## Duke Mathematical Journal

### Flops and the S-duality conjecture

Yukinobu Toda

#### Abstract

We prove the transformation formula of Donaldson–Thomas (DT) invariants counting two-dimensional torsion sheaves on Calabi–Yau 3-folds under flops. The error term is described by the Dedekind eta function and the Jacobi theta function, and our result gives evidence of a 3-fold version of the Vafa–Witten S-duality conjecture. As an application, we prove a blow-up formula of DT-type invariants on the total spaces of canonical line bundles on smooth projective surfaces. It gives an analogue of the similar blow-up formula in the original S-duality conjecture by Yoshioka, Li and Qin, and Göttsche.

#### Article information

Source
Duke Math. J., Volume 164, Number 12 (2015), 2293-2339.

Dates
Received: 20 December 2013
Revised: 15 October 2014
First available in Project Euclid: 16 September 2015

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1442364464

Digital Object Identifier
doi:10.1215/00127094-3129595

Mathematical Reviews number (MathSciNet)
MR3397387

Zentralblatt MATH identifier
1331.14055

#### Citation

Toda, Yukinobu. Flops and the S-duality conjecture. Duke Math. J. 164 (2015), no. 12, 2293--2339. doi:10.1215/00127094-3129595. https://projecteuclid.org/euclid.dmj/1442364464

