Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-25T05:49:58.908Z Has data issue: false hasContentIssue false
  • English
  • Français

The flow tree formula for Donaldson–Thomas invariants of quivers with potentials

Part of: Projective and enumerative geometry Representation theory of rings and algebras

Published online by Cambridge University Press:  19 December 2022

Hülya Argüz  and
Pierrick Bousseau
Hülya Argüz
Affiliation:
University of Georgia, Department of Mathematics, Athens, GA 30605, USA [email protected]
Pierrick Bousseau
Affiliation:
University of Georgia, Department of Mathematics, Athens, GA 30605, USA [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove the flow tree formula conjectured by Alexandrov and Pioline, which computes Donaldson–Thomas invariants of quivers with potentials in terms of a smaller set of attractor invariants. This result is obtained as a particular case of a more general flow tree formula reconstructing a consistent scattering diagram from its initial walls.

Keywords

quiver representationsDonaldson–Thomas invariantsscattering diagrams

MSC classification

Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants 16G20: Representations of quivers and partially ordered sets
Type
Research Article
Information
Compositio Mathematica , Volume 158 , Issue 12 , December 2022 , pp. 2206 - 2249
Check for updates
Copyright
© 2022 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alexandrov, S., Manschot, J. and Pioline, B., S-duality and refined BPS indices, Comm. Math. Phys. 380 (2020), 755–810; MR 4170291.CrossRefGoogle Scholar
Alexandrov, S. and Pioline, B., Attractor flow trees, BPS indices and quivers, Adv. Theor. Math. Phys. 23 (2019), 627–699; MR 4049073.CrossRefGoogle Scholar
Alexandrov, S. and Pioline, B., Black holes and higher depth mock modular forms, Comm. Math. Phys. 374 (2020), 549–625; MR 4072224.CrossRefGoogle Scholar
Alim, M., Cecotti, S., Córdova, C., Espahbodi, S., Rastogi, A. and Vafa, C., BPS quivers and spectra of complete $\mathcal {N}=2$ quantum field theories, Comm. Math. Phys. 323 (2013), 1185–1227; MR 3106506.CrossRefGoogle Scholar
Alim, M., Cecotti, S., Córdova, C., Espahbodi, S., Rastogi, A. and Vafa, C., $\mathcal {N}=2$ quantum field theories and their BPS quivers, Adv. Theor. Math. Phys. 18 (2014), 27–127; MR 3268234.CrossRefGoogle Scholar
Argüz, H. and Gross, M., The higher dimensional tropical vertex, Geom. Topol., to appear. Preprint (2020), arXiv:2007.08347.Google Scholar
Aspinwall, P., Bridgeland, T., Craw, A., Douglas, M., Gross, M., Kapustin, A., Moore, G., Segal, G., Szendröi, B. and Wilson, P., Dirichlet branes and mirror symmetry, Clay Mathematics Monographs, vol. 4 (American Mathematical Society, Providence, RI, 2009); MR 2567952.Google Scholar
Beaujard, G., Manschot, J. and Pioline, B., Vafa–Witten invariants from exceptional collections, Comm. Math. Phys. 385 (2021), 101–226; MR 4275783.CrossRefGoogle Scholar
Beĭlinson, A., Bernstein, J. and Deligne, P., Faisceaux pervers, in Analysis and topology on singular spaces, I (Luminy, 1981), Astérisque, vol. 100 (Société Mathématique de France, Paris, 1982), 5–171; MR 751966.Google Scholar
Bena, I., Berkooz, M., de Boer, J., El-Showk, S. and Van den Bleeken, D., Scaling BPS solutions and pure-Higgs states, J. High Energy Phys. 2012 (2012), paper 171; MR 3036440.CrossRefGoogle Scholar
Bousseau, P., The quantum tropical vertex, Geom. Topol. 24 (2020), 1297–1379; MR 4157555.CrossRefGoogle Scholar
Bridgeland, T., Stability conditions on triangulated categories, Ann. of Math. (2) 166 (2007), 317–345; MR 2373143 (2009c:14026).CrossRefGoogle Scholar
Bridgeland, T., Scattering diagrams, Hall algebras and stability conditions, Algebr. Geom. 4 (2017), 523–561; MR 3710055.CrossRefGoogle Scholar
Carl, M., Pumperla, M. and Siebert, B., A tropical view of Landau–Ginzburg models, Preprint (2022), arXiv:2205.07753.Google Scholar
Cecotti, S., Neitzke, A. and Vafa, C., R-twisting and 4d/2d correspondences, Preprint (2010), arXiv:1006.3435.Google Scholar
