Partition Functions of Chern-Simons Theory on Handlebodies by Radial Quantization

We use radial quantization to compute Chern-Simons partition functions on handlebodies of arbitrary genus. The partition function is given by a particular transition amplitude between two states which are defined on the Riemann surfaces that define the (singular) foliation of the handlebody. The final state is a coherent state while on the initial state the holonomy operator has zero eigenvalue. The latter choice encodes the constraint that the gauge fields must be regular everywhere inside the handlebody. By requiring that the only singularities of the gauge field inside the handlebody must be compatible with Wilson loop insertions, we find that the Wilson loop shifts the holonomy of the initial state. Together with an appropriate choice of normalization, this procedure selects a unique state in the Hilbert space obtained from a K\"ahler quantization of the theory on the constant-radius Riemann surfaces. Radial quantization allows us to find the partition functions of Abelian Chern-Simons theories for handlebodies of arbitrary genus. For non-Abelian compact gauge groups, we show that our method reproduces the known partition function at genus one.


Introduction
Chern-Simons gauge theory connects many different topics in mathematics and physics. On closed manifolds it is a topological theory that can be used to compute knot invariants [1], while on manifolds with boundaries it acquires additional boundary degrees of freedom that connect it to gravity in three dimensions [2,3,4,5] and to the theory of the fractional quantum Hall effect [6,7]. As remarked in [8], one intriguing feature distinguishes Chern-Simons theory from conventional topological field theories, such as topological Yang-Mills theories on Riemann surfaces or four-manifolds: the latter can be interpreted in terms of the cohomology ring of some classical moduli space of connections, while Chern-Simons, in general, cannot. In fact Chern-Simons theory is intrinsically a quantum theory that is best described by a Hilbert space.
When the three-manifold on which the theory is defined has special characteristics, the theory simplifies and may become computable. One remarkable example is the case of Seifert manifolds studied in [8]. Another case which could lead to exact computations is that of handlebodies [9].
The latter is interesting for various reasons. One of the most fascinating is that in order to test any conjectured holographic dualities relating pure gravity in three dimensions to a conformal field theory [10] (or an ensemble average thereof [11,12,13]), one would need to know the partition function of SL(2, C) Chern-Simons theory on a negatively curved manifold, whose boundary is a Riemann surface. Handlebodies are the simplest such manifolds for a fixed genus of the boundary [14].
The reason why one may think that a Chern-Simons theory may be exactly soluble on handlebodies is that these spaces are almost factorized as the topological product [0, R] × Σ.
We say "almost" because the closed Riemann surface Σ defining the foliation of the space becomes singular at one of the extrema of the interval [0, R]. The simplest example of this foliation is the solid torus handlebody, that is the direct product of a disk D 2 and a circle S 1 . Its singular foliation is D 2 × S 1 ≈ [0, R] × T 2 . The ≈ sign means that the two-torus leaf T 2 = S 1 × S 1 becomes singular at the end r = 0 of the interval [0, R], where one of the two S 1 cycles degenerates. By interpreting r ∈ [0, R] as time, we can quantize the theory and define a Hamiltonian that evolves in r. This allows us to rewrite the partition function of the theory as a transition amplitude between some initial state |i at r = 0 and some final state |f at r = R.
We will show in this paper that the condition that the initial state is a "shrunken," degenerate surface imposes a restriction on the initial state that, combined with the constraints descending from gauge invariance and the independence of the scalar product from the complex structure, completely fixes the partition function.
Let us describe now more precisely the procedure that we shall follow and the organization of this paper. We study partition functions of Chern-Simons theory of compact gauge groups on handlebodies using a radial quantization. First, we establish the equivalence between three quantities: Euclidean path integrals with holomorphic boundary condition, transition amplitudes under radial evolution with a coherent state as the final state, and wave functions integrated over the gauge orbit. Second, we map a Wilson loop inserted in a path integral to a "blown-up" operator defined on the Riemann surface, which in the radial quantization acts on a seed wave function and defines an initial state of definite holonomy along the contractible cycles. Together with an appropriate choice of normalization, this procedure singles out a unique vector in the Hilbert space obtained by a canonical quantization of Chern-Simons theory on the Riemann surface. Moreover, we find that requiring that such "blown-up" operator must be gauge-invariant corresponds to selecting a particular class of framings of the original Wilson loop. We are thus able to establish a precise state-operator correspondence associating each vector in the Hilbert space of the canonically quantized Chern-Simons theory on Σ to an explicitly computed partition function with insertions of Wilson loops. We first consider the Abelian U (1) gauge group on the solid torus, then on handlebodies of arbitrary genus and finally we study general compact simple groups on the solid torus.
In Section 2, we study the U (1) Chern-Simons theory, first on a torus handlebody and then on handlebodies defined by higher-genus Riemann surfaces. In Section 3, we move on to consider the case of a general non-Abelian simple compact Lie group on the torus handlebody.
Appendices A and B respectively summarize essential facts about the Riemann theta function and quadratic differentials on a Riemann surface.

