Born–Jordan and Weyl Quantizations of the 2D Anisotropic Harmonic Oscillator

Born–Jordan and Weyl Quantizations
of the 2D Anisotropic Harmonic Oscillator

Giovanni RASTELLI

G. Rastelli

Dipartimento di Matematica, Università di Torino, Torino, via Carlo Alberto
10, Italy<EMAIL_ADDRESS>
Received July 15, 2016, in final form August 15, 2016; Published online August
17, 2016

We apply the Born–Jordan and Weyl quantization formulas for polynomials in
canonical coordinates to the constants of motion of some examples of the
superintegrable 2D anisotropic harmonic oscillator. Our aim is to study the
behaviour of the algebra of the constants of motion after the different
quantization procedures. In the examples considered, we have that the Weyl
formula always preserves the original superintegrable structure of the system,
while the Born–Jordan formula, when producing different operators than the
Weyl’s one, does not.

Born–Jordan quantization; Weyl quantization; superintegrable systems; extended
systems

81S05; 81R12; 70H06

##  1 Introduction

Max Born and Pascual Jordan proposed a first model of quantization in [3],
restricted to one-dimensional systems, a model generalized to $n$-dimensional
systems, together with Werner Heisenberg, in [4]. Hermann Weyl introduced in
[14, 15] a general quantization scheme based on the Fourier transform formula.
The Born–Jordan and Weyl quantizations produce in general different operators
from the same classical function of momenta and coordinates. The two
quantization methods, however, coincide on natural Hamiltonians in
$n$-dimensional Euclidean spaces. This property and many others of the two
quantizations are studied in the recent book [9], that is our main source for
the following discussion.

We consider here the Born–Jordan and Weyl quantizations applied to the algebra
of constants of motion of the 2D anisotropic harmonic oscillator. This system
admits the maximal number of three functionally independent constants of
motion (and it is said to be superintegrable) whenever the ratio of the
parameters is a rational number. Otherwise, the independent constants of
motion are just two.

Our aim here is to check the behaviour of the Born–Jordan and Weyl
quantizations of the anisotropic harmonic oscillator with respect to the
integrability and superintegrability of the resulting quantum system for some
particular choice of the ratio of the parameters. This aspect of the two
quantization procedures is not considered in [9].

We use here the same expression for the classical 2D anisotropic harmonic
oscillator and its independent constants of motion that is employed in [7].
The nD anisotropic harmonic oscillator is there considered as an “extended
system”, a particular structure of some natural Hamiltonians that allows the
existence of polynomial constants of motion of higher degree, described in [7,
8]. The interest in using such a construction here comes from other our
studies in progress on the quantization of extended systems (see for example
[6]).

We consider briefly also the factorization in annihilation-creation operators
for the 2D anisotropic harmonic oscillator as given in [11], to check how the
corresponding classical integrals are quantized by applying again Born–Jordan
and Weyl procedures. The examples are confronted with those arising from the
extension procedure.

We can apply here the simpler formulas for Born–Jordan and Weyl quantizations
for monomials in coordinates and momenta discussed in [9].

The quantization procedures of classical quantities are not exhausted by
Born–Jordan’s and Weyl’s approaches. Several techniques have been developed
since the beginning of the quantum era, many of them specific for the
particular system considered.

Indeed, there is no unique way to assign quantum operators to classical
quantities in a meaningful way. Hermiticity of the operator is, usually, a
necessary requirement and it is obtained by some symmetrization procedure,
however not uniquely determined, see for example [13].

The problem of preserving the algebra of the constants of motion of a
Hamiltonian system after quantization is object of many recent studies, see
for example [1, 5, 10, 12] and references therein, for solutions in flat and
non-flat manifolds.

For superintegrability and quantization in classical and quantum systems see
also [12], in particular for the definition of quantum superintegrable
systems, and [5].

##  2 Born–Jordan and Weyl quantizations of monomials

In [3, 4, 9] the Born–Jordan quantization of monomials in coordinates
$(x_{i},p_{i})$ is determined by the following general rules,

$\displaystyle[\hat{x}_{i},\hat{p}_{j}]=i\hbar\delta_{ij},\qquad[\hat{x}_{i},\hat{x}_{j}]=0,\qquad[\hat{p}_{i},\hat{p}_{j}]=0,$

where $\delta_{ij}=1$ for $i=j$ and zero otherwise, for any quantization of
the coordinates $x_{i}\rightarrow\hat{x}_{i}$ and of the momenta
$p_{i}\rightarrow\hat{p}_{i}$, and by

$\displaystyle
x_{i}^{r}p_{i}^{s}\rightarrow\frac{1}{s+1}\sum_{k=0}^{s}\hat{p}_{i}^{s-k}\hat{x}_{i}^{r}\hat{p}_{i}^{k},$
(1)

for the monomials with same indices. When the indices are different, the
operators commute by the general quantization rules of above and their
quantization is therefore straightforward.

For the Weyl quantization [9, 14], the general rules are the same of above
and, for the monomials, we have instead

$\displaystyle
x_{i}^{r}p_{i}^{s}\rightarrow\frac{1}{2^{s}}\sum_{k=0}^{s}\binom{s}{k}\hat{p}_{i}^{s-k}\hat{x}_{i}^{r}\hat{p}_{i}^{k}.$
(2)

The standard realization of the operators $\hat{x}_{i}$ and $\hat{p}_{i}$, at
least for Cartesian coordinates, are

$\displaystyle\hat{x}_{i}\phi=x_{i}\phi,\qquad\hat{p}_{i}\phi=-i\hbar\frac{\partial}{\partial
x_{i}}\phi,$

for any function $\phi(x_{j})$. We employ in the following this standard
quantization of the canonical coordinates.

It is easy to check that for $r=s=1$ the Born–Jordan and Weyl quantizations of
monomials coincide, but differ for $r,s\geq 2$.

Many properties of the Born–Jordan quantization, generalized to any function
of coordinates and momenta, and of the Weyl quantization are considered into
details in [9] and the different characteristics are discussed.

We focus here our analysis on the effect of the two quantizations on the first
integrals of a particular superintegrable system. We check in some examples if
the quantized first integrals commute with the Hamiltonian operator, that is,
if the algebraic structure of the constants of motion is preserved by the
different formulas of quantization, an issue not considered in [9].

##  3 The superintegrable 2D anisotropic harmonic oscillator

In order to express the first integrals of the system, we can write the
Hamiltonian of the superintegrable 2D anisotropic harmonic oscillator in the
form of an extended Hamiltonian [7] as follows

$\displaystyle
H_{m,n}=\frac{1}{2}\left(p_{u}^{2}+\left(\frac{m}{n}\right)^{2}p_{x}^{2}\right)+\omega^{2}\left(\frac{m}{n}\right)^{2}\left(x^{2}+u^{2}\right),$
(3)

where $(x,u)$ are coordinates in the Euclidean plane, $\omega\in\mathbb{R}$
and $m,n\in\mathbb{N}\setminus\\{0\\}$.

Two independent first integrals of $H_{m,n}$ are $H_{m,n}$ itself and

$\displaystyle L=\frac{1}{2}p_{x}^{2}+\omega^{2}x^{2},$

that is associated with the separability of the Hamilton–Jacobi equation of
$H_{m,n}$ in coordinates $(u,x)$.

If we put

$\displaystyle
G_{n}=\sum_{k=0}^{\left[\frac{n-1}{2}\right]}\binom{n}{2k+1}\big{(}{-}2\omega^{2}\big{)}^{k}x^{2k+1}p_{x}^{n-2k-1},$
(4)

where $[a]$ denotes the integer part of $a$, and $X_{L}$ is the Hamiltonian
vector field of the function $L$, then, adapting to our case the more general
theorem proved in [7], we have

######  Proposition 1.

For any couple of positive integers $(m,n)$, the function $K_{m,n}$ is a first
integral of $H_{m,n}$, where

$\displaystyle K_{m,n}=P_{m,n}G_{n}+D_{m,n}X_{L}(G_{n}),$

with

$\displaystyle
P_{m,n}=\sum_{k=0}^{[m/2]}\binom{m}{2k}\left(-\frac{m}{n}u\right)^{2k}p_{u}^{m-2k}\big{(}{-}2\omega^{2}\big{)}^{k},$
$\displaystyle
D_{m,n}=\frac{1}{n}\sum_{k=0}^{[(m-1)/2]}\binom{m}{2k+1}\,\left(-\frac{m}{n}u\right)^{2k+1}p_{u}^{m-2k-1}\big{(}{-}2\omega^{2}\big{)}^{k},\qquad
m>1,$

and $D_{1,n}=-\frac{m}{n^{2}}u$.

We can introduce the usual Cartesian coordinates $(x,y)$ by leaving $x$,
$p_{x}$ unchanged and putting

$\displaystyle u=\frac{n}{m}y,\qquad p_{u}=\frac{m}{n}p_{y},$ (5)

so that

$\displaystyle
H_{m,n}=\left(\frac{m}{n}\right)^{2}\left(\frac{1}{2}\left(p_{x}^{2}+p_{y}^{2}\right)+\omega^{2}\left(x^{2}+\left(\frac{n}{m}\right)^{2}y^{2}\right)\right).$

In the following, we consider the $H_{m,n}$ of above by dropping the
negligible overall factor $\left(\frac{m}{n}\right)^{2}$.

The classical and quantum superintegrability of the $n$D anisotropic harmonic
oscillator has been studied in [11]. The quantization of the classical system
is there obtained by introducing creation and annihilation operators. This
technique is widely in use today (see for example [5] and [2], where the
anisotropic harmonic oscillator is generalized to 2D constant-curvature
manifolds obtaining new classical and quantum superintegrable systems) and can
have some application towards the quantization of extended systems.

The Jauch–Hill Hamiltonian of the anisotropic harmonic oscillator is

$\displaystyle
H_{JH}=\frac{1}{2}\left(\frac{p_{1}^{2}}{M_{1}}+\frac{p_{2}^{2}}{M_{2}}+M_{1}\omega_{1}^{2}q_{1}^{2}+M_{2}\omega_{2}^{2}q_{2}^{2}\right),$

and coincide with $H_{m,n}$ if we put

$\displaystyle x=\sqrt{M_{1}}q_{1},\qquad p_{1}=\sqrt{M_{1}}p_{x},\qquad
y=\sqrt{M_{2}}q_{2},\qquad p_{2}=\sqrt{M_{2}}p_{y},$

with

$\displaystyle m\omega_{2}=n\omega_{1},\qquad\omega_{1}=\sqrt{2}\omega.$

The classical first integrals given in [11] become

$\displaystyle
F_{1}(m,n)=\frac{1}{2}\big{(}b_{1}^{n}b_{2}^{*m}+b_{1}^{*n}b_{2}^{m}\big{)},\qquad
F_{2}(m,n)=-\frac{i}{2}\big{(}b_{1}^{n}b_{2}^{*m}-b_{1}^{*n}b_{2}^{m}\big{)},$
(6)

where

$\displaystyle
b_{1}=\frac{1}{\sqrt{2\omega_{1}}}\left(p_{x}-i\omega_{1}x\right),\qquad
b_{1}^{*}=\frac{1}{\sqrt{2\omega_{1}}}\left(p_{x}+i\omega_{1}x\right),$

and similarly $b_{2}$, $b_{2}^{*}$ in function of $y$, $p_{y}$, $\omega_{2}$.
The corresponding quantum operators are obtained simply by substituting
$p_{x}$, $p_{y}$ with $\hat{p}_{x}$, $\hat{p}_{y}$ in the expressions of
above.

