Advertisement

Communications in Mathematical Physics

, Volume 350, Issue 2, pp 421–442 | Cite as

Weyl n-Algebras

  • Nikita MarkarianEmail author
Article

Abstract

We introduce Weyl n-algebras and show how their factorization complex may be used to define invariants of manifolds. In the appendix, we heuristically explain why these invariants must be perturbative Chern–Simons invariants.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. AC1.
    Arone, G., Ching, M.: Manifolds, K-theory and the calculus of functors. arXiv:1410.1809 [math.AT]
  2. AC2.
    Arone G., Ching M.: Operads and chain rules for the calculus of functors. Astérisque 338, vi+158 (2011)MathSciNetzbMATHGoogle Scholar
  3. AS1.
    Axelrod, S., Singer, I.M.: Chern-Simons perturbation theory. In: Proceedings of the XXth International Conference on Differential Geometric Methods in Theoretical Physics, vol. 1, 2 (New York, 1991), pp. 3–45. World Sci. Publ., River Edge (1992)Google Scholar
  4. AS2.
    Axelrod S., Singer I.M.: Chern-Simons perturbation theory. II. J. Differ. Geom. 39(1), 173–213 (1994)MathSciNetzbMATHGoogle Scholar
  5. Bal.
    Balavoine D.: Homology and cohomology with coefficients, of an algebra over a quadratic operad. J. Pure Appl. Algebra 132(3), 221–258 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  6. BC.
    Bott R., Cattaneo A.S.: Integral invariants of 3-manifolds. J. Differ. Geom. 48(1), 91–133 (1998)MathSciNetzbMATHGoogle Scholar
  7. BD.
    Beilinson, A., Drinfeld, V.: Chiral algebras, volume 51 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence (2004)Google Scholar
  8. CPT+.
    Calaque, D., Pantev, T., Toen, B., Vaquie, M., Vezzosi, G.: Shifted poisson structures and deformation quantization. arXiv:1506.03699 [math.AG]
  9. Cur.
    Curry, J.: Sheaves, Cosheaves and Applications. arXiv:1303.3255 [math.AT]
  10. FFS.
    Feigin B., Felder G., Shoikhet B.: Hochschild cohomology of the Weyl algebra and traces in deformation quantization. Duke Math. J. 127(3), 487–517 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Fra.
    Francis J.: Factorization homology of topological manifolds. J. Topol. 8(4), 1045–1084 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. FT.
    Feĭgin, B.L., Tsygan, B.L.: Riemann–Roch theorem and Lie algebra cohomology. I. In: Proceedings of the Winter School on Geometry and Physics (Srní, 1988), number 21, pp. 15–52 (1989)Google Scholar
  13. Get.
    Getzler E.: Batalin–Vilkovisky algebras and two-dimensional topological field theories. Commun. Math. Phys. 159(2), 265–285 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. Get.
    Gwilliam, O., Haugseng, R.: Linear Batalin–Vilkovisky quantization as a functor of \({\infty}\)-categories. arXiv:1608.01290 [math.AT]
  15. Gin.
    Ginot, Gr.: Notes on factorization algebras, factorization homology and applications. In: Calaque, D., Strobl, T. (eds.) Mathamatical Aspects of Quantum Field Theories. Springer, Berlin (2015) arXiv:1307.5213 [math.AT]
  16. GJ.
    Getzler, E., Jones, J.D.S.: Operads, homotopy algebra and iterated integrals for double loop spaces. arXiv:hep-th/9403055v1
  17. GK.
    Ginzburg V., Kapranov M.: Koszul duality for operads. Duke Math. J. 76(1), 203–272 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  18. GTZ.
    Ginot G., Tradler T., Zeinalian M.: Higher Hochschild homology, topological chiral homology and factorization algebras. Commun. Math. Phys. 326(3), 635–686 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. Gwi.
    Gwilliam, O.: Factorization Algebras and Free Field Theories. ProQuest LLC, Ann Arbor, MI, 2012. Ph.D. Thesis, Northwestern UniversityGoogle Scholar
  20. JF.
    Johnson-Freyd T.: Homological perturbation theory for nonperturbative integrals. Lett. Math. Phys. 105(11), 1605–1632 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. Kon1.
    Kontsevich, M.: Formal (non)commutative symplectic geometry. In: The Gel'fand Mathematical Seminars, 1990–1992, pp. 173–187. Birkhäuser Boston, Boston (1993)Google Scholar
  22. Kon2.
    Kontsevich, M.: Feynman diagrams and low-dimensional topology. In: First European Congress of Mathematics, Vol. II (Paris, 1992), volume 120 of Progr. Math., pp. 97–121. Birkhäuser, Basel (1994)Google Scholar
  23. Lod.
    Loday, J.-L.: Cyclic homology, volume 301 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 2nd edn. Springer, Berlin (1998)Google Scholar
  24. Lur.
  25. Mar1.
    Markarian, N.: Manifoldic homology and Chern–Simons formalism. arXiv:1106.5352 [math.QA]
  26. Mar2.
    Markl M.: A compactification of the real configuration space as an operadic completion. J. Algebra 215(1), 185–204 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  27. QZ.
    Qiu J., Zabzine M.: Introduction to graded geometry, Batalin-Vilkovisky formalism and their applications. Arch. Math. 47(5), 415–471 (2011)MathSciNetzbMATHGoogle Scholar
  28. Sal.
    Salvatore, P.: Configuration spaces with summable labels. In: Cohomological Methods in Homotopy Theory (Bellaterra, 1998), volume 196 of Progr. Math., pp. 375–395. Birkhäuser, Basel (2001)Google Scholar
  29. TT.
    Tamarkin D., Tsygan B.: Noncommutative differential calculus, homotopy BV algebras and formality conjectures. Methods Funct. Anal. Topol. 6(2), 85–100 (2000)MathSciNetzbMATHGoogle Scholar
  30. Tur.
    Turaev V.G.: Euler structures, nonsingular vector fields, and Reidemeister-type torsions. Izv. Akad. Nauk SSSR Ser. Mat. 53(3), 607–643,672 (1989)MathSciNetGoogle Scholar
  31. Vey.
    Vey J.: Déformation du crochet de Poisson sur une variété symplectique. Comment. Math. Helv. 50(4), 421–454 (1975)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsNational Research University Higher School of EconomicsMoscowRussia

Personalised recommendations