Table Of Content5
Description of surfaces associated with
0
0
2 Grassmannian sigma models on Minkowski
n
a space
J
3
1
A. M. Grundland
∗
]
G Centre de Recherches Math´ematiques, Universit´e de Montr´eal,
D C. P. 6128, Succ. Centre-ville, Montr´eal, (QC) H3C 3J7, Canada
.
h Universit´e du Qu´ebec, Trois-Rivi`eres CP500 (QC) G9A 5H7, Canada
t
a and
m
ˇ
L. Snobl
[ †
Centre de Recherches Math´ematiques, Universit´e de Montr´eal,
1
v
C. P. 6128, Succ. Centre-ville, Montr´eal, (QC) H3C 3J7, Canada
0
0 Faculty of Nuclear Sciences and Physical Engineering,
2
1 Czech Technical University,
0
5 Bˇrehov´a 7, 115 19 Prague 1, Czech Republic
0
/
h
t
a
m
Abstract
:
v
i We construct and investigate smooth orientable surfaces in su(N)
X
algebras. The structural equations of surfaces associated with Grass-
r
a mannian sigma models on Minkowski space are studied using moving
frames adapted to the surfaces. The first and second fundamental
forms of these surfaces as well as the relations between them as ex-
pressedin the Gauss–Weingarten and Gauss–Codazzi–Ricci equations
are found. The scalar curvature and the mean curvature vector ex-
pressed in terms of a solution of Grassmanian sigma model are ob-
tained.
∗email address: [email protected]
†email address: Libor.Snobl@fjfi.cvut.cz
1
Keywords: Sigma models, structural equations of surfaces, Lie alge-
bras.
PACS numbers: 02.40.Hw, 02.20.Sv, 02.30.Ik
1 Introduction
Sigma models are of great interest in mathematical physics because a
significant number of physical systems can be reduced to these, rela-
tively simple, models, either on Euclidean or Minkowski space. One
such example is the string theory in which sigma models on space-
time and their supersymmetric extensions play a crucial role. Other
relevant applications of recent interest are in the areas of statistical
physics (for example reduction of self–dual Yang–Mills equations to
the Ernst model [1, 2]), phase transitions [3, 4] and the theory of fluid
membranes [5, 6].
The objective of this paper is to study geometric properties of
surfaces in Lie algebras associated with sigma models on Minkowski
space. Recently, we investigated surfaces in su(N) associated with
CPN−1 sigma models [7] and found a few examples [8]. In this paper
we extend this approach to more general models based on Grassman-
nian manifolds, i.e. the homogeneous spaces
SU(N)
G(m,n) = , N = m+n.
S(U(m) U(n))
×
Grassmannian sigma models are a generalization of CPN−1 sigma
models. Their important common feature is that the Euler–Lagrange
equations can be written in terms of projectors only [9]. They share a
lotofpropertieslikeinfinitenumberoflocaland/ornonlocalconserved
quantities,Hamiltonianstructure,completeintegrability,infinite–dimensional
symmetry algebra, existence of multisoliton solutions etc. The N N
×
projector matrix P for the complex Grassmannian sigma models has
ingeneralranklowerthanthecorrespondingonefortheCPN−1 sigma
model and consequently new phenomena can arise.
The generalization of our previous results [7, 8] to Grassmannian
sigma models seemed to be rather natural – in fact, it was in a sense
more straightforward than the generalization from CP1 to CPN−1,
provided one expressed the corresponding formulas in terms of the
projector (2.8). On the other hand, a different perspective obtained
2
in more general case allowed to write some of the results in more
compact and presumably more natural way.
The results can be of interest in the area of relativistic classical
andquantumfieldtheory[10,11],stringtheoryinwhichsigmamodels
on spacetime and their supersymmetric extensions play a crucial role
[12]. Other relevant applications of recent interest are in the areas of
nonlinearinteractionsinparticlephysics[13]. Theexplicitformsofthe
surfaces can serve to illuminate the role of the Kac–Moody algebras
in integrable models associated with the Grassmannian sigma models
[14, 15].
