Framed motivic Donaldson–Thomas invariants of small crepant resolutions

Abstract

For an arbitrary integer r≥1$r\ge 1$, we compute r‐framed motivic DT and PT invariants of small crepant resolutions of toric Calabi–Yau 3‐folds, establishing a “higher rank” version of the motivic DT/PT wall‐crossing formula. This generalises the work of Morrison and Nagao. Our formulae, in particular their relationship with the r=1$r=1$ theory, fit nicely in the current development of higher rank refined DT invariants.

Higher rank motivic Donaldson–Thomas invariants of via wall-crossing, and asymptotics

“…The fact that partition functions of rank r invariants factor as r copies of partition functions of rank 1 invariants, shifted just as in Formula (0·2), has also been observed in the context of K-theoretic DT theory of [14], as well as in string theory [29] and other motivic settings, such as [8, 23]. Recent work of Feyzbakhsh–Thomas [16] shows that general higher rank DT invariants can be linked back to genuine rank 1 DT invariants: we believe our Formula (0·2) is an explicit (motivic) shadow of this general principle.…”

“…This approach allows us us to express the invariants for , which we view as ‘ r -framed’ DT invariants, in terms of the universal series of the invariants of unframed representations of the 3-loop quiver in a critical chamber. These ideas can be employed to compute framed motivic DT invariants of small crepant resolutions of affine toric Calabi–Yau 3-folds [8], which also exhibit similar factorisation properties.…”

Equivariant Enumerative Geometry and Donaldson-Thomas Theory

“…The case r = 1 was computed in [11]. The general proof of Formula (4.5.3) is obtained in a similar fashion in [163,47], and via a wall-crossing technique in [49]. Moreover, it is immediate to verify that DT mot r satisfies a product formula analogous to the one proved in Theorem 4.5.9 for the K-theoretic invariants: we have (4.5.4)…”

“…Just as in the case of the naive motives of the Quot scheme [164], the resulting partition function DT mot r (Y, q) only depends on the motivic class [Y ] ∈ K 0 (Var C ) and on r = rk F . See also [49] for calculations of motivic higher rank DT and PT invariants in the presence of nonzero curve classes: the generating function DT mot r (Y, q), computed easily starting with Formula (4.5.3), is precisely the DT/PT wall-crossing factor. 4.6.…”

“…There are to date a number of examples of this phenomenon in Donaldson-Thomas theory, exhibiting a generating series of rank r invariants as a product of r (suitably shifted) generating series of rank 1 invariants: see for instance [9,165] for enumerative DT invariants, [76] for K-theoretic DT invariants, [48,49] for motivic DT invariants and [143,62] for the parallel pictures in string theory.…”