Exact renormalization group in Batalin-Vilkovisky theory
In this paper, inspired by the Costello’s seminal work , we present a general formulation of exact renormalization group (RG) within the Batalin-Vilkovisky (BV) quantization scheme. In the spirit of effective field theory, the BV bracket and Laplacian structure as well as the BV effective action (EA) depend on an effective energy scale. The BV EA at a certain scale satisfies the BV quantum master equation at that scale. The RG flow of the EA is implemented by BV canonical maps intertwining the BV structures at different scales. Infinitesimally, this generates the BV exact renormalization group equation (RGE). We show that BV RG theory can be extended by augmenting the scale parameter space R to its shifted tangent bundle T ℝ. The extra odd direction in scale space allows for a BV RG supersymmetry that constrains the structure of the BV RGE bringing it to Polchinski’s form . We investigate the implications of BV RG supersymmetry in perturbation theory. Finally, we illustrate our findings by constructing free models of BV RG flow and EA exhibiting RG supersymmetry in the degree −1 symplectic framework and studying the perturbation theory thereof. We find in particular that the odd partner of effective action describes perturbatively the deviation of the interacting RG flow from its free counterpart.
KeywordsBRST Quantization Differential and Algebraic Geometry Renormalization Group Topological Field Theories
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
- I.A. Batalin and G.A. Vilkovisky, Quantization of Gauge Theories with Linearly Dependent Generators, Phys. Rev. D 28 (1983) 2567 [Erratum ibid. D 30 (1984) 508] [INSPIRE].
- J. Polchinski, Renormalization and Effective Lagrangians, Nucl. Phys. B 231 (1984) 269 [INSPIRE].
- K.J. Costello, Renormalisation and Effective Field Theory, Mathematical Surveys and Monographs 170 (2011), http://bookstore.ams.org/surv-170.
- K.J. Costello and O. Gwilliam Factorization Algebras in Quantum Field Theory, vol. I and II, available at O. Gwilliam’s webpage http://people.mpim-bonn.mpg.de/gwilliam/.
- O. Gwilliam Factorization algebras and free field theories, Ph.D Thesis, Northwestern University, available at O. Gwilliam’s webpage http://people.mpim-bonn.mpg.de/gwilliam/.
- D. Friedan, Nonlinear Models in 2 + ϵ Dimensions, Phys. Rev. Lett. 45 (1980) 1057 [INSPIRE].
- D.H. Friedan, Nonlinear Models in 2 + ϵ Dimensions, Annals Phys. 163 (1985) 318 [INSPIRE].
- R. Grady and B. Williams, Homotopy RG fow and the non-linear σ model, arXiv:1710.05973.
- W.P. Thurston, Three-dimensiional geometry and topology, vol. I, S. Levy ed., Princeton Math. Series 35, Princeton University Press (1997), http://press.princeton.edu/titles/6086.html.