QUANTUM MECHANICS AND NON-ABELIAN THETA FUNCTIONS FOR THE

31 Pages·2013·0.49 MB·English

QUANTUM MECHANICS AND NON-ABELIAN THETA
FUNCTIONS FOR THE GAUGE GROUP SU(2)
RA˘ZVANGELCAANDALEJANDROURIBE
Abstract. In this paper we propose a direction of study of non-abelian theta
functions by establishing an analogy between the Weyl quantization of a one-
dimensional particle and the metaplectic representation on the one hand, and
thequantizationofthemodulispaceofflatconnectionsonasurfaceandtherep-
resentationofthemappingclassgrouponthespaceofnon-abelianthetafunctions
on the other. We examplify this with the cases of classical theta functions and
of the non-abelian theta functions for the gauge group SU(2). The emphasis of
thepaper isonthisanalogyandonthepossibilityofgeneralizingthis approach
to other gauge groups, and not on the results, of which some have appeared
elsewhere.
1. Introduction
The paper outlines a study of the non-abeliantheta functions that arisein Chern-
Simons theory by adapting the method used by Andr´e Weil for studying classical
theta functions [44]. The goal is to derive the constructs of Chern-Simons theory
from quantum mechanics, as oposed to quantum field theory. We exemplify with
the case of the gauge group SU(2). We envision two possible applications of our
method: the generalization to other gauge groups, including non-compact ones, and
the discovery of the analytical model for the quantization that corresponds to the
quantum group quantization of the moduli space of flat SU(2)-connections on the
torus. Thelatterisalongstandingproblemonwhichmodestprogresswasmade(see
[2], [14], [29]).
In Weil’s approach, classical theta functions come with an action of the finite
Heisenberg group and a projective representation of the mapping class group. By
analogy, the theory of non-abelian theta functions consists of:
•theHilbertspaceofnon-abelianthetafunctions,namelytheholomorphicsections
of the Chern-Simons line bundle;
• an irreducible representation on the space of theta functions of the algebra gen-
erated by quantized Wilson lines (i.e. of the quantizations of traces of holonomies of
simple closed curves);
•aprojectiverepresentationofthemappingclassgroupofthesurfaceonthespace
of non-abelian theta functions.
TherepresentationofthemappingclassgroupintertwinesthequantizedWilsonlines;
in this sense the two representations satisfy an exact Egorov identity.
Our prototype is the quantization of a one-dimensional particle. The paradigm is
thatthe quantumgroup quantization of the moduli space of flat SU(2)-connections on
1991 Mathematics Subject Classification. 81S10, 81R50,57R56,81T45,57M25.
Keywordsandphrases. Non-abelianthetafunctions,Reshetikhin-Turaevtheory,skeinmodules,
Weylquantization.
Researchofthefirstauthor supportedbytheNSF,awardNo. DMS0604694.
ResearchofthesecondauthorsupportedbytheNSF,awardNo. DMS0805878.
1
2 RA˘ZVANGELCAANDALEJANDROURIBE
asurfaceandtheReshetikhin-Turaevrepresentationofthemappingclassgrouparethe
analogues of the Schro¨dinger representation of the Heisenberg group and of the meta-
plectic representation. The Schr¨odinger representation arises from the quantization
of the position and the momentum of a one-dimensional free particle, and is a conse-
quence of a fundamental postulate in quantum mechanics. It is a unitary irreducible
representation of the Heisenberg group, and the Stone-von Neumann theorem shows
that it is unique. This uniqueness implies that linear changes of coordinates (which
act as outer automorphisms of the Heisenberg group) are also quantizable, and their
quantization yields an infinite dimensional representation of the metaplectic group.
Weil [44] observedthat a finite Heisenberg groupacts on classicaltheta functions,
and the action of the modular group is induced via a Stone-von Neumann theorem.
