Quantum \(D\)-modules and generalized mirror transformations.

*(English)*Zbl 1170.53071Givental conjectured that the semi-infinite cohomology of the free loop space of a symplectic manifold \(M\) with a natural \(D\)-module structure is isomorphic to the \(S^1\)-equivariant Floer cohomology with the small quantum \(D\)-module (QDM) structure. The author in an ealier work proved this conjecture for a complete intersection \(M\) such that \(c_1(M)\) of the tangent bundle is nef, by using a mirror theorem of Givental. The paper under review is to extend this earlier result to the situation when \(c_1(M)\) of the tangent bundle is not nef. The proof is similar by extending the small quantum \(D\)-module to the big quantum \(D\)-module and a mirror theorem of Coates and Givental.

When \(c_1(M)\) is not nef, the \(S^1\)-equivariant Floer cohomology is not isomorphic to the small QDM, and the ordinary mirror transformation does not work. Hence in section 2, the big and small QDM structures are introduced. Since the total cohomology ring of \(M\) consists only of the even degree part, one can define the twisted Gromov-Witten invariants satisfying almost all formal properties of the ordinary Gromov-Witten invariants, and the big QDM is constructed from the small QDM by adding contributions from cohomology classes even of degree \(\geq 4\). The big QDM is considered as a flat bundle over a formal neighborhood of the origin in \(C^r \times H^*(M, \mathbb C)\), where \(C^r\) is the partial compactification of \(H^2(M, \mathbb C^*)\) given by the Novikov variable dual to \(H^2(M, \mathbb Z)_{\text{free}}\). The small QDM is the restriction of the big one to \(\mathbb C^r \times \{0\}\). A fundamental solution to the big QDM is a formal section \(L\) of the endomorphism bundle given explicitly in section 2.3. The big \(J\)-function of \((M, V)\) is a cohomology valued formal function defined as \(L^{-1}p_0\) for a unit \(p_0\). The big QDM is generated by the unit section \(p_0\) as a \(D\)-module.

In section 3, the author reviews equivariant Floer theory for toric complete intersections. The construction is of algebraic models for the universal cover of the free loop space, and the equivariant Floer theory is considered as a Morse-Bott theory on this algebraic model, as a completion of semi-infinite cohomology with respect to certain filtration. The big abstract QDM and a generalized Mirror transformation are formulated in section 4. The main result and its proof are given in the section 5. The main theorem (Theorem 5.8) under a technique condition 5.6 is proved for the various cases. Examples are computed explicitly in the last section 6.

When \(c_1(M)\) is not nef, the \(S^1\)-equivariant Floer cohomology is not isomorphic to the small QDM, and the ordinary mirror transformation does not work. Hence in section 2, the big and small QDM structures are introduced. Since the total cohomology ring of \(M\) consists only of the even degree part, one can define the twisted Gromov-Witten invariants satisfying almost all formal properties of the ordinary Gromov-Witten invariants, and the big QDM is constructed from the small QDM by adding contributions from cohomology classes even of degree \(\geq 4\). The big QDM is considered as a flat bundle over a formal neighborhood of the origin in \(C^r \times H^*(M, \mathbb C)\), where \(C^r\) is the partial compactification of \(H^2(M, \mathbb C^*)\) given by the Novikov variable dual to \(H^2(M, \mathbb Z)_{\text{free}}\). The small QDM is the restriction of the big one to \(\mathbb C^r \times \{0\}\). A fundamental solution to the big QDM is a formal section \(L\) of the endomorphism bundle given explicitly in section 2.3. The big \(J\)-function of \((M, V)\) is a cohomology valued formal function defined as \(L^{-1}p_0\) for a unit \(p_0\). The big QDM is generated by the unit section \(p_0\) as a \(D\)-module.

In section 3, the author reviews equivariant Floer theory for toric complete intersections. The construction is of algebraic models for the universal cover of the free loop space, and the equivariant Floer theory is considered as a Morse-Bott theory on this algebraic model, as a completion of semi-infinite cohomology with respect to certain filtration. The big abstract QDM and a generalized Mirror transformation are formulated in section 4. The main result and its proof are given in the section 5. The main theorem (Theorem 5.8) under a technique condition 5.6 is proved for the various cases. Examples are computed explicitly in the last section 6.

Reviewer: Weiping Li (Stillwater)

##### MSC:

53D40 | Symplectic aspects of Floer homology and cohomology |

57R58 | Floer homology |

53C45 | Global surface theory (convex surfaces à la A. D. Aleksandrov) |