14 Jun 1996 | Andrew Strominger, Shing-Tung Yau, Eric Zaslow
This paper argues that mirror symmetry in string theory is equivalent to T-duality on 3-cycles of Calabi-Yau manifolds. It shows that every Calabi-Yau manifold X with a mirror Y admits a family of supersymmetric toroidal 3-cycles. The moduli space of these cycles, together with their flat connections, is precisely the mirror space Y. The geometry of this moduli space is discussed in a general framework, with several examples provided.
The paper begins by introducing the concept of mirror symmetry and its implications for BPS soliton states in Calabi-Yau manifolds. It then discusses the implications of quantum mirror symmetry, which implies that an identical theory is obtained by compactifying IIB string theory on the mirror Y of X. In this formulation, all BPS states arise from supersymmetric 3-branes wrapping 3-cycles in Y. These 3-cycles must have a specific topology, namely, they must be three-tori, since their moduli space is X.
The paper then considers the action of T-duality on the supersymmetric 3-tori. T-duality maps 0-branes to 3-branes and vice versa, and does not change the moduli space of the D-brane. This implies that mirror symmetry is equivalent to T-duality on the 3-tori.
The paper provides a detailed calculation of the metric on the moduli space of supersymmetric 3-branes, showing that it is equivalent to the metric on the mirror space Y. This is done by considering the tree-level contribution to the action of the 3-brane and computing the metric on the moduli space. The paper also discusses the role of instanton corrections in correcting the moduli space geometry.
The paper then discusses the geometry of the moduli space of special Lagrangian submanifolds, showing that it has a Kähler structure and is endowed with a holomorphic b₁-form. This is consistent with the conclusions of mirror symmetry.
The paper concludes by discussing the implications of these results for the construction of M-theory duals for a large class of N=1 heterotic string compactifications. It also notes that the relation between fiberwise T-duality and mirror symmetry is quite general and not restricted to the semi-flat limit.This paper argues that mirror symmetry in string theory is equivalent to T-duality on 3-cycles of Calabi-Yau manifolds. It shows that every Calabi-Yau manifold X with a mirror Y admits a family of supersymmetric toroidal 3-cycles. The moduli space of these cycles, together with their flat connections, is precisely the mirror space Y. The geometry of this moduli space is discussed in a general framework, with several examples provided.
The paper begins by introducing the concept of mirror symmetry and its implications for BPS soliton states in Calabi-Yau manifolds. It then discusses the implications of quantum mirror symmetry, which implies that an identical theory is obtained by compactifying IIB string theory on the mirror Y of X. In this formulation, all BPS states arise from supersymmetric 3-branes wrapping 3-cycles in Y. These 3-cycles must have a specific topology, namely, they must be three-tori, since their moduli space is X.
The paper then considers the action of T-duality on the supersymmetric 3-tori. T-duality maps 0-branes to 3-branes and vice versa, and does not change the moduli space of the D-brane. This implies that mirror symmetry is equivalent to T-duality on the 3-tori.
The paper provides a detailed calculation of the metric on the moduli space of supersymmetric 3-branes, showing that it is equivalent to the metric on the mirror space Y. This is done by considering the tree-level contribution to the action of the 3-brane and computing the metric on the moduli space. The paper also discusses the role of instanton corrections in correcting the moduli space geometry.
The paper then discusses the geometry of the moduli space of special Lagrangian submanifolds, showing that it has a Kähler structure and is endowed with a holomorphic b₁-form. This is consistent with the conclusions of mirror symmetry.
The paper concludes by discussing the implications of these results for the construction of M-theory duals for a large class of N=1 heterotic string compactifications. It also notes that the relation between fiberwise T-duality and mirror symmetry is quite general and not restricted to the semi-flat limit.