Warped Schwarzian theory

We consider the (twisted) warped Virasoro group Diff(S)⋉ C∞(S1) in the presence of its three cocycles. We compute the Kirillov-Kostant-Souriau symplectic 2-form on coadjoint orbits. We then construct the Euclidean action of the ‘warped Schwarzian theory’ associated to the orbit with SL(2,R)×U(1) stabilizer as the effective theory of the reparametrization over the base circle and evaluate the corresponding one-loop-exact path integral.


Introduction
The Sachdev-Ye-Kitaev (SYK) model [1][2][3] has attracted much attention recently as a toy model for AdS/CFT in lowest possible dimensions. It is an interacting statistical quantum mechanical model of N Majorana fermions with random couplings. In the regime of large coupling (low temperature) this model is perturbatively (in the 1/N-expansion) dual to a simple theory of two-dimensional dilaton-gravity known as the Jackiw-Teitelboim (JT) gravity [4,5]. 1 A universal feature emerging for all variety of these quantum mechanical models is their low energy dynamics which is governed by a solvable Euclidean theory known as the 'Schwarzian theory' [8] which turns out to be the appropriate boundary term defining our two dimensional dilaton-gravity theory. The dynamical field in this theory is a quasi-periodic field f (τ ) with τ being the imaginary time on the thermal circle 0 < τ < β and performing as the reparametrization on the thermal circle Diff(S 1 ). This effective theory is by construction defined on the symplectic manifold Diff(S 1 )/SL(2,R) or Diff(S 1 )/U(1) and its quantum mechanical realization has been shown to be one-loop exact [9]. These symplectic manifolds are technically known as coadjoint orbits of the Virasoro group [10][11][12][13]. The coadjoint orbit method [14][15][16][17] which we will benefit from in this work gives a geometric interpretation of the coadjoint representation of Lie groups and provides a systematic to construct field theory actions on a given Lie group orbit. For reviews and other applications and employment of the orbit method see [18][19][20][21][22][23][24][25][26].
One of the consequences of the exactness of the path integral of the Schwarzian theory is that the Hamiltonian function associated to the imaginary time evolution is in fact proportional to the Schwarzian action itself. In this paper we use this consequence as a base for constructing the 'warped Schwarzian theory' in section 4 based upon the coadjoint orbits of the warped Virasoro group [27] which is reviewed and developed for our application in section 2. We also calculate the Kirillov-Kostant-Souriau symplectic 2-form [14][15][16][17] on the corresponding coadjoint orbits in section 3 and then use it as a measure to evaluate the one-loop-exact path-integral of the theory in section 5. We then discuss the spectrum of the theory in terms of the density of states in section 6.

Warped Virasoro symmetry
The Virasoro group is the unique nontrivial central extension of the group of reparametrization on the circle. It has several generalizations but a simple extension is to make a semi-direct product of it with functions on the circle. We denote it as 'warped Virasoro group', a terminology inferred from the physics literature [28,29] and mostly studied in the context of two dimensional warped conformal field theories (WCFTs). 2 Here in this work we are interested in the one dimensional (Euclidean time) realization of this symmetry group. Recently it has been shown that the warped Virasoro algebra is the underling symmetry for the low energy effective action of the complex SYK model [33].

