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Renormalization theory for interacting crumpled manifolds

Abstract : We consider a continuous model of D-dimensional elastic (polymerized) manifold fluctuating in d-dimensional euclidean space, interacting with a single impurity via an attractive or repulsive $\delta$-potential (but without self-avoidance interactions). Except for $D$ = 1 (the polymer case), this model cannot be mapped onto a local field theory. We show that the use of intrinsic distance geometry allows for a rigorous construction of the high-temperature perturbative expansion and for analytic continuation in the manifold dimension D. We study the renormalization properties of the model for 0 < D < 2, and show that for bulk space dimension d smaller that the upper critical dimension , the perturbative expansion is ultraviolet finite, while ultraviolet divergences occur as pole at $d$ = $d^★$. The standard proof of perturbative renormalizability for local field theories (the Bogoliubov-Parasiuk-Hepp theorem) does not apply to this model. We prove perturbative renormalizability to all orders by constructing a subtraction operator R based on a generalization of the Zimmermann forests formalism, and which makes the theory finite at $d$ = $d^★$. This subtraction operation corresponds to a renormalization of the coupling constant of the model (strength of the interaction with the impurity). The existence of a Wilson function, of an ϵ-expansion à la Wilson-Fisher around the critical dimension, of scaling laws for $d$ < $d^★$ in the repulsive case, and of non-trivial critical exponents of the delocalization transition for $d$ > $d^★$ in the attractive case, is thus established. To our knowledge, this study provides the first proof of renormalizability for a model of extended objects, and should be applicable to the study of self-avoidance interactions for random manifolds.
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Submitted on : Friday, June 25, 2021 - 3:59:31 PM
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François David, Bertrand Duplantier, Emmanuel Guitter. Renormalization theory for interacting crumpled manifolds. Nuclear Physics B, Elsevier, 1993, 394 (3), pp.555-664. ⟨10.1016/0550-3213(93)90226-F⟩. ⟨cea-02008001⟩

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