Schwarzian quantum mechanics as a Drinfeld-Sokolov reduction of $BF$ theory

We give an interpretation of the holographic correspondence between two-dimensional $BF$ theory on the punctured disk with gauge group ${\rm PSL}(2, \mathbb{R})$ and Schwarzian quantum mechanics in terms of a Drinfeld-Sokolov reduction. The latter, in turn, is equivalent to the presence of certain edge states imposing a first class constraint on the model. The constrained path integral localizes over exceptional Virasoro coadjoint orbits. The reduced theory is governed by the Schwarzian action functional generating a Hamiltonian $S^1$-action on the orbits. The partition function is given by a sum over topological sectors, each of which is computed by the Duistermaat-Heckman integration formula. We recover the same results in the operator formalism.

mechanics on the boundary. This is an instance of an AdS 2 /CFT 1 correspondence and serves as a toy model for quantum gravity. Notably, the only dynamical variables of the theory are (orientation preserving) diffeomorphisms of the boundary. More recently, the partition function of JT gravity on two-dimensional compact surfaces of arbitrary genus and with arbitrary many boundaries has been computed exactly by describing it as a matrix model and employing topological recursion methods [24].
JT gravity is a two-dimensional dilatonic gravity theory [14,27]. It is well-known that the model admits a first order formulation in terms of a two-dimensional BF theory with gauge group SL(2, R) [15]. In fact, this is a special example of a more general construction of two-dimensional dilatonic gravity theories, which were classified in [12,25] by means of so-called Poisson sigma models.
Schwarzian quantum mechanics is a very rich and interesting theory in itself due to its strong ties with the Sachdev-Ye-Kitaev model [19] (and references therein) where it arises as a low energy limit. Notably, its partition function is one-loop exact and was calculated in [26]. In particular, the path integral is taken over the space of orientation preserving diffeomorphisms Diff + (S 1 ) modulo an SL(2, R)-action. Now, Diff + (S 1 )/SL(2, R) can be identified with an exceptional Virasoro orbit [2,28] on which the Schwarzian action functional generates a Hamiltonian S 1 -action by rotating the (source) circle. The one-loop exactness of the partition function is therefore a consequence of an analog of the Duistermaat-Heckman integration formula.
Motivated by the above observations, this article is devoted to the study of the holographic dual of two-dimensional BF theory on a punctured disk (or cylinder) with gauge group PSL(2, R). The holographic duality between the Schwarzian theory and BF theory on a disk for gauge group SL(2, R) was already explored in [3,6,7,21]. The duality was derived from the holographic correspondence of three-dimensional gravity and two-dimensional Liouville theory by dimensional reduction. It was shown that the configuration space of the BF theory on a disk reduces to the space of contractible based loops of the gauge group. Furthermore, constraining the gauge group to SL + (2, R) recovers the correct density of states [6] which was previously calculated in [26].
Our approach is different: In passing from the disk to the cylinder we are no longer restricted to contractible loops. Rather, we consider loops of arbitrary winding number n. In addition, the constraint we impose is different in nature as it keeps loops with n > 0 in the theory. Indeed, we find that the partition function picks up contributions from all corresponding topological sectors. For n = 1 we recover the results of [26].
In more detail: following [6], we implement the boundary conditions by adding a Hamiltonian on the boundary. In addition we assume that half of the fields vanish at the puncture and we restrict the holonomy of the gauge field to be trivial. After shortly recalling some background material in Section 2, we recall in Section 3 how integration over the scalar fields localizes the BF theory over the space of flat connections modulo gauge transformations which are trivial on the boundary. The latter can be naturally identified with the space of based maps from the boundary to PSL(2, R), while the action functional reduces to quantum mechanics of a free particle moving in the group manifold. Furthermore, we explain how the presence of edge states, in the sense of [11], constrains the path integral.
