Papers
Topics
Authors
Recent
2000 character limit reached

An open-closed string analogue of Hochschild cohomology (2410.20888v1)

Published 28 Oct 2024 in math.QA, hep-th, math-ph, math.AT, and math.MP

Abstract: We prove that every open-closed homotopy algebra, introduced by Kajiura and Stasheff (arXiv: archive/0410291), naturally gives rise to an open-closed version of Hochschild cochain complex whose cohomology admits a canonical Gerstenhaber algebra structure. We also develop the open-closed brace relations, provide a concise description of OCHAs, and establish an A-infinity structure that extends the open-closed Hochschild differential.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (57)
  1. Denis Auroux. Mirror symmetry and T-duality in the complement of an anticanonical divisor. Journal of Gökova Geometry Topology, 1:51–91, 2007.
  2. The cyclic Deligne conjecture and Calabi-Yau structures. arXiv preprint arXiv:2305.10323, 2023.
  3. Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds. Asian Journal of Mathematics, 10(4):773–814, 2006.
  4. Alain Connes. Non-commutative differential geometry. Publications Mathematiques de l’IHES, 62:41–144, 1985.
  5. Kevin Costello. Topological conformal field theories and Calabi–Yau categories. Advances in Mathematics, 210(1):165–214, 2007.
  6. Symmetrization of brace algebras. arXiv preprint math/0503464, 2005.
  7. The irreducibility of the space of curves of given genus. Publications Mathématiques de l’IHES, 36:75–109, 1969.
  8. Lagrangian Floer theory on compact toric manifolds, I. Duke Mathematical Journal, 151(1):23–175, 2010.
  9. Lagrangian intersection Floer theory: anomaly and obstruction, Part I, volume 1. American Mathematical Soc., 2010.
  10. Lagrangian intersection Floer theory: anomaly and obstruction, Part II, volume 2. American Mathematical Soc., 2010.
  11. Construction of Kuranishi structures on the moduli spaces of pseudo holomorphic disks: I. arXiv preprint arXiv:1710.01459, 2017.
  12. Spectral invariants with bulk, quasi-morphisms and Lagrangian Floer theory, volume 260. American Mathematical Society, 2019.
  13. Kuranishi Structures and Virtual Fundamental Chains. Springer, 2020.
  14. Homotopy Batalin–Vilkovisky algebras. Journal of Noncommutative Geometry, 6(3):539–602, 2012.
  15. On a variant of noncommutative differential geometry. In Doklady Akademii Nauk, volume 308, pages 1293–1297. Russian Academy of Sciences, 1989.
  16. Murray Gerstenhaber. The cohomology structure of an associative ring. Annals of Mathematics, pages 267–288, 1963.
  17. Ezra Getzler. Cartan homotopy formulas and the gauss-manin connection in cyclic homology. In Israel Math. Conf. Proc, volume 7, pages 65–78, 1993.
  18. Victor Ginzburg. Calabi-yau algebras. arXiv preprint math/0612139, 2006.
  19. Operads, homotopy algebra and iterated integrals for double loop spaces. arXiv preprint hep-th/9403055, 1994.
  20. Mirror symmetry: from categories to curve counts. arXiv preprint arXiv:1510.03839, 2015.
  21. Tensor constructions of open string theories (i) foundations. Nuclear Physics B, 505(3):569–624, 1997.
  22. Open-closed homotopy algebras and strong homotopy leibniz pairs through koszul operad theory. Letters in Mathematical Physics, 101:195–222, 2012.
  23. Eduardo Hoefel. OCHA and the swiss-cheese operad. Journal of Homotopy and Related Structures, 4(1):123–151, 2009.
  24. T. V. Kadeishvili. The structure of the A⁢(∞)𝐴A(\infty)italic_A ( ∞ )-algebra, and the Hochschild and Harrison cohomologies. Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR, 91:19–27, 1988. (In Russian).
  25. Hiroshige Kajiura. Homotopy algebra morphism and geometry of classical string field theories. Nuclear Physics B, 630(3):361–432, 2002.
  26. Ralph M Kaufmann. A proof of a cyclic version of Deligne’s conjecture via cacti. Mathematical Research Letters, 15(5):901–921, 2008.
  27. Maxim Kontsevich. Enumeration of rational curves via torus actions. In The moduli space of curves, pages 335–368. Springer, 1995.
  28. Maxim Kontsevich. Homological algebra of mirror symmetry. In Proceedings of the international congress of mathematicians, pages 120–139. Springer, 1995.
  29. Maxim Kontsevich. Operads and motives in deformation quantization. Letters in Mathematical Physics, 48:35–72, 1999.
  30. Deformations of algebras over operads and the deligne conjecture. In Conférence Moshé Flato 1999, volume 1, pages 255–307. Kluwer Acad. Publ. Dordrecht, 2000.
  31. Homotopy algebras inspired by classical open-closed string field theory. Communications in mathematical physics, 263(3):553–581, 2006.