The first integrals obtained with the two methods of above are different,
having for example, up to constant factors,

$\displaystyle K_{1,1}=xp_{y}-yp_{x},\qquad
F_{1}(1,1)=p_{x}p_{y}+\omega^{2}xy.$

##  4 The two quantizations of the constants of motion
of the oscillator

The Born–Jordan and Weyl quantizations of both $H_{m,n}$ and $L$ clearly
coincide, being in both cases

$\displaystyle
H_{m,n}\rightarrow\hat{H}_{m,n}=-\frac{\hbar^{2}}{2}\left(\frac{\partial^{2}}{\partial
x^{2}}+\frac{\partial^{2}}{\partial
y^{2}}\right)+\omega^{2}\left(x^{2}+\left(\frac{n}{m}\right)^{2}y^{2}\right),$
$\displaystyle\
L\rightarrow\hat{L}=-\frac{\hbar^{2}}{2}\frac{\partial^{2}}{\partial
x^{2}}+\omega^{2}x^{2}.$

The operators are clearly independent and commuting, so that the integrability
of the system is preserved by both the quantizations.

The Born–Jordan and Weyl quantizations of $K_{1,1}$, $K_{2,1}$ and $K_{3,1}$
coincide and commute with the corresponding Hamiltonian operators
$\hat{H}_{m,n}$. Things become different for $(m,n)=(4,1)$. Indeed

######  Proposition 2.

The first integral $K_{4,1}$ of $H_{4,1}$ is

$\displaystyle
K_{4,1}=256\left(xp_{y}^{4}-yp_{x}p_{y}^{3}-\frac{3}{4}\omega^{2}xy^{2}p_{y}^{2}+\frac{\omega^{2}}{8}y^{3}p_{x}p_{y}+\frac{\omega^{4}}{64}xy^{4}\right).$
(7)

By applying to $K_{4,1}$ the Weyl quantization formula (2), we have the Weyl
operator

$\displaystyle\hat{K}_{4,1}^{\rm
W}=256\left(\hbar^{4}\left(x\frac{\partial^{4}}{\partial
y^{4}}-y\frac{\partial^{4}}{\partial x\partial
y^{3}}\right)-\frac{3\hbar^{4}}{2}\frac{\partial^{3}}{\partial x\partial
y^{2}}+\frac{\hbar^{2}\omega^{2}}{8}\left(6xy^{2}\frac{\partial^{2}}{\partial
y^{2}}-y^{3}\frac{\partial^{2}}{\partial x\partial y}\right)\right.$
$\displaystyle\left.\hphantom{\hat{K}_{4,1}^{\rm
W}=}{}+\hbar^{2}\omega^{2}y\left(\frac{3}{2}x\frac{\partial}{\partial
y}-\frac{3}{16}y\frac{\partial}{\partial
x}\right)+\frac{\omega^{4}}{64}xy^{4}+\frac{3\hbar^{2}\omega^{2}}{8}x\right),$
(8)

and, by applying to $K_{4,1}$ the formula (1), we get the Born–Jordan operator

$\displaystyle\hat{K}_{4,1}^{\rm BJ}=\hat{K}_{4,1}^{\rm
W}+32\hbar^{2}\omega^{2}x.$ (9)

The computation of the commutators gives

$\displaystyle\big{[}\hat{H}_{4,1},\hat{K}^{\rm
W}_{4,1}\big{]}=0,\qquad\big{[}\hat{H}_{4,1},\hat{K}^{\rm
BJ}_{4,1}\big{]}=-32\hbar^{4}\omega^{2}\frac{\partial}{\partial x}.$ (10)

###### Proof.

We have from (4) $G_{1}=x$, therefore $X_{L}G_{1}=p_{x}$. The application of
Proposition 1 to (3) with $(m,n)=(4,1)$ followed by the transformation of
coordinates (5) gives

$\displaystyle
P_{4,1}=256p_{y}^{4}-192\omega^{2}y^{2}p_{y}^{2}+4\omega^{2}y,\qquad
D_{4,1}=-256yp_{y}^{3}+32\omega^{2}y^{3}p_{y},$

and we get (7). We observe now that, both for Born–Jordan and Weyl
quantizations, we have

$\displaystyle\hat{K}_{4,1}=\hat{P}_{4,1}\hat{x}+\hat{D}_{4,1}\hat{p}_{x},$

where the order of the operators is immaterial, because $P_{4,1}$ and
$D_{4,1}$ depend uniquely on $(p_{y},y)$. Therefore, we need to apply the two
quantizations to $P_{4,1}$ and $D_{4,1}$ only, since the quantizations
coincide on $x$ and $p_{x}$. By applying (1) and (2) to the monomials
$y^{2}p_{y}^{2}$, $yp_{y}^{3}$ and $y^{3}p_{y}$, we have

$\displaystyle\hat{P}_{4,1}=256\hat{p}_{y}^{4}-192\omega^{2}\hat{Q}_{1}+4\omega^{2}\hat{y},\qquad\hat{D}_{4,1}=-256\hat{Q}_{2}+32\omega^{2}\hat{Q}_{3},$

where

$\displaystyle\hat{Q}_{2}=i\frac{\hbar^{3}}{2}\left(2y\frac{\partial^{3}}{\partial
y^{3}}+3\frac{\partial^{2}}{\partial
y^{2}}\right),\qquad\hat{Q}_{3}=-i\frac{\hbar}{2}y^{2}\left(2y\frac{\partial}{\partial
y}+3\right),$

coincide for both quantizations, and, for the Weyl case,

$\displaystyle\hat{Q}_{1}=-\frac{\hbar^{2}}{2}\left(2y^{2}\frac{\partial^{2}}{\partial
y^{2}}+4y\frac{\partial}{\partial y}+1\right),$

while, for the Born–Jordan case,

$\displaystyle\hat{Q}_{1}=-\frac{\hbar^{2}}{3}\left(3y^{2}\frac{\partial^{2}}{\partial
y^{2}}+6y\frac{\partial}{\partial y}+2\right).$

We obtain in this way (8) and (9).

We can finally compute the commutators of $\hat{H}_{4,1}$ with
$\hat{K}_{4,1}^{\rm W}$ and $\hat{K}_{4,1}^{\rm BJ}$ with the help of the
formula

$\displaystyle\big{[}\hat{H}_{4,1},\hat{K}_{4,1}\big{]}=\hat{p}_{x}\big{(}{-}i\hbar\hat{P}_{4,1}+\big{[}\hat{H}_{4,1},\hat{D}_{4,1}\big{]}\big{)}+\hat{x}\big{(}2\omega^{2}i\hbar\hat{D}_{4,1}+\big{[}\hat{H}_{4,1},\hat{P}_{4,1}\big{]}\big{)},$

after making the suitable substitutions, obtaining the (10). ∎

We computed the quantizations for several other values of $(m,n)$, for example
$(5,1)$, $(6,1)$, $(1,4)$, $(3,4)$, such that $\hat{K}_{m,n}^{\rm
BJ}\neq\hat{K}_{m,n}^{\rm W}$, obtaining always commutation with
$\hat{H}_{m,n}$ for $\hat{K}^{\rm W}_{m,n}$ and no commutation with
$\hat{H}_{m,n}$ for $\hat{K}^{\rm BJ}_{m,n}$. In these last cases, it can be
observed that both $\omega$ and $\hbar$ always appear as factors in the
commutator, meaning that for $\omega=0$ and for $\hbar\rightarrow 0$ the
operators commute.

Analogous results are obtained from the quantizations of the first integrals
$F_{1}(m,n)$ or $F_{2}(m,n)$ given in (6) for several values of $(m,n)$: the
Weyl formula produces symmetry operators of the Hamiltonian, the Born–Jordan
one, when giving different operators, does not.

Actually, one can conjecture that the Weyl quantizations of $K_{m,n}$ and
$F_{i}(m,n)$ always commute with the Hamiltonian operator $\hat{H}_{m,n}$.

It can be observed, as one of the referees of this article pointed out, that
“for the Jauch–Hill approach the quantization using creation and annihilation
operators shows very easily that the quantum extensions of $F_{1}$ and $F_{2}$
are still integrals. It is less obvious to prove that this quantization is
nothing but Weyl’s one”.

######  Remark 3.

The failure of the Born–Jordan quantization formula in reproducing the algebra
of constants of motion at the quantum level is, actually, restricted to the
particular set of generators of the algebra that we choose. We do not know in
general if another choice of independent first integrals can lead to different
results.

######  Remark 4.

The failure in reproducing the algebra of the constants of motion after
quantization appears also in the case of natural Hamiltonian systems on curved
manifolds. In these cases, quantum corrections of the Hamiltonian operator are
necessary in order to preserve the integrable or superintegrable algebraic
structure [1, 10].

##  5 Conclusions

From the examples computed, it appears that the Born–Jordan quantization
formula fails in preserving the high-degree constants of motion of the 2D
anisotropic harmonic oscillator, and therefore its superintegrability, to the
quantum level, differently from the Weyl formula. A study of this problem in
full generality is then desirable.

### Acknowledgements

I am grateful to the referees of this article for their comments and
suggestions.

## References

  * [1] Ballesteros Á., Enciso A., Herranz F.J., Ragnisco O., Riglioni D., Quantum mechanics on spaces of nonconstant curvature: the oscillator problem and superintegrability, Ann. Physics 326 (2011), 2053–2073, arXiv:1102.5494.
  * [2] Ballesteros Á., Herranz F.J., Kuru Ş., Negro J., The anisotropic oscillator on curved spaces: a new exactly solvable model, Ann. Physics 373 (2016), 399–423, arXiv:1605.02384.
  * [3] Born M., Jordan P., Zur Quantenmechanik, Z. Phys. 34 (1925), 858–888.
  * [4] Born M., Heisenberg W., Jordan P., Zur Quantenmechanik. II, Z. Phys. 35 (1925), 557–615.
  * [5] Celeghini E., Kuru Ş., Negro J., del Olmo M.A., A unified approach to quantum and classical TTW systems based on factorizations, Ann. Physics 332 (2013), 27–37.
  * [6] Chanu C., Degiovanni L., Rastelli G., Generalizations of a method for constructing first integrals of a class of natural Hamiltonians and some remarks about quantization, J. Phys. Conf. Ser. 343 (2012), 012101, 15 pages, arXiv:1111.0030.
  * [7] Chanu C.M., Degiovanni L., Rastelli G., Extensions of Hamiltonian systems dependent on a rational parameter, J. Math. Phys. 55 (2014), 122703, 11 pages, arXiv:1310.5690.
  * [8] Chanu C.M., Degiovanni L., Rastelli G., The Tremblay–Turbiner–Winternitz system as extended Hamiltonian, J. Math. Phys. 55 (2014), 122701, 8 pages, arXiv:1404.4825.
  * [9] de Gosson M.A., Born–Jordan quantization. Theory and applications, Fundamental Theories of Physics, Vol. 182, Springer, Cham, 2016.
  * [10] Duval C., Valent G., Quantum integrability of quadratic Killing tensors, J. Math. Phys. 46 (2005), 053516, 22 pages, math-ph/0412059.
  * [11] Jauch J.M., Hill E.L., On the problem of degeneracy in quantum mechanics, Phys. Rev. 57 (1940), 641–645.
  * [12] Miller Jr. W., Post S., Winternitz P., Classical and quantum superintegrability with applications, J. Phys. A: Math. Theor. 46 (2013), 423001, 97 pages, arXiv:1309.2694.
  * [13] Post S., Winternitz P., General $N$th order integrals of motion in the Euclidean plane, J. Phys. A: Math. Theor. 48 (2015), 405201, 24 pages, arXiv:1501.00471.
  * [14] Weyl H., Quantenmechanik und Gruppentheorie, Z. Phys. 46 (1927), 1–46.
  * [15] Weyl H., The theory of groups and quantum mechanics, Dover, New York, 1950.