A detailed analysis of the constrained model is given in Section 4. In particular, we show that the constraint can be understood in terms of a Drinfeld-Sokolov reduction of the space of based loops in PSL(2, R). After the reduction, the path integral localizes to a sum of integrals over exceptional Virasoro coadjoint orbits. Each orbit is associated to a topologically distinct sector corresponding to different windings of the (based) loop. At the same time, the theory reduces to Schwarzian quantum mechanics whose action functional generates an S 1 -action on the orbits. Inspired by [22,26], we define the partition function in terms of an infinite-dimensional version of the Duistermaat-Heckman integration formula. It turns out that there is a unique fixed point of the S 1 -action and we find Furthermore, we show that in the presence of a fixed edge state the constrained partition function reduces to a sum of integrals over non-exceptional Virasoro coadjoint orbits. Again the reduced action generates a Hamiltonian S 1 -action. Once more, we compute the partition function by means of a Duistermaat-Heckman integration.
Since the holographic dual of the two-dimensional BF theory in the presence of edge states is (constrained) quantum mechanics, the partition function can be computed also in the operator formalism. This computation is carried out in Section 4.5. The Hamiltonian of the model coincides (up to a constant) with the quadratic Casimir operator of sl(2, R) and the Hilbert space is given by certain unitary irreducible representations of PSL(2, R). The constraint singles out a special eigenvalue of the generator of the noncompact subgroup of upper triangular matrices. The corresponding states were described by Lindblad and Nagel [18] and constitute the spectrum of a particle moving in a quantized magnetic field on the upper half plane [8]. Identifying the Hilbert space with the positive discrete series representation of PSL(2, R), we compute the constrained partition function in the operator formalism and find the same result as in the path integral approach.
Acknowledgements. The authors are grateful to Anton Alekseev and Samson Shatashvili for many insightful discussions and useful comments.

2.1.
Finite-dimensional Duistermaat-Heckman integration. Since the aim of this note is to calculate the partition function of a quantum mechanical system on the group manifold PSL(2, R) by means of an infinitedimensional version of Duistermaat-Heckman integration, we first want to give a non-exhaustive recollection of finite-dimensional Duistermaat-Heckman integration. For more details we refer the interested reader to [1,4,22].
Let (M, ω) be a compact symplectic 2n-dimensional manifold endowed with an action of S 1 . Suppose that this circle action is Hamiltonian, that is we assume that the action is generated by a vector field ξ and the existence of a smooth function H on M which satisfy the relation Moreover, suppose that H has only isolated critical points. Then, Duistermaat and Heckman showed in [10] that the integral n! localizes over the fixed points of H: , where the sum runs over all (isolated) critical points m of H and the w j (m) are the weights of the S 1 -action on the tangent space T m M of M at m. Finally, the integration measure is taken to be the Liouville measure ω n /n! defined by the symplectic form ω.

2.2.
Kac-Moody orbits. Let G be a semi-simple Lie group with Lie algebra g. Denote by LG = C ∞ (S 1 , G) the loop space of G. The space LG is itself a Lie group, whose Lie algebra Lg coincides with the algebra of smooth g-valued function on the circle. In the following, we will be interested in the central extension Lg. Elements of Lg are of the form (u(x), k), where u(x) is a g-valued function of S 1 and k ∈ R is central. The Lie bracket on Lg is defined by where [u, v] g denotes the bracket in the Lie algebra g and tr denotes the normalized Killing form on g. Elements of the dual space Lg * can be described equally by pairs (v(x), −k), where v(x) is again g-valued function on S 1 and k a real number. The number k is called the level. The pairing between Lg * and Lg is The coadjoint action of LG on Lg * is then and is isomorphic to the homogeneous space LG/ Stab(v). Since any coadjoint orbit is naturally symplectic, so is O v . The symplectic form on O v descends from the left-invariant pre-symplectic form on LG, which can be described as follows: For X, Y ∈ T e (LG) = Lg, one defines Using left translation in LG, the pre-symplectic form can then be defined at any g ∈ LG: where g −1 δg denotes the left-invariant Maurer-Cartan element of LG.
It is instructive to look at two examples in detail.
