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Ru Received: 26 December 2005 / Accepted: 7 April 2006 Published online: 13 October 2006 – © Springer-Verlag 2006 Abstract: We suggest an interpretation of mirror symmetry for toric varieties via an equivalence of two conformal field theories. The first theory is the twisted sigma model of a toric variety in the infinite volume limit (the A–model). The second theory is an intermediate model, which we call the I–model. The equivalence between the A–model and the I–model is achieved by realizing the former as a deformation of a linear sigma model with a complex torus as the target and then applying to it a version of the T –duality.

Traditionally the Landau-Ginzburg model is defined by adding to the action of the supersymmetric linear sigma model the term d 2 zd 2 θ W (Y )+ d 2 zd 2 θ W (Y ). Usually, one chooses W (Y ) to be complex conjugate of W (Y ). But in a type B twisted LandauGinzburg model there is an essential difference between the first and the second terms: Mirror Symmetry in Two Steps: A–I–B 45 while the integrand in the first one is a (1, 1)–form, the integrand in the second is a (0, 0)–form, and hence to integrate it one needs to pick a metric on the worldsheet.

6), sup (x,t)∈R×R+ eασ+ Ri L 2ξ ≤ cα,γ , |||gin |||. 8). With Lemma 14 and Propositions 1 and 2, our main theorem, Theorem 1, is proved. Acknowledgements. The research of the first author is supported by the NSC Grant 94-2115-M-001-006 and 094-2917-1-001-001. The research of the second author is supported in part by the Institute of Mathematics, Academia Sinica, Taipei, and NSC Grant 94-2115-M-001-006. The reserach of the third author is supported by the Competitive Earmarked Research Grant of Hong Kong Cityu 103005.

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