Integrality of relative BPS state counts of toric del Pezzo surfaces

Garrel, Michel van; Wong, Tony W. H.; Zaimi, Gjergji

doi:10.4310/cntp.2013.v7.n4.a3

Cited by 10 publications

(10 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Proof. The argument follows the same lines as the proof of the main result in [32] and we therefore do not reproduce it here.…”

Section: Multiple Covers and Loop Quiver Dt Invariantsmentioning

confidence: 96%

“…In [43,Proposition 6.1] (see (6.1)), the authors compute the contribution of multiple covers over rigid relative maps to the relative GW invariants. This leads them to define relative BPS state counts, which are shown to be integers for toric del Pezzo surfaces in [32]. One motivation for this work is that the enumerative meaning of relative BPS state counts are not clear in the context of a smooth divisor.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Log BPS numbers of log Calabi-Yau surfaces

Choi¹,

Garrel²,

Katz³

et al. 2020

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Let (S, E) be a log Calabi-Yau surface pair with E a smooth divisor. We define new conjecturally integer-valued counts of A 1 -curves in (S, E). These log BPS numbers are derived from genus 0 log Gromov-Witten invariants of maximal tangency along E via a formula analogous to the multiple cover formula for disk counts. A conjectural relationship to genus 0 local BPS numbers is described and verified for del Pezzo surfaces and curve classes of arithmetic genus up to 2. We state a number of conjectures and provide computational evidence.

show abstract

“…Proof. The argument follows the same lines as the proof of the main result in [32] and we therefore do not reproduce it here.…”

Section: Multiple Covers and Loop Quiver Dt Invariantsmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Log BPS numbers of log Calabi-Yau surfaces

Choi¹,

Garrel²,

Katz³

et al. 2020

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…From Corollary 1.13 we derive an enumerative meaning of the local BPS numbers n β (K X ) subject to Conjecture 1.8 ([21,Conjecture 1.3]). This is a BPS version of the log-local principle pursued in [10][11][12]45,56,58]. Notice that Conjecture 1.8 is proven for X = P 2 in [8] based on [9,23], and for any del Pezzo surface and classes of arithmetic genus ≤ 2 in [20,21].…”

Section: Now We Return To the Case Of Anmentioning

confidence: 99%

Sheaves of maximal intersection and multiplicities of stable log maps

Choi

Garrel

Katz

et al. 2021

Sel. Math. New Ser.

View full text Add to dashboard Cite

A great number of theoretical results are known about log Gromov–Witten invariants (Abramovich and Chen in Asian J Math 18:465–488, 2014; Chen in Ann Math (2) 180:455–521, 2014; Gross and Siebert J Am Math Soc 26: 451–510, 2013), but few calculations are worked out. In this paper we restrict to surfaces and to genus 0 stable log maps of maximal tangency. We ask how various natural components of the moduli space contribute to the log Gromov–Witten invariants. The first such calculation (Gross et al. in Duke Math J 153:297–362, 2010, Proposition 6.1) by Gross–Pandharipande–Siebert deals with multiple covers over rigid curves in the log Calabi–Yau setting. As a natural continuation, in this paper we compute the contributions of non-rigid irreducible curves in the log Calabi–Yau setting and that of the union of two rigid curves in general position. For the former, we construct and study a moduli space of “logarithmic” 1-dimensional sheaves and compare the resulting multiplicity with tropical multiplicity. For the latter, we explicitly describe the components of the moduli space and work out the logarithmic deformation theory in full, which we then compare with the deformation theory of the analogous relative stable maps.

show abstract

“…In [32,Proposition 6.1] (see (6.1)), the authors compute the contribution of multiple covers over rigid relative maps to the relative GW invariants. This leads them to define relative BPS state counts, which are shown to be integers for toric del Pezzo surfaces in [22]. One motivation for this work is that the enumerative meaning of relative BPS state counts are not clear in the context of a smooth divisor.…”

Section: Introductionmentioning

confidence: 99%