Then the stabilizer Stab(v) is the group G itself and the coadjoint orbit passing through 0 is isomorphic to the space of based loops and the symplectic form, at level k, reduces to Rotating the loop defines an S 1 -action on ΩG by 2 : g(x) → g(x + t), t ∈ S 1 . This action is Hamiltonian, see e.g. [4,23]. Its generating Hamiltonian, with respect to the symplectic form (7), is the energy function H : ΩG → R of the loop defined by Moreover, any closed subgroup H ⊂ G acts on ΩG by conjugation. This action is as well Hamiltonian with respect to (7). If H has Lie algebra h, then the moment map is defined by the projection of g g −1 ∈ Lg ∼ = Lg * to Lh * (see for instance [23]) Example 2.2. Let T ⊂ G be a maximal torus with Lie algebra t, and consider a constant regular element v 0 ∈ t * ⊂ Lg * . Here, regular means that the stabilizer subgroup of v 0 is a maximal torus. Then the orbit through v 0 is isomorphic to the homogeneous space LG/T . In this case, the symplectic form can be conveniently written in terms of the quasi-periodic element The S 1 -action which rotates the loop is again Hamiltonian, with

Virasoro orbits.
This section follows closely [2,28]. Let Vect(S 1 ) be the Lie algebra of the orientation preserving diffeomorphism group of the circle, denoted by Diff + (S 1 ). Elements φ ∈ Diff + (S 1 ) can be seen as increasing quasi-periodic maps, i.e. φ : R → R satisfying Let Vir be the central extension of Vect(S 1 ) by R. Elements of Vir are pairs (v, r) where v = v(x)∂ x is a vector field on S 1 and r ∈ R is central. The Lie bracket on Vir is defined by (12) [ In particular, if we define define the Virasoro algebra relation, which, in the literature, is more commonly written as Remark 1. Shifting L 0 by c 24 amounts to replacing m 3 by m 3 − m, which is found more often in the literature.
Elements of the dual space Vir * are pairs (b, c) where b = b(x)dx 2 is a quadratic differential on the circle and c ∈ R. The pairing between Vir and Vir * is given by The coadjoint action of Diff + (S 1 ) on Vir * is described as follows: infinitesimally, i.e. for v ∈ Vect(S 1 ), one has This integrates to We can identify the coadjoint orbit O b through the point (b, c) ∈ Vir * with (20) O where Stab(b) ⊂ Diff + (S 1 ) is the stabilizer subgroup of b under the coadjoint action (18). The stabilizers of a general quadratic differential b ∈ Vir * are hard to calculate. However, if b = b 0 dx 2 , b 0 ∈ R is constant, the calculation becomes feasible: A point in the orbit 3 O b 0 can be written as This shows that Stab(b 0 ) = S 1 such that the orbit is isomorphic to the homogeneous space Diff(S 1 )/S 1 . However, for exceptional values of b 0 one finds that the stabilizer subgroup of b 0 is larger than S 1 : choosing the vector field v to be L n = ie inx ∂ x in Equation (17), one has for constant b 0 Therefore, for the special values the stabilizer is generated by {L 0 , L ±n }. This integrates to the subgroup SL (n) (2, R) ⊂ Diff + (S 1 ) which is the n-fold cover of SL(2, R) [28]. Therefore, for the exceptional values (23), the orbits are identified with the homogeneous space Diff + (S 1 )/SL (n) (2, R): Being a coadjoint orbit, O b 0 is naturally symplectic. In terms of the leftinvariant Maurer-Cartan element Y of Diff + (S 1 ) [2], the symplectic form at any point φ is given by Explicitly, we have Y (φ) = δφ φ , where δ denotes the de Rham differential on Diff + (S 1 ). Then, In the following, we will be interested in an infinite-dimensional version of Duistermaat-Heckman integration over the orbits O b 0 . To this end, let us remark that there exists again an S 1 -action on O b 0 which rotates the source circle: φ(x) → φ(x + a). As it turns out, this S 1 -action is Hamiltonian with respect to the symplectic form (26). The Hamiltonian in this case is 3. BF theory on a punctured disk 3.1. Quantum mechanics in PSL(2, R) as a holographic dual. Let D * = D − {0} be the punctured unit disk with the origin removed and let G = PSL(2, R). We denote by g its Lie algebra and by tr the non-degenerate Killing form. The BF theory we are interested in is defined by a g-valued scalar field X and a g-valued one form 4 A. The action functional of the model is In order to have a well-defined Dirichlet problem for the variational principle, we need to specify boundary conditions. Following [6], we choose to implement these boundary conditions by adding a Hamiltonian on the boundary: where dx denotes a volume form on S 1 = ∂D * and the Hamiltonian tr X 2 is, up to a constant, the quadratic Casimir of PSL(2, R). Moreover, we assume that the scalar field X vanish at the puncture.

