Open Gromov-Witten invariants from the Fukaya category. (English) Zbl 1543.14041
J. Walcher [Commun. Math. Phys. 276, No. 3, 671–689 (2007; Zbl 1135.14030)] and D. R. Morrison and J. Walcher [Adv. Theor. Math. Phys. 13, No. 2, 553–598 (2009; Zbl 1166.81036)] proposed an extension of the classical picture of enumerative mirror symmetry by including open Gromov-Witten invariants of the real locus of the Fermat quintic, which were defined in [J. P. Solomon, “Intersection theory on the moduli space of holomorphic curves with Lagrangian boundary conditions”, Preprint, arXiv:math/0606429], and the normal function associated to a family of curves in \(Y\) [P. A. Griffiths, Ann. Math. (2) 90, 460–495, 496–541 (1969; Zbl 0215.08103)]. This extended enumerative mirror symmetry can be encapsulated by an isomorphism of extensions of variants of Hodge structures. Walcher used this to predict values for all genus \(0\) open Gromov-Witten invariants of the real locus of the quintic, which were later verified in [R. Pandharipande et al., J. Am. Math. Soc. 21, No. 4, 1169–1209 (2008; Zbl 1203.53086)]. This paper proposes a framework to show that homological mirror symmetry implies this extended enumerative mirror symmetry, having been inspried by J. P. Solomon and S. B. Tukachinsky [“Relative quantum cohomology”, Preprint, arXiv:1906.04795] of the relative quantum T-structure \(QH^{\ast}(X,L)\left[ \left[ u\right] \right] \), which is a modification of quantum cohomology additionally incorporating open Gromov-Witten invariants. The author additionally defines a connection in the \(u\)-direction, showing that this equips \(QH^{\ast}(X,L)\left[ \left[ u\right] \right] \) with a TE-structure. When the Lagrangian is null-homologous, this yields an extension of TE-structures:
\[
0\rightarrow\Lambda\left[ \left[ u\right] \right] \rightarrow QH^{\ast }(X,L;\Lambda)\left[ \left[ u\right] \right] \rightarrow QH^{\ast}(X;\Lambda)\left[ \left[ u\right] \right] \rightarrow\rightarrow0
\]
On the categorical side, given a unital \(A_{\infty}\)-category \(\mathcal{C}\) over a ring \(R\) and an object \(L\in\mathcal{C}\), the author constructs a version of negative cyclic homology called relative cyclic homology, denoted \(HC_{\ast}^{-}(\mathcal{C},L)\), which is equipped with a TE-structure. If \(L\) has zero Chern character \(ch(L)=\left[ \boldsymbol{e}_{L}\right] =0\in HC_{\ast}^{-}(\mathcal{C})\), it fits into an exact sequence of TE-structures:
\[
0\rightarrow HC_{\ast}^{-}(\mathcal{C})\rightarrow HC_{\ast }^{-}(\mathcal{C},L)\rightarrow R\left[ \left[ u\right] \right] \rightarrow0.
\]
The author proposes the following conjecture.
Conjecture. There exists a relative cyclic open-closed map \[ \mathcal{OC}_{L}^{-}:HC_{\ast}^{--}(Fuk(X),L)\rightarrow QH_{\ast}(X,L)\left[ \left[ u\right] \right] \] which is a morphism of TE-structures. When \(L\) is null-homogeneous, it fits into the commutative diagram: \[ \begin{array} [c]{ccccccccc} 0 & \rightarrow & HC_{\ast}^{--}(Fuk(X))& \rightarrow & HC_{\ast}^{--}(Fuk(X),L)& \rightarrow & \Lambda\left[ \left[ u\right] \right] & \rightarrow & 0\\ & & \downarrow\mathcal{OC}^{--} & & \downarrow\mathcal{OC}_{L}^{--} & & \parallel & & \\ 0 & \rightarrow & QH_{\ast}(X;\Lambda)\left[ \left[ u\right] \right] & \rightarrow & QH_{\ast}(X,L)\left[ \left[ u\right] \right] & \rightarrow & \Lambda\left[ \left[ u\right] \right] & \rightarrow & 0 \end{array} \]
This paper uses the same technical setup as J. P. Solomon and S. B. Tukachinsky [J. Symplectic Geom. 20, No. 4, 927–994 (2022; Zbl 1516.53077)], where a Fukaya \(A_{\infty}\)-algebra \(CF^{\ast}(L,L)\) was constructed using differential forms, to establish the above conjecture in this local setting (Theorem 5.6). The author proposes that \(HC_{\ast}^{--}(Fuk(X),L)\) can be used to extract open Gromov-Witten invariants of \(L\) from the Fukaya category, establishing the following theorem.
