arXiv:0704.0116v2  [math-ph]  21 May 2007
Stringy Jacobi fields in Morse theory
Yong Seung Cho∗
National Institute for Mathematical Sciences, 385-16 Doryong, Yuseong, Daejeon 305-340 Korea and
Department of Mathematics, Ewha Womans University, Seoul 120-750 Korea
Soon-Tae Hong†
Department of Science Education and Research Institute for Basic Sciences, Ewha Womans University, Seoul 120-750 Korea
(Dated: November 4, 2018)
We consider the variation of the surface spanned by closed strings in a spacetime manifold.
Using the Nambu-Goto string action, we induce the geodesic surface equation, the geodesic surface
deviation equation which yields a Jacobi field, and we define the index form of a geodesic surface as
in the case of point particles to discuss conjugate strings on the geodesic surface.
PACS numbers: 02.40.-k, 04.20.-q, 04.90.+e, 11.25.-w, 11.40.-q
Keywords: Nambu-Goto string action, geodesic surface, Jacobi field, index of geodesic surface, conjugate
strings
I. INTRODUCTION
It is well known that string theory [1, 2] is one of the
best candidates for a consistent quantum theory of grav-
ity to yield a unification theory of all the four basic forces
in nature. In D-brane models [2], closed strings represent
gravitons propagating on a curved manifold, while open
strings describe gauge bosons such as photons, or mat-
ter attached on the D-branes. Moreover, because the
two ends of an open string can always meet and con-
nect, forming a closed string, there are no string theories
without closed strings.
On the other hand, the supersymmetric quantum me-
chanics has been exploited by Witten [3] to discuss the
Morse inequalities [4, 5, 6]. The Morse indices for pair of
critical points of the symplectic action function have been
also investigated based on the spectral flow of the Hes-
sian of the symplectic function [7], and on the Hilbert
spaces the Morse homology [8] has been considered to
discuss the critical points associated with the Morse in-
dex [9]. The string topology was initiated in the seminal
work of Chas and Sullivan [10]. Using the Morse theoretic
techniques, Cohen in Ref. [11] constructs string topology
operations on the loop space of a manifold and relates
the string topology operations to the counting of pseudo-
holomorphic curves in the cotangent bundle. He also
speculates the relation between the Gromov-Witten in-
variant [12] of the cotangent bundle and the string topol-
ogy of the underlying manifold. Recently, the Jacobi
fields and their eigenvalues of the Sturm-Liouville oper-
ator associated with the particle geodesics on a curved
manifold have been investigated [13], to relate the phase
factor of the partition function to the eta invariant of
Atiyah [14, 15].
In this paper, we will exploit the Nambu-Goto string
∗Electronic address: yescho@ewha.ac.kr
†Electronic address: soonhong@ewha.ac.kr
action to investigate the geodesic surface equation and
the geodesic surface deviation equation associated with
a Jacobi field. The index form of a geodesic surface will
be also discussed for the closed strings on the curved
manifold.
In Section II, the string action will be introduced to
investigate the geodesic surface equation in terms of the
world sheet currents associated with τ and σ world sheet
coordinate directions. By taking the second variation of
the surface spanned by closed strings, the geodesic sur-
face deviation equation will be discussed for the closed
strings on the curved manifold. In Section III, exploiting
the orthonormal gauge, the index form of a geodesic sur-
face will be also investigated together with breaks on the
string tubes. The geodesic surface deviation equation in
the orthonormal gauge will be exploited to discuss the
Jacobi field on the geodesic surface.
