Geodesic
Monopole contributions to refined Vafa–Witten invariants
Laarakker, Ties1
1 Imperial College, London, United Kingdom
Geometry & topology, Tome 24 (2020) no. 6, pp. 2781-2828.

Voir la notice de l'article provenant de la source Mathematical Sciences Publishers

We study the monopole contribution to the refined Vafa–Witten invariant recently defined by Maulik and Thomas (work in progress). We apply the results of Gholampour and Thomas (to appear in Compos. Math.) to prove a universality result for the generating series of contributions of Higgs pairs with 1–dimensional weight spaces. For prime rank, these account for the entire monopole contribution by a theorem of Thomas. We use toric computations to determine part of the generating series and find agreement with the conjectures of Göttsche and Kool (Pure Appl. Math. Q. 14 (2018) 467–513) for ranks 2 and 3.

Classification : 14C05, 14D20, 14J80
Keywords: Vafa–Witten invariants, monopole contribution, VW invariants

Laarakker, Ties 1

1 Imperial College, London, United Kingdom 
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Laarakker, Ties. Monopole contributions to refined Vafa–Witten invariants. Geometry & topology, Tome 24 (2020) no. 6, pp. 2781-2828. http://geodesic.mathdoc.fr/item/GT_2020_24_6_a3/

[1] E Carlsson, A Okounkov, Exts and vertex operators, Duke Math. J. 161 (2012) 1797 | DOI

[2] G Ellingsrud, L Göttsche, M Lehn, On the cobordism class of the Hilbert scheme of a surface, J. Algebraic Geom. 10 (2001) 81

[3] B Fantechi, L Göttsche, Riemann–Roch theorems and elliptic genus for virtually smooth schemes, Geom. Topol. 14 (2010) 83 | DOI

[4] A Gholampour, A Sheshmani, S T Yau, Localized Donaldson–Thomas theory of surfaces, Amer. J. Math. 142 (2020) 405 | DOI

[5] A Gholampour, A Sheshmani, S T Yau, Nested Hilbert schemes on surfaces : virtual fundamental class, Adv. Math. 365 (2020) 107046 | DOI

[6] A Gholampour, R P Thomas, Degeneracy loci, virtual cycles and nested Hilbert schemes, I, Tunis. J. Math. 2 (2020) 633 | DOI

[7] A Gholampour, R P Thomas, Degeneracy loci, virtual cycles and nested Hilbert schemes, II, Compos. Math. 156 (2020) 1623 | DOI

[8] L Göttsche, The Betti numbers of the Hilbert scheme of points on a smooth projective surface, Math. Ann. 286 (1990) 193 | DOI

[9] L Göttsche, A conjectural generating function for numbers of curves on surfaces, Comm. Math. Phys. 196 (1998) 523 | DOI

[10] L Göttsche, M Kool, Refined SU(3) Vafa–Witten invariants and modularity, Pure Appl. Math. Q. 14 (2018) 467 | DOI

[11] L Göttsche, M Kool, Virtual refinements of the Vafa–Witten formula, Comm. Math. Phys. 376 (2020) 1 | DOI

[12] L Göttsche, H Nakajima, K Yoshioka, Instanton counting and Donaldson invariants, J. Differential Geom. 80 (2008) 343 | DOI

[13] L Göttsche, W Soergel, Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces, Math. Ann. 296 (1993) 235 | DOI

[14] T Graber, R Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999) 487 | DOI

[15] T Laarakker, Vertical Vafa–Witten invariants, (2019)

[16] Y P Lee, R Pandharipande, Algebraic cobordism of bundles on varieties, J. Eur. Math. Soc. 14 (2012) 1081 | DOI

[17] D Maulik, N Nekrasov, A Okounkov, R Pandharipande, Gromov–Witten theory and Donaldson–Thomas theory, I, Compos. Math. 142 (2006) 1263 | DOI

[18] T Mochizuki, Donaldson type invariants for algebraic surfaces, 1972, Springer (2009) | DOI

[19] T S Developers, SageMath : the Sage mathematics software system, version 6.10 (2015)

[20] Y Tanaka, R P Thomas, Vafa–Witten invariants for projective surfaces, I : Stable case, J. Algebraic Geom. 29 (2020) 603 | DOI

[21] R P Thomas, Equivariant K–theory and refined Vafa–Witten invariants, Comm. Math. Phys. (2020) | DOI

[22] C Vafa, E Witten, A strong coupling test of S–duality, Nuclear Phys. B 431 (1994) 3 | DOI