# Carrollian $\boldsymbol{c}$-functions and flat space holographic RG flows in
BMS3/CCFT2

Daniel Grumiller  Institute for Theoretical Physics, TU Wien, Wiedner
Hauptstrasse 8–10/136, A-1040 Vienna, Austria  Theoretical Sciences Visiting
Program, Okinawa Institute of Science and Technology Graduate University,
Onna, 904-0495, Japan<EMAIL_ADDRESS>Max Riegler  Institute for
Theoretical Physics, TU Wien, Wiedner Hauptstrasse 8–10/136, A-1040 Vienna,
Austria  Quantum Technology Laboratories, Clemens-Holzmeister-Straße 6/6, 1100
Austria, Vienna<EMAIL_ADDRESS>
###### Abstract

We discuss $c$-functions and their holographic counterpart for two-dimensional
field theories with Carrollian conformal fixed points in the UV and the IR.
Specifically, we construct asymptotically flat domain wall solutions of three-
dimensional Einstein-dilaton gravity that model holographic RG flows between
BMS3 invariant UV and IR fixed points. We prove three theorems for such flows:
1. for every holographic RG flow in AdS3, there is a corresponding one in flat
space, 2. the BMS central charge in the UV cannot be smaller than in the IR,
and 3. the UV/IR ratio of Virasoro central charges is identical to the UV/IR
ratio of corresponding BMS central charges. Finally, we tentatively propose a
Casini–Huerta-like $c$-functions for BMS3-invariant quantum field theories,
inspired by the AdS3/CFT2 relation between monotonicity of the $c$-function
and the quantum null energy condition.

###### Contents

  1. I Introduction
  2. II AdS3/CFT2 review
    1. II.1 Casini–Huerta $c$-function
    2. II.2 Relation to Quantum Null Energy Condition
    3. II.3 Domain walls in AdS3
  3. III BMS3/CCFT2 summary
    1. III.1 Gravity aspects of BMS3/CCFT2
    2. III.2 Field theory aspects
    3. III.3 Quantum energy conditions in CCFT2
  4. IV Domain walls in flat space
    1. IV.1 Geometric aspects of flat space domain walls
    2. IV.2 Flat space domain walls in Einstein-dilaton gravity
    3. IV.3 Flat space holographic RG flow example
  5. V Flat space holographic RG flow theorems
    1. V.1 Definitions
    2. V.2 Correspondence theorem
    3. V.3 Monotonicity theorem and central charge ratio equivalence
  6. VI Tentative proposal for Casini–Huerta-inspired Carrollian $\boldsymbol{c}$-function

##  I Introduction

Quantum field theories (QFT) adequately describe many systems in nature. A
prototypical scenario is a relativistic QFT with conformal fixed points in the
ultraviolet (UV) and the infrared (IR). The renormalization group (RG) flow
connecting these fixed points can be characterized by $c$-functions that obey
a $c$-theorem.

The latter guarantees the existence of some (positive) function,
$c(g_{i},\,\mu)$, depending on the coupling constants $g_{i}$ and the RG scale
$\mu$ with two key properties: 1. it decreases monotonically under RG flow
towards the IR, and 2. at the UV and IR fixed points the $c$-function is a
(finite) constant. This mathematical statement captures the physical intuition
that QFTs have more degrees of freedom in the UV than in the IR.

In two-dimensional (2d) QFTs, Zamolodchikov proved a $c$-theorem by explicitly
constructing a $c$-function composed of the energy-momentum tensor components
Zamolodchikov:1986gt . At the UV- and IR-fixed points, the value of this
$c$-function coincides with the respective values of the central charge that
characterize the corresponding fixed point conformal field theories (CFT),
$c^{\textrm{\tiny UV}}$ and $c^{\textrm{\tiny IR}}$. The $c$-theorem implies
$c^{\textrm{\tiny UV}}\geq c^{\textrm{\tiny IR}}$. The $c$-function
monotonically interpolates between these fixed point values.

After the advent of AdS/CFT Maldacena:1997re , it was natural to seek a
holographic version of RG flows and construct holographic versions of
$c$-functions. Domain wall solutions in AdS (reviewed below) provide a simple
geometric $c$-function Freedman:1999gp . Alternatively, Casini and Huerta (CH)
proposed a $c$-function Casini:2004bw  based on entanglement entropy (EE) and
one of its indispensable properties, strong subadditivity. The CH $c$-function
is tailor-made for AdS/CFT since EE is generally hard to compute in QFTs but
simple to compute on the gravity side in terms of minimal Ryu:2006bv  or
extremal Hubeny:2007xt  surfaces.

The main purpose of our work is to (holographically) construct $c$-functions
for 2d field theories with conformal Carrollian fixed points in the UV and IR.
The primary motivation for pursuing this goal is a desire for a better
understanding of flat space holography and Carrollian CFTs. The main tools we
shall employ are flat space domain walls (which we construct and discuss in
detail) and flat space holographic EE Bagchi:2014iea ; Basu:2015evh .

This paper is organized as follows. In section II, we review AdS3/CFT2-aspects
pertinent to (holographic) $c$-functions. In section III, we summarize
BMS3/CCFT2 results required for our constructions. In section IV, we construct
domain walls in flat space, discuss their geometric properties, and propose a
flat space holographic $c$-function. In section V, we prove the three theorems
mentioned in the abstract. Section VI concludes with a tentative CH-inspired
proposal for a Carrollian $c$-function.

##  II AdS3/CFT2 review

This section reviews AdS3/CFT2-aspects pertinent to (holographic)
$c$-functions.

In section II.1, we provide the definition and main properties of the CH
$c$-function. In section II.2, we recall the relation to the 2d quantum null
energy condition (QNEC2). Section II.3 summarizes a specific class of domain
wall solutions in AdS3 as an example for a holographic model with non-trivial
CH $c$-functions.

###  II.1 Casini–Huerta $\boldsymbol{c}$-function

The CH $c$-function Casini:2004bw ; Casini:2006es

$c(\ell)=3\ell\,\frac{\operatorname{d}\\!S_{0}}{\operatorname{d}\\!\ell}$ (1)

is constructed from ground state EE $S_{0}$ and depends on the size $\ell$ of
the entangling region. By construction, it is monotonic

$\frac{\operatorname{d}\\!c(\ell)}{\operatorname{d}\\!\ell}\leq 0$ (2)

as a consequence of strong subadditivity.

When the inequality (2) is saturated, integrating it twice using (1) yields
the result for ground state EE in a CFT2 on the plane Holzhey:1994we ;
Calabrese:2004eu ,

$S_{0}=\frac{c}{3}\,\ln\frac{\ell}{\varepsilon}$ (3)

where $c$ is the UV fixed-point value of $c(\ell)$ and the integration
constant $\varepsilon$ is interpreted as UV cutoff.

The limit of vanishing entangling region yields the UV-value of the central
charge,

$c^{\textrm{\tiny UV}}=\lim_{\ell\to 0}c(\ell)\,.$ (4)

Similarly, in cases where the theory flows to a CFT2 fixed point in the IR its
central charge is obtained in the limit of infinite entangling region.

$c^{\textrm{\tiny IR}}=\lim_{\ell\to\infty}c(\ell)\leq c^{\textrm{\tiny UV}}$
(5)

###  II.2 Relation to Quantum Null Energy Condition

Inserting the definition (1) into the monotonicity condition (2) yields an
inequality for up to second derivatives of EE.

$0\geq\frac{\operatorname{d}\\!^{2}S_{0}}{\operatorname{d}\\!\ell^{2}}-\frac{1}{\ell}\,\frac{\operatorname{d}\\!S_{0}}{\operatorname{d}\\!\ell}+\frac{6}{c(\ell)}\bigg{(}\frac{\operatorname{d}\\!S_{0}}{\operatorname{d}\\!\ell}\bigg{)}^{2}\geq\frac{\operatorname{d}\\!^{2}S_{0}}{\operatorname{d}\\!\ell^{2}}-\frac{1}{\ell}\,\frac{\operatorname{d}\\!S_{0}}{\operatorname{d}\\!\ell}+\frac{6}{c^{\textrm{\tiny
UV}}}\bigg{(}\frac{\operatorname{d}\\!S_{0}}{\operatorname{d}\\!\ell}\bigg{)}^{2}$
(6)

The last expression has an interpretation in terms of variations of EE with
respect to null deformations of the interval and can be rewritten as

$0\geq\frac{\operatorname{d}\\!^{2}S_{0}}{\operatorname{d}\\!\lambda^{2}}+\frac{6}{c^{\textrm{\tiny
UV}}}\,\bigg{(}\frac{\operatorname{d}\\!S_{0}}{\operatorname{d}\\!\lambda}\bigg{)}^{2}$
(7)

where $\lambda$ is the deformation parameter (see section 2.5 in Ecker:2020gnw
). The combinations of derivatives (7) is the right-hand side of QNEC2
Bousso:2015mna ; Bousso:2015wca ; Koeller:2015qmn ; Balakrishnan:2017bjg

$2\pi\,\langle
T\rangle\geq\frac{\operatorname{d}\\!^{2}S}{\operatorname{d}\\!\lambda^{2}}+\frac{6}{c^{\textrm{\tiny
UV}}}\,\bigg{(}\frac{\operatorname{d}\\!S}{\operatorname{d}\\!\lambda}\bigg{)}^{2}$
(8)

for the ground state EE $S_{0}$, while the left-hand side of QNEC2 contains
the expectation value of the null projection of the stress-energy tensor,
denoted here by $T$. Since the Poincaré-invariant ground state has $\langle
T\rangle=0$, the CH inequality (2) implies QNEC2 for the ground state.

This relation between QNEC2 and the CH $c$-function can guide our proposal for
$c$-functions in non-Lorentzian QFTs, provided that quantum energy
inequalities are available. In the context of flat space holography, this is
indeed the case Grumiller:2019xna .