Remark 2.
We can think of the punctured disk as a semi-infinite cylinder be the coordinate along the cylinder and x the angle coordinate. Then the origin, i.e., the puncture of the disk corresponds to t → −∞.
The assumption that X vanishes at the puncture of the disk is therefore equivalent to the assumption that X vanishes at infinity.
The equations of motion for X are Hence, integrating out the scalar field X, the path integral localizes to the moduli space of flat connection on D * . To determine the moduli space in question, we first point out that the action is gauge invariant only under those gauge transformations that are trivial on the boundary. Indeed, let G = C ∞ (D * , G) be the full gauge group, acting on the fields X and A as Then the action is invariant only up to a boundary term: However, the normal subgroup G 0 = {g ∈ G | g| ∂D * = Id} of gauge transformations which are trivial on the boundary leaves the action invariant. The path integral therefore localizes over the space of flat connections A flat modulo the aforementioned gauge transformations: M 0 = A flat /G 0 . Flat connections on D * are characterized by their holonomy around the boundary.
We restrict ourselves to flat connections A ∈ A 0 flat whose holonomy around the boundary is the identity. The moduli space M 0 = A 0 flat /G 0 can be identified with the space of based loops ΩG = {g ∈ C ∞ (S 1 , G) | g(1) = Id} as follows: let where P A (1 x) denotes the parallel transport defined by A from 1 ∈ D * to x ∈ D * . This map is well-defined and in particular independent of the chosen path 1 x. Indeed, since the connection is flat, the parallel transport map only depends on the homotopy type of the path. If now γ, γ : 1 x are any two paths connecting 1 and x, then their difference is a loop (which possibly winds several times around the puncture at the origin), c.f. Figure  1. This loop is always homotopic to a multiple of a loop winding around the boundary. Since the holonomy of the connection around the boundary is trivial, the parallel transport maps along γ and γ are the same. The inverse map is constructed as follows: given a based loop g ∈ ΩG, we can extend it to the punctured disk. Indeed, if r : D * → S 1 is a retraction, theñ g = g • r : D * → G is an extension of the based loop g to D * . We then get a flat connection by setting A = −dgg −1 .
The path integral thus localizes over ΩG and any connection is parametrized by an element g ∈ ΩG as A = −dgg −1 . Using the boundary equations of motion A| ∂D * = Xdx| ∂D * the boundary action becomes which describes a quantum mechanical free particle moving in the group manifold PSL(2, R). Notice that the boundary action coincides with the Hamiltonian (8).

3.2.
Edge states and larger gauge groups. Recall that due to the presence of the boundary the action was invariant only under the smaller gauge group G 0 ⊂ G of gauge transformations which are trivial on the boundary. Following [11] and reference therein, one can reinstate the missing gauge degrees of freedom by allowing so-called edge states: let Λ ∈ C ∞ (∂D * , G) which transforms under a gauge transformation as Λ g = gΛ. Then, one can add the following extra term to the action: The new action is indeed invariant under any gauge transformation: The action (33) admits a global G-symmetry which acts on the loops g by conjugation. Allowing the edge states Λ to take values only in a certain subgroup H ⊂ G gauges the global symmetry corresponding to the subgroup H, i.e. the gauge group is enhanced to Considering the action (34), integration over the scalar field X imposes the equations of motion Parametrizing A = −dgg −1 , the action on the boundary becomes where S ∂ (g) is given as in Equation (33).
In particular, if we choose an Iwasawa decomposition G = N AK, where N is the group of upper triangular matrices with ones on the diagonal, A diagonal with unit determinant and K = SO(2) a compact subgroup, one can restrict the edge state Λ to take values only in N : The path integral now localizes over the space of flat connections modulo the augmented gauge group G N = {g : D * → G | g| ∂D * ∈ N }. This moduli space can be identified with the space of based loops, modulo the action of based loops in N , i.e. the space of based loops in AK ⊂ G. Importantly, one finds S ∂ (Λ) = 0. The boundary action in the presence of edge states then becomes After an integration by parts, λ takes over the role of a Lagrangian multiplier which imposes the condition J − (g) = cst. We will see in the next section that the above constraint arises as a first class constraint in the theory without edge states and can be equivalently seen as a Drinfeld-Sokolov reduction of the moduli space M 0 ∼ = ΩG. Finally, we will show how the constraint leads to Schwarzian quantum mechanics.