II. STRINGY GEODESIC SURFACES IN
MORSE THEORY
In analogy of the relativistic action of a point parti-
cle, the action for a string is proportional to the area
of the surface spanned in spacetime manifold M by the
evolution of the string. In order to define the action
on the curved manifold, let (M, gab) be a n-dimensional
manifold associated with the metric gab. Given gab, we
can have a unique covariant derivative ∇a satisfying [6]
∇agbc = 0, ∇aω
b = ∂aω
b + Γbac ω
c and
(∇a∇b −∇b∇a)ωc = R
abc ωd. (2.1)
We parameterize the closed string by two world sheet
coordinates τ and σ, and then we have the correspond-
ing vector fields ξa = (∂/∂τ)a and ζa = (∂/∂σ)a. The
Nambu-Goto string action is then given by [1, 2, 16]
S = −
dτdσf(τ, σ) (2.2)
http://arxiv.org/abs/0704.0116v2
where the coordinates τ and σ have ranges 0 ≤ τ ≤ T
and 0 ≤ σ ≤ 2π respectively and
f(τ, σ) = [(ξ · ζ)2 − (ξ · ξ)(ζ · ζ)]1/2. (2.3)
We now perform an infinitesimal variation of the tubes
γα(τ, σ) traced by the closed string during its evolution in
order to find the geodesic surface equation from the least
action principle. Here we impose the restriction that the
length of the string circumference is τ independent. Let
the vector field ηa = (∂/∂α)a be the deviation vector
which represents the displacement to an infinitesimally
nearby tube, and let Σ denote the three-dimensional sub-
manifold spanned by the tubes γα(τ, σ). We then may
choose τ , σ and α as coordinates of Σ to yield the com-
mutator relations,
a = ξb∇bη
a − ηb∇bξ
a = 0,
a = ζb∇bη
a − ηb∇bζ
a = 0,
a = ξb∇bζ
a − ζb∇bξ
a = 0. (2.4)
Now we find the first variation as follows [17]
dτdσ ηb(ξ
τ + ζ
dσ P bτ ηb|
τ=0 −
dτ P bσηb|
σ=0 , (2.5)
where the world sheet currents associated with τ and σ
directions are respectively given by [17]
P aτ =
[(ξ · ζ)ζa − (ζ · ζ)ξa],
P aσ =
[(ξ · ζ)ξa − (ξ · ξ)ζa]. (2.6)
Using the endpoint conditions ηa(0) = ηa(T ) = 0 and pe-
riodic condition ηa(σ+2π) = ηa(σ), we have the geodesic
surface equation [17]
ξa∇aP
τ + ζ
σ = 0, (2.7)
and the constraint identities [17]
Pτ · ζ = 0, Pτ · Pτ + ζ · ζ = 0,
Pσ · ξ = 0, Pσ · Pσ + ξ · ξ = 0.
(2.8)
Let γα(τ, σ) denote a smooth one-parameter family
of geodesic surfaces: for each α ∈ R, the tube γα is
a geodesic surface parameterized by affine parameters τ
and σ. For an infinitesimally nearby geodesic surface in
the family, we then have the following geodesic surface
deviation equation
ξb∇b(η
τ ) + ζ
b∇b(η
+R abcd (ξ
bP dτ + ζ
bP dσ )η
c ≡ (Λη)a = 0. (2.9)
For a small variation ηa, our goal is to compare S(α) with
S(0) of the string. The second variation d2S/dα2(0) is
then needed only when dS/dα(0) = 0. Explicitly, the
second variation is given by
|α=0 = −
(ηc∇cP
τ )(ξ
a∇aηb)
+(ηc∇cP
a∇aηb)−R
acb (ξ
aP bτ + ζ
aP bσ)η
dσ P bτ η
a∇aηb|
τ=0 −
dτ P bση
a∇aηb|
σ=0 .
(2.10)
Here the boundary terms vanish for the fixed endpoint
and the periodic conditions, even though on the geodesic
surface we have breaks which we will explain later. After
some algebra using the geodesic surface deviation equa-
tion, we have
|α=0 =
dτdσ ηa(Λη)
a. (2.11)
III. JACOBI FIELDS IN ORTHONORMAL
GAUGE
The string action and the corresponding equations
of motion are invariant under reparameterization σ̃ =
σ̃(τ, σ) and τ̃ = τ̃ (τ, σ). We have then gauge degrees of
freedom so that we can choose the orthonormal gauge as
follows [17]
ξ · ζ = 0, ξ · ξ + ζ · ζ = 0, (3.1)
where the plus sign in the second equation is due to the
fact that ξ·ξ is timelike and ζ·ζ is spacelike. Note that the
gauge fixing (3.1) for the world sheet coordinates means
that the tangent vectors are orthonormal everywhere up
to a local scale factor [17]. In this parameterization the
world sheet currents (2.6) satisfying the constraints (2.8)
are of the form
P aτ = −ξ
a, P aσ = ζ
a. (3.2)
The geodesic surface equation and the geodesic surface
deviation equation read
− ξa∇aξ
b + ζa∇aζ
b = 0, (3.3)
−ξb∇b(ξ
a) + ζb∇b(ζ
−R abcd (ξ
bξd − ζbζd)ηc = (Λη)a = 0. (3.4)
We now restrict ourselves to strings on constant scalar
curvature manifold such as Sn. We take an ansatz that
on this manifold the string shape on the geodesic surface
γ0 is the same as that on a nearby geodesic surface γα at a
given time τ . We can thus construct the variation vectors
ηa(τ) as vectors associated with the centers of the string
of the two nearby geodesic surfaces at the given time
τ . We then introduce an orthonormal basis of spatial
vectors eai (i = 1, 2, ..., n−2) orthogonal to ξ
a and ζa and
parallelly propagated along the geodesic surface. The
geodesic surface deviation equation (3.4) then yields for
i, j = 1, 2, ..., n− 2
+ (R iτjτ −R
σjσ)η
j = 0. (3.5)
The value of ηi at time τ must depend linearly on the
initial data ηi(0) and dη
(0) at τ = 0. Since by con-
struction ηi(0) = 0 for the family of geodesic surfaces,
we must have
ηi(τ) = Aij(τ)
(0). (3.6)
Inserting (3.6) into (3.5) we have the differential equation
for Aij(τ)
d2Aij
+ (R iτkτ −R
σkσ)A
j = 0, (3.7)
with the initial conditions
Aij(0) = 0,
(0) = δij . (3.8)
Note that in (3.7) we have the last term originated from
the contribution of string property.
