##
**The structure of 2D semi-simple field theories.**
*(English)*
Zbl 1248.53074

Algebraic classifications of 2-dimensional family topological field theories (FTFT’s) are presented. It leads to the reconstruction of Gromov-Witten invariants from the quantum cup-product at a single semisimple point and the first Chern class of the manifold.

On a topological space \(X\), a FTFT is a symmetric monoidal transformation \(Z\) (i.e., multiplicative under disjoint unions) from \({\mathcal C}(X)\) to \({\mathcal F}(X)\), where objects of \({\mathcal C}\) and \({\mathcal F}\) are bundles of closed oriented 1-manifolds over \(X\) and flat complex vector bundles over \(X\), and morphisms are bundles of compact 2-bordisms, modulo boundary-fixing oriented homeomorphisms over \(X\), and the graded vector spaces \[ \text{Hom}_{{\mathcal F}(X)}(V, W)= H^*(X: \operatorname{Hom}_X(V, W)). \] In this paper, assuming \(A= Z(S^1)\) to be semisimple, classifications of the following four variants of FTFT’s are given.

1. The simplest case, where the surfaces have parametrized boundaries.

2. The case boundaries allow free rotations.

3. The case surfaces are allowed to degenerate modally into the Lefschetz fibrations of algebraic geometry.

4. Deligne-Mumford theories (DMT’s), which includes the Cohomological Field theories (CohFT’s). M. Kontsevich and Yu. Manin [Commun. Math. Phys. 164, No. 3, 525–562 (1994; Zbl 0853.14020)].

The transform from \({\mathcal C}(X)\) to \({\mathcal F}(X)\) of the types 1, 2 and 3, 4 are denoted by \(\widetilde Z\), \(Z\) and \(\overline Z\), respectively (§1).

Classifications of these theories are described in terms of the tautological classes on the moduli of surfaces. The key to the classifications are stability theorems of J. L. Harer [Ann. Math. (2) 121, 215–249 (1985; Zbl 0579.57005)], cf. [N. V. Ivanov, On stabilization of the homology of Teichmüller modular groups. Leningr. Math. J. 1, No. 3, 675–691 (1990); translation from Algebra Anal. 1, No. 3, 110–126 (1989; Zbl 0727.30036); Theorem 2.1] and Madsen and Weiss, which asserts \(H^*(M_g; \mathbb{Q})\cong\mathbb{Q}[\kappa_j]\), \(j= 1,2,\dots\), where \(\kappa_j\)’s are Morita-Mumford classes (Mumford conjecture [I. Madsen and M. Weiss, The stable mapping class group and stable homology theory. Zürich: European Mathematical Society (EMS). 283–307 (2005; Zbl 1082.55004)], Theorem 2.2).

In the limit of large genus surfaces, the sewing axiom; sewing boundary components leads to the composition of maps, becomes an equation in the complex cohomology of the stable mapping class groups, which is a power series ring in the tautological classes by Theorem 2.1. Semisimplicity of \(A\) retrieve the low-dimensional answer from high genus thanks to invertibility of the Euler class. According to this line, theories of case 1 are classified by a single, group-like class \(\widetilde Z^+\) in the \(A\)-valued cohomology of the stable mapping class group of surfaces (§2.1).

To classify theories of case 2 needs a new classification datum, a \(\mathbb{C}\)-linear map \(E: A\to A[[z]]\) with \(E= \text{Id}(\text{mod\,}z)\) (§3).

DMT’s (and the case 3) require an additional argument. The universal families of stable nodal surfaces are classified by orbifolds with a normal-crossing stratification. By the same argument as above, the classes \(Z\) is determined on each stratum. But there could be ambiguities in patching these classes together, However, the Euler classes of certain boundary strata involving large-genus surfaces are not zero-divisors in low-degree cohomology. This ensures the unique gluing of cohomology classes over suitably chosen strata. In this paper, enough strata to cover all Deligne-Mumford moduli orbifolds, are find and prove the unique patching of the \(Z\)-classes to a global class \(\overline Z\) (§4 and 5). The author says these are main parts of this paper. By this result, it is shown semi-simple DMT’s are uniquely determined by the nodal propagator \(D\), an incoming-outgoing pair of crossing disks, and by the associated free-boundary theory on smooth curves (§5).

