，揭示其与扭曲Atiyah类的联系。

关于TCFT中的高阶Leibniz恒等式

On higher order Leibniz identities in TCFT

利用非局部算子推广了拓扑共形域论中的局部可观量代数。这允许构造通过紧化模空间上的某些积分实现的依赖于参数的操作，满足莱布尼茨恒等式和高阶莱布尼茨恒等式的类似物。我们猜想，可以构建一个完整的一套这样的操作，导致参数依赖版本的Loday的同伦莱布尼茨代数。

一类辛Dirichlet能量的全局极小元

Some global minimizers of a symplectic Dirichlet energy

考虑泛函$F=\frac12\|ϕ^*ω\|_{L^2}^2$的变分问题，其中$ϕ:(M,g)\to (N,ω)$映射一个黎曼流形到一个辛流形.这个泛函在理论物理中作为Faddeev-Hopf能量的强耦合极限出现，并且可以被视为调和映射理论中熟悉的狄利克雷能量的辛模拟。已知Hopf纤维化$π:S^3\to S^2$是$F$的局部稳定临界点。本文证明了$π$实际上在其同伦类中极小化$F$，并将这一结果推广到了$S^3$是Berger球面度量的情形.证明了如果$ϕ^*ω$是余闭的，则$ϕ$是$F$的临界点，并且是$F$在其同伦类中的最小值.若$M$是紧Riemann曲面，则证明了$F$的每个临界点都有$ϕ^*ω$余闭.本文构造了一类到Hermite对称空间的全纯齐性投影，并证明了这类投影也在其同伦类中极小化$F$.

Symplectic embeddings, homotopy algebras and almost Poisson gauge symmetry

We formulate general definitions of semi-classical gauge transformations for noncommutative gauge theories in general backgrounds of string theory, and give novel explicit constructions using techniques based on symplectic embeddings of almost Poisson structures. In the absence of fluxes the gauge symmetries close a Poisson gauge algebra and their action is governed by a $P_\infty$-algebra which we construct explicitly from the symplectic embedding. In curved backgrounds they close a field dependent gauge algebra governed by an $L_\infty$-algebra which is not a $P_\infty$-algebra. Our techniqu…

Homotopy momentum sections on multisymplectic manifolds

We introduce a notion of a homotopy momentum section on a Lie algebroid over a pre-multisymplectic manifold. A homotopy momentum section is a generalization of the momentum map with a Lie group action and the momentum section on a pre-symplectic manifold, and is also regarded as a generalization of the homotopy momentum map on a multisymplectic manifold. We show that a gauged nonlinear sigma model with Wess-Zumino term with Lie algebroid gauging has the homotopy momentum section structure.

Strong homotopy Lie algebras, homotopy Poisson manifolds and Courant algebroids

We study Maurer-Cartan elements on homotopy Poisson manifolds of degree $n$. They unify many twisted or homotopy structures in Poisson geometry and mathematical physics, such as twisted Poisson manifolds, quasi-Poisson $\g$-manifolds, and twisted Courant algebroids. Using the fact that the dual of an $n$-term $L_\infty$-algebra is a homotopy Poisson manifold of degree $n-1$, we obtain a Courant algebroid from a $2$-term $L_\infty$-algebra $\g$ via the degree $2$ symplectic NQ-manifold $T^*[2]\g^*[1]$. By integrating the Lie quasi-bialgebroid associated to the Courant algebroid, we obtain a Lie…