Next we consider the second variation equation (2.10)
under the above restrictions
|α=0 =
− (R iτjτ −R
σjσ)η
(3.9)
We define the index form Iγ of a geodesic surface γ as
the unique symmetric bilinear form Iγ : Tγ × Tγ → R
such that
Iγ(V, V ) =
|α=0 (3.10)
for V ∈ Tγ . From (3.9) we can easily find
Iγ(V,W ) =
−(R mτjτ −R
σjσ )W
. (3.11)
If we have breaks 0 = τ0 < · · · < τk+1 = T , and the
restriction of γ to each set [τi−1, τi] is smooth, then the
tube γ is piecewise smooth. The variation vector field
V of γ is always piecewise smooth. However dV/dτ will
generally have a discontinuity at each break τi (1 ≤ i ≤
k). This discontinuity is measured by
(τi) =
(τ+i )−
(τ−i ), (3.12)
where the first term derives from the restrictions
γ|[τi, τi+1] and the second from γ|[τi−1, τi]. If γ and
V ∈ Tγ have the breaks τ1 < · · · < τk, we have
∫ τi+1
dτ = −
(3.13)
to yield
Iγ(V,W ) = −
dτdσ V m
(3.14)
+(R mτjτ −R
σjσ )W
dσ Vm∆
(τi). (3.15)
Here note that if we do not have the breaks, (3.9) yields
|α=0 = −
dτdσ ηi
+ (R iτjτ −R
σjσ)η
(3.16)
A solution ηa of the geodesic surface deviation equation
(3.5) is called a Jacobi field on the geodesic surface γ. A
pair of strings p, q ⊂ γ defined by the centers of the closed
strings on the geodesic surface is then conjugate if there
exists a Jacobi field ηa which is not identically zero but
vanishes at both strings p and q. Roughly speaking, p
and q are conjugate if an infinitesimally nearby geodesic
surface intersects γ at both p and q. From (3.6), q will be
conjugate to p if and only if there exists nontrivial initial
data: dηi/dτ(0) 6= 0, for which ηi = 0 at q. This occurs
if and only if detAij = 0 at q, and thus detA
j = 0
is the necessary and sufficient condition for a conjugate
string to p. Note that between conjugate strings, we have
detAij 6= 0 and thus the inverse of A
j exists. Using (3.7)
we can easily see that
Aik −Aij
= 0. (3.17)
In addition, the quantity in parenthesis of (3.17) vanishes
at p, since Aij(0) = 0. Along a geodesic surface γ, we
thus find
Aik −Aij
= 0. (3.18)
If γ is a geodesic surface with no string conjugate to p
between p and q, then Aij defined above will be nonsingu-
lar between p and q. We can then define Y i = (A−1)ijη
or ηi = AijY
j . From (3.16) and (3.18), we can easily
verify
|α=0 =
≥ 0. (3.19)
Locally γ minimizes the Nambu-Goto string action, if γ is
a geodesic surface with no string conjugate to p between
p and q.
