# Formal mathematical definition of renormalization group flow

I was watching some lectures by Huisken where he mentioned that one-loop renormalization group flow was in some analogous to mean curvature flow. I have tried reading up the exact definition of what this flow actually is, but could not find anything suitable and was wondering if anyone could explain it to me.

I have tried reading texts on QFT, but I don't really want the physics behind it and I don't want vague descriptions. I just want to know what the precise mathematical definition of the flow is, similar to the one for Ricci flow or mean curvature flow. Is there a manifold, what is the PDE involved etc.? I am familiar with the idea of loops in the context of Feynman diagrams and integrals if that helps, so I know what a one-loop Feynman diagram is.

• In what context do you want a formal definition? There are a lot of conceptually similar but different-in-detail things that go by the name of renormalization flow. – user1504 Aug 5 '19 at 14:16
• In the context of, how is it a geometric flow, if that makes sense? What is the manifold? What is the equation for the flow? – Hollis Williams Aug 5 '19 at 14:17
• Have you seen the corresponding nLab entry: ncatlab.org/nlab/show/renormalization+group+flow ? It is too technical but maybe you could find some interest in it. You can also take a look at these notes: physics.rutgers.edu/pages/friedan/talks/flows/…, from the BIRS workshop: Geometric flows in mathematics and physics, where the relation of RG flow with the Ricci flow is discussed. – Konstantinos Kanakoglou Aug 5 '19 at 15:45
• Very related: mathoverflow.net/questions/62770 – j.c. Aug 5 '19 at 16:16
• @KonstantinosKanakoglou Another nLab entry: ncatlab.org/nlab/show/… – AHusain Aug 5 '19 at 17:51

The renormalization group (RG) as a geometric flow (like the Ricci flow) is a very special case of the RG, namely, the one corresponding to the nonlinear sigma-model (NLSM) in two dimensions with values in a Riemannian manifold. Now the RG is much more general and applies to all sorts of models, not just the NLSM. In order to find satisfactory answers to your questions, I recommend first understanding how the RG works in general by specializing to a simpler model: the scalar field. This is explained below. Then, look at the NLSM and see how in this particular case, the Ricci flow emerges from the RG. For this second part, I think the two references given by Igor are spot on. I should however mention a common source of confusion about the RG. There are two different but related RG's (this distinction is not specific to the scalar field or the NLSM but holds across models): 1) the old Stueckelberg-Peterman-Gell-Mann-Low RG (SPGLRG), 2) the more recent Wilsonian RG (WRG). Below, I provide details which should hopefully explain the relation between the two. The short story is the WRG is a flow on the space of theories with a fixed ultraviolet cutoff (say at unit scale) whereas the SPGLRG only concerns theories in the continuum, after the removal of the ultraviolet cutoff. These continuum theories (points in some space) are parametrized by coordinates (renormalized couplings). When rescaling a theory by some factor, one changes the point and therefore one would like to know how the coordinates change. The answer to this question is the SPGLRG.

As far as I can tell, Igor's answer and the second (very nice) reference he gave by Carfora et al. only concerns the SPGLRG, because I did not see the word "cutoff" mentioned once. I think the first reference he gave by Nguyen would be the first place to go to understand the RG vs. Ricci flow connection (immediately after reading my nontechnical explanations below) because it articulates both RG's, the SPGLRG and the WRG, and thus provides a fuller picture. Another reference I could add is also a review written by Carfora for mathematicians about the RG vs. Ricci flow connection, which is available at https://arxiv.org/abs/1001.3595.

Below is the answer I gave at

https://physics.stackexchange.com/questions/372306/what-is-the-wilsonian-definition-of-renormalizability

which has more details than the one in the MO post mentioned by J. C.

This is a very good question which, however, shows the extent of the reigning confusion about renormalization even four decades after Wilson's Nobel Prize winning theory on the matter. I essentially answered the OP's question, and much more, about constructing continuum QFTs in Wilson's framework in my expository article "QFT, RG, and all that, for mathematicians, in eleven pages" but in a very condensed fashion (one needs to do computations on the side to follow what is being said). So let me give more details pertaining to the OP's specific question. I should preface this by saying that what follows is a "cartoon" for renormalization. I will oversimplify things by ignoring anomalous dimensions, marginal operators, and nonlocal terms generated by the RG. You will not find technical details but hopefully the conceptual picture and logical structure of renormalization will become clearer.

