CALT-TH-2022-030

Diffeomorphism Symmetry in Two Dimensions and Celestial Holography

John H<EMAIL_ADDRESS>
_Walter Burke Institute for Theoretical Physics
California Institute of Technology 452-48
Pasadena, CA 91125, USA_

Abstract

> Two-dimensional diffeomorphism symmetry can be described by an operator
> algebra extension of the well-known Virasoro algebra description of
> conformal symmetry. Utilizing this extension, this note explains why the
> conformal symmetry that appears in celestial holography should not be
> extended to diffeomorphism symmetry, a possibility that several authors have
> proposed. The description of the two-dimensional diffeomorphism algebra
> presented here might be useful for other purposes.

##  1 Introduction

Unlike in higher dimensions, the conformal group in two dimensions is infinite
dimensional. In the case of Euclidean signature, which is the main concern in
this work, the generators consist of the product of a holomorphic Virasoro
algebra and its complex conjugate antiholomorphic algebra. An enormous amount
of work has taken place over the past 50 years studying 2d conformal field
theories. It is a rich subject with many important applications.

Two-dimensional Euclidean space can be parametrized by a complex coordinate
$z$ and its complex conjugate $\bar{z}$. By means of stereographic projection,
these coordinates can also be used to parametrize a sphere. By definition, a
conformal operator $\Phi(z,\bar{z})$ has dimensions $(h,\bar{h})$ provided
that $\Phi(z,\bar{z})(dz)^{h}(d\bar{z})^{\bar{h}}$ is invariant under
conformal transformations. It is customary to define dimension
$\Delta=h+\bar{h}$ and spin $s=h-\bar{h}$. The mode expansion of such an
operator is

$\Phi(z,\bar{z})=\sum_{m,n}\frac{\Phi_{m,n}}{z^{m+h}\,{\bar{z}}^{n+\bar{h}}}.$
(1)

The range of $m$ and $n$ are such that $m+h$ and $n+\bar{h}$ take all integer
values.

Conformal transformations are generated by the operators

$T(z)=\sum_{m\in{\mathbbm{Z}}}\frac{L_{m}}{z^{m+2}}{\quad{\rm
and}\quad}\overline{T}(\bar{z})=\sum_{m\in{\mathbbm{Z}}}\frac{{\overline{L}}_{m}}{{\bar{z}}^{m+2}}.$
(2)

The modes satisfy the Virasoro algebra

$[L_{m},L_{n}]=(m-n)L_{m+n},\quad[{\overline{L}}_{m},{\overline{L}}_{n}]=(m-n){\overline{L}}_{m+n},\quad[L_{m},{\overline{L}}_{n}]=0.$
(3)

No central extension is required.222See [1] for the possible role of a central
extension in the context of celestial holography. Conformal transformations of
an operator $\Phi(z,\bar{z})$ having dimensions $(h,\bar{h})$ are generated by
the Virasoro operators

$[L_{k},\Phi_{m,n}]=((h-1)k-m)\Phi_{m+k,n},$ (4)
$[{\overline{L}}_{l},\Phi_{m,n}]=((\bar{h}-1)l-n)\Phi_{m,n+l}.$ (5)

These commutation relations correspond to the operator-product formulas (as
$z\to w$)

$T(z)\Phi(w,\bar{w})\sim
h\frac{\Phi(w,\bar{w})}{(z-w)^{2}}+\frac{\partial\Phi(w,\bar{w})}{z-w},$ (6)
$\overline{T}(\bar{z})\Phi(w,\bar{w})\sim\bar{h}\frac{\Phi(w,\bar{w})}{(\bar{z}-\bar{w})^{2}}+\frac{\bar{\partial}\Phi(w,\bar{w})}{\bar{z}-\bar{w}},$
(7)

where $\partial=\partial/\partial w$ and
$\bar{\partial}=\partial/\partial\bar{w}$.