On the other hand, if γ is a geodesic surface but has a
conjugate string r between strings p and q, then we have
a non-zero Jacobi field J i along γ which vanishes at p
and r. Extend J i to q by putting it zero in [r, q]. Then
dJ i/dτ(r−) 6= 0, since J i is nonzero. But dJ i/dτ(r+) = 0
to yield
(r) = −
(r−) 6= 0. (3.20)
We choose any ki ∈ Tγ such that
(r) = c, (3.21)
with a positive constant c. Let ηi be ηi = ǫki + ǫ−1J i
where ǫ is some constant, then we have
Iγ(η, η) = ǫ
2Iγ(k, k) + 2Iγ(k, J) + ǫ
−2Iγ(J, J). (3.22)
By taking ǫ small enough, the first term in (3.22) vanishes
and the third term also vanishes due to the definition
of the Jacobi field and (3.15). Substituting (3.21) into
(3.15) we have Iγ(k, J) = −2πc and thus
|α=0 = −4πc, (3.23)
which is negative definite. From the above arguments, we
conclude that given a smooth timelike tube γ connecting
two strings p, q ⊂ M , the necessary and sufficient con-
dition that γ locally minimizes the surface of the closed
string tube between p and q over smooth one parameter
variations is that γ is a geodesic surface with no string
conjugate to p between p and q. It is also interesting
to see that on Sn, the first non-minimal geodesic sur-
face has n − 1 conjugate strings as in the case of point
particle. Moreover, on the Riemannian manifold with
the constant sectional curvature K, the geodesic surfaces
have no conjugate strings for K < 0 or K = 0, while
conjugate strings occur for K > 0 [18].
IV. CONCLUSIONS
The Nambu-Goto string action has been introduced to
study the geodesic surface equation in terms of the world
sheet currents associated with τ and σ directions. By
constructing the second variation of the surface spanned
by closed strings, the geodesic surface deviation equation
has been discussed for the closed strings on the curved
manifold.
Exploiting the orthonormal gauge, the index form of
a geodesic surface has been defined together with breaks
on the string tubes. The geodesic surface deviation equa-
tion in this orthonormal gauge has been derived to find
the Jacobi field on the geodesic surface. Given a smooth
timelike tube connecting two strings on the manifold,
the condition that the tube locally minimizes the sur-
face of the closed string tube between the two strings
over smooth one parameter variations has been also dis-
cussed in terms of the conjugate strings on the geodesic
surface.
In the Morse theoretic approach to the string theory,
one could consider the physical implications associated
with geodesic surface congruences and their expansion,
shear and twist. It would be also desirable if the string
topology and the Gromov-Witten invariant can be in-
vestigated by exploiting the Morse theoretic techniques.
These works are in progress and will be reported else-
where.
Acknowledgments
The work of YSC was supported by the Korea Re-
search Council of Fundamental Science and Technol-
ogy (KRCF), Grant No. C-RESEARCH-2006-11-NIMS,
and the work of STH was supported by the Korea Re-
search Foundation (MOEHRD), Grant No. KRF-2006-
331-C00071, and by the Korea Research Council of Fun-
damental Science and Technology (KRCF), Grant No.
C-RESEARCH-2006-11-NIMS.
[1] M.B. Green, J.H. Schwarz and E. Witten, Superstring
Theory Vol. 1 (Cambridge Univ. Press, Cambridge,
1987).
[2] J. Polchinski, String Theory Vol. 1 (Cambridge Univ.
Press, Cambridge, 1999).
[3] E. Witten, J. Diff. Geom. 17, 661 (1982).
[4] M. Morse, The Calculus of Variations in the Large
(Amer. Math. Soc., New York, 1934).
[5] J. Milnor, Morse Theory (Princeton Univ. Press, Prince-
ton, 1963).
[6] R.M. Wald, General Relativity (The Univ. of Chicago
Press, Chicago, 1984).
[7] A. Floer, Comm. Pure Appl. Math. 41, 393 (1988).
[8] M. Schwarz, Morse Homology, Vol. 111 of Prog. Math.
(Birkhäuser, Basel, 1993).
[9] A. Abbondandolo, P. Majer, Comm. Pure Appl. Math.
54, 689 (2001).
[10] M. Chas and D. Sullivan, String Topology, to appear in
Ann. Math., math.GT/9911159.
[11] P. Biran, O. Cornea and F. Lalonde, Morse Theoretic
Methods in Nonlinear Analysis and in Symplectic Topol-
ogy Series II: Mathematics, Physics and Chemistry, Vol.
217 of NATO Sci. Series (Springer, New York, 2004).