Quantum Mechanics on PSL(2, R)
4.1. The constrained model. As before, let G = PSL(2, R). The considerations of the preceding section lead us to the study of quantum mechanics of a free particle moving in G (for compactified time). The action 5 of the model is given by (33) Under a small variation of the loop, g → g + δg the action changes by The equations of motion, therefore, imply that the current J(g) = g g −1 is conserved. To fix notation, let The action admits a global G symmetry acting by conjugation on the loops g ∈ ΩG. We will be interested in gauging part of this symmetry. Notice that LG acts on ΩG by The action satisfies In the case that h takes values in the subgroup N ⊂ G of upper triangular matrices, one finds Therefore, imposing the first class constraint J − (g) = 1, the action acquires a local LN symmetry.
Remark 3. Equivalently, according to the previous section, we could study the model in the presence of edge states taking values in N ⊂ G. Upon integration, the edge states impose the above constraint.
Let us parametrize elements g ∈ G by an Iwasawa decomposition of the form Here, n ∈ Z is called the degree (or winding number) of θ.
In terms of these functions, the action (35) is It is now straightforward to read off the conjugate momentum for F : Fixing π F to be a constant imposes a first class constraint: (40) 2π F = 1 ⇐⇒ a 2 θ = 1.
Notice in particular that π F = J − (g).
As we have discussed before, the first class constraint generates a gauge symmetry in F . Indeed, by setting a 2 θ = 1 in (38), F decouples from the action entirely, since it occurs as a total derivative. We are therefore interested in the following partition function: where the measure dλ(g) is taken to be the symplectic measure on ΩG. The above is a one-dimensional analog of the constrained WZW model studied by Bershadsky and Ooguri in [5]. Substituting F = 0, which fixes the gauge, and the constraint (40) into Equation (38), the action becomes where we used Equation (19) to write the action explicitly in terms of the Schwarzian derivative. As we have remarked before, θ is a based map from R to R satisfying θ(x + 2π) = θ(x) + 2πn for some n ∈ Z. The integer n is called the degree (or winding number) and classifies θ topologically. It counts how often the map winds around the target circle. The constraint (40) implies that θ > 0, i.e. θ is an increasing map and hence has strictly positive degree. Any such map can be parametrized by some diffeomorphism φ ∈ Diff + (S 1 ) by setting θ = nφ. The constrained path integral is therefore taken to be over Diff + (S 1 ).
The partition function then splits into a sum over topologically distinct sectors which are labeled by the winding number. Each sector is governed by an action (43) S n (φ) = k 4π This action, however, admits a residual SL (n) (2, R) symmetry which, as was shown in [2], acts on the diffeomorphisms φ by In particular, one needs to reduce the configuration space Diff + (S 1 ) by this global symmetry. The totally reduced configuration space, in each sector, will therefore be Diff + (S 1 )/SL (n) (2, R) which is isomorphic to the exceptional Virasoro orbit O n passing through b 0 = − cn 2 48π , c.f. (24). Now, the key observation is that the orbits O n are symplectic (see the discussion in Section 2.3) with symplectic structure Moreover, the actions S n are the Hamiltonians (27)  The constrained partition function is therefore the sum over all of the aforementioned topologically distinct sectors: where dλ n is the symplectic measure of the reduced configuration space O n . Indeed, notice that the symplectic form on ΩG reduces to the symplectic form on O n : Substituting the Iwasawa decomposition (37) into the symplectic form (7) yields: which, after imposing the constraint (40) and gauging F to zero, becomes With θ = nφ and c = 6k we therefore find that the symplectic form restricted to the (totally) reduced configuration space is given by which coincides with the symplectic form on O n , c.f. (26). In this way, the symplectic measure dλ(g) on ΩG gives rise to a symplectic measure dλ n on the reduced spaces O n . Finally, returning to (46), each of the Z n are integrals over an infinitedimensional symplectic manifold endowed with a circle action. In each case, the integrand is the exponential of the Hamiltonian which generates this circle action. In the case of a finite-dimensional symplectic manifold, this would be precisely the setup suited for Duistermaat-Heckman integration. Hence, by analogy, we define the Z n by the right hand side of the Duistermaat-Heckman integration formula, namely , with p being the fixed points and w j (p) the corresponding weights of the S 1 -action.

