Dispersionless bigraded Toda Hierarchy and its additional symmetry

In this paper, we firstly give the definition of dipersionless bigraded Toda hierarchy (dBTH) and introduce some Sato theory on dBTH. Then we define Orlov-Schulman’s $\M_L$, $\M_R$ operator and give the additional Block symmetry of dBTH. Meanwhile we give tau function of dBTH and some some related dipersionless bilinear equations.

💡 Research Summary

The paper introduces the dispersionless bigraded Toda hierarchy (dBTH), a non‑dispersive limit of the bigraded Toda hierarchy (BTH), and develops its full integrable‑system framework. After a brief motivation that situates dBTH among other dispersionless hierarchies (e.g., the dispersionless KP hierarchy) and highlights its relevance to complex geometry and random matrix models, the authors define the hierarchy in terms of two Lax operators (L) and (\bar L). Both operators are expressed as Laurent series in a spectral variable, with coefficients depending on an infinite set of time variables (t_{m,n}) (where (m,n\ge0)). The core of the definition is the Poisson‑bracket structure \