Table Of ContentCPN Models in General Coordinates
K.J.Barnes
Department of Physics and Astronomy, University of Southampton, Highfield,
Southampton, SO17 1BJ, United Kingdom.
2
0
0
2
Abstract
n
a
J An analysis of CPN models is given in terms of general coordinates or arbitrary in-
6 terpolating fields. Only closed expressions made from simple functions are involved.
1
Specialattention isgiven toCP2andCP4.Inthefirstof thesetheretrieval of stere-
1 ographic coordinates reveals the hermitian form of the metric. A similar analysis
v for the latter case allows comparison with the Fubini-Study metric.
3
2
1
Key words: Kahler, stereographic, complex projective
1
0 PACS: , 11.10 Lm, 11.25.-W, 11.30. Pb, 11.30. Rd
2
0
/
h
t
- Introduction
p
e
h Despite the central importance of CPN models [1], and their recent revival
:
v as arising in supersymmetric from from minimized linear models [2] following
i
X the revision of the underlying supersymmetry algebra of densities to include
r central terms, there appears to be no treatment of them in general coordinates
a
which would allow arbitrary field redefinitions for the interpolating Goldstone
Bosons. In this article just such an analysis is presented. The next section ex-
plains how this is achieved by embedding the necessary structure into a more
complicated one. Strangely, perhaps, nothing is needed but simple functions,
and a completely general solution is found in closed form. In the following
section the special case of CP2 and stereographic coordinates is presented.
Then the corresponding step is made for CP4 allowing the connection to the
Fubini-Study metric. Finally, there are brief conclusions and suggestions are
made for future work.
General Framework
Curiously this section begins by consideration of the embedding of the struc-
ture needed for the current problem into that of a larger system which has
Preprint submitted to Elsevier Preprint 1 February 2008
previously been solved in general coordinates leading to a closed form involv-
ing only simple functions [3]. The embedding is unique. Thus the starting
point is a review of this established larger system and its solution, in which
the liberty of changing notation (slightly) for convenience has been taken.
Consider then the Lie algebra of SU specified by taking as a basis the set of
n
(n2 1) traceless hermitian n n matrices λ with the product law
i
− ×
2
λ λ = δ +(d +if )λ (1)
i j ij ijk ijk k
n
as specified by Gell Mann [4]. When specific values of the structure constants
f , or the symmetric d tensors or the λ matrices themselves are needed
ijk ijk i
then the notation of reference [4] will be assumed. An element of the group
SU , g(θ) is specified in exponential form by a set of (n2 1) real parameters
n
−
θ , so that in infinitesimal form the transformations
i
i
q q θi(λ ) q (2)
A A i AB B
−→ − 2
and
M M +θ f M (3)
i i k ikj k
−→
specify the behaviour of the basic spinor q (quark) and adjoint vector M
A i
fields. Now define a traceless matrix M by
M = M (λ ) (4)
KL i i KL
so that
1
M = Tr.(Mλ ) (5)
i i
2
then a group element g(θ) which induces a unitary transformation
q U(θ) q (6)
A AB B
−→
on the basic spinors clearly induces an orthogonal transformation
1
M R M = Tr.(U−1λ Uλ )M (7)
i ij j i j j
−→ 2
on the adjoint representation.
2
The algebra of SU SU is spanned by two sets of (n2 1) orthogonal
n n
× −
elements L and R satisfying the commutation relations
i i
[L ,L ] = if L (8)
i j ijk k
[R ,R ] = if L (9)
i j ijk k
[L ,R ] = 0 (10)
i j
and the linear combinations
V = L +R (11)
i i i
A = L R (12)
i i i
−
are frequently used. Obviously the V generate a SU subgroup which is parity
i n
conserving. An element of the SU SU group may be specified by two sets
n n
×
of (n2 1) real parameters, and the alternative expressions
−
g = exp( i[θVV +θAA ]) (13)
− i i i i
and
g = exp( iθLL )exp( iθRR ) (14)
− i i − i i
will prove useful with
θL = θV +θA (15)
i i i
θR = θV θA (16)
i i − i
specifying the correspondence. Every element of the group can also be decom-
posed into a product of the form
g = exp( iφ Ai)exp( iθ Vi) (17)
i i
− −
which is unique in a neighbourhood of the identity element and this will play
a crucial role in the general nonlinear realization scheme. The linear transfor-
mation laws are best specified by giving the quarks a Dirac spinor index in
the usual manner and taking
i (1+γ ) i (1 γ )
q q θLλ 5 q θRλi − 5 q (18)
−→ − 2 i i 2 − 2 i 2
3
as the concrete infinitesimal form.