[12] D. McDuff and D. Salamon, J-holomorphic Curves and
Quantum Cohomology, Vol. 6 of Univ. Lecture Series
(Amer. Math. Soc., Providence, 1994).
[13] S.T. Hong, J. Geom. Phys. 48, 135 (2003).
[14] M.F. Atiyah, V. Patodi and I. Singer, Math. Proc. Camb.
Phil. Soc. 77, 43 (1975); Math. Proc. Camb. Phil. Soc.
78, 405 (1975); Math. Proc. Camb. Phil. Soc. 79, 71
(1976).
[15] E. Witten, Comm. Math. Phys. 121, 351 (1989).
[16] Y. Nambu, Lecture at the Copenhagen Symposium,
1970, unpublished; T. Goto, Prog. Theor. Phys. 46, 1560
(1971).
[17] J. Scherk, Rev. Mod. Phys. 47, 123 (1975); J. Govaerts,
Lectures given at Escuela Avanzada de Verano en Fisica,
Mexico City, Mexico (1986).
[18] J. Cheeger and D. Ebin, Comparison Theorems in Rie-
mannian Geometry (North-Holland, Amsterdam, 1975).

  We consider the variation of the surface spanned by closed strings in a
spacetime manifold. Using the Nambu-Goto string action, we induce the geodesic
surface equation, the geodesic surface deviation equation which yields a Jacobi
field, and we define the index form of a geodesic surface as in the case of
point particles to discuss conjugate strings on the geodesic surface.

arXiv:0704.0116v2  [math-ph]  21 May 2007
Stringy Jacobi fields in Morse theory
Yong Seung Cho∗
National Institute for Mathematical Sciences, 385-16 Doryong, Yuseong, Daejeon 305-340 Korea and
Department of Mathematics, Ewha Womans University, Seoul 120-750 Korea
Soon-Tae Hong†
Department of Science Education and Research Institute for Basic Sciences, Ewha Womans University, Seoul 120-750 Korea
(Dated: November 4, 2018)
We consider the variation of the surface spanned by closed strings in a spacetime manifold.
Using the Nambu-Goto string action, we induce the geodesic surface equation, the geodesic surface
deviation equation which yields a Jacobi field, and we define the index form of a geodesic surface as
in the case of point particles to discuss conjugate strings on the geodesic surface.
PACS numbers: 02.40.-k, 04.20.-q, 04.90.+e, 11.25.-w, 11.40.-q
Keywords: Nambu-Goto string action, geodesic surface, Jacobi field, index of geodesic surface, conjugate
strings
I. INTRODUCTION
It is well known that string theory [1, 2] is one of the
best candidates for a consistent quantum theory of grav-
ity to yield a unification theory of all the four basic forces
in nature. In D-brane models [2], closed strings represent
gravitons propagating on a curved manifold, while open
strings describe gauge bosons such as photons, or mat-
ter attached on the D-branes. Moreover, because the
two ends of an open string can always meet and con-
nect, forming a closed string, there are no string theories
without closed strings.
On the other hand, the supersymmetric quantum me-
chanics has been exploited by Witten [3] to discuss the
Morse inequalities [4, 5, 6]. The Morse indices for pair of
critical points of the symplectic action function have been
also investigated based on the spectral flow of the Hes-
sian of the symplectic function [7], and on the Hilbert
spaces the Morse homology [8] has been considered to
discuss the critical points associated with the Morse in-
dex [9]. The string topology was initiated in the seminal
work of Chas and Sullivan [10]. Using the Morse theoretic
techniques, Cohen in Ref. [11] constructs string topology
operations on the loop space of a manifold and relates
the string topology operations to the counting of pseudo-
holomorphic curves in the cotangent bundle. He also
speculates the relation between the Gromov-Witten in-
variant [12] of the cotangent bundle and the string topol-
ogy of the underlying manifold. Recently, the Jacobi
fields and their eigenvalues of the Sturm-Liouville oper-
ator associated with the particle geodesics on a curved
manifold have been investigated [13], to relate the phase
factor of the partition function to the eta invariant of
Atiyah [14, 15].
In this paper, we will exploit the Nambu-Goto string
∗Electronic address: yescho@ewha.ac.kr
†Electronic address: soonhong@ewha.ac.kr
action to investigate the geodesic surface equation and
the geodesic surface deviation equation associated with
a Jacobi field. The index form of a geodesic surface will
be also discussed for the closed strings on the curved
manifold.