The OP is right to point out that in the setting of ODEs and dynamical systems a first order equation can be run backwards in time. So let me start by recalling some important terminology from that area. Consider a first order nonautonomous ODE of the form $$\frac{dX}{dt}=f(t,X)\ .$$ It generates a flow (groupoid morphism from time pairs to diffeomorphisms of phase space) I will denote by $$U[t_2,t_1]$$ which sends the initial value $$X(t_1)$$ to the value of the solution at time $$t_2$$. It trivially satisfies $$\forall t, U[t,t]={\rm Id}$$ and the semigroup property $$\forall t_1,t_2,t_3,\ \ U[t_3,t_2]\circ U[t_2,t_1]=U[t_3,t_1]\ .$$ This time-dependent situation is to be distinguished from the autonomous ODE case $$\frac{dX}{dt}=f(X)$$ where $$U[t_2,t_1]=U[t_2-t_1,0]=:U[t_2-t_1]$$.

In Wilson's RG, time is scale or more precisely, $$t=-\log\Lambda$$ where the UV cutoff is implemented in momentum space by a condition like $$|p|\le\Lambda$$ or in position space by $$\Delta x\ge \Lambda^{-1}=e^t$$. The high energy physics literature usually works in a nonautonomous setting while it is essential to translate the equation to autonomous form for a proper understanding of Wilson's RG. The latter imported tools and concepts from dynamical system theory like fixed points, stable and unstable manifolds etc. It is possible to do some contortions to try to make sense of these concepts in the nonautonomous setting, but these truly are notions that are congenial to autonomous dynamical systems.

Let $$\mu=:\mu_{-\infty,\infty}$$ denote the probability measure corresponding to the free Euclidean theory. Its propagator is $$\int \phi(x)\phi(y)\ d\mu_{-\infty,\infty}(\phi)=\langle \phi(x)\phi(y)\rangle_{-\infty,\infty}= \int\frac{dp}{(2\pi)^D} \frac{e^{ip(x-y)}}{p^{D-2\Delta}}$$ where $$\Delta$$ is the scaling dimension of the field $$\phi$$. Normally, $$\Delta=\frac{D-2}{2}$$ but I will allow more general $$\Delta$$'s in this discussion. Now let me introduce a mollifier, i.e., a smooth function of fast decay $$\rho(x)$$ such that $$\int \rho(x)\ dx=\widehat{\rho}(0)=1$$. For any $$t$$, let me set $$\rho_t(x)=e^{-Dt}\rho(e^{-t}x)$$, so in particular $$\rho_0=\rho$$. Let $$\mu_{t,\infty}$$ be the law of the field $$\rho_t\ast\phi$$ where $$\phi$$ is sampled according to $$\mu_{-\infty,\infty}$$ and we used a convolution with the rescaled mollifier. In other words, $$\mu_{t,\infty}$$ is the free cutoff measure at $$\Lambda_H=e^{-t}$$ and propagator $$\int \phi(x)\phi(y)\ d\mu_{t,\infty}(\phi)=\langle \phi(x)\phi(y)\rangle_{t,\infty}= \int\frac{dp}{(2\pi)^D} \frac{|\widehat{\rho}_t(p)|^2\ e^{ip(x-y)}}{p^{D-2\Delta}}\ .$$ Note that $$\widehat{\rho}_t(p)=\widehat{\rho}(e^t p)$$ which we assume to have decreasing modulus with respect to $$t$$. We have $$\widehat{\rho}_{-\infty}=1$$ and $$\widehat{\rho}_{\infty}=0$$ and $$|\widehat{\rho}_{t_1}(p)|^2-|\widehat{\rho}_{t_2}(p)|^2\ge 0$$ whenever $$t_1\le t_2$$. One can thus define a more general family of modified free/Gaussian theories $$\mu_{t_1,t_2}$$ with $$t_1\le t_2$$ by the propagator $$\int \phi(x)\phi(y)\ d\mu_{t_1,t_2}(\phi)=\langle \phi(x)\phi(y)\rangle_{t_1,t_2}= \int\frac{dp}{(2\pi)^D} \frac{\left(|\widehat{\rho}_{t_1}(p)|^2-|\widehat{\rho}_{t_2}(p)|^2\right)\ e^{ip(x-y)}}{p^{D-2\Delta}}\ .$$ One has the semigroup property for convolution of (probability) measures $$\mu_{t_1,t_2}\ast\mu_{t_2,t_3}=\mu_{t_1,t_3}$$ when $$-\infty\le t_1\le t_2\le t_3\le \infty$$. This means that for any functional $$F(\phi)$$, $$\int F(\phi)\ d\mu_{t_1,t_3}=\int\int d\mu_{t_1,t_2}(\zeta)\ d\mu_{t_2,t_3}(\psi)\ F(\zeta+\psi)\ .$$ The other key players are scale transformations $$S_t$$. Their action on fields is given by $$(S_t \phi)(x)=e^{-\Delta t}\phi(e^{-t}x)$$ and obviously satisfies $$S_{t_1}\circ S_{t_2}=S_{t_1+t_2}$$. Using the notion of push-forward/direct image of measures, one has $$(S_t)_{\ast}\mu_{t_1,t_2}=\mu_{t_1+t,t_2+t}$$, i.e., $$\int d\mu_{t_1,t_2}(\phi)\ F(S_t\phi)=\int d\mu_{t_1+t,t_2+t}(\phi)\ F(\phi)\ .$$ Since these are centered Gaussian measures, it is enough to check the last property on propagators, i.e., $$F(\phi)=\phi(x)\phi(y)$$ where this follows from a simple change of momentum integration variable from $$p$$ to $$q=e^{-t}p$$ in the above formula for the propagator.
This also covers the infinite endpoint case with the conventions $$t+\infty=\infty$$, $$t-\infty=-\infty$$ for finite $$t$$.

