The cyclic open–closed map, u-connections and R-matrices

• Kai Hugtenburg ^[1]
  1. [1] Lancaster University

     Lancaster University

     Lancaster, Reino Unido

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 30, Nº. 2, 2024, 90 págs.
• Idioma: inglés
• DOI: 10.1007/s00029-024-00925-7
• Enlaces
  • Texto completo
• Resumen

  • This paper considers the (negative) cyclic open–closed map OC−, which maps the cyclic homology of the Fukaya category of a symplectic manifold to its S1-equivariant quantum cohomology. We prove (under simplifying technical hypotheses) that this map respects the respective natural connections in the direction of the equivariant parameter. In the monotone setting this allows us to conclude that OC− intertwines the decomposition of the Fukaya category by eigenvalues of quantum cup product with the first Chern class, with the Hukuhara–Levelt–Turrittin decomposition of the quantum cohomology. We also explain how our results relate to the Givental–Teleman classification of semisimple cohomological field theories: in particular, how the Rmatrix is related to OC− in the semisimple case; we also consider the non-semisimple case.

• Referencias bibliográficas
  • Amorim, L., Tu, J.: Categorical primitive forms of Calabi–Yau A∞-categories with semi-simple cohomology. Sel. Math. (NS) 28(3), 44–54 (2022)....
  • Barannikov, S.: Quantum periods, I: semi-infinite variations of Hodge structures. Int. Math. Res. Not. 2001(23), 1243–1264 (2001). ISSN: 1073-7928....
  • Bott, R., Tu, L.W.: Differential forms in algebraic topology. Graduate Texts in Mathematics, vol. 82, pp. xiv+331. Springer, Berlin (1982)....
  • Candelas, P. et al.: A pair of Calabi–Yau manifolds as an exactly soluble superconformal theory. In: Essays on Mirror Manifolds, pp. 31–95....
  • Cˇaldˇararu, A., Li, S., Junwu, T.: Categorical primitive forms and Gromov–Witten invariants of An singularities. Int. Math. Res. Not. IMRN...
  • Dubrovin, B.: Painlevé transcendents in two-dimensional topological field theory. In: The Painlevé Property, pp. 287–412. CRM Series Mathematics...
  • Fukaya, K. et al.: Lagrangian intersection Floer theory: anomaly and obstruction. Part I. AMS/IP Studies in Advanced Mathematics, vol. 46,...
  • Fukaya, K. et al.: Lagrangian Floer theory on compact toric manifolds. I. Duke Math. J. 151(1), 23–174 (2010). https://doi.org/10.1215/00127094-2009-062
  • Galkin, S., Golyshev, V., Iritani, H.: Gamma classes and quantum cohomology of Fano manifolds: gamma conjectures. Duke Math. J. 165(11), 2005–2077...
  • Ganatra, S.: Cyclic homology, S1-equivariant Floer cohomology, and Calabi–Yau structures. Geometry and Topology (2019). arXiv:1912.13510
  • Ganatra, S., Perutz, T., Sheridan, N.: The cyclic open-closed map and non-commutative Hodge structures (in preparation)
  • Ganatra, S., Perutz, T., Sheridan, N.: Mirror symmetry: from categories to curve counts (2015). arXiv:1510.03839
  • Ganatra, S.: Symplectic cohomology and duality for the wrapped Fukaya category. Thesis (Ph.D.)–Massachusetts Institute of Technology. ProQuest...
  • Gantmacher, F.R.: The theory of matrices, vol. 1, pp. x+374 (Translated from the Russian by K. A. Hirsch, Reprint of the 1959 translation)....
  • Georgieva, P., Zinger, A.: Enumeration of real curves in CP2n−1 and a Witten–Dijkgraaf–Verlinde–Verlinde relation for real Gromov–Witten invariants....
  • Getzler, E.: Cartan homotopy formulas and the Gauss–Manin connection in cyclic homology. In: Quantum Deformations of Algebras and Their Representations...
  • Givental, A.B.: Gromov–Witten invariants and quantization of quadratic Hamiltonians. In: Dedicated to the Memory of I. G. Petrovskii on the...
  • Gross, M.: Tropical geometry and mirror symmetry. CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the...
  • Hertling, C.: tt∗ geometry, Frobenius manifolds, their connections, and the construction for singularities. J. R. Angew. Math. 555, 77–161...
  • Hertling, C., Sevenheck, C.: Nilpotent orbits of a generalization of Hodge structures. J. R. Angew. Math. 609, 23–80 (2007). https://doi.org/10.1515/CRELLE.2007.060
  • Hukuhara, M.: Sur les points singuliers des équations différentielles linéaires III. Mem. Fac. Sci. Imp. Univ. A 2, 125–137 (1942)
  • Joyce, D.: On manifolds with corners. In: Advances in Geometric Analysis, vol. 21, pp. 225–258. Advance Lecture in Mathematics (ALM). Int....