The Abelian Case
To study Chern-Simons theory with gauge group U (1) on the genus-g handlebody M , we define a singular foliation on M as M = Σ × [0, R]. The constant-radius leaves are closed Riemann surfaces, Σ, and the initial surface Σ 0 at r = 0 is degenerate. The final surface Σ R is at r = R.
On Σ we specify the complex structure by giving the period matrix Ω, for which Σ has area det Im Ω, and which defines the basis {ω I |I = 1, . . . , g} of Abelian differentials and the local complex coordinate z on Σ. Since we will be considering either Abelian gauge fields at genus g or non-Abelian gauge fields at genus one, the period matrix will suffice to define the complex structure; we will not need to give explicit definitions of either Teichmüller or moduli space coordinates. We also use the notation ω I = ω I (z)dz, and when it can be done unambiguously we keep the index I implicit. The integration measure on Σ is normalized to d 2 x = dz ∧ dz/(−2i), so that Σ d 2 xω I (z)ω J (z) = (Im Ω) IJ .
One of our goals is to establish the equivalence of three different quantities. The first is a path integral, in which on the final surface Σ R we impose a holomorphic boundary condition that fixes the antiholomorphic part Azdz of the gauge connection A, while on the initial surface Σ 0 we fix the component of A along the contractible cycles. The second is a transition amplitude under radial evolution, from an initial state of definite holonomy along the contractible cycles to a coherent final state. The third is a wave function in a coherent state basis, obtained by integrating over the gauge orbit a seed wave function which is an eigenstate of the holonomy operator along the contractible cycles.
These quantities will be compared to the Chern-Simons partition functions that are identified with the wave functions obtained by a holomorphic quantization on the Riemann surface Σ [15][16] [17]. The basis wave functions spanning the gauge-invariant Hilbert space were explicitly given in [16] as The complex number u defines the harmonic part of the differential Azdz while the integervalued vector µ labels the independent vectors spanning the basis of the Hilbert space. Moreover, k is the Chern-Simons level, χ is a periodic function on Σ, and θ a b (u, Ω) is the Riemann theta function with characteristics [18], as defined in (A.1). F (Ω) 1 2 is the "holomorphic square root" of the scalar Laplace determinant on Σ [19], The obstruction to holomorphic factorization [20], S ZT L , is the nonholomorphic part of the Liouville action defined by Zograf and Takhtajan (see [19,21]). For genus one on the flat metric, F (Ω)

The torus case
As a warm-up, we first consider the case where M is the solid three-dimensional torus. On each constant-radius surface Σ = T 2 (which is a two-torus) the period matrix is the modular parameter τ ≡ τ 1 + iτ 2 and defines the global holomorphic coordinate z on the torus. From this, we can define local real coordinates x 1,2 by z ≡ x 1 + ix 2 , where x 1 ∼ x 1 + 1 parametrizes the contractible cycle on M . The restriction to T 2 of a one-form field A, .

