REVIEW 3 major objections 5 minor 77 references
Schwarzian functional integrals calculus
T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read The paper claims that every Schwarzian functional integral over the circle reduces to ordinary multiple integrals built from one explicit basic integral, giving finite two- and four-point correlation functions that do not factor.
desk verdict A serious functional-integration calculus for Schwarzian theories; the flagged Jacobian error in Eq. (26) is a false alarm, but the circle results rest on imported and under-derived steps. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the basic functional integral $E_\sigma(u,v)$ of eq. (24), the integral of the measure over $\mathrm{Diff}^1_+([0,1])$ with delta functions fixing the endpoint derivatives $\phi'(0)=u$, $\phi'(1)=v$; its explicit closed form (28) is the one formula that all later results call upon. The splitting rule (26)-(27) is the mechanism that propagates this input: after changing variables from the two half-interval diffeomorphisms to intermediate values $z_i=\phi(t_i)$ and derivative variables $x_0,y_i,x_1$, a functional integral becomes an ordinary multiple integral weighted by a product of $E_\sigma$ factors with rescaled variances. The quasi-invariance identity (42), with the one-parameter family $g_\alpha$ of eq. (43), is the second mechanism: it converts the singular $\alpha=\pi$ limit that defines the circle integral into a regulated integral $J(\alpha)$, and the renormalized functional integral is the limit (61) of the ratio of $J(\alpha)$ to the regularized volume of $\mathrm{SL}(2,\mathbb{R})$.
What would settle it
Evaluate the two-point function (70) numerically at fixed $\sigma$ and $t_1$ by direct quadrature, and compare with an independent computation of the same Schwarzian two-point function obtained by a spectral decomposition over eigenstates; any disagreement beyond numerical error would disprove the claimed equality of the functional integral and its ordinary-integral reduction. As a more local check, substitute the closed form (28) into both sides of the convolution identity (29) at an arbitrary split point $t_*$; equality for all $u,v,t_*$ is necessary for the splitting rule to be self-consistent.
Extended reading notes
Core claim
The paper's central claim is that functional integration over $\mathrm{Diff}^1_+(S^1)/\mathrm{SL}(2,\mathbb{R})$ for Schwarzian-type integrands is reducible to ordinary integration. The reduction has three parts. First, the measure $\mu_\sigma$ on $\mathrm{Diff}^1_+([0,1])$ is identified with the Wiener measure through the substitution $\phi(t)=\int_0^t e^{\xi(\tau)}d\tau/\int_0^1 e^{\xi(\eta)}d\eta$, eq. (9). Second, splitting the interval at every argument of the integrand converts any integral of the form (13), or its $k$-point generalization (27), into an ordinary multiple integral whose only functional input is the basic integral $E_\sigma(u,v)$ of eq. (24), evaluated in closed form in eq. (28). Third, for the circle the paper shows that the integration space factorizes as $\mathrm{SL}(2,\mathbb{R})\times\mathrm{Diff}^1_+(S^1)/\mathrm{SL}(2,\mathbb{R})$ for invariant integrands, and defines the renormalized integral (61) as the $\alpha\to\pi-0$ limit of a regulated integral divided by the regularized $\mathrm{SL}(2,\mathbb{R})$ volume. The resulting two-point correlator (70) and four-point correlators (77) are finite ordinary multiple integrals. The paper's distinctive physical conclusion is that, over the circle, gluing the interval ends destroys the Markov property of the underlying Wiener process: every part of the circle contributes to a given correlator, and neither the time-ordered nor the out-of-time-ordered four-point function factorizes into two two-point functions, unlike the real-line correlators of section III.
Load-bearing premise
The whole calculus stands on the imported closed form of the basic integral $E_\sigma(u,v)$ in (28), plus the asserted cancellation of the $(\pi-\alpha)^{-1}$ singularities in the numerator and denominator of the renormalization ratio (61); if either of these gives way, the finite ordinary-integral representations of the correlators do not follow.
Editorial extensions
If this is right
- Any $n$-point Schwarzian correlation function over the circle can be written as an ordinary multiple integral of the same type, so numerical evaluation becomes a finite-dimensional quadrature problem.
- The two-point function $G^n_2(0,t_1)$ in (70) is finite and explicitly computable for all $0<t_1<1$ once the two $\theta$-integrals and the $z_1$-integral are evaluated.