Thenitwasnoticedthatclassicalthetafunctions,theactionoftheHeisenberggroup,
and of the modular group arise from the Weyl quantizationof Jacobianvarieties. As
such, classical theta functions are the holomorphic sections of a line bundle over the
moduli space of flat u(1)-connections on a surface, and by analogy, the holomorphic
sections of the similar line bundle over the moduli space of flat g-connections over a
surface (where g is the Lie algebra of a compact simple Lie group) were called non-
abelianthetafunctions. Witten[46]placednon-abelianthetafunctionsinthecontext
of Chern-Simons theory, related them to the Jones polynomial [17] and conformal
fieldtheory,andgavenewmethodsforstudyingthem. We showhowwithinWitten’s
theory one can find the non-abelian analogues of Weil’s constructs.
The paper runs the parallel between the Schr¨odinger and metaplectic representa-
tions, the Weil representationof the Heisenberg groupand the actionof the modular
group on theta functions, and the quantum group quantization of the moduli space
of flat su(2)-connections on a surface and the Reshetikhin-Turaev representation.
Among the features we mention the translation of the quantum group quantization
of the moduli space into skein theoretical language, and the derivation strictly from
quantum mechanical considerations of the element Ω which is the building block of
the Witten-Reshetikhin-Turaev invariants,and of the Reshethikhin-Turaev represen-
tation of the mapping class group. Our analogy suggest that any analytical model
for the quantization of the moduli space of flat su(2)-connections should be similar
to Weyl quantization. It also establishes a programme for studying Chern-Simons
theory with general gauge group. For that reason, we present proofs of the results
which could allow generalizations.
2. The prototype
2.1. The Schro¨dinger representation. InthissectionwereviewbrieflytheSchr¨o-
dinger and the metaplectic representations. For more details see [28].
Considerparticleina1-dimensionalspace. ThephasespaceisR2,withcoordinates
thepositionxandthemomentumy,symplecticformω =dx∧dy andPoissonbracket
{f,g}= ∂f ∂g − ∂f ∂g. The symplectic form induces a nondegenerate antisymmetric
∂x∂y ∂y∂x
bilinear form on R2, also denoted by ω, given by ω((x,y),(x′,y′))=xy′−x′y.
TheLiealgebraofobservableshasasubalgebrageneratedbyQ(x,y)=x,P(x,y)=
y, and E(x,y)=1, called the Heisenberg Lie algebra. Abstractly, this algebra is de-
fined by [Q,P]=E,[P,E]=[Q,E]=0.
It is a postulate of quantum mechanics that the quantization of the position, the
momentum, and the constant functions is the representation of the Heisenberg Lie
algebra on L2(R,dx) defined by
~ d
Q→M , P → , E →i~Id.
x
i dx
QUANTUM MECHANICS AND THETA FUNCTIONS 3
Here M denotes the operator of multiplication by the variable: φ(x) → xφ(x) and
x
~=h/2π is the reduced Planck’s constant. This is the Schr¨odinger representationof
the Heisenberg Lie algebra.
By exponentiation one obtains the Schr¨odinger representation of the Heisenberg
group with real entries H(R):
exp(x Q+y P +tE)φ(x)=e2πi(x0Q+y0P+tE)φ(x)=e2πix0x+πihx0y0+2πitφ(x+hy ).
0 0 0
Using this representation, Hermann Weyl gave a method for quantizing functions
f ∈C∞(R2), using the Fourier transform
fˆ(ξ,η)= f(x,y)exp(−2πixξ−2πiyη)dxdy
ZZ
to define
Op(f)= fˆ(ξ,η)exp2πi(ξQ+ηP)dξdη,
ZZ
where for exp(ξQ+ηP) he used the Schr¨odinger representation.
Theorem (Stone-von Neumann) The Schr¨odinger representation of the Heisenberg
groupistheuniqueirreducibleunitaryrepresentationofthisgroupsuchthatexp(tE)
acts as e2πitId for all t∈R.