The high energy physics Wilsonian RG is the transformation of functionals $$RG[t_2,t_1]$$ for pairs $$t_1\le t_2$$ obtained as follows. Using the convolution semigroup property $$\int e^{-V(\phi)} d\mu_{t_1,\infty}(\phi)=\int e^{-V(\zeta+\psi)} d\mu_{t_1,t_2}(\zeta)\ d\mu_{t_2,\infty}(\psi)$$ $$=\int e^{-(RG[t_2,t_1](V))(\phi)} d\mu_{t_2,\infty}(\phi)$$ after renaming the dummy integration variable $$\psi\rightarrow\phi$$ and introducing the definition $$(RG[t_2,t_1](V))(\phi):=-\log \int e^{-V(\zeta+\phi)} d\mu_{t_1,t_2}(\zeta)\ .$$ If $$V$$ is the functional of $$\phi$$ corresponding to the bare action/potential with UV cutoff $$\Lambda_H=e^{-t_1}$$, then $$RG[t_2,t_1](V)$$ is the effective potential at momentum/mass scale $$\Lambda_L=e^{-t_2}$$. Trivially (Fubini plus associativity of convolution of probability measures) one has, for $$t_1\le t_2\le t_3$$, $$RG[t_3,t_2]\circ RG[t_2,t_1]=RG[t_3,t_1]$$ which is indicative of a nonautonomous dynamical system structure, to be remedied shortly. At this point one can already state the main goal of renormalization/taking continuum limits of QFTs: finding a correct choice of cutoff-dependent potentials/actions/integrated Lagrangians, $$(V_t^{\rm bare})_{t\in\mathbb{R}}$$ such that $$\forall t_2,\ \lim_{t_1\rightarrow -\infty} RG[t_2,t_1](V_{t_1}^{\rm bare})\ =:\ V_{t_2}^{\rm eff}\ {\rm exists}.$$ The OP's intuition is correct in seeing this as a backwards shooting problem: choosing the right initial condition at $$\Lambda_{H}$$ to arrive where we want at $$\Lambda_{L}$$. A difficulty here (related to scattering in classical dynamical systems) is this involves an IVP at $$t=-\infty$$ instead of finite time. Note that the continuum QFT, its correlations, etc. should be completely determined by the collection of its effective theories indexed by scales $$(V_{t}^{\rm eff})_{t\in\mathbb{R}}$$. This is most easily seen when considering correlations smeared with test functions with compact support in Fourier space and with a sharp cutoff $$\widehat{\rho}(p)$$ given by the indicator function of the condition $$|p|\le 1$$ (or at least one which satisfies $$\widehat{\rho}(p)=1$$ in a neighborhood of zero momentum).

