Posted: November 18th, 2015

Riemann Surface

Riemann Surface

Mathematics, Statistics and Applied Mathematics
Abstract
Monodromy considerations in Conformal Field Theory (CFT) on a genus g
hyperelliptic surface
Sg : y
2 = p(x) , deg p = n = 2g + 1 ,
naturally lead to the hypergeometric differential equation for the case g = 1. In
the latter case, S1 is the complex torus defined as a double cover of the Riemann
sphere P
1
C  C ? {8} which is ramified at the roots of the polynomial
p(x) = a0 x(x – X1)(x – 1) , a0 ? C ,
in C, and at the point at infinity.
1 The hypergeometric equation, for n = 3
1.1 Notations and conventions
• X1 = z is a free complex parameter, for which we use both notations z and X1
interchangeably. The other ramification points are fixed and ordered as follows:
X2 = 0, X3 = 1, and X4 = 8.
• h1i and h?(x)i are so-called zero- and one-point functions, respectively, of CFT.
These are functions of X1 (though this is not made explicit in the notation), and
h?(x)i depends in addition on x ? C.
• To enhance readibility of the formulae, we shall denote p(x),h?(x)i, . . . by px,h?xi, . . ..
Thus pX1
, p
0
X1
, . . . ,h?X1
i is the evaluation of px, p
0
x
, . . . ,h?xi at x = X1, where
p
0
x =
d
dx px.
• c = –
22
5
is a physical parameter (the central charge of the model considered).
1
1.2 Second order ODE
Previous study of a specific model of CFT on a g = 1 Riemann surface lead to the
following set of ODEs with regular singularities:
?
???????
d
dX1
–
c
8
X
3
i=2
1
X1 – Xi
?
???????
h1i = 2
h?X1
i
p
0
X1
,
?
???????
d
dX1
–
c
8
X
3
i=2
1
X1 – Xi
?
???????
h?X1
i
p
0
X1
=
1
40
?
??????
7c
8
?
?????
p
00
X1
p
0
X1
?
?????
2
–
13c
2
a0
p
0
X1
?
??????
h1i +
2
5
p
00
X1
p
0
X1
h?X1
i
p
0
X1
.
1.3 The hypergeometric differential equation
Every second-order ODE with at most three regular singular points can be transformed
into the hypergeometric differential equation
z(1 – z)
d
2w
dz2
+ [C – (A + B + 1)z]
dw
dz
– ABw = 0 (1)
(1) is an ODE with regular singularities at z = 0, 1, 8.
1.4 Proposition
1.4.1 Prove the following statement:
When g = 1,
w(z) :=
Y
3
i=2
(z – Xi)
-k-
c
8 h1i
solves the hypergeometric differential equation (1) in z = X1 provided k takes on one
of the two following values:
k = –
7
10
, –
11
10
.
The parameters of the hypergeometric differential equation (1) are
A, B =
?
???
???
3
10 , –
1
10 for k = –
7
10
7
10 ,
11
10 for k = –
11
10 ,
and
C =
?
???
???
3
5
for k = –
7
10
7
5
for k = –
11
10 .
1.4.2 Hints
Direct computation.
1. Use that
d
dx
+ fx = e
-Fx
d
dx
e
Fx
, F
0 = f .
so in Subsection 1.2, by going over to the functions e
FX1 h1i and e
FX1 h?X1
i, we
can transform the equations for the covariant derivative 
d
dX1
–
c
8
P3
i=2
1
X1-Xi

on
h1i and h?X1
i into equations involving the ordinary derivative d
dX1
only.
2
2. Set
gx :=
Y
3
i=2
(x – Xi) . (2)
Compare gX1
, g
0
X1
, g
00
X1
with the evaluation of px and its derivatives at X1. Reformulate
the set of first order ODEs given in Subsection 1.2 in terms of gX1
and its
derivatives.
3. Find the second order ODE for e
FX1 h1i which corresponds to the set of first order
ODEs in the formulation of the previous step. Make the ansatz
e
FX1 h1i =: g
k
X1
wX1
,
where gx is the function (2), and choose k such that the second order pole drops
out.
4. Determine A, B,C so that wX1
solves the hypergeometric differential equation (1)
in z = X1, and conclude.
3

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