(2.4)
Although these real coordinates are also valid locally on higher-genus Riemann surfaces, for those cases we will use a better description, given in terms of Strebel differentials [22].
In the next subsections, we establish the equivalence between the three quantities mentioned earlier: the partition function given as a path integral, the transition amplitude, and the gauge invariant wave function obtained from an appropriate "seed" wave function.

The path integral
We impose a holomorphic final condition, fixing Azdz| Σ R = ∂zχdz + iπuτ −1 2 ω, as in (2.2); on the torus, ω = dz. In addition, as initial condition we fix the component of A along the In (2.6), the boundary term 1 ; τ ) is in order. We are considering the gauge group U (1), not R. The distinction is that U (1) includes large gauge transformations defined on the boundary Σ R of the handlebody M . A large gauge transformation that has a non-trivial winding along a homotopy cycle of Σ R that is contractible in M cannot be extended smoothly to M . This implies that the partition function is a sum of terms that are not related by bulk gauge transformations.
Integrating out A r , the path integral imposes F 12 = 0 [17], so we get The standard procedure is to express (A 1 dx 1 + A 2 dx 2 ) as a flat connection, resulting in a chiral Wess-Zumino-Witten path integral on the final surface [17].

The transition amplitude
We turn now to the coherent state method. The first term in (2.7) (the bulk term) defines the symplectic structure of the theory, implying that A 1 and A 2 are conjugate variables and satisfy upon quantization the equal-radius canonical commutation relation: Here δ (2) (x, y) denotes the delta function with respect to the (x 1 , x 2 )-coordinates. Moreover, define the A 1 -eigenstate |A 1 as a translation from the A 1 = 0 eigenstate |0 effected by applying the conjugate momentumÂ 2 Here too C is a normalization constant, which we leave arbitrary for the time being. Using (2.9) together with (2.4), we can construct the wave function of the coherent state |A z ) in the |A 1basis, which satisfies the defining properties (with Az = A * z ), Let us consider the transition amplitude, from an A 1 -eigenstate |A (0) 1 on the initial surface Σ 0 , to a coherent state |A R z ) on the final surface Σ R , as we radially evolve the system with the Hamiltonian read off from (2.7): This is identical to the partition function (2.7), Z(Az| Σ R , A 1 ; τ ), with the boundary term f B [A 1 ] = 0, and A R z = Az| Σ R . In both cases, we have imposed the initial condition 1 . The equivalence between (2.7) and (2.14) holds for arbitrary genus because it only relies on a local decomposition of the complex coordinate z into real coordinates that is independent of the topology of the surface Σ. From now on, without ambiguity, we drop the superscript R from A R z . Next, we evaluate Eq. (2.14) and find out what it computes for the torus case. We parametrize the A 1,2 that solve the constraint F 12 = 0 by where λ 0 (r, x 1 , x 2 ) is a periodic function on Σ, and the shift in A 1 by 2πn with n ∈ Z comes from the large gauge transformations that are singular inside the bulk. Note that a shift in λ 0 (r, x 1 , x 2 ) by any x 1 -independent function f 2 (r, x 2 ) also solves F 12 = 0 and leaves the integrand of the path integral invariant, thus the x 1 -independent modes can be factored out of the path integral and consistently discarded 4 . On the other hand, shifting λ 0 (r, x 1 , x 2 ) by some f 1 (r, x 1 ) changes the boundary action, so these modes cannot be factored out from the path integral. We restrict our initial condition to A 1 | Σ 0 = a 1 (0) with a 1 (0) = constant-this is a natural choice since the initial surface is in fact degenerate, so λ 0 (r = 0, is independent of x 1 . The integration measure in (2.14) satisfies [17] i.e. the change of variables (2.15) has unit Jacobian. Here a prime denotes discarding x 1independent functions. Moreover, as in (2.2), Az = ∂zχ + iπuτ −1 2 . The amplitude (2.14) becomes In arriving at (2.17), we integrated out a 2 (r) to obtain an r-independent a 1 (r) = a 1 . Together with the initial condition A 1 (r = 0) = a 1 (0), this means a 1 (R) = a 1 (0). We also defined The path integral on λ 0 equals det −1/2 (− k 2π ∂z∂ 1 ). We are still free to choose the constant C. Besides removing ultraviolet divergences in the functional determinant, it can be further fixed by requiring that eq. (2.18) be a section of a projectively flat connection on the moduli space of complex structures [15]. This is simply the requirement that the scalar product of the base wave functions (2.18) must be independent of the complex structure. By making this choice we get 1/ F (Ω) genus g = 1 torus Σ = T 2 , they are given by (2.19) where Az = ∂zχ + iπuτ −1 2 , and µ = 0, 1, . . . , k − 1.
We cannot reabsorb this difference into a redefinition of the constant C without giving up one of the objectives of our paper, which is to establish a state-operator correspondence associating each state obtained by applying Wilson loops to the vacuum to the partition function of Chern-Simons on a solid torus containing the same Wilson loop. So, once we normalize the vacuum and the vacuum partition function, we cannot further normalize separately the other partition functions. What we can do is to understand where the discrepancy comes from and try to fix it by appropriately changing the definition of the Wilson loop operator.
To find the meaning of this discrepancy, we consider a different basis on the torus. We define global coordinates (φ, t) which both have unit period, so that z = φ + τ t, φ ∼ φ + 1, . (2.20) In particular, are related to the previous conjugate variables (A 1 , A 2 ) by a canonical transformation which simply shifts A 2 by a term linear in A 1 . The canonical commutation relation is Here δ (2) (x, y) is again the delta function in the (x 1 , x 2 ) coordinates. Similarly to (2.9), we define the A φ -eigenstate |A φ by translating the A φ = 0 eigenstate |0 ≡ |0 , but this time with the operatorÂ t , The eigenstates |A φ and |A φ are related by a pure phase, By using (2.20), we see that the wave function of the coherent state |A z ) in the |A φ -basis differs from (2.10) Repeating the same calculations as above, one finds that where we normalized C as in Eq. (2.18). This is exactly one of the (2.19) when we set a 1 (0)/2π = µ/k. Thus, we learn that to get an answer holomorphic in the complex structure τ we need a particular choice of canonical variables (A φ , A t ), or equivalently a particular choice of eigenstate |A φ . In terms of the path integral Z(Az| Σ R , a φ (0); τ ), this corresponds to a particular choice of the boundary term, namely:

The gauge-invariant wave function
Under a gauge transformation Here λ can include large gauge transformations. Thus, starting from any "seed" wave function Ψ 0 [Az] we can integrate over the gauge group to construct a gauge-invariant wave function: This formula includes a sum over large gauge transformations, so the most general λ is where λ 0 is periodic on the torus, while the multivalued large gauge parameter λ enters in the integral only through its derivatives, which are single-valued on the torus; they are given by If we take (Az|a 1 (0) to be a seed wave function Ψ 0 [Az] which is not necessarily gauge-invariant and integrate over all gauge transformations including the large transformations m, n ∈ Z, we reproduce the theta function in (2.19). To see this, we impose again the conditions (2.2) which is exactly Z(Az, µ; τ ) in (2.19). In arriving at (2.35), the constant C was fixed as in (2.18) using the normalization condition C det −1/2 (− k 2π ∂z∂ 1 ) = η(τ ) −1 and we used that k/2 ∈ Z >0 , m ∈ Z and ka 1 (0)/2π = µ ∈ Z, so the summand does not depend on m, Because of this, in the last line we discarded the infinite sum over m ∈ Z. Similarly, the same calculation done with (Az|a 1 (0) as seed wave function reproduces (2.17). Notice that discarding the sum over m means simply to remove identical gauge copies from the definition of the gauge-invariant wave function. This is a standard part of the construction of a gauge invariant, normalizable state or operator using an integral (and/or sum) over gauge transformations. Its analog in the context of three-dimensional gravity is explained for instance in [14].