Conformal algebras can be described either by commutation relations for modes
of operators and or by the identification of the singular terms in operator
product expansions (OPEs), as we have just demonstrated. The latter is more
succinct and elegant, and therefore it is usually preferred. One advantage of
considering the commutation relations of modes is that this makes it easier to
check consistency with Jacobi identities of the form
$[[A,B],C]+[[B,C],A]+[[C,A],B]=0$. There are cases where checking this
consistency is important, as will become clear later. These Jacobi identities
correspond to associativity of the operator algebra, which can be investigated
by introducing appropriate contour integrals.

One of the important precursors of celestial holography was the famous BMS
analysis in the early 1960s [2][3][4]. These authors analyzed the symmetries
of general relativity in asymptotically flat spacetime (AFS) geometries. They
concluded that the usual translation generators, i.e., momenta, are
supplemented by an infinite set of additional ones, which they called
supertranslations. Similarly, the Lorentz generators acquire an infinite
extension, which are called superrotations. These conclusions were reached by
analyzing the asymptotic behavior of the spacetime metric in appropriately
chosen coordinates, subject to assumptions about the asymptotic falloff of
various functions that appear in the analysis.

Eventually it was realized that the BMS algebra should be extended to include
the entire infinite-dimensional Virasoro algebra [5][6][7]. This implies that
the Lorentz symmetry together with the superrotations can be interpreted as
the infinite-dimensional conformal symmetry of the two-dimensional celestial
sphere.333Other possibilities considered in the literature include various
Carrollian groups. See [8] – [10], for example. Furthermore, there is
compelling evidence for the existence of a dual description of gravitational
physics in asymptotically flat four-dimensional spacetime in terms of a 2d
conformal field theory on the celestial sphere. Much of the pioneering work
has been done by Strominger, who has written a book on the subject [11]. There
are also a number of more recent reviews including [12]–[16].

A basic premise of celestial holography is that bulk fields in four-
dimensional asymptotically flat spacetime map to operators on the celestial
sphere. Furthermore, spacetime scattering amplitudes map to correlation
functions of these operators on the celestial sphere. Operator product
expansions of these operators on the celestial sphere play a central role in
the current research. They correspond to collinear limits of the bulk fields.

In recent years, various authors have revisited the BMS analysis. By making
slight changes to the assumptions some of them have been led to conclude that
the asymptotic symmetry is larger than the conformal group. Some of these
studies are [17]–[23]. In most cases they appear to conclude that the
conformal symmetry of celestial holography, Conf($S^{2}$), should be extended
to diffeomorphism symmetry, Diff($S^{2}$). It is difficult to decide which
conclusion is correct within the general relativity framework that is used in
these analyses. Some works, such as [24][25], are careful to distinguish
between asymptotic spacetime symmetries and symmetries of the celestial CFT,
even though they must be closely related. For previous studies of
Diff($S^{2}$) and related algebras see [26].

The goal of this paper is to answer this question by using considerations that
do not refer explicitly to a spacetime metric. Specifically, the plan is to
explore what is involved in extending the conformal symmetry algebra of a 2d
Euclidean conformal field theory to diffeomorphism symmetry and what its
physical implications are in the context of celestial holography. We will
conclude that the conformal symmetry in celestial holography should not be
extended to diffeomorphism symmetry.

##  2 The diffeomorphism algebra

The key fact is that the conformal symmetry generator $T(z)$ describes a
subset of the modes of a diffeomorphism generator $J(z,\bar{z})$. The complex
conjugate statement is that $\overline{T}(\bar{z})$ describes a subset of the
modes of $\overline{J}(z,\bar{z})$. The mode expansions of these operators
have the form given previously for operators of dimension $(2,1)$ and $(1,2)$,
respectively. Thus,

$J(z,\bar{z})=\sum_{mn}\frac{J_{m,n}}{z^{m+2}\,{\bar{z}}^{n+1}}{\quad{\rm
and}\quad}\overline{J}(z,\bar{z})=\sum_{mn}\frac{{\overline{J}}_{m,n}}{z^{m+1}\,{\bar{z}}^{n+2}}\quad
m,n\in{\mathbbm{Z}}.$ (8)

