The term “mirror symmetry” is used to refer to a wide array of
phenomena in mathematics and physics, and there is no consensus as to
its precise definition. In general, it refers to a correspondence that maps
objects of a certain type— manifolds, for example, or polynomials— to
objects of a possibly different type in such a way that the “A-model”
of the first object is exchanged with the “B-model” of its image. The
phrases “A-model” and “B-model” originate in physics, and the various
definitions of mirror symmetry arise from different ideas about the
mathematical data that these physical notions are supposed to capture.
The Calabi-Yau A-model, for example, encodes deformations of the
K¨ahler structure of a Calabi-Yau manifold, while the Calabi-Yau Bmodel
encodes deformations of its complex structure. There is also
a Landau-Ginzburg A-model and B-model, which are associated to
a polynomial rather than a manifold, and which are somewhat less
geometric in nature. The versions of mirror symmetry that we will
consider in this course are:
• The Batyrev construction, which interchanges the Calabi-Yau Amodel
of a manifold and the Calabi-Yau B-model of its mirror
manifold;
• The Hori-Vafa construction, which interchanges the Calabi-Yau
(or, more generally, semi-Fano) A-model of a manifold and the
Landau-Ginzburg B-model of its mirror polynomial;
• The Berglund-H¨ubsch-Krawitz construction, which interchanges
the Landau-Ginzburg A-model of a polynomial and the LandauGinzburg
B-model of its mirror polynomial.
In each case, mirror symmetry is a conjectural equivalence between
the sets of data encoded by the two models. In full generality it remains
a conjecture, but many cases are known to hold. The CalabiYau/Calabi-Yau
mirror symmetry, for example, has been proven whenever
the Calabi-Yau manifold X is a complete intersection in a toric
variety, and in some cases when X is a complete intersection in a more
general GIT quotient.
We should note that, in these notes, mirror symmetry will only be
discussed as an interchange of cohomology groups (or “state spaces”)
on the A-side and B-side. At least in the Calabi-Yau case, however,
both the A-model and the B-model are understood to capture much
more data than these vector spaces alone. The Calabi-Yau A-model,
for example, can be encoded in terms of Gromov-Witten theory.
The structure of the notes is as follows. In Chapter 0, we will review
the fundamentals of toric geometry, which are necessary to explain the
Batyrev construction. Chapters 1, 2, and 3 develop the three forms
of mirror symmetry outlined above. The Appendix reviews the basics
of Chen-Ruan cohomology, a cohomology theory for orbifolds that is
needed in order to define the state spaces of the Calabi-Yau A- and
B-model, and that also provides a useful parallel to the definition of
the states spaces in Landau-Ginzburg theory.