###  II.3 Domain walls in AdS3

Domain walls are a specific set of geometries describing a holographic RG flow
of a QFT2 from a UV to an IR CFT2-fixed point. The geometry dual to such a
flow has Poincaré invariant slices. In adapted coordinates

$\operatorname{d}\\!s^{2}=\operatorname{d}\\!\rho^{2}+e^{2A(\rho)}\,\big{(}-\operatorname{d}\\!t^{2}+\operatorname{d}\\!x^{2}\big{)}$
(9)

the function $A(\rho)$ characterizes the RG-flow. For any $\rho=\rm const.$ we
have Poincaré2-invariant slices, i.e., the metric (9) has the Killing vectors
$\partial_{t}$, $\partial_{x}$ and $x\partial_{t}+t\partial_{x}$. Each
$\rho=\rho_{0}=\rm const.$-slice thus induces a 2d flat-space metric
$\operatorname{d}\\!s^{2}_{(2)}=e^{2A(\rho_{0})}\,\big{(}-\operatorname{d}\\!t^{2}+\operatorname{d}\\!x^{2}\big{)}$.
There are infinitely many conformal Killing vectors for each such slice,
corresponding to the conformal symmetries generated in a CFT2. By convention,
the asymptotic region describing the UV is reached in the limit
$\rho\to\infty$.

Domain wall solutions arise, for instance, as solutions to Einstein-dilaton
gravity. The bulk action

$I=\frac{1}{16\pi
G_{N}}\int\operatorname{d}\\!^{3}x\sqrt{-g}\,\Big{(}R-\frac{1}{2}(\partial\phi)^{2}-V(\phi)\Big{)}$
(10)

with the potential (we use unit AdS-radius)

$V(\phi)=-2+\frac{1}{2}\,m^{2}\phi^{2}+\dots$ (11)

yields the equations of motion

$\displaystyle R_{\mu\nu}-\frac{1}{2}\,g_{\mu\nu}\,R$
$\displaystyle=\frac{1}{2}\,\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{4}\,(\partial\phi)^{2}g_{\mu\nu}-\frac{1}{2}\,V(\phi)\,g_{\mu\nu}$
(12a) $\displaystyle\nabla^{2}\Phi$ $\displaystyle=\frac{\partial
V(\phi)}{\partial\phi}$ (12b)

Rewriting the potential in terms of a superpotential $W$

$V(\phi)=-\frac{1}{2}\,W(\phi)^{2}+\frac{1}{2}\,W^{\prime}(\phi)^{2}$ (13)

reduces the equations of motion for domain wall solutions (9) to first order
equations

$\frac{\operatorname{d}\\!A(\rho)}{\operatorname{d}\\!\rho}=-\frac{1}{2}\,W(\phi(\rho))\qquad\qquad\frac{\operatorname{d}\\!\phi(\rho)}{\operatorname{d}\\!\rho}=\frac{\operatorname{d}\\!W(\phi(\rho))}{\operatorname{d}\\!\phi(\rho)}\,.$
(14)

An example is the superpotential

$W(\phi)=-2-\frac{1}{4}\,\phi^{2}-\frac{\alpha}{8}\,\phi^{4}$ (15)

corresponding to a mass $m^{2}=-\frac{3}{4}$. Integrating the equations (14)
for this superpotential yields the domain wall solution

$A(\rho)=\Big{(}1-\frac{1}{16\alpha}\Big{)}\,\rho-\frac{j^{2}}{16(e^{\rho}-\alpha
j^{2})}+\frac{\log\big{(}e^{\rho}-\alpha j^{2}\big{)}}{16\alpha}$ (16)

and the scalar field

$\phi(\rho)=\phi_{0}+\frac{je^{-\rho/2}}{\sqrt{1-\alpha j^{2}e^{-\rho}}}$ (17)

where $j$ and $\phi_{0}$ are integration constants.

For this example, the CH $c$-function was calculated for small $\ell$
Ecker:2020gnw

$c(\ell\ll 1)=c\,\Big{(}1-\frac{\pi\ell}{64}+{\cal O}(\ell^{2})\Big{)}$ (18)

and large $\ell$

$c(\ell\gg 1)=\frac{c}{1-\frac{1}{16\alpha}}+\dots$ (19)

assuming negative $\alpha$. Here $c=c^{\textrm{\tiny UV}}=3/(2G_{N})$ takes
the Brown–Henneaux value, and we have the relation

$c^{\textrm{\tiny IR}}=\frac{c^{\textrm{\tiny
UV}}}{1-\frac{1}{16\alpha}}<c^{\textrm{\tiny UV}}\,.$ (20)

Alternatively, there is a domain wall holographic $c$-function Freedman:1999gp

$c_{\textrm{\tiny dw}}(\rho)=\frac{c^{\textrm{\tiny UV}}}{A^{\prime}(\rho)}$
(21)

that does not require calculating EE but only uses the (derivative of the)
domain wall profile function $A(\rho)$ as input. Since
$\lim_{\rho\to\infty}A^{\prime}(\rho)=1$ and
$\lim_{\rho\to-\infty}A^{\prime}(\rho)=1-\frac{1}{16\alpha}$ we recover the
correct UV- and IR-values of the central charge. Moreover, $A^{\prime}(\rho)$
has the correct monotonicity for a $c$-function.

In our flat space construction below, we shall propose something analogous to
the domain wall $c$-function (21).

##  III BMS3/CCFT2 summary

In this section, we summarize BMS3/CCFT2 results required for our
constructions of flat space holographic $c$-functions in later sections.

In section III.1, we summarize gravity-aspects of BMS3-invariant QFTs, also
known as CCFT2. We collect corresponding field theory aspects in section
III.2. In section III.3, we state the quantum inequalities based on EE that
apply to these theories.

###  III.1 Gravity aspects of BMS3/CCFT2

We are interested in 2d QFTs invariant under CCFT2 symmetries generated by the
$\mathfrak{bms}_{3}$ algebra

$\displaystyle[{\tt L}_{n},{\tt L}_{m}]$ $\displaystyle=(n-m)\,{\tt
L}_{n+m}+\frac{c_{\textrm{L}}}{12}\,n(n^{2}-1)\,\delta_{n+m,\,0}$ (22a)
$\displaystyle[{\tt L}_{n},{\tt M}_{m}]$ $\displaystyle=(n-m)\,{\tt
M}_{n+m}+\frac{c_{\textrm{M}}}{12}\,n(n^{2}-1)\,\delta_{n+m,\,0}$ (22b)
$\displaystyle[{\tt M}_{n},{\tt M}_{m}]$ $\displaystyle=0\,.$ (22c)

The generators ${\tt L}_{n}$ yield a Virasoro subalgebra with central charge
$c_{\textrm{L}}$. They are sometimes referred to as “superrotations” in a
gravity context. The supertranslation generators ${\tt M}_{n}$ produce a
central charge $c_{\textrm{M}}$ in the mixed commutator.

The simplest gravity dual leading to (22) as asymptotic symmetries is Einstein
gravity with Barnich–Compère boundary conditions Barnich:2006av .

$\operatorname{d}\\!s^{2}={\cal
M}(\varphi)\,\operatorname{d}\\!u^{2}-2\operatorname{d}\\!u\operatorname{d}\\!r+r^{2}\,\operatorname{d}\\!\varphi^{2}\\\
+\big{(}2{\cal L}(\varphi)+u{\cal
M}^{\prime}(\varphi)\big{)}\,\operatorname{d}\\!u\operatorname{d}\\!\varphi+\dots$
(23)

The coordinate ranges are $u,r\in\mathbb{R}$ and either
$\varphi\sim\varphi+2\pi$ or $\varphi\in\mathbb{R}$. The ellipsis denotes
subleading terms in a large-$r$ expansion. The state-dependent functions $\cal
L,M$ appear as integrands in the boundary charges. Their (Fourier-) modes
generate the $\mathfrak{bms}_{3}$ algebra (22) as asymptotic symmetry algebra,
with central charges $c_{\textrm{L}}$, $c_{\textrm{M}}$ the values of which
depend on the gravity theory, see e.g. Bagchi:2012yk .

For Einstein gravity without cosmological constant, the Virasoro central
charge $c_{\textrm{L}}$ vanishes since there is no dimensionless coupling
constant, while the mixed central charge $c_{\textrm{M}}$ is non-zero
Barnich:2006av . We use a (standard) normalization of the generators ${\tt
M}_{n}$ where $c_{\textrm{M}}=3/G_{N}$.

The null orbifold (${\cal M}={\cal L}=0$) Cornalba:2002fi

$\operatorname{d}\\!s^{2}=-2\operatorname{d}\\!u\operatorname{d}\\!r+r^{2}\,\operatorname{d}\\!\varphi^{2}$
(24)

is dual to the ground state of the BMS3 invariant QFT in the same way that
Poincaré-patch AdS3 is the gravity dual of the ground state of a QFT on the
plane with a CFT2 fixed point in the UV. If $\varphi\sim\varphi+2\pi$, the
null orbifold has a singularity in the causal structure at $r=0$. For our
purposes, this is as irrelevant as the coordinate singularity in the Poincaré
patch horizon since we will construct domain wall solutions that only
asymptote to the null orbifold but do not exhibit its singular behavior in the
interior. Moreover, we can simply decompactify $\varphi$.

We later investigate what happens when the retarded time coordinate $u$ gets
rescaled by some factor $\lambda$ (absorbing such factors in the state-
dependent functions $\cal L,M$ when possible).

$\operatorname{d}\\!s^{2}={\cal
M}(\varphi)\,\operatorname{d}\\!u^{2}-2\lambda\,\operatorname{d}\\!u\operatorname{d}\\!r+r^{2}\,\operatorname{d}\\!\varphi^{2}\\\
+\big{(}2{\cal L}(\varphi)+u{\cal
M}^{\prime}(\varphi)\big{)}\,\operatorname{d}\\!u\operatorname{d}\\!\varphi+\dots$
(25)

The superrotation charges ${\tt L}_{n}$ are associated with asymptotic Killing
vectors $e^{in\varphi}\partial_{\varphi}$ and hence unaffected by a rescaling
of retarded time. By contrast, the supertranslation charges ${\tt M}_{n}$ are
associated with asymptotic Killing vectors $e^{in\varphi}\partial_{u}$ and
thus get rescaled by $1/\lambda$.

$[{\tt L}_{n},\,{\tt M}_{m}/\lambda]=(n-m)\,{\tt
M}_{n+m}/\lambda+\frac{c_{\textrm{M}}}{12}\,n(n^{2}-1)\,\delta_{n+m,\,0}$ (26)

Effectively, this rescales the BMS central charge by $\lambda$.

$[{\tt L}_{n},\,{\tt M}_{m}]=(n-m)\,{\tt
M}_{n+m}+\frac{\lambda\,c_{\textrm{M}}}{12}\,n(n^{2}-1)\,\delta_{n+m,\,0}$
(27)

This observation will be crucial for holographic RG flows modeled by flat
space domain walls, discussed in section IV.

###  III.2 Field theory aspects

We now summarize some aspects of CCFTs; see Bagchi:2019xfx ; deBoer:2023fnj
and refs. therein for more details.