The splitting result in §5 is also used to show that every cohomology class of \(\overline M^n_{n,\infty}\) is uniquely determined by its restriction to all the strata \(M_\gamma\) (Appendix of §5).

In §6, the classification of semisimple DMT’s are reformulated in terms of the action of the symplectic group on the cohomology of Deligne-Mumford spaces; cf. [Y. Chen, M. Kontsevich and A. Schwarz, Nucl. Phys., B 730, No. 3, 352–363 (2005; Zbl 1276.81096); L. Katzarkov, M. Kontsevich and T. Pantev, Hodge theoretic aspects of mirror symmetry, From Hodge theory to integrability and TQFT tt\(^*\)-geometry, Providence, RI: American Mathematical Society (AMS). Proceedings of Symposia in Pure Mathematics 78, 87–174 (2008; Zbl 1206.14009)]. Results are applied to Cohomological Field theory (CohFT; DMT with \(D=\text{Id}\), and show a semisimple CohFT satisfying the homogeneity constraint (stated in §1.5 (iii)) is uniquely and explicitly reconstructible from genus zero data. For homogeneous theories with flat vacuum, the Euler vector field and the Frobenius algebra sructure suffice for reconstruction (§1.7. Th.1 Proved in §6.6).

To reconstruct the theory from data, a family of DMT’s parametrized by a neighborhood \(U\) of \(0\in A\) is constructed in §7. When starting with a CohFT, the genus zero part of this family defines on \(U\) the structure of a Frobenius manifold (cf. Yu. Manin [Frobenius manifolds, quantum cohomology, and moduli spaces. Colloquium Publications. American Mathematical Society (AMS). 47. Providence, RI: American Mathematical Society (AMS). xiii, 303 p. (1999; Zbl 0952.14032)]). \(E_u\) and \(Y_u\), the vector whose entires are the eigenvalues of the operator of multiplication by \(\widetilde Z_u\) are shown to satisfy systems of ODE’s in \(u\) (Proposition 7.2). Explicit form of vacuum is also given (§7.2, (7.1)).

In §8, using canonical coordinates of Frobenius manifolds [B. Dubrovin, Geometry of 2D topological field theories, Integrable systems and quantum groups. Lectures given at the 1st session of the Centro Internazionale Matematico Estivo (CIME) held in Montecatini Terme, Italy, Berlin: Springer-Verlag. Lect. Notes Math. 1620, 120–348 (1996; Zbl 0841.58065)], a semisimple homogeneous CohFT is shown to be determined by the unique Euler-invariant solution \(E\) of the ODE in Proposition 7.2 and the vacuum (7.1) (Givental conjecture. Cor.8.1.) cf. [A. Givental, Int. Math. Res. Not. 2001, No. 23, 1265–1286 (2001; Zbl 1074.14532)]. The author says §8 is largely a review and adaptation of this paper).

The inhomogeneous theories corresponds to a given semisimple Frobenius manifold are related geometrically by Hodge bundle twists. By this argument and previous results all CohFT’s with flat vacuum based on a fixed semisimple, pointed Frobenius manifold are shown to be classified by matrices \(E\circ\exp h(z)\), with arbitrary \(h\) but same \(E\) (Proposition 8.3 (ii)). All possible rank one theories are also described (Manin-Zograf conjecture. Proposition 8.4 and (8.1)) (cf. [Yu. Manin and P. Zograf, Ann. Inst. Fourier 50, No. 2, 519–535 (2000; Zbl 1001.14008)]). Then after giving homogeneity condition of CohFT with flat vacuum (Proposition 8.5), this paper is concluded to show the Gromov-Witten classes \({\mathrm GW}^n_{g,d}\in H^{\mathrm ev}(\overline M^n_g)\) of a compact symplectic manifold are uniquely determined by its first Chern class and by the quantum multiplication law at any single semisimple point (Theorem 8.1).

