aainstitutetext: Université Libre de Bruxelles and International Solvay
Institutes, ULB-Campus Plaine CP231, B-1050 Brussels, Belgiumbbinstitutetext:
Collège de France, Université PSL, 11 place Marcelin Berthelot, 75005 Paris,
France

# Simplifying (super-)BMS algebras

Oscar Fuentealba a,b Marc Henneaux<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We show that the non-linear BMS5 symmetry algebra of asymptotically flat
Einstein gravity in five dimensions, as well as the super-BMS4 superalgebra of
asymptotically flat supergravity, can be redefined so as to take a direct sum
structure. In the new presentation of the (super-)algebra, angle-dependent
translations and angle-dependent supersymmetry transformations commute with
the (super-)Poincaré generators. We also explain in detail the structure and
charge-integrability of asymptotic symmetries with symmetry parameters
depending on the fields (through the charges themselves), a topic relevant for
nonlinear asymptotic symmetry algebras.

##  1 Introduction

In a recent paper Fuentealba:2022xsz , we showed how pure BMS4
supertranslations Bondi:1962px ; Sachs:1962wk ; Sachs:1962zza  could be made
to commute with all Poincaré transformations by introducing logarithmic
supertranslations and allowing non-linear redefinitions of the homogeneous
Lorentz charge-generators. In retrospect, this is perhaps not too surprizing
since Poincaré transformations and pure supertranslations have a different
origin when one traces them back to the linear theory Fuentealba:2020ghw :
while the Poincaré transformations are rigid symmetries of the linear theory,
with a non-trivial bulk contribution to their charge-generator, the pure
supertranslations appear already there as “improper Benguria:1976in  gauge
transformations”. This decoupling of the symmetries in the algebra, which
becomes a direct sum, is also in the spirit of the Coleman-Mandula theorem
Coleman:1967ad .

The redefinitions of the generators of the BMS4 algebra shed furthermore new
light on what is sometimes called the “angular momentum ambiguity” since they
provide a supertranslation independent definition of the Poincaré charges. It
was proved in Fuentealba:2023syb  that the redefinitions of Fuentealba:2022xsz
made at spatial infinity and those of Mirbabayi:2016axw ; Bousso:2017dny ;
Javadinezhad:2018urv ; Javadinezhad:2022hhl ; Chen:2021szm ; Chen:2021zmu ;
Compere:2021inq ; Compere:2023qoa  carried out at null infinity were
equivalent.

What makes the algebraic decoupling possible is that the generators of the
logarithmic supertranslations are canonically conjugate to those of the pure
supertranslations, allowing to eliminate by a Darboux-like procedure cross-
terms in the brackets between the generators of the pure supertranslations
with the other generators of the BMS4 algebra. The charges generating
logarithmic supertranslations were shown in Fuentealba:2023syb  to match the
null infinity potential $C$ for the electric part of the Bondi shear,
identified in Strominger:2013jfa ; Strominger:2017zoo  to be the Goldstone
boson of spontaneously broken supertranslation invariance. The importance of
the logarithmic branch to describe complete canonical pairs including the
Goldstone boson was derived in a different context, at null infinity, in
Donnay:2018neh .

The purpose of this article is to extend the construction of
Fuentealba:2022xsz  to BMS algebras in higher dimensions – specifically in the
explicit case of 5 dimensions Fuentealba:2021yvo ; Fuentealba:2022yqt  – as
well as to the BMS4 superalgebra of Fuentealba:2021xhn . What makes the
extension and the algebraic decoupling possible is that in these algebras, the
generators of pure supertranslations or pure angle-dependent supersymmetry
transformations have also canonical partners, just as in the case of the log-
BMS4 algebra. These conjugate variables can be used to perform the appropriate
redefinitions of the (super-)Poincaré generators, again along Darboux-like
lines. These redefinitions are nonlinear. For that reason, we also explain in
detail the structure of the corresponding nonlinear charges and the asymptotic
symmetry transformations that they generate.

Our paper is organized as follows. In Section 2, we review the definition of
asymptotic symmetries and recall how nonlinear redefinitions of the charges
fit into the construction in a manner that preserves the general local
structure (“weakly vanishing bulk term $+$ surface integral at infinity”) of
the generators of asymptotic symmetries. We consider next in Section 3 the
BMS5 algebra and show that one can algebraically decouple the Poincaré
subalgebra from the pure supertranslations and their subleading nontrivial
partners. In Section 4, we turn to supersymmetry and consider first the
superalgebra of Fuentealba:2021xhn , where we show how to decouple the super-
BMS generators in the superalgebra. By introducing the logarithmic
supertranslations in the superalgebra of Fuentealba:2021xhn , we define the
log-BMS4 superalgebra in Section 5 and achieve a complete algebraic
decoupling. We show that the log-BMS4 superalgebra can indeed be redefined so
as to take the direct sum structure of the finite-dimensional super-Poincaré
algebra with the infinite-dimensional superalgebra containing the pure BMS
supertranslations, the logarithmic supertranslations and their fermionic
partners. We close our paper with some comments on generalizations (Section 6)
and a technical appendix.

Finally, a note on notations: in the pure bosonic case, Poisson brackets will
be denoted by $\\{,\\}$. When we deal with supersymmetric theories, however,
we shall use a mixed notation ($[,]$ for the Poisson bracket between a bosonic
variable and a bosonic or fermionic one, $\\{,\\}$ for the Poisson bracket
between two fermionic variables) to emphasize the symmetry properties.

##  2 Asymptotic symmetries and nonlinear redefinitions

###  2.1 Asymptotic symmetries

Asymptotic symmetries are gauge transformations that preserve the boundary
conditions and the action. They are thus canonical transformations with well-
defined generators, which we call equivalently “charge-generators”,
“generators” or “charges”. Denoting the (first class) constraints by
$\mathcal{H}_{\alpha}\approx 0$ and the gauge parameters by $\xi^{\alpha}$,
the generators of asymptotic symmetries take the form

$G[\xi^{\alpha}]=\int d^{d}x\xi^{\alpha}\mathcal{H}_{\alpha}+B_{\xi}\,,$ (1)

where $B_{\xi}=\oint_{S^{d-1}_{\infty}}d^{d-1}yf$ is an appropriate surface
term evaluated on the $(d-1)$-sphere $S^{d-1}_{\infty}$ at infinity, which is
necessary to make the key formula
$\iota_{X_{\xi}}\Omega=-d_{V}G[\xi^{\alpha}]$ hold. The phace space vector
field $X_{\xi}$ is the Hamiltonian vector field associated with the generator
$G[\xi^{\alpha}]$, the phase space $2$-form $\Omega$ is the symplectic form
and $d_{V}$ is the phase space exterior derivative, not to be confused with
the exterior derivative in space(time) $d$ (see Henneaux:2018gfi  for more
information). The coordinates $y^{A}$ are coordinates on the $(d-1)$-sphere,
and polar coordinates are $(x^{k})=(r,y^{A})$. The integrand $f$ of the
surface integral giving $B_{\xi}$ depends on the asymptotic values of the
gauge parameters and of the fields.

If the gauge parameters decrease fast enough at infinity, in a way that makes
the boundary term $B_{\xi}$ equal to zero, the generator $G[\xi^{\alpha}]$
weakly vanishes, $G[\xi^{\alpha}]\approx 0$, and the corresponding
transformation is a proper gauge transformation implying a redundancy in the
description of the system Benguria:1976in . If the boundary term $B_{\xi}$
does not vanish, $G[\xi^{\alpha}]$ generates an improper gauge transformation
with physical content which should not be factored. Two generators
$G[\xi^{\alpha}]$ and $G[\xi^{\prime\alpha}]$ that differ by constraint terms,
i.e., which are weakly equal,

$G[\xi^{\alpha}]\approx G[\xi^{\prime\alpha}]$ (2)

generate transformations that coincide at infinity and differ only in the
bulk, by proper gauge transformations. They are thus physically equivalent.
The physical content of asymptotic symmetries is entirely captured by their
asymptotic behaviour, hence the terminology. Note that the generators
$G[\xi^{\alpha}]$ are invariant under proper gauge transformations, even if
the asymptotic symmetry algebra is non-abelian.

We will carry on the redundancy (2) and will thus work below “up to proper
gauge transformations”, i.e., up to weakly vanishing terms for the gauge-
invariant generators (even if not explicitly stated). For that reason, we
shall often only write down explicitly the asymptotic expression of the
charges, i.e., focus only on the relevant boundary terms. Equivalently, one
could fix the proper gauge by proper gauge fixing conditions and introduce the
Dirac bracket. As it is known, the Dirac brackets of (proper) gauge-invariant
quantities coincide with their Poisson brackets (see e.g. Henneaux:1992ig ),
so the two methods are indeed equivalent.

The asymptotic symmetries depend on the asymptotic values of the gauge
parameters at infinity, which are parametrized in the general case by constant
parameters $U^{s}$ (e.g., Poincaré transformations) and functions on the
sphere at infinity, which we denote by $T^{a}(y)$ (e.g., supertranslations).
One has, for the gauge parameters $\xi^{\alpha}$, the asymptotic expansion

$\xi^{\alpha}(r,y)\rightarrow_{r\rightarrow\infty}\overset{\circ}{\xi}\,^{\alpha}(r,y^{A},U^{s},T^{a},\partial_{A}T^{a},\cdots)+\textrm{``more''}\,,$
(3)

where $\overset{\circ}{\xi}\,^{\alpha}$ is the leading term which takes a
specific form (at leading order) in terms of $r$, $y^{A}$, of the asymptotic
values of the fields as well as of the asymptotic symmetry parameters $U^{s}$
and $T^{a}(y)$ and their derivatives up to some order. The parameters and
their derivatives occur linearly. The term “more” in (3) denotes subleading
terms irrelevant for the surface integrals at infinity. The charge-generators
of the gauge transformations then take the form

$G[\xi^{\alpha}]=\int d^{d}x\xi^{\alpha}\mathcal{H}_{\alpha}+U^{s}\oint
d^{d-1}y\mathcal{Q}_{s}+\oint d^{d-1}yT^{a}\mathcal{G}_{a}\,.$ (4)

We assume that the parametrization of the asymptotic symmetries is chosen such
that $U^{s}$ and $T^{a}(y)$ do not depend on the fields and that the charge-
generators $G[\xi^{\alpha}]$ given by (4) have therefore well-defined
functional derivatives under that condition. We will investigate below what
happens when one makes redefinitions involving the fields (through the
charges).

The asymptotic charges are, modulo unwritten weakly vanishing bulk terms,

$Q_{s}=\oint d^{d-1}y\mathcal{Q}_{s}\,,\qquad Q_{a}(y)=\mathcal{G}_{a}(y)\,.$
(5)

The Poisson brackets $\\{Q_{s},Q_{r}\\}$, $\\{Q_{s},Q_{a}(y)\\}$ and
$\\{Q_{a}(y),Q_{b}(y^{\prime})\\}$ of the charges are on general grounds
functions of the charges, which might be nonlinear. Nonlinear algebras are in
fact the rule rather than a fancy exception (see e.g. deBoer:1995cqx ).
Explicit examples in the context of asymptotic symmetries and charges have
been derived in Henneaux:1999ib ; Henneaux:2010xg ; Campoleoni:2010zq ;
Afshar:2013vka ; Gonzalez:2013oaa ; Henneaux:2015ywa ; Fuentealba:2015wza ;
Henneaux:2015tar ; Fuentealba:2020zkf .

It must be stressed that the nonlinear functions of the charges occuring in
the brackets or in the redefinitions made below are still given by expressions
such as (1) with appropriate gauge parameters and boundary terms and not, say,
by nonlocal expressions such as $(\int
d^{d}x\xi^{\alpha}\mathcal{H}_{\alpha}+B_{\xi})^{2}$. Since there might be
some confusion on that point, we shall develop it further.

###  2.2 Equations obeyed by the charges

We denote the canonically conjugate fields by $(\phi^{\Gamma},\pi_{\Gamma})$
and assume for definiteness and simplicity that the symplectic structure is

$\Omega=\int d^{d}x\,d_{V}\pi_{\Gamma}\wedge d_{V}\phi^{\Gamma}.$ (6)

In that case, the condition $\iota_{X_{M}}\Omega=-d_{V}M$ for $M=\int
d^{d}x\mathcal{M}+\oint d^{d-1}y\,m$ implies the condition of Regge:1974zd
that $M$ should have well defined functional derivatives, i.e., that

$d_{V}M=\int
d^{d}x\Big{(}L_{\Gamma}d_{V}\phi^{\Gamma}+N^{\Gamma}d_{V}\pi_{\Gamma}\Big{)}\,,$

for some $L_{\Gamma}$ and $N^{\Gamma}$ (without extra surface terms since
there is no differentiated $d_{V}\pi_{\Gamma}$ or $d_{V}\phi^{\Gamma}$ in
$\Omega$), which are identified as the functional derivatives of $M$,

$\frac{\delta M}{\delta\phi^{\Gamma}(x)}=L_{\Gamma}(x)\,,\quad\frac{\delta
M}{\delta\pi_{\Gamma}(x)}=N^{\Gamma}(x)\,.$

The corresponding phase space Hamiltonian vector field has components

$X_{M}=(N^{\Gamma},-L_{\Gamma})\quad\Leftrightarrow\quad X_{M}=\int
d^{d}x\left(\frac{\delta
M}{\delta\pi_{\Gamma}(x)}\frac{\delta}{\delta\phi^{\Gamma}(x)}-\frac{\delta
M}{\delta\phi^{\Gamma}(x)}\frac{\delta}{\delta\pi_{\Gamma}(x)}\right)\,.$

With a more general symplectic structure having for instance additional
surface contributions, the condition $\iota_{X_{M}}\Omega=-d_{V}M$ still holds
but implies conditions that generalize those of Regge:1974zd , as explained in
Henneaux:2018gfi .