Coadjoint orbits of the warped Virasoro group
The warped Virasoro groupĜ is the universal central extension of the Lie group G of all orientation preserving diffeomorphism of the circle acting naturally on smooth functions of the circle; For f ∈ Diff(S 1 ) and σ ∈ C ∞ (S 1 ) the group operation is (2.15) as 3 This group operation is inferred from the semi-direct product structure of the group [27,34]. The corresponding Lie algebraĝ, called the warped Virasoro algebra is defined as the unique (up to isomorphism) nontrivial central extension of the Lie algebra g of first order differential operators on with ǫ(τ + β) = ǫ(τ ) and σ(τ + β) = σ(β) where β is the circumference of S 1 . This Lie algebra admits three central extensions [30,[35][36][37]. We denote each element ofĝ = g ⊕ R 3 by a pair (v(τ ), a) with a = a 1 e 1 + a 2 e 2 + a 3 e 3 ∈ R 3 where e i can be considered as orthonormal basis of R 3 . The commutator in the centrally extended Lie algebraĝ is defined as where we have introduced the notation S 1 ≡ β 0 dτ . Note that at this point β is just a parameter specifying the size of the circle S 1 , later in our physical setup it will denote the inverse temperature of our quantum mechanical system on S 1 .
Corresponding to the adjoint vectors v(τ ) ∈ g, we have coadjoint vectors b(τ ) ∈ g * which maps adjoint vectors to numbers. In the case of warped Virasoro algebra these covectors can be represented in terms of a quadratic density and a one-form on the circle; (2.7) A coadjoint vector of the warped Virasoro algebraĝ is a pair (2.8) The groupĜ acts on vectors by the adjoint action Ad (f,σ) and on covectors by the coadjoint action Ad * (f,σ) . The pairing (2.8) should be invariant under the group action so the coadjoint representation Ad * of the groupĜ is defined by 4 The coadjoint action leaves the central elements invariant but transforms the b(τ ) covector into a new covector in the coadjoint orbit where is the natural action of the group G =Diff(S 1 )⋉C ∞ (S 1 ) on the dual space of its Lie algebra g * while S(f, σ) term can be considered as the contribution of the central extension of the group;

Vacuum orbit on the cylinder
Each coadjoint orbit is parametrized by f and σ and identified uniquely w.r.t. an orbit repre- In order to determine the orbit representative we visualize the warped Virasoro group as the following infinitesimal transformation on the R 2 plane with coordinates (τ, x); , and x → x + σ(τ ) . (2.14) Upon integrating the infinitesimal transformations (2.14), one obtains finite diffeomorphisms of the plane R 2 given by . The reparametrization f (τ ) and the translation g(τ ) are respectively an orientation preserving conformal transformation and an arbitrary function on a line forming the group Diff(R) ⋉ C ∞ (R). In our physics setup, τ is parametrizing the temperature of the system whereas the pure imaginary line x (not to be confused by any spacetime coordinate) parametrizes a direction along a new chemical potential µ as we will see later on.
Associated to these group elements, the pair of currents (T (τ ), P (τ )) generate infinitesimal general coordinate transformations along Euclidean time τ and translation along x as in (2.14) respectively. This group admits three central extensions (c, κ, k), therefore these currents are anomalous which can be observed in their transformation under the coadjoint representation of G as we briefly reviewed above and more elaborated in [27].
In order to consider the theory at finite temperature T = β −1 we replace R by S 1 with τ ∼ τ + β. In this case g(τ ) is an arbitrary function on the circle and f (τ ) is a diffeomorphism of the circle; Relative to the line, the vacuum energy on the circle is shifted. We require the vacuum on the circle to be SL(2,R)×U(1) invariant so we can determine these vacuum state values which correspond to the orbit representative uniquely by mapping between the cylinder and the plane using the following set of finite warped conformal transformation; where α is an arbitrary pure imaginary tilt [29] quantifying the chemical potential µ = iα in the WCFT context [38]. We have the following thermal identification; In our one dimensional (quantum) mechanical system one should not consider x as a real space coordinate but as the coordinate on the phase space parametrizing a new chemical potential µ.
The coadjoint representation (2.10) provide the finite transformation law of the currents T and P under a finite transformation (2.15). These transformation rules map the pair (T, P ) to a new pair (T,P ) at f (τ ) whose explicit form can be read from (2.10) by setting c 1 = c, c 2 = 4κ and c 3 = −k according to the notation in [27]; denotes the Schwarzian derivative and prime refers to derivative w.r.t. τ . The infinitesimal form of these transformation can be obtained by expanding f and g to linear order of ǫ and σ; (2.19) and setting the vacuum values on the plane to zero we obtain the vacuum values on the cylinder; If we choose the exponential map by f −1 = e −2πiτ /β we are led to the same expression as in (2.21) with β → −β. This means that the Z 2 -symmetry of interchanging between degenerate vacua |β and | − β is broken by the presence of the κ-term. This map amounts to choosing only one of these vacuum on the cylinder and as a consequence our final result will be sensitive to this. This symmetry is however restored if we simultaneously change κ → −κ. The linearin-τ choice of g(τ ) in (2.17) has essentially been made such that the (T, P ) are transformed only by constant values which amounts to a shift of only zero modes. It is important to note that for having a well-defined quantum theory defined on the circle we have α and g(τ ) being pure imaginary. Otherwise the Energy ground state value T vac will have imaginary part on the circle. This is consistent with the holographic constraints on WCFTs [29,39].
By assigning vacuum values (2.21) to (T,P ) in (2.19) we generate all vacuum-representatives of the SL(2,R)×U(1) invariant warped Virasoro coadjoint orbit on the cylinder; with the property tan π β f ; τ = {f ; τ } + 2π 2 β 2 f ′2 . Subject to boundary conditions (2.16) on fields f (τ ) and g(τ ) the Fourier mode generators of (2.22) on the cylinder satisfy the twisted warped Virasoro algebra; β inτ , and the fact that It is clear form the algebra (2.24) that the global part of this symmetry algebra is four dimensional corresponding to the invariance of the vacuum under (L ±1,0 , P 0 ) or under (L 1,0 , P 0,−1 ).