In Section II, the string action will be introduced to
investigate the geodesic surface equation in terms of the
world sheet currents associated with τ and σ world sheet
coordinate directions. By taking the second variation of
the surface spanned by closed strings, the geodesic sur-
face deviation equation will be discussed for the closed
strings on the curved manifold. In Section III, exploiting
the orthonormal gauge, the index form of a geodesic sur-
face will be also investigated together with breaks on the
string tubes. The geodesic surface deviation equation in
the orthonormal gauge will be exploited to discuss the
Jacobi field on the geodesic surface.
II. STRINGY GEODESIC SURFACES IN
MORSE THEORY
In analogy of the relativistic action of a point parti-
cle, the action for a string is proportional to the area
of the surface spanned in spacetime manifold M by the
evolution of the string. In order to define the action
on the curved manifold, let (M, gab) be a n-dimensional
manifold associated with the metric gab. Given gab, we
can have a unique covariant derivative ∇a satisfying [6]
∇agbc = 0, ∇aω
b = ∂aω
b + Γbac ω
c and
(∇a∇b −∇b∇a)ωc = R
abc ωd. (2.1)
We parameterize the closed string by two world sheet
coordinates τ and σ, and then we have the correspond-
ing vector fields ξa = (∂/∂τ)a and ζa = (∂/∂σ)a. The
Nambu-Goto string action is then given by [1, 2, 16]
S = −
dτdσf(τ, σ) (2.2)
http://arxiv.org/abs/0704.0116v2
where the coordinates τ and σ have ranges 0 ≤ τ ≤ T
and 0 ≤ σ ≤ 2π respectively and
f(τ, σ) = [(ξ · ζ)2 − (ξ · ξ)(ζ · ζ)]1/2. (2.3)
We now perform an infinitesimal variation of the tubes
γα(τ, σ) traced by the closed string during its evolution in
order to find the geodesic surface equation from the least
action principle. Here we impose the restriction that the
length of the string circumference is τ independent. Let
the vector field ηa = (∂/∂α)a be the deviation vector
which represents the displacement to an infinitesimally
nearby tube, and let Σ denote the three-dimensional sub-
manifold spanned by the tubes γα(τ, σ). We then may
choose τ , σ and α as coordinates of Σ to yield the com-
mutator relations,
a = ξb∇bη
a − ηb∇bξ
a = 0,
a = ζb∇bη
a − ηb∇bζ
a = 0,
a = ξb∇bζ
a − ζb∇bξ
a = 0. (2.4)
Now we find the first variation as follows [17]
dτdσ ηb(ξ
τ + ζ
dσ P bτ ηb|
τ=0 −
dτ P bσηb|
σ=0 , (2.5)
where the world sheet currents associated with τ and σ
directions are respectively given by [17]
P aτ =
[(ξ · ζ)ζa − (ζ · ζ)ξa],
P aσ =
[(ξ · ζ)ξa − (ξ · ξ)ζa]. (2.6)
Using the endpoint conditions ηa(0) = ηa(T ) = 0 and pe-
riodic condition ηa(σ+2π) = ηa(σ), we have the geodesic
surface equation [17]
ξa∇aP
τ + ζ
σ = 0, (2.7)
and the constraint identities [17]
Pτ · ζ = 0, Pτ · Pτ + ζ · ζ = 0,
Pσ · ξ = 0, Pσ · Pσ + ξ · ξ = 0.
(2.8)
Let γα(τ, σ) denote a smooth one-parameter family
of geodesic surfaces: for each α ∈ R, the tube γα is
a geodesic surface parameterized by affine parameters τ
and σ. For an infinitesimally nearby geodesic surface in
the family, we then have the following geodesic surface
deviation equation
ξb∇b(η
τ ) + ζ
b∇b(η
+R abcd (ξ
bP dτ + ζ
bP dσ )η
c ≡ (Λη)a = 0. (2.9)
For a small variation ηa, our goal is to compare S(α) with
S(0) of the string. The second variation d2S/dα2(0) is
then needed only when dS/dα(0) = 0. Explicitly, the
second variation is given by
|α=0 = −
(ηc∇cP
τ )(ξ
a∇aηb)
+(ηc∇cP
a∇aηb)−R
acb (ξ
aP bτ + ζ
aP bσ)η
dσ P bτ η
a∇aηb|
τ=0 −
dτ P bση
a∇aηb|
σ=0 .
