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Is there someone can tell me some papers or books about the basic material of Spectral flow? I want to know, what is spectral flow and how to use it to geometry.

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MathOverflow is not the appropriate place to ask somebody to write an expository article for you. If you want somebody to write an article about some subject, you should make a stub on Wikipedia, make a query block on nLab, or make a request on PlanetMath. –  noobcake Nov 29 '10 at 15:22
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I think maybe the OP just asks for a reference... –  Pietro Majer Nov 29 '10 at 15:31
    
You could try Atiyah-Patodi-Singer (if I remember correctly, spectral flow is discussed in part III of their series). –  Tim Perutz Nov 29 '10 at 15:52
    
@Tim Perutz,Thank you! –  Chen Nov 30 '10 at 2:56

2 Answers 2

up vote 4 down vote accepted

A good start could be to read this paper by Philips which was recommended to me, when I was looking for an overview:

http://cms.math.ca/cmb/v39/p460

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A discussion of the spectral flow not only in the context of pure supersymmetric conformal field theory, but also in the context of their associated geometries, can be found in a recent book by Ralph Blumenhagen and Erik Plauschinn, "Introduction to conformal field theory, with applications to string theory", Lecture Notes in Physics 779, Springer Verlag, 2009. In this reference the spectral flow is also discussed in the context of Gepner models, exactly solvable conformal field theories that describe the underlying field theory of certain types of Calabi-Yau varieties. This provides a link between CFTs and CYs.

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