Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-28T03:39:24.896Z Has data issue: false hasContentIssue false
  • English
  • Français

Quasimaps and stable pairs

Part of: Cycles and subschemes Projective and enumerative geometry

Published online by Cambridge University Press:  12 April 2021

Henry Liu Open the ORCID record for Henry Liu [Opens in a new window]
Henry Liu*
Affiliation:
Department of Mathematics, Columbia University, 2990 Broadway, New YorkNY10027, USA; E-mail: hliu@math.columbia.edu

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove an equivalence between the Bryan-Steinberg theory of $\pi $-stable pairs on $Y = \mathcal {A}_{m-1} \times \mathbb {C}$ and the theory of quasimaps to $X = \text{Hilb}(\mathcal {A}_{m-1})$, in the form of an equality of K-theoretic equivariant vertices. In particular, the combinatorics of both vertices are described explicitly via box counting. Then we apply the equivalence to study the implications for sheaf-counting theories on Y arising from 3D mirror symmetry for quasimaps to X, including the Donaldson-Thomas crepant resolution conjecture.

MSC classification

Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants
Secondary: 14C05: Parametrization (Chow and Hilbert schemes)
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 9 , 2021 , e32
Check for updates
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited
Copyright
© The Author(s), 2021. Published by Cambridge University Press