(2.10)
Here the boundary terms vanish for the fixed endpoint
and the periodic conditions, even though on the geodesic
surface we have breaks which we will explain later. After
some algebra using the geodesic surface deviation equa-
tion, we have
|α=0 =
dτdσ ηa(Λη)
a. (2.11)
III. JACOBI FIELDS IN ORTHONORMAL
GAUGE
The string action and the corresponding equations
of motion are invariant under reparameterization σ̃ =
σ̃(τ, σ) and τ̃ = τ̃ (τ, σ). We have then gauge degrees of
freedom so that we can choose the orthonormal gauge as
follows [17]
ξ · ζ = 0, ξ · ξ + ζ · ζ = 0, (3.1)
where the plus sign in the second equation is due to the
fact that ξ·ξ is timelike and ζ·ζ is spacelike. Note that the
gauge fixing (3.1) for the world sheet coordinates means
that the tangent vectors are orthonormal everywhere up
to a local scale factor [17]. In this parameterization the
world sheet currents (2.6) satisfying the constraints (2.8)
are of the form
P aτ = −ξ
a, P aσ = ζ
a. (3.2)
The geodesic surface equation and the geodesic surface
deviation equation read
− ξa∇aξ
b + ζa∇aζ
b = 0, (3.3)
−ξb∇b(ξ
a) + ζb∇b(ζ
−R abcd (ξ
bξd − ζbζd)ηc = (Λη)a = 0. (3.4)
We now restrict ourselves to strings on constant scalar
curvature manifold such as Sn. We take an ansatz that
on this manifold the string shape on the geodesic surface
γ0 is the same as that on a nearby geodesic surface γα at a
given time τ . We can thus construct the variation vectors
ηa(τ) as vectors associated with the centers of the string
of the two nearby geodesic surfaces at the given time
τ . We then introduce an orthonormal basis of spatial
vectors eai (i = 1, 2, ..., n−2) orthogonal to ξ
a and ζa and
parallelly propagated along the geodesic surface. The
geodesic surface deviation equation (3.4) then yields for
i, j = 1, 2, ..., n− 2
+ (R iτjτ −R
σjσ)η
j = 0. (3.5)
The value of ηi at time τ must depend linearly on the
initial data ηi(0) and dη
(0) at τ = 0. Since by con-
struction ηi(0) = 0 for the family of geodesic surfaces,
we must have
ηi(τ) = Aij(τ)
(0). (3.6)
Inserting (3.6) into (3.5) we have the differential equation
for Aij(τ)
d2Aij
+ (R iτkτ −R
σkσ)A
j = 0, (3.7)
with the initial conditions
Aij(0) = 0,
(0) = δij . (3.8)
Note that in (3.7) we have the last term originated from
the contribution of string property.
Next we consider the second variation equation (2.10)
under the above restrictions
|α=0 =
− (R iτjτ −R
σjσ)η
(3.9)
We define the index form Iγ of a geodesic surface γ as
the unique symmetric bilinear form Iγ : Tγ × Tγ → R
such that
Iγ(V, V ) =
|α=0 (3.10)
for V ∈ Tγ . From (3.9) we can easily find
Iγ(V,W ) =
−(R mτjτ −R
σjσ )W
. (3.11)
If we have breaks 0 = τ0 < · · · < τk+1 = T , and the
restriction of γ to each set [τi−1, τi] is smooth, then the
tube γ is piecewise smooth. The variation vector field
V of γ is always piecewise smooth. However dV/dτ will
generally have a discontinuity at each break τi (1 ≤ i ≤
k). This discontinuity is measured by
(τi) =
(τ+i )−
(τ−i ), (3.12)
where the first term derives from the restrictions
γ|[τi, τi+1] and the second from γ|[τi−1, τi]. If γ and
V ∈ Tγ have the breaks τ1 < · · · < τk, we have
∫ τi+1
dτ = −
(3.13)
to yield
Iγ(V,W ) = −
dτdσ V m
(3.14)
+(R mτjτ −R
σjσ )W
dσ Vm∆
(τi). (3.15)
Here note that if we do not have the breaks, (3.9) yields
|α=0 = −
dτdσ ηi
+ (R iτjτ −R
σjσ)η
(3.16)
A solution ηa of the geodesic surface deviation equation
(3.5) is called a Jacobi field on the geodesic surface γ. A
pair of strings p, q ⊂ γ defined by the centers of the closed
strings on the geodesic surface is then conjugate if there
exists a Jacobi field ηa which is not identically zero but
vanishes at both strings p and q. Roughly speaking, p
and q are conjugate if an infinitesimally nearby geodesic
surface intersects γ at both p and q. From (3.6), q will be
conjugate to p if and only if there exists nontrivial initial
data: dηi/dτ(0) 6= 0, for which ηi = 0 at q. This occurs
if and only if detAij = 0 at q, and thus detA
j = 0
is the necessary and sufficient condition for a conjugate
string to p. Note that between conjugate strings, we have
detAij 6= 0 and thus the inverse of A
j exists. Using (3.7)
we can easily see that
Aik −Aij
= 0. (3.17)
In addition, the quantity in parenthesis of (3.17) vanishes
at p, since Aij(0) = 0. Along a geodesic surface γ, we
thus find
Aik −Aij
= 0. (3.18)
If γ is a geodesic surface with no string conjugate to p
between p and q, then Aij defined above will be nonsingu-
lar between p and q. We can then define Y i = (A−1)ijη
or ηi = AijY
j . From (3.16) and (3.18), we can easily
verify
|α=0 =
≥ 0. (3.19)
Locally γ minimizes the Nambu-Goto string action, if γ is
a geodesic surface with no string conjugate to p between
p and q.