On a topological space \(X\), a FTFT is a symmetric monoidal transformation \(Z\) (i.e., multiplicative under disjoint unions) from \({\mathcal C}(X)\) to \({\mathcal F}(X)\), where objects of \({\mathcal C}\) and \({\mathcal F}\) are bundles of closed oriented 1-manifolds over \(X\) and flat complex vector bundles over \(X\), and morphisms are bundles of compact 2-bordisms, modulo boundary-fixing oriented homeomorphisms over \(X\), and the graded vector spaces \[ \text{Hom}_{{\mathcal F}(X)}(V, W)= H^*(X: \operatorname{Hom}_X(V, W)). \] In this paper, assuming \(A= Z(S^1)\) to be semisimple, classifications of the following four variants of FTFT’s are given.

1. The simplest case, where the surfaces have parametrized boundaries.

2. The case boundaries allow free rotations.

3. The case surfaces are allowed to degenerate modally into the Lefschetz fibrations of algebraic geometry.

4. Deligne-Mumford theories (DMT’s), which includes the Cohomological Field theories (CohFT’s). M. Kontsevich and Yu. Manin [Commun. Math. Phys. 164, No. 3, 525–562 (1994; Zbl 0853.14020)].

The transform from \({\mathcal C}(X)\) to \({\mathcal F}(X)\) of the types 1, 2 and 3, 4 are denoted by \(\widetilde Z\), \(Z\) and \(\overline Z\), respectively (§1).

Classifications of these theories are described in terms of the tautological classes on the moduli of surfaces. The key to the classifications are stability theorems of J. L. Harer [Ann. Math. (2) 121, 215–249 (1985; Zbl 0579.57005)], cf. [N. V. Ivanov, On stabilization of the homology of Teichmüller modular groups. Leningr. Math. J. 1, No. 3, 675–691 (1990); translation from Algebra Anal. 1, No. 3, 110–126 (1989; Zbl 0727.30036); Theorem 2.1] and Madsen and Weiss, which asserts \(H^*(M_g; \mathbb{Q})\cong\mathbb{Q}[\kappa_j]\), \(j= 1,2,\dots\), where \(\kappa_j\)’s are Morita-Mumford classes (Mumford conjecture [I. Madsen and M. Weiss, The stable mapping class group and stable homology theory. Zürich: European Mathematical Society (EMS). 283–307 (2005; Zbl 1082.55004)], Theorem 2.2).

In the limit of large genus surfaces, the sewing axiom; sewing boundary components leads to the composition of maps, becomes an equation in the complex cohomology of the stable mapping class groups, which is a power series ring in the tautological classes by Theorem 2.1. Semisimplicity of \(A\) retrieve the low-dimensional answer from high genus thanks to invertibility of the Euler class. According to this line, theories of case 1 are classified by a single, group-like class \(\widetilde Z^+\) in the \(A\)-valued cohomology of the stable mapping class group of surfaces (§2.1).

To classify theories of case 2 needs a new classification datum, a \(\mathbb{C}\)-linear map \(E: A\to A[[z]]\) with \(E= \text{Id}(\text{mod\,}z)\) (§3).

DMT’s (and the case 3) require an additional argument. The universal families of stable nodal surfaces are classified by orbifolds with a normal-crossing stratification. By the same argument as above, the classes \(Z\) is determined on each stratum. But there could be ambiguities in patching these classes together, However, the Euler classes of certain boundary strata involving large-genus surfaces are not zero-divisors in low-degree cohomology. This ensures the unique gluing of cohomology classes over suitably chosen strata. In this paper, enough strata to cover all Deligne-Mumford moduli orbifolds, are find and prove the unique patching of the \(Z\)-classes to a global class \(\overline Z\) (§4 and 5). The author says these are main parts of this paper. By this result, it is shown semi-simple DMT’s are uniquely determined by the nodal propagator \(D\), an incoming-outgoing pair of crossing disks, and by the associated free-boundary theory on smooth curves (§5).

The splitting result in §5 is also used to show that every cohomology class of \(\overline M^n_{n,\infty}\) is uniquely determined by its restriction to all the strata \(M_\gamma\) (Appendix of §5).