- The time-ordered and out-of-time-ordered four-point functions over the circle, given in (77), are not products of two two-point functions; this distinguishes the circle theory from the real-line theory of section III.
- Regions of the circle beyond the operator positions contribute to the correlators, so the Markov property of the Wiener representation is lost after gluing the ends; the present feels the future.
- For $\mathrm{Diff}^1_+(\mathbb{R})$, the same method reproduces the known two- and four-point correlators, showing that the line and circle prescriptions define different theories rather than different regularizations of the same one.
Reading between the lines
- A direct numerical self-check follows from the convolution rule (29): substituting the closed form (28) into both sides at an arbitrary split point $t_*$ should give equality for all $u,v$; failure would expose an inconsistency in the splitting calculus. (Editorial extension.)
- The nonfactorization of the out-of-time-ordered four-point function suggests that a chaos diagnostic computed from this circle theory may not be reducible to two-point data; one could extract the Lyapunov exponent directly from the full expression (77) and compare it with the two-point-based bound. (Editorial extension.)
- The same reduction should extend to higher-point correlators and to Schwarzian actions with additional local terms built from $\phi'$, yielding analogous finite multiple integrals; this is a testable prediction of the method, not a result proven in the paper. (Editorial extension.)
- The sharp difference between the line and circle results implies that computations in Schwarzian quantum mechanics must specify both the diffeomorphism group and the end-gluing prescription before comparison with spectral or gravitational results. (Editorial extension.)
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper develops a calculus for functional integrals of Schwarzian type over the groups Diff^1_+([0,1]), Diff^1_+(R), and Diff^1_+(S^1). The central technical step is the substitution of the Wiener measure (Section II), which reduces functional integrals over Diff^1_+([0,1]) with integrands depending on finitely many values of the diffeomorphism and its derivative to ordinary multiple integrals involving a single basic integral E_sigma(u,v), Eq. (24), whose closed form (28) is taken from the authors' earlier work [2]. Section III applies the method to Diff^1_+(R) and reproduces two- and four-point correlators previously obtained in [26], [28], and [42]. Sections IV-V introduce a regularization and a renormalization prescription for the SL(2,R) zero modes of the circle theory, and Section VI presents the resulting finite two- and four-point correlation functions over Diff^1_+(S^1)/SL(2,R) as ordinary multiple integrals. The paper claims that the circle correlators differ structurally from the real-line ones, in particular that neither the time-ordered nor the out-of-time-ordered four-point function factorizes into a product of two two-point functions.
Significance. If the renormalization procedure and the imported integral (28) are accepted, the paper provides a direct functional-integral framework that reduces Schwarzian correlators on the circle to explicit ordinary multiple integrals. The real-line sector is a genuine strength: the derivation is detailed and reproduces known results from independent approaches, providing an external check. The paper is also explicit about the measure-theoretic input, namely quasi-invariance under Diff^3_+(S^1). I checked the change of variables in Section II: the determinant of (22)-(23) is [t1(1-t1)]^2 y1/[z1^3(1-z1)^3], which is exactly the factor used in Eq. (26); the stress-test concern about a missing y1 factor does not land. The main weaknesses are that the renormalized limit (61) is not actually computed, the key integral (28) is imported without proof, and the non-factorization claim is asserted rather than demonstrated. These issues are load-bearing for the paper's central new claims, so the manuscript needs revision before the circle-sector results can be regarded as established.
major comments (3)
- [Section V, Eqs. (61)-(64)] The renormalized correlator is defined as lim_{alpha -> pi^-} J(alpha)/V_SL(2,R)(alpha), and it is asserted that the (pi-alpha)^{-1} singularities in numerator and denominator cancel. However, the paper never extracts the leading singularity of J(alpha), nor does it compute the constant ratio that remains after the cancellation. The finite values in Eqs. (65), (70), and (77) therefore do not follow from the displayed derivation: one needs the alpha -> pi expansion of the regularized integral (63), or an explicit citation of the result in [2] that supplies the missing residue. Because the circle correlators are the main new output of the paper, this gap is load-bearing and should be fixed.
- [Section VI, Eqs. (74)-(77)] The paper concludes that neither the TO nor the OTO four-point correlation function factorizes into a product of two two-point functions. The displayed integrands are indeed not literally products of the corresponding two-point integrands, but this does not rule out a possible factorization after the multiple integrations are performed. Since the non-factorization claim is presented as one of the main physical results of the circle theory, the authors should provide a proof, or at least quantitative numerical evidence at representative parameter values, that the final integrals do not satisfy the product relation.