Theorem. Let \(X\) be any Calabi-Yau and \(L\subset X\) any Lagrangian brane with \(\left[ L\right] =0\in H_{n}(X)\). Then, assuming the above conjecture for \((X,L)\), the Fukaya category of \(X\) determines the total open Gromov-Witten invariant: \[ OGW_{\bigstar,0}(\_)=\sum\limits_{d\in H_{2}(X,L)}q^{w(d)}OGW_{d,0}(\_):H^{n-1}(X)\rightarrow\Lambda \] where \(OGW_{d,0}:H^{n-1}(X)\rightarrow\mathbb{R}\) denote the genus \(0\) open Gromov-Witten invariants in degree \(d\in H_{2}(X,L)\) with no boundary marked points, as constructed in [J. P. Solomon and S. B. Tukachinsky, “Relative quantum cohomology”, Preprint, arXiv:1906.04795; J. Eur. Math. Soc. (2023)].
In the Fano setting the author can say more about the extension, setting the Novikov parameter equal to \(1\) as well as all bulk parameters equal to \(0\), so that one works over \(\mathbb{C}\), only considering an E-structure.
Conjecture. There exists a relative cyclic open-closed map \[ \mathcal{OC}_{L}^{-}:HC_{\ast}^{--}(Fuk(X),L)\rightarrow QH_{\ast}(X,L)\left[ \left[ u\right] \right] \] which is a morphism of TE-structures. When \(L\) is null-homogeneous, it fits into the commutative diagram: \[ \begin{array} [c]{ccccccccc} 0 & \rightarrow & HC_{\ast}^{--}(Fuk(X))& \rightarrow & HC_{\ast}^{--}(Fuk(X),L)& \rightarrow & \Lambda\left[ \left[ u\right] \right] & \rightarrow & 0\\ & & \downarrow\mathcal{OC}^{--} & & \downarrow\mathcal{OC}_{L}^{--} & & \parallel & & \\ 0 & \rightarrow & QH_{\ast}(X;\Lambda)\left[ \left[ u\right] \right] & \rightarrow & QH_{\ast}(X,L)\left[ \left[ u\right] \right] & \rightarrow & \Lambda\left[ \left[ u\right] \right] & \rightarrow & 0 \end{array} \]
This paper uses the same technical setup as J. P. Solomon and S. B. Tukachinsky [J. Symplectic Geom. 20, No. 4, 927–994 (2022; Zbl 1516.53077)], where a Fukaya \(A_{\infty}\)-algebra \(CF^{\ast}(L,L)\) was constructed using differential forms, to establish the above conjecture in this local setting (Theorem 5.6). The author proposes that \(HC_{\ast}^{--}(Fuk(X),L)\) can be used to extract open Gromov-Witten invariants of \(L\) from the Fukaya category, establishing the following theorem.
Theorem. Let \(X\) be any Calabi-Yau and \(L\subset X\) any Lagrangian brane with \(\left[ L\right] =0\in H_{n}(X)\). Then, assuming the above conjecture for \((X,L)\), the Fukaya category of \(X\) determines the total open Gromov-Witten invariant: \[ OGW_{\bigstar,0}(\_)=\sum\limits_{d\in H_{2}(X,L)}q^{w(d)}OGW_{d,0}(\_):H^{n-1}(X)\rightarrow\Lambda \] where \(OGW_{d,0}:H^{n-1}(X)\rightarrow\mathbb{R}\) denote the genus \(0\) open Gromov-Witten invariants in degree \(d\in H_{2}(X,L)\) with no boundary marked points, as constructed in [J. P. Solomon and S. B. Tukachinsky, “Relative quantum cohomology”, Preprint, arXiv:1906.04795; J. Eur. Math. Soc. (2023)].
In the Fano setting the author can say more about the extension, setting the Novikov parameter equal to \(1\) as well as all bulk parameters equal to \(0\), so that one works over \(\mathbb{C}\), only considering an E-structure.
