Vafa-Witten invariants for projective surfaces I: stable case

Tanaka, Yuuji; Thomas, Richard

doi:10.1090/jag/738

Authors: Yuuji Tanaka and Richard P. Thomas
Journal: J. Algebraic Geom. 29 (2020), 603-668
DOI: https://doi.org/10.1090/jag/738
Published electronically: October 23, 2019
Full-text PDF

Abstract | References | Additional Information

Abstract:

On a polarised surface, solutions of the Vafa-Witten equations correspond to certain polystable Higgs pairs. When stability and semistability coincide, the moduli space admits a symmetric obstruction theory and a $\mathbb {C}^*$ action with compact fixed locus. Applying virtual localisation we define invariants constant under deformations.

When the vanishing theorem of Vafa-Witten holds, the result is the (signed) Euler characteristic of the moduli space of instantons. In general there are other, rational, contributions. Calculations of these on surfaces with positive canonical bundle recover the first terms of modular forms predicted by Vafa and Witten.

References

Luis Álvarez-Cónsul and Oscar García-Prada, Hitchin-Kobayashi correspondence, quivers, and vortices, Comm. Math. Phys. 238 (2003), no. 1-2, 1–33. MR 1989667, DOI https://doi.org/10.1007/s00220-003-0853-1
E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. MR 770932
Kai Behrend, Donaldson-Thomas type invariants via microlocal geometry, Ann. of Math. (2) 170 (2009), no. 3, 1307–1338. MR 2600874, DOI https://doi.org/10.4007/annals.2009.170.1307
Kai Behrend, Jim Bryan, and Balázs Szendrői, Motivic degree zero Donaldson-Thomas invariants, Invent. Math. 192 (2013), no. 1, 111–160. MR 3032328, DOI https://doi.org/10.1007/s00222-012-0408-1
K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), no. 1, 45–88. MR 1437495, DOI https://doi.org/10.1007/s002220050136

D. Boccalini, Homology of the three flag Hilbert scheme, arXiv:1609.05005, 2016.

A. Bondal and D. Orlov, Semiorthogonal decomposition for algebraic varieties, alg-geom/9506012, 1995.

Dennis Borisov and Dominic Joyce, Virtual fundamental classes for moduli spaces of sheaves on Calabi-Yau four-folds, Geom. Topol. 21 (2017), no. 6, 3231–3311. MR 3692967, DOI https://doi.org/10.2140/gt.2017.21.3231

K. Buzzard and N. Elkies, http://mathoverflow.net/questions/244808/, 2016.

Damien Calaque, Shifted cotangent stacks are shifted symplectic, Ann. Fac. Sci. Toulouse Math. (6) 28 (2019), no. 1, 67–90 (English, with English and French summaries). MR 3940792, DOI https://doi.org/10.5802/afst.1593

Y. Cao and N.C. Leung, Donaldson-Thomas theory for Calabi-Yau 4-folds, preprint, arXiv:1407.7659, 2014.

Stefan Cordes, Gregory Moore, and Sanjaye Ramgoolam, Large $N$ $2$D Yang-Mills theory and topological string theory, Comm. Math. Phys. 185 (1997), no. 3, 543–619. MR 1463054, DOI https://doi.org/10.1007/s002200050102

Kevin Costello, Notes on supersymmetric and holomorphic field theories in dimensions 2 and 4, Pure Appl. Math. Q. 9 (2013), no. 1, 73–165. MR 3126501, DOI https://doi.org/10.4310/PAMQ.2013.v9.n1.a3

Ben Davison, The critical CoHA of a quiver with potential, Q. J. Math. 68 (2017), no. 2, 635–703. MR 3667216, DOI https://doi.org/10.1093/qmath/haw053

Robbert Dijkgraaf and Gregory Moore, Balanced topological field theories, Comm. Math. Phys. 185 (1997), no. 2, 411–440. MR 1463049, DOI https://doi.org/10.1007/s002200050097

R. Dijkgraaf, J-S. Park, and B. Schroers, $N=4$ supersymmetric Yang-Mills theory on a Kähler surface, hep-th/9801066, 1998.