There are two other important realizations of the irreducible representation that
this theorem characterizes. One comes from the quantization of the plane in a holo-
morphicpolarization. TheHilbertspaceistheBargmannspaceofholomorphicsquare
integrable entire functions with respect to the measure e−2π|Im z|2dzdz¯, with the
Heisenberg group acting by
exp(x Q+y P +tE)f(z)=eπih(y0+ix0)+2πix0z+2πitf(z+h(y +ix )).
0 0 0 0
For the other, choose a Lagrangian subspace L of RP +RQ. Then exp(L+RE)
is a maximal abelian subgroup of the Heisenberg group. Consider the character of
this subgroup defined by χL(exp(l +tE)) = e2πit,l ∈ L. The Hilbert space of the
quantization, H(L), is defined as the space of functions φ(u) on H(R) satisfying
φ(uu′)=χ (u′)−1φ(u) for all u′ ∈exp(L+RE)
L
andsuchthatu→|φ(u)|isasquareintegrablefunctionontheleftequivalenceclasses
modulo exp(L+RE). The representation of the Heisenberg group is given by
u φ(u)=φ(u−1u).
0 0
ChoosinganalgebraiccomplementL′ ofLandwritingRP+RQ=L+L′ =R+R,
H(L)isrealizedasL2(L′)∼=L2(R). ForL=RP andL′ =RQ,onegetsthestandard
Schr¨odinger representation.
2.2. The metaplectic representation. By the Stone-von Neumann theorem, if
we change coordinates by a linear symplectomorphism and then quantize, we get a
unitarily equivalent representation of the Heisenberg group. Hence linear symplecto-
morphisms can be quantized by unitary operators. Schur’s lemma implies that these
operators are unique up to a multiplication by a constant. So we have a projective
representationρofthe linearsymplecticgroupSL(2,R)onL2(R). Thiscanbe made
into a true representation by passing to the double cover of SL(2,R), the metaplec-
tic group Mp(2,R). The representation of the metaplectic group is known as the
metaplectic representation or the Segal-Shale-Weil representation.
4 RA˘ZVANGELCAANDALEJANDROURIBE
The fundamental symmetry of the Weyl quantization is
Op(f ◦h−1)=ρ(h)Op(f)ρ(h)−1,
for every observable f ∈C∞(R2) and h∈Mp(2,R), where Op(f) is the operator as-
sociatedto f throughWeylquantization. Forotherquantizationmodels this relation
holds only mod O(~), (Egorov’s theorem). When satisfied with equality, as in our
case, it is called an exact Egorov identity.
ThemetaplecticrepresentationcanbedefinedusingthethirdversionoftheSchr¨odinger
representationin§2.1,whichidentifies itasaFouriertransform(see [28]). Leth be a
linear symplectomorphism of the plane, L a Lagrangian subspace of RP +RQ and
1
L =h(L ). The quantization of h is ρ(h):H(L )→H(L ),
2 1 1 2
(ρ(h)φ)(u)=ZexpL2/exp(L1∩L2)φ(uu2)χL2(u2)dµ(u2),
where dµ is the measure induced on the space of equivalence classes by the Haar
measure on H(R).
For explicit formulas for ρ(h) one needs to choose the algebraic complements L′
1
and L′ of L and L and unfold the isomorphism L2(L′)∼=L2(R). For example, for
2 1 2
0 1
S = ,
(cid:18) −1 0 (cid:19)
ifwesetL =RP withvariabley andL =S(L )=RQwithvariablexandL′ =L
1 2 1 1 2
and L′ =S(L′)=L , then
2 1 1
ρ(S)f(x)= f(y)e−2πixydy,
ZR
is the usualFouriertransform,whichestablishes the unitaryequivalence between the
position and the momentum representations. Similarly, for
1 a
T = ,
a (cid:18) 0 1 (cid:19)
if we set L =L =RP =, L′ =RQ, and L′ =R(P +Q), then
1 2 1 2
ρ(T )f(x)=e2πix2af(x).
a
The cocycle of the projective representation of the symplectic group is
cL(h′,h)=e−i4πτ(L,h(L),h′◦h(L))
where τ is the Maslov index. This means that
ρ(h′h)=cL(h′,h)ρ(h′)ρ(h)
for h,h′ ∈SL(2,R).
