We revisit the Virasoro constraints and explore the relation to the Hirota bilinear equations. We furthermore investigate and provide the solution to non-homogeneous Virasoro constraints, namely those coming from matrix models whose domain of integration has boundaries. In particular, we provide the example of Hermitean matrices with positive eigenvalues in which case one can find a solution by induction on the rank of the matrix model.

中文翻译：

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重新审视 Virasoro 约束

我们重新审视 Virasoro 约束并探索与 Hirota 双线性方程的关系。我们进一步研究并提供了非齐次 Virasoro 约束的解决方案，即那些来自集成域有边界的矩阵模型的约束。特别是，我们提供了具有正特征值的 Hermite 矩阵的示例，在这种情况下，可以通过对矩阵模型的秩进行归纳来找到解决方案。