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Correlation functions in the Schwarzian theory

A mathematically correct approach to study theories with infinite-dimensional groups of symmetries is presented. It is based on quasi-invariant measures on the groups. In this paper, the properties of the measure on the group of diffeomorphisms are used to evaluate the functional integrals in the Schwarzian theory. As an important example of the application of the new technique, we explicitly evaluate the correlation functions in the Schwarzian theory.

I.
The action of the theory is where is the Schwarzian derivative.
The presence of the term (ϕ (t)) 2 in the action (1) makes the functional integral look as if it is the integral over the Wiener measure analytically continued to the point κ = i σ √ 2π . However, the attempt to treat (3) as the integral over the Wiener measure is a misleading one. The point is that the Wiener measure is concentrated on the trajectories that are nondifferentiable almost everywhere. The set of smooth, or even differentiable at a point, functions has zero Wiener measure. Nevertheless, the formal representation (4) is correct and very useful if the derivative ϕ (t) is considered in a generalized sense [9]. However, the Schwarzian derivative cannot be understood in this way.
To evaluate correlation functions in SYK it was proposed [10] to map the theory onto Liouville quantum mechanics, and to use spectral decomposition to represent functional integrals as the sum over quantum states. Although the approach turned out to be very profound and qualitatively helpful (see, also, [11], [12], [13]), it is desirable to develop a technique of functional integration in the theory.
Mathematically correct approach is based on the quasi-invariant measure on the group of diffeomorphisms (see the next section) And the functional integrals in the Schwarzian theory should be considered as the integrals over the measure (5).
In [14], the integral for the partition function in the Schwarzian theory was explicitly evaluated with the result thereby confirming the conjecture about the exactness of the one-loop result [3], [6].
In this paper, we evaluate the functional integrals assigning correlation functions in the Schwarzian theory. As the technique of functional integration over non-Wiener measures is not common knowledge, we try to make the presentation maximally explicit.
Section II, and appendices A and B contain the relevant mathematical apparatus. In Section III, the explicit evaluation of the correlation functions is presented. In addition, some relevant technical results are obtained in appendices C and D. In Section IV, we give the concluding remarks.
The quasi-invariance of the Wiener measure (4) under the shifts of the argument of the measure by a differentiable function is the simplest example [9].
The Wiener measure (4) turns out to be quasi-invariant under the group of diffeomorphisms. The proof of the quasi-invariance and the explicit form of the Radon-Nikodim derivative was first obtained in [16] (see a more simple derivation of the result, e.g., in [17]).
The evaluation of the functional integrals considered in this paper is based on the equation: It is a consequence of the quasi-invariance of the Wiener measure with respect to the action of the group of diffeomorphisms Dif f 3 + ([0, 1]) . In [14], we postponed the proof of the equation (8) till the next paper, "Hanc marginis exiguitas non caparet." (P. Fermat). Now, the explicit proof of the more general formula is given in [18] ( see, also, appendix B of the present paper).
Consider the function ξ ∈ C 0 ([0, 1]) . That is, ξ(t) is a continuous function on the interval satisfying the boundary condition ξ(0) = 0 . Then The integral over the group Dif f 1 (S 1 ) can be transformed into the integral over the group Dif f 1 ([0, 1]) . Note that if we fix a point t = 0 on the circle S 1 then it is necessary "to glue the ends of the interval". That is, to putφ(0) =φ(1) or ξ(0) = ξ(1) = 0 . In this case, the function ξ is a Brownian bridge, and we denote the corresponding functional space by Now, the integral over Dif f 1 (S 1 ) turns into the the integral over Dif f 1 ([0, 1]) as follows: Here, we have used the equation The quasi-invariance of the measure and the explicit form of the Radon-Nikodim derivative can be used to evaluate nontrivial functional integrals. For the measure µ on the interval [0, 1] in particular, the Radon-Nikodim derivative is where Here, the well known property of the Schwarzian derivative : has been used.
Thus, for functional integrals over the measure µ , we have In what follows, we assume the function f to be In this case, and the equation (15) looks like: Generally speaking, the functional integrals (18) converge for 0 ≤ α < π , and diverge for The Schwarzian action is invariant under the noncompact group SL(2, R). Therefore, integrating over the quotient space Dif f 1 ([0, 1])/SL(2, R) we exclude the infinite volume of the group SL(2, R) and get the finite results for functional integrals in the Schwarzian theory. In our approach, we evaluate regularized (α < π) functional integrals over the group Dif f 1 ([0, 1]) and then normalize them to the corresponding integrals over the group SL(2, R) .
In particular, in [14], to get the partition function (36), we first evaluated the regularized and then divided it by the regularized volume of the group SL(2, R) Note that the functional measure in the equation (19) and the Haar measure dµ H on the group SL(2, R) in the equation (20) are regularized in the same manner.
For the Schwarzian partition function, we take the limit In the next section, the quasi-invariance of the measure (5) is used to evaluate the functional integrals assigning the correlation functions in the Schwarzian theory.

