arXiv:hep-th/9206092v1 24 Jun 1992
FINITE AND INFINITE SYMMETRIES IN
(2+1)-DIMENSIONAL FIELD THEORY*
Center for Theoretical Physics
Laboratory for Nuclear Science
and Department of Physics
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139U.S.A.
350 Commonwealth Avenue
Boston, Massachusetts 02115
Dedicated to Franco Iachello on his 50th Birthday
Recent Problems in Mathematical Physics, Salamanca, Spain, June 1992; XIX International
Colloquium on Group Theoretical Methods in Physics, Salamanca, Spain, July 1992, Con-
densed Matter and High Energy Physics, Cagliari, Italy, September 1992.
Typeset in TEX by Roger L. GilsonCTP#2110
BU HEP 92-21
* This work is supported in part by funds provided by the U. S. Department of Energy
(D.O.E.) under contract #DE-AC02-76ER03069 (RJ), and #DE-AC02-89ER40509(S-YP).
These days, Franco Iachello is the eminent practitioner applying classical and finite groups
to physics. In this he is following a tradition at Yale, established by the late Feza Gursey,
and succeeding Gursey in the Gibbs chair; Gursey in turn, had Pauli as a mentor. Iachello’s
striking achievement has been to find an actual realization of arcane supersymmetry within
mundane adjacent even-odd nuclei. Thus far this is the only physical use of supersymmetry,
and its fans surely must be surprised at the venue. Here we describe the role of SO(2,1)
conformal symmetry in non-relativistic Chern–Simons theory: how it acts, how it controls
the nature of solutions, how it expands to an infinite group on the manifold of static solutions
thereby rendering the static problem completely integrable. Since Iachello has also used the
SO(2,1) group in various contexts, this essay is presented to him on the occasion of his fiftieth
We shall discuss finite- and infinite-dimensional conformal symmetries of field theories
with non-relativistic kinematics. Such field theories also describe the second quantization of
non-relativistic particle mechanics. Particle mechanics, with its second order in time dynam-
ics, has the structure of a relativistic field theory in one time and zero space dimensions, and
a relativistic field theory in any dimension can enjoy conformal symmetry. Thus there are
family relationships between the conformal symmetries of non-relativistic field theory, non-
relativistic particle mechanics and relativistic field theory, and our first task is to describe
A conformal transformation in (D + 1)-dimensional relativistic field theory changes the
independent variables, viz. the space-time coordinates xµof the fields (fields are dependent
variables), and infinitesimally reads
where fµis a conformal Killing vector, i.e. fµsatisfies the conformal Killing equation.
D + 1gµν∂αfα
Here gµν is the Minkowski metric tensor with signature (1,−1,−1,...) and D is the spatial
As is well-known, Eq. (1.2) has the finite number of1
2(D+2)(D+3) solutions for D > 1,
and conformal transformations form an SO(2,D + 1) group. The solutions to (1.2) comprise
D + 1 space-time translationsfµ(x) = aµ,aµconstant (1.3a)
2D(D + 1) space-time rotationsfµ(x) = ωµνxν,ωµν= −ωνµ
a single scale transformationfµ(x) = axµ,a constant(1.3c)
D + 1 special conformal transformationsfµ(x) = 2c · xxµ− cµx2,cµconstant
The finite versions of these are, respectively,
xµ→ xµ+ aµ
xµ→ Λµνxµ,ΛµαΛνβgµν= gαβ
1 − 2c · x + c2x2
The last, the finite special conformal transformation, can also be seen as an inversion, xµ→
xµ/x2, followed by a translation and another inversion, i.e. a translation in the inverted
At D = 1 there exists an infinite number of solutions to (1.2) corresponding to arbitrary
redefinition of x±=
√2(x0±x1) and forming an infinite parameter group. Infinitesimally we
δfx±= −f±(x±) ,f±arbitrary(1.5)
while the finite version reads
x±→ X±(x±) ,X±arbitrary(1.6)
A linear conformal transformation on a space-time multiplet of Lorentz covariant rela-
tivistic fields ϕ, i.e. on the dependent variables, can be taken as
δfϕ = fα∂αϕ + ∂αfβ
D + 1gαβ+1
One factor of the Jacobian disappears when changing spatial variables in the integration, and
the symplectic form?d2riψ∗∂tψ is invariant. But the Hamiltonian density H remains with
one factor J, hence the total Lagrangian is not in general invariant, and neither is the action,
the time integral of L — because t is not changed in the present transformation rules [in
contrast to (2.12) and (2.14)]. [It does not appear possible to find a transformation of time
that would restore invariance.]