References

Aganagic, M. and Okounkov, A., ‘Elliptic stable envelopes’, 2016, arXiv:1604.00423.Google Scholar
Beentjes, S. V., Calabrese, J. and Vold Rennemo, J., ‘A proof of the Donaldson-Thomas crepant resolution conjecture’, 2018, arXiv:1810.06581.Google Scholar
Bridgeland, T., King, A. and Reid, M., ‘The McKay correspondence as an equivalence of derived categories’, J. Amer. Math. Soc. 14(3) (2001), 535–554.CrossRefGoogle Scholar
Bryan, J., Cadman, C. and Young, B., ‘The orbifold topological vertex’, Adv. Math. 229(1) (2012), 531–595.CrossRefGoogle Scholar
Bryan, J. and Steinberg, D., ‘Curve counting invariants for crepant resolutions’, Trans. Amer. Math. Soc. 368(3) (2016), 1583–1619.CrossRefGoogle Scholar
Carlsson, E. and Okounkov, A., ‘Exts and vertex operators’, Duke Math. J. 161(9) (2012), 1797–1815.CrossRefGoogle Scholar
Cautis, S. and Logvinenko, T., ‘A derived approach to geometric McKay correspondence in dimension three’, J. Reine Angew. Math. 636 (2009), 193–236.Google Scholar
Ciocan-Fontanine, I. and Kapranov, M., ‘Virtual fundamental classes via dg-manifolds’, Geom. Topol. 13(3) (2009), 1779–1804.CrossRefGoogle Scholar
Ciocan-Fontanine, I., Kim, B. and Maulik, D., ‘Stable quasimaps to GIT quotients’, J. Geom. Phys. 75 (2014), 17–47.CrossRefGoogle Scholar
de Boer, J., Hori, K., Ooguri, H., Oz, Y. and Yin, Z., ‘Mirror symmetry in three-dimensional gauge theories, $\text{SL}\left(2,\mathbb{Z}\right)$and D-brane moduli spaces’, Nuclear Phys. B 493(1–2) 1997, 148–176.CrossRefGoogle Scholar
Dinkins, H. and Smirnov, A., ‘Characters of tangent spaces at torus fixed points and $3d$-mirror symmetry’, 2019, arXiv:1908.01199.CrossRefGoogle Scholar
Dinkins, H. and Smirnov, A., ‘Quasimaps to zero-dimensional ${A}_{\infty }$-quiver varieties’, 2019, arXiv:1912.04834.Google Scholar
Fernandez, J., ‘Hodge structures for orbifold cohomology’, Proc. Amer. Math. Soc. 134(9) (2006), 2511–2520.CrossRefGoogle Scholar
Gansner, E. R., ‘The Hillman-Grassl correspondence and the enumeration of reverse plane partitions’, J. Combin. Theory Ser. A 30(1) (1981), 71–89.CrossRefGoogle Scholar
Ginzburg, V., ‘Lectures on Nakajima’s quiver varieties’, in Geometric Methods in Representation Theory. I, Vol. 24 of Sémin. Congr. (Soc. Math. France, Paris, 2012), 145–219.Google Scholar
Givental, A. B., ‘Equivariant Gromov-Witten invariants’, Internat. Math. Res. Notices (13) (1996), 613–663.CrossRefGoogle Scholar
Gonzalez-Sprinberg, G. and Verdier, J.-L., ‘Construction géométrique de la correspondance de McKay’, Ann. Sci. École Norm. Sup. (4) 16(3) (1984), 409–449.CrossRefGoogle Scholar
Gothen, P. B. and King, A. D., ‘Homological algebra of twisted quiver bundles’, J. London Math. Soc. (2) 71(1) (2005), 85–99.CrossRefGoogle Scholar
Intriligator, K. and Seiberg, N., ‘Mirror symmetry in three-dimensional gauge theories’, Phys. Lett. B 387(3) (1996), 513–519.CrossRefGoogle Scholar
Iqbal, A. and Kashani-Poor, A.-K., ‘The vertex on a strip’, Adv. Theor. Math. Phys. 10(3) (2006), 317–343.CrossRefGoogle Scholar
Ishii, S., Introduction to Singularities (Springer, Tokyo, 2014).Google Scholar
James, G. and Kerber, A., The Representation Theory of the Symmetric Group, Vol. 16 of Encyclopedia of Mathematics and Its Applications (Addison-Wesley Publishing Co., Reading, MA, 1981). With a foreword by Cohn, P. M.. With an introduction by Gilbert de B. Robinson.Google Scholar
Kapranov, M. and Vasserot, E., ‘Kleinian singularities, derived categories and Hall algebras’, Math. Ann. 316(3) (2000), 565–576.CrossRefGoogle Scholar
Kuznetsov, A., ‘Quiver varieties and Hilbert schemes’, Mosc. Math. J. 7(4) (2007), 673–697, 767.CrossRefGoogle Scholar
Lee, Y.-P., ‘Quantum $K$-theory. I. Foundations’, Duke Math. J. 121(3) (2004), 389–424.CrossRefGoogle Scholar
Lian, B. H., Liu, K. and Yau, S.-T., ‘Mirror principle. I’, in Surveys in Differential Geometry: Differential Geometry Inspired by String Theory, Vol. 5 of Surv. Differ. Geom. (International Press, Boston, MA, 1999), 405–454.Google Scholar
Maulik, D., Nekrasov, N., Okounkov, A. and Pandharipande, R.’, ‘Gromov-Witten theory and Donaldson-Thomas theory. I’, Compos. Math. 142(5) (2006), 1263–1285.CrossRefGoogle Scholar
Maulik, D., Nekrasov, N., Okounkov, A. and Pandharipande, R.’, ‘Gromov-Witten theory and Donaldson-Thomas theory. II’, Compos. Math. 142(5) (2006), 1286–1304.CrossRefGoogle Scholar
Maulik, D., Oblomkov, A., Okounkov, A. and Pandharipande, R.’, ‘Gromov-Witten/Donaldson-Thomas correspondence for toric 3-folds’, Invent. Math. 186(2) (2011), 435–479.CrossRefGoogle Scholar
Nagao, K.’, ‘Quiver varieties and Frenkel-Kac construction’, J. Algebra 321(12) (2009), 3764–3789.CrossRefGoogle Scholar
Nakajima, H., ‘Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras’, Duke Math. J. 76(2) (1994), 365–416.CrossRefGoogle Scholar
Nakajima, H., Lectures on Hilbert Schemes of Points on Surfaces, Vol. 18 of University Lecture Series (American Mathematical Society, Providence, RI, 1999).CrossRefGoogle Scholar
Nakajima, H., ‘Sheaves on ALE spaces and quiver varieties’, Mosc. Math. J. 7(4) (2007), 699–722, 767.CrossRefGoogle Scholar
Nekrasov, N. and Okounkov, A., ‘Membranes and sheaves’, Algebr. Geom. 3(3) (2016), 320–369.CrossRefGoogle Scholar
Oberdieck, G. and Pixton, A., ‘Holomorphic anomaly equations and the Igusa cusp form conjecture’, Invent. Math. 213(2) (2018), 507–587.CrossRefGoogle Scholar
Okounkov, A., ‘Lectures on K-theoretic computations in enumerative geometry’, in Geometry of Moduli Spaces and Representation Theory, Vol. 24 of IAS/Park City Math. Ser. (American Mathematical Society, Providence, RI, 2017), 251–380.CrossRefGoogle Scholar
Okounkov, A., ‘Enumerative symplectic duality’, presented at MSRI workshop Structures in Enumerative Geometry, March 2018.Google Scholar
Okounkov, A., Reshetikhin, N., and Vafa, C., ‘Quantum Calabi-Yau and classical crystals’, in The Unity of Mathematics, Vol. 244 of Progr. Math., (Birkhäuser, Boston, MA, 2006), 597–618.Google Scholar
Pandharipande, R. and Thomas, R. P., ‘The 3-fold vertex via stable pairs’, Geom. Topol. 13(4) (2009), 1835–1876.CrossRefGoogle Scholar
Pandharipande, R. and Thomas, R. P., ‘Curve counting via stable pairs in the derived category’, Invent. Math. 178(2) (2009), 407–447.CrossRefGoogle Scholar
Rimányi, R., Smirnov, A., Varchenko, A. and Zhou, Z., ‘Three-dimensional mirror self-symmetry of the cotangent bundle of the full flag variety’, SIGMA Symmetry Integrability Geom. Methods Appl. 15 (2019), 22.Google Scholar
Ross, D., ‘Donaldson-Thomas theory and resolutions of toric $A$-singularities’, Selecta Math. (N.S.) 23(1) (2017), 15–37.CrossRefGoogle Scholar
Smirnov, A., ‘Rationality of capped descendent vertex in $K$-theory’, 2016,arXiv:1612.01048.Google Scholar
Toda, Y., ‘Curve counting theories via stable objects I. DT/PT correspondence’, J. Amer. Math. Soc. 23(4) (2010), 1119–1157.CrossRefGoogle Scholar