3. Classical theta functions
3.1. Classical theta functions from the quantization of the torus. For an
extensive treatment of theta functions the reader canconsult [30], [28], [31]. Here we
only consider the simplest situation, that of theta functions on the Jacobian variety
ofa 2-dimensionalcomplex torus T2. Our discussionis sketchy;details canbe found,
for all closed Riemann surfaces, in [15].
Given the complex torus and oriented simple closed curves a and b with algebraic
intersection number 1, which define a canonical basis of H (T2,R) (or equivalently
1
of π (T2)), take a holomorphic 1-form ζ such that ζ = 1. The complex number
1 a
τ = ζ, which depends on the complex structure, hRas positive imaginary part. The
b
R
QUANTUM MECHANICS AND THETA FUNCTIONS 5
JacobianvarietyofT2,J(T2),isa2-dimensionaltoruswithcomplexstructuredefined
by τ (as an element in its Teichmu¨ller space). Equivalently, J(T2)=C/Z+Zτ. We
introducerealcoordinates(x,y)onJ(T2)bysettingz =x+τy. Inthesecoordinates,
J(T2) is the quotient of R2 by Z2. J(T2) is endowed with the symplectic form
ω = dx∧dy, which is a generator of H2(T2,Z). J(T2) with its complex structure
and this symplectic form is a K¨ahler manifold.
Classical theta functions and the action of the Heisenberg group can be obtained
byapplyingWeylquantizationtoJ(T2)intheholomorphicpolarization. Thetafunc-
tions are obtained by geometric quantization. We start by setting Planck’s constant
h= 1, N a positive even integer.
N
The Hilbert space of the quantization consists of the classical theta functions,
which are the holomorphic sections of a line bundle over the Jacobian variety. This
line bundle is the tensor product of a line bundle of curvature −2πiNω and a half-
density. By pulling back the line bundle to C, we can view these sections as entire
functions with some periodicity. The line bundle with curvature 2πiNω is unique up
totensoringwithaflatbundle. Choosingthelatterappropriately,wecanensurethat
the periodicity conditions are
f(z+m+nτ)=e−2πiN(τn2+2nz)f(z).
An orthonormal basis of the space of classical theta functions is given by the theta
series
(3.1) θτ(z)= e2πiNhτ2(Nj+n)2+z(Nj+n)i, j =0,1,2,...,N −1.
j n∈Z
P
It is convenient to extend this definition to all indices j by the periodicity condition
θτ (z)=θτ(z), namely to take indices modulo N.
j+N j
Let us turn to the operators. The only exponentials on the plane that are doubly
periodic, and therefore give rise to functions on the torus, are
f(x,y)=exp2πi(mx+ny), m,n∈Z.
Since the torus is a quotient of the plane by a discrete group, we can apply the Weyl
quantization procedure. In the complex polarization Weyl quantization is defined as
follows(see[9]): Afundamentaldomainofthetorusistheunitsquare[0,1]×[0,1](this
is done in the (x,y) coordinates,in the complex plane it is actually a parallelogram).
The value of a theta function is completely determined by its values on this unit
square. The Hilbert space of classical theta functions can be isometrically embedded
into L2([0,1]×[0,1]) with the inner product
1 1
hf,gi=(−iN(τ −τ¯))1/2 f(x,y)g(x,y)eiN(τ−τ¯)πy2dxdy.
Z Z
0 0
For a proof of the following result see [15].
Proposition 3.1. The Weyl quantization of the exponentials in the momentum rep-
resentation is given by
Op e2πi(px+qy) θjτ(z(=e−πNipq−2Nπijqθjτ+p(z).
(cid:16) (cid:17)
The Weyl quantization of the exponentials gives rise to the Schr¨odinger represen-
tation of the Heisenberg group with integer entries H(Z) onto the space of theta
functions. This Heisenberg group is Z3, with multiplication
(p,q,k)(p′,q′,k′)=(p+p′,q+q′,k+k′+(pq′−qp′)).