Let us compute $d_{V}G[\xi^{\alpha}]$, with $d_{V}U^{s}=d_{V}T^{a}(y)=0$. We
get

$d_{V}G[\xi^{\alpha}]=\int d^{d}x(d_{V}\xi^{\alpha})\mathcal{H}_{\alpha}+\int
d^{d}x\xi^{\alpha}d_{V}\mathcal{H}_{\alpha}+U^{s}\oint
d^{d-1}yd_{V}\mathcal{Q}_{s}+\oint d^{d-1}yT^{a}d_{V}\mathcal{G}_{a}\,,$ (7)

where we allow for a field-dependence of $\xi^{\alpha}$ in the bulk
(compatible with its asymptotic behaviour). Now, $\int
d^{d}x\xi^{\alpha}d_{V}\mathcal{H}_{\alpha}$ can be transformed by integration
by parts into the bulk form $\int d^{d}xA_{\xi}^{\Gamma}d_{V}\pi_{\Gamma}-\int
d^{d}xd_{V}\phi^{\Gamma}B_{\xi,\Gamma}$ with undifferentiated
$d_{V}\phi^{\Gamma}$’s and $d_{V}\pi_{\Gamma}$’s, plus surface terms. Keeping
all terms - i.e., making no use of the boundary conditions at this stage -,
yields an identity of the form

$\int d^{d}x\xi^{\alpha}d_{V}\mathcal{H}_{\alpha}=\int
d^{d}xA_{\xi}^{\Gamma}d_{V}\pi_{\Gamma}-\int
d^{d}xd_{V}\phi^{\Gamma}B_{\xi,\Gamma}+\oint d^{d-1}y\mathcal{V}\,,$ (8)

for some $A_{\xi}^{\Gamma}$ and $B_{\xi,\Gamma}$. Here, the integrand
$\mathcal{V}$ of the surface integral at infinity involves the asymptotic
value of $\xi_{\alpha}$, as well as $d_{V}\phi^{\Gamma}$, $d_{V}\pi_{\Gamma}$
and possibly their derivatives. The gauge parameters $U^{s}$, $T^{a}(y)$ occur
linearly through $\overset{\circ}{\xi}\,^{\alpha}$. By integrating by parts on
the sphere, one can remove the derivatives of $T_{a}$ (if any), so that $\oint
d^{d-1}y\mathcal{V}$ is identical to,

$\oint d^{d-1}y\mathcal{V}=U^{s}\oint d^{d-1}yk_{s}+\oint
d^{d-1}yT^{a}s_{a}\,.$ (9)

Similarly, one gets another integration-by-part identity,

$\int d^{d}x(d_{V}\xi^{\alpha})\mathcal{H}_{\alpha}=\int
d^{d}x{A^{\prime}}_{\xi}^{\Gamma}d_{V}\pi_{\Gamma}-\int
d^{d}xd_{V}\phi^{\Gamma}B^{\prime}_{\xi,\Gamma}+\oint
d^{d-1}y\mathcal{V}^{\prime}\,,$ (10)

for some ${A^{\prime}}_{\xi}^{\Gamma}$, $B^{\prime}_{\xi,\Gamma}$ and
$\mathcal{V}^{\prime}$, which weakly vanish since they are proportional to the
constraints. One has again

$\oint d^{d-1}y\mathcal{V}^{\prime}=U^{s}\oint d^{d-1}yk^{\prime}_{s}+\oint
d^{d-1}yT^{a}s^{\prime}_{a}\,.$ (11)

The fact that $G[\xi^{\alpha}]$ has well-defined functional derivatives, i.e.,
$d_{V}G[\xi^{\alpha}]$ reduces to the bulk terms, is equivalent to the
condition

$\oint d^{d-1}y\mathcal{V}^{\prime}+\oint d^{d-1}y\mathcal{V}+U^{s}\oint
d^{d-1}yd_{V}\mathcal{Q}_{s}+\oint d^{d-1}yT^{a}d_{V}\mathcal{G}_{a}=0\,.$
(12)

While the relations (8)-(11) are identities, the equation (12) is in general
not identically satisfied but holds only if use is made of the boundary
conditions. One can rewrite it as

$U^{s}\oint d^{d-1}y(d_{V}\mathcal{Q}_{s}+k_{s}+k^{\prime}_{s})+\oint
d^{d-1}yT^{a}(d_{V}\mathcal{G}_{a}+s_{a}+s^{\prime}_{a})=0\,.$ (13)

Since the $U^{s}$’s and the $T^{a}$’s are arbitrary, this is equivalent
to111If the $T^{a}(y)$’s are not arbitrary functions on the sphere but are
restricted, say, by parity conditions, the conditions (14) should be
understood as “modulo terms that project to zero when enforcing the relevant
functional space restrictions”.

$d_{V}Q_{s}+\oint d^{d-1}y(k_{s}+k^{\prime}_{s})=0\,,\qquad
d_{V}\mathcal{G}_{a}+s_{a}+s^{\prime}_{a}=0\,.$ (14)

Note that in many cases, the boundary conditions enforce that the constraints
decrease sufficiently fast at infinity that they do not contribute to the
relevant surface integrals, so that $k^{\prime}_{s}=s^{\prime}_{a}(y)=0$, but
this is not necessary for our discussion.

###  2.3 Nonlinear charges

The question we examine in this subsection is how to determine the complete
(bulk + surface terms) form of the generators of asymptotic symmetries when
one prescribes the surface terms to be some specific function of the charges.
That is, what is the $\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\xi$\kern 0.0pt}}}^{\alpha}$ to be included in
$G[\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\xi$\kern
0.0pt}}}^{\alpha}]=\int d^{d}x\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\xi$\kern
0.0pt}}}^{\alpha}\mathcal{H}_{\alpha}+F[Q_{s},Q_{a}(y)]$ where
$F[Q_{s},Q_{a}(y)]$ is some given functional of the charges?

We define

$\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$U$\kern
0.0pt}}}^{s}=\frac{\partial F}{\partial Q_{s}}\,,\qquad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$T$\kern
0.0pt}}}^{a}(y)=\frac{\delta F}{\delta Q_{a}(y)}\,,$ (15)

and consider now gauge parameters $\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\xi$\kern 0.0pt}}}^{\alpha}$ that go asymptotically
to

$\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\xi$\kern
0.0pt}}}^{\alpha}\quad\rightarrow_{r\rightarrow\infty}\quad\overset{\circ}{\xi}\,^{\alpha}(r,y^{A},\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$U$\kern
0.0pt}}}^{s},\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$T$\kern 0.0pt}}}^{a},\partial_{A}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$T$\kern 0.0pt}}}^{a},\cdots)$ (16)

in which the asymptotic gauge parameters $U^{s}$, $T^{a}$ are replaced by
$\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$U$\kern
0.0pt}}}^{s}$, $\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$T$\kern 0.0pt}}}^{a}$. Such gauge parameters $\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\xi$\kern 0.0pt}}}^{\alpha}$ are
not unique but the ambiguity corresponds to a proper gauge transformation and
is physically irrelevant. We claim that the expression

$G[\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\xi$\kern
0.0pt}}}^{\alpha}]=\int d^{d}x\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\xi$\kern
0.0pt}}}^{\alpha}\mathcal{H}_{\alpha}+F[Q_{s},Q_{a}(y)]\,,$ (17)

with that choice of $\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\xi$\kern 0.0pt}}}^{\alpha}$, is a well-defined generator, with charge
clearly (weakly) equal to $F$.

To establish this fact, we note that one has

$d_{V}G[\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\xi$\kern 0.0pt}}}^{\alpha}]=\int d^{d}x(d_{V}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\xi$\kern
0.0pt}}}^{\alpha})\mathcal{H}_{\alpha}+\int d^{d}x\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\xi$\kern
0.0pt}}}^{\alpha}d_{V}\mathcal{H}_{\alpha}+d_{V}F[Q_{s},Q_{a}(y)]\,.$ (18)

Using the identities (8)-(11), one can bring the first two terms of the right-
hand side to the form

$\displaystyle\int d^{d}x(d_{V}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\xi$\kern
0.0pt}}}^{\alpha})\mathcal{H}_{\alpha}+\int d^{d}x\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\xi$\kern
0.0pt}}}^{\alpha}d_{V}\mathcal{H}_{\alpha}$ $\displaystyle=\int
d^{d}x\left((A_{\hbox{\vbox{\hrule height=0.5pt\kern 0.90417pt\hbox{\kern
0.0pt$\xi$\kern 0.0pt}}}}^{\Gamma}+{A^{\prime}}_{\hbox{\vbox{\hrule
height=0.5pt\kern 0.90417pt\hbox{\kern 0.0pt$\xi$\kern
0.0pt}}}}^{\Gamma})d_{V}\pi_{\Gamma}-d_{V}\phi^{\Gamma}(B_{\hbox{\vbox{\hrule
height=0.5pt\kern 0.90417pt\hbox{\kern 0.0pt$\xi$\kern
0.0pt}}},\Gamma}+B^{\prime}_{\hbox{\vbox{\hrule height=0.5pt\kern
0.90417pt\hbox{\kern 0.0pt$\xi$\kern 0.0pt}}},\Gamma})\right)$
$\displaystyle+\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$U$\kern 0.0pt}}}^{s}\oint d^{d-1}y(k_{s}+k^{\prime}_{s})+\oint
d^{d-1}y\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$T$\kern 0.0pt}}}^{a}(s_{a}+s^{\prime}_{a})\,,$ (19)

which, using the equality (14), can be rewritten as

$\displaystyle\int d^{d}x(d_{V}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\xi$\kern
0.0pt}}}^{\alpha})\mathcal{H}_{\alpha}+\int d^{d}x\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\xi$\kern
0.0pt}}}^{\alpha}d_{V}\mathcal{H}_{\alpha}$ $\displaystyle=\int
d^{d}x\left((A_{\hbox{\vbox{\hrule height=0.5pt\kern 0.90417pt\hbox{\kern
0.0pt$\xi$\kern 0.0pt}}}}^{\Gamma}+{A^{\prime}}_{\hbox{\vbox{\hrule
height=0.5pt\kern 0.90417pt\hbox{\kern 0.0pt$\xi$\kern
0.0pt}}}}^{\Gamma})d_{V}\pi_{\Gamma}-d_{V}\phi^{\Gamma}(B_{\hbox{\vbox{\hrule
height=0.5pt\kern 0.90417pt\hbox{\kern 0.0pt$\xi$\kern
0.0pt}}},\Gamma}+B^{\prime}_{\hbox{\vbox{\hrule height=0.5pt\kern
0.90417pt\hbox{\kern 0.0pt$\xi$\kern 0.0pt}}},\Gamma})\right)$
$\displaystyle-\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$U$\kern 0.0pt}}}^{s}d_{V}Q_{s}-\oint d^{d-1}y\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$T$\kern
0.0pt}}}^{a}d_{V}\mathcal{G}_{a}\,.$ (20)

On the other hand, $d_{V}F[Q_{s},Q_{a}(y)]$ is equal to

$d_{V}F[Q_{s},Q_{a}(y)]=\frac{\partial F}{\partial Q_{s}}d_{V}Q_{s}+\oint
d^{d-1}y\frac{\delta F}{\delta
Q_{a}(y)}d_{V}\mathcal{G}_{a}=\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$U$\kern 0.0pt}}}^{s}d_{V}Q_{s}+\oint
d^{d-1}y\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$T$\kern 0.0pt}}}^{a}d_{V}\mathcal{G}_{a}\,,$ (21)

so that $d_{V}G[\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\xi$\kern 0.0pt}}}^{\alpha}]$ reduces to the bulk term with
undifferentiated $d_{V}\phi^{\Gamma}$ and $d_{V}\pi_{\Gamma}$,

$d_{V}G[\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\xi$\kern 0.0pt}}}^{\alpha}]=\int d^{d}x\left((A_{\hbox{\vbox{\hrule
height=0.5pt\kern 0.90417pt\hbox{\kern 0.0pt$\xi$\kern
0.0pt}}}}^{\Gamma}+{A^{\prime}}_{\hbox{\vbox{\hrule height=0.5pt\kern
0.90417pt\hbox{\kern 0.0pt$\xi$\kern
0.0pt}}}}^{\Gamma})d_{V}\pi_{\Gamma}-d_{V}\phi^{\Gamma}(B_{\hbox{\vbox{\hrule
height=0.5pt\kern 0.90417pt\hbox{\kern 0.0pt$\xi$\kern
0.0pt}}},\Gamma}+B^{\prime}_{\hbox{\vbox{\hrule height=0.5pt\kern
0.90417pt\hbox{\kern 0.0pt$\xi$\kern 0.0pt}}},\Gamma})\right)\,,$ (22)

as it should. The on-shell value of $G[\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\xi$\kern 0.0pt}}}^{\alpha}]$ is furthermore
clearly equal to $F$.

