QUANTUM MECHANICS AND NON-ABELIAN THETA FUNCTIONS ...

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22RĂZVAN GELCA AND ALEJANDRO URIBEThe Hilbert space can be obtained using the method of geometric quantizationas the space of sections of a line bundle over M g . The generalprocedure is to obtain the line bundle as the tensor product of a line bundleL with curvature Nω and a half-density [23]. In our case the half-densityis the square root of the canonical line bundle. Because the moduli spacehas a natural complex structure, it is customary to work in the holomorphicpolarization, in which case the Hilbert space consists of the holomorphicsections of the line bundle.Let us briefly recall how each complex structure on the surface inducesa complex structure on the moduli space. The tangent space to M g ata nonsingular point A is the first cohomology group H 1 A (Σ g,ad P) of thecomplex of g-valued formsΩ 0 (Σ g ,ad P) dA → Ω 1 (Σ g ,ad P) dA → Ω 2 (Σ g ,ad P).Here P denotes the trivial principal G-bundle over Σ g . Each complex structureon Σ g induces a Hodge ∗-operator on the space of connections on Σ g ,hence a ∗-operator on HA 1(Σ g,ad P). The complex structure on M g isI : HA 1(Σ g,ad P) → HA 1(Σ g,ad P), IB = − ∗ B. For more details we referthe reader to [12]. This complex structure turns the smooth part of M g intoa complex manifold. It is important to point out that the complex structureis compatible with the symplectic form ω, in the sense that ω(B,IB) ≥ 0for all B.As said, the Hilbert space consists of the holomorphic sections of the linebundle of the quantization. These are the non-abelian theta functions. Thereason for the name is that in the simplest case, when G = U(1) and thesurface has genus one, the Hilbert space has dimension N and its elementsare the classical theta functions. To specify a basis of this vector space,i.e. to obtain the analogues of the theta series, one has to consider a rigidstructure on the original surface and use the relationship of the Hilbert spaceto conformal field theory established in [29].The analogue of the group ring of the finite Heisenberg group is the algebraof operators that are the quantizations of Wilson lines. They arise in thetheory of the Jones polynomial [11] as outlined by Witten [29], being definedheuristically in the framework of quantum field theory. They are integraloperators with kernel∫< A 1 |op(W γ,n )|A 2 >= e iNL(A) W γ,n (A)DA,M A1 A 2where A 1 ,A 2 are conjugacy classes of flat connections on Σ g modulo thegauge group, A is a conjugacy class under the action of the gauge group onΣ g × [0,1] such that A Σg×{0} = A 1 and A Σg×{1} = A 2 , andL(A) = 14π∫Σ g×[0,1]tr(A ∧ dA + 2 )3 A ∧ A ∧ Ais the Chern-Simons Lagrangian. As seen above, the operator quantizinga Wilson line is defined by Feynman path integrals, which does not have
QUANTUM MECHANICS AND GENERALIZED THETA FUNCTIONS 23a rigorous mathematical formulation. It is thought as an average of theWilson line computed over all connections that interpolate between A 1 andA 2 .It is the framework of Chern-Simons theory that motivates the skein theoreticapproach to classical theta functions outlined in Section 3.2. Let usexplain this in more detail.The moduli space of flat U(1)-connections on the torus is itself a torus,the Jacobian. The moduli space is canonically homeomorphic to the originaltorus in the following way. Let p 0 be the point on the original torus that isthe image of the origin under the quotient map C → C/Z + Zτ. Thinkingof the Jacobian as the moduli space of holomorphic topologically trivial linebundles, the homeomorphism in question maps a point p on the originaltorus to the line bundle O(p − p 0 ). Or, when viewing the Jacobian as acharacter variety, this homeomorphism maps the point (x,y) on the originaltorus to the U(1)-representation of the fundamental group that maps (1,0)to e 2πix and (0,1) to e 2πiy .The tangent space to this torus is two dimensional; viewing it as givenby cohomology classes of u(1)-valued 1-forms, it has a basis consisting ofdx and dy, which can be identified with the tangent vectors to the torus∂∂x and ∂ ∂y. Under this identification, we see that a complex structure onthe original torus induces exactly the same complex structure on the modulispace.A rigid structure on the original torus gives rise to a decomposition of thetangent space to the moduli space into a sum of two Lagrangian subspaces.To fit in the framework of section 3.2, we can exponentiate these Lagrangiansubspaces to obtain a rigid structure on the moduli space, which happensto be the same as the rigid structure on the original torus.The Chern-Simons line bundle over the moduli space is just the one fromSection 3.1, so in this case the theta functions are the classical theta functions.The Wilson lines can be quantized either by using one of the classicalmethods for quantizing the torus, or they can be quantized using the Feynmanpath integrals as above. The Feyman path integral approach allows thelocalization of the computations to small balls, in which a single crossingshows up. Witten [29] has explained that in each such ball skein relationshold, in this case the skein relations from Figure 2, which compute the linkingnumber. As such the path integral quantization gives rise to the skeintheoretic model.On the other hand, Witten’s quantization is symmetric under the actionof the mapping class group of the torus, a property shared by Weyl quantization.And indeed, we have seen in section 3.2 that Weyl quantizationand the skein theoretic quantization are the same. For the group U(1), thecase of higher genus surfaces can be done in the same framework, and willbe explained in future work.4.2. Non-abelian theta functions from quantum groups. More thanjust the quantization of moduli spaces of connections on surfaces, Witten
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