We start by recalling the definition of the Carrollian conformal weights,
analogous to conformal weights:

${\tt
L}_{0}|h_{\textrm{L}},h_{\textrm{M}}\rangle=h_{\textrm{L}}|h_{\textrm{L}},\,h_{\textrm{M}}\rangle\qquad\;\,{\tt
M}_{0}|h_{\textrm{L}},h_{\textrm{M}}\rangle=h_{\textrm{M}}|h_{\textrm{L}},h_{\textrm{M}}\rangle$
(28)

While the interpretation of the Virasoro central charge $c_{\textrm{L}}$ in
the $\mathfrak{bms}_{3}$ algebra (22) is analogous to the corresponding CFT2
interpretation, the interpretation of the mixed central charge
$c_{\textrm{M}}$ is more subtle. At first glance, its precise value seems
irrelevant, since a change of basis ${\tt M}_{n}\to\lambda{\tt M}_{n}$ is an
automorphism of the $\mathfrak{bms}_{3}$ algebra upon rescaling
$c_{\textrm{M}}$. Nevertheless, there is a Cardy-like entropy formula
Barnich:2012xq ; Bagchi:2012xr  (see also Riegler:2014bia ; Fareghbal:2014oba
), which for $c_{\textrm{L}}=0$ (and $h_{\textrm{M}}>0$) reads

$S=2\pi\,h_{\textrm{L}}\,\sqrt{\frac{c_{\textrm{M}}}{24h_{\textrm{M}}}}\,.$
(29)

The reason there is no contradiction between the appearance of
$c_{\textrm{M}}$ in the entropy formula (29) and the fact that its value can
be rescaled to an arbitrary (positive) number is decisive to understanding our
RG flow results presented in later sections. The simple point is that whenever
$c_{\textrm{M}}$ appears in dimensionless ratios, there is a meaning to this
ratio. In the Cardy-like formula (29), the combination
$c_{\textrm{M}}/h_{\textrm{M}}$ is dimensionless. Therefore, this formula
makes sense. (Another way to come to the same conclusions is to note that both
$h_{\textrm{M}}$ and $c_{\textrm{M}}$ scale in the same way under the
automorphism ${\tt M}_{n}\to\lambda\,{\tt M}_{n}$, so that the entropy (29) is
invariant under it.) The lesson for later is that the value of
$c_{\textrm{M}}$ by itself is physically irrelevant, but dimensionless ratios
involving $c_{\textrm{M}}$ can be physically relevant.

We focus now on the main observable of interest, EE. It was calculated in
Bagchi:2014iea  for the null orbifold, the global flat space vacuum, and for
thermal states, and in Grumiller:2019xna  for any vacuum-like state, including
arbitrary BMS3-descendants of the vacuum and of thermal states. We shall need
only the result for the null orbifold.

$S_{\textrm{EE}}=S_{\textrm{L}}+S_{\textrm{M}}$ (30)

with

$S_{\textrm{L}}=\frac{c_{\textrm{L}}}{6}\log\frac{\Delta\varphi}{\epsilon_{\varphi}}\qquad\qquad
S_{\textrm{M}}=\frac{c_{\textrm{M}}}{6}\,\bigg{(}\frac{\Delta
u}{\Delta\varphi}-\frac{\epsilon_{u}}{\epsilon_{\varphi}}\bigg{)}$ (31)

where $\Delta\varphi$ ($\Delta u$) is the spatial (temporal) extent of the
entangling region, $\epsilon_{\varphi}$, $\epsilon_{u}$ are UV cut-offs, and
$c_{\textrm{L}}$, $c_{\textrm{M}}$ are the central charges of the
$\mathfrak{bms}_{3}$ algebra (22). The result (31) was confirmed in
holographic calculations Basu:2015evh ; Jiang:2017ecm ; Apolo:2020bld ;
Apolo:2020qjm . For $c_{\textrm{M}}=0$ (31) coincides with one chiral half of
ground state EE in a CFT2 (3). The UV cut-offs $\epsilon_{\varphi}$,
$\epsilon_{u}$ drop out in the quantum inequalities and our proposal for the
$c$-functions discussed below, so we do not discuss them further.

###  III.3 Quantum energy conditions in CCFT2

By analogy to AdS3/CFT2, our main interests are quantum energy conditions
Grumiller:2019xna . We define the expectation values

$2\pi\langle{\cal T}_{M}\rangle=\frac{c_{\textrm{M}}}{24}{\cal
M}\qquad\;2\pi\langle{\cal T}_{L}\rangle=\frac{c_{\textrm{L}}}{24}{\cal
M}+\frac{c_{\textrm{M}}}{24}\big{(}2{\cal L}+u{\cal M}^{\prime}\big{)}$ (32)

with conventional normalizations (primes denote $\varphi$-derivatives). The
quantum energy condition for theories with $c_{\textrm{M}}=0$

$2\pi\,\langle{\cal T}_{L}\rangle\geq
S_{L}^{\prime\prime}+\frac{6}{c_{\textrm{L}}}\,S^{\prime\,2}_{L}$ (33)

is a chiral half of the QNEC2 inequalities and essentially equivalent to (8).
However, we are more interested in the opposite case, when $c_{\textrm{L}}=0$
but $c_{\textrm{M}}\neq 0$. In that case, the quantum energy condition is
(dots denote $u$-derivatives)

$2\pi\,\langle{\cal
T}_{M}\rangle\geq\dot{S}_{M}^{\prime}+\frac{6}{c_{\textrm{M}}}\,\dot{S}^{2}_{M}\,.$
(34)

The quantum inequalities above inspire the CH-like proposal of $c$-functions
in CCFT2 at the end of our paper.

##  IV Domain walls in flat space

In this section, we construct domain wall solutions in 3d flat space,
intending to generate holographic RG-flows analogous to the ones discussed in
section II.3.

In section IV.1, we set the stage by deriving the possible geometries of flat
space domain walls. Section IV.2 focuses on domain wall solutions in flat
space Einstein-dilaton gravity. In section IV.3, we pick specific solutions
that allow a holographic RG flow interpretation.

###  IV.1 Geometric aspects of flat space domain walls

In AdS3, domain walls are constructed by requiring Poincaré2 invariance on
each slice; alternatively, we could have demanded that the conformal Killing
vectors of each 2d slice generate CFT2 symmetries. We follow the second
approach to construct flat space domain walls and demand that the degenerate
(Carrollian) induced metric has conformal Killing vectors that generate BMS3
symmetries.

Looking at the asymptotic expansion (23), it is suggestive to consider as
ansatz the degenerate (Carrollian) induced metric

$\operatorname{d}\\!s^{2}_{(2)}=g_{\mu\nu}^{(2)}\,\operatorname{d}\\!x^{\mu}\operatorname{d}\\!x^{\nu}=e^{2A(r)}\,\operatorname{d}\\!\varphi^{2}+0\cdot\operatorname{d}\\!u^{2}+0\cdot\operatorname{d}\\!u\operatorname{d}\\!\varphi\,.$
(35)

This ansatz ensures that even at finite values of the radial coordinate $r$,
all slices have the same features as in the asymptotic limit $r\to\infty$. The
function $A(r)$ is arbitrary at this stage. Since the induced metric (35) is
degenerate, we make sure not to use its inverse in any of our considerations.

To verify this ansatz, we solve the conformal Killing equation

$\xi^{\mu}\partial_{\mu}g_{\alpha\beta}^{(2)}+g_{\alpha\mu}^{(2)}\partial_{\beta}\xi^{\mu}+g_{\beta\mu}^{(2)}\partial_{\alpha}\xi^{\mu}=g_{\alpha\beta}^{(2)}\,\partial_{\mu}\xi^{\mu}$
(36)

for the vector field $\xi$ using the degenerate metric (35). The result

$\xi=\big{(}\xi_{M}(\varphi)+u\,\xi_{L}^{\prime}(\varphi)\big{)}\,\partial_{u}+\xi_{L}(\varphi)\,\partial_{\varphi}$
(37)

shows that the conformal Killing vectors (37) indeed generate centerless
$\mathfrak{bms}_{3}$ as Lie-bracket algebra (compare e.g. with Oblak:2016eij
).

$\displaystyle\big{[}\xi\big{(}\xi_{M}^{(1)},\,\xi_{L}^{(1)}\big{)},\,\xi\big{(}\xi_{M}^{(2)},\,\xi_{L}^{(2)}\big{)}\big{]}_{\textrm{\tiny
Lie}}=$ (38)
$\displaystyle\xi\big{(}\xi_{M}^{(1)}\xi_{L}^{(2)\,\prime}+\xi_{L}^{(1)}\xi_{M}^{(2)\,\prime}-\xi_{M}^{(2)}\xi_{L}^{(1)\,\prime}-\xi_{L}^{(2)}\xi_{M}^{(1)\,\prime},\,\xi_{L}^{(1)}\xi_{L}^{(2)\,\prime}-\xi_{L}^{(2)}\xi_{L}^{(1)\,\prime}\big{)}$

Therefore, $r=\rm const.$ slices in flat space domain walls only contain the
term $e^{2A(r)}\,\operatorname{d}\\!\varphi^{2}$.

The 3d metric describing flat space domain walls

$\boxed{\phantom{\Big{(}}\operatorname{d}\\!s^{2}=-e^{A(r)}\,2\operatorname{d}\\!u\operatorname{d}\\!r+e^{2A(r)}\,\operatorname{d}\\!\varphi^{2}\phantom{\Big{)}}}$
(39)

depends on one arbitrary function111One could add another function $B(r)$ in
the first term, but we have eliminated it by fixing the diffeomorphisms $r\to
f(r)$ suitably. The gauge choice (39) ensures that both metric coefficients
remain bounded and never change sign, provided the function $A(r)$ remains
bounded. of the radial coordinate, $A(r)$. The additional assumption implicit
in (39) is that we keep Eddington–Finkelstein gauge in the interior of the
bulk. This is analogous to keeping Gaussian normal coordinates in the bulk of
AdS3 domain walls (9).

Let us now address curvature invariants. The Ricci tensor has a single non-
zero component.

$R_{rr}=-A^{\prime\prime}$ (40)

Regardless of the choice of $A(r)$, all geometries (39) have vanishing scalar
curvature invariants and vanishing Cotton tensor. This means these geometries
are not only locally conformally flat, but it also implies we need some Page-
like curvature invariants Page:2008yb  if we want to characterize these
geometries.

An example of such an invariant is

$P=\frac{(R_{\mu\nu}k^{\mu}k^{\nu})^{2}}{\big{(}\nabla_{\mu}\nabla_{\nu}R_{\alpha\beta}\big{)}k^{\mu}k^{\nu}k^{\alpha}k^{\beta}}$
(41)

where $k^{\mu}$ is any vector with non-vanishing $r$-component.222In Page’s
construction, $k^{\mu}$ had to be null, and the scalar invariants were the
maximum and minimum with respect to changes of directions of $k^{\mu}$. In our
case, the Ricci tensor is so simple that the quantity $P$ is constant not only
under changes of direction but also under changes of the signature of
$k^{\mu}$ from light-like to time-like or space-like; the only requirement
that $k^{\mu}$ has to fulfill is that $R_{\mu\nu}k^{\mu}k^{\nu}$ does not
vanish unless $R_{\mu\nu}=0$, which in our coordinates implies $k^{r}\neq 0$.
However, in the following subsection, we identify an even simpler and more
useful scalar invariant, namely the matter scalar, so we will not employ (41).