- [Section II, Eq. (28)] The basic integral E_sigma(u,v) is imported from the authors' previous paper [2] without re-derivation. Every subsequent reduction, including the general rule (27) and the circle correlators (65), (70), (74), and (77), depends on this closed form. Given the manuscript's claim to be mathematically rigorous and to contain no unproved conjectures, the derivation of (28) should either be reproduced in the present paper or stated as an imported theorem with a precise reference to the proof in [2]. As written, the central reduction rests on an input whose proof is not available in the manuscript.
minor comments (5)
- [Section II, before Eq. (26)] The displayed Jacobian appears to be misprinted: the exponent of [z1(1-z1)] should be -3, with y1 in the numerator, matching the correct determinant computed from (22)-(23). The subsequent formula (26) is consistent with the correct Jacobian.
- [Sections V and VI] The word 'nominator' should be 'numerator' in the sentences about cancellation of singularities.
- [Section VI, Eq. (65)] The notation G_n^2 is introduced without definition; in Section III the same symbol G_2 is used for the n=2 case. Please define the index n explicitly when G_n^2 first appears.
- [References] Reference [68] is described as unpublished work; the text later gives its arXiv number (1811.11863v3), which should be included in the reference itself.
- [Section II, Eq. (29)] The convolution identity for E_sigma is stated without proof. Since it is used to justify independence of the splitting point, a one-line derivation from (26) would improve readability.
Circularity Check
No circularity found: the circle correlators are derived from the measure rules plus the independently anchored formula (28), and the Jacobian objection is a correctness matter, not a circular reduction.
full rationale
The derivation chain is not circular. In Section II the general rules (26) and (27) are obtained by a change of variables from the interval-wise Wiener representation (9)-(17), not by assuming the correlators they later produce. The one imported non-elementary input is the closed form (28) for the basic functional integral E_sigma(u,v), taken from the authors' previous paper [2]. This is a parameter-free exact formula, and it is not defined in terms of the Section VI outputs; moreover, Section III uses (28) to reproduce the known real-line correlators of [26], [28], and [42], so the imported formula has an external anchor. The quasi-invariance results of [65]-[67] and [71] are separate published mathematical theorems about measures on diffeomorphism groups; they do not encode the claimed factorization properties of the four-point functions. The factorization (59)-(60) is argued in the text rather than merely imported from [73]. No fitted parameter is renamed as a prediction, no ansatz is smuggled in via citation, and no uniqueness claim by the authors is used to force a choice. The claims that the TO and OTO four-point functions are not products of two-point functions follow from the explicit multiple-integral expressions (74)-(77), which are outputs of the calculation rather than inputs. The skeptic's Jacobian objection to (26), if correct, would be a mathematical error in the reduction, and the asserted cancellation of (pi-alpha)^-1 singularities in (61)-(64) is not demonstrated; both are correctness risks, not circular equivalences, so under this pass's remit they do not raise the circularity score.
Assumptions & free parameters
assumptions (3)
- domain assumption The measure mu_sigma on Diff^1_+(S^1) exists and is quasi-invariant under Diff^3_+(S^1), with Radon-Nikodym derivative given by eq. (42).
- domain assumption The basic functional integral E_sigma(u,v) has the explicit closed form (28).
- ad hoc to paper The renormalized correlation function is defined by the limit of J(alpha)/V_{SL(2,R)} as alpha tends to pi from below, eq. (61).
Cite this review
Pith. "Pith review of Schwarzian functional integrals calculus." pith.science (2026). https://pith.science/paper/TO4MQ5XM
@misc{pith2026190810387,
author = {Pith},
title = {Pith review of: Schwarzian functional integrals calculus},
year = {2026},
howpublished = {\url{https://pith.science/paper/TO4MQ5XM}},
note = {Machine review of arXiv:1908.10387}
}
abstract
We derive the general rules of functional integration in the theories of Schwarzian type, thus completing the elaboration of Schwarzian functional integrals calculus initiated in \cite{(BShExact)}, \cite{(BShCorrel)}. Our approach is mathematically rigorous and does not contain any unproved conjectures. It is based on the analysis of the properties of the measures on the groups of diffeomorphisms, and does not appeal for the experience from other physical models. Its great merit consists in reducing a problem of functional integration to that of the only functional integral (\ref{E}) that is calculated explicitly with the result written in the form of the ordinary integral. We evaluate two-point and four-point correlation functions defined as functional integrals over the groups $Diff^{1}_{+}(\textbf{R}) $ and $Diff^{1}_{+}(S^{1})\,,$ and discuss the difference between the results in the two cases.
