Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 014, 28 pages      arXiv:1909.13588      https://doi.org/10.3842/SIGMA.2020.014
Contribution to the Special Issue on Algebra, Topology, and Dynamics in Interaction in honor of Dmitry Fuchs

Short Star-Products for Filtered Quantizations, I

Pavel Etingof and Douglas Stryker
Department of Mathematics, MIT, Cambridge, MA 02139, USA

Received October 01, 2019, in final form March 01, 2020; Published online March 11, 2020

Abstract
We develop a theory of short star-products for filtered quantizations of graded Poisson algebras, introduced in 2016 by Beem, Peelaers and Rastelli for algebras of regular functions on hyperKähler cones in the context of 3-dimensional $N=4$ superconformal field theories [Beem C., Peelaers W., Rastelli L., Comm. Math. Phys. 354 (2017), 345-392]. This appears to be a new structure in representation theory, which is an algebraic incarnation of the non-holomorphic ${\rm SU}(2)$-symmetry of such cones. Using the technique of twisted traces on quantizations (an idea due to Kontsevich), we prove the conjecture by Beem, Peelaers and Rastelli that short star-products depend on finitely many parameters (under a natural nondegeneracy condition), and also construct these star products in a number of examples, confirming another conjecture by Beem, Peelaers and Rastelli.

Key words: star-product; quantization; hyperKähler cone; symplectic singularity.

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References

  1. Beem C., Peelaers W., Rastelli L., Deformation quantization and superconformal symmetry in three dimensions, Comm. Math. Phys. 354 (2017), 345-392, arXiv:1601.05378.
  2. Bellamy G., Schedler T., Symplectic resolutions of quiver varieties and character varieties, arXiv:1602.00164.
  3. Braden T., Licata A., Proudfoot N., Webster B., Quantizations of conical symplectic resolutions II: category $\mathcal O$ and symplectic duality, Astérisque 384 (2016), 75-179, arXiv:1407.0964.
  4. Brylinski R., Instantons and Kähler geometry of nilpotent orbits, in Representation Theories and Algebraic Geometry (Montreal, PQ, 1997), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 514, Kluwer Acad. Publ., Dordrecht, 1998, 85-125, arXiv:math.SG/9811032.
  5. Carlson F., Über ganzwertige Funktionen, Math. Z. 11 (1921), 1-23.
  6. de Siebenthal J., Sur les groupes de Lie compacts non connexes, Comment. Math. Helv. 31 (1956), 41-89.
  7. Dedushenko M., Fan Y., Pufu S.S., Yacoby R., Coulomb branch operators and mirror symmetry in three dimensions, J. High Energy Phys. 2018 (2018), no. 4, 037, 111 pages, arXiv:1712.09384.
  8. Dedushenko M., Fan Y., Pufu S.S., Yacoby R., Coulomb branch quantization and abelianized monopole bubbling, J. High Energy Phys. 2019 (2019), no. 10, 179, 96 pages, arXiv:1812.08788.
  9. Dedushenko M., Pufu S.S., Yacoby R., A one-dimensional theory for Higgs branch operators, J. High Energy Phys. 2018 (2018), no. 3, 138, 83 pages, arXiv:1610.00740.
  10. Etingof P., Ginzburg V., Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2002), 243-348, arXiv:math.AG/0011114.
  11. Etingof P., Schedler T., Poisson traces and $D$-modules on Poisson varieties, Geom. Funct. Anal. 20 (2010), 958-987, arXiv:0908.3868.
  12. Gaiotto D., Neitzke A., Tachikawa Y., Argyres-Seiberg duality and the Higgs branch, Comm. Math. Phys. 294 (2010), 389-410, arXiv:0810.4541.
  13. Ginzburg V., On primitive ideals, Selecta Math. (N.S.) 9 (2003), 379-407, arXiv:math.RT/0202079.
  14. Gordon I.G., Losev I., On category $\mathcal{O}$ for cyclotomic rational Cherednik algebras, J. Eur. Math. Soc. (JEMS) 16 (2014), 1017-1079, arXiv:1109.2315.
  15. Herbig H.-C., Schwarz G.W., Seaton C., Symplectic quotients have symplectic singularities, Compos. Math. 156 (2020), 613-646, arXiv:1706.02089.
  16. Hitchin N.J., Karlhede A., Lindström U., Roček M., Hyper-Kähler metrics and supersymmetry, Comm. Math. Phys. 108 (1987), 535-589.
  17. Kaledin D., Symplectic singularities from the Poisson point of view, J. Reine Angew. Math. 600 (2006), 135-156, arXiv:math.AG/0310186.
  18. Kraft H., Procesi C., On the geometry of conjugacy classes in classical groups, Comment. Math. Helv. 57 (1982), 539-602.
  19. Losev I., Deformations of symplectic singularities and orbit method for semisimple Lie algebras, arXiv:1605.00592.
  20. Losev I., Bernstein inequality and holonomic modules, Adv. Math. 308 (2017), 941-963, arXiv:1501.01260.
  21. Losev I., On categories $\mathcal O$ for quantized symplectic resolutions, Compos. Math. 153 (2017), 2445-2481, arXiv:1502.00595.
  22. Losev I., Quantizations of regular functions on nilpotent orbits, Bull. Inst. Math. Acad. Sin. (N.S.) 13 (2018), 199-225, arXiv:1505.08048.
  23. Lusztig G., Character sheaves on disconnected groups. I, Represent. Theory 7 (2003), 374-403, Errata, Represent. Theory 8 (2004), 179, arXiv:math.RT/0305206.
  24. Nakajima H., Towards a mathematical definition of Coulomb branches of 3-dimensional ${\mathcal N}=4$ gauge theories, I, Adv. Theor. Math. Phys. 20 (2016), 595-669, arXiv:1503.03676.
  25. Namikawa Y., Poisson deformations of affine symplectic varieties, II, Kyoto J. Math. 50 (2010), 727-752, arXiv:0902.2832.
  26. Namikawa Y., Fundamental groups of symplectic singularities, in Higher Dimensional Algebraic Geometry - in Honour of Professor Yujiro Kawamata's Sixtieth Birthday, Adv. Stud. Pure Math., Vol. 74, Editors K. Oguiso, C. Birkar, S. Ishii, S. Takayama, Math. Soc. Japan, Tokyo, 2017, 321-334, arXiv:1301.1008.
  27. Namikawa Y., A characterization of nilpotent orbit closures among symplectic singularities, Math. Ann. 370 (2018), 811-818, arXiv:1707.02825.
  28. Pólya G., Über gewisse notwendige Determinantenkriterien für die Fortsetzbarkeit einer Potenzreihe, Math. Ann. 99 (1928), 687-706.
  29. Soergel W., The Hochschild cohomology ring of regular maximal primitive quotients of enveloping algebras of semisimple Lie algebras, Ann. Sci. École Norm. Sup. (4) 29 (1996), 535-538.
  30. Steinberg R., Conjugacy classes in algebraic groups, Lecture Notes in Math., Vol. 366, Springer-Verlag, Berlin - New York, 1974.
  31. Swann A., Hyper-Kähler and quaternionic Kähler geometry, Math. Ann. 289 (1991), 421-450.

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