III.1. Mean value of ϕ
First we recall the main steps of the evaluation of the partition function in the Schwarzian theory [14].
If we take the function F in the equation (18) to be and note that Now the regularized partition function has the form Under the substitution (11) it turns into To evaluate the functional integral explicitly we use the equation (8). Instead of β , we should substitute a solution of the equation We take the following one: As the result, we obtain with the asymptotic form at α → π The asymptotic form of the α−regularized volume of the group SL(2, R) (see appendix According to the equation (21), the Schwarzian partition function has the form Consider now the α−regularized mean value of ϕ Note that it is ϕ , but not ϕ, that is the dynamical variable in the theory given by the action (1).
After the substitution (11), it is written as Having in mind the equations (8), (26), and (27), we get Asymptotically, it looks like The asymptotic form of the Φ α on the group SL(2, R) (see appendix D) is Now the normalized mean value of ϕ has the form Φ = lim

III.2. Two-point correlation function
Define the α−regularized two-point correlation function as By the special choice of the function F in (18), we identify the integrands in (37) and in the right-hand side of the equation (18).
Represent the function F in the form where To use the equation (18), it is necessary to find F i (ϕ) . F 1 (ϕ) was found above to be To find F 2 (ϕ) and F 3 (ϕ) , note that for , and F 4 (ϕ) looks more complicated: For the function y = f α (x) given by the equation (16), And finally, Thus for the correlation function, we have The substitution (11) in the above equation gives the correlation function in terms of the Wiener integral: transforms the Wiener integral over the measure w σ (dξ) into the two Wiener integrals over the measures To verify this statement, note that To return to the integrals over the group of diffeomorphisms, consider the functions The useful relations can be obtained directly from the above definitions Therefore, the two-point correlation function has the form of the double functional inte- Define the function E σ (u, v) by the equation We can rewrite the equation (46) as In terms of variables the correlation function looks like In appendix C, we study the properties of the function E and, in particular, obtain the equations Now, using the equations (50), (51) and the substitution z = (2x − 1) tan α 2 , we get the following representation for the correlation function in terms of the ordinary integrals: The substitutions V 1 = ρ sin 2 ω , V 2 = ρ cos 2 ω , and y = tan ω reduce the equation (53) The asymptotics of the two-point correlation function at α → π has the form Define the normalized two-point correlation function G 2 (0, t) as the limit Here, the correlation function on the group SL(2, R) at the symmetrical points is chosen as the normalizing factor. It is evaluated in appendix D with the result G As 2; SL(2,R) 0, 1 2 = −π log(π − α) . × sin 4π τ σ 2 t sin 4π θ σ 2 (1 − t) sinh(τ ) sinh(θ) (cosh(τ ) + cosh(θ)) 4 dτ dθ .

Thus we have
(57) From the above equation, it follows that the correlation function is singular at t → 0, and t → 1 . Its form is presented at Fig.1.

III.3. N-point correlation functions
The method described in detail in the above subsection can be used to evaluate the N-point correlation function given by the functional integral and denote ∆t n = t n − t n−1 , ∆x n = x n − x n−1 ; χ α (x n ) = 2 α tan α 2 1 + tan 2 α 2 (2x n − 1) 2 , n = 1, ...N . Then the N-point correlation function is written as Now we can perform functional integration in the functions E , and obtain The analysis of the dependence of the function G α N on the regularization parameter α as well as the study of possible relations between different correlation functions will be given in another paper. Here, we only note that G N (0, t 1 , ...t N −1 ) is singular if there is a pair of coinciding arguments, that is, when some ∆t n = 0 , similarly to the equation (57).