Blowing up Wilson loops
One can insert into the path integral a gauge-invariant Wilson loop operator defined along a loop C on M , asŴ (2.37) P means path-ordering, and the U (1) charge µ is integer-valued such thatŴ µ [C] is invariant under large gauge transformations defined on C. We restrict C to be a path that runs along the non-contractible cycle of M , and without loss of generality put it at the origin r = 0 of the solid torus.
We would like to mapŴ µ [C] to a "blown-up" operator in radial quantization, which acts on a state defined on Σ. To this end, recall that the A 1 -eigenstate |A 1 is the translation of the A 1 = 0 eigenstate |0 by the operatorÂ 2 given in (2.9), The initial surface Σ 0 is degenerate but the Wilson loop operatorŴ µ [C 2 ], with C 2 at r = 0 running along the x 2 -direction, can be "blown-up" and identified with the translation operator defined in (2.38) acting on the Hilbert space on Σ, Alternatively, choosing A t as the conjugate momentum from (2.22) we have We can also define a "blown-up" version of the Wilson loop operatorŴ µ [C t ], with C t at r = 0 running along the t-direction, and identify it with the translation operator in (2.41), (2.43)

Gauge invariance and framing
Both C 2 and C t trace the same closed loop at the origin, though with twists differing by τ 1 .
One may wish to assign a framing to this loop by defining a vector field on it [1,24], thereby extending this loop into a ribbon. Such a vector field must be periodic under the global identification (x 1 , x 2 ) ∼ (x 1 + τ 1 , x 2 + τ 2 )-now that we are away from the degenerate r = 0 surface. The simplest choice is that corresponding to C t , while that corresponding to C 2 does not respect the periodicity.
In the language of the "blowing-up" procedure, this fact translates to demanding that Note that theŴ µ [Σ, t], µ ∈ Z, are not the only gauge-invariant operators. The most general gauge invariant "blown-up" operator on Σ with constant coefficients 6 takes the form  6 We will drop this restriction in the higher-genus cases.

Higher genus
For partition functions on higher-genus handlebodies, it is convenient to make use of certain special quadratic differentials on Riemann surfaces, reviewed in Appendix B. Specifically, we pick a Strebel differential ϕ, which is a quadratic differential on the Riemann surface Σ, holomorphic in the complex structure; locally, ϕ = h(z)dz 2 where h(z) is holomorphic. The existence of such differentials is proven in [22]. We do not need to know their precise form.
All we need from a Strebel differential is the fact that it foliates the Riemann surface Σ into horizontal trajectories, which are closed curves given by The Strebel differential ϕ also defines a metric on Σ, which takes the form This metric may have zeros or singularities, which define the singular points of the foliation.
We define next a vector field v of unit norm with respect to (2.52), whose integral curves are Figure 1: A schematic illustration of a Strebel differential on a genus-two Riemann surface, which defines horizontal trajectories denoted by blue loops. The vector field v that generates the horizontal trajectories is denoted by red arrows.
the horizontal trajectories so that We use the vector field v to define cycles on a higher-genus Riemann surface Σ, that are contractible on the corresponding handlebody M . On the torus, v =v = 1 and ∂ h = ∂ 1 .
The square root in (2.53) can cause generically an obstruction to defining a global holomorphic vector field on Σ. On the other hand, we do not need v to be holomorphic, so we can always rescale v by a common factor: v → v with a smooth real function. The equations that we will find in the next subsections depend only on the ratiov/v, which is not affected by the rescaling. The function can even vanish on subsets of measure zero that are transverse to the horizontal trajectories without altering the ratiov/v. By making vanish somewhere on Σ, a nonholomorphic vector field can be defined everywhere on Σ.
The horizontal trajectories and the vector field v that generates them are illustrated schematically in Figure 1.

