On the Toda hierarchy for the stationary theory of the tube with negative contact orders

Abstract

We construct a master $2$-Toda tau sequence for the stationary relative theory of $(\PP^1,0,\infty)$ with any fixed finite lists of negative contacts. The negative-contact blocks are encoded by finite products of exponentials of infinite-wedge Lie algebra elements, with the block at $\infty$ taken as the adjoint of the entire corresponding left block. After adjoining independent positive-profile Toda times, the resulting charged-vacuum matrix elements form a genuine $2$-Toda tau sequence. Following the standard $Q,u$ specialization, its positive signed-degree coefficients recover the stationary relative Gromov--Witten invariants with the prescribed positive and negative contacts, which occur in and are constrained by the induced coefficientwise Hirota equations. The master tau sequence is larger than the enumerative sector: it generally contains nonzero, non-vacuum signed-degree-zero coefficients, as well as auxiliary nonzero lattice components, for which no enumerative interpretation is asserted. Consequently the Hirota equations form a coupled master system and do not, without additional input, close on the positive-degree invariants alone. The construction uses the large-root definition of negative contact, the Johnson--Kumaran--Wu operator formula for the tube, and the Kyoto/Pl\"ucker construction of the $2$-Toda hierarchy.