Thepaperisorganizedasfollows. InSection2werecallsomebasic
notions anddefinitions dealingwith thecomplex Grassmanniansigma
models and their Euler–Lagrange equations. In Section 3 we perform
the analysis of two–dimensional surfaces immersed in the su(N) alge-
bra, associated with these models. The geometric properties of sur-
faces and the construction of moving frames are discussed in detail in
Sections 4,5. Finally, we summarize our results.
2 Grassmannian sigma models and their
Euler–Lagrange equations
As a starting point let us present some basic formulae and notation
for complex Grassmannian sigma models definedon Minkowski space.
Weadapttooursignaturethenotation introducedin[9]forEuclidean
Grassmannian sigma models.
The Grassmannian manifold is defined as homogeneous space
SU(N)
G(m,n) = , N = m+n. (2.1)
S(U(m) U(n))
×
We expresselements G(m,n)usingtheequivalence classes of elements
g SU(N) as
∈
U 0
[g] = g.ψ ψ = m , U U(m),U U(n),detψ = 1 .
{ | 0 Un m ∈ n ∈ }
(cid:18) (cid:19)
(2.2)
We decompose g SU(N) into submatrices X,Y
∈
g = (φ ,...,φ )= (X,Y), X = (φ ,...,φ ), Y = (φ ,...,φ )
1 N 1 m m+1 N
(2.3)
3
and from g†g = 1, i.e. φ†φ = δ we find
j k jk
X†X = 1 , X†Y = 0, Y†X = 0, Y†Y = 1 .
m×m n×n
From these orthogonality relations and (2.2) we realize that on the
subset of G(m,n) such that the lower square n n submatrix of Y is
×
nonsingular,X itselfissufficienttodetermine[g](sinceU canbeused
n
to bring the lower square part of Y to 1 and the remaining entries
n×n
in Y are fully determined by the orthogonality properties). In the
followingweshallassumethatweareworkinginsuchchart. Evidently
they cover the whole G(m,n) up to lower dimensional submanifolds.
We shall denote the equivalence classes either [X] or [g] depending
on circumstances. Note that there is still some freedom in the choice
SU(m) 0
of X, namely X and X.h, h give rise to the same
∈ 0 1
(cid:18) (cid:19)
equivalenceclass[X] = [Xh]. Therefore,onecannotidentifyX = [X].
Let ξ0, ξ1 be the standard Minkowski coordinates in R2, with the
metric
(ds)2 = (dξ0)2 (dξ1)2.
−
In what follows we suppose that ξ = ξ0 +ξ1, ξ = ξ0 ξ1 are the
L R
−
light–cone coordinates in R2, i.e.
(ds)2 = dξ dξ . (2.4)
L R
We shall denote by ∂ and ∂ the derivatives with respect to ξ and
L R L
ξ , respectively.
R
Let us assume that Ω is an open, connected and simply connected
subsetin R2 with Minkowski metric (2.4). We definecovariant deriva-
tives D acting on maps X : Ω G(m,n) by
µ
→
DµX = ∂µX XX†∂µX, ∂µ ∂ξµ, µ = 0,1. (2.5)
− ≡
In the study of Grassmannian sigma models we are interested in
maps X : Ω G(m,n) which are stationary points of the action
→
functional
= tr (D X)†(DµX) dξ0dξ1. (2.6)
µ
S { }
ZΩ
The Lagrangian density can be further developed to get
= tr (D X)†(DµX) = tr ∂µX(∂ X)†P (2.7)
µ µ
L { } { }
where
P = 1 XX† (2.8)
−
4
is an orthogonal projector, i.e. P2 = P, P† = P satisfying PX =
0, X†P = 0.
