On quantum gravity coupled to a model ^{†}^{†}thanks: Talk given by H. Nicolai at the 29th International Symposium Ahrenshoop, Buckow, Germany, August 29  September 2, 1995.
1 Introduction
This contribution is a review of the method of isomonodromic quantization of dimensionally reduced gravity developed in [1, 2, 3]. Our approach is based on the complete separation of variables in the isomonodromic sector of the model and the related “twotime” Hamiltonian structure. This allows an exact quantization in the spirit of the scheme developed in the framework of integrable systems [4]. Possible ways to identify a quantum state corresponding to the Kerr black hole are discussed. In addition, we briefly describe the relation of this model with Chern Simons theory.
2 The model
The Lagrangian of gravity coupled to a model is
(1) 
where the metric has been brought into conformal gauge
(2) 
is the Gaussian curvature of the worldsheet, and is the dilaton. The equation of motion for derived from (1)
(3) 
is solved by
(4) 
where is a (locally) holomorphic function. Now we can further specialize the gauge by identifying with the worldsheet coordinate. Then the equation of motion for coming from (1) is
(5) 
Using the following parameterization of an arbitrary valued matrix:
(6) 
in terms of the complexvalued function , we can rewrite (5) in the familiar form of the Ernst equation [5]:
(7) 
To get from (1) the remaining equations of motion for the conformal factor , we have to temporarily relax the conformal gauge and to vary (1) with respect to the offdiagonal elements of the metric. Restoring the conformal gauge then yields
(8) 
3 Deformation equations and function
Consider the following system of differential equations for matrices and the functions with :
(9)  
(10)  
These (compatible) equations are solved by
where are constants of integration; in the sequel we shall assume to be defined by (3). One can easily check that the system (9) is always compatible if (10) holds.
Next define the function associated with (9) by
(12) 
where the differential is to be taken with respect to the variables . Equivalently it can be computed with respect to the variables , which gives as a function of the parameters . Notice that the form on the r.h.s. of (12) is always closed as a consequence of (9).
It is easy to check that , and are integrals of motion of the system (9).
Our purpose will be to exhibit the link between the system (9) with given by (3) and the equations of motion (5) and (8) of the previous section. A partial answer is given by
Theorem 1
To understand the precise correspondence between the solutions of (9) and the original model, one has to ensure the coset and reality conditions and . To this aim we define the rational function by
(15) 
The proof of the following theorem may be found in [3].
Theorem 2
Let be a solution of the system (9) satisfying the following additional conditions:

Reality:
(16) 
Asymptotic regularity:
(17) 
Invariance of the function with respect to the involution :
(18) with some constant .
Then the constants of integration in (8) and (13) may be chosen in such a way that and .
At this point the relation between (9) and the original model may still appear obscure; it will be clarified in section 6. Let us just emphasize that the variables in the system (9) have been completely separated; thus we can treat the “left” () and “right” () moving sectors as completely independent.
4 An example: the KerrNUT solution
The general solution of the system (9) for arbitrary values of and the parameters is certainly not possible. However, one can try to understand which solutions of (9) correspond to known solutions of (7). For example, the KerrNUT solution of (7) corresponds to ,
(20) 
with and . The integrals of motion should equal (since , this means that the eigenvalues of are equal to ). It is not difficult to show that the solution of (9) satisfying these conditions and the constraints given by Thm.2 corresponds to the Ernst potential
(21) 
where
(22) 
with
are prolate ellipsoidal coordinates; are complex constants satisfying . This is nothing but the KerrNUT solution; the Kerr solution itself corresponds to .
5 Twotime Hamiltonian structure
We adopt here a “twotime” Hamiltonian formalism with the two “times” corresponding to the lightcone coordinates and . One major advantage of this procedure is that the quantum theory is manifestly covariant under coordinate transformations, a feature which is far from obvious (and possibly not even true) for the ADM formulation of canonical quantum gravity (see e.g. [8] for a recent discussion). Moreover, we must to treat the “times” and as phase space variables because they are really fields in a special gauge; then, according to the general canonical procedure, the related total Hamiltonians should weakly vanish, i.e. should be considered as firstclass constraints.