To illustrate the construction, consider the case where $F$ is quadratic,

$F=\frac{1}{2}g^{rs}Q_{r}Q_{s}+Q_{r}\oint
d^{d-1}yg^{ra}(y)Q_{a}(y)+\frac{1}{2}\oint d^{d-1}y\oint
d^{d-1}y^{\prime}g^{ab}(y,y^{\prime})Q_{a}(y)Q_{b}(y^{\prime})\,,$ (23)

where $g^{ab}(y,y^{\prime})$ involves the delta function and its derivatives,
e.g.,
$g^{ab}(y,y^{\prime})=h^{ab}(y)\delta(y,y^{\prime})+h^{abA}(y)\partial_{A}\delta(y,y^{\prime})$,
so that the double integral in $F$ reduces to the single integral

$\frac{1}{2}\oint
d^{d-1}yQ_{a}\left(h^{ab}Q_{b}+h^{abA}\partial_{A}Q_{b}\right).$

The coefficients $g^{rs}$ and $g^{ab}(y,y^{\prime})$ are symmetric by
construction,

$\displaystyle g^{rs}$ $\displaystyle=g^{sr}\,,$ (24) $\displaystyle
g^{ab}(y,y^{\prime})$
$\displaystyle=g^{ba}(y^{\prime},y)\quad\Leftrightarrow\quad
h^{ba}=h^{ab}-\partial_{A}{h^{abA}},\;h^{abA}=-h^{baA}\,.$ (25)

The asymptotic symmetry parameters are then

$\displaystyle\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$U$\kern 0.0pt}}}^{s}$ $\displaystyle=\frac{\partial F}{\partial
Q_{s}}=g^{rs}Q_{r}+\oint d^{d-1}yg^{sa}(y)Q_{a}(y)\,,$ (26)
$\displaystyle\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$T$\kern 0.0pt}}}^{a}(y)$ $\displaystyle=\frac{\delta F}{\delta
Q_{a}(y)}=g^{sa}(y)Q_{s}+h^{ab}(y)Q_{b}(y)+h^{abA}(y)\partial_{A}Q_{b}(y)\,.$
(27)

Because $g^{ab}(y,y^{\prime})$ is symmetric, if the $a$-component of the
parameter of the asymptotic symmetry involves the $b$-component of the charge,
the $b$-component of the parameter of the asymptotic symmetry will involve in
turn the $a$-component of the charge.

We can draw two conclusions from the above analysis.

  * •

There is no difficulty in handling non-linear expressions in the asymptotic
symmetry charges, and these take the standard form of an asymptotic symmetry
generator, i.e., are the sum of a standard bulk term proportional to the
constraints and of the appropriate surface integral. Furthermore, the
asymptotic redefinition determines everything modulo physically irrelevant
proper gauge symmetries.

  * •

Integrability of the charges is manifest in the analysis since we work with
the integrated charges themselves throughout. The corresponding transformation
(Hamiltonian vector field) is derived by taking the Poisson bracket.

Alternatively, one might ask the following question: given a transformation
that preserves the boundary conditions on the fields, with gauge parameters
$\xi^{\alpha}$ going asymptotically to

$\overset{\circ}{\xi}\,^{\alpha}(r,y^{A},\tilde{U}^{s},\tilde{T}^{a},\partial_{A}\tilde{T}^{a},\cdots)$

where $\tilde{U}^{s}$ and $\tilde{T}^{a}$ are a priori allowed to depend on
the fields. What are the restrictions on that dependence that must be imposed
in order to guarantee the existence of a generator? This is equivalent to
requesting invariance of the action under the gauge transformation which has
an asymptotic behaviour dictated by $\tilde{U}^{s}$ and $\tilde{T}^{a}$, i.e.,
to the demand that that gauge transformation defines a true Hamiltonian vector
field. Repeating the above steps, one finds that the generator
$G[\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\xi$\kern
0.0pt}}}^{\alpha}]=\int d^{d}x\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\xi$\kern
0.0pt}}}^{\alpha}\mathcal{H}_{\alpha}+\tilde{F}$, if it exists, must fulfill

$d_{V}\tilde{F}=\tilde{U}^{s}d_{V}Q_{s}+\oint
d^{d-1}y\tilde{T}^{a}d_{V}\mathcal{G}_{a}\,,$ (28)

i.e., $\tilde{U}^{s}d_{V}Q_{s}+\oint
d^{d-1}y\tilde{T}^{a}d_{V}\mathcal{G}_{a}$ must be integrable. Since this
differential involves $d_{V}Q_{s}$ and $d_{V}\mathcal{G}_{a}$ only, the same
must be true for $\tilde{U}^{s}$, $\tilde{T}^{a}$ and $\tilde{F}$.
Furthermore, integrability imposes that they obey the relations

$\tilde{U}^{s}=\frac{\partial\tilde{F}}{\partial
Q_{s}}\,,\qquad\tilde{T}^{a}(y)=\frac{\delta\tilde{F}}{\delta Q_{a}(y)}\,.$
(29)

Under the condition of integrability, the functions of the charges exhaust all
possibilities.

Finally, we note that in the case of gravity, a definite choice of
$\xi^{\alpha}$ corresponds to a definite choice of slicing of spacetime by
spacelike hypersurfaces. One sometimes speaks therefore of “integrable
slicings” for the choice of $\xi^{\alpha}$ leading to well-defined charges
Adami:2021nnf .

##  3 Redefinition of the BMS5 algebra

###  3.1 Structure of the BMS5 algebra

In 5 spacetime dimensions, the generators of the BMS algebra include, besides
the Poincaré generators, two types of supertranslations, namely, the “pure
supertranslations” and the “subleading supertranslations”. The pure
supertranslations are diffeomorphisms of order $\mathcal{O}(1)$ as
$r\rightarrow\infty$, with no $\ell\leq 1$ spherical harmonic components,
which correspond to the standard translations. They are the natural
generalization to higher dimensions of the pure supertranslations in 4
spacetime dimensions. The subleading supertranslations are diffeomorphisms of
order $\mathcal{O}(r^{-1})$, which define improper gauge symmetries in $D=5$
dimensions thanks to the boundary conditions of Fuentealba:2021yvo ;
Fuentealba:2022yqt  which make them relevant. They also do not involve
$\ell\leq 1$ spherical harmonic components.

While at different orders in $r$ at $D=5$, the two supertranslation branches
are at the same order $\mathcal{O}(1)$ at $D=4$. Because of this coincidence,
the second branch acquires a logarithm. In higher dimensions, the second
branch decays as $\mathcal{O}(r^{-(D-4)})$ and is independent from the
$\mathcal{O}(1)$ branch.

Denoting the generators of the homogeneous Lorentz transformations, of the
ordinary translations, of the pure supertranslations and of the subleading
supertranslations respectively by $M_{a}$, $T_{i}$, $S_{\alpha}$ and
$L^{\beta}$, one finds that the BMS5 algebra has the structure
Fuentealba:2021yvo ; Fuentealba:2022yqt

$\displaystyle\\{M_{a},M_{b}\\}$ $\displaystyle=f_{ab}^{c}M_{c}\,,$ (30)
$\displaystyle\\{M_{a},T_{i}\\}$
$\displaystyle=R_{ai}^{\phantom{ai}j}T_{j}\,,$ (31)
$\displaystyle\\{M_{a},S_{\alpha}\\}$
$\displaystyle=G_{a\alpha}^{\phantom{a\alpha}i}T_{i}+G_{a\alpha}^{\phantom{a\alpha}\beta}S_{\beta}+U_{a\alpha\beta\gamma}L^{\beta}L^{\gamma}\,,\quad
U_{a\alpha\beta\gamma}=U_{a\alpha(\beta\gamma)}\,,$ (32)
$\displaystyle\\{M_{a},L^{\alpha}\\}$
$\displaystyle=-G_{a\beta}^{\phantom{a\alpha}\alpha}L^{\beta}\,,$ (33)
$\displaystyle\\{L^{\alpha},S_{\beta}\\}$
$\displaystyle=\delta^{\alpha}_{\beta}\,.$ (34)

It is a non-linear algebra, with quadratic terms appearing in the bracket
$\\{M_{a},S_{\alpha}\\}$. The relation (34) expresses that the generators of
the subleading supertranslations $L^{\alpha}$ and the generators of the pure
supertranslations $S_{\beta}$ are canonically conjugate. It is this key
feature that enables one to simplify the algebra. The subleading
supertranslations play thus in $D=5$ the role played in $D=4$ by the
logarithmic supertranslations.

The explicit expressions of the charges $M_{a}$, $T_{i}$, $S_{\alpha}$ and
$L^{\beta}$ generating the asymptotic symmetries are given in
Fuentealba:2021yvo ; Fuentealba:2022yqt . Since the asymptotic symmetries are
diffeomorphisms, each charge-generator takes the form

$\int
d^{4}x\left(\xi\mathcal{H}+\xi^{k}\mathcal{H}_{k}\right)+\oint_{S^{3}_{\infty}}d^{3}x\mathcal{B}\,,$
(35)

where $\mathcal{H}\approx 0$ and $\mathcal{H}_{k}\approx 0$ are the
gravitational constraints and where $\oint_{S^{3}_{\infty}}d^{3}x\mathcal{B}$
is a surface integral on the three-sphere $S^{3}_{\infty}$ at infinity,
involving the coefficients of the metric, of its conjugate momentum and of the
vector field $(\xi,\xi^{k})$ in the expansion near infinity. The vector field
$(\xi,\xi^{k})$ tends to a homogeneous Lorentz transformation for $M_{a}$, to
an ordinary spacetime translation for $T_{i}$, to a pure supertranslation for
$S_{\alpha}$ and to a logarithmic supertranslation for $L^{\beta}$.

As recalled above, the canonical generator (35) is the sum of (i) a bulk
contribution proportional to the constraints multiplied by the components of
the vector field parametrizing the corresponding diffeomorphism and (ii) a
surface term involving the asymptotic form of this diffeomorphism, such that
the key Hamiltonian relation $\iota_{X}\Omega=-d_{V}F_{X}$ relating charges
$F_{X}$ and phase space Hamiltonian vector fields $X$ through the symplectic
form $\Omega$, does indeed hold. These expressions will not be needed here
since our derivations rely only on the form (30)-(34) of the algebra of the
asymptotic symmetry generators, without involving the detailed form of the
charges.

Jacobi identities constrain the structure constants. Among these, the ones
that give useful relations for the subsequent construction involve the Lorentz
generators and the pure supertranslations. From

$\\{M_{a},\\{M_{b},S_{\alpha}\\}\\}+\\{M_{b},\\{S_{\alpha},M_{a}\\}\\}+\\{S_{\alpha},\\{M_{a},M_{b}\\}\\}=0\,,$
(36)

we get

$\displaystyle
G_{b\alpha}^{\,\,\,\,\,j}R_{aj}^{\,\,\,\,\,i}+G_{b\alpha}^{\,\,\,\,\,\beta}G_{a\beta}^{\,\,\,\,\,i}-\left(a\leftrightarrow
b\right)$ $\displaystyle=f_{ab}^{c}G_{c\alpha}^{\,\,\,\,\,i}\,,$ (37)
$\displaystyle
G_{b\alpha}^{\,\,\,\,\,\beta}G_{a\beta}^{\,\,\,\,\,\gamma}-\left(a\leftrightarrow
b\right)$ $\displaystyle=f_{ab}^{c}G_{c\alpha}^{\,\,\,\,\,\gamma}\,,$ (38)
$\displaystyle
G_{b\alpha}^{\,\,\,\,\,\beta}U_{a\beta\gamma\delta}-2G_{a\delta}^{\,\,\,\,\,\beta}U_{b\alpha\gamma\beta}-\left(a\leftrightarrow
b\right)$ $\displaystyle=f_{ab}^{c}U_{c\alpha\gamma\delta}\,.$ (39)

Similarly, the identity

$\\{M_{a},\\{S_{\alpha},S_{\beta}\\}\\}+\\{S_{\alpha},\\{S_{\beta},M_{a}\\}\\}+\\{S_{\beta},\\{M_{a},S_{\alpha}\\}\\}=0\,,$
(40)

implies that

$U_{a[\alpha\beta]\gamma}=0\qquad\Leftrightarrow\qquad
U_{a\alpha\beta\gamma}=U_{a(\alpha\beta)\gamma}\,.$ (41)

Consequently, since $U_{a\alpha[\gamma\delta]}=0$, the structure constants
$U_{a\alpha\gamma\delta}$ are totally symmetric in their Greek indices,

$U_{a\alpha\gamma\delta}=U_{a(\alpha\beta\gamma)}=0\,.$ (42)

###  3.2 Redefinitions of the homogeneous Lorentz generators

In our paper Fuentealba:2022yqt , we tried to simplify the nonlinear BMS5
algebra by considering a special type of redefinitions where only the
supertranslations $S_{\alpha}$ were modified by quadratic terms. We proved the
existence of obstructions within this restricted context.