Reviewer: Hirokazu Nishimura (Tsukuba)
MSC:
| 14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |
| 14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |
| 14J33 | Mirror symmetry (algebro-geometric aspects) |
| 14C30 | Transcendental methods, Hodge theory (algebro-geometric aspects) |
| 53D45 | Gromov-Witten invariants, quantum cohomology, Frobenius manifolds |
| 53D37 | Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category |
References:
| [1] | Amorim, L.; Tu, J., Categorical primitive forms of Calabi-Yau \(A_\infty \)-categories with semi-simple cohomology, Sel. Math. New Ser., 28, 3, Article 54 pp., 2022, (ISSN 1420-9020) · Zbl 1492.18016 |
| [2] | Barannikov, S., Quantum periods, I: semi-infinite variations of Hodge structures, Int. Math. Res. Not., 1243-1264, Jan. 2001 · Zbl 1074.14510 |
| [3] | Candelas, P., A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, (Essays on Mirror Manifolds, 1992, Int. Press: Int. Press Hong Kong), 31-95 · Zbl 0826.32016 |
| [4] | Carlson, J. A., The geometry of the extension class of a mixed Hodge structure, (Algebraic Geometry, Bowdoin, 1985. Algebraic Geometry, Bowdoin, 1985, Brunswick, Maine, 1985. Algebraic Geometry, Bowdoin, 1985. Algebraic Geometry, Bowdoin, 1985, Brunswick, Maine, 1985, Proc. Sympos. Pure Math., vol. 46, 1987, Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 199-222 · Zbl 0637.14007 |
| [5] | Deligne, P., Equations Différentielles à Points Singuliers Réguliers, Lecture Notes in Mathematics, vol. 163, 1970, Springer-Verlag: Springer-Verlag Berlin · Zbl 0244.14004 |
| [6] | Dubrovin, B., Painlevé transcendents in two-dimensional topological field theory, (The Painlevé Property. The Painlevé Property, CRM Ser. Math. Phys., 1999, Springer: Springer New York), 287-412 · Zbl 1026.34095 |
| [7] | Fukaya, K., Lagrangian Intersection Floer Theory: Anomaly and Obstruction. Part I, AMS/IP Studies in Advanced Mathematics, vol. 46, 2009, American Mathematical Society, Providence, RI, International Press: American Mathematical Society, Providence, RI, International Press Somerville, MA, pp. xii+396 · Zbl 1181.53002 |
| [8] | Ganatra, S., Symplectic Cohomology and Duality for the Wrapped Fukaya Category, 2012, Massachusetts Institute of Technology. ProQuest LLC: Massachusetts Institute of Technology. ProQuest LLC Ann Arbor, MI, (no paging) |
| [9] | Ganatra, S., Cyclic homology, \( S^1\)-equivariant Floer cohomology and Calabi-Yau structures, Geom. Topol., 27, 9, 3461-3584, 2023, (ISSN 1364-0380) · Zbl 07777465 |
| [10] | Getzler, E., Cartan homotopy formulas and the Gauss-Manin connection in cyclic homology, (Quantum Deformations of Algebras and Their Representations (Ramat-Gan, 1991/1992; Rehovot, 1991/1992), vol. 7. Israel Math. Conf. Proc. Bar-Ilan Univ., Ramat Gan, 1993), 65-78 · Zbl 0844.18007 |
| [11] | S. Ganatra, T. Perutz, N. Sheridan, The cyclic open-closed map and non-commutative Hodge structures, in preparation. |
| [12] | Ganatra, S.; Perutz, T.; Sheridan, N., Mirror symmetry: from categories to curve counts, 2015 |
| [13] | Griffiths, P. A., On the periods of certain rational integrals. I, II, Ann. Math. (2), 90, 460-495, 1969; Griffiths, P. A., Ann. Math. (2), 90, 496-541, 1969 · Zbl 0215.08103 |
| [14] | Hertling, C., \(t t^\ast\) geometry, Frobenius manifolds, their connections, and the construction for singularities, J. Reine Angew. Math., 555, 77-161, 2003 · Zbl 1040.53095 |
| [15] | Hugtenburg, K., The cyclic open-closed map, u-connections and R-matrices, 2022, To appear in Selecta Math. (N.S.) |
| [16] | Hukuhara, M., Sur les points singuliers des équations différentielles linéaires. III, Mem. Fac. Sci. Kyūsyū Imp. Univ. A, 2, 125-137, 1942 · Zbl 0061.19603 |
| [17] | Kaledin, D., Spectral sequences for cyclic homology, (Algebra, Geometry, and Physics in the 21st Century. Algebra, Geometry, and Physics in the 21st Century, Progr. Math., vol. 324, 2017, Birkhäuser/Springer: Birkhäuser/Springer Cham), 99-129 · Zbl 1390.14011 |