Symplectic form
A coadjoint orbit M ξ with ξ ∈ g * is a symplectic submanifold of M which carries a natural symplectic structure ω ξ denoted as Kirillov-Kostant-Souriau symplectic form [14][15][16] defined by where X(ξ) is the vector field at ξ ∈ g * generated by the coadjoint action of X ∈ g; Let us apply the definition (3.2) to the coadjoint action of the warped Virasoro algebraĝ on its dual spaceĝ * . Using the vectors (2.3) and covectors (2.7) we have; where δT and δP are defined in (2.20).
Now by having the knowledge of section 2.1 we are ready to calculate the value of the symplectic form ω (3.1) for the warped Virasoro coadjoint orbit with constant representative at the point (b, c) ∈ĝ * on the pair of vector fields X 1 (ξ) and X 2 (ξ) on the orbit, where in (3.4) we used the covector (2.7) and the vector (2.3) and used the pairing (2.8). In (3.5) the τ -dependence of variables is implicit and for brevity we introducedg(τ ) = g(τ ) + αf (τ ).
In transition from (3.4) to (3.5) we inserted the expression for P (τ ) and T (τ ) from (2.22) for the vacuum orbit on the cylinder which is globally SL(2,R)×U(1) invariant. Now we rephrase (3.5) in the form language; Here d acts on fields and not on coordinates and it essentially symbolizes a one-form on the coadjoint orbit. The symplectic form in (3.6) is closed dω = 0 by construction which can also be examined directly. It is invariant under the action of the twisted warped group (2.1) and it is nondegenerate on the coadjoint orbit. In the following we find the finite form of the Symplectic 2-form.
According to the group operation (2.2) it is important to note that, f as an element of the (twisted) warped Virasoro group (2.1), transforms under Diff(S 1 ) while σ transforms both under Diff(S 1 ) and C ∞ (S 1 ). So we have; Using (3.7) one can rewrite (3.6) in a more abstract way as; The first term in (3.8) is the famous Kirillov-Kostant-Souriau symplectic form for the Virasorogroup [10,11] at central charge c while the last term in (3.8) is the contribution from the infinite dimensional Heisenberg group (U(1) Kac-Moody) at level k/2 [11,17] or the k-cocycle of the warped Virasoro group. The symplectic form of the twisted warped Virasoro group has an offdiagonal κ-contribution which is shown in the middle term which to the best of our knowledge is new. If k = 0 we can diagonalize (3.8) and rewrite it in a more compact form It is essential to notice that at least classically when k = 0, we could hide the κ-contribution from the vacuum orbit into the Virasoro and the Heisenberg orbits by doing shifts in (3.10) and the coadjoint orbit does not change in this case. However the price one pays is that the field g eff unlike the fieldg is no more pure imaginary and has a real part which is 2κ k log f ′ . This field redefinition also will happen at the level of the action (4.5) and field equations (4.9). When the level k is zero this change of variable is no more valid and we should directly work with (3.6) or (3.8) at k = 0.