Geir Ellingsrud, Lothar Göttsche, and Manfred Lehn, On the cobordism class of the Hilbert scheme of a surface, J. Algebraic Geom. 10 (2001), no. 1, 81–100. MR 1795551

William Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323

A. Gholampour, A. Sheshmani, and S.-T. Yau, Nested Hilbert schemes on surfaces: Virtual fundamental class, arXiv:1701.08899, 2017.

A. Gholampour, A. Sheshmani, and S.-T. Yau, Localized Donaldson-Thomas theory of surfaces, arXiv:1701.08902, 2017.

A. Gholampour and R. P. Thomas, Degeneracy loci, virtual cycles and nested Hilbert schemes, arXiv:1709.06105, 2017.

W. D. Gillam, Deformation of quotients on a product, arXiv:1103.5482, 2011.

L. Göttsche and M. Kool, Virtual refinements of the Vafa-Witten formula, arXiv:1703.07196, 2017.

Lothar Göttsche, Hiraku Nakajima, and K\B{o}ta Yoshioka, Instanton counting and Donaldson invariants, J. Differential Geom. 80 (2008), no. 3, 343–390. MR 2472477

T. Graber and R. Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999), no. 2, 487–518. MR 1666787, DOI https://doi.org/10.1007/s002220050293

Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157

Daniel Huybrechts and Richard P. Thomas, Deformation-obstruction theory for complexes via Atiyah and Kodaira-Spencer classes, Math. Ann. 346 (2010), no. 3, 545–569. MR 2578562, DOI https://doi.org/10.1007/s00208-009-0397-6

Luc Illusie, Complexe cotangent et déformations. I, Lecture Notes in Mathematics, Vol. 239, Springer-Verlag, Berlin-New York, 1971 (French). MR 0491680

Yunfeng Jiang, Note on MacPherson’s local Euler obstruction, Michigan Math. J. 68 (2019), no. 2, 227–250. MR 3961215, DOI https://doi.org/10.1307/mmj/1548817530

Yunfeng Jiang and Richard P. Thomas, Virtual signed Euler characteristics, J. Algebraic Geom. 26 (2017), no. 2, 379–397. MR 3607000, DOI https://doi.org/10.1090/S1056-3911-2016-00690-2

Dominic Joyce and Yinan Song, A theory of generalized Donaldson-Thomas invariants, Mem. Amer. Math. Soc. 217 (2012), no. 1020, iv+199. MR 2951762, DOI https://doi.org/10.1090/S0065-9266-2011-00630-1

Anton Kapustin and Edward Witten, Electric-magnetic duality and the geometric Langlands program, Commun. Number Theory Phys. 1 (2007), no. 1, 1–236. MR 2306566, DOI https://doi.org/10.4310/CNTP.2007.v1.n1.a1

Young-Hoon Kiem and Jun Li, Localizing virtual cycles by cosections, J. Amer. Math. Soc. 26 (2013), no. 4, 1025–1050. MR 3073883, DOI https://doi.org/10.1090/S0894-0347-2013-00768-7

Young-Hoon Kiem and Jun Li, A wall crossing formula of Donaldson-Thomas invariants without Chern-Simons functional, Asian J. Math. 17 (2013), no. 1, 63–94. MR 3038725, DOI https://doi.org/10.4310/AJM.2013.v17.n1.a4

M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435, 2008.

Martijn Kool, Fixed point loci of moduli spaces of sheaves on toric varieties, Adv. Math. 227 (2011), no. 4, 1700–1755. MR 2799810, DOI https://doi.org/10.1016/j.aim.2011.04.002

Martijn Kool and Richard Thomas, Reduced classes and curve counting on surfaces I: theory, Algebr. Geom. 1 (2014), no. 3, 334–383. MR 3238154, DOI https://doi.org/10.14231/AG-2014-017

Martijn Kool and Richard P. Thomas, Stable pairs with descendents on local surfaces I: the vertical component, Pure Appl. Math. Q. 13 (2017), no. 4, 581–638. With an appendix by Aaron Pixton and Don Zagier. MR 3903061, DOI https://doi.org/10.4310/PAMQ.2017.v13.n4.a2