The action (2.6) has the local (gauge) SU(m) symmetry
SU(m) 0
X(ξ ,ξ ) X(ξ ,ξ ).h(ξ ,ξ ), h(ξ ,ξ ) (2.9)
L R → L R L R L R ∈ 0 1
(cid:18) (cid:19)
provingthatthemodeldoesn’tdependonthechoiceofrepresentatives
X of elements [X] of G(m,n); and the SU(N) global symmetry
X gX, g SU(N). (2.10)
→ ∈
It is also invariant under the conformal transformations
ξ α(ξ ), ξ β(ξ ), (2.11)
L L R R
→ →
where α,β : R R are arbitrary 1–to–1 maps such that ∂ α(ξ ) =
L L
→ 6
0, ∂ β(ξ )= 0, as well as under the parity transformation
R R
6
ξ ξ , ξ ξ . (2.12)
L R R L
→ →
Let us note that the invariance properties (2.9)–(2.12) are naturally
reproduced on the level of Euler–Lagrange equations.
By variation of the action (2.6) respecting the constraint
X†X = 1, i.e. δX†X +X†δX = 0, ∂ X†X +X∂ δX = 0 (2.13)
µ µ
and assuming that due to suitable boundary conditions the boundary
terms vanish we find the Euler–Lagrange equations
P(∂ ∂ X 2∂ XX†∂µX) = 0. (2.14)
L R µ
−
They can be also expressed in the matrix form
[∂ ∂ P,P] = 0 (2.15)
L R
or in the form of a conservation law
∂ [∂ P,P]+∂ [∂ P,P] = 0. (2.16)
L R R L
Methods for finding special solutions of (2.14), e.g. soliton solutions,
are known [16, 17].
Byexplicitcalculationonecancheckthatthereal–valuedfunctions
J = tr(∂ X∂ X†P), J = tr(∂ X∂ X†P) (2.17)
L L L R R R
satisfy
∂ J = ∂ J = 0 (2.18)
L R R L
for any solution X of the Euler–Lagrange equations (2.14). The func-
tionsJ ,J areinvariantunderlocalSU(m)andglobalSU(N)trans-
L R
formations (2.9) and (2.10).
5
3 Surfaces obtained from Grassman-
nian sigma model
Let us now discuss the analytical description of a two–dimensional
smooth orientable surface immersed in the su(N) algebra, associ-
F
ated with the Grassmannian sigma model (2.14). We shall construct
an exact su(N)–valued 1–form whose “potential” 0–form defines the
surface . Next, we shall investigate the geometric characteristics of
F
the surface .
F
Let us introduce a scalar product
1
(A,B) = trAB
−2
on su(N) and identify the (N2 1)–dimensional Euclidean space with
−
the su(N) algebra
RN2−1 su(N).
≃
We denote
M = [∂ P,P], M = [∂ P,P]. (3.1)
L L R R
It follows from (2.16) that if X is a solution of the Euler–Lagrange
equations (2.14) then
∂ M +∂ M = 0. (3.2)
L R R L
We identify tangent vectors to the surface with the matrices M
L
F
and M , as follows
R
= M , = M . (3.3)
L L R R
Z Z −
Equation (3.2) implies there exists a closed su(N)–valued 1–form on
Ω
= dξ + dξ , d = 0.
L L R R
Z Z Z Z
Because is closed and Ω is connected and simply connected, is
Z Z
also exact. In other words, there exists a well–defined su(N)–valued
function Z on Ω such that = dZ. The matrix function Z is unique
Z
up to addition of any constant element of su(N) and we identify the
components of Z with the coordinates of the sought–after surface
in RN2−1. Consequently, we get F
∂ Z = , ∂ Z = . (3.4)
L L R R
Z Z
6
The map Z is called the Weierstrass formula for immersion. In prac-
tice, the surface is found by integration
F
: Z(ξ ,ξ )= (3.5)
L R
F Z
Zγ(ξL,ξR)
along any curve γ(ξ ,ξ ) in Ω connecting the point (ξ ,ξ ) Ω with
L R L R
∈
an arbitrary chosen point (ξ0,ξ0) Ω.