However, as we will see, this statement does not adequately describe their
algebra. The embedding of $T$ and $\overline{T}$ is given by

$J_{m,0}=L_{m}{\quad{\rm and}\quad}{\overline{J}}_{0,n}={\overline{L}}_{n}.$
(9)

Aside from a possible central extension, the natural extension of the
conformal symmetry algebra to the diffeomorphism algebra is

$[J_{k,l},J_{m,n}]=(k-m)J_{k+m,l+n}$ (10)
$[{\overline{J}}_{k,l},{\overline{J}}_{m,n}]=(l-n){\overline{J}}_{k+m,l+n}$
(11) $[J_{k,l},{\overline{J}}_{m,n}]=lJ_{k+m,l+n}-m{\overline{J}}_{k+m,l+n}.$
(12)

The coefficients have been determined by setting various indices to zero and
implementing the correct conformal subalgebra. As we will show, these terms
are correct, but another term involving a new operator needs to be added.

The fact that this is essentially the correct algebra for the generators of
diffeomorphisms, also known as general coordinate transformations, is shown by
representing them by differential operators

$J_{k,l}\sim z^{1-k}{\bar{z}}^{-l}\frac{\partial}{\partial z}\quad{\rm
and}\quad{\overline{J}}_{m,n}\sim
z^{-m}{\bar{z}}^{1-n}\frac{\partial}{\partial\bar{z}}.$ (13)

Operators of dimension $(h,\bar{h})$ satisfy the same algebra as conjugate
operators of dimensions $(1-h,1-\bar{h})$. (The conjugate operator is called
the “shadow transform.”) That is why the $(2,1)$ and $(1,2)$ operators $J$ and
$\overline{J}$ satisfy the same algebra as the $(-1,0)$ and $(0,-1)$
differential operators that transform the coordinates $z$ and $\bar{z}$. It is
also why we used the $\sim$ symbol here. These differential operators will not
be used in the subsequent analysis.

If $\Phi$ is an operator with dimensions $(h,\bar{h})$, but not $J$ or
$\overline{J}$, diffeomorphisms are encoded in the commutation relations of
its modes

$[J_{k,l},\Phi_{m,n}]=((h-1)k-m)\Phi_{k+m,l+n}$ (14)
$[{\overline{J}}_{k,l},\Phi_{m,n}]=((\bar{h}-1)l-n)\Phi_{k+m,l+n}.$ (15)

These formulas play a crucial role in the subsequent analysis, so we should
say more about them. The first formula is certainly correct for $l=0$, since
$J_{k,0}=L_{k}$, so any modification should be proportional to $l$. Similarly,
any modification of the second formula should be proportional to $k$. So let
us be safe and (temporarily) write

$[J_{k,l},\Phi_{m,n}]=((h-1)k-m+\alpha l)\Phi_{k+m,l+n}$ (16)
$[{\overline{J}}_{k,l},\Phi_{m,n}]=((\bar{h}-1)l-n+\beta k)\Phi_{k+m,l+n},$
(17)

where $\alpha$ and $\beta$ are allowed to depend on $h$ and $\bar{h}$.

Diffeomorphism invariance is not sufficient by itself to describe theories
containing operators with nonzero conformal spin. To demonstrate the issue let
us examine the Jacobi identity involving $J$, $\overline{J}$, and an operator
$\Phi$ that has dimensions $(h,\bar{h})$. (There is no problem with the
$JJ\Phi$ and $\overline{J}\overline{J}\Phi$ Jacobi identities.) All of the
requisite commutators have been presented. Using them we find that

$[[J_{k,l},{\overline{J}}_{m,n}],\Phi_{p,q}]+[[{\overline{J}}_{m,n},\Phi_{p,q}],J_{k,l}]+[[\Phi_{p,q},J_{k,l}],{\overline{J}}_{m,n}]$
$=[(h-\bar{h})lm+\alpha ln-\beta km]\,\Phi_{k+m+p,l+n+q},$ (18)

which shows that the algebra is inconsistent as it stands. The only way to
rectify it is to set $\alpha=\beta=0$ and to add an additional term to the
$[J,\overline{J}]$ equation.