It turns that these obstructions are absent, and that the nonlinear BMS5
algebra can be linearized and transformed into a direct sum algebra, if one
allows redefinitions of the Lorentz generators which involve both quadratic
and cubic terms in the supertranslations and subleading supertranslation
generators - a possibility not explored in Fuentealba:2022yqt .

The searched-for redefinitions are simply

$\displaystyle\tilde{M}_{a}$
$\displaystyle=M_{a}-G_{a\beta}^{\,\,\,\,\,i}L^{\beta}T_{i}-G_{a\beta}^{\,\,\,\,\,\gamma}L^{\beta}S_{\gamma}-\frac{1}{3}U_{a\beta\gamma\delta}L^{\beta}L^{\gamma}L^{\delta}$
(43)
$\displaystyle=M_{a}-L^{\beta}\\{M_{a},S_{\beta}\\}+\frac{2}{3}U_{a\beta\gamma\delta}L^{\beta}L^{\gamma}L^{\delta}\,,$
(44)

(other generators unchanged). Note that the coefficients
$U_{a\alpha\gamma\delta}$ are totally symmetrized in their Greek indices when
multiplied by $L^{\beta}L^{\gamma}L^{\delta}$ in (44), but since they are
totally symmetric, no information on the $U$’s is lost in the expression,
i.e., one can completely read off $U_{a\alpha\gamma\delta}$ from (44).

One can easily prove that the brackets of these new Lorentz generators with
ordinary translations remain the same and that the brackets with all
supertranslations vanish:

$\displaystyle\\{\tilde{M}_{a},T_{i}\\}$
$\displaystyle=R_{ai}^{\,\,\,\,\,j}T_{j}\,,$ (45)
$\displaystyle\\{\tilde{M}_{a},S_{\alpha}\\}$
$\displaystyle=L^{\beta}\\{\\{M_{a},S_{\beta}\\},S_{\alpha}\\}+2U_{a\beta\gamma\delta}\\{L^{\beta},S_{\alpha}\\}L^{\gamma}L^{\delta}$
$\displaystyle=-2U_{a\beta\gamma\delta}L^{\beta}\\{L^{\gamma},S_{\alpha}\\}L^{\delta}+2U_{a\beta\gamma\delta}\\{L^{\beta},S_{\alpha}\\}L^{\gamma}L^{\delta}$
$\displaystyle=0\,,$ (46) $\displaystyle\\{\tilde{M}_{a},L^{\alpha}\\}$
$\displaystyle=\\{M_{a},L^{\alpha}\\}-G_{a\beta}^{\,\,\,\,\,\gamma}L^{\beta}\\{S_{\gamma},L^{\alpha}\\}$
$\displaystyle=\\{M_{a},L^{\alpha}\\}+G_{a\beta}^{\,\,\,\,\,\alpha}L^{\beta}$
$\displaystyle=0\,.$ (47)

Using the consequences of the Jacobi identities, one can also prove that the
redefinition of the Lorentz generators defines a nonlinear automorphism of the
Lorentz algebra, i.e., that the $\tilde{M}_{a}$’s obey the same algebra as the
$M_{a}$’s,

$\displaystyle\\{\tilde{M}_{a},\tilde{M}_{b}\\}$
$\displaystyle=\\{M_{a}-G_{a\beta}^{\,\,\,\,\,i}L^{\beta}T_{i}-G_{a\beta}^{\,\,\,\,\,\gamma}L^{\beta}S_{\gamma},M_{b}-G_{b\mu}^{\,\,\,\,\,j}L^{\mu}T_{j}-G_{b\mu}^{\,\,\,\,\,\nu}L^{\mu}S_{\nu}\\}$
(48)
$\displaystyle=\\{M_{a},M_{b}\\}+\left(G_{b\delta}^{\,\,\,\,\,\beta}G_{a\beta}^{\,\,\,\,\,\gamma}-G_{a\delta}^{\,\,\,\,\,\beta}G_{b\beta}^{\,\,\,\,\,\gamma}\right)L^{\delta}S_{\gamma}$
$\displaystyle\quad-\left[G_{b\gamma}^{\,\,\,\,\,i}R_{ai}^{\,\,\,\,\,j}+G_{b\gamma}^{\,\,\,\,\,\beta}G_{a\beta}^{\,\,\,\,\,j}-G_{a\gamma}^{\,\,\,\,\,i}R_{bi}^{\,\,\,\,\,j}-G_{a\gamma}^{\,\,\,\,\,\beta}G_{b\beta}^{\,\,\,\,\,j}\right]L^{\gamma}T_{j}$
$\displaystyle\quad+\left(U_{b\gamma\mu\nu}G_{a\beta}^{\,\,\,\,\,\gamma}-U_{a\gamma\mu\nu}G_{b\beta}^{\,\,\,\,\,\gamma}\right)L^{\beta}L^{\mu}L^{\nu}$
(49)
$\displaystyle=f_{ab}^{c}\left(M_{c}-G_{c\beta}^{\,\,\,\,\,i}L^{\beta}T_{i}-G_{c\beta}^{\,\,\,\,\,\gamma}L^{\beta}S_{\gamma}-\frac{1}{3}U_{c\beta\gamma\delta}L^{\beta}L^{\gamma}L^{\delta}\right)$
$\displaystyle=f_{ab}^{c}\tilde{M}_{c}\,.$ (50)

The new form of the algebra is therefore

$\displaystyle\\{\tilde{M}_{a},\tilde{M}_{b}\\}$
$\displaystyle=f_{ab}^{c}\tilde{M}_{c}\,,$ (51)
$\displaystyle\\{\tilde{M}_{a},T_{i}\\}$
$\displaystyle=R_{ai}^{\,\,\,\,\,j}T_{j}\,,$ (52)
$\displaystyle\\{\tilde{M}_{a},S_{\alpha}\\}$ $\displaystyle=0\,,$ (53)
$\displaystyle\\{\tilde{M}_{a},L^{\alpha}\\}$ $\displaystyle=0\,,$ (54)
$\displaystyle\\{L^{\alpha},S_{\beta}\\}$
$\displaystyle=\delta^{\alpha}_{\beta}\,,$ (55)

which explicitly exhibits the direct sum structure “Poincaré $\oplus$
Supertranslations” (including the subleading supertranslations). This is
exactly the same structure as found in four spacetime dimensions
Fuentealba:2022xsz .

###  3.3 Redefined transformations

The redefined Lorentz generators lead to redefined Lorentz transformations.
The variations of the fields under the new Lorentz transformations are simply
obtained by taking their bracket with the new charges. Since the redefinitions
of the charges are nonlinear, the added terms to the original transformations
will correspond to supertranslations and subleading supertranslations with
coefficients that depend on the fields through the charges (see Section 2).

This yields, for any phase space function $F$,

$\displaystyle\\{F,\tilde{M}_{a}\\}=$
$\displaystyle\\{F,M_{a}\\}-G_{a\beta}^{\,\,\,\,\,i}L^{\beta}\\{F,T_{i}\\}-G_{a\beta}^{\,\,\,\,\,\gamma}L^{\beta}\\{F,S_{\gamma}\\}$
$\displaystyle-\left(G_{a\beta}^{\,\,\,\,\,i}T_{i}-G_{a\beta}^{\,\,\,\,\,\gamma}S_{\gamma}-U_{a\beta\gamma\delta}L^{\gamma}L^{\delta}\right)\\{F,L^{\beta}\\}$
(56)

This expression explicitly shows that in addition to the original Lorentz
variation $\\{F,M_{a}\\}$, the transformation of $F$ involves an ordinary
translation with coefficient $-G_{a\beta}^{\,\,\,\,\,i}L^{\beta}$, a pure
supertranslation with coefficient $-G_{a\beta}^{\,\,\,\,\,\gamma}L^{\beta}$
and a subleading supertranslation with coefficient
$-(G_{a\beta}^{\,\,\,\,\,i}T_{i}-G_{a\beta}^{\,\,\,\,\,\gamma}S_{\gamma}-U_{a\beta\gamma\delta}L^{\gamma}L^{\delta})$.

By substituting in (56) the explicit form of the charges given in
Fuentealba:2022yqt , one obtains the expression of the compensating field-
dependent diffeomorphisms (in particular, their relevant asymptotic behaviour)
that must be added to the Lorentz transformations to give to the algebra the
direct sum structure.

##  4 Redefinition of the BMS4 superalgebra

###  4.1 Algebra and Jacobi identities

We now turn to supergravity. We start with the version of the super-BMS
algebra found in Fuentealba:2021xhn  where the asymptotic symmetries are the
following:

  * •

Bosonic asymptotic symmetries (with respective charge-generators $M_{a}$,
$T_{i}$ and $S_{\alpha}$):

    * –

diffeomorphisms asymptotically behaving at infinity as homogeneous Lorentz
transformations

    * –

ordinary translations and

    * –

pure supertranslations.

They evidently form a subalgebra isomorphic to the BMS4 algebra.

  * •

Fermionic asymptotic symmetries: The fermionic asymptotic symmetries are of
three different types. First, there are the local supersymmetry
transformations with parameters asymptotically approaching a Killing spinor
(i.e., becoming asymptotically constant in asymptotically cartesian frames ).
Their charge-generators are denoted $Q_{I}$. Then there are the local
supersymmetry transformations with parameters asymptotically approaching
higher spherical harmonics modes. The corresponding supercharges are in
infinite number and denoted $q_{A}$. Finally, there are the conjugate
transformations acting on surface variables, with charge-generators $s^{A}$
Fuentealba:2021xhn ; Fuentealba:2020aax .

The explicit expressions are given in Fuentealba:2021xhn  and are not needed
here, because our derivation relies solely on the structure of the charge
algebra, which is,

$\displaystyle[M_{a},M_{b}]$ $\displaystyle=f_{ab}^{c}M_{c}\,,$ (57)
$\displaystyle{}[M_{a},T_{i}]$ $\displaystyle=R_{ai}^{j}T_{j}\,,$ (58)
$\displaystyle{}[M_{a},S_{\alpha}]$
$\displaystyle=G_{a\alpha}^{i}T_{i}+G_{a\alpha}^{\beta}S_{\beta}\,,$ (59)
$\displaystyle{}[M_{a},Q_{I}]$
$\displaystyle=g_{aI}^{J}Q_{J}+V_{aIB}^{i}s^{B}T_{i}+V_{aIB}^{\alpha}s^{B}S_{\alpha}\,,$
(60) $\displaystyle{}[M_{a},q_{A}]$
$\displaystyle=h_{aA}^{B}q_{B}+U_{aAB}^{i}s^{B}T_{i}+U_{aAB}^{\alpha}s^{B}S_{\alpha}\,,$
(61) $\displaystyle{}[M_{a},s^{B}]$ $\displaystyle=-h_{aC}^{B}s^{C}\,,$ (62)
$\displaystyle\\{s^{A},q_{B}\\}$ $\displaystyle=\delta_{B}^{A}\,,$ (63)
$\displaystyle\\{Q_{I},q_{A}\\}$
$\displaystyle=d_{IA}^{i}T_{i}+d_{IA}^{\alpha}S_{\alpha}\,,$ (64)
$\displaystyle\\{q_{A},q_{B}\\}$
$\displaystyle=d_{AB}^{i}T_{i}+d_{AB}^{\alpha}S_{\alpha}\,,$ (65)
$\displaystyle\\{Q_{I},Q_{J}\\}$ $\displaystyle=d_{IJ}^{i}T_{i}\,.$ (66)

The remaining brackets involving translations and supertranslations vanish

$\displaystyle[T_{i},T_{j}]$
$\displaystyle=[T_{i},S_{\alpha}]=[S_{\alpha},S_{\beta}]=0\,,$ (67)
$\displaystyle[S_{\alpha},Q_{I}]$ $\displaystyle=[T_{i},Q_{I}]=0\,,$ (68)
$\displaystyle[S_{\alpha},q_{A}]$ $\displaystyle=[T_{i},q_{A}]=0\,,$ (69)
$\displaystyle[S_{\alpha},s^{B}]$ $\displaystyle=[T_{i},s^{B}]=0\,,$ (70)

and we have also

$\\{Q_{I},s^{A}\\}=0=\\{s^{A},s^{B}\\}\,.$ (71)

The asymptotic symmetry superalgebra is nonlinear since it involves quadratic
terms in the brackets of the Lorentz generators with the supercharges
$Q_{I},q_{A}$.