The vacuum partition function
We would like to generalize the third approach used in the torus case: start from a non-gauge As for genus one, here the constant C may depend on the complex structure and is fixed by properly normalizing the vacuum partition function. We show next that integrating it over the gauge orbit does result in a particular vector in the Hilbert space (2.1) obtained from Kähler quantization, holomorphic in the complex structure Ω. In particular, we will see that, after integration, the vector v appears only in the Weyl anomaly.
Given any seed wave function Ψ 0 , the gauge integral is given by a generalization of (2.29) on the torus. It reads We begin by decomposing where both χ, λ 0 are single-valued on Σ. The multivalued function λ appears everywhere only through its derivatives, which are also single-valued on Σ. Next we evaluate the integrand: (2.60) (On the torus, this reduces to (2.33) with a 1 (0) = 0). Let us look now at the terms involving Thus, we see that the vector field v indeed drops out, except in the fluctuation term.
Substituting the definitions (2.58) into Ψ[Az] and repeating the same calculation done in the torus case, we arrive at Note that the constant C also reabsorbs a term that contains a Weyl anomaly and therefore, because of (2.53), a dependence on v. We also used k/2 ∈ Z and m, n ∈ Z g , and discarded the trivial sum over m, that is the sum over large gauge transformations that can be extended to the bulk and under which the wave function is invariant.

Wilson loops
On a higher-genus handlebody, besides Wilson loops that can be regarded as "world histories of mesons," there is also another class of gauge-invariant observables, which correspond to the "world histories of baryons" running along the non-contractible cycles; see [24]. For the Abelian case that we have considered here, however, the fusion rule is trivial, so those "baryon world histories" can be decomposed into disjoint Wilson loops running along the non-contractible cycles of the handlebody. Therefore, it suffices to consider only standard Wilson loops.
We would like to generalize the "blowing-up" of Wilson loops that we studied on the torus in Section 2.1.1 to higher genus. Consider the loops C I running along the g non-contractible cycles of M and endowed with charges µ I ∈ Z k . The resulting Wilson loops are then "blown up" into operatorsŴ [Σ, w] on Σ, parametrized by real one-forms (2.70a) Here η is a real single-valued function on Σ, and λ is a large gauge transformation (2.58). As discussed in Section 2.
Using again the Baker-Campbell-Hausdorff formula (2.46), one gets We repeat the calculation done in the last subsection to evaluate the gauge integral (2.56). The integrand is The terms containing λ 0 arẽ The saddle point λ 0,cl at whichS is extremal satisfies the equation of motion So, once again, v drops out of the action-except in the fluctuation term which also gives the Weyl anomaly. Moreover, the function η in w drops out as well. Evaluating the path integral in the same way as before, we finally get Here used the same normalization for C as in eq. (2.67) together with k/2 ∈ Z, m, n ∈ Z g and µ ∈ Z, and discarded the trivial sum over m.
Since η drops out eventually, we can repeat the analysis we performed in the torus case.
Namely, In the special case N I = N IJ µ J for some symmetric matrix N with integer entries, this phase exp(+(iπ/k)µN µ) is naturally interpreted as the framing anomaly.

The non-Abelian Case
We consider now the non-Abelian case, with a compact, simply-connected and simple Lie group G on a solid torus. In this section M is always the torus handlebody and Σ = T 2 . By generalizing the equation (2.24) found in the Abelian case, we will consider the A φ -eigenstate |A φ translated by the conjugate momentum A t . In the coherent state basis, it reads Solving the constraint F 12 = 0 by [17] A On the torus a φ (r) and a t (r) commute so they are elements of the Cartan subalgebra h of g; this is not true in general for higher genus. Moreover, by integrating out a t (r) we get a φ (r) = a φ (0), so the amplitude (3.3) becomes With an appropriate, Az-independent choice of C, this is the chiral Wess-Zumino-Witten path integral. For a φ (0) = 2πµ/k where µ is an integral weight of G and Az = iuτ −1 2 , ref. [26] shows that the path integral gives the Weyl-Kac character χ µ,k (u, τ ): where ρ and h ∨ are respectively the Weyl vector and the dual Coxeter number of g. The Weyl-odd theta function is defined as where W is the Weyl group of G and (w) is the signature of w ∈ W . θ µ,k (u, τ ) is the level-k theta function for the Lie algebra g, whose definition is recalled in (A.3).