The appearance of the spin of $\Phi$, $h-\bar{h}$, suggests the need to
incorporate a spin operator in the formulas. For this purpose let us introduce
a dimension $(1,1)$ spin operator $S(z,\bar{z})$, with the commutation
relation

$[S_{k,l},\Phi_{m,n}]=(h-\bar{h})\Phi_{k+m,l+n}.$ (19)

Then we add another term to eq. (12) giving

$[J_{k,l},{\overline{J}}_{m,n}]=lJ_{k+m,l+n}-m{\overline{J}}_{k+m,l+n}-lmS_{k+m,l+n},$
(20)

which repairs the Jacobi identity. Additional Jacobi identities require that

$[J_{k,l},S_{m,n}]=-mS_{k+m,l+n},$ (21)
$[{\overline{J}}_{k,l},S_{m,n}]=-nS_{k+m,l+n},$ (22) $[S_{k,l},S_{m,n}]=0.$
(23)

Note that $J$ and $\overline{J}$ are special and do not follow the general
rule for operators of dimension $(2,1)$ and $(1,2)$. In particular, eq. (19)
does not apply to them. Instead, they transform $S$ in accordance with their
general rule in eqs. (14) and (15). This conclusion was reached by studying
Jacobi identities.

We can now convert the various commutation relations of modes into equivalent
operator product expansions. Typically what happens when one evaluates a
commutator of two operators, such as $[A(z,\bar{z}),B(w,\bar{w})]$ by
inserting mode expansions for each of them is that one encounters ill-defined
series of the form $\sum_{m}w^{m}/z^{m+1}$ times some operator $C(w)$. The
ill-defined series, as well as its derivatives and complex conjugate, need to
be interpreted. These series determine the poles in the corresponding OPE,
which carry all of the relevant information. A convenient mnemonic that allows
one to convert between commutators of modes and singular OPEs in either
direction uses the correspondence

$\sum_{m}\frac{w^{m}}{z^{m+1}}\leftrightarrow\frac{1}{z-w},$ (24)

as well as derivatives and complex conjugates of this correspondence. This
rule reproduces the results obtained by the contour integral methods mentioned
earlier.

We are now ready to re-express the complete algebra of the $J$,
$\overline{J}$, and $S$ symmetry operators, combining diffeomorphism symmetry
with the spin operator, as well as their action on an arbitrary conformal
operator $\Phi$ of dimensions $(h,\bar{h})$. The OPEs of various $\Phi$
operators among themselves addresses the dynamics of specific theories and
(with one exception that will appear later) is beyond the scope of this note.

Using the procedure sketched above to deduce the OPEs from commutation
relations, we find the following444In order to save space, we do not display
the $w$ and $\bar{w}$ dependence of the operators appearing on the right-hand
side of the OPEs.

$S(z,\bar{z})S(w,\bar{w})\sim 0.$ (25)
$J(z,\bar{z})S(w,\bar{w})\sim\frac{1}{|z-w|^{2}}\left(\frac{S}{z-w}+\partial
S\right)$ (26)
$\overline{J}(z,\bar{z})S(w,\bar{w})\sim\frac{1}{|z-w|^{2}}\left(\frac{S}{\bar{z}-\bar{w}}+\bar{\partial}S\right)$
(27)
$J(z,\bar{z})J(w,\bar{w})\sim\frac{1}{|z-w|^{2}}\left(\frac{2J}{z-w}+\partial
J\right)$ (28)
$\overline{J}(z,\bar{z})\overline{J}(w,\bar{w})\sim\frac{1}{|z-w|^{2}}\left(\frac{2\overline{J}}{\bar{z}-\bar{w}}+\bar{\partial}\overline{J}\right)$
(29)
$J(z,\bar{z})\overline{J}(w,\bar{w})\sim\frac{1}{|z-w|^{2}}\left(\frac{S}{|z-w|^{2}}+\frac{\partial
S+J}{\bar{z}-\bar{w}}+\frac{\overline{J}}{z-w}+\partial\overline{J}\right).$
(30)
$\overline{J}(z,\bar{z})J(w,\bar{w})\sim\frac{1}{|z-w|^{2}}\left(\frac{S}{|z-w|^{2}}+\frac{\bar{\partial}S+\overline{J}}{z-w}+\frac{J}{\bar{z}-\bar{w}}+\bar{\partial}J\right).$
(31)