6 RA˘ZVANGELCAANDALEJANDROURIBE
The proposition implies that
(p,q,k)7→ the Weyl quantization of eπNikexp2πi(px+qy)
is a group morphism. This is the Schr¨odinger representation.
The Schr¨odinger representation of H(Z) is far from faithful. Because of this we
factoritoutbyitskernel. Thekernelisthesubgroupconsistingoftheelementsofthe
form (p,q,k)N, with k even [15]. Let H(Z ) be the finite Heisenberg groupobtained
N
by factoring H(Z) by this subgroup, and let exp(pP +qQ+kE) be the image of
(p,q,k) in it.
The following is an analogue of the Stone-von Neumann theorem.
Theorem 3.2. The Schro¨dinger representation of H(Z ) is the unique irreducible
N
unitary representation of this group with the property that exp(kE) acts as eπNikId for
all k ∈Z.
The Schr¨odinger representationof the finite Heisenberg groupcan be extended by
linearity to a representation of the group algebra with coefficients in C of the finite
Heisenberg group, C[H(Z )]. Since the elements of exp(ZE) act as multiplications
N
by constants, this is in fact a representationof the algebra A obtained by factoring
N
C[H(ZN)] by the relations exp(kE)−eπNik for all k ∈ Z. By abuse of language, we
call this the Schr¨odinger representation as well. The Schr¨odinger representation of
A defines the quantizations of trigonometric polynomials on the torus.
N
Proposition3.3. a) The algebra of Weyl quantizations of trigonometric polynomials
contains all linear operators on the space of theta functions.
b) The Schro¨dinger representation of A on theta functions is faithful.
N
Proof. For a) see [15]. Part b) follows from the fact that exp(pP + qQ), p,q =
0,1,...,N −1, form a basis of A as a vector space. (cid:3)
N
As explained in [15], the Schr¨odinger representation can be described as the left
regular action of the group algebra of the finite Heisenberg group on a quotient of
itself. The construction is like for the Schr¨odinger representation in the abstract
setting in §2.2.
3.2. Classical theta functions from a topological perspective. In[15]the the-
ory of classical theta functions was shown to admit a reformulation in purely topo-
logical language. Let us recall the facts.
LetM be asmoothorientedcompact3-manifold. Aframedlink inM is a smooth
embedding of a disjoint union of circles, with the framing of each link component
defined by a vector field orthogonal to it. We can view the framed link as an em-
bedding of severalannuli, each having a specified boundary component (which is the
actuallink component). We drawalldiagramsinthe blackboardframing,sothatthe
framing is parallel to the plane of the paper.
Consider the free C[t,t−1]-module with basis the set of isotopy classes of framed
oriented links in M, including the empty link ∅. Factor it by all equalities of the
form shown in Figure 1. In each diagram, the two links are identical except for an
embedded ball in M, inside of which they look as shown. Thus in a link we can
smoothen a crossing provided that we add a coefficient of t or t−1, and trivial link
components can be ignored. These are called skein relations. The skein relations are
considered for all possible embeddings of a ball. When strands are joined, framings
should agree. The result of the factorization is the linking number skein module of
M, denoted L (M). These skein modules were first introduced by Przytycki in [33].
t
QUANTUM MECHANICS AND THETA FUNCTIONS 7
If M = S3, then each link L is, as an element of L (S3), equivalent to the empty
t
linkwiththecoefficientequaltotraisedtothesumofthelinkingnumbersofordered
pairs of components and the writhes of the components, hence the name.
t ; t−1
t ; t−1
Figure 1
For a fixed even positive integer N we define the reduced linking number skein
module of M, denoted by Lt(M), as the quotient of Lt(M) by t = eiNπ and γN = ∅
for every framed link component γ, where γN denotes N parallel copies of γ. As a
e
rule,inaskeinmoduletisafreevariable,whileinareducedskeinmoduleitisaroot
of unity.
