Abstract
We apply Weyl 𝑛-algebras to prove formality theorems for higher Hochschild cohomology. We present two approaches: via propagators and via the factorization complex. It is shown that the second approach is equivalent to the first one taken with a new family of propagators we introduce.
Funding statement: The study has been funded within the framework of the HSE University Basic Research Program and the Russian Academic Excellence Project “5-100”.
Acknowledgements
I am grateful to D. Calaque, V. Dotsenko, B. Feigin, A. Khoroshkin, S. Merkulov, B. Shoikhet, D. Tamarkin and A. Voronov for fruitful discussions. I thank the referee for useful recommendations and for pointing out many relevant references.
Communicated by: Frederick R. Cohen
References
[1] S. Axelrod and I. M. Singer, Chern–Simons perturbation theory, Proceedings of the 20th International Conference on Differential Geometric Methods in Theoretical Physics. Vol. 1, 2 (New York 1991), World Scientific, River Edge (1992), 3–45. 10.4310/jdg/1214454681Search in Google Scholar
[2] D. Ayala, J. Francis and H. L. Tanaka, Factorization homology of stratified spaces, Selecta Math. (N. S.) 23 (2017), no. 1, 293–362. 10.1007/s00029-016-0242-1Search in Google Scholar
[3]
C. Berger,
Combinatorial models for real configuration spaces and
[4] S. Chmutov, S. Duzhin and J. Mostovoy, Introduction to Vassiliev Knot Invariants, Cambridge University, Cambridge, 2012. 10.1017/CBO9781139107846Search in Google Scholar
[5]
J. Francis,
The tangent complex and Hochschild cohomology of
[6] M. Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math. (2) 78 (1963), 267–288. 10.2307/1970343Search in Google Scholar
[7] E. Getzler and J. D. S. Jones, Operads, homotopy algebra and iterated integrals for double loop spaces, preprint (1994), https://arxiv.org/abs/hep-th/9403055. Search in Google Scholar
[8] G. Ginot, Notes on factorization algebras, factorization homology and applications, Mathematical Aspects of Quantum Field Theories, Math. Phys. Stud., Springer, Cham (2015), 429–552. 10.1007/978-3-319-09949-1_13Search in Google Scholar
[9]
G. Ginot, T. Tradler and M. Zeinalian,
Higher Hochschild cohomology, Brane topology and centralizers of
[10] G. Ginot, T. Tradler and M. Zeinalian, Higher Hochschild homology, topological chiral homology and factorization algebras, Comm. Math. Phys. 326 (2014), no. 3, 635–686. 10.1007/s00220-014-1889-0Search in Google Scholar
[11] V. Ginzburg and M. Kapranov, Koszul duality for operads, Duke Math. J. 76 (1994), no. 1, 203–272. 10.1215/S0012-7094-94-07608-4Search in Google Scholar
[12] E. Hoefel, M. Livernet and A. Quesney, On the deformation complex of homotopy affine actions, Adv. Math. 358 (2019), Article ID 106857. 10.1016/j.aim.2019.106857Search in Google Scholar
[13] R. M. Kaufmann, Moduli space actions on the Hochschild co-chains of a Frobenius algebra. II. Correlators, J. Noncommut. Geom. 2 (2008), no. 3, 283–332. 10.4171/JNCG/22Search in Google Scholar
[14] R. M. Kaufmann, Open/closed string topology and moduli space actions via open/closed Hochschild actions, SIGMA Symmetry Integrability Geom. Methods Appl. 6 (2010), Paper No. 036. 10.3842/SIGMA.2010.036Search in Google Scholar
[15] M. Kontsevich, Operads and motives in deformation quantization, Lett. Math. Phys. 48 (1999), 35–72. 10.1023/A:1007555725247Search in Google Scholar
[16] M. Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003), no. 3, 157–216. 10.1023/B:MATH.0000027508.00421.bfSearch in Google Scholar
[17] M. Kontsevich and Y. Soibelman, Deformations of algebras over operads and the Deligne conjecture, Conférence Moshé Flato 1999, Vol. I, Math. Phys. Stud. 21, Kluwer Academic, Dordrecht (2000), 255–307. Search in Google Scholar
[18] J.-L. Loday, Cyclic Homology, 2nd ed., Grundlehren Math. Wiss. 301, Springer, Berlin, 1998. 10.1007/978-3-662-11389-9Search in Google Scholar
[19] J. Lurie, Higher algebra, preprint (2017), http://www.math.harvard.edu/~lurie/papers/HigherAlgebra.pdf. Search in Google Scholar
[20] N. Markarian, Weyl 𝑛-algebras and the Kontsevich integral of the unknot, J. Knot Theory Ramifications 25 (2016), no. 12, Article ID 1642008. 10.1142/S0218216516420086Search in Google Scholar
