Geometry & Topology

Motivic Donaldson–Thomas invariants for the one-loop quiver with potential

Ben Davison and Sven Meinhardt

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/gt.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We compute the motivic Donaldson–Thomas invariants of the one-loop quiver, with an arbitrary potential. This is the first computation of motivic Donaldson–Thomas invariants to use in an essential way the full machinery of μ̂–equivariant motives, for which we prove a dimensional reduction result similar to that of Behrend, Bryan and Szendrői in their study of degree-zero motivic Donaldson–Thomas invariants. Our result differs from theirs in that it involves nontrivial monodromy.

Article information

Source
Geom. Topol., Volume 19, Number 5 (2015), 2535-2555.

Dates
Received: 30 January 2013
Revised: 26 August 2014
Accepted: 30 September 2014
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1510858844

Digital Object Identifier
doi:10.2140/gt.2015.19.2535

Mathematical Reviews number (MathSciNet)
MR3416109

Zentralblatt MATH identifier
06503548

Subjects
Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]
Secondary: 14D23: Stacks and moduli problems

Keywords
motivic Donaldson–Thomas theory vanishing cycles quivers

Citation

Davison, Ben; Meinhardt, Sven. Motivic Donaldson–Thomas invariants for the one-loop quiver with potential. Geom. Topol. 19 (2015), no. 5, 2535--2555. doi:10.2140/gt.2015.19.2535. https://projecteuclid.org/euclid.gt/1510858844


Export citation

References

  • K Behrend, Donaldson–Thomas type invariants via microlocal geometry, Ann. of Math. 170 (2009) 1307–1338
  • K Behrend, J Bryan, B Szendrői, Motivic degree zero Donaldson–Thomas invariants, Invent. Math. 192 (2013) 111–160
  • A Białynicki-Birula, Some theorems on actions of algebraic groups, Ann. of Math. 98 (1973) 480–497
  • A Białynicki-Birula, Some properties of the decompositions of algebraic varieties determined by actions of a torus, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 24 (1976) 667–674
  • J Denef, F Loeser, Geometry on arc spaces of algebraic varieties, from: “European Congress of Mathematics, Volume I”, (C Casacuberta, R M Miró-Roig, J Verdera, S Xambó-Descamps, editors), Progr. Math. 201, Birkhäuser, Basel (2001) 327–348
  • J Denef, F Loeser, Lefschetz numbers of iterates of the monodromy and truncated arcs, Topology 41 (2002) 1031–1040
  • T Ekedahl, The Grothendieck group of algebraic stacks
  • J Engel, M Reineke, Smooth models of quiver moduli, Math. Z. 262 (2009) 817–848
  • D Joyce, Configurations in abelian categories, I: Basic properties and moduli stacks, Adv. Math. 203 (2006) 194–255
  • D Joyce, Configurations in abelian categories, II: Ringel–Hall algebras, Adv. Math. 210 (2007) 635–706
  • D Joyce, Configurations in abelian categories, III: Stability conditions and identities, Adv. Math. 215 (2007) 153–219
  • D Joyce, Motivic invariants of Artin stacks and “stack functions”, Q. J. Math. 58 (2007) 345–392
  • D Joyce, Configurations in abelian categories, IV: Invariants and changing stability conditions, Adv. Math. 217 (2008) 125–204
  • D Joyce, Y Song, A theory of generalized Donaldson–Thomas invariants, Mem. Amer. Math. Soc. 1020, Amer. Math. Soc. (2012)
  • M Kontsevich, Y Soibelman, Stability structures, motivic Donaldson–Thomas invariants and cluster transformations
  • M Kontsevich, Y Soibelman, Motivic Donaldson–Thomas invariants: Summary of results, from: “Mirror symmetry and tropical geometry”, (R Castaño-Bernard, Y Soibelman, I Zharkov, editors), Contemp. Math. 527, Amer. Math. Soc. (2010) 55–89
  • M Kontsevich, Y Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson–Thomas invariants, Commun. Number Theory Phys. 5 (2011) 231–352
  • A Kresch, Cycle groups for Artin stacks, Invent. Math. 138 (1999) 495–536
  • Q T Lê, Proofs of the integral identity conjecture over algebraically closed fields, Duke Math. J. 164 (2015) 157–194
  • D Maulik, Motivic residues and Donaldson–Thomas theory, preprint, available from the author
  • A Morrison, S Mozgovoy, K Nagao, B Szendrői, Motivic Donaldson–Thomas invariants of the conifold and the refined topological vertex, Adv. Math. 230 (2012) 2065–2093
  • S Mozgovoy, On the motivic Donaldson–Thomas invariants of quivers with potentials, Math. Res. Lett. 20 (2013) 107–118
  • S Mozgovoy, Wall-crossing formulas for framed objects, Q. J. Math. 64 (2013) 489–513
  • R P Thomas, A holomorphic Casson invariant for Calabi–Yau $3$–folds, and bundles on $K3$ fibrations, J. Differential Geom. 54 (2000) 367–438