L R ∈
By computation of traces of . , B,D = L,R we find the
B D
Z Z
components of the induced metric on the surface
F
G , G
G = LL LR = (3.6)
G , G
LR RR
(cid:18) (cid:19)
J tr ∂LX∂RX†+∂RX∂LX†P
= L − 2 .
tr ∂LX∂RX†+∂RX∂LX†P (cid:16) J (cid:17)
− 2 R
(cid:16) (cid:17)
Thefirstfundamentalformofthesurface takessurprisinglycompact
F
form
I = J (dξ )2 2G dξ dξ +J (dξ )2
L L LR L R R R
−
= (2δ 1)tr(∂ X∂ X†P)dξ dξ (3.7)
B,D B D B D
−
where summation over repeated indices B,D = L,R applies and
δ = 1 if B = D and 0 otherwise.
B,D
In order to establish conditions on a solution X of the Euler–
Lagrangeequations(2.14)underwhichthesurfaceexists,weintroduce
a scalar product on the space of N m matrices X
×
(b,a) = tr(a.b†), a,b CN×m
∈
and employ the Schwarz inequality, i.e.
tr(ab†A)2 tr(aa†A)tr(bb†A) (3.8)
| | ≤
valid for any positive hermitean operator A, namely for P : P(a) =
P.a. We may write
J =tr(∂ X∂ X†P) 0, D = L,R (3.9)
D D D
≥
and
2
detG = tr(∂ X∂ X†P)tr(∂ X∂ X†P) tr(∂ X∂ X†P) 0
L L R R L R
− ℜ ≥
(cid:16) (cid:17)(3.10)
7
since
tr(∂ X∂ X†P) tr(∂ X∂ X†P) tr(∂ X∂ X†P)2
L L R R L R
≥ | |
2
tr(∂ X∂ X†P) .
L R
≥ ℜ
(cid:16) (cid:17)
Therefore the first fundamental form I defined by (3.7) is positive for
any solution X of the Euler–Lagrange equations (2.14).
Analyzing the cases when equalities in Schwarz inequality hold
we find that I is positive definite in the point (ξ0,ξ0) either if the
L R
inequality
tr(∂ X∂ X†P) = 0 (3.11)
L R
ℑ 6
holds in (ξ0,ξ0) or if the matrices
L R
∂ X(ξ0,ξ0),∂ X(ξ0,ξ0),X(ξ0,ξ0) (3.12)
L L R R L R L R
arelinearly independent. Thereforeanyoftheconditions (3.11),(3.12)
is a sufficient condition for the existence of the surface associated
F
with the solution X of the Euler–Lagrange equations (2.14) in the
vicinity of the point (ξ0,ξ0).
L R
Using (3.6) we can write the formula for Gaussian curvature [18]
as
1 ∂ G 1G ∂ (lnJ )
K = ∂ L LR− 2 LR L L . (3.13)
R
J J G2 J J G2
L R − LR L R− LR
q q
4 The Gauss–Weingarten equations
Now we may formally determine a moving frame on the surface
F
and write the Gauss–Weingarten equations. Let X be a solution of
the Euler–Lagrange equations (2.14) such that det(G) is not zero in a
neighborhood of a regular point (ξ0,ξ0) in Ω. Assume also that the
L R
surface (3.5), associated with these equations is described by the
F
moving frame
~τ = (∂LZ,∂RZ,n3,...,nN2−1)T,
where the vectors ∂LZ,∂RZ,n3,...,nN2−1 satisfy the normalization
conditions
(∂ Z,∂ Z)= J , (∂ Z,∂ Z) = G , (∂ Z,∂ Z) = J ,
L L L L R LR R R R
8
(∂ Z,n ) = (∂ Z,n ) = 0, (n ,n )= δ . (4.1)
L k R k j k jk