Reference graph
Works this paper leans on
-
[26]
and in the present paper obtained in the different ways have exa ctly the same behaviour. Thus we reproduce the results for correlation functions obtained in [26], although in a slightly different form. 12 IV. QUASI-INV ARIANCE OF THE MEASURE AS A KEY TO FUNCTIONAL INTEGRA TION As an important step in evaluating functional integrals over the gro up of diffeo...
-
[28]
D. Bagrets, A. Altland and A. Kamenev, Sachdev-Ye-Kitaev Model as Liouville Quantum Mechanics, Nucl. Phys. B 911 (2016) 191, [arXiv:1607.00694]
arXiv 2016
-
[42]
Rosenhaus, An introduction to the SYK model, arXiv:1807.03334
V. Rosenhaus, An introduction to the SYK model, arXiv:1807.03334
-
[2]
V. V. Belokurov and E. T. Shavgulidze, Correlation functions in the Schwarzian theory, JHEP 11 (2018) 036 , [arXiv:1804.00424]
work page Pith review arXiv 2018
-
[65]
V. V. Belokurov and E. T. Shavgulidze, Unusual view of the Schwarzian theory, Mod. Phys. Lett. A 33 (2018) 1850221, [arXiv:1806.05605]
work page Pith review arXiv 2018
-
[67]
E. T. Shavgulidze, An example of a measure quasi-invariant with respect to the a ction of a group of diffeomorphisms of the circle, Funct. Anal. Appl. 12 (1978) 203
work page 1978
-
[1]
V. V. Belokurov and E. T. Shavgulidze, Exact solution of the Schwarzian theory, Phys. Rev. D 96 (2017) 101701(R), [arXiv:1705.02405]
work page Pith review arXiv 2017
-
[3]
S. Sachdev and J. Ye, Gapless spin fluid ground state in a random quantum Heisenber g magnet, Phys. Rev. Lett. 70 (1993) 3339, arXiv:cond-mat/9212030 [cond-mat]
arXiv 1993
Show all 77 references
-
[4]
Kitaev, Hidden correlations in the Hawking radiaion and thermal noi se, Talk at KITP , http://online.kitp.ucsb.edu/online/joint98/kitaev/, Febrary, 2015
A. Kitaev, Hidden correlations in the Hawking radiaion and thermal noi se, Talk at KITP , http://online.kitp.ucsb.edu/online/joint98/kitaev/, Febrary, 2015. 23
2015
-
[5]
A. Kitaev, A simple model of quantum holography, Talks at KITP, http://online.kitp.ucsb.edu/online/entangled15/kit aev/ and http://online.kitp.ucsb.edu/online/entangled15/kitaev2/, April and May, 2015
2015
-
[6]
Maldacena and D
J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94(2016) 106002, [arXiv:1604.07818]
2016 arXiv
-
[7]
Polchinski and V
J. Polchinski and V. Rosenhaus, The spectrum in the Sachdev-Ye-Kitaev model, JHEP 04 (2017) 001, [arXiv:1601.06768]
2017 arXiv
-
[8]
Jevicki, K
A. Jevicki, K. Suzuki and J. Yoon, Bi-local holography in the SYK model, JHEP 07 (2016) 007, [arXiv:1603.06246]
2016 arXiv
-
[9]
Jevicki and K
A. Jevicki and K. Suzuki, Bi-local holography in the SYK model: perturbations, JHEP 11 (2016) 046, [arXiv:1608.07567]
2016 arXiv
-
[10]
D. J. Gross and V. Rosenhaus, A Generalization of Sachdev-Ye-Kitaev, JHEP 02 (2017) 093, [arXiv:1610.01569]
2017 arXiv
-
[11]
D. J. Gross and V. Rosenhaus, The bulk dual of SYK: cubic couplings, JHEP 05 (2017) 092, [arXiv:1702.08016]
2017 arXiv
-
[12]
S.R. Das, A. Jevicki and K. Suzuki, Three dimensional view of the SYK/AdS duality, JHEP 09 (2017) 017, [arXiv:1704.07208]
2017 arXiv
-
[13]
Das, A.Ghosh, A