The Hamiltonian structure which gives the complete set of equations of motion in terms of the variables , , and is described by the following
Theorem 4
This theorem can be verified by direct calculation. The weak vanishing of and implies the equations (8) relating the gravitational and matter degrees of freedom. Commutativity of the Hamiltonian constraints may be obtained by use of the general relation
(27) 
which is valid for arbitrary and . The commutativity of the flows generated by and is equivalent to the decoupling of the classical equations of motion in (8) and (9) and may be viewed as a direct consequence of the compatibility of the system (8), (9). In terms of the standard “one time” canonical formalism with as Euclidean time, the combination corresponds to the Hamiltonian or WheelerDeWitt constraint while corresponds to the diffeomorphism constraint.
The “time evolutions” of any functional are generated as usual by commutation with the total Hamiltonian constraints and , i.e.
(28) 
On the l.h.s. here we have the total derivatives with respect to , ; the first term of or generates the partial derivatives with respect to the coordinates and the second term takes care of the dependence of . Observe that we have .
Defining
(29) 
where are the generators of , and inserting (15) into (23), we get
where are the structure constants of .
Observables in the sense of Dirac are by definition all those functionals on phase space which weakly commute with the constraints and , but do not vanish on the constraint hypersurface , i.e.
(30) 
By (28) the observables are independent of the coordinates and therefore highly nonlocal objects as one would expect on general grounds [9, 10]. First of all, the parameters trivially belong to this class since they commute with everything. Second, and more importantly, the monodromies of the connection defined by
(31) 
where the contour starts at and encircles the point , are also observables for arbitrary . For a discussion of this fact, see [3]. All observables can be generated from the set
(32) 
by taking products and linear combinations. In this sense, constitutes a complete set of classical (and quantum) observables for arbitrary . These are the conserved “nonlocal charges” of dimensionally reduced gravity.
Notice also that the constraints mentioned in Thm.2 are in fact first class constraints with respect to our Poisson structure. In particular, the constraint which closes into the algebra is nothing but the conserved charge which generates the Ehlers transformations with a constant matrix .
6 Link to Chern Simons theory and the linear system
It is known that the Ernst equation can be obtained as the compatibility of a linear system [11, 12]. The interpretation of the linear system as a zero curvature condition suggests a link with Chern Simons theory whose equations of motion also state the vanishing of some curvature. The new feature here is that the Chern Simons gauge connection lives on a space locally parameterized simultaneously by the spectral parameter and the true space time coordinate.
The relevant Chern Simons action (at level 1) reads
(33) 
where plays the role of time, is a time dependent connection 1form on the Riemann surface locally coordinatized by , and is the curvature 2form. The time component appears as a Lagrangian multiplier for the first class constraints of vanishing curvature :
(34) 
In the usual treatment is gauged to zero which leads to static components and . In particular, the singularities of this connection are then timeindependent and treated by inserting static Wilsonlines in the action (33). Alternatively, we consider the gauge
(35) 
The residual gauge freedom corresponding to (34) is fixed by demanding
(36) 
on the whole surface except for some set of zero measure. Because of (34) and the remaining component then becomes holomorphic up to poles. To allow such singularities in as in the previous section, it is clear that (36) cannot be imposed everywhere because the singularities arising via the relation would spoil the constraint (34). Instead one should think of as being localized on some string with endpoints at the singularities of .
The remaining equation of motion is
(37) 
The constraints can now be treated by introducing Dirac brackets. The original Poissonbracket that comes from the action (33):
(38) 
is thereby changed to a bracket between the remaining meromorphic components [14, 15]:
(39) 
This may be translated into a bracket structure on the coefficients of the poles of , which — together with the positions of the poles — now parameterize the phase space:
(40) 
for
It coincides with the Poisson structure introduced in Thm. 4 of the previous section. The equations of motion (37) give rise to equations (9) and (10).
Among the surviving first class constraints is the total sum of the residues:
as well as the Chern Simons Hamiltonian
(41) 
which generates the equations of motion for the holomorphic component of the connection:
Splitting the Hamiltonian (41) it is now possible to identify its parts with the expressions obtained in the previous section. A short calculation reveals
such that defining
we have
(42) 
in agreement with (25) if is given by (15). All equations of motion are now generated by
(43) 
In this way the Poisson structure as well as the Hamiltonian and the constraints have a natural explanation in the context of Chern Simons theory. Similar considerations lead to the analogous results for the sector. However, further work is required to embed this twotime treatment in one unified canonical approach.