Cecotti, S. and Vafa, C., Classification of complete N=2 supersymmetric theories in 4 dimensions, in Surveys in differential geometry, Geometry and Topology, vol. 18 (International Press, Somerville, MA, 2013), 19–101; MR 3087917.Google Scholar
Cheung, M.-W. and Mandel, T., Donaldson–Thomas invariants from tropical disks, Selecta Math. (N.S.) 26 (2020), paper 57; MR 4131036.CrossRefGoogle Scholar
Davison, B. and Mandel, T., Strong positivity for quantum theta bases of quantum cluster algebras, Invent. Math. 226 (2021), 725–843; MR 4337972.CrossRefGoogle Scholar
Davison, B. and Meinhardt, S., Donaldson–Thomas theory for categories of homological dimension one with potential, Preprint (2015), arXiv:1512.08898.Google Scholar
Davison, B. and Meinhardt, S., Cohomological Donaldson–Thomas theory of a quiver with potential and quantum enveloping algebras, Invent. Math. 221 (2020), 777–871; MR 4132957.CrossRefGoogle Scholar
de Boer, J., El-Showk, S., Messamah, I. and Van den Bleeken, D., Quantizing $N=2$ multicenter solutions, J. High Energy Phys. 5 (2009), paper 002; MR 2511440.Google Scholar
Deligne, P. and Katz, N., Groupes de monodromie en géométrie algébrique (SGA VII, 2), Lecture Notes in Mathematics, vol. 340 (Springer, 1973).CrossRefGoogle Scholar
Denef, F., Supergravity flows and D-brane stability, J. High Energy Phys. 8 (2000), paper 50; MR 1792870 (2002c:81154).Google Scholar
Denef, F., Quantum quivers and Hall/hole halos, J. High Energy Phys. 10 (2002), paper 023; MR 1952307 (2004b:81158).Google Scholar
Denef, F., Greene, B. and Raugas, M., Split attractor flows and the spectrum of BPS D-branes on the quintic, J. High Energy Phys. 5 (2001), paper 12; MR 1845419.CrossRefGoogle Scholar
Denef, F. and Moore, G., Split states, entropy enigmas, holes and halos, J. High Energy Phys. 2011(11) (2011), paper 129; MR 2913216.CrossRefGoogle Scholar
Derksen, H., Weyman, J. and Zelevinsky, A., Quivers with potentials and their representations. I. Mutations, Selecta Math. (N.S.) 14 (2008), 59–119; MR 2480710.CrossRefGoogle Scholar
Donaldson, S. and Thomas, R., Gauge theory in higher dimensions, in The geometric universe (Oxford, 1996) (Oxford University Press, Oxford, 1998), 31–47; MR 1634503.Google Scholar
Ferrara, S., Kallosh, R. and Strominger, A., $N=2$ extremal black holes, Phys. Rev. D (3) 52 (1995), R5412–R5416; MR 1360416.CrossRefGoogle ScholarPubMed
Filippini, S. and Stoppa, J., Block–Göttsche invariants from wall-crossing, Compos. Math. 151 (2015), 1543–1567; MR 3383167.CrossRefGoogle Scholar
Fiol, B., The BPS spectrum of $N=2$ $SU(N)$ SYM, J. High Energy Phys. 2 (2006), paper 065; MR 2219440.Google Scholar
Ginzburg, V., Calabi–Yau algebras, Preprint (2006), arXiv:0612139.Google Scholar
Gross, M., Mirror symmetry for $\mathbb {P}^2$ and tropical geometry, Adv. Math. 224 (2010), 169–245; MR 2600995.CrossRefGoogle Scholar
Gross, M., Hacking, P., Keel, S. and Kontsevich, M., Canonical bases for cluster algebras, J. Amer. Math. Soc. 31 (2018), 497–608; MR 3758151.CrossRefGoogle Scholar
Gross, M., Hacking, P. and Siebert, B., Theta functions on varieties with effective anti-canonical class, Mem. Amer. Math. Soc. 278 (2022); MR 4426709.Google Scholar
Gross, M. and Pandharipande, R., Quivers, curves, and the tropical vertex, Port. Math. 67 (2010), 211–259; MR 2662867 (2011g:14122).CrossRefGoogle Scholar
Gross, M., Pandharipande, R. and Siebert, B., The tropical vertex, Duke Math. J. 153 (2010), 297–362; MR 2667135 (2011f:14093).CrossRefGoogle Scholar
Gross, M. and Siebert, B., From real affine geometry to complex geometry, Ann. of Math. (2) 174 (2011), 1301–1428; MR 2846484.CrossRefGoogle Scholar
Joyce, D., On counting special Lagrangian homology 3-spheres, in Topology and geometry: commemorating SISTAG, Contemporary Mathematics, vol. 314 (American Mathematical Society, Providence, RI, 2002), 125–151; MR 1941627.CrossRefGoogle Scholar
Joyce, D. and Song, Y., A theory of generalized Donaldson–Thomas invariants, Mem. Amer. Math. Soc. 217 (2012); MR 2951762.Google Scholar
Keel, S. and Yu, T. Y., The Frobenius structure theorem for affine log Calabi–Yau varieties containing a torus, Preprint (2019), arXiv:1908.09861.Google Scholar