  • Kaledin, D.: Spectral sequences for cyclic homology. In: Algebra, Geometry, and Physics in the 21st Century, vol. 324, pp. 99–129. Progress...
  • Katzarkov, L., Kontsevich, M., Pantev, T.: Hodge theoretic aspects of mirror symmetry. In: From Hodge Theory to Integrability and TQFT tt-Geometry*,...
  • Kontsevich, M., Soibelman, Y.: Notes on A∞-algebras, A∞-categories and non-commutative geometry. In: Homological Mirror Symmetry, vol. 757,...
  • Kontsevich, M.: Homological algebra of mirror symmetry. Proceedings of the International Congress of Mathematicians, vol. 1, 2, pp. 120–139...
  • Levelt, A.H.M.: Jordan decomposition for a class of singular differential operators. Ark. Mat. 13, 1–27 (1975). https://doi.org/10.1007/BF02386195
  • Malgrange, B.: La classification des connexions irrégulières à une variable. In: Mathematics and Physics (Paris, 1979/1982), vol. 37, pp....
  • McDuff, D., Salamon, D.: J-Holomorphic Curves and Symplectic Topology, vol. 52, pp. xii+669. American Mathematical Society Colloquium...
  • McDuff, D., Salamon, D.: Introduction to Symplectic Topology. Third Oxford Graduate Texts in Mathematics, pp. xi+623. Oxford University...
  • Ohta, H., Sanda, F.: Meromorphic connections in filtered A∞ categories. Pure Appl. Math. Q. 16(3), 515–556 (2020). https://doi.org/10.4310/PAMQ.2020.v16.n3.a7
  • Pandharipande, R., Pixton, A., Zvonkine, D.: Relations on Mg,n via 3-spin structures. J. Am. Math. Soc. 28(1), 279–309 (2015). https://doi.org/10.1090/S0894-0347-2014-00808-0
  • Ritter, A.F., Smith, I.: The monotone wrapped Fukaya category and the open–closed string map. Sel. Math. (NS) 23(1), 533–642 (2017). https://doi.org/10.1007/s00029-016-0255-9
  • Ritter, A.F., Smith, I.: The monotone wrapped Fukaya category and the open–closed string map. Sel. Math. (NS) 23(1), 533–642 (2017). https://doi.org/10.1007/s00029-016-0255-9
  • Robbin, J.W., Ruan, Y., Salamon, D.A.: The moduli space of regular stable maps. Math. Z. 259(3), 525–574 (2008). https://doi.org/10.1007/s00209-007-0237-x
  • Sabbah, C.: Isomonodromic deformations and Frobenius manifolds. In: An Introduction, pp. xiv+279. Springer, London, EDP Sciences, Les...
  • Saito, K.: Period mapping associated to a primitive form. Publ. Res. Inst. Math. Sci. 19(3), 1231–1264 (1983). https://doi.org/10.2977/prims/1195182028
  • Saito, K.: The higher residue pairings for a family of hypersurface singular points. In: Singularities, Part 2 (Arcata, California, 1981),...
  • Saito, K., Takahashi, A.: From primitive forms to Frobenius manifolds. In: From Hodge Theory to Integrability and TQFT tt-Geometry, vol. 78....
  • Seidel, P.: Fukaya categories and Picard–Lefschetz theory. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), pp....
  • Sheridan, N.: On the Fukaya category of a Fano hypersurface in projective space. Publ. Math. Inst. Ht. Études Sci. 124, 165–317 (2016). https://doi.org/10.1007/s10240-016-0082-8
  • Sheridan, N.: Formulae in noncommutative Hodge theory. J. Homotopy Relat. Struct. 15(1), 249–299 (2020). https://doi.org/10.1007/s40062-019-00251-2
  • Smith, I.: Floer cohomology and pencils of quadrics. Invent. Math. 189(1), 149–250 (2012). https://doi.org/10.1007/s00222-011-0364-1
  • Solomon, J., Tukachinsky, S.B.: Relative quantum cohomology. Eng. J. Eur. Math. Soc. JEMS (2023). ISSN: 1435-9855
  • Solomon, J.P.: Intersection theory on the moduli space of holomorphic curves with Lagrangian boundary conditions. Thesis (Ph.D.)—Massachusetts...
  • Solomon, J.P., Tukachinsky, S.B.: Differential forms on orbifolds with corners. J. Topol. Anal. (2020). https://doi.org/10.48550/arXiv.2011.10030
  • Solomon, J.P., Tukachinsky, S.B.: Differential forms, Fukaya A∞ algebras, and Gromov–Witten axioms. J. Symplectic Geom. 20(4), 927–994 (2022)....
  • Teleman, C.: The structure of 2D semi-simple field theories. Invent. Math. 188(3), 525–588 (2012). https://doi.org/10.1007/s00222-011-0352-5
  • Turrittin, H.L.: Convergent solutions of ordinary linear homogeneous differential equations in the neighborhood of an irregular singular point....
  • Venugopalan, S., Woodward, C.T., Xu, G.: Fukaya categories of blowups (2020). https://doi.org/10.48550/arXiv.2006.12264
  • Wehrheim, K., Woodward, C.: Orientations for pseudoholomorphic quilts (2015). https://doi.org/10.48550/arXiv.1503.07803