Physics Letters B 770 (2017) 252–256
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Creating 3, 4, 6 and 10-dimensional spacetime from W3 symmetry
J. Ambjørn
a,b,∗
, Y. Watabiki
c
a
The Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark
b
Institute for Mathematics, Astrophysics and Particle Physics (IMAPP), Radbaud University Nijmegen, Heyendaalseweg 135, 6525 AJ, Nijmegen, The Netherlands
c
Tokyo Institute of Technology, Dept. of Physics, High Energy Theory Group, 2-12-1 Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan
a r t i c l e i n f o a b s t r a c t
Article history:
Received
28 March 2017
Accepted
24 April 2017
Available
online 26 April 2017
Editor: J.
Hisano
We describe a model where breaking of W3 symmetry will lead to the emergence of time and
subsequently of space. Surprisingly the simplest such models which lead to higher dimensional
spacetimes are based on the four “magical” Jordan algebras of 3 × 3 Hermitian matrices with real,
complex, quaternion and octonion entries, respectively. The simplest symmetry breaking leads to
universes with spacetime dimensions 3, 4, 6, and 10.
© 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
String field theory is notoriously complicated, but there is a
baby version, namely non-critical string field theory [1–3]. Non-
critical
string theory describes two-dimensional quantum gravity
coupled to a conformal field theory with a central charge c < 1 and
the corresponding string field theory aims to describe the dynam-
ics
of merging and splitting of such strings. For c = 0the situation
is particularly simple. One has creation and annihilation operators
†
(L) and (L) for spatial universes of length L. An even simpler
string field theory exists, CDT string field theory [4]. The starting
point is the continuum limit of 2d “causal dynamical triangula-
tions”
(CDT) [5,6], a limit which in the case of trivial spacetime
topology is 2d quantum Horava–Lifshitz gravity [7] (for higher di-
mensional
CDT which might also be related to Horava–Lifshitz
gravity [8], see e.g. [9,10]). CDT string field theory describes the
dynamics of topology changes of spacetime. The Hamiltonian in-
volves
terms like
†
(L
1
)
†
(L
2
)(L
1
+ L
2
),
†
(L
1
+ L
2
)(L
2
)(L
1
), (1)
which describe the annihilation of a universe of length L
1
+ L
2
and the creation of two universes of lengths L
1
and L
2
, or the
reverse process [4]. The interaction term in the string field Hamil-
tonian
thus contains products of three annihilation and creation
operators. The same is true for standard non-critical string theory
*
Corresponding author.
E-mail
addresses: ambjorn@nbi.dk (J. Ambjørn), watabiki@th.phys.titech.ac.jp
(Y. Watabiki).
which is related to a special kind of W
(3)
symmetry which en-
sures
that the partition function can be viewed as a τ -function
of certain coupling constants [11]. This led us to realize that one
can obtain the CDT string field theory starting from a so-called
W
(3)
Hamiltonian by symmetry breaking [12]. The W
(3)
Hamilto-
nian
has a natural, so-called absolute vacuum and offers no obvious
spacetime interpretation, but breaking the W
(3)
-symmetry led to
a so-called physical vacuum and the emergence of time and one
spatial dimension. The purpose of this article is to generalize the
construction such that one can create universes with one time
direction and higher dimensional spaces. The simplest symmetry
breaking leads to spacetime dimensions 2 + 1, 3 + 1, 5 + 1 and
9 +1.
In
Sec. 2 we shortly review the W
(3)
Hamiltonian formalism
introduced in [12] and in Sec. 3 we generalize it to include in-
ternal
degrees of freedom. This leads to the introduction of the
so-called magical Jordan algebras and it is the structures of these
algebras which result in spacetime dimensions 2 + 1, 3 + 1, 5 +1
and
9 +1.
2. The W
(3)
Hamiltonian
The formal definition of a W
(3)
algebra in terms of operators
α
n
satisfying
[α
m
, α
n
]=m δ
m+n,0
, (2)
is the following
α(z) =
n∈Z
α
n
z
n+1
, W
(3)
(z) =
1
3
:α(z )
3
:=
n∈Z
W
(3)
n
z
n+3
. (3)
http://dx.doi.org/10.1016/j.physletb.2017.04.051
0370-2693/
© 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.