If M =T2×[0,1],the topologicaloperationof gluing a cylinder on top of another
induces a multiplication in L (T2 × [0,1]) turning L (T2 × [0,1]) into an algebra,
t t
the linking number skein algebra of the cylinder over the torus. The multiplication
descends to L (T2×[0,1]). We explicate its structure.
t
For p and q coprime integers, orient the curve (p,q) by the vector from the origin
e
to the point (p,q), and frame it so that the annulus is parallelto the torus. Call this
thezeroframing,ortheblackboard framing. Anyotherframingofthecurve(p,q)can
berepresentedbyanintegerk,where|k|isthenumberoffulltwiststhatareinserted
on this curve, with k positive if the twists are positive, and k negative otherwise. In
L (T2×[0,1]), (p,q) with framing k is equivalent to tk(p,q).
t
If p and q are not coprime and n is their greatest common divisor, let (p,q) =
(p/n,q/n)nFinally, ∅=(0,0)is the empty link,the multiplicativeidentity ofL (T2×
t
[0,1]).
Theorem 3.4. [15] The algebra L (T2 ×[0,1]) is isomorphic to the group algebra
t
C[H(Z)], with the isomorphism induced by
tk(p,q)→(p,q,k).
This map descends to an isomorphism between L (T2×[0,1]) and the algebra A of
t N
Weyl quantizations of trigonometric polynomials.
e
Identifying the group algebra of the Heisenberg group with integer entries with
C [U±1,V±1], we conclude thatthe linking numberskeinalgebraofthe cylinder over
t
the torus is isomorphic to the ring of trigonometric polynomials in the noncommuta-
tive torus.
Let us look at the skeinmodule of the solidtorus L (S1×D2). Letα be the curve
t
thatisthecoreofthesolidtorus,withacertainchoiceoforientationandframing. The
reduced linking number skein module L (S1×D2) has basis αj, j =0,1,...,N −1.
t
Let h be a homeomorphism of the torus to the boundary of the solid torus that
0 e
mapsthefirstgeneratorofthefundamentalgrouptoacurveisotopictoα(alongitude)
andthesecondgeneratortothecurveontheboundaryofthe solidtorusthatbounds
a disk in the solid torus (a meridian). The operation of gluing T2 ×[0,1] to the
boundary of S1×D2 via h induces a left action of L (T2×[0,1]) onto L (S1×D2).
0 t t
This descends to a left action of L (T2×[0,1]) onto L (S1×D2).
t t
e e
8 RA˘ZVANGELCAANDALEJANDROURIBE
Observe that L (S1×D2) and L (S1×D2) are quotients of L (T2×[0,1]) respec-
t t t
tivelyL (S1×D2),withtwoframedcurvesequivalentonthetorusiftheyareisotopic
t e
in the solid torus.
Theorem3.5. [15]Thereisanisomorphism thatintertwinestheactionofthealgebra
ofWeylquantizationsoftrigonometricpolynomials onthespaceofthetafunctionsand
therepresentationofL (T2×[0,1])ontoL (S1×D2), andwhichmapsthethetaseries
t t
θτ(z) to αj for all j =0,1,...,N −1.
j e e
Remark 3.6. The choice of generators of π (T2) completely determines the homeo-
1
morphism h , allowing us to identify the Hilbert space of the quantization with the
0
vector space with basis α0 = ∅,α,...,αN−1. As we have seen above, these basis
elements are the theta series.
3.3. The discrete Fourier transform for classical theta functions from a
topological viewpoint. Thesymmetriesofclassicaltheta functionsareaninstance
of the Fourier transform. We put them in a topological perspective (see [15]). An
element
a b
(3.2) h= ∈SL(2,Z)
(cid:18) c d (cid:19)
definesanactionofthemappingclassgroupontheWeylquantizationsofexponentials
given by
h·exp(pP +qQ+kE)=exp[(ap+bq)P +(cp+dq)Q+kE].
This action is easy to describe in the skein theoretical setting, it just maps every
framed link γ on the torus to h(γ).