The following Jacobi identities are particularly useful:

  * •

From

$[M_{a},[M_{b},S_{\alpha}]]+[M_{b},[S_{\alpha},M_{a}]]+[S_{\alpha},[M_{a},M_{b}]]=0\,,$
(72)

we derive the relations

$\displaystyle
G_{b\alpha}^{j}R_{aj}^{i}+G_{b\alpha}^{\beta}G_{a\beta}^{i}-\left(a\leftrightarrow
b\right)$ $\displaystyle=f_{ab}^{c}G_{c\alpha}^{i}\,,$ (73) $\displaystyle
G_{b\alpha}^{\beta}G_{a\beta}^{\gamma}-\left(a\leftrightarrow b\right)$
$\displaystyle=f_{ab}^{c}G_{c\alpha}^{\gamma}\,.$ (74)

  * •

Analogously, from

$[M_{a},[M_{b},T_{i}]]+[M_{b},[T_{i},M_{a}]]+[T_{i},[M_{a},M_{b}]]=0\,,$ (75)

follows the relation

$R_{bj}^{k}R_{ak}^{i}-\left(a\leftrightarrow b\right)=f_{ab}^{c}R_{cj}^{i}\,.$
(76)

  * •

The identity

$[M_{a},\\{Q_{I},q_{A}\\}]-\\{q_{A},[M_{a},Q_{I}]\\}+\\{Q_{I},[q_{A},M_{a}]\\}=0\,,$
(77)

enables one to express the coefficients $V$’s of the quadratic terms in terms
of the other structure constants,

$\displaystyle V_{aIB}^{i}$
$\displaystyle=d_{IB}^{\alpha}G_{a\alpha}^{i}+d_{IB}^{j}R_{aj}^{i}-g_{aI}^{J}d_{JB}^{i}-h_{aB}^{C}d_{IC}^{i}\,,$
(78) $\displaystyle V_{aIB}^{\beta}$
$\displaystyle=d_{IB}^{\alpha}G_{a\alpha}^{\beta}-g_{aI}^{J}d_{JB}^{\beta}-h_{aB}^{C}d_{IC}^{\beta}\,.$
(79)

  * •

Similarly, from

$[M_{a},\\{q_{A},q_{B}\\}]-\\{q_{B},[M_{a},q_{A}]\\}+\\{q_{A},[q_{B},M_{a}]\\}=0\,,$
(80)

we can express the $U$’s,

$\displaystyle 2U_{a(AB)}^{i}$
$\displaystyle=d_{AB}^{\,\,\,\,\,\alpha}G_{a\alpha}^{i}+d_{AB}^{j}R_{aj}^{i}-h_{aB}^{C}d_{AC}^{i}-h_{aA}^{C}d_{BC}^{i}\,,$
(81) $\displaystyle 2U_{a(AB)}^{\beta}$
$\displaystyle=d_{AB}^{\alpha}G_{a\alpha}^{\beta}-h_{aB}^{C}d_{AC}^{\beta}-h_{aA}^{C}d_{BC}^{\beta}\,.$
(82)

  * •

Finally, the identity

$[M_{a},[M_{b},q_{A}]]+[M_{b},[q_{A},M_{a}]]+[q_{A},[M_{a},M_{b}]]=0\,,$ (83)

implies the relations

$\displaystyle h_{bA}^{C}h_{aC}^{B}-\left(a\leftrightarrow b\right)$
$\displaystyle=f_{ab}^{c}h_{cA}^{B}\,,$ (84) $\displaystyle
h_{bA}^{C}U_{aCB}^{i}-h_{aB}^{\,\,\,\,\,C}U_{bAC}^{i}+R_{aJ}^{i}U_{bAB}^{j}+G_{a\alpha}^{i}U_{bAB}^{\alpha}-\left(a\leftrightarrow
b\right)$ $\displaystyle=f_{ab}^{c}U_{cAB}^{i}\,,$ (85) $\displaystyle
h_{bA}^{\,\,\,\,\,C}U_{aCB}^{\alpha}-h_{aB}^{C}U_{bAC}^{\alpha}+G_{a\beta}^{\alpha}U_{bAB}^{\beta}-\left(a\leftrightarrow
b\right)$ $\displaystyle=f_{ab}^{c}U_{cAB}^{\alpha}\,.$ (86)

###  4.2 Algebraic decoupling of the fermionic charges

In order to decouple the infinite-dimensional set of BMS conjugate
supercharges $q_{A}$ and $s_{A}$ in the algebra, we must redefine not only the
Lorentz generators but also the supercharges $Q_{I}$ and $q_{A}$. The
following nonlinear redefinitions achieve the requested task:

$\displaystyle\tilde{Q}_{I}$
$\displaystyle=Q_{I}-d_{IB}^{i}s^{B}T_{i}-d_{IB}^{\alpha}s^{B}S_{\alpha}\,,$
(87) $\displaystyle\tilde{q}_{A}$
$\displaystyle=q_{A}-\frac{1}{2}d_{AB}^{i}s^{B}T_{i}-\frac{1}{2}d_{AB}^{\alpha}s^{B}S_{\alpha}\,,$
(88) $\displaystyle\tilde{M}_{a}$
$\displaystyle=M_{a}+h_{aA}^{B}s^{A}q_{B}+\frac{1}{2}\left(U_{aAB}^{i}-h_{aA}^{C}d_{BC}^{i}\right)s^{A}s^{B}T_{i}+\frac{1}{2}\left(U_{aAB}^{\alpha}-h_{aA}^{C}d_{BC}^{\alpha}\right)s^{A}s^{B}S_{\alpha}\,.$
(89)

Using the above consequences of the Jacobi identities, it is indeed
straightforward, although somewhat tedious, to check that:

  * •

The new supercharges $\tilde{Q}_{I}$ obey the same algebra as the $Q_{I}$’s,

$\\{\tilde{Q}_{I},\tilde{Q_{J}}\\}=\\{Q_{I},Q_{J}\\}=d_{IJ}^{i}T_{i}\,.$ (90)

and furthermore have vanishing Poisson bracket with the $\tilde{q}_{A}$’s

$\\{\tilde{Q}_{I},\tilde{q}_{A}\\}=0\,.$ (91)

In addition one has

$\\{\tilde{q}_{A},\tilde{q}_{B}\\}=0\,,$ (92)

and of course, the brackets

$\\{\tilde{Q}_{I},s^{A}\\}=0=\\{s^{A},s^{B}\\},\qquad\,,\\{s^{A},\tilde{q}_{B}\\}=\delta_{B}^{A}$
(93)

are unchanged.

  * •

The brackets between the fermionic and Lorentz generators get simplified to

$[\tilde{M}_{a},\tilde{Q}_{I}]=g_{aI}^{J}\tilde{Q}_{J}\,,\qquad[\tilde{M}_{a},\tilde{q}_{C}]=0=[\tilde{M}_{a},s^{C}]\,.$
(94)

  * •

The new Lorentz generators obey the Lorentz algebra,

$[\tilde{M}_{a},\tilde{M}_{b}]=f_{ab}^{c}\tilde{M}_{c}\,.$ (95)

The derivation of all these relations is direct and not particularly
illuminating. To give an idea of the arguments being used, we provide the
explicit verification of (95) in the appendix.

We have thus proved that after the redefinitions (87)-(89) are made, the
super-BMS algebra reduces to

$\displaystyle[\tilde{M}_{a},\tilde{M}_{b}]$
$\displaystyle=f_{ab}^{c}\tilde{M}_{c}\,,$ (96)
$\displaystyle{}[\tilde{M}_{a},T_{i}]$ $\displaystyle=R_{ai}^{j}T_{j}\,,$ (97)
$\displaystyle{}[\tilde{M}_{a},\tilde{Q}_{I}]$
$\displaystyle=g_{aI}^{J}\tilde{Q}_{J}\,,$ (98)
$\displaystyle{}\\{\tilde{Q}_{I},\tilde{Q}_{J}\\}$
$\displaystyle=d_{IJ}^{i}T_{i}\,,\quad[\tilde{Q}_{I},T_{i}]=0\,,\quad[T_{i},T_{j}]=0\,,$
(99)

(super-Poincaré algebra),

$\displaystyle[S_{\alpha},S_{\beta}]$
$\displaystyle=0\,,\quad[S_{\alpha},\tilde{q}_{A}]=0=[S_{\alpha},s^{B}]$ (100)
$\displaystyle\\{s^{A},\tilde{q}_{B}\\}$ $\displaystyle=\delta_{B}^{A}\,,$
(101) $\displaystyle\\{\tilde{q}_{A},\tilde{q}_{B}\\}$
$\displaystyle=0=\\{s^{A},s^{B}\\}\,,$ (102)

(infinite-dimensinal abelian superalgebra with a central extension in the
fermionic sector). The brackets between the two algebra all vanish, except the
ones between the Lorentz generators and the pure supertranslations, which are
unchanged,

$[\tilde{M}_{a},S_{\alpha}]=G_{a\alpha}^{i}T_{i}+G_{a\alpha}^{\beta}S_{\beta}\,.$
(103)

We have thus not reached the direct sum structure yet. We will now show how to
achieve this task.

##  5 The log-BMS4 superalgebra

###  5.1 Log-BMS4 superalgebra and Jacobi identities

In the preceding developments, we did not include logarithmic
supertranslations. This is because the boundary conditions of
Fuentealba:2021xhn  froze those transformations, which did not appear
therefore as asymptotic symmetries.

Due to the absence of logarithmic supertranslations in the super-BMS4 algebra
of Fuentealba:2021xhn , one can only disentangle the higher order
supersymmetric transformations from the algebra, but not the ordinary
supertranslations which do not have their canonically conjugate partners.

In order to complete the direct sum decomposition of the superalgebra, we need
to bring in the logarithmic supertranslations. This is achieved by relaxing
the boundary conditions of Fuentealba:2021xhn  by allowing terms that are
generated by diffeomorphisms involving asymptotically $\log r$, as in
Fuentealba:2022xsz .

We will not go through the tedious procedure of constructing these boundary
conditions and the asymptotic symmetries and charges, but will give instead
directly the resulting (nonlinear) algebra, which is the minimal extension
that respects Jacobi identities. It reads

$\displaystyle[M_{a},M_{b}]$ $\displaystyle=f_{ab}^{c}M_{c}\,,$ (104)
$\displaystyle{}[M_{a},T_{i}]$ $\displaystyle=R_{ai}^{j}T_{j}\,,$ (105)
$\displaystyle{}[M_{a},S_{\alpha}]$
$\displaystyle=G_{a\alpha}^{i}T_{i}+G_{a\alpha}^{\beta}S_{\beta}\,,$ (106)
$\displaystyle{}[M_{a},L^{\alpha}]$
$\displaystyle=-G_{a\beta}^{\alpha}L^{\beta}\,,$ (107)
$\displaystyle{}[L^{\alpha},S_{\beta}]$
$\displaystyle=\delta_{\beta}^{\alpha}\,,$ (108)
$\displaystyle{}[M_{a},Q_{I}]$
$\displaystyle=g_{aI}^{J}Q_{J}+V_{aIB}^{i}s^{B}T_{i}+V_{aIB}^{\alpha}s^{B}S_{\alpha}\,,$
(109) $\displaystyle{}[M_{a},q_{A}]$
$\displaystyle=h_{aA}^{B}q_{B}+U_{aAB}^{i}s^{B}T_{i}+U_{aAB}^{\alpha}s^{B}S_{\alpha}\,,$
(110) $\displaystyle{}[M_{a},s^{B}]$ $\displaystyle=-h_{aC}^{B}s^{C}\,,$ (111)
$\displaystyle\\{s^{A},q_{B}\\}$ $\displaystyle=\delta_{B}^{A}\,,$ (112)
$\displaystyle\\{Q_{I},q_{A}\\}$
$\displaystyle=d_{IA}^{i}T_{i}+d_{IA}^{\alpha}S_{\alpha}\,,$ (113)
$\displaystyle\\{q_{A},q_{B}\\}$
$\displaystyle=d_{AB}^{i}T_{i}+d_{AB}^{\alpha}S_{\alpha}\,,$ (114)
$\displaystyle\\{Q_{I},Q_{J}\\}$ $\displaystyle=d_{IJ}^{i}T_{i}\,.$ (115)

Jacobi identities $\left(L^{\alpha},q_{A},q_{B}\right)$ and
$\left(L^{\alpha},q_{A},Q_{I}\right)$ require the presence of the additional
non-vanishing brackets

$\displaystyle[L^{\alpha},q_{A}]$
$\displaystyle=n_{AB}^{\alpha}s^{B}\,,\quad[L^{\alpha},Q_{I}]=n_{IB}^{\alpha}s^{B}\,,$
(116)

with

$\displaystyle d_{IA}^{\alpha}$ $\displaystyle=n_{IA}^{\alpha}\,,\qquad
d_{AB}^{\alpha}=n_{AB}^{\alpha}+n_{BA}^{\alpha}\,.$ (117)

The remaining brackets vanish

$[T_{i},Q_{I}]=[T_{i},q_{A}]=[S_{\alpha},q_{A}]=[L^{\alpha},Q_{I}]=[T_{i},s^{B}]=[S_{\alpha},s^{B}]=\\{Q_{I},s^{A}\\}=0\,.$
(118)

The following Jacobi identities are particularly useful:

