Conformal blocks attached to twisted groups

Twisted loop groups and their affine flag varieties

We develop a theory of affine flag varieties and of their Schubert varieties for reductive groups over a Laurent power series local field k((t)) with k a perfect field. This can be viewed as a generalization of the theory of affine flag varieties for loop groups to a "twisted case"; a consequence of our results is that our construction also includes the flag varieties for Kac-Moody Lie algebras of affine type. We also give a coherence conjecture on the dimensions of the spaces of global sections of the natural ample line bundles on the partial flag varieties attached to a fixed group over k((t…

与某些非退化函数相关的指数扭曲上同调的弱可容许性

Weak admissibility of exponentially twisted cohomology associated with some nondegenerate functions

该文研究与非退化函数相关的指数扭曲上同调对应的滤过Φ-模，提出其弱可容许性的推测，并借助Adolphson和Sperber的理论在特定假设下证明该推测成立，丰富了p进霍奇理论相关研究。

Automorphisms and twisted forms of the N = 1, 2, 3 Lie conformal superalgebras

We classify the N = 1, 2, 3 superconformal Lie algebras of Schwimmer and Seiberg by means of differential non-abelian cohomology, and describe the general philosophy behind this new technique. The structure of the group (functor) of automorphisms of the corresponding Lie conformal superalgebra is a key ingredient of the proof.

扭曲共形块及其维数

Twisted conformal blocks and their dimension

本文证明有限群作用下扭曲共形块空间可约化为图自同构情形，并在Teleman消没定理的扭曲类比下推导出Kac-Walton公式和Verlinde公式的一般化，用于计算其维数。

Conformal blocks and the cohomology of configuration spaces of curves

We realize any space of conformal blocks attached to a punctured curve inside the cohomology of a configuration space of that curve and compare the WZW connection with the Gauss-Manin connection.

Orbifold conformal blocks and the stack of pointed G-covers

Starting with a vertex algebra $V$, a finite group $G$ of automorphisms of $V$, and a suitable collection of twisted $V$--modules, we construct (twisted) $D$--modules on the stack of pointed $G$--covers, introduced by Jarvis, Kaufmann, and Kimura. The fibers of these sheaves are spaces of orbifold conformal blocks defined in joint work with Edward Frenkel. The key ingredient is a $G$--equivariant version of the Virasoro uniformization theorem.

Twisted Modules over Vertex Algebras on Algebraic Curves

We extend the geometric approach to vertex algebras developed by the first author to twisted modules, allowing us to treat orbifold models in conformal field theory. Let $V$ be a vertex algebra, $H$ a finite group of automorphisms of $V$, and $C$ an algebraic curve such that $H \subset \on{Aut}(C)$. We show that a suitable collection of twisted $V$--modules gives rise to a section of a certain sheaf on the quotient $X=C/H$. We introduce the notion of conformal blocks for twisted modules, and analyze them in the case of the Heisenberg and affine Kac-Moody vertex algebras. We also give a chiral …

代数曲线伽罗瓦覆盖的共形块

Conformal blocks for Galois covers of algebraic curves

该文研究有限群Γ作用下带标记轨道的代数曲线的扭曲共形块空间，证明其满足传播与因子分解定理、相关层局部自由并具平坦射影联络，还证实了Pappas-Rapoport猜想，建立扭曲共形块与广义θ函数的等价性。

Galois representations attached to Q-curves and the generalized Fermat equation A^4 + B^2 = C^p

We show that the mod p Galois representations attached to a Q-curve E of degree d over an imaginary quadratic number field K are surjective for all p larger than some constant M_{K,d}, if E has potentially multiplicative reduction at any prime not dividing 6. The proof uses Mazur's formal immersion method, a result of Darmon and Merel on non-split modular curves, and an analytic argument showing that Jacobians of certain twisted modular curves admit quotients with Mordell-Weil rank 0. In combination with a previous theorem of the author and Skinner on modularity of Q-curves, the surjectivity r…

临界水平之外共形块的Hecke修正

Hecke modifications of conformal blocks outside the critical level

该研究定义了非临界水平下仿射李代数共形块的Hecke修正，证明其等价于在额外标记点插入扭真空模，并给出坐标变换下的显式描述。

扭曲的 $\mathbb Q$-同源类中的 Faltings 椭圆曲线

Faltings elliptic curves in twisted $\mathbb Q$-isogeny classes

本文研究二次扭曲后 $\mathbb Q$-同源类图中的最小 Faltings 高度曲线，确定其对应顶点，并给出顶点被选中的概率及其与模函数 $p$-进赋值的关系。