Turning now to the action of these operators on $\Phi$, we find

$S(z,\bar{z})\Phi(w,\bar{w})\sim\frac{(h-\bar{h})\Phi}{|z-w|^{2}}$ (32)
$J(z,\bar{z})\Phi(w,\bar{w})\sim\frac{1}{|z-w|^{2}}\left(\frac{h\Phi}{z-w}+\partial\Phi\right)$
(33)
$\overline{J}(z,\bar{z})\Phi(w,\bar{w})\sim\frac{1}{|z-w|^{2}}\left(\frac{\bar{h}\Phi}{\bar{z}-\bar{w}}+\bar{\partial}\Phi\right).$
(34)

##  3 Implications for Celestial Holography

The preceding analysis of diffeomorphism symmetry has an important implication
for celestial holography. In celestial holography the helicity of a massless
particle in four-dimensional spacetime is represented by the two-dimensional
conformal spin of the corresponding operator on the celestial sphere. Thus,
since the assumption of two-dimensional diffeomorphism symmetry implies
conservation of conformal spin, the holographic implication is conservation of
four-dimensional helicity in the interactions of massless particles such as
gravitons and photons. This is definitely not what we want. Thus, we are
forced to conclude that the conformal symmetry of the holographic theory on
the celestial sphere must not be extended to diffeomorphism symmetry.

To make perfectly clear what we are talking about, let us consider a specific
example – the OPE of two positive-helicity graviton operators derived in [27]:

$G^{+}_{\Delta_{1}}(z_{1},{\bar{z}}_{1})G^{+}_{\Delta_{2}}(z_{2},{\bar{z}}_{2})\sim
E_{+}(\Delta_{1},\Delta_{2})\frac{{\bar{z}}_{12}}{z_{12}}G^{+}_{\Delta_{1}+\Delta_{2}}(z_{2},{\bar{z}}_{2}),$
(35)

where $z_{12}=z_{1}-z_{2}$ and

$E_{+}(\Delta_{1},\Delta_{2})=-\frac{\kappa}{2}B(\Delta_{1}-1,\Delta_{2}-1).$
(36)

The left-hand side of the OPE has conformal spin $s=2+2=4$ and the right-hand
side has conformal spin $s=2$. Thus, conformal spin is not conserved. As we
have explained, this implies that diffeomorphism symmetry is violated as well.

Celestial holography may require all sorts of symmetries in addition to
conformal symmetry, such as supertranslations and a $w_{1+\infty}$ loop
algebra [28], but Diff($S^{2}$) is not one of them. There are classical field
theories that do conserve helicity, such as theories of scalars only or
Born–Infeld theory [29], not coupled to gravity, but they are unlikely to be
fully consistent as quantum theories. Even if there were consistent helicity-
conserving quantum theories, they would not be expected to have a dual
celestial realization with infinite-dimensional conformal symmetry, let alone
diffeomorphism symmetry, since they do not contain gravity.

It may be interesting to make precise the implications of this conclusion for
the asymptotic metric analysis of GR in asymptotically flat spacetime.
Although diffeomorphism symmetry is not appropriate for celestial holography,
the description of the two-dimensional diffeomorphism algebra presented here
might be useful for other purposes.

## Acknowledgments

I am grateful to Clifford Cheung for reading a draft of this manuscript and
making helpful comments. This work was supported in part by the Walter Burke
Institute for Theoretical Physics at Caltech and by U.S. DOE grant DE-
SC0011632. It was performed in part at the Aspen Center for Physics, which is
supported by NSF grant PHY-1607611.

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