Rubber tori in the boundary of expanded stable maps

Abstract

We investigate torus actions on logarithmic expansions in the context of enumerative geometry. Our main result is an intrinsic and coordinate-free description of the higher-rank rubber torus appearing in the boundary of the space of expanded stable maps. The rubber torus is constructed canonically from the tropical moduli space, and its action on each stratum of the expanded target is encoded in a linear tropical position map. The presence of 2-morphisms in the universal target forces expanded stable maps differing by the rubber action to be identified. This provides the first step towards a recursive description of the boundary of the expanded moduli space, with future applications including localisation and rubber calculus.

Logarithmic Gromov–Witten theory with expansions

“…A parallel logarithmic theory of Donaldson-Thomas invariants has been developed, via the construction of a moduli space of higher-rank expansions [MP20]. The methods in [MP20] can be applied to construct the spaces in Theorem A in a manner that is logically independent from the foundational papers in logarithmic Gromov-Witten theory [AC14, Che14b, GS13], and this is explained by Carocci-Nabijou [CN21]. On the other hand, punctured Gromov-Witten theory gives rise to a different solution to gluing problems for logarithmic maps [ACGS20b].…”

“…Superficially, maps to expansions are different beasts than punctured maps, but we expect that the gluing formulas will be similar. A version of the expanded theory for non-rigid targets can be constructed, which form the analogue of punctured maps; see [CN21].…”

Tropical expansions and toric variety bundles

“…The present paper is a companion to [CN21], which describes the higher-rank rubber torus action on the tropical expansions which appear as targets in the moduli space of expanded stable maps. Taken together, these works provide an avenue for probing the recursive structure of these moduli spaces.…”