However, for static solutions we know that E =
?d2rH vanishes. If this vanishing is
due to the local vanishing of H, as is true at g = 1/mκ, then the static critical points of the
action are invariant. This then shows that static solutions with zero H will be mapped into
each other by spatial conformal transformations — the dilation (2.12) expands to an infinite
symmetry group on the solutions, but there are no new constants of motion.
[Since in the non-Abelian generalization, with matter in the adjoint representation, equa-
tions are dimensional reductions of self-dual Yang–Mills equations in four dimensions,4the
finite-dimensional conformal invariance of the latter8is seen to survive the dimensional re-
duction, and in two dimensions expands to the infinite-dimensional conformal group.]
On the other hand, in the effective field theories for the eikonal regime (1.29), (1.31),
where there is no Hamiltonian to begin with, the transformations (2.38), (2.39) are symmetries
of the action, and also τ may be arbitrarily reparametrized. Note that owing to the derivative
relation (1.30) between Ω±and ψ; ψ =√2
∂z∗ (Ω+− iΩ−), the transformation law for Ω±is
without the weight factor,
Ω′±(r′) = Ω±(r)(2.47)
which arises for ψ, as in (2.39), when the derivative is taken.
III. CONCLUSION AND SUGGESTIONS FOR FURTHER RESEARCH
The rigid scale invariance of the action for non-relativistic (2+1)-dimensional field theory
with quartic self-interaction and coupling to a Chern–Simons gauge field, expands at the static
critical points of the action to the infinite conformal group on the plane. The scale symmetry
allows establishing the important result that static solutions carry zero energy, and the infinite
conformal symmetry “explains” why the static system is completely integrable. The kinetic
action of effective eikonal field theories also possesses the infinite symmetry.
The Chern–Simons model at g = 1/mκ is the bosonic partner of an N = 2 supersymmet-
ric theory with fermions and the invariance of the extended action against the supersymmetric
generalization of the bosonic symmetries (2.6) – (2.15) has been established.9While the in-
variances of the static critical points in the supersymmetric action have not been explicitly
checked, they too presumably enjoy an infinite conformal symmetry, because the supersym-
metric static equations retain the form of the bosonic equations.
In our considerations, the possibility of quantum symmetry breaking anomalies has been
ignored. It is known that the quartic self-interaction, which as we have seen is formally
scale invariant, suffers from quantum scale anomalies.10This is particularly clear in the first
quantized framework, where the two-dimensional δ-function potential, while scaling classically
as r−2, does not give rise to energy-independent phase shifts, as is required by scale invariance
and is explicitly realized by the scale invariant 1/r2potential. There is a quantum scale
anomaly — the simplest example of the anomaly phenomenon.11On the other hand, anomalies
in the theory with both quartic self-coupling and Chern–Simons interaction have thus far
not been assessed; in fact there is some indication of anomaly cancellation, even without
supersymmetry.12Further research on this question would be interesting.13
1. For a discussion of non-relativistic Chern–Simons theory and its relation through second
quantization to the particle mechanics of (1.22), see e.g. R. Jackiw and S.-Y. Pi, Phys.
Rev. D 42, 3500 (1990).
2. For gravity: H. Verlinde and E. Verlinde, Nucl. Phys. B371, 246 (1992); for Maxwell
theory: R. Jackiw, D. Kabat and M. Ortiz, Phys. Lett. B 277, 148 (1992).
3. For Abelian Chern–Simons interactions: R. Jackiw and S.-Y. Pi, Phys. Rev. Lett. 64,
2969 (1990); (C) 66, 2682 (1992) and Ref. .
4. For non-Abelian Chern–Simons interactions: B. Grossman, Phys. Rev. Lett. 65, 3230
(1990); G. Dunne, R. Jackiw, S.-Y. Pi and C. Trugenberger, Phys. Rev. D 43, 1332
(1991); G. Dunne, Commun. Math. Phys. (in press).
5. S. Takagi, Prog. Theor. Phys. 84, 1019 (1990), 85, 463, 723 (1991), 86, 783 (1991).
6. D. Freedman and A. Newell (unpublished).
7. Z. Ezawa, M. Hotta and Z. Iwazaki, Phys. Rev. Lett. 67, 441 (1991); Phys. Rev. D 44,
452 (1991); R. Jackiw and S.-Y. Pi, Phys. Rev. Lett. 67, 415 (1991) and Phys. Rev. D
44, 2524 (1991).
8. R. Jackiw and C. Rebbi, Phys. Rev. D 14, 517 (1977).
9. M. Leblanc, G. Lozano and H. Min, Ann. Phys. (NY) (in press).
10. O. Bergman, MIT preprint CTP#2045 (1991).
11. R. Jackiw, in M. A. B. B´ eg Memorial Volume, A. Ali and P. Hoodbhoy, eds. (World
Scientific, Singapore, 1991).
12. G. Lozano, Phys. Lett. B (in press).
13. O. Bergman, in preparation.