S.R. Das, A.Ghosh, A. Jevicki and K. Suzuki, Space-time the SYK model, JHEP 07 (2018) 184, [arXiv:1712.02725]
2018 arXiv
-
[14]
Almheiri and J
A. Almheiri and J. Polchinski, Models of AdS2 backreaction and holography, JHEP 11 (2015) 014, [arXiv:1402.6334]
2015 arXiv
-
[15]
Jensen, Chaos in AdS2 holography, Phys
K. Jensen, Chaos in AdS2 holography, Phys. Rev. Lett. 117 (2016) 111601, [arXiv:1605.06098]
2016 arXiv
-
[16]
Maldacena, D
J. Maldacena, D. Stanford and Z. Yang, Conformal symmetry and its breaking in two dimen- sional Nearly Anti-de-Sitter space, PTEP 2016, no. 12, 12C104 (2016), [arXiv:1606.01857]
2016 arXiv
-
[17]
Engelsoy, T
J. Engelsoy, T. G. Mertens, and H. L. Verlinde, An investigation of AdS2 backreaction and holography, JHEP 07 (2016) 139, [arXiv:1606.03438]
2016 arXiv
-
[18]
Cotler, X.-L
Y. Cotler, X.-L. Qi, and D. Stanford, Local criticality, diffusion and chaos in generalized Sachdev-Ye-Kitaev models, JHEP 05 (2017) 125, [arXiv:1609.07832]
2017 arXiv
-
[19]
Cotler, G
J.S. Cotler, G. Gur-Ari, M. Hanada, J. Polchinski, P. Sa ad, S.H. Shenker, D. Stanford, A. Streicher and M. Tezuka, Black Holes and Random Matrices, JHEP 05 (2017) 118, [arXiv:1611.04650]. 24
2017 arXiv
-
[20]
A. M. Garcia-Garcia and J. J. M. Verbaarschot, Analytical Spectral Density of the Sachdev- Ye-Kitaev Model at finite N, Phys. Rev. D 96 (2017) 066012, [arXiv:1701.06593]
2017 arXiv
-
[21]
G. J. Turiaci and H. L. Verlinde, Towards a 2d QFT Analog of the SYK Model, JHEP 10 (2017) 167, [arXiv:1701.00528]
2017 arXiv
-
[22]
Mandal, P
G. Mandal, P. Nayak and S. R. Wadia, Coadjoint orbit action of Virasoro group and two-dimentional quantum gravity dual SYK/tensor model s, JHEP 11 (2018) 046, [arXiv:1702.04266]
2018 arXiv
-
[23]
Kitaev and S
A. Kitaev and S. J. Suh, The soft mode in the Sachdev-Ye-Kitaev model and its gravity d ual, JHEP 05 (2018) 183, [arXiv:1711.08467]
2018 arXiv
-
[24]
Sarosi, AdS2 holography and the SYK model, PoS (Modave2017)001 (2018), [arXiv:1711.08482]
G. Sarosi, AdS2 holography and the SYK model, PoS (Modave2017)001 (2018), [arXiv:1711.08482]
2018 arXiv
-
[25]
Maldacena, S.H
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08(2016) 106, [arXiv:1503.01409]
2016 arXiv
-
[27]
Bagrets, A
D. Bagrets, A. Altland and A. Kamenev, Power-law out of time order correlation functions in the SYK model, Nucl. Phys. B 921 (2017) 727, [arXiv:1702.08902]
2017 arXiv
-
[29]
D. J. Gross and V. Rosenhaus, All point correlation functions in SYK, JHEP 12 (2017) 148, [arXiv:1710.08113]
2017 arXiv
-
[30]
T. G. Mertens, G. J. Turiaci and H. L. Verlinde, Solving the Schwarzian via the Conformal Bootstrap, JHEP 08 (2017) 136, [arXiv:1705.08408]
2017 arXiv
-
[31]
T. G. Mertens, The Schwarzian Theory - Origins, JHEP 05 (2018) 036, [arXiv:1801.09605]
2018 arXiv
-
[32]
Stanford and E
D. Stanford and E. Witten, Fermionic Localization of the Schwazian Theory, JHEP 10 (2017) 008, [arXiv:1703.04612]
2017 arXiv
-
[33]
Blommaert, T
A. Blommaert, T. G. Mertens and H. Verschelde, The Schwarzian Theory - A Wilson Line Perspective, arXiv:1806.07765
-
[34]