Since the matrices
(1+γ )
5
P = (19)
L
2
and
(1 γ )
5
P = − (20)
R
2
act as a standard set of projection operators, the treatment of linearly trans-
forming multiplets of SU SU now follows trivially.
n n
×
To treat the nonlinear realizations of SU SU in full generality the (n2 1)
n n
× −
hermitian components M of the adjoint vector of SU must be considered in
i n
more detail. In the terminology of Michel and Radicati [5], the vector is said
to be generic (or to belong to the generic stratum) if all eigenvalues of M are
distinct. For the generic case the minimal polynomial for the matrix is the
characteristic polynomial satisfying the equation
n
(M m ) = 0 (21)
A
−
A=1
Y
where the m are the eigenvalues which satisfy
A
n
m = 0 (22)
A
m=1
X
if the matrix is traceless. Thus the (n - 1) vectors with components given by
powers of the matrix in the form
1
Mα = Tr.([M]αλ ) [α = 1,2,...,(n 1)] (23)
i i
2 −
are a linearly independent set, and the quantities
n n
T = Tr.([M]A) = [m ]A m (24)
A B AB
≡
B=1 B=1
X X
are (n 1) independent SU invariants. (S is identically zero.) At once it is
n 1
−
clear that the general vector which can be constructed from the M has the
i
4
form
ξ = F M (25)
i α αi
where the F are functions of the (n 1) independent SU invariants. This
α n
−
freedom has been discussed at length by Gasiorowicz and Geffen [6]. From the
pointofviewoffieldtheoryitcorrespondstofreedomofchoiceofinterpolating
fields. Provided that F (0) is taken to be unity, and parity is respected, then
1
all ξ so defined are equally good interpolating fields. From a geometrical
i
viewpoint the ξ may be regarded as coordinates of points of the (n2 1)
i
−
dimensional coset space manifold formed by the quotient of SU SU by
n n
×
the vector SU subgroup. The freedom is then viewed as the ability to change
n
coordinates within a local patch near the origin.
An arbitrary point on the manifold is parameterized by
iθ
exp( iξ A ) L(M) exp − M λ [P P ] (26)
i i i i L R
− ≡ ≡ 2φ − !
where the first form corresponds with equation (17) and the second form
represents the appropriate expression when equation (25) has been used so
that the M are regarded as the coordinates, and φ2 = M M .
i i i
The general theory is well described by Coleman, Wess and Zumino [7] and
Callan, Coleman, Wess and Zumino [8], and the geometrical approach by
Isham [9]. With the decomposition given in equation (17) the action of a
general element g of the full group may be written as
gexp( iξ A ) = exp( iξ′A )exp( iη V ) (27)
− i i − i i − i i
L(M′)exp( iη V ) (28)
i i
≡ −
where M′ and η both depend on M and g. Then the primary result of the
i i i
general theory is that
g : M M′ (29)
i −→ i
gives a nonlinear realization of the algebra which is linear on the SU vector
n
subgroup. Moreover if h is an element of the vector subgroup and
h : Ψ D(h) Ψ (30)
Ω ΩΓ Γ
−→
5
is a linear (unitary) representation of that subgroup, then
g : Ψ D[exp( iη Ψ )] Ψ (31)
Ω i i ΩΓ Γ
−→ −
gives a realization of the full group. Notice that this latter transformation is
linear in Ψ but nonlinear (through η ) in the M when g is not in the vector
i i
subgroup. Fields which transform according to equation (31) are called stan-
dard fields, and it is important to understand that by a suitable redefinition
of coordinates any nonlinear realization of SU SU which is linear on the
n n
×
vector subgroup can be brought into this standard form. In practice the most
useful result is that, if one has a linear irreducible (unitary) representation of
SU SU such that
n n
×
g : N D[g] N (32)
Ω ΩΓ Γ
−→
then
Ψ (M) = D[L−1(M)] N (33)
Ω ΩΓ Γ
transform as the components of standard fields.
It is now clear that there are just three classes of fields to consider:
(1) Linear representations which may be built up in the usual way as mul-
tispinors with transformation laws defined by equation (18). These will
not be treated in more detail.
(2) Vectors M transforming as the adjoint representation of SU with a
i n
nonlinear transformation law under chiral action specified by equation
(27). These will allow a description of the massless Goldstone Bosons
(pions etc.) corresponding to the axial degrees of freedom spontaneously
violated. The specification of invariants constructed (nonlinearly) from
these is most important and will be exhibited later.
(3) Standard fields which appear linearly in their transformation laws, but
with nonlinear functions of the M induced according to equations (31)
i
and (28). These are important in describing matter (eg nucleons) in-
teracting with the Goldstone Bosons as chiral matter. Once more, the
specification of the corresponding invariants is most important and will
be given later.
The technical problem of finding the invariants is solved in reference [3]. A
crucial step is the resolution of the powers of the matrix M in the form
[M]A = [m ]AP m P (34)
B B AB B
≡
6
where the P are n hermitian matrices, each n n, with the properties
B
×
P P = δ P (no sum) (35)
A B AB B
Tr.(P ) = 1 (36)
A
and
n
P = 1 (37)
A
A=1
X
where this 1 is the unit (n n) matrix. Although the P are not in general
A
×
diagonal,theaboveprojectionoperatorpropertiesmakecalculationstractable.
Now define
1
P = Tr.(P λ ) (38)
Ai A i
2
and
1
(P ) = P (λ ) + δ (39)
A MN Ai i MN MN
n
where because the P are complete it follows that
A
n
P = 0 (40)
Ai
A=
X
and, introducing
p = √2[P (1+√n)−1]P (41)
Ai Ai ni
−
with
1
√2P = p + p (42)
Ai Ai ni
√n
establishes that p for µ = 1,2,...,(n 1) are orthonormal.
