Skip to main content
SpringerLink
Log in

Batalin–Vilkovisky Quantization and Supersymmetric Twists

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We show that a family of topological twists of a supersymmetric mechanics with a Kähler target exhibits a Batalin–Vilkovisky quantization. Using this observation we make a general proposal for the Hilbert space of states after a topological twist in terms of the cohomology of a certain perverse sheaf. We give several examples of the resulting Hilbert spaces including the categorified Donaldson–Thomas invariants, Haydys–Witten theory and the 3-dimensional A-model.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Anselmi, D., Fré, P.: Topological \(\sigma \)-models in four dimensions and triholomorphic maps. Nuclear Phys. B 416(1), 255–300 (1994)

    ADS  MathSciNet  MATH  Google Scholar 

  2. Arinkin, D., Gaitsgory, D.: Singular support of coherent sheaves and the geometric Langlands conjecture. Selecta Math. (NS) 21(1), 1–199 (2015)

    MathSciNet  MATH  Google Scholar 

  3. Abouzaid, M., Manolescu, C.: A sheaf-theoretic model for SL(2, C) Floer homology. J. Eur. Math. Soc. (JEMS) 22(11), 3641–3695 (2020)

    MathSciNet  MATH  Google Scholar 

  4. Anderson, L.: Five-dimensional topologically twisted maximally supersymmetric Yang–Mills theory. J. High Energy Phys. 2, 131 (2013)

    ADS  MathSciNet  MATH  Google Scholar 

  5. Acharya, B.S., O’Loughlin, M., Spence, B.: Higher-dimensional analogues of Donaldson–Witten theory. Nuclear Phys. B 503(3), 657–674 (1997). arXiv:hep-th/9705138

    ADS  MathSciNet  MATH  Google Scholar 

  6. Baptista, J.M.: Twisting gauged non-linear sigma-models. J. High Energy Phys. 2, 09628 (2008)

    MathSciNet  Google Scholar 

  7. Bullimore, M., Dimofte, T., Gaiotto, D.: The Coulomb branch of 3d N = 4 theories. Commun. Math. Phys. 354(2), 671–751 (2017)

    ADS  MathSciNet  MATH  Google Scholar 

  8. Ben-Bassat, O., Brav, C., Bussi, V., Joyce, D.: A ‘Darboux theorem’ for shifted symplectic structures on derived Artin stacks, with applications. Geom. Topol. 19(3), 1287–1359 (2015)

    MathSciNet  MATH  Google Scholar 

  9. Behrend, K., Fantechi, B.: Gerstenhaber and Batalin–Vilkovisky structures on Lagrangian intersections. Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I. Vol. 269. Progr. Math. Birkhäuser Boston, Boston, MA (2009), pp. 1–47

  10. Bullimore, M., Ferrari, A., Kim, H.: Supersymmetric Ground States of 3d N = 4 Gauge Theories on a Riemann Surface (May 2021). arXiv:2105.08783 [hep-th]

  11. Bezrukavnikov, R., Kaledin, D.: Fedosov quantization in algebraic context. Mosc. Math. J. 4(3), 559–592 (2004)

    MathSciNet  MATH  Google Scholar 

  12. Brav, C., Bussi, V., Dupont, D., Joyce, D., Szendrõi, B.: Symmetries and stabilization for sheaves of vanishing cycles. J. Singul. 11 (2015). With an appendix by Jörg Schürmann, pp. 85–151. arXiv:1211.3259 [math.AG]

  13. Bressler, P., Soibelman, Y.: Homological mirror symmetry, deformation quantization and noncommutative geometry. J. Math. Phys. 45(10), 3972–3982 (2004). arXiv: hep-th/0202128

    ADS  MathSciNet  MATH  Google Scholar 

  14. Bussi, V.: Categorification of Lagrangian intersections on complex symplectic manifolds using perverse sheaves of vanishing cycles (2014). arXiv:1404.1329 [math.AG]

  15. Calaque, D., Pantev, T., Toën, B., Vaquié, M., Vezzosi, G.: Shifted poisson structures and deformation quantization. J. Topol. 10(2), 483–584 (2017)

    MathSciNet  MATH  Google Scholar 

  16. Costello, K., Gwilliam, O.: Factorization algebras in quantum field theory. Vol. 2. (2018). https://people.math.umass.edu/~gwilliam

  17. Corlette, K.: Flat G-bundles with canonical metrics. J. Differ. Geom. 28(3), 361–382 (1988)

    MathSciNet  MATH  Google Scholar 

  18. Deligne, P., Freed, D. S.: Supersolutions. Quantum fields and strings: a course for mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997). Amer. Math. Soc., Providence, RI, (1999), pp. 227–355. arXiv: hep-th/9901094

  19. Dimofte, T., Garner, N., Geracie, M., Hilburn, J.: Mirror symmetry and line operators. J. High Energy Phys. 2, 075147 (2020)