Theorem 3.7. 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
h·exp(pP +qQ+kE)=ρ(h)exp(pP +qQ+kE)ρ(h)−1.
Moreover, for every h, ρ(h) is unique up to multiplication by a constant.
Proof. We will exhibit two proofs of this well-known result, to which we will refer
when discussing non-abelian Chern-Simons theory.
Proof 1: The map that associates to exp(pP +qQ+kE) the operator that acts on
theta functions as
θτ →exp[(ap+bq)P +(cp+dq)Q+kE]θτ
j j
is alsoa unitaryirreducible representationofthe finite Heisenberggroupwhichmaps
exp(kE) to multiplication by eiNπ. By the Stone-von Neumann theorem, this repre-
sentation is unitarily equivalent to the Schr¨odinger representation. This proves the
existence of ρ(h) satisfying the exact Egorov identity. By Schur’s lemma, the map
ρ(h)isuniqueuptomultiplicationbyaconstant. Hence,ifhandh′ aretwoelements
of the mapping class group, then ρ(h′ ◦h) is a constant multiple of ρ(h′)ρ(h). It
follows that ρ defines a projective representation.
Proof 2: The map exp(pQ + qQ + kE) → h · exp(pP + qQ + kE) extends to an
automorphism of the algebra C[H(Z)]. Because the ideal by which we factor to ob-
tainA is invariantunder the actionofthe mapping classgroup,this automorphism
N
induces an automorphism Φ : A → A , which maps each scalar multiple of the
N N
identity to itself. Since, by Proposition 3.3, A is the algebra of all linear operators
N
QUANTUM MECHANICS AND THETA FUNCTIONS 9
on the N-dimensional space of theta functions, Φ is inner [42], meaning that there is
ρ(h):A →A such that Φ(x)=ρ(h)xρ(h)−1. In particular
N N
h·exp(pP +qQ+kE)=ρ(h)exp(pP +qQ+kE)ρ(h)−1.
The Schr¨odinger representation of A is obviously irreducible, so again we apply
N
Schur’s lemma and conclude that ρ(h) is unique up to multiplication by a constant
and h→ρ(h) is a projective representation. (cid:3)
The representation ρ is the well-known action of the modular group given by dis-
crete Fourier transforms.
AsaconsequenceofProposition3.3,foranyelementhofthemappingclassgroup,
the linear map ρ(h) is in L (T2×[0,1]), hence it canbe representedby a skeinF(h).
t
This skein satisfies
e
h(σ)F(h)=F(h)σ
forallσ ∈L (T2×[0,1]). MoreoverF(h)isuniqueuptomultiplicationbyaconstant.
t
We recall the formula for F(h) derived [15].
e
Every3-dimensionalmanifoldistheboundaryofa4-dimensionalmanifoldobtained
by adding 2-handlesD2×D2 to a 4-dimensionalballalongthe solidtoriD2×S1. On
the boundary S3 of the ball, the operation of adding handles gives rise to surgery on
a framed link. Thus any given 3-dimensional manifold can be obtained as follows:
Start with a suitable framed link L ⊂ S3. Take a regular neighborhood of L made
out of disjoint solid tori, each with a framing curve on the boundary such that the
coreofthe solidtorusandthis curvedetermine the framingofthe correspondinglink
component. Remove these tori, then glue them back in so that meridians are glued
toframingcurvesinasuitableway. The resultisthe desired3-dimensionalmanifold.
Sliding one2-handleoveranothercorrespondsto slidingone link componentalong
another using a Kirby band-sum move [22]. A slide of K along K, denoted by
1
K #K,is obtainedasby cutting openthe two knotsandthen joiningthe ends along
1
the opposite sides of an embedded rectangle. The band sum is not unique.
An element h of the mapping class group of the torus can also be described by
surgery along a framed link L in the cylinder over the torus. Surgery still yields a
cylinder over the torus, but the homeomorphism to the original cylinder is identity
on T2×{0} and h on T2×{1}.
We introduce the element
N−1
Ω =N−1/2 αj ∈L (S1×D2).
