NON-ABELIAN THETA FUNCTIONS A LA ANDRE WEIL´ R˘azvan Gelca

27 Pages·2010·0.27 MB·English

´
NON-ABELIAN THETA FUNCTIONS A LA ANDRE WEIL
R˘azvan Gelca Alejandro Uribe
Texas Tech University University of Michigan
WE WILL SHOW THAT THE CONSTRUCTS OF THE THEORY OF
CLASSICAL THETA FUNCTIONS IN THE REPRESENTATION THEORETIC
´
POINT OF VIEW OF ANDRE WEIL HAVE ANALOGUES FOR THE NON-
ABELIAN THETA FUNCTIONS OF THE WITTEN-RESHETIKHIN-TURAEV
THEORY
1
THE PROTOTYPE:
• the Schr¨odinger representation
• the metaplectic representation
~
They arise when quantizing a free 1-dimensional particle ( = 1).
Phase space has coordinates q: position, p: momentum.
2 R
Quantization: phase space 7→ L ( ),
q 7→ Q = multiplication by q,
1 d
p 7→ P = .
i dq
Add to this the operator E = Id.
Heisenberg uncertainty principle:
~
QP − P Q = i E
2
Weyl quantization:
ZZ
ˆ
f(ξ, η) = f(x, y) exp(−2πixξ − 2πiyη)dxdy
and then defining
ZZ
ˆ
Op(f) = f(ξ, η) exp 2πi(ξQ + ηP )dξdη.
Notation:
2πi(xP+yQ+tE)
exp(xP + yQ + tE) = e .
EXAMPLE:
exp(x P )φ(x) = φ(x + x ),
0 0
2πixy
exp(y Q)φ(x) = e 0φ(x).
0
3
R
The elements exp(xP + yQ + tE), x, y, t ∈ form the Heisenberg group
R
with real entries H( ):
0 0 0
exp(xP + yQ + tE) exp(x P + y Q + t E)
1
0 0 0 0 0
= exp[(x + x )P + (y + y )Q + (t + t + (xy − yx ))E]
2
R 2 R
The action of H( ) on L ( ) is called the Schr¨odinger representation.
THEOREM (Stone-von Neumann) The Schr¨odinger representation is the
R
unique irreducible unitary representation of H( ) that maps exp(tE) to mul-
2πit R
tiplication by e for all t ∈ .
COROLLARY Linear (symplectic) changes of coordinates can be quantized.
4
Recall that the linear maps that preserve the classical mechanics are
(cid:26)(cid:18) (cid:19) (cid:27)
a b
R
SL(2, ) = , ad − bc = 1 .
c d
R 0 0
If h ∈ SL(2, ), and h(x, y) = (x , y ) then
0 0
exp(xP + yQ + tE) ◦ φ = exp(x P + y Q + tE)φ
is another representation, which by the Stone-von Neumann theorem is unitary
equivalent to the Schr¨odinger representation.
2 R 2 R
Hence there is a unitary map ρ(h) : L ( ) → L ( ) such that the exact
Egorov identity is satisfied:
0 0 −1
exp(x P + y Q + tE) = ρ(h) exp(xP + yQ + tE)ρ(h) .
R 2 R
The map h → ρ(h) is a projective representation of SL(2, ) on L ( ).
One can make this into a true representation by passing to a double cover
R R
M(2, ) of SL(2, ). This is the metaplectic representation.
5
EXAMPLE: If
(cid:18) (cid:19) (cid:18) (cid:19)
0 1 1 a
S = , T =
−1 0 0 1
then
Z
−2πixy
ρ(S)φ(x) = φ(y)e dy
R
2
2πix a
ρ(T )φ(x) = e φ(x).
The metaplectic representation can be interpreted as a general Fourier trans-
form. It is a Fourier-Mukai transform.
In general, a Heisenberg group is a U(1) (or cyclic) extension of a locally
compact abelian group. It has an associated Schr¨odinger representation and
Fourier-Mukai transform. The two are related by the exact Egorov identity.
6
A. Weil’s representation theoretic point of view:
~
Weyl quantization of a particle with periodic position and momentum ( =
1/N, N an even integer) .
Quantization: phase space 7→ space of theta functions which has an or-
thonormal basis consisting of the theta series
(cid:20) (cid:21)
(cid:16) (cid:17)2 (cid:16) (cid:17)
i j j
2πiN +n +z +n
X
2 N N
θ (z) = e , j = 0, 1, 2, . . . , N − 1.
j
Z
n∈
The quantizations of the exponential functions on the torus generate a finite
Z Z Z Z
Heisenberg group H( ) which is a -extension of × :
N 2N N N
πi 2πi πi
− pq− jq+ k Z
exp(pP + qQ + kE)θ = e θ , p, q, k ∈ .
N N N
j j+p
7
Z
THEOREM (Stone-von Neumann) The Schr¨odinger representation of H( )
N
is the unique irreducible unitary representation of this group with the property
πi
k Z
that exp(kE) acts as e Id for all k ∈ .
N
The linear maps on the torus that preserve classical mechanics are
(cid:26)(cid:18) (cid:19) (cid:27)
a b
Z Z
SL(2, ) = , a, b, c, d ∈ , ad − bc = 1 ,
c d
i.e. the mapping class group of the torus.
COROLLARY There is a projective representation ρ of the mapping class
group of the torus on the space of theta functions that satisfies the exact
Egorov identity
0 0 −1
exp(p P + q Q + kE) = ρ(h) exp(pP + qQ + kE)ρ(h)
0 0
where (p , q ) = h(p, q). Moreover, for every h, ρ(h) is unique up to multi-
plication by a constant.
This action of the mapping class group is called the Hermite-Jacobi action.
8
Non-abelian theta functions for the group SU(2)
Arise from the quantization of the moduli space of flat su(2)-connections
~ 1
on a genus g surface Σ ( = , N = 2r, r an integer).
2N
SU(2)
M = {A | A : su(2) − connection}/G
g
= {ρ : π (Σ) −→ SU(2)}/conjugation
1
SU(2)
Quantization: M 7→ space of non-abelian theta functions.
g
Non-abelian theta series can be parametrized by coloring the core of a genus
C
g handlebody by irreducible representations of U (SL(2, )), the quantum
~
group of SU(2).
C 1 2 r−1
U (SL(2, )) has irreducible representations V , V , . . . , V . They sat-
~
isfy a Clebsch-Gordan theorem
m n p
V ⊗ V = ⊕ V
p
where |m − n| + 1 ≤ p ≤ min(m + n − 1, 2r − 2 − m − n).
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1 2 r−1
The theta series are the colorings of the core by V , V , . . . , V such that
at each vertex the conditions from the Clebsch-Gordan theorem are satisfied.
The analogues of the exponentials are the Wilson lines
W (A) = tr hol (A)
γ,n V n γ
where γ is a simple closed curve on the surface.
The operator associated to W , denoted by op(W ), has a matrix whose
γ,n γ,n
“entries” are the Reshetikhin-Turaev invariants of diagrams of the form:
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