scholarly journals Some applications of Ricci flow in physics

2008 ◽  
Vol 86 (4) ◽  
pp. 645-651 ◽  
Author(s):  
E Woolgar
Keyword(s):  
General Relativity ◽  
Renormalization Group ◽  
Ricci Flow ◽  
Sigma Model ◽  
Final Section ◽  
Higher Dimensions ◽  
Ricci Soliton ◽  
Two Dimensions ◽  
Nonlinear Sigma Model

I discuss certain applications of the Ricci flow in physics. I first review how it arises in the renormalization group (RG) flow of a nonlinear sigma model. I then review the concept of a Ricci soliton and recall how a soliton was used to discuss the RG flow of mass in two dimensions. I then present recent results obtained with Oliynyk on the flow of mass in higher dimensions. The final section discusses how Ricci flow may arise in general relativity, particularly for static metrics.PACS Nos.: 02.40Ky, 02.30Ik, 04.20.–q

STOCHASTIC QUANTIZATION AND 1/N EXPANSION

1993 ◽  
Vol 08 (02) ◽  
pp. 115-128
Author(s):  
J.C. BRUNELLI ◽  
R.S. MENDES
Keyword(s):  
Perturbation Theory ◽  
Sigma Model ◽  
Functional Approach ◽  
Two Dimensions ◽  
Field Theories ◽  
Nonlinear Sigma Model ◽  
Stochastic Quantization ◽  
Quantization Method ◽  
Systematic Procedure

We study the 1/N expansion of field theories in the stochastic quantization method of Parisi and Wu using the supersymmetric functional approach. This formulation provides a systematic procedure to implement the 1/N expansion which resembles the ones used in the equilibrium. The 1/N perturbation theory for the nonlinear sigma-model in two dimensions is worked out as an example.

scholarly journals DILATONIC GRAVITY NEAR TWO DIMENSIONS AS A STRING THEORY

1995 ◽  
Vol 10 (27) ◽  
pp. 2001-2008 ◽  
Author(s):  
E. ELIZALDE ◽  
S.D. ODINTSOV
Keyword(s):  
Fixed Point ◽  
Renormalization Group ◽  
String Theory ◽  
Sigma Model ◽  
Scalar Fields ◽  
Two Dimensions ◽  
Gravitational Coupling ◽  
Dilaton Coupling

Using the renormalization group formalism, a sigma model of a special type — in which the metric and the dilaton depend explicitly on one of the string coordinates only — is investigated near two dimensions. It is seen that dilatonic gravity coupled to N scalar fields can be expressed in this form, using a string parametrization, and that it possesses the usual uv fixed point. However, in this stringy parametrization of the theory the fixed point for the scalar-dilaton coupling turns out to be trivial, while that for the gravitational coupling remains the same as in previous studies being, in particular, nontrivial.

RENORMALIZATION GROUP FLOW AND OPEN STRING DYNAMICS

1988 ◽  
Vol 03 (18) ◽  
pp. 1797-1805 ◽  
Author(s):  
NAOHITO NAKAZAWA ◽  
KENJI SAKAI ◽  
JIRO SODA
Keyword(s):  
Renormalization Group ◽  
Sigma Model ◽  
Open String ◽  
Tree Level ◽  
Nonlinear Sigma Model ◽  
Renormalization Group Flow ◽  
Point Solution ◽  
Β Function ◽  
Group Flow ◽  
Model Approach

The renormalization group flow in the nonlinear sigma model approach is explicitly solved to the fourth order in the case of an open string propagating in the tachyon background. Using a regularization different from the original one used by Klebanov and Susskind (K-S), we show that its fixed point solution produces the tree-level 5-point tachyon amplitude. Furthermore we prove K-S’s conjecture, i.e., the equivalence between the vanishing β-function defined by our regularization and the equation of motion arising from the effective action, up to all orders.

scholarly journals Statistical inference and string theory

2015 ◽  
Vol 30 (26) ◽  
pp. 1550160 ◽  
Author(s):  
Jonathan J. Heckman
Keyword(s):  
String Theory ◽  
Statistical Inference ◽  
Sigma Model ◽  
Two Dimensions ◽  
Nonlinear Sigma Model ◽  
Dimensional Manifold ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Lorentzian Signature ◽  
Stability Under Perturbations

In this paper, we expose some surprising connections between string theory and statistical inference. We consider a large collective of agents sweeping out a family of nearby statistical models for an [Formula: see text]-dimensional manifold of statistical fitting parameters. When the agents making nearby inferences align along a [Formula: see text]-dimensional grid, we find that the pooled probability that the collective reaches a correct inference is the partition function of a nonlinear sigma model in [Formula: see text] dimensions. Stability under perturbations to the original inference scheme requires the agents of the collective to distribute along two dimensions. Conformal invariance of the sigma model corresponds to the condition of a stable inference scheme, directly leading to the Einstein field equations for classical gravity. By summing over all possible arrangements of the agents in the collective, we reach a string theory. We also use this perspective to quantify how much an observer can hope to learn about the internal geometry of a superstring compactification. Finally, we present some brief speculative remarks on applications to the AdS/CFT correspondence and Lorentzian signature space–times.

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