The String Coffee Table

2D superconformal sigma models with Calabi-Yau targets have enhanced supersymmetry, as we all know. Originally this was of interest to physicist because using targets that are products of 4-dimensional Minkowski space $M^4$ with a Calabi-Yau yields effective field theories on $M^4$ that have $N=1$ supersymmetry. Once upon a time everybody was pretty certain that this is phenomenologically attractive.

Anyway, even though the study of 2D SCFTs with Calabi-Yau targets was originally motivated by phenomenology, its extremely rich structure has given rise to a whole field of purely formal high energy physics and mathematical investigations.

In fact, the SCFT that plays a role here is really too hard for mathematicians, on the whole. But twisting it one way or another gives rise to 2D TFTs that are tractable - and still mind-bogglingly rich in structure.

As Paul Aspinwall says somewhere (p. 3, actually) in his review ($\to$)

P. Aspinwall
D-Branes on Calabi-Yau Manifolds
hep-th/0403166,

given the richness of these twisted theories one can only marvel at how abstruse the full theory must be.

And - mind you - when I say “full theory” here I am just talking about the full perturbative theory.

So, there is a lot to be explored here.

On the math side, the largest impact has come from the abstract formulation of mirror symmetry ($\to$). This is an instance of T-duality ($\to$) for Calabi-Yau manifolds, as explained in

Andrew Strominger, Shing-Tung Yau, Eric Zaslow
Mirror Symmetry is T-Duality
hep-th/9606040.

Since it is a T-duality, it comes from a Fourier-Mukai transformation ($\to$):

Claudio Bartocci, Ugo Bruzzo, Daniel Hernandez Ruiperez, José M. Munoz Porras
Mirror symmetry on K3 surfaces via Fourier-Mukai transform
alg-geom/9704023.

This connects mirror symmetry (in 2D, coming from S-duality of 4D SYM) to the geometric Langlands duality ($\to$), as explained in

A. Kapustin & E. Witten
Electric-Magnetic Duality And the Geometric Langlands Program
hep-th/0604151.

This relation between mirror symmetry and geometric Langlands is amplified in particular in

Tamas Hausel, Michael Thaddeus
Mirror symmetry, Langlands duality, and the Hitchin system
math/0205236 .

In terms of the topological A- and B-model truncations of the full 2D SCFT, mirror symmetry relates the A-model on one CY to the B-model on another. It is known ($\to$, $\to$, $\to$) that the boundary condition (“D-branes”, “B-branes”) for the B-model are objects in the derived category of coherent sheaves on the underlying CY, while those of the A-model live in something called a Fukaya category, or a generalization thereof, associated to the mirror CY.

Formal physics suggests that these two categories are equivalent.

In 1994 Maxim Kontsevich stated the homological mirror symmetry conjecture ($\to$), which states that this is indeed the case. See p. 18 of

Maxim Kontsevich
Homological Algebra of Mirror Symmetry
alg-geom/9411018 .

Of course, when Kontsevich stated this conjecture nobody knew that it could be interpreted in terms of branes of topological strings.

There are plenty of technical details hidden behind this statement. In particular, Kontsevich usually formulates it in a version that involves $A_\infty$-categories. A definition of these can be found for instance in section 4 of the paper

Maxim Kontsevich, Yan Soibelman
Homological mirror symmetry and torus fibrations
math.SG/0011041,

which connects homological mirror symmetry with the T-duality picture found by Strominger, Zaslow and Yau, mentioned above.

Kevin Costello builds in his work on 2D “TCFT” ($\to$) a lot on these notions. A good definition of $A_\infty$-category can be found in section 6.1 of his paper

Kevin J. Costello
Topological conformal field theories and Calabi-Yau categories
math.QA/0412149 .

The best introduction to homological mirror symmetry and related concepts that I have come across is

Anton Kapustin, Dmitri Orlov
Lectures on Mirror Symmetry, Derived Categories, and D-branes
math.AG/0308173 .

This also discusses the relation to Floer homology ($\to$) that goes back to

Kenji Fukaya
Floer Homology and Mirror Symmetry I
(pdf).

A nice overview of the field of mirror symmetry in the math community is given on the website of a mirror symmetry workshop that took place at the Fields institute in 2001

Workshop on Arithmetic, Geometry and Physics around Calabi-Yau Varieties and Mirror Symmetry
Fields Institute
July 23-29, 2001
(website).

You can see from the text on that website that mirror symmetry is related to a lot of hot topics, not the least to Wiles’ proof of the Taniyama-Shimura-Weil conjecture.

Blogwise, Andrew Neitzke once reported on a talk by K. Hori related to mirror symmetry.