Exact renormalization group and effective action: A Batalin–Vilkovisky algebraic formulation

Abstract

In the present paper, which is a mathematical follow-up of [16] taking inspiration from [11], we present an abstract formulation of exact renormalization group (RG) in the framework of Batalin-Vilkovisky (BV) algebra theory.In the first part, we work out a general algebraic and geometrical theory of BV algebras, canonical maps, flows and flow stabilizers. In the second part, relying on this formalism, we build a BV algebraic theory of the RG. In line with the graded geometric outlook of our approach, we adjoin the RG scale with an odd parameter and analyse in depth the implications of the resulting RG supersymmetry and find that the RG equation (RGE) takes Polchinski's form [3]. Finally, we study abstract purely algebraic odd symplectic free models of RG flow and effective action (EA) and the perturbation theory thereof to illustrate and exemplify the general theory.MSC: 81T13 81T20 81T45

Double Copy from Homotopy Algebras

“…For aspects regarding renormalisation in this context, see e.g. [204,205] and in particular. [206,207] (iv) Since both classical and quantum minimal models can be computed recursively by the homological perturbation lemma, [208,209] see Section 4.3 for details, we obtain Berends-Giele-type recursion relations for amplitudes in any BV quantisable field theory both at the tree and loop levels.…”

Exact renormalization group in Batalin-Vilkovisky theory

“…The present paper aims to present the field theoretic foundations of our BV formulation of RG theory. In a more mathematical oriented paper [23], we shall reformulate the results obtained here in the framework of the abstract theory of BV algebras and manifolds.…”

“…while the A-deformed BV Laplacian reads In the calculations carried out below, we use repeatedly a host of basic identities, which we collect here for convenience in index free form and whose proof will be given in [23]. A part of these involve vector fields of the basic form Another part concern quadratic functions of FunpE 0 q of the form The action of the deformed Laplacian on them is also simple enough, glp1|1q structures, which we review briefly in this subsection, are recurrent in differential geometry, supersymmetric quantum mechanics and topological sigma models.…”

L∞‐Algebras of Classical Field Theories and the Batalin–Vilkovisky Formalism

“…For many purposes including quantisation, however, we require an off-shell description. This can be obtained by a further 25 We shall describe this function in more detail in Section 3.5.…”