Irreversibility of world-sheet renormalization group flow. (English) Zbl 1247.81310
Summary: We demonstrate the irreversibility of a wide class of world-sheet renormalization group (RG) flows to first order in \(\alpha'\) in string theory. Our techniques draw on the mathematics of Ricci flows, adapted to asymptotically flat target manifolds. In the case of somewhere-negative scalar curvature (of the target space), we give a proof by constructing an entropy that increases monotonically along the flow, based on Perelman’s Ricci flow entropy. One consequence is the absence of periodic solutions, and we are able to give a second, direct proof of this. If the scalar curvature is everywhere positive, we instead construct a regularized volume to provide an entropy for the flow. Our results are, in a sense, the analogue of Zamolodchikov’s c-theorem for world-sheet RG flows on noncompact spacetimes (though our entropy is not the Zamolodchikov C-function).
MSC:
81T17 | Renormalization group methods applied to problems in quantum field theory |
81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |
53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) |
53Z05 | Applications of differential geometry to physics |
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