Switching to an autonomous setting involves some twisting by the scaling maps $$S_t$$. Given a potential V (bare or effective) which "lives at" scale $$t_1$$, one has $$\int e^{-V(\phi)}\ d\mu_{t_1,\infty}(\phi)=\int e^{-V(S_{t_1}\phi)}\ d\mu_{0,\infty}(\phi)= \int e^{-(S_{-t_1}V)(\phi)}\ d\mu_{0,\infty}(\phi)$$ where we now define the action of rescaling maps on functionals by $$(S_t V)(\phi):=V(S_{-t}\phi)\ .$$ As maps on functionals, one has the identity $$RG[t_2,t_1]=S_{t_1}\circ RG[t_2-t_1,0]\circ S_{-t_1}\ .$$

Wilson's Wilsonian RG is $$WRG[t]:=S_{-t}\circ RG[t,0]$$, for $$t\ge 0$$. It acts on the space of "unit lattice theories" (I put quotes because I am using Fourier rather than lattice cutoffs). Thus the previous identity becomes $$RG[t_2,t_1]=S_{t_2}\circ WRG[t_2-t_1]\circ S_{-t_1}\ .$$ The identity can be derived as follows (note the orgy of parentheses due to the increase of abstraction from functions to functionals, then to maps on functionals): $$[(RG[t_2-t_1,0]\circ S_{-t_1})(V)](\phi)=-\log\int d\mu_{0,t_2-t_1}(\zeta) \exp[-(S_{-t_1}V)(\phi+\zeta)]$$ $$=-\log\int d\mu_{0,t_2-t_1}(\zeta) \exp[-V(S_{t_1}\phi+S_{t_1}\zeta)]$$ $$=-\log\int d\mu_{t_1,t_2}(\xi) \exp[-V(S_{t_1}\phi+\xi)]$$ where we changed variables to $$\xi=S_{t_1}\zeta$$. From this one gets $$[(S_{t_1}\circ RG[t_2-t_1,0]\circ S_{-t_1})(V)](\phi)=[(RG[t_2,t_1,]\circ S_{-t_1})(V)](S_{t_1}\phi)$$ and the identity follows from the trivial fact $$S_{t_1}(S_{-t_1}\phi)=\phi$$.

Note that $$(V_t)_{t\in\mathbb{R}}$$ is trajectory of $$RG$$, i.e., $$\forall t_1\le t_2,\ V_{t_2}=RG[t_2,t_1](V_{t_1})$$ if and only if $$W_t:=S_{-t}V_t$$ is a trajectory of $$WRG$$, i.e., $$\forall t_1\le t_2,\ W_{t_2}=WRG[t_2-t_1](W_{t_1})\ .$$ The semigroup property for $$RG$$ readily implies that for $$WRG$$, namely, $$\forall t_1, t_2\ge 0,\ WRG[t_1+t_2]=WRG[t_1]\circ WRG[t_2]\ .$$ Now define $$W_{t}^{\rm start}:=S_{-t} \circ V_t^{\rm bare}$$. Then assuming continuity of all these RG maps one has $$V_{t_2}^{\rm eff}=\lim_{t_1\rightarrow -\infty} RG[t_2,t_1](V_{t_1}^{\rm bare})=S_{t_2}(W_{t_2}^{\rm eff})$$ where $$W_{t_2}^{\rm eff}:=\lim_{t_1\rightarrow -\infty} WRG[t_2-t_1](W_{t_1}^{\rm start})\ .$$ The definiteness of the continuum QFT can also be rephrased as the existence of the potentials $$W_{t}^{\rm eff}$$. A common source of confusion is the failure to see that while $$(W_{t}^{\rm eff})_{t\in\mathbb{R}}$$ is (by definition, the semigroup property and continuity) a trajectory of $$WRG$$, the family of bare potentials $$(W_{t}^{\rm bare})_{t\in\mathbb{R}}$$ is not. The same statement is true, by undoing the "moving frame change of coordinates", when replacing $$W$$'s with $$V$$'s and $$WRG$$ with $$RG$$.

