Annales Mathematiques Blaise Pascal 0, 0-0 (2001)

An Example of an Article for this Journal

Timothy Kohl^{1} Steven Rosenberg^{2} Charles Zed

Résumé

This is a demonstration of the abstract for an article. Recall that quantum cohomology gives a finite dimensional integrable system via the Dubrovin connection. Motivated by Givental’s work on mirror symmetry, we use gauge theory techniques and the Frobenius Integrability Theorem to find flat sections for the Dubrovin connection

1 Introduction

The work of Givental [3] and Liu-Lian-Yau [6] on mirror symmetry relates Gromov-Witten invariants of a quintic hypersurface in ℙ^{3} to period integrals of Kähler structures on the mirror manifold. Givental’s method uses detailed calculations of equivariant GW invariants to produce flat sections of the Dubrovin connection on the tangent bundle to the even cohomology of the hypersurface, which are then related to solutions of the Picard-Fuchs ODE for the periods on the mirror.

We would like to thank Charles Kay for several helpful conversations. Throughout the paper, we use the following notation:

Notation: Let ℤ denote the set of integers, and let A denote the set of all even integers which cannot be written as the sum of two primes. Finally, let B = A ∩ [0,200].

1The first author is partially supported by NSF Grant XYZPDQ

2The first author is partially supported by NSF Grant #12345567

T. Kohl, S. Rosenberg, C. Zed

2 The next section

Let ∗ be an associative, commutative product on a complex vector space H. The associated Dubrovin connection on T H is

∇_{Y} X = dX(Y) + \sqrt{−1}Y ∗ X  (2.1)

with connection one-form ω(Y)(X) = \sqrt{−1}Y ∗ X. If \{T_{0}, …,T_{m}\} is a basis of H with

T_{i} ∗ T_{j} = Γ_{ij}^{k}T_{k},  (2.2)

then ω_{j}^{i} = \sqrt{−1}Γ_{ℓj}^{i} dt^{ℓ} (where a typical α ∈ H is α = t_{i}T^{i}). It is fundamental that ∇ is flat because the product is associative and commutative. Note that ∇ stays flat if we replace \sqrt{−1} in (2.1) by ℏ ∈ ℂ.

Proposition 2.1: Let ∗ be a commutative, associative product on a complex vector space H. Define matrices Γ_{i}, i = 1, …,dim H by T_{i} ∗ T_{j} = (Γ_{i})_{j}^{k}T_{k} for a basis {T_{k}} of H. Then a basis of the flat sections of the associated flat Dubrovin connection on T H is

\{exp[−\sqrt{−1}t^{ℓ}Γ_{ℓ}]\}_{ℓ=0}^{m} .

Lemme 2.2: Except in characteristic one, 0 ≠ 1.

Théorème 2.3: Even in characteristic one, 1 + 1 = 2.

Preuve: The proof of this theorem is left for the reader.

Corollaire 2.4: In all characteristics, we have

2 + 2 = 4.  (2.3)

The second paper in this series will be devoted to applications of (2.3).

Exemple: Let A consist of a set of two elements and B a set of two elements. If A ∩ B = ∅, then |A ∪ B| = 4.

Remarque: The reader can easily generalize this example to the case of three sets.

An Example of an Article

More precisely, the small quantum product is defined on i^{∗}T H, where i : H^{2} → H is the inclusion. For {T_{1}, …,T_{k}} a basis of H^{2}, the small quantum product is defined as in (2.2) by Γ_{jr}^{i}(t^{1}, …,t^{k}) = \sqrt{−1}ψ_{jrℓ}h^{ℓi}, where (h_{αβ}) = (T_{α} · T_{β}) is the intersection matrix, (h^{αβ}) its inverse, and

ψ_{jrℓ} = T_{j} · T_{r} · T_{ℓ} + ∑_{β} exp [∑_{s=1}^{k} t^{s} ∫_{β} T_{s}] I_{β}(T_{j},T_{r},T_{ℓ})  (2.4)

[2, §2.1]. In (2.4), I_{β}(T_{j},T_{r},T_{ℓ}) is the three-pointed Gromov-Witten invariant, and β ranges over H_{2}(M,ℤ) \ {0}.

Définition: A Hopf algebra H is co-commutative if Δ = τ ∘ Δ where Δ is the co-multiplication and τ is the switch map defined by τ(h_{1} ⊗ h_{2}) = h_{2} ⊗ h_{1}.

Théorème 2.5: All non-empty sets have at least one element.

Corollaire 2.6: The set consisting of the authors of this article has at least one element.

For other related references, see [1], [2], [4], [5], [7], [8], [9, Vol. I, §2.3], [10], [11].

Références

[1] M. F. Atiyah. The Geometry and Physics of Knots. Cambridge University Press, Cambridge, 1990.

[2] G. Bini, C. De Concini, M. Polito, and C. Procesi. On the work of Givental relative to mirror symmetry. Appunti dei Corsi Tenuti da Docenti della Scuola, Scuola Normale Superiore, Pisa, 1998. math.AG/9805097.

[3] A. Givental. Equivariant Gromov-Witten invariants. Internat. Math. Res. Notices, 13:613–663, 1996.

[4] C. King and C.-L. Terng. Submanifolds in path space. In K. Uhlenbeck, editor, Global Analysis in Modern Mathematics, pages 253–282. Publish or Perish, Inc., Houston, 1993.

[5] J. A. Lappo-Danilevsky. Mémoires sur la Théorie des Systèmes des Èquations Différentieles Linéaires. AMS Chelsea Pub., New York, 1953.

[6] B. Lian, K. Liu, and S.-T. Yau. Mirror principle I. Asian J. Math., 4:729–763, 1997.

[7] X. Liu and G. Tian. Virasoro constraints for quantum cohomology. J. Differential Geometry, 50:537–590, 1998.

[8] R. Pandharipande. Rational curves on hypersurfaces (after A. Givental). Séminaire Bourbaki, 252:307–340–55, 1998.

[9] M. Spivak. A Comprehensive Introduction to Differential Geometry. Publish or Perish, Boston, 1975.

[10] Moss Sweedler. Hopf Algebras. W.A. Benjamin, New York, 1969.

[11] M. Vajiac. Gauge theory techniques in quantum cohomology. Ph.D. thesis, Boston University, 2000.

An Example of an Article

Timothy Kohl  Steven Rosenberg
Boston University  Boston University
Department of Mathematics  Department of Mathematics
and Statistics  and Statistics
111 Cummington St.  111 Cummington St.
Boston, MA 02215  Boston, MA 02215
USA  USA

tkohl@bu.edu                        sr@math.bu.edu

Charles Zed Université Blaise Pascal Laboratoire de Mathématiques Pures Les Cézeaux 63177 AUBIERE CEDEX FRANCE

Francois.Dumas@math.univ-bpclermont.fr