###  IV.2 Flat space domain walls in Einstein-dilaton gravity

Above, we discussed the kinematics of flat space domain walls. Here, we focus
on the dynamics of these domain walls. As for AdS3, we consider Einstein-
dilaton gravity (10) with field equations (12). Remarkably, the field
equations hold for any choice of the function $A(r)$ in the flat space domain
wall (39) provided the scalar field potential vanishes, $V(\phi)=0$, and the
scalar field obeys the ordinary differential equation

$\frac{1}{2}\,\phi^{\prime\,2}=-A^{\prime\prime}\,.$ (42)

The geometric reason for this surprising result is that both
$(\partial\phi)^{2}$ and $\nabla^{2}\phi$ vanish on flat space domain wall
backgrounds (39) for any scalar field $\phi$ that is independent of $u$ and
$\varphi$.

This result implies that on-shell, the combination appearing in the Ricci
tensor (40) is related to (the derivative of) the scalar field. Thus, we can
use the scalar field $\phi$ as a scalar invariant that fully characterizes our
geometry. (The additive integration constant contained in $\phi$ does not play
any role for geometric properties and can be chosen conveniently; we fix it by
demanding $\lim_{r\to\infty}\phi(r)\to 0$.) Moreover, we can either provide
the function $A$ as input and determine $\phi$ by integrating once (42), or we
provide $\phi$ as input and determine $A$ by integrating twice (42).

Demanding compatibility with asymptotic flatness requires the expansion

$A(r\gg 1)=r-r_{0}+o(1)$ (43)

for the remaining function $A(r)$. Introducing the new radial coordinate
$\rho=e^{r-r_{0}}$ leads to the desired asymptotic expansion of the metric

$\operatorname{d}\\!s^{2}=-2\operatorname{d}\\!u\operatorname{d}\\!\rho+\rho^{2}\,\operatorname{d}\\!\varphi^{2}+\dots$
(44)

In the interior the bulk metric

$\operatorname{d}\\!s^{2}=-2e^{A(r)}\,\operatorname{d}\\!u\operatorname{d}\\!r+e^{2A(r)}\,\operatorname{d}\\!\varphi^{2}$
(45)

is free from singularities as long as the function $A(r)$ remains finite; in
particular, the null orbifold singularity at $r=0$ is absent since the factor
$e^{2A(r)}$ always is finite in the interior.

The asymptotic expansion for the scalar field compatible with (43) follows
from integrating the equations of motion (42). Since both the leading and the
first subleading terms in (43) drop out in $A^{\prime\prime}$, only the terms
that decay at $r\to\infty$ contribute to the scalar field.

$\phi(r\gg 1)=\phi_{0}+o(1)$ (46)

Without loss of generality, we set the integration constant to zero,
$\phi_{0}=0$. For example, if the sub-subleading term in $A$ scales like
$e^{-r}$, then the first term in the large-$r$ expansion of $\phi$ decays like
$e^{-r/2}$.

We highlight an important subtlety. Reality of the field configuration
requires the inequality

$A^{\prime\prime}\leq 0$ (47)

in the whole range of definition of the function $A$. Thus, when designing
flat space domain walls by choosing some function $A(r)$, it is crucial to
obey the concavity condition (47) for all values of the radial coordinate $r$.

If we slightly change the asymptotic behavior (43),

$A(r\gg
1)=\lambda^{-1}\,\big{(}r-r_{0}\big{)}+o(1)\qquad\lambda\in\mathbb{R}^{+}$
(48)

and use the radial coordinate $\rho=e^{(r-r_{0})/\lambda}$, the asymptotic
expansion of the metric

$\operatorname{d}\\!s^{2}=-2\lambda\,\operatorname{d}\\!u\operatorname{d}\\!\rho+\rho^{2}\,\operatorname{d}\\!\varphi^{2}+\dots$
(49)

shows that the first term is rescaled by $\lambda$. We will exploit this
property in the next subsection to establish holographic RG flows.

###  IV.3 Flat space holographic RG flow example

We are finally able to model holographic RG flows for BMS3 invariant QFTs.

One possibility to generate flat space domain walls is to take any AdS3 domain
wall for some potential, take the scalar field appearing in that solution as
input, and construct the function $A$ by integrating twice (42). Any such
choice generates a legitimate flat space domain wall; a subset of them
generates an associated holographic RG flow between BMS3 invariant UV and IR
fixed points. We will be more precise and general about this in section V. For
now, we focus on a specific example.

Figure 1: Scalar field (17) and bulk energy
$\frac{1}{2}\,\phi^{\prime}(r)^{2}$ for $\phi_{0}=0$, $j=54$ and $-\alpha
j^{2}=1$

In our first example, we pick the same scalar field as for AdS3 domain walls
(17) (renaming the radial $\rho$ into $r$). In Fig. 1, we plot the scalar
field $\phi(r)$ and the associated bulk energy
$\frac{1}{2}\,\phi^{\prime}(r)^{2}$ for the choices $\phi_{0}=0$, $j=54$ and
$-\alpha j^{2}=1$. The blue curve depicts the scalar field and has a clear
kink-like structure, interpolating between two different asymptotic values.
The orange curve shows that the bulk energy is localized in the interior of
the bulk. Its maximum is at $r=\ln 2$ [for general $\alpha,j$ the maximum is
at $r=\ln(-2\alpha j^{2})$].

Integrating twice (42) yields

$A(r)=A_{1}\,r+A_{0}-\frac{j^{2}}{16}\,\bigg{(}\frac{1}{e^{r}-\alpha
j^{2}}+\frac{r-\ln\big{(}e^{r}-\alpha j^{2}\big{)}}{\alpha j^{2}}\bigg{)}$
(50)

with two integration constants $A_{1}$, $A_{0}$. To obtain the desired
asymptotics (43) we fix $A_{1}=1$, yielding

$A(r\to\infty)=r+A_{0}-\frac{j^{2}}{8}\,e^{-r}+{\cal O}(e^{-2r})\,.$ (51)

Depending on the sign of $\alpha$, there are different possibilities. In the
AdS case, we needed negative $\alpha$ to generate domain walls with a CFT2
fixed point in the IR. We check now whether something analogous is true for
the corresponding flat space domain wall.

For negative $\alpha$, there is no singularity in $A(r)$ for any finite value
of $r$. Therefore, the coordinate range of this domain wall is
$(-\infty,\,\infty)$, and we obtain a second asymptotic region at
$r\to-\infty$. In this limit, the function (50) expands as

$A(r\to-\infty)=\Big{(}1-\frac{1}{16\alpha}\Big{)}\,r+A_{0}+\frac{1+\ln\big{(}-\alpha
j^{2}\big{)}}{16\alpha}+{\cal O}(e^{2r})\,.$ (52)

Denoting the UV central charge by $c_{\textrm{M}}^{\textrm{\tiny UV}}$,
comparison of the two asymptotic expansions (51) and (52) yields a result for
the IR central charge

$c_{\textrm{M}}^{\textrm{\tiny IR}}=\frac{c_{\textrm{M}}^{\textrm{\tiny
UV}}}{1-\frac{1}{16\alpha}}<c_{\textrm{M}}^{\textrm{\tiny UV}}$ (53)

according to the discussion at the end of section III.1. Note that the ratio
$c_{\textrm{M}}^{\textrm{\tiny IR}}/c_{\textrm{M}}^{\textrm{\tiny UV}}\leq 1$
is dimensionless, and hence equation (53) is meaningful [compare with the
discussion after the Cardy-like formula (29)].

Figure 2: Plot of $1/A^{\prime}(r)$ for $j=1$ and $-\alpha=\frac{1}{16}$ shows
it behaves like a $c$-function

As evident from the plot in Fig. 2, the function

$c_{\textrm{\tiny dw}}(r):=\frac{c_{\textrm{M}}^{\textrm{\tiny
UV}}}{A^{\prime}(r)}$ (54)

is a $c$-function for this domain wall solution since it approaches the
correct UV and IR values and is monotonically decreasing towards the IR.

The result (53) is precisely the same relation as for the corresponding
holographic RG flow in AdS3 (with the Virasoro central charge replaced by the
$\mathfrak{bms}_{3}$ central charge $c_{\textrm{M}}$), see (21). In the next
section, we shall prove that this is not a coincidence but a generic feature
relating AdS3 and flat space domain walls and their corresponding RG flow
interpretations.

Before generalizing our results, consider the case of positive $\alpha$. In
AdS3/CFT2 such “domain walls” do not model an RG flow from a UV to an IR fixed
point, but rather an RG flow from a UV fixed point to the IR, but without CFT2
fixed point in the IR. As we now show, something comparable happens in flat
space. Indeed, for positive $\alpha$ the IR boundary is at finite value of
$r$,

$r_{\textrm{\tiny IR}}=\ln\big{(}\alpha j^{2}\big{)}\,.$ (55)

The scalar field and the Ricci tensor are singular at the IR boundary, so in
this case, the flow ends at a naked singularity on the gravity side, and there
is no BMS3 field theory interpretation in the IR.

Finally, we consider the limiting case of vanishing $\alpha$. Here the
$c$-function tends to zero in the IR (which is again obtained in the limit
$r\to-\infty$); see Fig. 3. In this case, the IR fixed point is a trivial
BMS3-invariant QFT with vanishing central charge $c_{\textrm{M}}=0$. All these
features are analogous to corresponding AdS3/CFT2 features, see e.g. the
discussion in Ecker:2020gnw .

Figure 3: Plot of $1/A^{\prime}(r)$ for $j=1$ and $\alpha=0$ shows the
$c$-function vanishes in the IR

##  V Flat space holographic RG flow theorems

In this section, we state and prove three theorems for RG flows modeled by
domain wall solutions in 3d flat space Einstein-dilaton gravity.

In section V.1, we collect some definitions used in all theorems. In section
V.2, we state and prove a correspondence theorem, relating all AdS3 domain
walls to corresponding flat space domain walls. In section V.3, we state and
prove a monotonicity theorem, showing that bulk unitarity implies a
monotonically decreasing $c$-function. Putting together both theorems, we
prove a third one that shows the equivalence of the UV/IR ratios of Virasoro
and $\mathfrak{bms}_{3}$ central charges.

###  V.1 Definitions

In this whole section, we are solely concerned with holographic RG flows
generated by flat space domain wall solutions described in section IV. For
these domain walls, we found a $c$-function (54). While this definition was
based on a single example studied in section IV.3, it is natural to define
generically the flat space holographic $c$-function

$\boxed{\phantom{\Big{(}}c_{\textrm{\tiny
dw}}(r):=\frac{c_{\textrm{M}}^{\textrm{\tiny
UV}}}{A^{\prime}(r)}\qquad\quad\textrm{fixing\;}\lim_{r\to\infty}A(r)=r+{\cal
O}(1)\,.\phantom{\Big{)}}}$ (56)

This formally coincides with the AdS domain wall $c$-function discovered in
the seminal work Freedman:1999gp .