[21] N. Markarian, Weyl 𝑛-algebras, Comm. Math. Phys. 350 (2017), no. 2, 421–442. 10.1007/s00220-017-2835-8Search in Google Scholar
[22] N. Markarian and H. L. Tanaka, Factorization homology in 3-dimensional topology, Mathematical Aspects of Quantum Field Theories, Math. Phys. Stud., Springer, Cham (2015), 213–231. 10.1007/978-3-319-09949-1_7Search in Google Scholar
[23] M. Markl, A compactification of the real configuration space as an operadic completion, J. Algebra 215 (1999), no. 1, 185–204. 10.1006/jabr.1998.7709Search in Google Scholar
[24] C. A. Rossi and T. Willwacher, P. Etingof’s conjecture about Drinfeld associators, preprint (2014), https://arxiv.org/abs/1404.2047. Search in Google Scholar
[25] P. Salvatore, Configuration spaces with summable labels, Cohomological Methods in Homotopy Theory (Bellaterra 1998), Progr. Math. 196, Birkhäuser, Basel (2001), 375–395. 10.1007/978-3-0348-8312-2_23Search in Google Scholar
[26] P. Salvatore and N. Wahl, Framed discs operads and Batalin–Vilkovisky algebras, Q. J. Math. 54 (2003), no. 2, 213–231. 10.1093/qmath/hag012Search in Google Scholar
[27] J. Thomas, Kontsevich’s Swiss cheese conjecture, Geom. Topol. 20 (2016), no. 1, 1–48. 10.2140/gt.2016.20.1Search in Google Scholar
[28] A. A. Voronov, The Swiss-cheese operad, Homotopy Invariant Algebraic Structures (Baltimore 1998), Contemp. Math. 239, American Mathematical Society, Providence (1999), 365–373. 10.1090/conm/239/03610Search in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Lower bounds for regular genus and gem-complexity of PL 4-manifolds with boundary
- Free cyclic group actions on highly-connected 2n-manifolds
- The distinction problems for Sp4 and SO3,3
- Affine cones over cubic surfaces are flexible in codimension one
- Permutations of zero-sumsets in a finite vector space
- On the finiteness of solutions for polynomial-factorial Diophantine equations
- Galois action on Fuchsian surface groups and their solenoids
- On a Lévy process pinned at random time
- Borsuk–Ulam theorem for filtered spaces
- A non-commutative differential module approach to Alexander modules
- Generalized fractal dimensions of invariant measures of full-shift systems over compact and perfect spaces: generic behavior
- Quantum modularity of partial theta series with periodic coefficients
- Lyapunov-type inequalities for partial differential equations with 𝑝-Laplacian
- Zeros of GL2 𝐿-functions on the critical line
- Weighted boundedness of multilinear Calderón commutators
- Two characterizations of central BMO space via the commutators of Hardy operators
- Weyl 𝑛-algebras and the Swiss cheese operad
- Syzygies in equivariant cohomology in positive characteristic
- Epsilon factors of symplectic type characters in the wild case
Articles in the same Issue
- Frontmatter
- Lower bounds for regular genus and gem-complexity of PL 4-manifolds with boundary
- Free cyclic group actions on highly-connected 2n-manifolds
- The distinction problems for Sp4 and SO3,3
- Affine cones over cubic surfaces are flexible in codimension one
- Permutations of zero-sumsets in a finite vector space
- On the finiteness of solutions for polynomial-factorial Diophantine equations
- Galois action on Fuchsian surface groups and their solenoids
- On a Lévy process pinned at random time
- Borsuk–Ulam theorem for filtered spaces
- A non-commutative differential module approach to Alexander modules
- Generalized fractal dimensions of invariant measures of full-shift systems over compact and perfect spaces: generic behavior
- Quantum modularity of partial theta series with periodic coefficients
- Lyapunov-type inequalities for partial differential equations with 𝑝-Laplacian
- Zeros of GL2 𝐿-functions on the critical line
- Weighted boundedness of multilinear Calderón commutators
- Two characterizations of central BMO space via the commutators of Hardy operators
- Weyl 𝑛-algebras and the Swiss cheese operad
- Syzygies in equivariant cohomology in positive characteristic
- Epsilon factors of symplectic type characters in the wild case