µi
−
The second-rank tensors defined by the M are conveniently handled by an
i
extension of these ideas, and fall into two classes. One such class is formed by
the n(n 1) independent tensors defined by
−
1
(P ) P Tr.(P λ P λ ) (A = B) (43)
AB ij AiBj A i B j
≡ ≡ 2 6
7
and
1
I = Tr.(P λ P λ ), (44)
ij A i A j
2
which have the properties
II = I (45)
IP = 0 = P I (46)
AB AB
and
P P = δ δ P (no sum) (47)
AB CD AC BD AB
in terms of the matrix notation of the last section. Moreover, these are all
hermitian matrices and the trace of each P is unity. Since it is easy to show
AB
also that
′
P = 1 I (48)
AB
−
A6=B
X
where the sum is over all A and B but excluding terms with A = B, this gives
a projection operator resolution in one sector of the space of these second-
rank tensors and so I (with trace [n 1]) will decompose further. The second
−
class of tensors may be identified with the (n 1)2 independent matrices with
−
components
(p ) p p (49)
αβ ij αi βj
≡
which span the subspace of (n2 1) (n2 1) matrices projected out on
− × −
multiplication by I from both sides and which are therefore orthogonal to the
subspace in which the P lie. Since the p are orthonormal, the multiplica-
AB αi
tion law for the p is
αβ
p p = δ p (50)
αβ γδ βγ αδ
It has been established by Barnes and Delbourgo [10] that all the independent
second-rank tensors which can be constructed from the M are spanned by the
i
(n 1)(2n 1) independent p and P .
αβ AB
− −
8
The most general unitary unimodular matrix U constructed from the M may
i
be written in the form
i n
U = U P = exp − θ P where θ = 0 (51)
A A A A A
2
(cid:20) (cid:21) A=1
X
but the θ are otherwise completely arbitrary independent functions of the
A
independent SU invariants S subject to the considerations of parity and
n A
weakfieldlimitsasmentionedbefore.These(n 1)effectivearbitraryfunctions
−
of the (n 1) invariants are characteristic of the general solution and will
−
persist throughout this work.
It has been conventional to define
√2φ = m (1+√n)−1m (52)
A A n
−
with
m = √2(φ +n−21φ ) (53)
A A n
so that, extending the notation used previously,
M = φ p (54)
i α αi
and
φ = p (55)
α,i αi
follow immediately. Similarly, defining
√2ψ = θ (1+n12)−1θ (56)
A A n
−
with
θ = √2(ψ +n−12ψ ) (57)
A A n
the ψ may be treated as (n 1) independent (arbitrary) functions of the φ
α α
−
which then serve as the (n 1) independent invariants.
−
The transformation laws for all realizations are now given in reference [3]
in closed form and in terms of simple functions. Restricting attention to first-
order derivatives of the fields with respect to space and time, and also restrict-
ing attention to a study of the Goldstone Boson fields M and the standard
i
9
fields the results can be given in terms of the general analysis of references [7]
and [8]. There are two important results. First, although ∂ M and ∂ Ψ do
µ i µ Γ
not transform as standard fields, the covariant derivatives
D M = A (58)
µ i µi
and
D Ψ = ∂ Ψ iν (T ) Ψ (59)
µ Γ µ Γ µi i ΓΩ Ω
−
where under SU
n
Ψ Ψ iθ (T ) Ψ (60)
Γ Γ i i ΓΩ Ω
−→ −
and where
2iL−1(M)∂ L(M) = exp( iξ A )∂ exp( iξ A ) = νiV +aiA (61)
µ − i i µ − j j µ i µ i
have precisely this property. Secondly, they show that the most general La-
grangian of the type under consideration maybe written as a function of the
standard fields Ψ, D Ψ and D M only; that is the M will not appear ex-
µ µ i i
plicitly, and the Goldstone Bosons will be massless. It then follows that the
Lagrangian so formed will be invariant under SU SU if and only if it
n n
×
is constructed to be invariant under the SU vector subgroup. This latter
n
requirement is, of course, achieved by index saturation once more.
The result given in reference [3] (now dropping the chiral projectors and nor-
malizing for this problem) takes the concrete form
∂ψβ ′ √2 ψA ψB
D M = (P ) + sin − (P +P ) (∂ M )(62)
µ i γβ ik AB BA ik µ k
(cid:26)∂φγ AX6=B (φA −φB) " √2 # (cid:27)
and represent a complete specification of the required Lagrangian in simple
closed form.Using the geometric formulationof Isham [9] gives thecoset space
metric in the form related to the covariant derivatives as
g (∂ M )(∂µM ) = (D M )(DµM ) (63)
ij µ i j µ i j
and we have normalized g to δ in the limit of zero fields. In matrix notation
ij ij
this yields
1 ∂ψ ∂ψ 2 ψ ψ
g = p α α + ′ (P +P )sin2 A − B (64)
βλ AB BA
4 ∂φβ ∂φλ AX6=B (φA −φB)2 " √2 #
10