Drinfeld-Sokolov reduction.
Before proceeding with the calculation of the partition function by means of an infinite-dimensional analog of Duistermaat-Heckman integration, it is instructive to take a step back and to analyze the geometric meaning of the constraint (40). According to the discussion in Section 2, the space of based loops ΩG can be identified with the Kac-Moody coadjoint orbit LG/G passing through 0 ∈ Lg * and is therefore naturally symplectic. We recall that its symplectic form is given by The action (35) is then the Hamiltonian for the S 1 -action which rotates the loop: Let now N ⊂ G be the subgroup of upper triangular matrices. Recall from (9), that the coadjoint action of LN on LG/G is Hamiltonian. Let n be the Lie algebra of N . Then the moment map µ : LG/G → Ln * is given by projecting g g −1 ∈ Lg ∼ = Lg * onto Ln * ∼ = C ∞ (S 1 ). Consider the preimage of any real positive number q ∈ R + ⊂ Ln * : in terms of the Iwasawa decomposition (37), elements of µ −1 (q) satisfy which gives the condition 1 2 a 2 θ = q, q > 0 which, in turn, is equivalent to the constraint (40). Notice that the constraint is fixed under the action of LN . Thus, the constrained theory, whose space of fields is µ −1 (q), exhibits a gauge symmetry, corresponding to a shift of F . The reduced configuration space is µ −1 (q)/LN which can be seen as the symplectic reduction of LG/G with respect to the aforementioned moment map. In this special case, the reduction is known as Drinfeld-Sokolov reduction [9]. Notably, any class [g] ∈ µ −1 (q)/LN has a unique representative of the form 0 − 1 2q θ 2 2 + {θ, x} q 0 which is obtained by using the gauge freedom to set F = 1 q a a in (52) and using the constraint q = 1 2 a 2 θ to express a in terms of θ. As before, the constraint imposes θ > 0 and we can parametrize θ = nφ by diffeomorphisms φ ∈ Diff + (S 1 ). Therefore, if g has winding number n, the orbit [g] ∈ µ −1 (q)/LN can be mapped to a point n 2 2 φ 2 + {φ, x} in the exceptional Virasoro coadjoint orbit O n . To see that this establishes a one-to-one correspondence, let us show that φ transforms correctly under right translations of G: By the Iwasawa decomposition, any g ∈ G can be written as where one can identify tan θ/2 = γ/δ. If we parametrize θ = nφ one has that tan(nφ/2) transforms by fractional linear transformations which indeed coincides with the PSL(2, R)-action on Diff + (S 1 ).
In conclusion, one has that the reduced configuration space of the constrained theory is a disjoint union of Virasoro orbits: µ −1 (q)/LN ∼ = n 1 O n . , where the sum runs over all fixed points p of the S 1 -action (rotating the loop) on O n and w j (p) denote the weights (at p) of the aforementioned circle action. We therefore have to compute two things: the fixed points p and the weights w j (p).
Recall that O n is the left quotient of the group Diff + (S 1 ) by its subgroup SL (n) (2, R). The S 1 -action, which rotates the source circle, corresponds to the right multiplication by S 1 ⊂ SL (n) (2, R): t · φ(x) = φ(x + t). The fixed points on O n correspond therefore to (the cosets of) elements φ ∈ Diff + (S 1 ), s.t. for every t ∈ S 1 there exists χ ∈ SL (n) (2, R) satisfying φ • t = χ • φ. Equivalently, Since I φ is a homomorphism, its image I φ (S 1 ) is a torus in SL (n) (2, R), and therefore can be brought to the torus S 1 ⊂ SL (n) (2, R) by conjugation, i.e.