U(1) t
Xj=0
e
The index stands for U(1) Chern-Simons theory (see § 4.1). There is a well-known
analogueforthegroupSU(2),tobediscussedin§6.1. ForaframedlinkLwedenote
byΩ (L)the skeinobtainedbyreplacingeverylinkcomponentbyΩ suchthat
U(1) U(1)
α becomes the framing.
Theorem 3.8. [15] Let h be an element of the mapping class group of the torus
obtainedbyperformingsurgeryonaframedlinkL inT2×[0,1]. ThediscreteFourier
h
transform ρ(h):L (S1×D2)→L (S1×D2) is given by
t t
e ρ(he)β =ΩU(1)(Lh)β.
Remark 3.9. This result was proved using the exact Egorov identity. For a framed
curve γ on the torus, h(γ) is obtained by sliding γ along the components of L . The
h
exact Egorov identity for Ω (L ) means that we are allowed to perform slides in
U(1) h
10 RA˘ZVANGELCAANDALEJANDROURIBE
the cylinder over the torus along curves colored by Ω . This points to a surgery
U(1)
formula for U(1)-quantum invariants of 3-manifolds [15].
Like for the metaplectic representation, the representation of the mapping class
group can be made into a true representation by passing to an extension of the
mapping class group of the torus. While a Z -extension would suffice, we consider
2
a Z-extension instead, in order to show the similarity with the non-abelian theta
functions.
Let L be a subspace of H (T2,R) spanned by a simple closed curve. Define the
1
Z-extension of the mapping class group of the torus by the multiplication rule on
SL(2,Z)×Z,
(h′,n′)◦(h,n)=(h′◦h,n+n′−τ(L,h(L),h′◦h(L)).
where τ is the Maslov index [28]. Standard results in the theory of theta functions
show that the Hermite-Jacobi action lifts to a representation of this group.
4. Non-abelian theta functions from geometric considerations
4.1. Non-abelian theta functions from geometric quantization. Let G be a
compactsimpleLiegroup,gitsLiealgebra,andΣ aclosedorientedsurfaceofgenus
g
g ≥1. The moduli space of g-connections on Σ is the quotient of the affine space of
g
all g-connections on Σ (or rather on the trivial principal G-bundle P on Σ ) by the
g g
group G of gauge transformations A → φ−1Aφ+φ−1dφ, with φ : Σ → G a smooth
g
function. The space of all connections has a symplectic 2-form given by
ω(A,B)=− tr(A∧B),
Z
Σg
where A and B are connection forms in its tangent space. The groupof gaugetrans-
formations acts on the space of connections in a Hamiltonian fashion, with moment
map the curvature. The moduli space of flat g-connections,
M ={A|A: flat g−connection}/G,
g
arisesas the symplectic reductionof the space ofconnections modulo gaugetransfor-
mations. This space is the same as the moduli space ofsemi-stable G-bundles onΣ ,
g
and the character variety of G-representations of the fundamental group of Σ . It is
g
an affine algebraic set over the reals, and its smooth part is a symplectic manifold.
Eachcurveγ onthesurfaceandirreduciblerepresentationV ofGdefineaclassical
observable on M , W = tr hol (A), called Wilson line, by taking the trace of
g γ,V V γ
the holonomy of the connection along γ in the representation V. Wilson lines are
regular functions on M . For G = SU(2) let the Wilson line for the n-dimensional
g
irreducible representation be W , with W = W . The W ’s span the algebra of
γ,n γ γ,2 γ
regular functions on M .
g
TheformωinducesaPoissonbracket,whichforG=SU(2)wasfoundbyGoldman
[16] to be
1
{Wα,Wβ}= 2 sgn(x)(Wαβx−1 −Wαβx)
x∈Xα∩β
where αβ and αβ−1 are computed as elements of the fundamental group with base
x x
point x (see Figure 2), and sgn(x) is the signature of the crossing; positive if the
frame given by the tangent vectors to α and β is positively oriented with respect to
the orientation of Σ , and negative otherwise.
g