Modular Forms and Elliptic Genera for ALE Spaces

Multi instanton calculus on ALE spaces

We study SYM gauge theories living on ALE spaces. Using localization formulae we compute the prepotential (and its gravitational corrections) for SU(N) supersymmetric ${\cal N}=2, 2^*$ gauge theories on ALE spaces of the $A_n$ type. Furthermore we derive the Poincaré polynomial describing the homologies of the corresponding moduli spaces of self-dual gauge connections. From these results we extract the ${\cal N}=4$ partition function which is a modular form in agreement with the expectations of $SL(2,\Z)$ duality.

Noncompact Gepner Models for Type II Strings on a Conifold and an ALE Instanton

We construct modular invariant partition functions for type II strings on a conifold and a singular Eguchi-Hanson instanton by means of the SL(2,R)/U(1) version of Gepner models. In the conifold case, we find an extra massless hypermultiplet in the IIB spectrum and argue that it may be identified as a soliton. In the Eguchi-Hanson case, our formula is new and different from the earlier result, in particular does not contain graviton. The lightest IIB fields are combined into a six-dimensional (2,0) tensor multiplet with a negative mass square. We give an interpretation to it as a doubleton-lik…

Liouville Field, Modular Forms and Elliptic Genera

When we describe non-compact or singular Calabi-Yau manifolds by CFT, continuous as well as discrete representations appear in the theory. These representations mix in an intricate way under the modular transformations. In this article, we propose a method of combining discrete and continuous representations so that the resulting combinations have a simpler modular behavior and can be used as conformal blocks of the theory. We compute elliptic genera of ALE spaces and obtain results which agree with those suggested from the decompactification of K3 surface. Consistency of our approach is assur…

M-弦轨形理论的ALE配分函数

The ALE Partition Functions of M-String Orbifolds

本文计算M-弦轨形SCFT的ALE配分函数，通过几何工程联系高阶Donaldson-Thomas不变量，发现冻结BPS弦携带分数瞬子荷的新特征。

$S$-对偶、边界态以及ALE空间上的高阶形式对称性

$S$-duality, boundary states, and higher-form symmetries on ALE spaces

研究ALE空间上Maxwell理论的阿贝尔$S$-对偶，通过将路径积分分解为theta函数块并解释为边界态，揭示了向量值模协变性，并探讨了1-形式对称性背景下的混合反常。

S and T matrices for the super $U(1,1)$ WZW model. Application to surgery and 3-manifold invariants based on the Alexander Conway polynomial

We carry on the study of the Alexander Conway invariant from the quantum field theory point of view started in \cite{RS91}. We first discuss in details $S$ and $T$ matrices for the $U(1,1)$ super WZW model and obtain, for the level $k$ an integer, new finite dimensional representations of the modular group. These have the remarkable property that some of the $S$ matrix elements are infinite. Moreover, typical and atypical representations as well as indecomposable blocks are mixed: truncation to maximally atypical representations, as advocated in some recent papers, is not consistent. The main …