  * •

From

$[M_{a},[M_{b},S_{\alpha}]]+[M_{b},[S_{\alpha},M_{a}]]+[S_{\alpha},[M_{a},M_{b}]]=0\,,$
(119)

we obtain the relations

$\displaystyle
G_{b\alpha}^{j}R_{aj}^{i}+G_{b\alpha}^{\beta}G_{a\beta}^{i}-\left(a\leftrightarrow
b\right)$ $\displaystyle=f_{ab}^{c}G_{c\alpha}^{i}\,,$ (120) $\displaystyle
G_{b\alpha}^{\beta}G_{a\beta}^{\gamma}-\left(a\leftrightarrow b\right)$
$\displaystyle=f_{ab}^{c}G_{c\alpha}^{\gamma}\,,$ (121)

  * •

Analogously, from

$[M_{a},[M_{b},T_{i}]]+[M_{b},[T_{i},M_{a}]]+[T_{i},[M_{a},M_{b}]]=0\,,$ (122)

we get the following relation

$R_{bj}^{k}R_{ak}^{i}-\left(a\leftrightarrow b\right)=f_{ab}^{c}R_{cj}^{i}\,,$
(123)

  * •

The identity

$[M_{a},\\{Q_{I},q_{A}\\}]-\\{q_{A},[M_{a},Q_{I}]\\}+\\{Q_{I},[q_{A},M_{a}]\\}=0\,,$
(124)

enables one again to express the quadratic structure constants $V_{aIB}^{i}$
and $V_{aIB}^{\beta}$ in terms of the other structure constants,

$\displaystyle V_{aIB}^{i}$
$\displaystyle=d_{IB}^{\alpha}G_{a\alpha}^{i}+d_{IB}^{j}R_{aj}^{i}-g_{aI}^{J}d_{JB}^{i}-h_{aB}^{C}d_{IC}^{i}\,,$
(125) $\displaystyle V_{aIB}^{\beta}$
$\displaystyle=d_{IB}^{\alpha}G_{a\alpha}^{\beta}-g_{aI}^{J}d_{JB}^{\beta}-h_{aB}^{C}d_{IC}^{\beta}\,,$
(126)

  * •

And similarly, the identity

$[M_{a},\\{q_{A},q_{B}\\}]-\\{q_{B},[M_{a},q_{A}]\\}+\\{q_{A},[q_{B},M_{a}]\\}=0\,,$
(127)

leads to

$\displaystyle 2U_{a(AB)}^{i}$
$\displaystyle=d_{AB}^{\alpha}G_{a\alpha}^{i}+d_{AB}^{j}R_{aj}^{i}-h_{aB}^{C}d_{AC}^{i}-h_{aA}^{C}d_{BC}^{i}\,,$
(128) $\displaystyle 2U_{a(AB)}^{\beta}$
$\displaystyle=d_{AB}^{\alpha}G_{a\alpha}^{\beta}-h_{aB}^{C}d_{AC}^{\beta}-h_{aA}^{C}d_{BC}^{\beta}\,,$
(129)

  * •

The identity

$[M_{a},[M_{b},q_{A}]]+[M_{b},[q_{A},M_{a}]]+[q_{A},[M_{a},M_{b}]]=0\,,$ (130)

implies

$\displaystyle h_{bA}^{C}h_{aC}^{B}-\left(a\leftrightarrow b\right)$
$\displaystyle=f_{ab}^{c}h_{cA}^{B}\,,$ (131) $\displaystyle
h_{bA}^{C}U_{aCB}^{i}-h_{aB}^{C}U_{bAC}^{i}+R_{aj}^{i}U_{bAB}^{j}+G_{a\alpha}^{i}U_{bAB}^{\alpha}-\left(a\leftrightarrow
b\right)$ $\displaystyle=f_{ab}^{c}U_{cAB}^{i}\,,$ (132) $\displaystyle
h_{bA}^{C}U_{aCB}^{\alpha}-h_{aB}^{C}U_{bAC}^{\alpha}+G_{a\beta}^{\alpha}U_{bAB}^{\beta}-\left(a\leftrightarrow
b\right)$ $\displaystyle=f_{ab}^{c}U_{cAB}^{\alpha}\,.$ (133)

  * •

Finally, from the identities involving the sets of generators
$\left(M_{a},L^{\alpha},q_{A}\right)$ and
$\left(M_{a},S_{\alpha},q_{A}\right)$, we can express the structure constants
$U_{aAB}^{\alpha}$ and $V_{aIB}^{\alpha}$ in terms of the others,

$\displaystyle U_{aAB}^{\alpha}$
$\displaystyle=n_{AB}^{\beta}G_{a\beta}^{\alpha}-h_{aA}^{C}n_{CB}^{\alpha}-h_{aB}^{C}n_{AC}^{\alpha}\,,$
(134) $\displaystyle V_{aIB}^{\alpha}$
$\displaystyle=n_{IB}^{\beta}G_{a\beta}^{\alpha}-g_{aI}^{J}n_{JB}^{\alpha}-h_{aB}^{C}n_{IC}^{\alpha}\,.$
(135)

###  5.2 Algebraic decoupling

The redefinitions that enable one to algebraically decouple the infinite-
dimensional set of charges $\left(S_{\alpha},q_{A},L^{\beta},s^{B}\right)$
from the super-Poincaré generators in the log-BMS4 superalgebra are direct
generalizations of those of the previous sections. We define

$\displaystyle\tilde{Q}_{I}$
$\displaystyle=Q_{I}-d_{IB}^{i}s^{B}T_{i}-d_{IB}^{\alpha}s^{B}S_{\alpha}\,,$
(136) $\displaystyle\tilde{q}_{A}$
$\displaystyle=q_{A}-\frac{1}{2}d_{AB}^{i}s^{B}T_{i}-\frac{1}{2}d_{AB}^{\alpha}s^{B}S_{\alpha}\,,$
(137) $\displaystyle\tilde{M}_{a}$
$\displaystyle=M_{a}-G_{a\beta}^{i}L^{\beta}T_{i}-G_{a\beta}^{\gamma}L^{\beta}S_{\gamma}+h_{aA}^{B}s^{A}q_{B}$
$\displaystyle+\frac{1}{2}\left(U_{aAB}^{i}-h_{aA}^{C}d_{BC}^{i}-n_{AB}^{\beta}G_{a\beta}^{i}\right)s^{A}s^{B}T_{i}-h_{aA}^{C}n_{CB}^{\alpha}s^{A}s^{B}S_{\alpha}\,.$
(138) $\displaystyle\tilde{L}^{\alpha}$
$\displaystyle=L^{\alpha}+\frac{1}{2}n_{[AB]}^{\alpha}s^{A}s^{B}\,.$ (139)

Using the above consequences of the Jacobi identities, we find successively
that:

  * •

The brackets of the fermionic generators $\\{\tilde{Q}_{I},\tilde{Q}_{J}\\}$,
$\\{\tilde{Q}_{I},\tilde{q}_{A}\\}$ and $\\{\tilde{q}_{A},\tilde{q}_{B}\\}$
simplify to

$\\{\tilde{Q}_{I},\tilde{Q}_{J}\\}=d_{IJ}^{i}T_{i}\,,\qquad\\{\tilde{Q}_{I},\tilde{q}_{A}\\}=0\,,\qquad\\{\tilde{q}_{A},\tilde{q}_{B}\\}=0\,,$
(140)

while the brackets

$\\{\tilde{Q}_{I},s^{A}\\}=0=\\{s^{A},s^{B}\\}\,,\qquad\\{s^{A},\tilde{q}_{B}\\}=\delta_{B}^{A}\,,$
(141)

are unchanged.

  * •

The brackets between the fermionic and Lorentz generators reduce to

$\displaystyle[\tilde{M}_{a},\tilde{Q}_{I}]$
$\displaystyle=g_{aI}^{J}\tilde{Q}_{J}\,,$ (142)
$\displaystyle[\tilde{M}_{a},\tilde{q}_{C}]$
$\displaystyle=0\,,\quad[\tilde{M}_{a},s^{C}]=0\,.$ (143)

  * •

The brackets of the new logarithmic supertranslation generators
$\tilde{L}^{\beta}$ with the super-Poincaré generators $\tilde{M}_{a}$,
$T_{i}$ and $\tilde{Q}_{I}$ are zero:

$\displaystyle[\tilde{M}_{a},\tilde{L}^{\alpha}]$ $\displaystyle=0\,,$ (144)
$\displaystyle[\tilde{L}^{\alpha},\tilde{Q}_{I}]$ $\displaystyle=0\,.$ (145)

Similarly,

$[\tilde{L}^{\alpha},\tilde{q}_{A}]=0\,.$ (146)

  * •

Finally one verifies that the Lorentz algebra is fulfilled,

$[\tilde{M}_{a},\tilde{M}_{b}]=f_{ab}^{c}\tilde{M}_{c}\,.$ (147)

Again, the derivation of all these relations is direct but rather tedious. We
provide the explicit verification of (147) in the appendix.

Therefore, after the redefinitions given in this section are performed, the
log-BMS4 superalgebra takes the direct sum structure of “super-Poincaré”
$\oplus$ “Abelian superalgebra with central extension” (Heisenberg algebra).
The latter is infinite-dimensional and contains the pure supertranslations and
the angle-dependent supersymmetries, which are thus decoupled from the super-
Poincaré generators in the superalgebra.

The non-vanishing brackets are

$\displaystyle[\tilde{M}_{a},\tilde{M}_{b}]$
$\displaystyle=f_{ab}^{c}\tilde{M}_{c}\,,$ (148)
$\displaystyle{}[\tilde{M}_{a},T_{i}]$ $\displaystyle=R_{ai}^{j}T_{j}\,,$
(149) $\displaystyle{}[\tilde{M}_{a},\tilde{Q}_{I}]$
$\displaystyle=g_{aI}^{J}\tilde{Q}_{J}\,,$ (150)
$\displaystyle{}\\{\tilde{Q}_{I},\tilde{Q}_{J}\\}$
$\displaystyle=d_{IJ}^{i}T_{i}\,,$ (151)

(super-Poincaré algebra) and

$\displaystyle[L^{\alpha},S_{\beta}]$
$\displaystyle=\delta_{\beta}^{\alpha}\,,$ (152)
$\displaystyle{}\\{s^{A},\tilde{q}_{B}\\}$ $\displaystyle=\delta_{B}^{A}\,,$
(153)

(infinite-dimensional Heisenberg algebra).

##  6 Conclusions

In this paper, we have shown that the method of Fuentealba:2022xsz  can be
extended to the BMS algebra in 5 dimensions and to BMS superalgebras. In all
cases, we have been able to redefine (super-)Poincaré generators that are free
from pure supertranslation ambiguities and angle-dependent supergauge
ambiguities. Namely, we have exhibited new (super-)Poincaré generators that
have vanishing Poisson brackets with the charge-generators of the pure
supertranslations and of the angle-dependent supersymmetry transformations.

What makes this construction possible is the key property that the BMS
supertranslations and BMS supersymmetry transformations have canonically
conjugate charges, which play the same role as the logarithmic
supertranslation charges of Fuentealba:2022xsz . Since the presence of
canonical partners to the pure BMS transformations is expected to hold in
higher dimensions and for extended supersymmetries (we verified this to be the
case in the linear case), similar redefinitions are then also expected to be
possible in those generalizations.

The redefinitions of the generators necessary for bringing the algebra to the
desired direct sum form are nonlinear. The nonlinear context is natural for
discussing charges and symmetries and is compatible with the general structure
$\int d^{d}x\xi^{\alpha}\mathcal{H}_{\alpha}+B_{\xi}$ of generators of
improper gauge transformations: the nonlinearities occur through the
asymptotic nonlinear dependence of the gauge parameters on the charges.

We note that in the new presentations of the superalgebras, there is no square
root of the pure supertranslations. We also note that since the BMS
(super-)algebras can be written as direct sums of the (super-)Poincaré algebra
with infinite-dimensional Heisenberg superalgebras, the theory of their
representations takes a simpler form. We hope to return to this question in
the future.

## Acknowledgments

We thank Cédric Troessaert for important discussions. O.F. is grateful to the
Collège de France for kind hospitality while this work was completed. This
work was partially supported by a Marina Solvay Fellowship (O.F.) and by FNRS-
Belgium (conventions FRFC PDRT.1025.14 and IISN 4.4503.15), as well as by
funds from the Solvay Family.

##  Appendix A Brackets of new Lorentz generators

In this appendix, we provide the detailed derivation of the relation (147). If
one sets the logarithmic supertranslation generators equal to zero, this
yields also a derivation of (95).

