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GW/PT Descendent Correspondence via Vertex Operators

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Abstract

We propose an explicit formula for the \({{\mathsf {GW}}}/{\mathsf {PT}}\) descendent correspondence in the stationary case for nonsingular connected projective threefolds. The formula, written in terms of vertex operators, is found by studying the 1-leg geometry. We prove the proposal for all nonsingular projective toric threefolds. An application to the Virasoro constraints for the stationary descendent theory of stable pairs will appear in a sequel.

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Notes

  1. We take singular cohomology always with \({\mathbb {C}}\)-coefficients.

  2. The terminology agrees with the definition of stationary descendents in case X is a curve [22].

  3. Here, \(\tau _{i_1}\tau _{i_2}\dots \tau _{i_k}\) has degree k.

  4. If X has odd cohomology, then supercommutative. For simplicity, our analysis will restricted to commutative case. The modifications for odd cohomology are not significant and are left to the reader.

  5. Every operator in Sect. 2 is assumed to be an element of \(\mathfrak {gl}(V)\) but not \(\mathfrak {gl}(\Lambda ^{\infty /2} V)\).

  6. In [24], the notation \(\mathrm {A}_k={\mathcal {A}}_{k+1}\) is used.

  7. The equivariant cohomology of U is generated over \({\mathbb {Q}}[s,t]\) by the class \({\mathsf {p}}\) of the fixed point.

  8. See Section 2.3 of [7].

  9. \(H=\sum _{k>0} \alpha _{-k}\alpha _{k}\).

  10. For shorter formulas, we now drop \(Q\) from the notation.

  11. The existence of a \({\mathsf {T}}\)-equivariant \({{\mathsf {GW}}}/{{\mathsf {PT}}}\) descendent correspondence is proven in [29], but closed formulas are not known.

  12. For fun, we had originally termed these Frankenstein series.

  13. \(n=0\) term is defined by renormalization.

References

  1. Aganagic, M., Okounkov, A.: Quasimap counts and Bethe eigenfunctions. arXiv:1704.08746

  2. Behrend, K., Fantechi, B.: The intrinsic normal cone. Invent. Math. 128, 45–88 (1997)

    Google Scholar 

  3. Bridgeland, T.: Hall algebras and curve-counting invariants. JAMS 24, 969–998 (2011)

    Google Scholar 

  4. Bryan, J., Pandharipande, R.: Local Gromov–Witten theory of curves. JAMS 21, 101–136 (2008)

    Google Scholar 

  5. Fulton, W., Pandharipande, R.: Notes on stable maps and quantum cohomology. In: Proceedings of Algebraic Geometry—Santa Cruz (1995), Proceedings of Symposium on Pure Mathematics, vol. 62, pp. 45–96

  6. Getzler, E.: The equivariant Toda lattice. Publ. Res. Inst. Math. Sci. 40(2), 507–536 (2004)

    Google Scholar 

  7. Getzler, E., Pandharipande, R.: Virasoro constraints and Chern classes of the Hodge bundle. Nucl. Phys. B 530, 701–714 (1998)

    Google Scholar 

  8. Givental, A.: Gromov–Witten invariants and quantization of quadratic Hamiltonians. Mosc. Math. J. 1(4), 551–568 (2001)

    Google Scholar 

  9. Graber, T., Pandharipande, R.: Localization of virtual classes. Invent. Math. 135(2), 487–518 (1999)

    Google Scholar 

  10. Kassel, Ch.: Quantum Groups. Graduate Texts in Mathematics, vol. 155. Springer, New York (1995)

    Google Scholar 

  11. Li, J., Tian, G.: Virtual moduli cycles and Gromov–Witten invariants of algebraic varieties. JAMS 11, 119–174 (1998)

    Google Scholar 

  12. Li, Wei-ping, Qin, Zhenbo, Wang, Weiqiang: Vertex algebras and the cohomology ring structure of Hilbert schemes of points on surfaces. Math. Ann. 324(1), 105–133 (2002)

    Google Scholar 

  13. Maulik, D., Nekrasov, N., Okounkov, A., Pandharipande, R.: Gromov–Witten theory and Donaldson–Thomas theory. I. Compos. Math. 142(5), 1263–1285 (2006)

    Google Scholar 

  14. Maulik, D., Nekrasov, N., Okounkov, A., Pandharipande, R.: Gromov–Witten theory and Donaldson–Thomas theory. II. Compos. Math. 142(5), 1286–1304 (2006)

    Google Scholar 

  15. Maulik, D., Oblomkov, A.: DT theory of \(A_n\times {\mathbf{P}}^1\). Compos. Math. 145(5), 1249–1276 (2009)

    Google Scholar 

  16. Maulik, D., Oblomkov, A., Okounkov, A., Pandharipande, R.: GW/DT correspondence for toric varieties. Invent. Math. 186(2), 435–479 (2011)

    Google Scholar 

  17. Maulik, D., Pandharipande, R., Thomas, R.: Curves on K3 surfaces and modular forms. With an appendix by A. Pixton. J. Topol. 3(4), 937–996 (2010)

    Google Scholar 

  18. Moreira, M., Oblomkov, A., Okounkov, A., Pandharipande, R.: Virasoro constraints for stable pairs on toric 3-folds (in preparation)