On the other hand, if γ is a geodesic surface but has a
conjugate string r between strings p and q, then we have
a non-zero Jacobi field J i along γ which vanishes at p
and r. Extend J i to q by putting it zero in [r, q]. Then
dJ i/dτ(r−) 6= 0, since J i is nonzero. But dJ i/dτ(r+) = 0
to yield
(r) = −
(r−) 6= 0. (3.20)
We choose any ki ∈ Tγ such that
(r) = c, (3.21)
with a positive constant c. Let ηi be ηi = ǫki + ǫ−1J i
where ǫ is some constant, then we have
Iγ(η, η) = ǫ
2Iγ(k, k) + 2Iγ(k, J) + ǫ
−2Iγ(J, J). (3.22)
By taking ǫ small enough, the first term in (3.22) vanishes
and the third term also vanishes due to the definition
of the Jacobi field and (3.15). Substituting (3.21) into
(3.15) we have Iγ(k, J) = −2πc and thus
|α=0 = −4πc, (3.23)
which is negative definite. From the above arguments, we
conclude that given a smooth timelike tube γ connecting
two strings p, q ⊂ M , the necessary and sufficient con-
dition that γ locally minimizes the surface of the closed
string tube between p and q over smooth one parameter
variations is that γ is a geodesic surface with no string
conjugate to p between p and q. It is also interesting
to see that on Sn, the first non-minimal geodesic sur-
face has n − 1 conjugate strings as in the case of point
particle. Moreover, on the Riemannian manifold with
the constant sectional curvature K, the geodesic surfaces
have no conjugate strings for K < 0 or K = 0, while
conjugate strings occur for K > 0 [18].
IV. CONCLUSIONS
The Nambu-Goto string action has been introduced to
study the geodesic surface equation in terms of the world
sheet currents associated with τ and σ directions. By
constructing the second variation of the surface spanned
by closed strings, the geodesic surface deviation equation
has been discussed for the closed strings on the curved
manifold.
Exploiting the orthonormal gauge, the index form of
a geodesic surface has been defined together with breaks
on the string tubes. The geodesic surface deviation equa-
tion in this orthonormal gauge has been derived to find
the Jacobi field on the geodesic surface. Given a smooth
timelike tube connecting two strings on the manifold,
the condition that the tube locally minimizes the sur-
face of the closed string tube between the two strings
over smooth one parameter variations has been also dis-
cussed in terms of the conjugate strings on the geodesic
surface.
In the Morse theoretic approach to the string theory,
one could consider the physical implications associated
with geodesic surface congruences and their expansion,
shear and twist. It would be also desirable if the string
topology and the Gromov-Witten invariant can be in-
vestigated by exploiting the Morse theoretic techniques.
These works are in progress and will be reported else-
where.
Acknowledgments
The work of YSC was supported by the Korea Re-
search Council of Fundamental Science and Technol-
ogy (KRCF), Grant No. C-RESEARCH-2006-11-NIMS,
and the work of STH was supported by the Korea Re-
search Foundation (MOEHRD), Grant No. KRF-2006-
331-C00071, and by the Korea Research Council of Fun-
damental Science and Technology (KRCF), Grant No.
C-RESEARCH-2006-11-NIMS.
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