We define the term proper domain wall solution of AdS3-Einstein-dilaton
gravity to mean an exact solution of the equations of motion (12) with some
scalar potential of the form (11) such that in domain wall coordinates (9)
metric and scalar field have the following properties:

  1. 1.

for $\rho\to\infty$ the metric asymptotes to Poincaré patch AdS3 with unit
AdS-radius and the scalar field approaches zero

  2. 2.

for $\rho\to-\infty$ the metric asymptotes to Poincaré patch AdS3 with AdS-
radius smaller than one and the scalar field approaches a constant (that can
be zero)

  3. 3.

for finite values of $\rho$ the metric function $A(\rho)$ and the scalar field
$\phi(\rho)$ are bounded real functions; moreover, $A(\rho)$ is at least
$C^{2}$ and $\phi(\rho)$ at least $C^{1}$

Similarly, we define the term proper flat-space domain wall solution to mean
an exact solution of the equations of motion (12) with vanishing scalar
potential, $V(\phi)=0$, such that in flat space domain wall coordinates (39)
metric and scalar field have the following properties:

  1. 1.

for $r\to\infty$ the metric asymptotes to the null orbifold (44) and the
scalar field approaches zero

  2. 2.

for $r\to-\infty$ the metric asymptotes to the null orbifold, with a possible
rescaling of the first term as in (49) (with some positive $\lambda$) and the
scalar field approaches a constant (that can be zero)

  3. 3.

for finite values of $r$ the metric function $A(r)$ and the scalar field
$\phi(r)$ are bounded real functions; moreover, $A(r)$ is at least $C^{2}$ and
$\phi(r)$ at least $C^{1}$

By UV (IR), we always mean the limits $\rho,r\to\infty$ ($\rho,r\to-\infty$)
in the domain wall coordinates referred to above.

The Virasoro central charges appearing in domain wall solutions of
AdS3-Einstein-dilaton gravity are, therefore, denoted by $c^{\textrm{\tiny
UV}}$ at the UV boundary and by $c^{\textrm{\tiny IR}}$ at the IR boundary.
The Zamolodchikov $c$-theorem implies

$c^{\textrm{\tiny IR}}\leq c^{\textrm{\tiny UV}}\,.$ (57)

Finally, note that the definitions above imply that domain walls always
connect UV and IR fixed points, i.e., cases where we do not have an IR fixed
point, such as the one discussed at the end of section IV.3 (positive
$\alpha$), are excluded by our definitions of proper domain walls.

###  V.2 Correspondence theorem

Equipped with the definitions of section V.1, we can now formulate our first
theorem. It allows to translate any proper AdS3 domain wall solution into a
corresponding proper flat domain wall solution.

######  Theorem 1 (AdS3/flat space domain wall correspondence)

Given a proper domain wall solution of AdS3-Einstein-dilaton gravity, there is
a corresponding proper flat-space domain wall solution with the following
properties:

  * (I.)

In the UV, the flat space asymptotic symmetries generate a
$\mathfrak{bms}_{3}$ algebra with central charge
$c_{\textrm{M}}^{\textrm{\tiny UV}}=\frac{3}{G_{N}}$.

  * (II.)

In the IR, the flat space asymptotic symmetries generate a
$\mathfrak{bms}_{3}$ algebra with a central charge
$c_{\textrm{M}}^{\textrm{\tiny IR}}$ that in general differs from
$c_{\textrm{M}}^{\textrm{\tiny UV}}$.

Proof. Start with some scalar field $\phi(\rho)$ that generates a proper
domain wall solution of AdS3-Einstein-dilaton gravity and define $\phi(r)$ to
be the scalar field of the corresponding proper flat space domain wall. Since
by assumption, the AdS3 domain wall is proper, also the flat space domain wall
is proper, meaning there are no singularities at finite values of $r$.
Therefore, we need to consider only the UV and IR limits of the scalar field
and the metric. By definition we have the expansions $\phi(r\to\infty)=o(1)$
and $\phi(r\to-\infty)=\phi_{1}+o(1)$. Since $\phi$ is differentiable we have
$\phi^{\prime}(r\to\infty)=o(1/r)=\phi^{\prime}(r\to-\infty)$. Integrating the
equation of motion (42) yields $A(r\to\infty)=A_{1}r+A_{0}+o(1)$ and
$A(r\to-\infty)=A_{2}r+A_{3}+o(1)$. The quantities $A_{i}$ are integration
constants; only two of them can be chosen independently. Without loss of
generality, we fix $A_{1}=1$ and set $A_{0}=-r_{0}$, thereby recovering the
expansion (43). According to the discussion at the beginning of section III.1,
we then recover the $\mathfrak{bms}_{3}$ algebra as asymptotic symmetry
algebra in the UV with the usual central charge $c_{\textrm{M}}=3/G_{N}$.
Similarly, we recover a $\mathfrak{bms}_{3}$ algebra as asymptotic symmetry
algebra in the IR, but with a value of the central charge that depends on
$A_{2}$, according to the discussion in the second half of section III.1.
$\square$

The correspondence theorem 1 could be extended to non-proper domain walls
(those with no BMS3 fixed point in the IR and instead terminate in a naked
singularity), but we refrain from doing so.

In the final subsection, we prove two additional theorems and start by
addressing the paramount issue of monotonicity of the flat space domain wall
$c$-function.

###  V.3 Monotonicity theorem and central charge ratio equivalence

######  Theorem 2 (Monotonicity of $\boldsymbol{c}$-function)

The $c$-function associated with any flat space domain wall solution obtained
through the correspondence theorem 1 is a monotonically decreasing function
when flowing from the UV to the IR.

Proof. Since the scalar field is real, bounded and $C^{1}$ the metric function
$A(r)$ must obey the concavity inequality (47) for all values of $r$. Denoting
some fiducial radius as $r_{\textrm{\tiny UV}}$ and another, smaller, fiducial
radius as $r_{\textrm{\tiny IR}}<r_{\textrm{\tiny UV}}$, integrating the
concavity condition $A^{\prime\prime}(r)\leq 0$ from $r_{\textrm{\tiny IR}}$
to $r_{\textrm{\tiny UV}}$ implies

$A^{\prime}(r_{\textrm{\tiny UV}})\leq A^{\prime}(r_{\textrm{\tiny IR}})\,.$
(58)

Inserting this inequality into the definition of the $c$-function (54),

$c_{\textrm{\tiny dw}}(r_{\textrm{\tiny
UV}})=\frac{c_{\textrm{M}}^{\textrm{\tiny UV}}}{A^{\prime}(r_{\textrm{\tiny
UV}})}\geq\frac{c_{\textrm{M}}^{\textrm{\tiny
UV}}}{A^{\prime}(r_{\textrm{\tiny IR}})}=c_{\textrm{\tiny
dw}}(r_{\textrm{\tiny IR}})$ (59)

establishes that the $c$-function is a monotonically decreasing function when
flowing from the UV to the IR, $c_{\textrm{\tiny dw}}(r_{\textrm{\tiny
UV}})\geq c_{\textrm{\tiny dw}}(r_{\textrm{\tiny IR}})$. $\square$

The theorem 2 shows that (54) is, indeed, a BMS3 $c$-function for proper flat
space domain wall solutions. Note that one can consider theorem 2 to be a
consequence of bulk unitarity; indeed, if we drop the assumption of the scalar
field being real and allow for a purely imaginary scalar field, we can
circumvent theorem 2; the price for this is effectively a switched sign in the
kinetic term of the scalar field, which violates bulk unitarity.

We can be more quantitative and combine both theorems to show that the ratio
between IR and UV Virasoro central charges is equivalent to the corresponding
ratio of $\mathfrak{bms}_{3}$ central charges.

$1\leq\frac{c^{\textrm{\tiny UV}}}{c^{\textrm{\tiny
IR}}}=\frac{c_{\textrm{M}}^{\textrm{\tiny UV}}}{c_{\textrm{M}}^{\textrm{\tiny
IR}}}\geq 1$ (60)

The first inequality follows from Zamolodchikov’s $c$-theorem
Zamolodchikov:1986gt . The last inequality is the statement of theorem 2 that
we just proved. What remains to be shown is the equality in the middle. This
central charge ratio equivalence is guaranteed by the third theorem.

######  Theorem 3 (CFT/CCFT central charge ratio equivalence)

Given the assumptions of theorem 1, the ratio of UV/IR central charges obeys
the equality in (60).

Proof. For proper flat space domain walls, the UV/IR ratio of
$\mathfrak{bms}_{3}$ central charges is given by

$\frac{c_{\textrm{M}}^{\textrm{\tiny UV}}}{c_{\textrm{M}}^{\textrm{\tiny
IR}}}=\frac{A^{\prime}(r\to-\infty)}{A^{\prime}(r\to\infty)}=A_{2}\geq 1\,.$
(61)

The equalities follow from the proof of theorem 1 (and the discussion in
section III.1). The quantity $A_{2}$ was also defined in the proof of theorem
1. The inequality in (61) follows from theorem 2. For proper AdS3 domain
walls, the UV/IR ratio of Virasoro central charges is given by the ratio of
UV/IR AdS radii. The AdS radii follow from the UV and IR behavior of the
function $A(\rho)$ appearing in domain wall coordinates (9). Since by
assumption we set the AdS radius to unity in the UV, we must have the
expansion $A(\rho\to\infty)=\rho+\tilde{A}_{0}+o(1)$. Without loss of
generality, we equate $\tilde{A}_{0}=A_{0}$ by a constant shift of $\rho$. In
the IR we have the expansion
$A(\rho\to-\infty)=\tilde{A}_{2}\rho+\tilde{A}_{3}+o(1)$. Therefore, the UV/IR
ratio of Virasoro central charges is given by

$\frac{c^{\textrm{\tiny UV}}}{c^{\textrm{\tiny
IR}}}=\frac{A^{\prime}(\rho\to-\infty)}{A^{\prime}(\rho\to\infty)}=\tilde{A}_{2}\geq
1\,.$ (62)

What remains to be shown is $A_{2}=\tilde{A}_{2}$. Since $A(r)$ and $A(\rho)$
have the same leading and next-to-leading order terms in the UV, it is
sufficient to show that both of them obey the same second-order differential
equation $A^{\prime\prime}=-\frac{1}{2}\,(\phi^{\prime})^{2}$. For flat space
domain walls, this follows from (42). For AdS3 domain walls, this follows from
differentiating the left equation (14) with respect to $\rho$ and, using the
chain rule, insert the right equation (14) on the right-hand side of the left
equation, viz.,
$\operatorname{d}\\!^{2}A/\operatorname{d}\\!\rho^{2}=-\frac{1}{2}\,\operatorname{d}\\!W/\operatorname{d}\\!\phi\cdot\operatorname{d}\\!\phi/\operatorname{d}\\!\rho=-\frac{1}{2}\,(\operatorname{d}\\!\phi/\operatorname{d}\\!\rho)^{2}$.
Since $A(\rho)$ and $A(r)$ obey the same second-order differential equation
and have the same linear and constant terms in the UV, these two functions
must coincide for all values of the radial coordinates. This implies, in
particular, $\tilde{A}_{2}=A_{2}$. $\square$

In conclusion, the three theorems proven in this section provide Carrollian
$c$-functions (56) of domain wall solutions (39) to 3d Einstein-dilaton
gravity that describe flat space holographic RG flows from a Carrollian UV
fixed point to a Carrollian IR fixed point. Bulk unitarity guarantees the
monotonicity of our domain wall $c$-functions. Moreover, to every AdS3 domain
wall solution (reviewed in section II.3), there is a corresponding flat space
domain wall solution (discussed in section IV) with the same radial profile
and the same UV/IR-ratios of central charges. The principal difference is that
AdS3 domain walls require a scalar field potential for support, whereas flat
space domain walls demand vanishing potential.

