Gravity and Strings

Appendix G
The harmonic operator on R 3 × S 1
This section is based on [476].
We want to relate the solutions of the Laplace equation in R 3 × S 1 and in R 3 .Wedenote
the corresponding Laplacians by (4) and (3) and we have
(4) = (3) + ∂ 2 z . (G.1)
The Laplacian, being a local operator, has the same form in R 3 × S 1 and in R 4 .Clearly,
the difference is in the periodicity conditions that the solutions must satisfy in the first case.
This observation will help us to construct them starting with harmonic functions on R 4 .
The solution of the Laplace equation (more precisely, it is the Green function of the
Laplacian) in R 4 is 1/|x4 −x4(0)| 2 , where x4 = (x 1 , x 2 , x 3 , x 4 ).Inparticular, it satisfies
(4)
1
|x4 −x4(0)| 2 =−4π 2 δ (4) (x4 −x4(0)). (G.2)
This harmonic function has a singularity at x4 =x4(0). Weare in general interested in
harmonic functions that go to 1 at infinity and with a different coefficient for the pole (h):
h
HR4 = 1 +
. (G.3)
|x4 −x4(0)|
2
Since the Laplacian is linear, we can combine linearly harmonic functions to construct
one with singularities placed at regular intervals on the x 4 axis. The resulting harmonic
function will have the periodicity required for it to be a harmonic function on R 3 × S 1 .
More explicitly,
This series can be summed:
HR3 ×S1 = 1 +
h
|x3 −x3(0)| 2 . (G.4)
+ (z − z(0) − 2πnℓ) 2
H R 3 ×S 1 = 1 +
n∈Z
h
2ℓ|x3 −x3(0)| 2
sinh |x3 −x3(0)|/ℓ
. (G.5)
cosh |x3 −x3(0)|/ℓ − cos(z − z(0))/ℓ
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