#### References

• [1] K. Behrend, Donaldson-Thomas invariants via microlocal geometry, Ann. of Math. 170 (2009), 1307–1338.
• [2] T. Bridgeland, Flops and derived categories, Invent. Math. 147 (2002), 613–632.
• [3] T. Bridgeland, Stability conditions on triangulated categories, Ann. of Math. 166 (2007), 317–345.
• [4] T. Bridgeland, Hall algebras and curve-counting invariants, J. Amer. Math. Soc. 24 (2011), 969–998.
• [5] J. Bryan, S. Katz, and N. C. Leung, Multiple covers and integrality conjecture for rational curves on Calabi-Yau threefolds, J. Algebraic Geom. 10 (2001), 549–568.
• [6] J. Calabrese, Donaldson-Thomas invariants on flops, preprint, arXiv:1111.1670v5 [math.AG].
• [7] A. Căldăraru, The Mukai pairing, II: The Hochschild-Kostant-Rosenberg isomorphism, Adv. in Math. 194 (2005), 34–66.
• [8] J.-C. Chen, Flops and equivalences of derived categories for three-folds with only Gorenstein singularities, J. Differential Geom. 61 (2002), 227–261.
• [9] F. Denef and G. Moore, Split states, entropy enigmas, holes and halos, J. High Energy Phys. 11 (2011), 129.
• [10] W. Donovan and M. Wemyss, Noncommutative deformations and flops, preprint, arXiv:1309.0698v3 [math.AG].
• [11] M. Eichler and D. Zagier, The Theory of Jacobi Forms, Progr. in Math. 55, Birkhäuser, Boston, 1985.
• [12] W. Fulton, Intersection Theory, 2nd ed., Ergeb. Math. Grenzgeb. 3, Folge 2, Springer, Berlin.
• [13] D. Gaiotto, A. Strominger, and X. Yin, The M5-brane elliptic genus: Modularity and BPS states, J. High Energy Phys. 8 (2007), 070.
• [14] D. Gaiotto and X. Yin, Examples of M5-brane elliptic genera, J. High Energy Phys. 11 (2007), 004.
• [15] A. Gholampour and A. Sheshmani, Donaldson-Thomas invariants of 2-dimensional sheaves inside threefolds and modular forms, preprint, arXiv:1309.0050v2 [math.AG].
• [16] A. Gholampour and A. Sheshmani, Generalized Donaldson-Thomas invariants of 2-dimensional sheaves on local $\mathbb{P}^{2}$, preprint, arXiv:1309.0056v3 [math.AG].
• [17] A. Gholampour, A. Sheshmani, and R. P. Thomas, Counting curves on surfaces in Calabi-Yau 3-folds, Math. Ann. 360 (2014), 67–78.
• [18] L. Göttsche, Theta functions and Hodge numbers of moduli spaces of sheaves on rational surfaces, Comm. Math. Phys. 206 (1999), 105–136.
• [19] L. Göttsche, Invariants of moduli spaces and modular forms, Rend. Istit. Mat. Univ. Trieste 41 (2009), 55–76.
• [20] D. Happel, I. Reiten, and S. O. Smalø, Tilting in abelian categories and quasitilted algebras, Mem. Amer. Math. Soc. 120 (1996), no. 575.
• [21] J. Hu and W. P. Li, The Donaldson-Thomas invariants under blowups and flops, J. Differential Geom. 90 (2012), 391–411.
• [22] D. Huybrechts and M. Lehn, “Geometry of moduli spaces of sheaves” in Aspects in Mathematics, E31, Vieweg, Braunschweig, 1997.
• [23] D. Joyce, Configurations in abelian categories, II: Ringel-Hall algebras, Adv. Math. 210 (2007), 635–706.
• [24] D. Joyce, Configurations in abelian categories, IV: Invariants and changing stability conditions, Adv. in Math 217 (2008), 125–204.
• [25] D. Joyce and Y. Song, A theory of generalized Donaldson-Thomas invariants, Mem. Amer. Math. Soc. 217, Amer. Math. Soc., Providence, 2012.
• [26] S. Katz, Genus zero Gopakumar-Vafa invariants of contractible curves, J. Differential Geom. 79 (2008), 185–195.
• [27] S. Katz and D. R. Morrison, Gorenstein threefold singularities with small resolutions via invariant theory for Weyl groups, J. Algebraic Geom. 1 (1992), 449–530.
• [28] Y. Kawamata, Flops connect minimal models, Publ. Res. Inst. Math. Sci. 44 (2008), 419–423.
• [29] M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, preprint, arXiv:0811.2435v1 [math.AG].
• [30] A. Langer, Semistable sheaves in positive characteristic, Ann. of Math. 159 (2004), 251–276.
• [31] A. Langer, Moduli spaces of sheaves and principal $G$-bundles, Proc. Sympos. Pure Math. 80 (2009), 273–308.
• [32] W. P. Li and Z. Qin, On blowup formulae for the $S$-duality conjecture of Vafa and Witten, Invent. Math. 136 (1999), 451–482.
• [33] M. Lieblich, Moduli of complexes on a proper morphism, J. Algebraic Geom. 15 (2006), 175–206.
• [34] K. Nagao, Donaldson–Thomas theory and cluster algebras, Duke Math. J. 162 (2013), 1313–1367.
• [35] K. Nagao and H. Nakajima, Counting invariant of perverse coherent sheaves and its wall-crossing, Int. Math. Res. Not. IMRN (2011), no. 17, 3885–3938.
• [36] T. Nishinaka, Multiple D4-D2-D0 on the conifold and wall-crossing with the flop, J. High Energy Phys. 6 (2011), 065.
• [37] T. Nishinaka and S. Yamaguchi, Wall-crossing of D4-D2-D0 and flop of the conifold, J. High Energy Phys. 9 (2010), 026.
• [38] H. Ooguri, A. Strominger, and C. Vafa, Black hole attractors and the topological string, Phys. Rev. D 70 (2004), no. 10, 106007.
• [39] M. Reid, “Minimal models of canonical 3-foldings” in Algebraic Varieties and Analytic Varieties (S. Iitaka, ed.), Adv. Stud. Pure Math., Kinokuniya, Tokyo, and North-Holland, Amsterdam 1, 131–180.
• [40] R. P. Thomas, A holomorphic Casson invariant for Calabi-Yau 3-folds and bundles on ${K3}$-fibrations, J. Differential Geom. 54 (2000), 367–438.
• [41] Y. Toda, Birational Calabi-Yau 3-folds and BPS state counting, Comm. Number Theory and Physics 2 (2008), 63–112.
• [42] Y. Toda, Moduli stacks and invariants of semistable objects on K3 surfaces, Adv. in Math. 217 (2008), 2736–2781.
• [43] Y. Toda, Curve counting theories via stable objects, I: DT/PT correspondence, J. Amer. Math. Soc. 23 (2010), 1119–1157.
• [44] Y. Toda, Bogomolov-Gieseker type inequality and counting invariants, J. Topology 6 (2013), 217–250.
• [45] Y. Toda, Curve counting theories via stable objects, II: DT/ncDT flop formula, J. Reine Angew. Math. 675 (2013), 1–51.
• [46] Y. Toda, Stability conditions and extremal contractions, Math. Ann. 357 (2013), 631–685.
• [47] Y. Toda, Multiple cover formula of generalized DT invariants, I: Parabolic stable pairs, Adv. Math. 257 (2014), 476–526.
• [48] Y. Toda, Multiple cover formula of generalized DT invariants, II: Jacobian localizations, preprint, arXiv:1108.4993v1 [math.AG].
• [49] Y. Toda, S-duality for surfaces with $A_{n}$-type singularities, preprint, arXiv:1312.2300v2 [math.AG].
• [50] C. Vafa and E. Witten, A strong coupling test of S-duality, Nucl. Phys. B 431 (1994), 3–7.
• [51] M. Van den Bergh, Three-dimensional flops and noncommutative rings, Duke Math. J. 122 (2004), 423–455.
• [52] K. Yoshioka, Chamber structure of polarizations and the moduli space of rational elliptic surfaces, Int. J. Math. 7 (1996), 411–431.