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BOSONIC NAMBU-GOTO STRINGS AND THE EULER CHARACTERISTIC
SUNG LEE
In string theory, elementary particles are considered to be tiny vibrating strings in spacetime. A string evolves in time while sweeping a surface, a so-called world- sheet , in spacetime. Hence, string worldsheets are timelike surfaces. Moreover, it can be shown that string worldsheets are indeed timelike minimal surfaces^1 in spacetime. The following discussion is not restricted to ( 1 + 3 )-dimensions. So, one can assume any higher (1+D)-dimensional spacetime. Here we consider only classical strings. In case of quantized strings, our universe is, without super- symmetry, a 26-dimensional spacetime and it is a 10-dimensional spacetime with supersymmetry. We consider string worldsheets parametrised by τ and σ , where τ is time parameter in ( 1 + 1 )-dimensions. The motion of bosonic strings in spacetime is described by the Nambu-Goto string action
(1) S = − T
M
(− det hab )^1 /^2 dτdσ ,
where T is tension and hab is the metric tensor of the worldsheet ϕ : M −→ R1, D. Using the Einstein’s convention , hab is given by
hab = ∂aϕμ∂bϕν^ ημν ,
where ημν is the metric tensor of the flat Minkowski ( 1 + D )-spacetime with signature (−, +, · · · , +). Clearly, dA := (− det hab )^1 /^2 dτdσ is the area element of the string worldsheet ϕ.
Let us denote ˙ ϕ := ∂ ϕ ∂ τ and ϕ
′ := ∂ ϕ ∂ σ. The Lagrangian L of the string motion is
(2) L ( ϕ ˙, ϕ
′ ; σ , τ ) = − T (− det hab )^1 /^2.
By variational principle, δ S = 0, subject to the condition that the initial and final configurations of the string are kept fixed, implies the Euler-Lagrange equation for the string action (1) is
(3)
∂ τ
∂ L
∂ ϕ ˙ μ^
∂ σ
∂ L
∂ ϕ ′ μ =^0
Date : September 10, 2009. (^1) Surface area minimising surfaces. 1
2 SUNG LEE
with ∂ L ∂ ϕ ˙ μ^
∂ L
∂ ϕ ′ μ =^ 0 on^ ∂^ M^.
Note that the action (1) is invariant under conformal scaling of the worldsheet metric. Physicists call it Weyl invariance and it is an important symmetry of the action (1) along with worldsheet reparametrisations and Lorentz/Poincaré symme- tries. With conformal gauge fixing
−〈 ϕτ , ϕτ 〉 = 〈 ϕσ , ϕσ 〉 = eω , 〈 ϕτ , ϕσ 〉 = 0,
one can easily show that the Euler-Lagrange equation (3) is equivalent to the homogeneous wave equation
ϕ = −
∂^2 ϕ ∂ τ^2
∂^2 ϕ ∂ σ^2
which is also the equation of timelike minimal surfaces. The Gaußian curvature K plays an important role in string theory, especially when results from different orders of string perturbation theory are compared and when string interactions are considered. In order to write a more familiar expression in string theory, we introduce a new parameter α
′ which is defined by
α
′
2 πT
This parameter α ′ is known as the slope of Regge trajectories in physics. We refer to [ 2 ], [ 3 ] for physical background and details regarding Regge trajectories. As is well-known in variational theory, one can add more terms to action functionals as constraints. One physically interesting extra term is the Einstein-Hilbert action
(4) χ :=
4 πα ′
M
R (− det hab )^1 /^2 dτdσ +
2 πα ′
∂ M
kds ,
where R is the Ricci scalar (or scalar curvature) on the worldsheet M and k is the geodesic curvature on ∂ M. Note that the Einstein-Hilbert action (4) is invariant under the conformal transformation (Weyl transformation) hab −→ eωhab. Now, the full string action is given by
S
′ = −
2 πα ′
M
(− det hab )^1 /^2 dτdσ + λ
4 πα ′
M
R (− det hab )^1 /^2 dτdσ
2 πα ′
∂ M
kds
where λ is a coupling parameter. This full string action resembles 2-dim gravity coupled with bosonic matter fields and the equations of motion is given by the following Einstein’s field equation:
Rab −
habR = Tab.