For concreteness, we need coordinates on the space where the RG acts. Assume the bare potential $$V_t^{\rm bare}$$ is determined by a collection of coordinates or couplings $$(g_i)_{i\in I}$$ via $$V_{t}^{\rm bare}(\phi)=\sum_{i\in I} g_i^{\rm bare}(t)\ \int \mathcal{O}_i(x)\ dx$$ for local operators of the form $$\mathcal{O}_i(x)= :\partial^{\alpha_1}\phi(x)\cdots \partial^{\alpha_k}\phi(x):_t\ .$$ The Wick/normal ordering is with respect to the free cutoff measure $$\mu_{t,\infty}$$. More precisely, for every functional $$F$$, $$:F(\phi):_t\ \ :=\exp\left[-\frac{1}{2} \int dxdy\ \frac{\delta}{\delta\phi(x)}\ C_{t,\infty}(x,y)\ \frac{\delta}{\delta\phi(y)} \right]\ F(\phi)$$ where we denoted the propagator by $$C_{t,\infty}(x,y):=\langle\phi(x)\phi(y)\rangle_{t,\infty}$$. Note that changing $$-\frac{1}{2}$$ to $$+\frac{1}{2}$$ followed by setting $$\phi=0$$ is integration with respect to $$\mu_{t,\infty}$$. For instance $$:\phi(x)^2:_t=\phi(x)^2-C_{t,\infty}(x,x)$$ and $$:\phi(x)^4:_t=\phi(x)^4-6C_{t,\infty}(x,x)\phi(x)^2+3C_{t,\infty}(x,x)^2$$. An easy change of variables $$y=e^{-t}x$$ shows that $$(S_{-t}V_{t}^{\rm bare})(\phi)=\sum_{i\in I} g_i^{\rm start}(t) \int :\partial^{\alpha_1}\phi(y)\cdots \partial^{\alpha_k}\phi(y):_0\ dy$$ where $$g_i^{\rm start}(t):=e^{(D-\Delta_i)t}\ g_i^{\rm bare}(t)$$ and I used the notation $$\Delta_i=k\Delta+|\alpha_1|+\cdots+|\alpha_k|$$ for the scaling dimension of the local operator $$\mathcal{O}_i$$. The switch $$g_i^{\rm bare}\rightarrow g_i^{\rm start}$$ corresponds to that from dimensionful to dimensionless coupling constants. The indexing set splits as $$I=I_{\rm rel}\cup I_{\rm mar}\cup I_{\rm irr}$$, respectively corresponding to the three possibilities for operators: $$D-\Delta_i>0$$ or relevant, $$D-\Delta_i=0$$ or marginal, $$D-\Delta_i<0$$ or irrelevant.

$$W=0$$ is a fixed point of the autonomous dynamical system $$WRG$$. The behavior near this (trivial/Gaussian/free) fixed point is governed by the linearization or differential at $$W=0$$, i.e., the maps $$\mathcal{D}WRG[t]$$ given by $$[\mathcal{D}WRG[t](W)](\phi):=\int W(S_t\phi+\zeta)\ d\mu_{0,t}(\zeta)$$ as follows from the definition $$[WRG[t](W)](\phi)=-\log \int e^{-W(S_t\phi+\zeta)}\ d\mu_{0,t}(\zeta)$$ and the crude approximations $$e^z\simeq 1+z$$ and $$\log(1+z)\simeq z$$. If $$W$$ has coordinates $$(g_i)_{i\in I}$$ (with $$:\bullet :_0$$ Wick ordering), then one can show (good not so trivial exercise) that $$\mathcal{D}WRG[t](W)$$ has coordinates given exactly by $$(e^{(D-\Delta_i)t}g_i)_{i\in I}$$, in the same frame, i.e., with the same $$t=0$$ Wick ordering. If instead of flows one prefers talking in terms of the vector field $$\mathcal{V}$$ generating the dynamics, then a trajectory $$(W_t)_{t\in\mathbb{R}}$$ of $$WRG$$ satisfies $$\frac{dW_t}{dt}=\mathcal{V}(W_t)$$ with $$\mathcal{V}:=\left.\frac{d}{dt} WRG[t]\right|_{t=0}$$ admitting a linear plus nonlinear splitting $$\mathcal{V}=\mathcal{D}+\mathcal{N}$$. The linear part, in coordinates, is $$\mathcal{D}(g_i)_{i\in I}=((D-\Delta_i) g_i)_{i\in I}\ .$$ Assume the existence of $$W_{\rm UV}:=\lim_{t\rightarrow -\infty} W_{t}^{\rm eff}$$, the UV fixed point, and $$W_{\rm IR}:=\lim_{t\rightarrow \infty} W_{t}^{\rm eff}$$, the infrared fixed point (they have to be fixed points by continuity). The discussion of perturbative renormalizability always refers to the situation where $$W_{\rm UV}=0$$ corresponding to continuum QFTs obtained as perturbations of the free CFT $$\mu_{-\infty,\infty}$$. By definition, the QFT or the trajectory $$(W_t)_{t\in\mathbb{R}}$$ of its "unit lattice"-rescaled effective theories lies on the unstable manifold $$\mathcal{W}^{\rm u}$$ of the $$W=0$$ fixed point. In what follows I will assume for simplicity there are no marginal operators so the fixed point is hyperbolic and there are no subtleties due to center manifolds. The tangent space $$T\mathcal{W}^{\rm u}$$ is then spanned by functionals $$\phi\longmapsto \int \mathcal{O}_i$$, for $$i$$ in $$I_{\rm rel}$$ which is typically finite.

