Developing local RG: quantum RG and BFSS
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Springer
Received: November Revised: April Accepted: April Published: May
26, 22, 27, 13,
2019 2020 2020 2020
Jo˜ ao F. Meloa and Jorge E. Santosa,b a
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, U.K. b Institute for Advanced Study, Princeton, NJ 08540, U.S.A.
E-mail: [email protected], [email protected] Abstract: In this paper we study various forms of RG and apply these to the BFSS model of N coincident D0-branes. Firstly, as a warm-up, we perform standard Wilsonian RG, investigating the conditions under which supersymmetry is preserved along the flow. Next, we develop a local RG scheme such that the cutoff is spacetime dependent, which could have further applications to studying QFT in curved spacetime. Finally, we test the conjecture put forward in [1] that the method of quantum RG could be the mechanism responsible for the gauge/gravity duality by applying it to the BFSS model, which has a known gravitational dual. Although not entirely conclusive some questions are raised about the applicability of quantum RG as a description of the AdS/CFT correspondence. Keywords: Gauge-gravity correspondence, Renormalization Group, M(atrix) Theories, Supersymmetry Breaking ArXiv ePrint: 1910.09559
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP05(2020)063
JHEP05(2020)063
Developing local RG: quantum RG and BFSS
Contents 1 Introduction
1
2 Renormalisation group flow of BFSS model 2.1 Overview of the model 2.2 RG with a hard momentum cutoff 2.3 RG with smooth regulators
3 3 5 9 12 13 14
4 Quantum renormalisation group 4.1 Overview of QRG 4.2 Overview of the holographic dual to BFSS 4.3 QRG of BFSS 4.3.1 Single operator 4.3.2 Two operators
16 17 18 21 21 22
5 Discussion
25
A Details for the 1-loop calculation
25
1
Introduction
In its most precise form, the AdS/CFT correspondence is an equality of partition functions, where sources in the field theory side correspond to boundary conditions on the dynamical fields of the gravity side [2–5]. In the large N limit on the field theory side, and in the classical limit on the gravity side, we get, roughly, Z d (0) −SSuGra exp d x Oφ =e . (1.1) ∆−d =φ(0) (x) ¯ limz→0 φ(z,x)z
QFT
We can then use this to calculate correlation functions on both sides. However, this is not the whole story. If we try to evaluate the classical action as it stands, with boundary conditions precisely on the boundary of AdS, we would get infinity. As is standard in QFT calculations, the way to deal with this infinity is to do renormalisation, i.e. introducing counterterms to absorb the infinities. This procedure has been extensively developed, and is now a very standard technique under the name of Holographic Renormalisation [6–10]. There are many interesting peculiarities with this idea. Firstly, it seems that what would normally be the UV divergences in standard QFT are in fact IR divergences in the
–1–
JHEP05(2020)063
3 Local renormalisation group 3.1 Smooth regulat
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