I was heard (by an expert) that, in mirror symmetry, we have constructed a Quantum Master Equation associated to topological B model, and a solution to it. But I can't find any material about this. Is there anyone know anything about this, so you can give me some help? Is that Kontsevich and Bananikov's paper (for genus 0 topological B model on Calabi-Yau)?? And is the sullivan's paper "Sigma model and string topology" related to this to some extent?(but I can't see what the useness is of this one).Thank you very much for your help!
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1$\begingroup$ What do people mean when they refer to a 'master equation'? $\endgroup$– Minhyong KimCommented May 26, 2010 at 23:02
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2$\begingroup$ That's the master equation associated to a (dg)BV algebra. $\endgroup$– HYYYCommented May 27, 2010 at 2:43
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2 Answers
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This is the subject of this paper by Kevin Costello -- he constructs a solution to a master equation associated to a Calabi-Yau category, which one could take to be the category of branes in the topological B-model. (See the article for references to work of Sen and Zwiebach in the physics literature). String topology can be thought of as a kind of simplified B-model, so Sullivan's paper is also related I presume but haven't looked..
answered May 26, 2010 at 22:22
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$\begingroup$ Thanks!(I once assumed that's the paper by Kontsevich and Bananikov) Can we get the same thing for A model?(using his method). That's what exactly Costello once guided me, there is no such construction for A model currently. That's what I am confused here. But I do construct similar things for A model recently. $\endgroup$– HYYYCommented May 27, 2010 at 2:42
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Lets briefly explain the whole of picture:
A-model and B-model is study of twisted conformal field theory
If you want to study the invariants of complex manifolds , then you will face with two notion due to Edward Witten philosophy
A) Closed string B-model which is study of Hodge structure or periods of holomorphic differential forms
B-model on Calabi-Yau three-fold can be described by a Kodaira-Spencer gauge theory(due to Bershadsky-Cecotti-Ooguri-Vafa). sometimes called BCOV theory.
B) Open string B-model is study of Category of coherent sheaves on complex manifold(for example complex submanifolds equiped with holomorphic vector bundles)
If you want to study the invariants of symplectic manifolds, then we are facing with two object
A) Closed string A-model , which is study of Gromov-Witten invariants about counting holomorphic curves
B) Open string A-model which is Fukaya category (object is lagrangian submanifolds and morphisms is Floer chain )which corresponding to the Calabi–Yau category
From homological mirror symmetry point of view (due to M. Kontsevich conjecture), the A-model is dual to the B-model. $$D^\pi\mathcal F(M)\cong D^b Coh(M^\vee)$$ which called Meta-Mirror symmetry
Moreover, in the language of moduli spaces if $M$ and $M^\vee$ are two mirror to each other then $$\mathcal M_{sym}(M)\cong \mathcal M_{cpx}(M^\vee)$$ where $\mathcal M_{sym}(M)$ and $\mathcal M_{cpx}(M)$ are moduli spaces of symplectic structures and moduli spaces of complex structures respectively. This gives an equivalence of A-module and B-module for families
For your question see the Kontsevich paper
M. Alexandrov, M. Kontsevich, A. Schwarz, O. Zaboronsky, around page 19 in The geometry of the master equation and topological quantum field theory, Int. J. Modern Phys. A 12(7):1405–1429, 1997
and this thesis