Note that, in principle, knowing a QFT is the same as knowing a trajectory $$(W_t^{\rm eff})_{t\in\mathbb{R}}$$ and thus the same as knowing just one point of that trajectory say $$W_0^{\rm eff}$$ (if the $$t=0$$ IVP is well-posed forwards and backwards in time, which is another delicate issue as explained in Arnold's answer). The point $$W_0^{\rm eff}$$ can be made to sweep the unstable manifold which can be identified with the space of continuum QFTs obtained by perturbing the $$W=0$$ fixed point. On the other hand our control parameter is the choice of cut-off dependent starting points $$(W_t^{\rm start})_{t\in\mathbb{R}}$$. These belong to the bare surface $$T\mathcal{W}^{\rm u}$$. This is why when considering say the $$\phi^4$$ model only a small finite number of terms are put in the bare Lagrangian, otherwise we would be talking about some other (family of) model(s) like $$\phi^6$$, $$\phi^8$$, etc. So after all these explanations, it should be clear that renormaliztion in Wilson's framework can be seen as a parametrization of the nonlinear variety $$\mathcal{W}^{\rm u}$$ by the linear subspace $$T\mathcal{W}^{\rm u}$$. If we denote the stable manifold by $$\mathcal{W}^{\rm s}$$ and its tangent space by $$T\mathcal{W}^{\rm s}$$ then, assuming hyperbolicity of the trivial fixed point, the full space where the RG acts should be $$T\mathcal{W}^{\rm u}\oplus T\mathcal{W}^{\rm s}$$. The stable manifold theorem gives a representation of $$\mathcal{W}^{\rm u}$$ as the graph of a map from $$T\mathcal{W}^{\rm u}$$ into $$T\mathcal{W}^{\rm s}$$.

The main problem is to find $$(W_t^{\rm start})_{t\in\mathbb{R}}$$ so that the limit $$W_0^{\rm eff}=\lim_{t\rightarrow -\infty} WRG[-t](W_t^{\rm start})$$ exists. The stable manifold theorem is the $$t=-\infty$$ case of a mixed boundary problem where on a trajectory one imposes conditions (on coordinates) of the form $$g_i^{\rm start}(t)=0$$, $$i\in I_{\rm irr}$$, and $$g_i^{\rm eff}(0)=\lambda_{i}^{\rm R}$$, $$i\in I_{\rm rel}$$. Irwin's proof is a nice way to slove this and it works even if the RG is not reversible. This method can be applied for finite negative $$t$$, and this should produce a collection $$(W_t^{\rm })_{t<0}$$ (all that is needed in fact) dependent on the renormalized couplings $$\lambda_{i}^{\rm R}$$. Let us assume for instance that $$I_{\rm rel}=\{1,2\}$$ and $$I_{\rm irr}=\{3,4,\ldots\}$$. Consider the map $$P_t$$ given by $$(\lambda_{1}^{\rm B},\lambda_{2}^{\rm B})\longmapsto (g_i\{WRG[-t](\lambda_{1}^{\rm B}, \lambda_{2}^{\rm B},0,0,\ldots)\})_{i=1,2}$$ where $$g_i\{W\}$$ denotes the $$i$$-th coordinate of $$W$$. A possible choice of starting points is thus $$W_t^{\rm start}:=(P_t^{-1}(\lambda_{1}^{\rm R},\lambda_{2}^{\rm R}),0,0,\ldots)_{i\in I}\ .$$