In §6, the classification of semisimple DMT’s are reformulated in terms of the action of the symplectic group on the cohomology of Deligne-Mumford spaces; cf. [Y. Chen, M. Kontsevich and A. Schwarz, Nucl. Phys., B 730, No. 3, 352–363 (2005; Zbl 1276.81096); L. Katzarkov, M. Kontsevich and T. Pantev, Hodge theoretic aspects of mirror symmetry, From Hodge theory to integrability and TQFT tt\(^*\)-geometry, Providence, RI: American Mathematical Society (AMS). Proceedings of Symposia in Pure Mathematics 78, 87–174 (2008; Zbl 1206.14009)]. Results are applied to Cohomological Field theory (CohFT; DMT with \(D=\text{Id}\), and show a semisimple CohFT satisfying the homogeneity constraint (stated in §1.5 (iii)) is uniquely and explicitly reconstructible from genus zero data. For homogeneous theories with flat vacuum, the Euler vector field and the Frobenius algebra sructure suffice for reconstruction (§1.7. Th.1 Proved in §6.6).

To reconstruct the theory from data, a family of DMT’s parametrized by a neighborhood \(U\) of \(0\in A\) is constructed in §7. When starting with a CohFT, the genus zero part of this family defines on \(U\) the structure of a Frobenius manifold (cf. Yu. Manin [Frobenius manifolds, quantum cohomology, and moduli spaces. Colloquium Publications. American Mathematical Society (AMS). 47. Providence, RI: American Mathematical Society (AMS). xiii, 303 p. (1999; Zbl 0952.14032)]). \(E_u\) and \(Y_u\), the vector whose entires are the eigenvalues of the operator of multiplication by \(\widetilde Z_u\) are shown to satisfy systems of ODE’s in \(u\) (Proposition 7.2). Explicit form of vacuum is also given (§7.2, (7.1)).

In §8, using canonical coordinates of Frobenius manifolds [B. Dubrovin, Geometry of 2D topological field theories, Integrable systems and quantum groups. Lectures given at the 1st session of the Centro Internazionale Matematico Estivo (CIME) held in Montecatini Terme, Italy, Berlin: Springer-Verlag. Lect. Notes Math. 1620, 120–348 (1996; Zbl 0841.58065)], a semisimple homogeneous CohFT is shown to be determined by the unique Euler-invariant solution \(E\) of the ODE in Proposition 7.2 and the vacuum (7.1) (Givental conjecture. Cor.8.1.) cf. [A. Givental, Int. Math. Res. Not. 2001, No. 23, 1265–1286 (2001; Zbl 1074.14532)]. The author says §8 is largely a review and adaptation of this paper).

The inhomogeneous theories corresponds to a given semisimple Frobenius manifold are related geometrically by Hodge bundle twists. By this argument and previous results all CohFT’s with flat vacuum based on a fixed semisimple, pointed Frobenius manifold are shown to be classified by matrices \(E\circ\exp h(z)\), with arbitrary \(h\) but same \(E\) (Proposition 8.3 (ii)). All possible rank one theories are also described (Manin-Zograf conjecture. Proposition 8.4 and (8.1)) (cf. [Yu. Manin and P. Zograf, Ann. Inst. Fourier 50, No. 2, 519–535 (2000; Zbl 1001.14008)]). Then after giving homogeneity condition of CohFT with flat vacuum (Proposition 8.5), this paper is concluded to show the Gromov-Witten classes \({\mathrm GW}^n_{g,d}\in H^{\mathrm ev}(\overline M^n_g)\) of a compact symplectic manifold are uniquely determined by its first Chern class and by the quantum multiplication law at any single semisimple point (Theorem 8.1).

Reviewer: Akira Asada (Takarazuka)

### MSC:

53D45 | Gromov-Witten invariants, quantum cohomology, Frobenius manifolds |

57R56 | Topological quantum field theories (aspects of differential topology) |

81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |

### Keywords:

2D family topology field theory; Lefschetz fibration; Deligne-Mumford theory; cohomological field theory; Frobenius manifold; Gromov-Witten invariant### Citations:

Zbl 0853.14020; Zbl 0579.57005; Zbl 0727.30036; Zbl 1082.55004; Zbl 1206.14009; Zbl 0952.14032; Zbl 0841.58065; Zbl 1074.14532; Zbl 1001.14008; Zbl 1276.81096### References:

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