Replacing φ by g −1 •φ (they belong to the same equivalence class) we obtain a homomorphism I φ : S 1 → S 1 , which implies that I φ (t) = nt for some n ∈ Z. Equivalently, Differentiating with respect to s we see that φ is a constant function and thus φ is necessarily a (constant) rotation. Since rotations belong to the coset of the identity, the only fixed point is the identity φ(x) = x.
Remark 5. The same argument holds for the non-exceptional Virasoro orbits O b passing through (b, c) ∈ Vir * with b constant. Hence, the S 1action on O b which rotates the source circle has as well only one fixed point corresponding to the identity.
To compute the weights of the S 1 -action on T id O n , note that the latter is naturally identified with Vect(S 1 )/ L 0 , L ±n . The S 1 -action is generated by L 0 , which acts on L −m ∈ Vect(S 1 ) by [L 0 , L −m ] = mL −m , i.e. with integer weights. Therefore, the denominator in the Duistermaat-Heckman formula (1) is given by the infinite product where the product is understood in the zeta-regularized sense. With

4.4.
Sources of conserved charges and non-exceptional Virasoro orbits. It turns out that one can treat more general actions than the one for a free particle moving in G by the same methods. Suppose that we fix a particular edge state Λ taking values in K ⊂ G: Then, the BF action (28) in the presence of Λ becomes The boundary equations of motion for X are Integrating out X, the path integral localizes over the moduli space of flat connections and, parametrizing A by A = g −1 dg, the boundary action becomes is the current corresponding to the compact subgroup K. In particular, v plays the role of a source for the charge Indeed, taking derivatives of the partition function with respect to v, one obtains correlation functions of powers of the charge. For example, one has ∂Z(v) ∂v v=0 = dµ(g) e iS(g)/ Q K (g) = Q K (g) .

Remark 6.
By choosing more general v 0 , one obtains an action in presence of sources for more general conserved charges.
Motivated by the above considerations, let us now fix v 0 with v > 0 and consider the action where we introduced the quasi-periodic element h = g exp( 4π k v 0 x). For any element g ∈ LG, we have an Iwasawa decomposition for h (c.f (37)): where F , a and θ are, as before, real-valued periodic (respectively quasiperiodic) functions on R. Note that if θ has winding number n ∈ Z, thenθ satisfies the quasi-periodicity condition (59)θ(x + 2π) =θ(x) + 2π n + 4πv k .
Proceeding as in the previous section, we impose the first class constraint a 2θ = 1. In particular,θ is an increasing map,θ > 0. Equation (59) therefore implies that n + 4πv/k > 0. Any such map can be parametrized by an orientation preserving diffeomorphism φ ∈ Diff + (S 1 ): where n v is defined as the shift of n by the integer part of 4πv k and κ v denotes the corresponding fractional part. Since φ(x + 2π) = φ(x) + 2π,θ satisfies indeed the correct periodicity condition (59). In fact, sinceθ > 0, only thoseθ with n v 0 will contribute to the constrained path integral.

Remark 7.
Let us point out that the path integral reduces to an integral over the Kac-Moody orbit LG/K, c.f. Example 2.2. As before, the action coincides with the Hamiltonian generating the S 1 -action on the orbit. Again, the constraint a 2θ = 1 corresponds to a Drinfeld-Sokolov reduction for the LN -action on LG/K.