J. M. F. Labastida and Carlos Lozano, The Vafa-Witten theory for gauge group ${\rm SU}(N)$, Adv. Theor. Math. Phys. 3 (1999), no. 5, 1201–1225. MR 1796678, DOI https://doi.org/10.4310/ATMP.1999.v3.n5.a1

Jun Li and Gang Tian, Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11 (1998), no. 1, 119–174. MR 1467172, DOI https://doi.org/10.1090/S0894-0347-98-00250-1

D. Maulik and R. P. Thomas, Sheaf counting on local K3 surfaces, arXiv:1806.02657, 2018.

T. Mochizuki, A theory of the invariants obtained from the moduli stacks of stable objects on a smooth polarized surface, http://arxiv.org/abs/math/0210211, 2002.

A. Neguţ, AGT relations for sheaves on surfaces, arXiv:1711.00390, 2017.

Timo Schürg, Bertrand Toën, and Gabriele Vezzosi, Derived algebraic geometry, determinants of perfect complexes, and applications to obstruction theories for maps and complexes, J. Reine Angew. Math. 702 (2015), 1–40. MR 3341464, DOI https://doi.org/10.1515/crelle-2013-0037

Carlos T. Simpson, Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, J. Amer. Math. Soc. 1 (1988), no. 4, 867–918. MR 944577, DOI https://doi.org/10.1090/S0894-0347-1988-0944577-9

Richard P. Stanley, Enumerative combinatorics. Volume 1, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 2012. MR 2868112

Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. MR 1676282

Yuuji Tanaka, Stable sheaves with twisted sections and the Vafa-Witten equations on smooth projective surfaces, Manuscripta Math. 146 (2015), no. 3-4, 351–363. MR 3312450, DOI https://doi.org/10.1007/s00229-014-0706-6

Yuuji Tanaka, On the singular sets of solutions to the Kapustin-Witten equations and the Vafa-Witten ones on compact Kähler surfaces, Geom. Dedicata 199 (2019), 177–187. MR 3928797, DOI https://doi.org/10.1007/s10711-018-0344-3

Yuuji Tanaka and Richard P. Thomas, Vafa-Witten invariants for projective surfaces II: semistable case, Pure Appl. Math. Q. 13 (2017), no. 3, 517–562. MR 3882207, DOI https://doi.org/10.4310/pamq.2017.v13.n3.a6

R. P. Thomas, A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on $K3$ fibrations, J. Differential Geom. 54 (2000), no. 2, 367–438. MR 1818182

Bertrand Toën and Michel Vaquié, Moduli of objects in dg-categories, Ann. Sci. École Norm. Sup. (4) 40 (2007), no. 3, 387–444 (English, with English and French summaries). MR 2493386, DOI https://doi.org/10.1016/j.ansens.2007.05.001

Cumrun Vafa and Edward Witten, A strong coupling test of $S$-duality, Nuclear Phys. B 431 (1994), no. 1-2, 3–77. MR 1305096, DOI https://doi.org/10.1016/0550-3213%2894%2990097-3

Additional Information

Yuuji Tanaka
Affiliation: Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, United Kingdom
MR Author ID: 836351
Email: tanaka@maths.ox.ac.uk

Richard P. Thomas
Affiliation: Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom
MR Author ID: 636321
Email: richard.thomas@imperial.ac.uk

Received by editor(s): January 19, 2018
Received by editor(s) in revised form: November 21, 2018
Published electronically: October 23, 2019
Additional Notes: The first author was partially supported by JSPS Grant-in-Aid for Scientific Research numbers JP15H02054 and JP16K05125, and a Simons Collaboration Grant on “Special holonomy in Geometry, Analysis and Physics”. He thanks Seoul National University, NCTS at National Taiwan University, Kyoto University, and BICMR at Peking University for their support and hospitality during visits in 2015–17 when part of this work was done.
Article copyright: © Copyright 2019 University Press, Inc.