    MathSciNet  MATH  Google Scholar 

  20. De Wilde, M., Lecomte, P.B.A.: Existence of star-products and of formal deformations of the Poisson Lie algebra of arbitrary symplectic manifolds. Lett. Math. Phys. 7(6), 487–496 (1983)

    ADS  MathSciNet  MATH  Google Scholar 

  21. Elliott, C., Safronov, P., Williams, B.: A taxonomy of twists of supersymmetric Yang–Mills theory (2020). arXiv:2002.10517 [math-ph]

  22. Fedosov, B.V.: A simple geometrical construction of deformation quantization. J. Differ. Geom. 40(2), 213–238 (1994)

    MathSciNet  MATH  Google Scholar 

  23. Figueroa-O’Farrill, J.M., Köhl, C., Spence, B.: Supersymmetric Yang–Mills, octonionic instantons and triholomorphic curves. Nuclear Phys. B 521(3), 419–443 (1998). arXiv:hep-th/9710082

    ADS  MathSciNet  MATH  Google Scholar 

  24. Ginzburg, V., Rozenblyum, N.: Gaiotto’s Lagrangian subvarieties via derived symplectic geometry. Algebr. Represent. Theory 21(5), 1003–1015 (2018)

    MathSciNet  MATH  Google Scholar 

  25. Gualtieri, M.: Generalized complex geometry. Ann. Math. (2) 174(1), 75–123 (2011)

    MathSciNet  MATH  Google Scholar 

  26. Gukov, S., Witten, E.: Branes and quantization. Adv. Theor. Math. Phys. 13(5), 1445–1518 (2009)

    ADS  MathSciNet  MATH  Google Scholar 

  27. Halpern-Leistner, D.: Remarks on theta-stratifications and derived categories (2015). arXiv:1502.03083 [math.AG]

  28. Hitchin, N.J., Karlhede, A., Lindström, U., Roèek, M.: Hyper-Kähler metrics and supersymmetry. Commun. Math. Phys. 108(4), 535–589 (1987)

    ADS  MATH  Google Scholar 

  29. Joyce, D., Tanaka, Y., Upmeier, M.: On orientations for gauge-theoretic moduli spaces. Adv. Math. 362, 106957 (2020)

    MathSciNet  MATH  Google Scholar 

  30. Joyce, D., Upmeier, M.: On spin structures and orientations for gauge-theoretic moduli spaces. Adv. Math. 381, 107630 (2021)

    MathSciNet  MATH  Google Scholar 

  31. Joyce, D., Upmeier, M.: Orientation data for moduli spaces of coherent sheaves over Calabi–Yau 3-folds. Adv. Math. 381, 107627 (2021)

    MathSciNet  MATH  Google Scholar 

  32. Kapustin, A.: Topological strings on noncommutative manifolds. Int. J. Geom. Methods Mod. Phys. 1(1–2), 49–81 (2004). arXiv:hep-th/0310057

    MathSciNet  MATH  Google Scholar 

  33. Kapustin, A.: A-branes and Noncommutative Geometry (2005). arXiv:hep-th/0502212

  34. Kapustin, A.: Chiral de Rham complex and the half-twisted sigma-model (2005). arXiv:hep-th/0504074

  35. Khudaverdian, H.M.: Semidensities on odd symplectic supermanifolds. Commun. Math. Phys. 247(2), 353–390 (2004). arXiv:math/0012256

    ADS  MathSciNet  MATH  Google Scholar 

  36. Kinjo, T.: Dimensional reduction in cohomological Donaldson–Thomas theory (2021). arXiv:2102.01568 [math.AG]

  37. Kiem, Y.-H., Li, J.: Categorification of Donaldson–Thomas invariants via perverse sheaves (2012). arXiv:1212.6444 [math.AG]

  38. Kontsevich, M.: Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66(3), 157–216 (2003)

    ADS  MathSciNet  MATH  Google Scholar 

  39. Kontsevich, M., Soibelman, Y.: Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson–Thomas invariants. Commun. Number Theory Phys. 5(2), 231–352 (2011)

    MathSciNet  MATH  Google Scholar 

  40. Kapustin, A., Vyas, K.: A-models in three and four dimensions (2010). arXiv:1002.4241 [hep-th]

  41. Kapustin, A., Witten, E.: Electric-magnetic duality and the geometric Langlands program. Commun. Number Theory Phys. 1(1), 1–236 (2007). arXiv:hep-th/0604151

    MathSciNet  MATH  Google Scholar 

  42. Manin, Y. I.: Gauge field theory and complex geometry. Second. Vol. 289. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Translated from the 1984 Russian original by N. Koblitz and J. R. King, With an appendix by Sergei Merkulov. Springer, Berlin, (1997), pp. xii+346