Keller, B., Calabi–Yau triangulated categories, in Trends in representation theory of algebras and related topics, EMS Series of Congress Reports (European Mathematical Society, Zürich, 2008), 467–489; MR 2484733.CrossRefGoogle Scholar
King, A., Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. (2) 45 (1994), 515–530; MR 1315461 (96a:16009).CrossRefGoogle Scholar
Kontsevich, M. and Soibelman, Y., Affine structures and non-Archimedean analytic spaces, in The unity of mathematics, Progress in Mathematics, vol. 244 (Birkhäuser, Boston, MA, 2006), 321–385; MR 2181810.CrossRefGoogle Scholar
Kontsevich, M. and Soibelman, Y., Stability structures, motivic Donaldson–Thomas invariants and cluster transformations, Preprint (2008), arXiv:0811.2435.Google Scholar
Kontsevich, M. and Soibelman, Y., Wall-crossing structures in Donaldson–Thomas invariants, integrable systems and mirror symmetry, in Homological mirror symmetry and tropical geometry, Lecture Notes of the Unione Matematica Italiana, vol. 15 (Springer, Cham, 2014), 197–308; MR 3330788.CrossRefGoogle Scholar
Lee, S.-J., Wang, Z.-L. and Yi, P., BPS states, refined indices, and quiver invariants, J. High Energy Phys. 2012 (2012), paper 094; MR 3033854.Google Scholar
Lee, S.-J., Wang, Z.-L. and Yi, P., Quiver invariants from intrinsic Higgs states, J. High Energy Phys. 2012 (2012), paper 169; MR 2967676.Google Scholar
Mandel, T., Scattering diagrams, theta functions, and refined tropical curve counts, J. Lond. Math. Soc. (2) 104 (2021), 2299–2334; MR 4368677.CrossRefGoogle Scholar
Manschot, J., Wall-crossing of D4-branes using flow trees, Adv. Theor. Math. Phys. 15 (2011), 1–42; MR 2888006.CrossRefGoogle Scholar
Manschot, J., Pioline, B. and Sen, A., Wall crossing from Boltzmann black hole halos, J. High Energy Phys. 2011 (2011), paper 59; MR 2875965.CrossRefGoogle Scholar
Manschot, J., Pioline, B. and Sen, A., From black holes to quivers, J. High Energy Phys. 2012 (2012), paper 023; MR 3036499.CrossRefGoogle Scholar
Manschot, J., Pioline, B. and Sen, A., On the Coulomb and Higgs branch formulae for multi-centered black holes and quiver invariants, J. High Energy Phys. 2013 (2013), paper 166; MR 3080495.CrossRefGoogle Scholar
Manschot, J., Pioline, B. and Sen, A., Generalized quiver mutations and single-centered indices, J. High Energy Phys. 2014 (2014), paper 1.CrossRefGoogle Scholar
Meinhardt, S. and Reineke, M., Donaldson–Thomas invariants versus intersection cohomology of quiver moduli, J. Reine Angew. Math. 754 (2019), 143–178; MR 4000572.CrossRefGoogle Scholar
Mou, L., Scattering diagrams of quivers with potentials and mutations, Preprint (2019), arXiv:1910.13714.Google Scholar
Mozgovoy, S., Operadic approach to wall-crossing, J. Algebra 596 (2022), 53–88; MR 4366305.CrossRefGoogle Scholar
Mozgovoy, S. and Pioline, B., Attractor invariants, brane tilings and crystals, Preprint (2020), arXiv:2012.14358.Google Scholar
Nishinou, T. and Siebert, B., Toric degenerations of toric varieties and tropical curves, Duke Math. J. 135 (2006), 1–51; MR 2259922.CrossRefGoogle Scholar
Pioline, B., Mathematica Package CoulombHiggs (2020), https://www.lpthe.jussieu.fr/pioline/computing.html.Google Scholar
Reineke, M., Poisson automorphisms and quiver moduli, J. Inst. Math. Jussieu 9 (2010), 653–667; MR 2650811 (2011h:16021).CrossRefGoogle Scholar
Reineke, M., Cohomology of quiver moduli, functional equations, and integrality of Donaldson–Thomas type invariants, Compos. Math. 147 (2011), 943–964; MR 2801406 (2012i:16031).CrossRefGoogle Scholar
Saito, M., Mixed Hodge modules, Publ. Res. Inst. Math. Sci. 26 (1990), 221–333; MR 1047415.CrossRefGoogle Scholar
Strominger, A., Macroscopic entropy of $N=2$ extremal black holes, Phys. Lett. B 383 (1996), 39–43; MR 1402863.CrossRefGoogle Scholar
Thomas, R., A holomorphic Casson invariant for Calabi–Yau 3-folds, and bundles on $K3$ fibrations, J. Differential Geom. 54 (2000), 367–438; MR 1818182 (2002b:14049).CrossRefGoogle Scholar
Thomas, R. and Yau, S.-T., Special Lagrangians, stable bundles and mean curvature flow, Comm. Anal. Geom. 10 (2002), 1075–1113; MR 1957663.CrossRefGoogle Scholar