Mirror symmetry

About: Mirror symmetry is a research topic. Over the lifetime, 2422 publications have been published within this topic receiving 90786 citations.

Torus knots and mirror symmetry

Abstract: We propose a spectral curve describing torus knots and links in the B-model. In particular, the application of the topological recursion to this curve generates all their colored HOMFLY invariants. The curve is obtained by exploiting the full Sl(2;Z) symmetry of the spectral curve of the resolved conifold, and should be regarded as the mirror of the topological D-brane associated to torus knots in the large N Gopakumar{Vafa duality. Moreover, we derive the curve as the large N limit of the matrix model computing torus knot invariants.

Homological Geometry and Mirror Symmetry

Abstract: A homogeneous polynomial equation in five variables determines a quintic 3-fp;d in ℂP4. Hodge numbers of a nonsingular quintic are know to be: h p, p = 1, p = 0, 1, 2, 3 (Kahler form and its powers), h3, 0 = h0,3 = 1 (a quintic happens to bear a holomorphic volume form), h2,1 = h1, 2 = 101 = 126 - 25 (it is the dimension of the space of all quintics modulo projective transformations, and h2,1 is responsible here for infinitesimal variations of the complex structure) and all the other h p,q = 0.

Large Complex Structure Limits of K3 Surfaces

Abstract: Motivated by the picture of mirror symmetry suggested by Strominger, Yau and Zaslow, we made a conjecture concerning the Gromov-Hausdorff limits of Calabi-Yau n-folds (with Ricci-flat Kahler metric) as one approaches a large complex structure limit point in moduli; a similar conjecture was made independently by Kontsevich, Soibelman and Todorov. Roughly stated, the conjecture says that, if the metrics are normalized to have constant diameter, then this limit is the base of the conjectural special lagrangian torus fibration associated with the large complex structure limit, namely an n-sphere, and that the metric on this S^n is induced from a standard (singular) Riemannian metric on the base, the singularities of the metric corresponding to the discriminant locus of the fibration. This conjecture is trivially true for elliptic curves; in this paper we prove it in the case of K3 surfaces. Using the standard description of mirror symmetry for K3 surfaces and the hyperkahler rotation trick, we reduce the problem to that of studying Kahler degenerations of elliptic K3 surfaces, with the Kahler class approaching the wall of the Kahler cone corresponding to the fibration and the volume normalized to be one. Here we are able to write down a remarkably accurate approximation to the Ricci-flat metric -- if the elliptic fibres are of area $\epsilon >0$, then the error is $O(e^{-C/\epsilon})$ for some constant $C>0$. This metric is obtained by gluing together a semi-flat metric on the smooth part of the fibration with suitable Ooguri-Vafa metrics near the singular fibres. For small $\epsilon$, this is a sufficiently good approximation that the above conjecture is then an easy consequence.