  32. Open-closed homotopy algebra in mathematical physics. Journal of mathematical physics, 47(2), 2006.
  33. Notes on A∞subscript𝐴A_{\infty}italic_A start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT-algebras, A∞subscript𝐴A_{\infty}italic_A start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT-categories and non-commutative geometry. In Homological Mirror Symmetry: New Developments and Perspectives, pages 1–67. Springer, 2008.
  34. Symmetric brace algebras. Applied Categorical Structures, 13(4):351–370, 2005.
  35. Luc Menichi. Batalin-Vilkovisky algebra structures on Hochschild cohomology. Bulletin de la Société Mathématique de France, 137(2):277–295, 2009.
  36. A solution of Deligne’s Hochschild cohomology conjecture. Contemporary Mathematics, 293:153–194, 2002.
  37. Toshio Nakatsu. Classical open-string field theory: A∞subscript𝐴A_{\infty}italic_A start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT-algebra, renormalization group and boundary states. Nuclear Physics B, 642(1-2):13–90, 2002.
  38. On the Lie enveloping algebra of a pre-Lie algebra. Journal of K-theory, 2(1):147–167, 2008.
  39. James Dillon Stasheff. Homotopy associativity of H-spaces, I, II. Transactions of the American Mathematical Society, 108(2):275, 293, 1963.
  40. Dmitry Tamarkin. Another proof of M. Kontsevich formality theorem. arXiv preprint math/9803025, 1998.
  41. Dmitry E Tamarkin. Formality of chain operad of little discs. Letters in Mathematical Physics, 66, 2003.
  42. Thomas Tradler. The Batalin-Vilkovisky algebra on Hochschild cohomology induced by infinity inner products. In Annales de L’institut Fourier, volume 58, pages 2351–2379, 2008.
  43. Noncommutative differential calculus, homotopy BV algebras and formality conjectures. Methods of Functional Analysis and Topology, 6(02):85–100, 2000.
  44. Cyclic formality and index theorems. Letters in Mathematical Physics, 56:85–97, 2001.
  45. The ring of differential operators on forms in noncommutative calculus. In Proceedings of Symposia in Pure Mathematics, volume 73. Citeseer, 2005.
  46. On the cyclic Deligne conjecture. Journal of Pure and Applied Algebra, 204(2):280–299, 2006.
  47. Alexander A Voronov. The Swiss-cheese operad. Contemporary Mathematics, 239:365–374, 1999.
  48. Alexander A Voronov. Homotopy gerstenhaber algebras. In Conférence Moshé Flato 1999: Quantization, Deformations, and Symmetries Volume II, pages 307–331. Springer, 2000.
  49. Hang Yuan. Family Floer program and non-archimedean SYZ mirror construction. arXiv preprint arXiv: 2003.06106, 2020.
  50. Hang Yuan. Disk counting and wall-crossing phenomenon via family Floer theory. arXiv preprint arXiv:2101.01379, 2021.
  51. Hang Yuan. Family Floer superpotential’s critical values are eigenvalues of quantum product by c1subscript𝑐1c_{1}italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. arXiv preprint arXiv:2112.13537, 2021.
  52. Hang Yuan. Family Floer mirror space for local SYZ singularities. arXiv preprint arXiv:2206.04652, 2022.
  53. Hang Yuan. Family Floer SYZ singularities for the conifold transition. arXiv preprint arXiv:2212.13948, 2022.
  54. Hang Yuan. Family Floer SYZ conjecture for Ansubscript𝐴𝑛A_{n}italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT singularity. arXiv preprint arXiv:2305.13554, 2023.
  55. Hang Yuan. Non-archimedean analytic continuation of unobstructedness. arXiv preprint arXiv:2401.02577, 2024.
  56. Barton Zwiebach. Closed string field theory: Quantum action and the batalin-vilkovisky master equation. Nuclear Physics B, 390(1):33–152, 1993.
  57. Barton Zwiebach. Oriented open-closed string theory revisited. Annals of Physics, 267(2):193–248, 1998.

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now
Dice Question Streamline Icon: https://streamlinehq.com

Open Problems

  1. G_infty structure on open-closed Hochschild cochains
Lightbulb Streamline Icon: https://streamlinehq.com

Continue Learning

We haven't generated follow-up questions for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Generate Now

Authors (1)

  1. Hang Yuan
List To Do Tasks Checklist Streamline Icon: https://streamlinehq.com

Collections

Sign up for free to add this paper to one or more collections.

User Circle Single Streamline Icon: https://streamlinehq.com Sign Up
X Twitter Logo Streamline Icon: https://streamlinehq.com

Tweets

This paper has been mentioned in 1 tweet and received 2 likes.

Upgrade to Pro to view all of the tweets about this paper:

Start a free 7-day Pro trial