In terms of the fields (F, a,θ), the action reads: After imposing the constraint, and using the resulting gauge symmetry to set F to zero, the partition function splits again into distinct sectors, each governed by an action of the form As before, there exists residual symmetry, this time, however, it is only a S 1 symmetry which acts by constant shifts of φ: φ(x) → φ(x) + t, for t ∈ S 1 . Hence, the (totally) reduced configuration space is isomorphic to the more general Virasoro orbit passing through the point b 0 = −c(n v + κ v ) 2 /48π, which in turn is isomorphic to Diff + (S 1 )/S 1 . The reduced configuration space is therefore again a symplectic space. Setting c = 6k, the actions S nv again coincide with the Hamiltonians generating the S 1 -action (which rotates the source circle) on the aforementioned Virasoro coadjoint orbits. By the same arguments as in the previous section, the partition function splits into a sum of integrals over these more general Virasoro orbits which again are defined by the right hand side of the Duistermaat-Heckman integration formula (1): where the sum runs over all fixed points p and the w j (p) are the weights of the S 1 -action on the tangent space at p. As discussed previously, there is only one fixed point, namely φ = id at which the action takes the value S(id) = c 24 (n v + κ v ) 2 . By the same argument as before, the denominator is given by the zeta-regularized product Therefore, the partition function is Remark 8. It is intriguing that the limit of taking v to zero does not give back the constrained partition function Z red 0 . On the other hand, in this limit we recognize that (Diff + (S 1 )/S 1 , ω 0 ) is only a pre-symplectic space. The kernel of ω 0 , in the n-th topological sector, is generated precisely by the Virasoro vector fields L ±n . It is the emergence of these zero modes, which spoils the naive limit of the partition functions. In fact, we can fix the zero modes by considering an additional reduction, namely by taking the quotient with respect to ker ω 0 . The resulting spaces are exactly the exceptional Virasoro orbits O n ∼ = Diff + (S 1 )/SL (n) (2, R), whose partition function has been calculated in Equation (46). 4.5. Operator formalism. The partition function of the quantum mechanical model corresponding to the exceptional Virasoro orbits can also be calculated within the operator formalism. The description of the model in first order formalism is given in Equation (28) which, after integrating the scalar field in the bulk gives the boundary action The Hamiltonian is therefore (up to a constant) the quadratic Casimir operator 1 2 tr X 2 of PSL(2, R). If one parametrizes X as In order to canonical quantize the theory, we need to determine the conjugate momenta. Using the Iwasawa decomposition (37) and setting a = e ϕ one finds From the above one can derive an expression of X 0 , X ± in terms of the conjugate momenta π θ , π ϕ and π F : Canonical quantization is done by enforcing the canonical commutation relations: [x i , π i ] = i where x i ∈ {ϕ, θ, F } and π i ∈ {π ϕ , π θ , π F }. With these commutation relations one finds Remark 9. The canonical commutation relations should not be surprising: the phase space of the free particle moving in G can be identified with T * G ∼ = g * × G. Now, X 0 , X ± are linear functions on g * and hence their Poisson bracket is given by the Lie bracket on g, which up on quantization give the commutation relations (65). Now, the partition function can be defined as the trace over the Hilbert space H of the exponential of the Hamiltonian: Of course, one first needs to identify the Hilbert space of the model which in the case at hand coincides with unitary irreducible representations of PSL(2, R). Unitary irreducible representations of PSL(2, R) are labeled by the eigenvalue j(j − 1) of the quadratic Casimir. The corresponding states are, in addition, labeled by the eigenvalue of one of the generators [18]. The standard choice is to take the generator corresponding to the compact subgroup K ⊂ G. It is convenient define linear combinations of the generators X 0 , X ± : Then, J 0 corresponds to the generator of the compact subgroup K ⊂ G. Moreover, these generators satisfy the following commutation relations: Let now |j, m denote a standard basis 6 such that For a detailed account of the representation theory of PSL(2, R) we refer the interested reader to [17,18]. The first class constraints π F = cst, which we were imposing before, is now equivalent to restricting the representations to those, for which π F acts constantly. Since π F = X − = J 0 + J 1 , it is therefore necessary to diagonalize the generator of the subgroup N ⊂ PSL(2, R) rather than J 0 . These representations were studied in [8,18]. Let |j, q) be the basis that diagonalizes J 0 + J 1 , i.e., H|j, q) = j(j − 1)|j, q), (J 0 + J 1 )|j, q) = q|j, q).
The states |j, q) can be expanded in the standard basis |j, m , in which they are expressed in terms of Whittaker functions W a,b (x) [18]: j, m | j, q) = Γ(m − j) Γ(m + j + 1) where ρ (n; D + ) = (2π) −2 (n − 1/2) is the Plancherel measure of the discrete principal series of PSL(2, R) [17]. Here, tr (j) q=1 denotes the trace in the subspace of Hilbert space of the representation j which is singled out by a fixed eigenvalue q = 1 of J 0 + J 1 . The partition function (68) matches with the previous answer (55) which was obtained by the path integral method.