World-sheet instanton and string duality

1997年，江口，与Hori和Xiong一起研究了目标空间M上的拓扑弦理论，并得到了一个无穷微分算子序列，它们构成了一个中心荷为c = X（M）的Viarsoro代数（X（M）表示M的欧拉数）。1998年，江口、金善吉和熊研究了Virasoro算子的自由场实现，证明了M的每一个偶（奇）上同调类都存在1+1维玻色（费米子）自由场。Eguchi和Xiong利用二维引力的结果进一步导出了在任何亏格都有效的拓扑递归关系，并验证了Viraosoro算子在亏格2，3中也能正确预测全纯曲线的个数.这些结果现在被称为量子上同调的Virasoro猜想。在弦理论中，众所周知的是，Calabi-Yau流形的几何可以用Landau-Ginzburg理论（CY/LG对应）来描述。在规范线性（σ）模型中，通过改变超势的参数，可以在目标流形的经典几何区域和超势极值的局部几何区域之间进行插值。1999年江口和金善二研究了CY/LG对应，证明了即使M不是Calabi-Yau流形，只要它是自旋流形，对应也成立。当Calabi-Yau流形退化，它的某些圈消失时，弦理论中会出现非微扰现象，如规范对称性增强。因此，奇异卡-丘流形上的弦理论的分析是特别感兴趣的。江口和菅原一起研究了奇异卡-丘流形上的弦传播，并明确构造了模不变配分函数。模不变量根据孤立奇异点的A-D-E模式进行分类。

In 1997 Eguchi, together with Hori and Xiong studied the topological string theory on a target space M and obtained an infinite sequence of differential operators which form a Viarsoro algebra with a central charge c = X(M) (X(M) denotes the Euler number of M). In 1998 Eguchi, together with Jinzenji and Xiong studied the free field realization of Virasoro operators and have shown that there exist a bosonic (fermionic) free field in 1+1 dimension for each even (odd) cohomology class of M. Eguchi together with Xiong further derived topological recursion relations valid in any genus using the result of 2-dimensional gravity and verified that the Viraosoro operators predict the correct nunber of holomorphic curves also in genus 2, 3. These results are now called Virasoro conjecture of quantum cohomology.It is well-known in string theory that the geometry of Calabi-Yau manifold may be described by Landau-Ginzburg theory (CY/LG correspondence). In the gauged linear (σ model it is possible to interpole between the regime of the classical geometry of a target manifold and the regime of local geometry of extrema of some superpotential by varying its parameter. 1999 Eguchi together with Jinzenji studied CY/LG correspondence and have shown that the correspondence holds even when M is not a Calabi-Yau manifold as far as it is a spin manifold.When the Calabi-Yau manifold degenerates and some of its cycle vanishes, there occur non-perturbative phenomena in string theory like gauge symmetry enhancement. Thus the analysis of string theory in singular Calabi-Yau manifold is of particular interest. Eguchi together with Sugawara studied the string propagation on singular Calabi-Yau manifolds and explicitly constructed modular invariant partition functions. Modular invariants are classified according the A-D-E pattern of isolated sigularities.