Our aim is to prove the relation

$[\tilde{M}_{a},\tilde{M}_{b}]=f_{ab}^{c}\tilde{M}_{c}\,.$ (154)

The explicit computation goes as follows. After manipulation of the indices
and using the algebra, we obtain that

$\displaystyle[\tilde{M}_{a},\tilde{M}_{b}]$
$\displaystyle=f_{ab}^{c}\tilde{M}_{c}+\left[-f_{ab}^{d}h_{dB}^{C}-h_{bD}^{C}h_{aB}^{D}+h_{aD}^{C}h_{bB}^{D}\right]s^{B}q_{C}$
(155)
$\displaystyle\quad-\left[\frac{1}{2}f_{ab}^{d}\left(U_{dBC}^{i}-h_{dB}^{A}d_{CA}^{i}-n_{BC}^{\beta}G_{d\beta}^{i}\right)+h_{aB}^{D}h_{bC}^{E}d_{DE}^{i}\right]s^{B}s^{C}T_{i}$
$\displaystyle\quad+\Big{[}h_{bB}^{D}U_{aDC}^{i}+\frac{1}{2}\left(U_{bBC}^{\alpha}-h_{bB}^{D}d_{CD}^{\alpha}-G_{b\beta}^{\alpha}n_{BC}^{\beta}\right)G_{a\alpha}^{i}$
$\displaystyle\quad+\frac{1}{2}\left(U_{bBC}^{j}-h_{bB}^{A}d_{CA}^{j}-n_{BC}^{\beta}G_{b\beta}^{j}\right)R_{aj}^{i}-\left(a\leftrightarrow
b\right)\Big{]}s^{B}s^{C}T_{i}$ (156)
$\displaystyle\quad-\left[f_{ab}^{d}\left(-h_{dB}^{D}n_{DC}^{\alpha}\right)+h_{aB}^{D}h_{bC}^{E}d_{DE}^{\alpha}\right]s^{B}s^{C}S_{\alpha}$
$\displaystyle\quad+\left[h_{bB}^{D}U_{aDC}^{\alpha}+\left(-h_{bB}^{D}n_{DC}^{\alpha}\right)G_{a\beta}^{\alpha}-\left(a\leftrightarrow
b\right)\right]s^{B}s^{C}S_{\alpha}\,.$ (157)

The second term at the right hand side of (155) vanishes by virtue of the
identity (131). Using the identity (128), all the terms in the line above
(156) together with the terms in the line (156) reduce to

$\displaystyle\frac{1}{2}\left(f_{ab}^{d}h_{dB}^{D}+h_{bF}^{D}h_{aB}^{F}-h_{aF}^{D}h_{bB}^{F}\right)d_{CD}^{i}s^{B}s^{C}T_{i}$
$\displaystyle-\frac{1}{2}f_{ab}^{d}U_{d[AB]}^{i}s^{A}s^{B}T_{i}+\left[2h_{bA}^{C}U_{a[CB]}^{i}+U_{b[AB]}^{j}R_{aj}^{i}+U_{b[AB]}^{\alpha}G_{a\alpha}^{i}-\left(a\leftrightarrow
b\right)\right]s^{A}s^{B}T_{i}$
$\displaystyle\frac{1}{2}n_{AB}^{\beta}f_{ab}^{d}G_{d\beta}^{i}s^{A}s^{B}T_{i}+\frac{1}{2}n_{AB}^{\beta}\left[-G_{b\beta}^{\alpha}G_{a\alpha}^{i}-G_{b\beta}^{j}R_{aj}^{i}-\left(a\leftrightarrow
b\right)\right]s^{A}s^{B}T_{i}\,.$

The first line of the above expression is zero by virtue of the identity
(131). The second one vanishes by using the identity (132), while the third
one is zero after considering indentity (120).

For the remaining terms in the line above (157) and those in (157), we make
use of the identity (134). These terms reduce then to

$\displaystyle\left(f_{ab}^{d}h_{dB}^{D}+h_{bF}^{D}h_{aB}^{F}-h_{aF}^{D}h_{bB}^{F}\right)n_{DC}^{\alpha}s^{B}s^{C}S_{\alpha}\,,$

which vanishes by virtue of the identity (131).

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# Note on the conjugacy classes of elements and their centralizers for the
free product of two groups

Dan Burghelea  Department of Mathematics, The Ohio State University, Columbus,
OH 43210,USA. Email<EMAIL_ADDRESS>
###### Abstract

We describe the conjugacy classes of the elements of the free product of two
groups and their centralizers and, as a consequence, we correct the
calculation of the cyclic and periodic cyclic homology of the group ring of
the free product of two groups given in a previous paper.

##  1 Introduction

This Note was prompted by a mistake pointed out by Markus Land about the
cyclic and periodic cyclic homology of the group ring of the free product,
precisely Propositions II and IIp in the paper The cyclic homology of the
group rings  published in Comment. Math. Helv., 60 (1985) no 3, 354-365 . The
mistake was the result of a miscalculation of the centralizers of the
conjugacy classes of elements of the free product of two groups. In this note
we provide a correct description of them and, as a consequence, correct the
statements of Proposition II and IIp in [1].

In consistency with the notation in [1] for a group $G$ and an element $x\in
G$ one denotes by $G_{x}:=\\{y\in G\mid y\cdot x=x\cdot y\\}$ the centralizer
of $x,$ by $\\{x\\}$ the subgroup generated by the element $x$ and by $N_{x}$
the quotient group $N_{x}:=G_{x}/\\{x\\}.$ These groups remain isomorphic for
all $x$ in the same conjugacy class. Denote by $\langle G\rangle$ the set of
conjugacy classes of elements of $G$ and for $x\in G$ write $\hat{x}$ for the
conjugacy class of $x$. For the groups $H$ and $G$ one denotes the nonzero
elements by $h$ and $g$ and the neuter elements by $e_{H}$ and $e_{G}.$

Consider the free product $P=H\ast G.$ Any element of $x\in P$ is
representable (not uniquely) by a word $s_{1}s_{2}\cdots s_{r}$ with $s_{i}\in
H\sqcup G.$ The product of the elements $x$ and $x^{\prime},$ represented by
the words $s_{1}s_{2}\cdots s_{r}$ and $s^{\prime}_{1}s^{\prime}_{2}\cdots
s^{\prime}_{r^{\prime}}$ is representable by the concatenation
$s_{1}s_{2}\cdots s_{r}s^{\prime}_{1}s^{\prime}_{2}\cdots
s^{\prime}_{r^{\prime}}.$ Modifying a word representation of an element $x\in
P$ by

i) removing all elements $s_{i}$ of the form $e_{H}$ and $e_{G},$

ii) replacing consecutive elements $\cdots s_{i}s_{i+1}\cdots$ by their
product when in either $H$ or $G,$

leads to a smaller word representations of $x\in P,$ the reduced word
representation, which is unique. (The empty word is the reduced representation
of $e_{P}$).

The reduced word representation $s_{1}s_{2}\cdots s_{r}$ for the element $x\in
P$ is characterized by

a) $s_{i}\in(H\setminus e_{H})\sqcup(G\setminus e_{G})$

b) consecutive $s_{i}$ and $s_{i+1}$ belong to different groups.

Consequently, a nontrivial elements of $x\in P$ has an unique reduced word
representation of one of the following seven types:

Type 1: $w=\ \ h_{1}g_{1}h_{2}g_{2}\cdots h_{k}g_{k},$ $k\geq 1,$

Type 2: $w=\ \ h_{1}g_{1}h_{2}g_{2}\cdots h_{k}g_{k}\ h,$ $k\geq 1,$

Type 3: $w=g\ h_{1}g_{1}h_{2}g_{2}\cdots h_{k}g_{k},$ $k\geq 1,$

Type 4: $w=g\ h_{1}g_{1}h_{2}g_{2}\cdots h_{k}g_{k}\ h,$ $k\geq 1,$

Type 5: $w=g,$

Type 6: $w=h,$

Type 7: $w=gh.$

Because of the unicity of the reduced word representation the following facts
hold true:

  1. 1.

if two nontrivial elements commute then they belong to the same type; as a
consequence if $x$ is represented by a reduced word of type 5 or type 6 then
$P_{x}=G_{x}$ or $P_{x}=H_{x}.$

  2. 2.

any nontrivial element $x\in P$ of type different from type 5 and type 6 is
conjugate to an element of type 1.

  3. 3.

the following proposition holds true

###### Proposition 1.1

Suppose $w^{\prime}=h_{1}\ g_{1}\ h_{2}\ g_{2}\cdots h_{r}\ g_{r}$ and
$w^{\prime}=h^{\prime}_{1}\ g^{\prime}_{1}\ h^{\prime}_{2}\
g^{\prime}_{2}\cdots h^{\prime}_{r}\ g^{\prime}_{r^{\prime}}$ are two reduced
words of type 1, representing elements $x$ and $x^{\prime}$ in $P$ s.t.
$x\cdot x^{\prime}=x^{\prime}\cdot x$ and suppose $c$ is the greatest common
divisor of $r$ and $r^{\prime}.$ Then there exists a reduced word of type 1,
$w_{0}=h_{1}^{\prime\prime}\ g_{1}^{\prime\prime}\ h_{2}^{\prime\prime}\
g_{2}^{\prime\prime}\cdots h_{c}^{\prime\prime}\ g_{c}^{\prime\prime},$ such
that $w$ is the concatenation of $r/c$ copies of $w_{0}$ and $w^{\prime}$ is
the concatenation of $r^{\prime}/c$ copies of $w_{0}.$

###### Observation 1.2

Items 1 and 2 above imply that $\langle P\rangle=e_{P}\sqcup(\langle
H\rangle\setminus e_{H})\sqcup(\langle G\rangle\setminus e_{G})\sqcup U$ with
$U=\\{\hat{x}\in\langle P\rangle\mid\hat{x}\cap(e_{H}\ast
G)=\emptyset,\hat{x}\cap(H\ast e_{G})=\emptyset\\}$ and the centralizers in
$P$ of the elements in $e_{H}\ast G\subset P$ resp. in $H\ast e_{G}\subset P$
remain the same as the centralizers in $G$ and $H$ resp..

Item 3 ( i.e. Proposition (1.1) shows that for $x\in\hat{x}\in U,$ the pair
group-subgroup $(P_{x},\\{x\\})$ is isomorphic to the pair
$(\mathbb{Z},k(x)\mathbb{Z}),$ hence $N_{x}\simeq\mathbb{Z}_{k(x)}.$ Here
$k(x)$ is the largest integer $k$ s.t. $x=y^{k}.$

##  2 Proof of Proposition 1.1

Let $\mathbb{S}$ be a set of symbols. Let
$\mathcal{S}:=\\{s_{1},s_{2},\cdots,s_{r}\cdots,s_{n}\\}$ be an ordered set of
symbols with $s_{i}\in\mathbb{S}$ (i.e. an word with letters in $\mathbb{S}$),
$p<n$ and $d=n-p.$

###### Lemma 2.1

Suppose that the collection $\mathcal{S}$ satisfies:

  1. (i)

$s_{i}=s_{d+i}$ for $i\leq p,$

  2. (ii)

$s_{i}=s_{i+p}$ for $i\leq d.$

1\. If $n$ and $p$ are relatively prime then all $s_{i}$ are equal.

2\. If $c$ is the greatest common divisor of $n$ and $p$ and
$\mathcal{S}^{\prime}=s_{1},s_{2},\cdots s_{c}$ is the ordered set of the
first $c$ symbols of $\mathcal{S}$ then $\mathcal{S}$ is the concatenation of
$n/c$ copies of $\mathcal{S}^{\prime}.$

###### Proof.

For any $r=1,2,\cdots,d$ define the subsets of
$\mathcal{S}(r)\subset\mathcal{S},$ consisting of all elements of
$\mathcal{S}$ indexed by $r+kd$ for $k=0,1,\cdots,$ namely

$\mathcal{S}(r):=\\{s_{r},s_{r+d},s_{r+2d},\cdots s_{r+kd}\cdots\\}.$

Note that the sets $\mathcal{S}(r)$ are disjoint and their union is
$\mathcal{S}.$

Proof of item 1: For each $\mathcal{S}(r)$ let $k_{r}$ be the unique integer
such that $r+(k_{r}-1)d\leq p<r+k_{r}d.$ The first inequality guaranties that

$r+k_{r}d-p\leq d.$ (1)

In view of the hypothesis (i) all elements of $\mathcal{S}(r)$ are equal and
in view of the hypothesis (ii) the elements of the collections $S(r)$ and
$\mathcal{S}(r+k_{r}d-p)$ are equal;

Consider the pairs of integers $(r_{i},\kappa_{i})$ with $1\leq i\leq d$
defined inductively by:

a) $r_{1}=1$ and $\kappa_{1}:=k_{1}$

b) $r_{i+1}=1+(\kappa_{1}+\kappa_{2}+\cdots\kappa_{i})d-ip$ and
$\kappa_{i+1}=k_{r_{i+1}}$

Note that in view of inequality (1) all $r_{i}\leq d.$

Let $\mathcal{S}_{i}=:\mathcal{S}(r_{i}).$ Since $p$ and $d$ are relatively
prime $\mathcal{S}_{i}$ and $\mathcal{S}_{j}$ for $i\neq j$ can never be the
same since $d$ can not divide $i-j.$ Then the sets
$\mathcal{S}_{i},i=1,2,\cdots d$ provide a permutation of the sets
$\mathcal{S}(r),r=1,2,\cdots d.$ Hypothesis (ii) implies that the elements of
$\mathcal{S}_{i}$ and $\mathcal{S}_{i+1}$ are equal for any $i$ hence all
elements of $\mathcal{S}$ are equal.