We now show that the moving frame satisfies the Gauss–Weingarten
equations
∂ ∂ Z = AL∂ Z +AL∂ Z +QLn ,
L L L L R R j j
∂ ∂ Z = H n ,
L R j j
∂ n = αL∂ Z +βL∂ Z +sL n ,
L j j L j R jk k
∂ ∂ Z = H n ,
R L j j
∂ ∂ Z = AR∂ Z +AR∂ z+QRn ,
R R L L R R j j
∂ n = αR∂ Z +βR∂ Z +sRn , (4.2)
R j j L j R jk k
where sL +sL = 0, sR +sR = 0, j,k = 3,...,N2 1,
jk kj jk kj −
H G QLJ QLG H J
αL = j LR− j R, βL = j LR − j L,
j detG j detG
QRG H J H G QRJ
αR = j LR− j R, βR = j LR − j L,
j detG j detG
and AL,AL (AR,AR have similar form which can be obtained by ex-
L R L R
change L R) are written as
↔
1
AL = (J (∂ ∂ Z,∂ Z) G (∂ ∂ Z,∂ Z))
L detG R L L L − LR L L R
1
AL = (J (∂ ∂ Z,∂ Z) G (∂ ∂ Z,∂ Z)) (4.3)
R detG L L L R − LR L L L
where
1
(∂ ∂ Z,∂ Z) = tr (∂ ∂ X∂ X† +∂ X∂ ∂ X†)P ,
L L L L L L L L L
2
1 (cid:16) (cid:17)
(∂ ∂ Z,∂ Z) = tr (∂ ∂ X∂ X† +∂ X∂ ∂ X†)P
L L R L L R R L L
−2
(cid:16)
+ 2∂ X∂ X†(X∂ X† +∂ XX†) . (4.4)
L L R R
(cid:17)
Note that in fact we can write it in a compact way
1
(∂ ∂ Z,∂ Z) = (δ )tr (∂ ∂ X∂ X† +∂ X∂ ∂ X†)P
B B D B,D B B D D B B
− 2
(cid:16)
+ 2∂ X∂ X†(X∂ X†+∂ XX†) . (4.5)
B B D D
(cid:17)
9
The explicit form of the coefficients H ,QD (where D = L,R; j =
j j
3,...,N2 1)dependsonthechosenorthonormalbasis n3,...,nN2−1
− { }
of the normal space to the surface at the point X(ξ0,ξ0). Partial
F L R
information about them will be obtained in (5.6).
Indeed, if ∂ Z,∂ Z are defined by (3.4) for an arbitrary solution
L R
X of the Euler–Lagrange equations (2.14), then by straightforward
calculation using (2.15) one finds that
∂ ∂ Z = ∂ ∂ Z = [∂ P,∂ P] =
L R R L L R
= λ λXX† XX†λ+X(∂ X†∂ X ∂ X†∂ X)X†
L R R L
− − −
where
λ = ∂ X∂ X† ∂ X∂ X†.
L R R L
−
By computing
tr (∂ ∂ Z.∂ Z)= tr([∂ P,∂ P].[∂ P,P]) = 0, D = L,R (4.6)
L R D L R D
±
we conclude that ∂ ∂ Z is perpendicular to the surface and conse-
L R
F
quently it has the form given in (4.2).
The remaining relations in (4.2) and (4.3) follow as differential
consequences from the assumed normalizations of the normals (4.1),
e.g.
(n ,n )= 0, j = k
j k
6
which gives
0= (∂ n ,n )+(∂ n ,n )= sL +sL .
L j k L k j jk kj
Similarly
(n ,∂ Z)= 0, (n ,∂ Z)= 0
j L j R
by differentiation leads to
(∂ n ,∂ Z)+(n ,∂ ∂ Z)= 0, (∂ n ,∂ Z)+(n ,∂ ∂ Z)= 0
R j L j L R R j R j R R
implying
J αR+G βR +H = 0, G αR+J βR+QR = 0.
L j LR j j LR j R j j
Consequently, αR,βR can be determined in terms of H ,QR and of
j j j j
the components of the induced metric G. The remaining coefficients
αL,βL are derived in an analogous way by exchanging indices L R
j j ↔
in the successive differentiations.
10