| [18] | Katzarkov, L.; Kontsevich, M.; Pantev, T., Hodge theoretic aspects of mirror symmetry, (From Hodge Theory to Integrability and TQFT tt*-Geometry. From Hodge Theory to Integrability and TQFT tt*-Geometry, Proc. Sympos. Pure Math., vol. 78, 2008, Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 87-174 · Zbl 1206.14009 |
| [19] | Kontsevich, M.; Soibelman, Y., Notes on \(A_\infty \)-algebras, \( A_\infty \)-categories and non-commutative geometry, (Homological Mirror Symmetry. Homological Mirror Symmetry, Lecture Notes in Phys., vol. 757, 2009, Springer: Springer Berlin), 153-219 · Zbl 1202.81120 |
| [20] | Levelt, A. H.M., Jordan decomposition for a class of singular differential operators, Ark. Mat., 13, 1-27, 1975 · Zbl 0305.34008 |
| [21] | Morrison, D. R., Compactifications of moduli spaces inspired by mirror symmetry, (218. Journées de Géométrie Algébrique d’Orsay. 218. Journées de Géométrie Algébrique d’Orsay, Orsay, 1992, 1993), 243-271 · Zbl 0824.14007 |
| [22] | Morrison, D. R.; Walcher, J., D-branes and normal functions, Adv. Theor. Math. Phys., 13, 2, 553-598, 2009 · Zbl 1166.81036 |
| [23] | Pandharipande, R.; Solomon, J.; Walcher, J., Disk enumeration on the quintic 3-fold, J. Am. Math. Soc., 21, 4, 1169-1209, 2008 · Zbl 1203.53086 |
| [24] | Sabbah, C., Isomonodromic deformations and Frobenius manifold, (Universitext, 2007, Springer-Verlag London, Ltd., London; EDP Sciences: Springer-Verlag London, Ltd., London; EDP Sciences Les Ulis), ISBN 978-2-7598-0047-6, pp. xiv+279 · Zbl 1131.14004 |
| [25] | Sheridan, N., Homological mirror symmetry for a Calabi-Yau hypersurface in projective space, 2012, Massachusetts Institute of Technology. ProQuest LLC: Massachusetts Institute of Technology. ProQuest LLC Ann Arbor, MI, (no paging) |
| [26] | Sheridan, N., Formulae in noncommutative Hodge theory, J. Homotopy Relat. Struct., 15, 1, 249-299, 2020 · Zbl 1506.14025 |
| [27] | Shklyarov, D., Non-commutative Hodge structures: towards matching categorical and geometric examples, Trans. Am. Math. Soc., 366, 6, 2923-2974, 2014 · Zbl 1354.16009 |
| [28] | Solomon, J. P., Intersection theory on the moduli space of holomorphic curves with Lagrangian boundary conditions, 2006, Massachusetts Institute of Technology. ProQuest LLC: Massachusetts Institute of Technology. ProQuest LLC Ann Arbor, MI, (no paging) |
| [29] | Solomon, J. P.; Tukachinsky, S. B., Differential forms on orbifolds with corners, 2020, To appear in J. Topol. Anal. |
| [30] | Solomon, J. P.; Tukachinsky, S. B., Point-like bounding chains in open Gromov-Witten theory, Geom. Funct. Anal., 31, 5, 1245-1320, 2021 · Zbl 1494.53098 |
| [31] | Solomon, J. P.; Tukachinsky, S. B., Differential forms, Fukaya \(A_\infty\) algebras, and Gromov-Witten axioms, J. Symplectic Geom., 20, 4, 927-994, 2022, (ISSN 1540-2347) · Zbl 1516.53077 |
| [32] | Solomon, J.; Tukachinsky, S. B., Relative quantum cohomology, J. Eur. Math. Soc., 2023 |
| [33] | Turrittin, H. L., Convergent solutions of ordinary linear homogeneous differential equations in the neighborhood of an irregular singular point, Acta Math., 93, 27-66, 1955 · Zbl 0064.33603 |
| [34] | Voisin, C., Hodge Theory and Complex Algebraic Geometry. II, Cambridge Studies in Advanced Mathematics, vol. 77, 2003, Cambridge University Press: Cambridge University Press Cambridge, pp. x+351. Translated from the French by Leila Schneps · Zbl 1032.14002 |
| [35] | Walcher, J., Opening mirror symmetry on the quintic, Commun. Math. Phys., 276, 3, 671-689, 2007 · Zbl 1135.14030 |
| [36] | Wasow, W., Asymptotic Expansions for Ordinary Differential Equations, Pure and Applied Mathematics, vol. XIV, 1965, Interscience Publishers John Wiley & Sons, Inc.: Interscience Publishers John Wiley & Sons, Inc. New York, London, Sydney, pp. ix+362 · Zbl 0169.10903 |
| [37] | Zucker, S., Generalized intermediate Jacobians and the theorem on normal functions, Invent. Math., 33, 185-222, Oct. 1976 · Zbl 0329.14008 |
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