Kontsevich - Miwa Transform of the Virasoro Constraints as Null-Vector Decoupling Equations

Details Kontsevich - Miwa Transform of the Virasoro Constraints as Null-Vector Decoupling Equations Kontsevich - Miwa Transform of the Virasoro Constraints as Null-Vector Decoupling Equations

1991-11-25

arxiv.org/abs/hep-th/9111051v2

Physics

High Energy Physics

High Energy Physics - Theory

17 pages

Scientific paper

We use the Kontsevich--Miwa transform to relate the Virasoro constraints on integrable hierarchies with the David-Distler-Kawai formalism of gravity-coupled conformal theories. The derivation relies on evaluating the energy-momentum tensor on the hierarchy at special values of the spectral parameter. We thus obtain in the Kontsevich parametrization the `master equations' which implement the Virasoro constraints and at the same time coincide with null-vector decoupling equations in an `auxiliary' conformal field theory on the complex plane of the spectral parameter. This gives the operators their gravitational scaling dimensions (for one out of four possibilities to choose signs), with the $\alpha_+$ being equal to the background charge $Q$ of an abstract $bc$ system underlying the structure of the Virasoro constraints. The formalism also generalizes to the $N$-KdV hierarchies.

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