The above is more like a road map for what needs to be done but it does not quite provide a recipe for doing it. In the perturbative setting, one trades numbers in $$\mathbb{R}$$ for formal power series in $$\mathbb{R}[[\hbar]]$$. The propagators of the $$\mu$$ measures get multiplied by $$\hbar$$ and there is now $$\frac{1}{\hbar}$$ in front of the $$V$$'s or $$W$$'s in the exponential. All the couplings $$g_i$$ now also become elements of $$\mathbb{R}[[\hbar]]$$. The invertibility of $$P_t$$ in this setting is easy and follows by analogues of the implicit/inverse function theorem for formal power series (e.g. in Bourbaki, Algebra II, Chapters 4-7, Berlin, Springer-Verlag, 1990). All the work is in showing that for $$i\ge 3$$, the quantities $$f_i(\lambda_{1}^{\rm R},\lambda_{2}^{\rm R}):=\lim_{t\rightarrow -\infty} g_i\{WRG[-t](P_t^{-1}(\lambda_{1}^{\rm R},\lambda_{2}^{\rm R}),0,0,\ldots)\}$$
converge to finite values. This gives the wanted parametrization $$(\lambda_{1}^{\rm R},\lambda_{2}^{\rm R}) \mapsto(\lambda_{1}^{\rm R},\lambda_{2}^{\rm R},f_3(\lambda_{1}^{\rm R},\lambda_{2}^{\rm R}),f_4(\lambda_{1}^{\rm R},\lambda_{2}^{\rm R}),\ldots)$$ of $$\mathcal{W}^{\rm u}$$ by $$T\mathcal{W}^{\rm u}$$. There are two ways of showing the above convergence statement. Underlying both ways is the fact (see Bourbaki above) the formal power series $$P_t^{-1}(\lambda_{1}^{\rm R},\lambda_{2}^{\rm R})\in \mathbb{R}[[\hbar]]^2$$ exist and are unique.

Fans of combinatorics would prefer a two-step procedure consisting in 1) finding an explicit formula for $$WRG[-t](P_t^{-1}(\lambda_{1}^{\rm R},\lambda_{2}^{\rm R}),0,0,\ldots)$$ for finite $$t$$; then 2), with this formula in hand, analyze the limit $$t\rightarrow -\infty$$. The explicit formula in 1) is Zimmermann's forest formula. See this article by Hairer for a recent take on the delicate analytical estimates needed for step 2).

For those who abhor combinatorics, there is another method which avoids explicit formulas. Change the scale $$0$$ in the mixed boundary problem to an arbitrary scale $$s>t$$. Namely, impose $$g_i(t)=0$$ for $$i\ge 3$$ and $$g_i(s)=\lambda_i^{\rm R}$$ for $$i=1,2$$ and study the variation of $$s$$ from $$s=t$$ to $$s=0$$ by ODE techniques. This is the Wilson-Polchinski approach. The best rigorous account that I know for this second approach is in the book "Renormalization: An Introduction" by Salmhofer.

Finally, one could ask what would happen if one used $$W_{s}^{\rm eff}$$, for some fixed $$s\neq 0$$, to parametrize the QFTs instead of $$W_{0}^{\rm eff}$$. The answer is obtained by noticing that the maps $$W_s^{\rm eff}\mapsto {\rm QFT}$$ intertwine the action of $$WRG$$ on $$\mathcal{W}^{\rm u}$$ and that of the scaling maps $$S_t$$ on QFTs (simply rescale correlations, i.e., do $$\phi\rightarrow S_t\phi$$ inside correlations). This is the relation to the old Stueckelberg-Peterman-Gell-Mann-Low RG (i.e., change of scale can be absorbed in a change of renormalized coupling constants). In other words, the restriction of the nonreversible $$WRG$$ to the finite dimensional manifold $$\mathcal{W}^{\rm u}$$ should be reversible since $$S_t$$'s (on collections of correlations) are, or because of the remark I made about Irwin's proof being applicable even for noninvertible (discrete) dynamical systems.