  43. Marcus, N.: The other topological twisting of N = 4 Yang–Mills. Nuclear Phys. B 452(1–2), 331–345 (1995). arXiv:hep-th/9506002

    ADS  MathSciNet  MATH  Google Scholar 

  44. Nakajima, H.: Towards a mathematical definition of Coulomb branches of 3-dimensional N = 4 gauge theories, I. Adv. Theor. Math. Phys. 20(3), 595–669 (2016)

    MathSciNet  MATH  Google Scholar 

  45. Nest, R., Tsygan, B.: Deformations of symplectic Lie algebroids, deformations of holomorphic symplectic structures, and index theorems. Asian J. Math. 5(4), 599–635 (2001). arXiv:math/9906020

    MathSciNet  MATH  Google Scholar 

  46. Pantev, T., Toën, B., Vaquié, M., Vezzosi, G.: Shifted symplectic structures. Publ. Math. Inst. Hautes Études Sci. 117, 271–328 (2013)

    MathSciNet  MATH  Google Scholar 

  47. Pridham, J.P.: Shifted Poisson and symplectic structures on derived N-stacks. J. Topol. 10(1), 178–210 (2017)

    MathSciNet  MATH  Google Scholar 

  48. Pridham, J.P.: Deformation quantisation for (-1)-shifted symplectic structures and vanishing cycles. Algebr. Geom. 6(6), 747–779 (2019)

    MathSciNet  MATH  Google Scholar 

  49. Polesello, P., Schapira, P.: Stacks of quantization-deformation modules on complex symplectic manifolds. Int. Math. Res. Not. 49, 2637–2664 (2004). arXiv:math/0305171

    MathSciNet  MATH  Google Scholar 

  50. Qiu, J., Zabzine, M.: On twisted N = 2 5D super Yang–Mills theory. Lett. Math. Phys. 106(1), 1–27 (2016)

    ADS  MathSciNet  MATH  Google Scholar 

  51. Rothstein, M.: The structure of supersymplectic supermanifolds. Differential geometric methods in theoretical physics (Rapallo, Vol. 375. Lecture Notes in Phys, vol. 1991, 331–343. Springer, Berlin (1990)

  52. Ševera, P.: On the origin of the BV operator on odd symplectic supermanifolds. Lett. Math. Phys. 78(1), 55–59 (2006). arXiv:math/0506331

    ADS  MathSciNet  MATH  Google Scholar 

  53. Sabbah, C., Saito, M.: Kontsevich’s conjecture on an algebraic formula for vanishing cycles of local systems. Algebr. Geom. 1(1), 107–130 (2014)

    MathSciNet  MATH  Google Scholar 

  54. Stacey, A.: The differential topology of loop spaces (2005). arXiv:math/0510097

  55. Solomon, J.P., Verbitsky, M.: Locality in the Fukaya category of a hyperkähler manifold. Compos. Math. 155(10), 1924–1958 (2019)

    MathSciNet  MATH  Google Scholar 

  56. Tsygan, B.: Oscillatory modules. Lett. Math. Phys. 88(1–3), 343–369 (2009)

    ADS  MathSciNet  MATH  Google Scholar 

  57. Vezzosi, G.: Basic structures on derived critical loci. Differ. Geom. Appl. 71, 101635 (2020)

    MathSciNet  MATH  Google Scholar 

  58. Walpuski, T.: A compactness theorem for Fueter sections. Comment. Math. Helv. 92(4), 751–776 (2017)

    MathSciNet  MATH  Google Scholar 

  59. Witten, E.: Fivebranes and knots. Quantum Topol. 3(1), 1–137 (2012)

    MathSciNet  MATH  Google Scholar 

  60. Witten, E.: Supersymmetry and Morse theory. J. Differ. Geometry 17(4), 661–692 (1982)

    MathSciNet  MATH  Google Scholar 

  61. Witten, E.: Topological sigma models. Commun. Math. Phys. 118(3), 411–449 (1988)

    ADS  MathSciNet  MATH  Google Scholar 

  62. Witten, E.: A note on the antibracket formalism. Modern Phys. Lett. A 5(7), 487–494 (1990)

    ADS  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We would like to thank Mathew Bullimore, Tudor Dimofte, and Sam Gunningham for useful conversations.

Author information

Authors and Affiliations

  1. School of Mathematics, University of Edinburgh, Edinburgh, UK

    Pavel Safronov

  2. Department of Mathematics and Statistics, Boston University, Boston, USA

    Brian R. Williams

Authors
  1. Pavel Safronov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Brian R. Williams
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Brian R. Williams.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Safronov, P., Williams, B.R. Batalin–Vilkovisky Quantization and Supersymmetric Twists. Commun. Math. Phys. 402, 35–77 (2023). https://doi.org/10.1007/s00220-023-04721-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-023-04721-w

Search

Navigation