Proof of item 2: Consider the set of symbols
$\mathbb{S}^{\prime}=\mathbb{S}\times\mathbb{S}\times\cdots\mathbb{S},$ the
$c-$ fold cartesian product of $\mathbb{S};$ clearly
$\mathcal{S}^{\prime}\in\mathbb{S}^{\prime}.$ Interpret $\mathcal{S}$ as an
ordered set of $n/c$ symbols of $\mathbb{S}^{\prime}.$ Clearly item 1. implies
item 2.

∎

To prove Proposition 1.1 consider the set of symbols $\mathbb{S}=(H\setminus
e_{H})\times(G\setminus e_{G})$ and write $w=s_{1}\ s_{2}\ \cdots\ s_{n},$ and
$w^{\prime}=s^{\prime}_{1}\ s^{\prime}_{2}\ \cdots\ s^{\prime}_{p},$ Since the
concatenation $ww^{\prime}$ and $w^{\prime}w$ are the same one has

  1. 1.

$s^{\prime}_{i}=s_{i}$ for $i\leq p$

  2. 2.

$s^{\prime}_{i}=s_{d+i}$ for $i\leq p$

  3. 3.

$s_{i}=s_{i+p}$ for $i\leq n-p=d$

which implies

$s_{i}=s_{d+i}$ for all $i\leq p$ and

$s_{i}=s_{p+i}$ for $i\leq d,$

which in view of Lemma (2.1) implies that $w$ is the concatenation exactly
$n/c$ times of $w_{0}=s_{1}s_{2}\cdots,s_{c}$ and in view of the equality
$s_{i}=s^{\prime}_{i}$ for $i\leq d,$ $w^{\prime}$ is the concatenation of
exactly $p/c$ copies of $w_{0}.$

##  3 Cyclic resp. periodic cyclic homology of the group-ring $R[P],$

Let $R$ be a commutative ring with unit and $G$ a group. Recall that the
reduced cyclic resp. periodic cyclic homology, $\tilde{H}C_{\ast}(R[G])$ resp.
$\tilde{PHC}_{\ast}(R[G]),$ of the group ring $R[G]$ is the co-kernel of the
split injective map $HC_{\ast}(R[e_{g}])=HC_{\ast}(R)\to HC_{\ast}(R[G])$
resp. $PHC_{\ast}(R[e_{G}])=PHC_{\ast}(R)\to PHC_{\ast}(R[G])$ induced by the
inclusion of the trivial subgroup $e_{G}$ to $G.$ One refers to the cyclic
resp. periodic cyclic homology of the group ring $R[G]$ as the unreduced
version of these homologies. Clearly, the unreduced version is the direct sum
of the reduced version with one copy of the cyclic resp. periodic cyclic
homology of $R.$

As shown in [1] all these homologies, reduced or unreduced, say
$\mathcal{H}_{\ast}(R[G]),$ are graded $R-$modules which are direct sums of
graded $R-$modules $\mathcal{H}_{\ast}(R[G])_{\hat{x}}$ indexed by the
conjugacy classes $\hat{x}\in\langle G\rangle,$ referred to as the
contribution of $\hat{x},$

$\mathcal{H}_{\ast}(R[G])=\oplus_{\hat{x}\in\langle
G\rangle}\mathcal{H}(R[G])_{\hat{x}}.$

For each $\hat{x}\neq e_{G}$ the contribution to the reduced and unreduced
version are the same but for $e_{G}$ the unreduced version is equal to the
reduced version direct sum $\mathcal{H}_{\ast}(R).$

For each conjugacy class $\hat{x}$ one defines $n(\hat{x}):=n(x)$ the order of
the element $x$ and $k(\hat{x}):=\kappa(x)$ the largest $k$ s.t. $x=y^{k}$;
clearly $n(x)$ and $\kappa(x)$ are the same for all $x$ in the same conjugacy
class.

Recall from [1] the notations:

  1. 1.

$K_{\ast}(R[G]):=\begin{cases}\oplus_{n\geq 0}H_{2n}(BG;R)\ \rm{if}\
\ast=\rm{even}\\\ \oplus_{n\geq 0}H_{2n+1}(BG;R)\ \rm{if}\
\ast=\rm{odd}\end{cases}\ \
\tilde{K}_{\ast}(R[G]):=\begin{cases}\oplus_{n>0}H_{2n}(BG;R)\ \rm{if}\
\ast=\rm{even}\\\ \oplus_{n\geq 0}H_{2n+1}(BG;R)\ \rm{if}\
\ast=\rm{odd}\end{cases}$

  2. 2.

for $x\in G$ with $n(x)=\infty$

$T_{\ast}(\hat{x};R)=T_{\ast}(x;R):=\lim(\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
6.75pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
30.75pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
30.75pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{H_{\ast+2n}(BN_{x}:r)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
105.98802pt\raise 5.39166pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.39166pt\hbox{$\scriptstyle{S}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 121.72684pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
121.72684pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{H_{\ast+2n-2}(BN_{x};R)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
219.14005pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
219.14005pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\cdots}$}}}}}}}\ignorespaces}}}}\ignorespaces$

with $S$ the Gysin homomorphism of the fibration $B\\{x\\}=S^{1}\to BG_{x}\to
BN_{x}$ where $N_{x}=G_{x}/\\{x\\},$ which up to isomorphism depends only on
the conjugacy class of $x$ and then denoted by $T(\hat{x};R).$

Recall from [1] that the contribution of $\hat{x}$ when $0\neq
n(\hat{x})<\infty$ is

$HC_{\ast}(R[G])_{\hat{x}}=H_{\ast}(B(N_{\hat{x}})\times BS^{1}\times
K(\mathbb{Z}_{n(\hat{x})},1);R),\quad
PHC_{\ast}(R[G])_{\hat{x}}=K_{\ast}(R[N_{\hat{x}}])$

and when $n(\hat{x})=\infty$ is

$HC_{\ast}(R[G])_{\hat{x}}=H_{\ast}(B(N_{x});R),\quad
PHC_{\ast}(R[G])_{\hat{x}}=T_{\ast}(\hat{x};R)$

while for $n(\hat{x})=0,$ hence $\hat{x}=e_{G},$ is

$HC_{\ast}(R[G])_{e_{G}}=H_{\ast}(B(G)\times BS^{1};R),\quad
PHC_{\ast}(R[G])_{e_{G}}=K_{\ast}(R[G])$

and

$\tilde{H}C_{\ast}(R[G])_{e_{G}}=H_{\ast}(B(G)\times BS^{1}/\ast\times
BS^{1};R).\quad\tilde{PHC}_{\ast}(R[G])_{e_{G}}=\tilde{K}_{\ast}(R[G])$

In particular one has

###### Proposition 3.1

$\tilde{PHC}_{\ast}(R[G])=\tilde{K}_{\ast}(R[G])\bigoplus(\oplus_{\hat{x}\in(\langle
G\rangle^{\prime}\setminus{e_{G}})}K_{\ast}(BN_{\hat{x}};R))\bigoplus(\oplus_{\hat{x}\in\langle
G\rangle^{\prime\prime}}T_{\ast}(\hat{x};R))$

where $\langle G\rangle^{\prime}:=\\{\hat{x}\in\langle G\rangle\mid
n(\hat{x})<\infty\\}$ and $\langle
G\rangle^{\prime\prime}=\\{\hat{x}\in\langle G\rangle\mid
n(\hat{x})=\infty\\}.$

An equivalent form of this proposition is stated in [1] for a field of
characteristic zero as Theorem 1’.

Let $H$ and $G$ be two groups and $P=H\ast G$ their free product. Recall that
$B(H\ast G)=BH\vee BG$ the base point union of the spaces $BH$ and $BG.$

As an immediate consequence of Observation (1.2) (description of conjugacy
classes of elements of $P$ and of their centralizers) in the previous section
one has

###### Proposition 3.2

$\tilde{H}C_{\ast}(R[H\ast
G])=\tilde{H}C_{\ast}(R[H])\bigoplus\tilde{H}C_{\ast}(R[G])\bigoplus(\oplus_{\hat{x}\in
U}H_{\ast}(B\mathbb{Z}_{k(\hat{x})};R))$ $\tilde{PHC}_{\ast}(R[H\ast
G])=\tilde{PHC}_{\ast}(R[H])\bigoplus\tilde{PHC}_{\ast}(R[G])\
\bigoplus(\oplus_{\hat{x}\in U}T_{\ast}(\hat{x};R))$

where

$H_{\ast}(B(\mathbb{Z}_{k});R)=\begin{cases}R\ \rm{for}\ \ast=0\\\
H_{1}(B(\mathbb{Z}_{k});R)\ \rm{for}\ \ast\rm{odd}\\\
H_{2}(B(\mathbb{Z}_{k});R)\ \rm{for}\ast\ \rm{even}\neq 0\end{cases}$

and

$T_{\ast}(\hat{x};R)=\begin{cases}H_{1}(B(\mathbb{Z}_{k(x)});R)\ \rm{for}\
\ast\rm{odd}\\\ H_{2}(B(\mathbb{Z}_{k(x)});R)\ \rm{for}\ast\
\rm{even}\end{cases}$

Note that if $R$ is an algebra over a field of characteristic zero then
$H_{\ast}(B(\mathbb{Z}_{k});R)$ is concentrated in degree zero and isomorphic
to $R$ and for any $\hat{x}\in U,$ $T_{\ast}(\hat{x};R)$ vanishes.

To correct all inaccuracies in [1] we insert the following errata to [1]

##  4 Errata to the paper The cyclic homology of the group rings, Comment.
Math. Helv., 60, 1985, no. 3, 354-365

In the paper The cyclic homology of the group rings  published in Comment.
Math. Helv., 60 (1985) no 3, 354-365 Propositions II and IIp, straightforward
consequences of the main result,Theorem I, are not true as stated. The
statements become correct provided the cyclic resp. periodic cyclic homology
are replaced by their reduced versions and the ring $R$ is a
$\mathbb{Q}-$algebra, for instance a field of characteristic zero. The reduced
cyclic resp. periodic cyclic homology of $R[G]$ is quotient of the obvious
split injective maps, $i_{G}:HC_{\ast}(R)\to HC_{\ast}(R[G])$ resp.
$i_{G}:PHC_{\ast}(R)\to PHC(R[G])$ induced by the inclusion $e_{G}\in G.$ Also
on line 1 page 363 to make the statement correct one shall replace
”$P_{x}\neq\\{x\\}$” by ”$N_{x}\neq\mathbb{Z}_{k(x)}$ with $\mathbb{Z}_{k}$
denoting the finite cyclic group of order $k,$ and $k(x)$ the largest integer
$k$ s.t. $x=y^{k}.$

I thank Markus Land for bringing this to my attention and suggesting the use
of reduced cyclic and periodic cyclic homology for group rings.

For an arbitrary commutative ring with unit $R,$ Propositions II and IIp
should be corrected as follows

  1. 1.

In Proposition II, $HC_{\ast}$ has to be replaced by the reduced version
$\tilde{H}C_{\ast}$ and the sentence ” $\ R_{\hat{\alpha}}=R$ regarded as a
graded module concentrated in the degree zero” by
”$R_{\hat{\alpha}}=H_{\ast}(B(\mathbb{Z}_{k(x)});R),x\in\hat{\alpha}.$”

  2. 2.

In Proposition IIp, $PHC_{\ast}$ has to be replaced by its reduced version,
$\tilde{PHC}_{\ast},$ and to the right side of the equality one should add
$\oplus_{\hat{x}\in U}T_{\ast}(\hat{x};R),$ with $U=\\{\hat{x}\in\langle H\ast
G\rangle\mid\hat{x}\cap(e_{H}\ast G)=\emptyset,\hat{x}\cap(H\ast
e_{G})=\emptyset\\}$ where $\langle\Gamma\rangle$ denotes the set of conjugacy
classes of elements of the group $\Gamma$ and $e_{\Gamma}$ the neuter element
of $\Gamma.$

Note that

$H_{\ast}(B(\mathbb{Z}_{k});R)=\begin{cases}R\ \rm{for}\ \ast=0\\\
H_{1}(B(\mathbb{Z}_{k});R)\ \rm{for}\ \ast\rm{odd}\\\
H_{2}(B(\mathbb{Z}_{k});R)\ \rm{for}\ast\ \rm{even}\neq 0\end{cases}$

and for $\hat{x}\in U,$ and $x\in\hat{x}$ the pair of group - subgroup
$(G_{x},\\{x\\})$ is isomorphic to the pair $(\mathbb{Z},k(x)\mathbb{Z}),$
hence $N_{x}=\mathbb{Z}_{k(x)},$ and

$T_{\ast}(\hat{x};R)=\begin{cases}H_{1}(B(\mathbb{Z}_{k(x)});R)\ \rm{for}\
\ast\rm{odd}\\\ H_{2}(B(\mathbb{Z}_{k(x)});R)\ \rm{for}\ast\
\rm{even}\end{cases}.$

## References

  * [1] Dan Burghelea, The cyclic homology of the group rings  Comment. Math. Helv., 60 (1985) no 3, 354-365