Abstract
We propose a quantization algebra of the Loday-Ronco Hopf algebra \(k[Y^\infty ]\), based on the Topological Recursion formula of Eynard and Orantin. We have shown in previous works that the Loday-Ronco Hopf algebra of planar binary trees is a space of solutions for the genus 0 version of Topological Recursion, and that an extension of the Loday Ronco Hopf algebra as to include some new graphs with loops is the correct setting to find a solution space for arbitrary genus. Here we show that this new algebra \(k[Y^\infty ]_h\) is still a Hopf algebra that can be seen in some sense to be made precise in the text as a quantization of the Hopf algebra of planar binary trees, and that the solution space of Topological Recursion \(\mathcal {A}^h_{\text {TopRec}}\) is a subalgebra of a quotient algebra \(\mathcal {A}_{\text {Reg}}^h\) obtained from \(k[Y^\infty ]_h\) that nevertheless doesn’t inherit the Hopf algebra structure. We end the paper with a discussion on the cohomology of \(\mathcal {A}^h_{\text {TopRec}}\) in low degree.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abe, E.: Hopf algebras, Cambridge Tracts in Mathematics, Cambridge University Press, 9780521604895, (2004) https://books.google.pt/books?id=D0AIcewz5-8C
Aguiar, M., Sottile, F.: Structure of the Loday-Ronco Hopf algebra of trees. J. Algebra 295(2), 473–511 (2006). https://doi.org/10.1016/j.jalgebra.2005.06.021
Atiyah, M.F.: Topological quantum field theory. Publications Mathématiques de l’IHÉS 68, 175–186 (1988)
Di Francesco, P., Ginsparg, P.H., Zinn-Justin, J.: 2-D Gravity and random matrices, Phys. Rept., 254, 1–133, (1995). https://www.sciencedirect.com/science/article/abs/pii/037015739400084G?via%3Dihub
incollection, Dumitrescu, O., Mulase, M.: Lectures on the topological recursion for higgs bundles and quantum curves: the geometry, topology and physics of moduli spaces of higgs bundles. World Scientific, 103–198 (2018). https://www.worldscientific.com/doi/abs/10.1142/9789813229099_0003
incollection, Dumitrescu, O, Mulase, M, Safnuk, B, Sorkin, A, The spectral curve of the Eynard-Orantin recursion via the Laplace transform: algebraic and geometric aspects of integrable systems and random matrices. Contemp. Math. Amer. Math. Soc., Providence, RI, 593, 263–315 (2013). https://doi.org/10.1090/conm/593/11867
incollection, Esteves, JN.: Hopf algebras and topological recursion: Topological recursion and its influence in analysis, geometry, and topology. In: Liu, C.-C.M., Mulase, M. (eds.) Am. Math. Soc. Proc. Sympos. Pure Math, pp. 333–357. (2018)
Esteves, JN.: Hopf algebras and topological recursion, J. Phys. A Math. Theor., 48(44), 445205 (2015) http://stacks.iop.org/1751-8121/48/i=44/a=445205
Eynard, B., Orantin, N.: Invariants of algebraic curves and topological expansion. Commun. Number Theory Phys. 1(2), 347–452 (2007). https://doi.org/10.4310/CNTP.2007.v1.n2.a4
Eynard, B., Orantin, N.: Computation of open gromov-witten invariants for toric calabi-yau 3-folds by topological recursion, a proof of the bkmp conjecture. Commun. Math. Phys. 337(2), 483–567 (2015). https://doi.org/10.1007/s00220-015-2361-5
Eynard, B.: The geometry of integrable systems. tau functions and homology of spectral curves. Perturbative definition (2019), arXiv:1706.04938
Frabetti, A.: Simplicial properties of the set of planar binary trees. J. Algebraic Combin. 13(1), 41–65 (2001). https://doi.org/10.1023/A:1008723801201
Iliopoulos, J., Itzykson, C., Martin, A.: Functional methods and perturbation theory. Rev. Mod. Phys. 47, 165–192 (1975). https://link.aps.org/doi/10.1103/RevModPhys.47.165
Liu, C.C.M., Mulase, M.: Topological recursion and its influence in analysis, geometry, and topology. Proceedings of Symposia in Pure Mathematics, American Mathematical Society, 9781470435417 (2018) https://books.google.pt/books?id=39V7DwAAQBAJ
Loday, J.-L., Ronco, M.O.: Hopf algebra of the planar binary trees. Adv. Math. 139(2), 293–309 (1998). https://doi.org/10.1006/aima.1998.1759
Milnor, J.W., Moore, J.C.: On the structure of hopf algebras. Annals Math. 81(2), 211–264, 0003486X (1965). http://www.jstor.org/stable/1970615
Mulase, M., Sulkowski, P.: Spectral curves and the schrödinger equations for the eynard-orantin recursion. Adv. Theor. Math. Phys. 19(5), 9551015 (2015). https://www.intlpress.com/site/pub/pages/journals/items/atmp/content/vols/0019/0005/a002/
Wang, Z., Zhou, J.: A formalism of abstract quantum field theory of summation of fat graphs (2021), arXiv:2108.10498
Funding
Open access funding provided by FCT|FCCN (b-on). Partially supported by Fundação para a Ciência e a Tecnologia, Portugal through projects PTDC/MAT-PUR/31089/2017 and UID/MAT/04459/2020.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Competing Interests
Not applicable.
Ethical Approval
Not applicable.
Additional information
Presented by: Milen Yakimov
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Esteves, J.N. A Quantization of the Loday-Ronco Hopf Algebra. Algebr Represent Theor 27, 1177–1201 (2024). https://doi.org/10.1007/s10468-024-10253-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-024-10253-1