Wilson loops
The Wilson loop operator of the representation generated by the integral highest weight µ of G, along a loop C of constant radius in M , iŝ (3.10) In the last equality, we stripped off the pure gauge part ofÂ (recall the definition A i = g −1 a i g + g −1 ∂ i g) due to the trace in the definition ofŴ µ [C], so we only need to look at the equal-radius canonical commutation relation ofâ i (r), which we read off from (3.5): Here we have expanded a φ,t (r) = rank(g) j=1 a j φ,t (r)H j in the Cartan-Weyl basis {H j } of the Cartan subalgebra h of g, where j, l = 1, . . . , rank(g). For the loop C t at r = 0 running along the t- (3.12) As in the Abelian case, we map (3.12) to a "blown up" gauge-invariant operatorŴ µ [Σ] defined on Σ, which is to be identified with the translation operator by the conjugate momentum a j t , acting on the a φ = 0 eigenstate |0 . Sinceâ t is constant on Σ, the Wilson loop is simply given The first equality is the character ofâ t as an element of h. This is expressed as a Weyl character in the second equality. We recall the latter's definition: (3.14) where µ are the weights in the weight system Ω µ of the highest weight µ, which span a highestweight representation of G. By the Weyl character formula, (3.14) can be written as a ratio of sums over the Weyl group W of G: Because the radial evolution is linear in the initial state, this identity holds if the corresponding identity is true for the Weyl character of the Lie algebra, µ ∈Ωµ This should be understood as an equality in terms of the Weyl character formula (3.15). Intuitively, this identity should hold due to the fact that all the weights (µ + ρ) with µ ∈ Ω µ , except for the highest weight µ, pair up under simple Weyl transformations.

Partition function as a gauge-invariant wave function
Here we proceed in the same way as in the Abelian case. A wave function transforms as Since we consider a simply-connected group G, the gauge group G is connected. Similarly, starting from a wave function Ψ 0 [Az] that is not gauge-invariant, we can construct a gaugeinvariant wave function by integrating over the gauge group: Taking Ψ 0 [Az] = (Az|a φ as the seed wave function in (3.1) and after a quick calculation, one recovers the chiral Wess-Zumino-Witten path integral (3.6). In other words, radially evolving the wave function is equivalent to integrating over the gauge group, which results in a gaugeinvariant wave function.

B Quadratic Differentials
We summarize essential facts about quadratic differentials on a Riemann surface from Strebel [22] and Hubbard & Masur [27].
Consider a compact Riemann surface Σ of genus g and n punctures, endowed with a complex structure which defines a local complex coordinate denoted by z. A (meromorphic) quadratic differential ϕ on Σ is a (2, 0)-meromorphic differential; it locally takes the form where h(z) is meromorphic, and under a holomorphic change of coordinate z →z(z), it transforms by the chain rule as z →z(z), h(z) →h(z) = dz dz 2 h(z), so that ϕ =h(z)dz 2 = h(z)dz 2 .

(B.2)
When h(z) is holomorphic, then ϕ is a holomorphic quadratic differential. On a closed genus-g > 1 Riemann surface without punctures, the complex dimension of the space of all holomorphic quadratic differentials is (3g − 3), as a result of the Riemann-Roch theorem.
Quadratic differentials find applications in physics, especially in conformal field theory and string field theory (see e.g. [28,29,30,31]), because they provide a convenient foliation for a Riemann surface Σ. Given a meromorphic quadratic differential ϕ, a horizontal trajectory is a non-self-intersecting continuous loop on which ϕ is real and positive, while a vertical trajectory is a non-self-intersecting continuous loop on which ϕ is real and negative. Equivalently, on a local patch U of Σ with complex coordinate z, and a base point p 0 ∈ U , we can define a local natural complex coordinate w on p ∈ U by w(p) ≡ Then, on a horizontal (vertical) trajectory, w has constant imaginary (real) part. A critical point of a ϕ meromorphic on Σ is a zero or a pole of ϕ, while all other points on Σ are called regular points. A critical trajectory is a horizontal trajectory that joins critical points. In general, a zero of order n is the endpoint of some (n + 2) critical trajectories.
A quadratic differential ϕ defines a metric on Σ, which is locally given by For g = 1, i.e. the torus, a Strebel differential ϕ obviously exists: ϕ = dz 2 .