  19. Okounkov, A.: Descendent correspondence. Talk at Compositio Prize Festivity, June 18, 2010, https://burttotaro.wordpress.com/2010/06/18/2nd-july-compositio-prize-festivity/

  20. Okounkov, A.: Hilbert schemes and multiple q-zeta values. Funct. Anal. Appl. 48(2), 138–144 (2014)

    Google Scholar 

  21. Okounkov, A., Olshanski, G.: Shifted Schur functions. St. Petersb. Math. J. 9(2), 239–300 (1998)

    Google Scholar 

  22. Okounkov, A., Pandharipande, R.: GW theory, Hurwitz theory and completed cycles. Ann. Math. (2) 163(2), 517–560 (2006)

    Google Scholar 

  23. Okounkov, A., Pandharipande, R.: The equivariant Gromov–Witten theory of \({\mathbf{P}}^1\). Ann. Math. (2) 163(2), 561–605 (2006)

    Google Scholar 

  24. Okounkov, A., Pandharipande, R.: Virasoro constraints for target curves. Invent. Math. 163(1), 47–108 (2006)

    Google Scholar 

  25. Pandharipande, R.: Descendents for stable pairs on 3-folds, modern geometry: a celebration of the work of Simon Donaldson. Proc. Symp. Pure Math. 99, 251–288 (2018). arXiv:1703.01747

    Google Scholar 

  26. Pandharipande, R., Pixton, A.: Descendents on local curves: rationality. Comput. Math. 149, 81–124 (2013)

    Google Scholar 

  27. Pandharipande, R., Pixton, A.: Descendents on local curves: stationary theory. In: Geometry and Arithmetic, pp. 283–307, EMS Series of Congress Reports, European Mathematical Society, Zürich (2012)

  28. Pandharipande, R., Pixton, A.: Descendent theory for stable pairs on toric 3-folds. J. Math. Soc. Jpn. 65, 1337–1372 (2013)

    Google Scholar 

  29. Pandharipande, R., Pixton, A.: Gromov–Witten/pairs descendent correspondence for toric 3-folds. Geom. Topol. 18, 2747–2821 (2014)

    Google Scholar 

  30. Pandharipande, R., Pixton, A.: Gromov–Witten/pairs correspondence for the quintic. JAMS 30, 389–449 (2017)

    Google Scholar 

  31. Pandharipande, R., Thomas, R.P.: Curve counting via stable pairs in the derived category. Invent. Math. 178, 407–447 (2009)

    Google Scholar 

  32. Pandharipande, R., Thomas, R.P.: The 3-fold vertex via stable pairs. Geom. Topol. 13, 1835–1876 (2009)

    Google Scholar 

  33. Pandharipande, R., Thomas, R.P.: 13/2 ways of counting curves. In: Brambila-Paz, L. (ed.) Moduli Spaces. LMS Lecture Note Series, vol. 411, pp. 282–333. Cambridge University Press, Cambridge (2014)

    Google Scholar 

  34. Pixton, A.: Gromov–Witten theory of an elliptic curve and quasi-modular forms. Senior Thesis, Princeton University (2009)

  35. Smirnov, A.: Rationality of capped descendent vertex in K-theory. arXiv:1612.01048

  36. Toda, Y.: Curve counting theories via stable objects I: DT/PT correspondence. JAMS 23, 1119–1157 (2010)

    Google Scholar 

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Acknowledgements

We are very grateful to D. Maulik, M. Moreira, N. Nekrasov, G. Oberdieck, A. Pixton, J. Shen, R. Thomas, and Q. Yin for many conversations about descendents and descendent correspondences. A. Ob. was partially supported by NSF CAREER Grant DMS-1352398. This paper is based upon work supported by the National Science Foundation under Grant No. 1440140, while the first two authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California during the Spring semester of 2018. A. Ok. was partially supported by the Simons Foundation as a Simons Investigator. A. Ok. gratefully acknowledges funding by the Russian Academic Excellence Project ‘5-100’ and RSF Grant 16-11-10160. R. P. was partially supported by SNF-200021143274, SNF-200020162928, ERC-2012-AdG-320368-MCSK, SwissMAP, and the Einstein Stiftung. This project has received funding from the European Research Council (ERC) under the European Union Horizon 2020 research and innovation program (Grant Agreement No. 786580).

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Massachusetts, Amherst, USA

    A. Oblomkov

  2. Department of Mathematics, Columbia University, New York, USA

    A. Okounkov

  3. Institute for Problems of Information Transmission, Moscow, Russia

    A. Okounkov

  4. Laboratory of Representation Theory and Mathematical Physics, HSE, Moscow, Russia

    A. Okounkov

  5. Department of Mathematics, ETH Zürich, Zurich, Switzerland

    R. Pandharipande

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  1. A. Oblomkov
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Correspondence to A. Oblomkov.

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Communicated by N. Nekrasov

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Oblomkov, A., Okounkov, A. & Pandharipande, R. GW/PT Descendent Correspondence via Vertex Operators. Commun. Math. Phys. 374, 1321–1359 (2020). https://doi.org/10.1007/s00220-020-03686-4

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