H.T Lam, T. G. Mertens, G. J. Turiaci and H. L. Verlinde, Shockwave S-matrix from Schwarzian Quantum Mechanics, arXiv:1804.09834
-
[35]
Aref’eva and I
I. Aref’eva and I. Volovich, Notes on the SYK model in real time, Theor. Math. Phys. 25 197(2018) 1650, [arXiv:1801.08118]
2018 arXiv
-
[36]
Gaikwad, L
A. Gaikwad, L. K. Joshi, G. Mandal and S. R. Wadia, Holographic dual to charged SYK from 3D Gravity and Chern-Simons, arXiv:1802.07746
-
[37]
D. A. Roberts, D. Stanford and A. Streicher, Operator grouth in the SYK model, arXiv:1802.02633
-
[38]
Nayak, A
P. Nayak, A. Shukla, R. M. Soni, S. P. Trivedi and V. Visha l, On the Dynamics of Near- Extremal Black Holes , JHEP 09 (2018) 048, [arXiv:1802.09547]
2018 arXiv
-
[39]
Gur-Ari, R
G. Gur-Ari, R. Mahajan and A. Vaezi, Does the SYK model have a spin glass phase?, JHEP 11 (2018) 070, [arXiv:1806.10145]
2018 arXiv
-
[40]
Saad, S.H
P. Saad, S.H. Shenker and D. Stanford, A semiclassical ramp in SYK and in gravity, arXiv:1806.06840
-
[41]
Cotler and K
J. Cotler and K. Jensen, A theory of reparametrization for AdS2 gravity, arXiv:1808.03263
-
[43]
Banerjee, A
A. Banerjee, A. Kundu and R. P. Poojary, String, Brains, Schwarzian Action and Maximal Chaos, arXiv:1809.02090
-
[44]
are written as the sums over the eigenstates of the Hamiltonian of the certain physical model. At the special limiting values of parameters, the action of the model is reduced to 11 Considering t as a time variable, one could say that ”the present” is influenced by ” the future...
-
[45]
Kitaev and S
A. Kitaev and S. J. Suh, Statistical mechanics of a two-dimensional black hole, JHE P 05 (2019) 198, [ arXiv:1808.07032]
2019 arXiv
-
[46]
Yang, The Quantum Gravity Dynamics of Near Extremal Black Holes, JH EP 05 (2019) 205, [ arXiv:1809.08647]
Zh. Yang, The Quantum Gravity Dynamics of Near Extremal Black Holes, JH EP 05 (2019) 205, [ arXiv:1809.08647]
2019 arXiv
-
[47]
Berkooz, M
M. Berkooz, M. Isachenkov, V. Narovlansky and G. Torren ts, Towards a full solution of the large N double-scaled SYK model , arXiv:1811.02584
-
[48]
Aref’eva, M
I. Aref’eva, M. Khramtsov, M. Tikhanovskaya and I. Volo vich, Replica-nondiagonal solutions in the SYK model , arXiv:1811.04831
-
[49]
H. Wang, D. Bagrets, A. L. Chudnovskiy and A. Kamenev, On the replica structure of Sachdev- Ye-Kitaev model, arXiv:1812.02666
-
[50]
Blommaert, T
A. Blommaert, T. G. Mertens, and H. Verschelde, Fine Structure of Jackiw-Teitelboim Quan- tum Gravity, arXiv:1812.00918
-
[51]
A. M. Garcia-Garcia, T. Nosaka, D. Rosa, and J. J. M. Verb aarschot, Quantum chaos transi- tion in a two-site SYK model dual to an eternal traversable wo rmhole, arXiv:1901.06031
1901 arXiv
-
[52]
Sachdev, Universal low temperature theory of charged black holes wit h AdS 2 horisons, arXiv:1902.04078
S. Sachdev, Universal low temperature theory of charged black holes wit h AdS 2 horisons, arXiv:1902.04078
1902 arXiv
-
[53]
Blommaert, T
A. Blommaert, T. G. Mertens, and H. Verschelde, Clocks and Rods in Jackiw-Teitelboim Quantum Gravity, arXiv:1902.11194. 26
1902 arXiv
-
[54]
L. V. Iliesiu, S. S. Pufu, H. Verlinde, and Y. Wang, An exact quantization of Jackiw-Teitelboim gravity, arXiv:1905.02726