IV. CONCLUDING REMARKS
In this paper, we propose a new approach to study theories invariant under the infinitedimensional groups of diffeomorphisms and evaluate functional integrals assigning correlation functions in the Schwarzian theory.
Since the Schwarzian theory appears as a limiting theory of various theoretical models, it is of interest to study how the results obtained above could be used in these models.
Having in mind the higher-dimensional as well as supersymmetric versions of the models leading to the Schwarzian theory, it would be desirable to generalize the proposed approach and to make it applicable in the corresponding problems.

Consider the map
The map A identifies the spaces Dif f 1 + ([0, 1]) and C 0 ([0, 1]) . In this case, A detailed proof of the quasi-invariance of the measure µ σ is given in [21]). Here, we present a scheme of the proof. Consider It is written in the form where h = A(f ), that is, Note that, if ϕ ∈ Dif f 3 + ([0, 1]) , then h ∈ C 2 0 ([0, 1]) . The Jacobian of the map (63) at the point ξ found in the [21]) does not depend on ξ and is equal to 1 f (1) . For continuously differentiable function η, we obtain Let P be a continuous bounded functional on C 0 ([0, 1]) . Then Note that if ξ were continuously differentiable, then by the integration by parts we would However, the Wiener process ξ(t) is nonsmooth, and there appear the additional terms ( [23]) that can be evaluated in the discrete version of the theory by the correct passage to the continuous limit (see ([21])).
As the result, we have Note that The map A −1 gives the equation (15) Dif Thus, the measure µ σ on Dif f 1 where the Radon -Nikodim derivative has the form (14)

B. Proof of the basic formula
To evaluate the basic Wiener integral (9) we use the quasi-invariance of the Wiener measure under the action of the operator K f ≡ A L f A −1 : The Radon-Nikodim derivative is In this case, the Radon-Nikodim derivative has the form and At the same time, δ (η(1) + 2 log(β + 1) − x) w σ (dη) Here, we have used the equation and the following relation: or ξ(1) = η(1) + 2 log(β + 1) .
Therefore, the basic formula is proven.
C. Properties of the function E σ (u, v) To study the properties of the function E σ (u, v) given by the equation (47) consider the functions ϕ, ψ ∈ Dif f 1 ([0, 1]) connected by the diffeomorphism g λ : In this case, the equation (15) gives Therefore, we have In particular, for the above equation has the symmetric form (50): Now we perform the functional integration in E σ (u, u) . First we write it in the form of the integrals over the Wiener measure Taking the Fourier transform of the first δ−function in (71), we get It is convenient to rewrite the above equation as To use the equation (8), note that a solution of the equation Due to the identity log y + y 2 − 1 = arc cosh y , the equation (8) gives Therefore, the function E σ (u, u) is written as Having in mind the properties of the function arc cosh, we turn the integration contour with the result After the substitution τ = arc cosh x − 2 2 , x = 2 + 2 cosh τ , the function E σ (u, u) takes the form (51) The forms of the functions E σ (x, x) and E σ (x, y) at σ = 1 are presented at Fig.2 and at In this appendix, we evaluate the asymptotic form (at π − α → 0) of the regularized integrals over the Haar measure dν on the group SL(2, R) . These integrals are used to normalize the corresponding functional integrals over the group Dif f 1 considered in this paper. We are interesting in the asymptotic form of the following integrals: Here, To perform the integration over the group SL(2, R) we choose the representation [24] ϕ z (t) = − i 2π log e i2πt + z z e i2πt + 1 , z = ρe iθ , ρ < 1 .
Now we find the asymptotic form of the integral G α 2; SL(2,R) 0, The method widely used to study the asymptotic behavior of Feynman diagrams (see, e.g., [25] and refs. therein) is very helpful here. Namely, we consider the Mellin transform of the integrand and rewrite (88) as follows: