**Title:** Elliptic genus test for the mirror of some complete intersection

**Speaker:** Professor Vassily Gorbounov

**Speaker Info:** University of Kentucky

**Brief Description:**

**Special Note**:

**Abstract:**

Understanding the mathematics behind Quantum and in particular Conformal Field Theory has been a challenge for more then twenty five years. The usual ground for mathematical interpretations of Quantum Field Theory predictions has been the Topological Quantum Field Theory, which is a certain reduction of the honest Quantum Field Theory. The mathematical advances in this area over these years are significant, studying Mirror symmetry certainly is among them. The "explicit" construction of the mirror partner for a given manifold has been the central task. There is a vast number of work done in this direction. We will use relatively recent results of Hori and Vafa. They showed that the mirror partner of a large class of manifold turns out to be a Landau-Ginsburg model of some kind or its orbifold. The mathematical statements implied by mirror symmetry in the Topological Quantum Field Theory is a reduction of stronger statements in the original Quantum Field Theory. The rigorous mathematical structure behind Quantum Field Theory is not known at the present time so exploring mathematical consequences of mirror symmetry at this higher level is difficult. There are some objects in mathematics which have a counterpart in the Quantum Theory. One of them is the elliptic genus. Discovered in algebraic topology by Ochanine it was almost immediately rooted into quantum physics, later shown to be identical for mirror partners and therefore providing another test for a pair of manifolds to be mirror partners. Moreover it was explained by Witten that there is a physics counterpart of the elliptic genus in a large class of Conformal Field Theories, in particular in Landau-Ginsburg models and their orbifolds, and this physics elliptic genus should coincide with the one defined in topology for the sigma model. Taking on Witten's ideas a number of physicists came up with new type of formulas for the elliptic genus of some classes of manifolds in terms of their mirror Landau-Ginsburg partner. For about ten years no mathematical proof that this formulas indeed hold was produced. The first work where such a proof was given in the case of Calabi-Yau hypersurfaces was by Gorbounov and Malikov. The scope of this paper was much broader and the result about the elliptic genus fell out as a simple consequence of a much deeper connection found between the Landau-Ginsburg model and the sheaf Chiral Operators on a hypersurface. The purpose of this talk is to prove by more or less elementary means that the Landau-Ginsburg mirror parners found by Hori and Vafa for complete intersection have the physics elliptic genus identical to the topological elliptic of appropriate complete intersections. It is interesting to note that our result besides being an extra test for the constructions, also illuminates and refines conditions obtained by Hori and Vafa for the existence of Landau-Ginsburg theories as a mirror partners for complete intersections.

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