It is quite instructive to see how the wellknown auxiliary linear system [11, 12] arises in this framework. The analogous treatment of Chern Simons model in space with the gauge choice
(44) 
gives the equation of motion supplementing (37):
(45) 
The vanishing of the curvatures (37) and (45) implies the existence of a gauge transformation such that
(46) 
Substituting (35) and (44) into the last two equations and using (13) we get just the linear system of [12] with playing the role of the spectral parameter. The solutions of (5) for which can be represented as in (15) are called isomonodromic; in particular, they contain all known solutions such as multisoliton solutions [12] and the algebro geometrical solutions of [13], as well as many others. Of course in assuming (15) we truncate the total phase space of the original model. We would expect that there exists a topology on the space of solutions for which the isomonodromic solutions constitute a dense subset of the phase space of “all solutions” (notice that the Poisson structure given by (23), (24) and (42) is independent of the ansatz (15)).
7 Quantization
To quantize the model, we replace the Poisson brackets (23) by commutators in the usual fashion:
(47) 
(48)  
Suppose now that all are imaginary (i.e. ); then by Thm.2 we should require all elements of to be real at the classical level. Quantum mechanically we get
(49) 
where are the antihermitian Chevalley generators of obeying the standard commutation relations
(50)  
Unitary representations of (50) with Casimir operator
equal to are given by
(51)  
where are the arguments of the functions spanning the representation space , which may belong to the principal, supplementary or discrete series of .
According to (48) one can choose
(52) 
Thus the wave function of a given isomonodromic sector with should depend on and live in the direct product
of unitary representation spaces of . This means that can be realized as a function
8 WheelerDeWitt equations and KnizhnikZamolodchikov system for
The WheelerDeWitt equations now take the form
(53) 
or, equivalently,
(54) 
which can be written out by use of the explicit form of the constraints and given in (26) (49),(51) and (52):
(55)  
where
According to Thm.2, the wave functionals satisfying the coset constraints should be symmetric with respect to the involution and satisfy the constraint
(57) 
The general solution of the system (55) is not known. However, these equations turn out to be intimately related to the KnizhnikZamolodchikov system for [16, 17]:
(58) 
with an valued function .
Theorem 5
The full set of quantum observables is related to the algebra of monodromies for the KZ equations (58) which is well understood only for where it gives rise to certain quantum groups [18].
The only solutions of KZ equations for the noncompact group known so far are solutions corresponding to the unitary discrete series representations (either positive for all or negative for all ) all of which possess a ground (lowest weight) state in . However, it is possible to show [3] that solutions of this kind cannot satisfy the constraint (57). Moreover, a simple analysis of the sign of the Casimir operator shows that on order to construct wave functions corresponding to physically interesting classical solutions (such as KerrNUT) one would have to consider representations of the continuous series. Namely, for all known classical solutions (including KerrNUT) we have . However, in the quantum case,
For the discrete series, when is real and integer, this is always negative. For the continuous series we have with , and the eigenvalue of is positive.
In the next section we shall briefly discuss how one might go about constructing a quantum state whose semiclassical limit would reduce to the KerrNUT solution.
9 Towards a quantum Kerr solution
According to the previous section, the desired solution of (58) for is a function of the four positions of the poles defined by (20) and of the four auxiliary variables , on which the algebra is represented.
The constraints (57) have to annihilate this function which hence is invariant. Therefore it essentially depends only on the ratio
just as in conformal field theory where the conformal Ward identities reduce the correlation functions to a function of a single variable [20]. Moreover, the validity of the KZ equations implies an analogous reduction of the dependence.
The quantum state then reduces to the following form:
(60) 
with
The remaining KZ equation for the function can be obtained by a lengthy but straightforward calculation which gives
(61) 
with
An equivalent form of this equation appeared in the study of fourpoint correlationfunctions in Liouville theory [19].
Equation (61) is very similar to the standard hypergeometric equation, where and are just two matrices. The singular points have a very definite physical meaning from the point of view of the classical Kerr solution. Namely, we can express the variable in terms of prolate ellipsoidal coordinates (22) as follows:
This shows that corresponds classically to the spatial infinity and the part of the symmetry axis outside of the event horizon. The value corresponds to the poles of the event horizon, and corresponds to the surface of the event horizon.
The analysis of equation (61) should give asymptotical expansions of the wave functional at these singular points and allow us to relate them. This would then enable us to understand the behavior of physically interesting expectation values at these points and to clarify the meaning and the fate of the classical singularities in the quantum theory.
The classical limit leading to the KerrNUT solution should look like
If this limit is equal to an integer , the related classical solution should be the th member of the TomimatsuSato hierarchy.
Acknowledgments
The work of D. Korotkin was supported by DFG contract Ni 290/51; H. Samtleben was supported by Studienstiftung des deutschen Volkes.
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