1905 arXiv
-
[55]
P. Saad, S. H. Shenker, and D. Stanford, JT gravity as a matrix integral, arXiv:1903.11115
1903 arXiv
-
[56]
Stanford and E
D. Stanford and E. Witten, JT Gravity and the Ensembles of Random Matrix Theory, arXiv:1907.03363
1907 arXiv
-
[57]
D. J. Gross, J. Kruthoff, A. Rolph, and E. Shaghoulian, T T in AdS2 and Quantum Mechanics, arXiv:1907.04873
1907 arXiv
-
[58]
Weil, L’integration dans les groupes topologiques et ses applicat ions, Actual
A. Weil, L’integration dans les groupes topologiques et ses applicat ions, Actual. Scient. et Ind. , 869, Paris: Herman, 1940
1940
-
[59]
A. M. Polyakov, Gauge Fields and Strings, Contemp. Concepts Phys. 3 (1987)
1987
-
[60]
Alekseev and S
A. Alekseev and S. L. Shatashvili, Path integral quantization of the coadjoint orbits of the Virasoro group and 2D gravity, Nucl. Phys. B 323 (1989) 719
1989
-
[61]
Hui-Hsiung Kuo, Gaussian Measures in Banach Spaces, Springer, Berlin-Heid elberg-NY, 1975
1975
-
[62]
Teschner, Liouville theory revisited, Class
J. Teschner, Liouville theory revisited, Class. Quant. Grav. 18 (2001) R153, [ hep-th/0104158]
2001 arXiv
-
[63]
Comtet and P
A. Comtet and P. J. Houston, Effective Action on the Hyperbolic Plane in a Constant Externa l Field, J. Math. Phys. 25 (1985) 185
1985
-
[64]
Comtet, On the Landau Levels on the Hyperbolic Plane, Ann
A. Comtet, On the Landau Levels on the Hyperbolic Plane, Ann. Phys. 173 (1987) 185
1987
-
[66]
V. V. Belokurov and E. T. Shavgulidze, Polar decomposition of the Wiener measure: Schwarzian theory versus conformal quantum mechanics, Theo r. Math. Phys. 200(3) (2019) 1324, [arXiv:1812.04039]
2019 arXiv
-
[68]
E. T. Shavgulidze, A measure quasi-invariant with respect to the action of a gro up of diffeo- morphisms of a finite-dimensional manifold, Sov. Math. Dokl . 38 (1988) 622
1988
-
[69]
E. T. Shavgulidze, Some Properties of Quasi-Invariant Measures on Groups of Diff eomor- phisms of the Circle, Russ. J. Math. Phys. 7 (2000) 464
2000
-
[70]
V. V. Belokurov and E. T. Shavgulidze, Simple rules of functional integration in the Schwarzian theory: SYK correlators, arXiv:1811.11863v3
-
[71]
Lang, ” SL2(R)”
S. Lang, ” SL2(R)”. Addison-Wesley Publishing. 1975. 27
1975
-
[72]
A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series. V. 1. Elementary Functions, Gordon Breach: New York-London , 1986
1986
-
[73]
L. A. Shepp, Radon - Nikodym Derivatives of Gaussian measures, Ann. Math . Statistics, 37 (1966) 321
1966
-
[74]
V. V. Belokurov and E. T. Shavgulidze, Extraordinary Properties of Functional Integrals and Groups of Diffeomorphisms, Phys. Part. Nucl. 48 (2017) 267
2017
-
[75]
V. V. Belokurov and E. T. Shavgulidze, Functional integration over the factor-space Dif f1 +(S1)/SL (2, R), arXiv:1912.07841
1912 arXiv
-
[76]
L. V. Iliesiu, J. Kruthoff, G.J. Turiaci and H. Verlinde, JT gravity at finite cutoff, arXiv:2004.07242
2004 arXiv
-
[77]
Stanford and Zh
D. Stanford and Zh. Yang, Finite-cutoff JT Gravity and self-avoiding loops, arXiv:2004.08005. 28
2004 arXiv
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