The drawback of our $c$-functions (56) is that we need some bulk dual, which
may not always be available. Thus, it would be satisfying to have an intrinsic
construction for a Carrollian $c$-function without recourse to holography,
either along the lines of Zamolodchikov’s original design Zamolodchikov:1986gt
or the CH construction reviewed in sections II.1.

We tentatively follow the latter path in our final section, inspired by the
relation between the CH $c$-function and QNEC2 recapitulated in section II.2.
We are guided by the considerations of sections II.1, II.2, III.2, and III.3.

##  VI Tentative proposal for Casini–Huerta-inspired Carrollian
$\boldsymbol{c}$-function

Without further ado, here is our tentative proposal for the Carrollian
$c$-functions in CCFT2:

$\boxed{\phantom{\Big{(}}c_{\textrm{L}}(\Delta
u,\,\Delta\varphi):=6\Delta\varphi\,S_{L}^{\prime}\qquad c_{\textrm{M}}(\Delta
u,\,\Delta\varphi):=6\Delta\varphi\,\dot{S}_{M}\phantom{\Big{)}}}$ (63)

Prime denotes $\varphi$-derivatives and dot $u$-derivatives.

A sanity check that our proposal is not ruled out immediately is to consider
the special case $c_{\textrm{M}}=0$, $c_{\textrm{L}}\neq 0$ corresponding to a
chiral half of a CFT2. In this case, the identity

$\frac{1}{6\Delta\varphi}\,c_{\textrm{L}}^{\prime}=S_{L}^{\prime\prime}+\frac{6}{c_{\textrm{L}}}\,S^{\prime\,2}_{L}$
(64)

recovers the expected QNEC2 combination, see the discussion in section II.2.
Thus, for $c_{\textrm{M}}=0$, $c_{\textrm{L}}\neq 0$, we recover the CH
$c$-function for a chiral half of a CFT2.

Another consistency check is that our definitions are independent of the UV
cut-offs, as expected on physical grounds.

Finally, even in the more interesting case $c_{\textrm{L}}=0$,
$c_{\textrm{M}}\neq 0$, the $c$-function $c_{\textrm{M}}$ reproduces the
quantum energy combination of terms (34).

$\frac{1}{6\Delta\varphi}\,c_{\textrm{M}}^{\prime}=\dot{S}_{M}^{\prime}+\frac{6}{c_{\textrm{M}}}\,\dot{S}^{2}_{M}$
(65)

Thus, if the $c$-function $c_{\textrm{M}}$ is monotonic, it implies the
quantum energy condition (34) for the ground state.

The arguments above are neither proof of the quantum energy conditions nor
proof that $c_{\textrm{M}}$ is a $c$-function; they merely show that our
putative $c$-function in (63) is consistent with the CCFT2 quantum energy
conditions Grumiller:2019xna . We leave applications and scrutiny of our
proposal (63) to future work.

###### Acknowledgements.

We thank Luis Apolo, Arjun Bagchi, Rudranil Basu, Jacqueline Caminiti, Rob
Myers, and Wei Song for discussions on flat space holographic EE. This work
was supported by the Austrian Science Fund (FWF), projects P 32581, P 33789,
and P 36619. Some of our results were presented in March 2021 at the virtual
workshop “Flat Asymptotia” organized by Aritra Banerjee, Sudip Ghosh, Slava
Lysov, and Yasha Neiman. This research was supported in part by Perimeter
Institute for Theoretical Physics. Research at Perimeter Institute is
supported by the Government of Canada through the Department of Innovation,
Science and Economic Development and by the Province of Ontario through the
Ministry of Colleges and Universities. The final part of this research was
conducted while DG was visiting the Okinawa Institute of Science and
Technology (OIST) through the Theoretical Sciences Visiting Program (TSVP) in
July/August 2023.

## Note added

After posting our paper on the arXiv we were apprised that two of our main
results and one of our theorems were published in Fareghbal:2015bxd .

In particular, the flat space domain wall geometries (39) correspond to their
(4.11) upon redefining our radial coordinate as $\operatorname{d}\\!r\to
e^{-A(r)}\operatorname{d}\\!r$, the domain wall $c$-function (56) essentially
corresponds to their (4.16), and theorem 2 corresponds to their statement
after (4.17).

New results in our work not contained in Fareghbal:2015bxd  include the proof
that flat space domain walls are solutions to Einstein-dilaton gravity without
scalar field potential, a discussion of their curvature invariants, the flat
space holographic RG flow example, theorems 1 and 3, and the tentative
proposal for the CH $c$-function.

## References

  * (1) A. B. Zamolodchikov, “Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory,” JETP Lett. 43 (1986) 730–732.
  * (2) J. M. Maldacena, “The large $N$ limit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys. 2 (1998) 231–252, hep-th/9711200.
  * (3) D. Z. Freedman, S. S. Gubser, K. Pilch, and N. P. Warner, “Renormalization group flows from holography supersymmetry and a c theorem,” Adv. Theor. Math. Phys. 3 (1999) 363–417, hep-th/9904017.
  * (4) H. Casini and M. Huerta, “A Finite entanglement entropy and the c-theorem,” Phys. Lett. B 600 (2004) 142–150, hep-th/0405111.
  * (5) S. Ryu and T. Takayanagi, “Holographic derivation of entanglement entropy from AdS/CFT,” Phys.Rev.Lett. 96 (2006) 181602, hep-th/0603001.
  * (6) V. E. Hubeny, M. Rangamani, and T. Takayanagi, “A Covariant holographic entanglement entropy proposal,” JHEP 0707 (2007) 062, 0705.0016.
  * (7) A. Bagchi, R. Basu, D. Grumiller, and M. Riegler, “Entanglement entropy in Galilean conformal field theories and flat holography,” Phys.Rev.Lett. 114 (2015), no. 11, 111602, 1410.4089.
  * (8) R. Basu and M. Riegler, “Wilson Lines and Holographic Entanglement Entropy in Galilean Conformal Field Theories,” Phys. Rev. D 93 (2016), no. 4, 045003, 1511.08662.
  * (9) H. Casini and M. Huerta, “A c-theorem for the entanglement entropy,” J. Phys. A 40 (2007) 7031–7036, cond-mat/0610375.
  * (10) C. Holzhey, F. Larsen, and F. Wilczek, “Geometric and renormalized entropy in conformal field theory,” Nucl.Phys. B424 (1994) 443–467, hep-th/9403108.
  * (11) P. Calabrese and J. L. Cardy, “Entanglement entropy and quantum field theory,” J.Stat.Mech. 0406 (2004) P06002, hep-th/0405152.
  * (12) C. Ecker, D. Grumiller, H. Soltanpanahi, and P. Stanzer, “QNEC2 in deformed holographic CFTs,” JHEP 03 (2021) 213, 2007.10367.
  * (13) R. Bousso, Z. Fisher, S. Leichenauer, and A. C. Wall, “Quantum focusing conjecture,” Phys. Rev. D 93 (2016), no. 6, 064044, 1506.02669.
  * (14) R. Bousso, Z. Fisher, J. Koeller, S. Leichenauer, and A. C. Wall, “Proof of the Quantum Null Energy Condition,” Phys. Rev. D 93 (2016), no. 2, 024017, 1509.02542.
  * (15) J. Koeller and S. Leichenauer, “Holographic Proof of the Quantum Null Energy Condition,” Phys. Rev. D 94 (2016), no. 2, 024026, 1512.06109.
  * (16) S. Balakrishnan, T. Faulkner, Z. U. Khandker, and H. Wang, “A General Proof of the Quantum Null Energy Condition,” JHEP 09 (2019) 020, 1706.09432.
  * (17) D. Grumiller, P. Parekh, and M. Riegler, “Local quantum energy conditions in non-Lorentz-invariant quantum field theories,” Phys. Rev. Lett. 123 (2019), no. 12, 121602, 1907.06650.
  * (18) G. Barnich and G. Compère, “Classical central extension for asymptotic symmetries at null infinity in three spacetime dimensions,” Class.Quant.Grav. 24 (2007) F15–F23, gr-qc/0610130.
  * (19) A. Bagchi, S. Detournay, and D. Grumiller, “Flat-Space Chiral Gravity,” Phys.Rev.Lett. 109 (2012) 151301, 1208.1658.
  * (20) L. Cornalba and M. S. Costa, “A New cosmological scenario in string theory,” Phys.Rev. D66 (2002) 066001, hep-th/0203031.
  * (21) A. Bagchi, A. Mehra, and P. Nandi, “Field Theories with Conformal Carrollian Symmetry,” JHEP 05 (2019) 108, 1901.10147.
  * (22) J. de Boer, J. Hartong, N. Obers, W. Sybesma, and S. Vandoren, “Carroll stories,” 2307.06827.
  * (23) G. Barnich, “Entropy of three-dimensional asymptotically flat cosmological solutions,” JHEP 1210 (2012) 095, 1208.4371.
  * (24) A. Bagchi, S. Detournay, R. Fareghbal, and J. Simon, “Holography of 3d Flat Cosmological Horizons,” Phys. Rev. Lett. 110 (2013) 141302, 1208.4372.
  * (25) M. Riegler, “Flat space limit of higher-spin Cardy formula,” Phys.Rev. D91 (2015), no. 2, 024044, 1408.6931.
  * (26) R. Fareghbal and A. Naseh, “Rindler/Contracted-CFT Correspondence,” JHEP 1406 (2014) 134, 1404.3937.
  * (27) H. Jiang, W. Song, and Q. Wen, “Entanglement Entropy in Flat Holography,” JHEP 07 (2017) 142, 1706.07552.
  * (28) L. Apolo, H. Jiang, W. Song, and Y. Zhong, “Swing surfaces and holographic entanglement beyond AdS/CFT,” JHEP 12 (2020) 064, 2006.10740.
  * (29) L. Apolo, H. Jiang, W. Song, and Y. Zhong, “Modular Hamiltonians in flat holography and (W)AdS/WCFT,” JHEP 09 (2020) 033, 2006.10741.
  * (30) B. Oblak, BMS Particles in Three Dimensions.  PhD thesis, U. Brussels (ULB), 2016.  1610.08526.
  * (31) D. N. Page, “Nonvanishing Local Scalar Invariants Even in VSI Spacetimes with All Polynomial Curvature Scalar Invariants Vanishing,” Class. Quant. Grav. 26 (2009) 055016, 0806.2144.
  * (32) R. Fareghbal, A. Naseh, and S. Rouhani, “Aspects of Ultra-Relativistic Field Theories via Flat-space Holography,” Phys. Lett. B 771 (2017) 189–193, 1511.01774.