• I will point out that there was a suggested edit where the intention seems to be to correct a typo - although I am not sure to able to judge whether it should be "$(W_{t}^{\rm bare})_{t\in\mathbb{R}}$" or "$(V_{t}^{\rm bare})_{t\in\mathbb{R}}$". If you have already seen it, feel free to ping me in a comment or in chat to let me know that I can deleted this comment. – Martin Sleziak May 25 at 6:11
• The suggested edit was incorrect and frivolous. Thank you to the moderators for rejecting it. – Abdelmalek Abdesselam May 25 at 13:54

Classical field theories (Lagrangian variational principles) sometimes come in families. The families may be finite dimensional, or also infinite dimensional. One could even take the family to consist of all possible Lagrangians, say with the fields (dependent variables) as well the source and target manifolds (domains of the independent and dependent variables, respectively) fixed. The notion of a symmetry of a single theory is straightforward: a local transformation of the fields that leaves the Lagrangian fixed (up to total derivative terms). The notion of a symmetry of a family of theories is similar: the action of the transformation of the fields must keep any Lagrangian from the family within the same family. When quantizing the family of classical field theories, lets just take it as given that one can lift field transformations from the classical to the quantum level (this involves technical subtleties and is not automatic, but that is not the point here). Ideally, one would like to preserve the symmetries of classical theories, but in general the quantum lift of the classical symmetry may not be a symmetry of the quantum theory. Strictly speaking, one should then say that the symmetry is anomalous. However, when considering quantization as a deformation problem (parameter $$\hbar=0$$ corresponds to classical, while $$\hbar \ne 0$$ to quantum), it is natural to also to allow quantum corrections ($$\hbar$$-parametrized deformations) of the lifted classical symmetries. So the term anomalous symmetry is reserved for those symmetries whose quantum lifts cannot even be quantum corrected. Even more confusingly, the quantization procedure is not unique (in the context of perturbative quantization, specific quantization procedures are referred to as renormalization schemes). Thus, if one can change the quantization procedure to make an anomalous symmetry into a non-anomalous one, then one says that the anomaly in the symmetry can be cancelled.

Now, getting more specific, suppose that the family under consideration has a scaling symmetry (basically, an action by the multiplicative positive reals $$\mathbb{R}_+^\times$$). Lets call the infinitesimal version of this action the classical scaling flow. Obviously, the scaling flow has an action on the parameters of our family of theories. The quickest definition of renormalization group flow is that it is the quantum corrected lift of the classical scaling flow (provided that the quantization procedure has been chosen to cancel potential anomalies in the scaling symmetry). One could consider specifically the action of the renormalization group flow on the parameters of our family of theories, and call that renormalization group flow as well. The latter meaning is the one encountered most often in the literature and the one appearing in the OP.

To make matters a bit more muddy, a given quantum lift of a classical symmetry strictly speaking depends on the quantization procedure. So the quantum lift changes when the quantization procedure changes (of course restricting ourselves to changes for which the symmetry remains non-anomalous). So some people say that the renormalization group flow (or its restriction to some parameters) is non-trivial only if there is no choice of quantization procedure for which no quantum corrections are needed.

Of course, the discussion I gave in the first paragraph is quite heuristic. When reading about mathematically rigorous treatment of the renormalization group flow, most of the details deal with making what I described mathematically precise. There are different approaches to doing that and the number of technical obstacles is not small, which is what makes such treatments difficult reading for outsiders to the field.

Finally coming back to Ricci flow, one could say that it coincides with the renormalization group flow of a 2-dimensional Euclidean non-linear sigma model (quantized in a reasonable perturbative way), when restricted to the target space-metric, as a parameter of the Lagrangian). One can find at least two mathematically precise approaches to making the above statement rigorous:

Nguyen, Timothy, Quantization of the nonlinear sigma model revisited, J. Math. Phys. 57, No. 8, 082301, 40 p. (2016). ZBL1351.81089. arXiv:1408.4466

Mauro Carfora, Claudio Dappiaggi, Nicolò Drago, Paolo Rinaldi, Ricci Flow from the Renormalization of Nonlinear Sigma Models in the Framework of Euclidean Algebraic Quantum Field Theory, Commun. Math. Phys. (2019). arXiv:1809.07652