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  Is there a mirror of the Rozansky-Witten theory?

+ 6 like - 0 dislike

In http://arxiv.org/abs/hep-th/9612216 , Rozansky and Witten have associated to every holomorphic symplectic manifold $X$ a 3d TQFT (three dimensional Topological Quantum Field Theory), by topological twisting of a 3d sigma model of target $X$. The 3d sigma model is not renormalizable and so not well-defined in the UV but nevertheless, the 3d topological theory is well-defined. It is a kind of odd Chern-Simons theory.

One can show that if we take $X = T^*Y$ for $Y$ a complex manifold ( $T^*Y$ is naturally holomorphic symplectic), then the dimensional reduction over a circle of the 3d Rozansky-Witten TQFT of target $X$ is the B-model 2d TQFT of target $Y$. The question is: is there a similar story for the A-model?

In other words: Let $Z$ be a (real) symplectic manifold, the A-model of target $Z$ is a 2d TQFT. Is there a 3d TQFT constructed from $Z$ whose dimensional reduction over a circle is the A-model 2d TQFT of target $Z$ ?

This question is motivated by the fact that the existence of the Rozansky-Witten TQFT in some sense "explains" the existence of a (derived) tensor product over the derived category of coherent sheaves $D(Y)$ of a complex manifold $Y$. If one thinks about $D(Y)$ as the category of branes of the B-model, this structure seems to have no physical meaning ($D(Y)$ is what is associated to a point by the TQFT but there is no smooth cobordism between 2 points on the one and and 1 point on the other hand). But in the Rozansky-Witten theory on $X=T^*Y$, $D(Y)$ is the category of line defects, it is what is associated to a circle by the TQFT and the tensor product becomes natural because there exists a cobordism between 2 circles on the one hand and 1 circle and the other hand (a pair of pants). Mathematically, the (derived) tensor product is a rather obvious operation on $D(Y)$. If $Z$ is a symplectic manifold, the category of branes of the A-model of target $Z$ is the Fukaya category $Fuk(Z)$ of $Z$. But now, mathematically, there is no obvious tensor product on $Fuk(Z)$. Nevertheless, such a thing should exist if mirror symmetry is true, i.e. if there exists a complex manifold $Y$ such that $Fuk(Z)=D(Y)$. Maybe that if the A-model analogue of the Rozansky-Witten theory exists, it will be easier to see what is the tensor product like operation on the Fukaya category.  

asked Jul 31, 2014 in Theoretical Physics by 40227 (5,140 points) [ revision history ]

1 Answer

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There is the 3d A model formulated by A. Kapustin and K. Vyas in http://arxiv.org/pdf/1002.4241.pdf .

The Fukaya category of $T^*X$ occurs as the category of line operators between a Dirichlet and a Neumann boundary condition for the 3d A model on $X$.

Of course, $Fuk(T^*X)$ (actually its triangulated envelope) is equivalent to the (derived) category of constructible sheaves on $X$, by Nadler and Zaslow by http://arxiv.org/abs/math/0612399 . This has a natural tensor product... maybe they're the same?

EDIT: Actually it seems Sh(X) has two tensor products... Maybe this is because of the 4d A model?

answered Jul 31, 2014 by Ryan Thorngren (1,925 points) [ revision history ]

Thanks for your answer. The paper by Kapustin and Vyas is exactly the kind of things I was looking for. I find strange that to have a 3d interpretation of $D(X)$, we have to go to a bigger space ($T^*X$) whereas to have a 3d interpretation of $Fuk(Z)$, we have to go to a smaller space (a $W$ such that $Z=T^*W$). Do you have any intuitive explanation of this fact ? Another question: you answered my original question for $Z$ being a cotangent space: do you think there is a analogue for general $Z$ (for example a compact symplectic manifold)?

I think your two questions are related. The RW model compactified on a circle gives the B-model on the same target. I assume the same should be true of the "correct" 3d A-model. Something is strange with the 3d A-model since it can only describe the A-model of a cotangent bundle. You should think of the cotangent bundle as the local picture of a symplectic manifold _near a Lagrangian_. So it seems to me what is missing is some way of gluing the A-models of cotangent spaces together into the A-model on an arbitrary symplectic manifold. Then perhaps one can look again at the 3d A-model to figure out what to do.

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