Pfaffian
We may evaluate the symplectic form on the coadjoint orbit around the identity element f = τ of Diff(S 1 ) ⋉ C ∞ (S 1 ) such that f (τ ) = τ + ǫ(τ ) and g(τ ) = σ(τ ) by Fourier expanding the fluctuations as; In terms of these Fourier modes the symplectic form of the coadjoint orbit at identity becomes We assume as a consequence of f being essentially a real angle and g being a pure imaginary function of S 1 . We restricted the domain of sums in (3.12) such that the symplictic form remains non-degenerate for generic values of couplings. Of course there are special values where the simplectic matrix becomes reducible and we should restrict it more. We will consider these cases separately below.
The natural volume form (or measure of the integral) in an M-dimensional symplectic manifold M with coordinates ξ 1 , · · · , ξ M which is equipped with the antisymmetric symplectic matrix ω ij is vol M = Pf(ω) dξ 1 · · · dξ M . (3.14) In our case the matrix ω ij has the following block form; The rank of matrices ω ǫǫ , ω ǫσ and ω σσ are generically 2N where N is the dimensionality of the vector space which is a large number da 1 · · · da N and is the upper limit of sums in (3.12). In order to calculate the Pfaffian of the matrix (3.15) we can factorize it as ω = BΩB T with the non-singular matrix B and use the following identity; Pf(BΩB T ) = Pf(Ω)det(B) . In our discussion this decomposition is made such that B is upper/lower triangular. We keep c being non-zero all the time, depending on other cocycles being zero or not we have the following three cases: • In the case where κ = 0 the symplectic form is reducible to with dσ n = dσ n + α dǫ n . The Pfaffian of the antisymmetric symplectic matrix becomes In this case ω ǫǫ and ω σσ are (2N − 2) × (2N − 2) and 2N × 2N matrices and both should be non-degenerate in order to have a well-defined volume. This amounts to having the full SL(2, R)×U(1) global identification on the orbit. In other words, in the absence of the κ cocycle the zero modes of the algebra which form the maximal finite subalgebra are ǫ 0,±1 and σ 0 and they form the SL(2,R)×U(1).
• In the case where k = 0, the symplectic form is also reducible and since the matrix ω σσ is identically zero the matrix ω ǫǫ does not play any role in the Pfaffian of the matrix ω; In this case it is enough to have the ω ǫσ , which is a (2N − 1) × (2N − 1) matrix, being non-degenerate for having a well-defined symplectic volume in the phase space. As a consequence in the presence of κ while k = 0 the zero modes defining our orbit are ǫ 0,1 and σ −1,0 and they obey the centrally extended so(1,1) algebra [27].
• In the most general case where all cocycles are non-zero we can find the Pfaffian using (3.12)-(3.16); where c eff = c − 24κ 2 /k. Again we have restricted the domain of n such that the determinant of the symplectic matrix is non-zero. In this case non-degeneracy of ω again leads to the SL(2, R)×U(1) as the stabilizer of the orbit.
This last result in the list is compatible with the diagonalization of the symplectic form in its finite form (3.9).

Warped Schwarzian theory
In general the Hamiltonian corresponding to the imaginary time evolution τ → τ + ǫ 0 is associated to the zero-mode of T (τ ): (4.1) Since we are considering the Euclidean theory the action is equal to its imaginary-time Hamiltonian up to normalization. This is a consequence of applying the Duistermaat-Heckman formula which means the corresponding partition function in one-loop exact [9]. This application is based on the fact that these theories arise from coadjoint orbits which are symplectic manifolds and possess a U(1) imaginary-time translation generator L 0 . In fact this fact has been used to express the exponential of the Schwarzian action as the evolution operator in the quantum theory [40]. In the most general case we consider the following one dimensional classical action; where S (c,k,κ) correspond to different contributions of the three cocycles (c, k, κ) of the twisted warped Virasoro group respectively. Before going any further for simplicity we first make the following field redefinition;g to eliminate the contribution of α terms form the action (4.2). This however changes the periodicity property in g asg(τ +β) =g(τ )+αβ. After we drop the total derivative contributions we have; If k = 0 we can always absorb the contribution of S (κ) into S (c) and S (k) up to total derivative terms, leading to the following action; with effective values defined in (3.10). As obviously seen from (4.5) the action for the nonvanishing U(1) level corresponds to the Schwarzian action of the vacuum Virasoro coadjoint orbit as diffeomorphisms of the circle with an improvement in the central charge decoupled from a free scalar g eff . We expect that the measure of the integration remains invariant under these shifts in g; as it can be considered as part of the group action on g. However the periodicity property of g will be twisted as a consequence of multiple shifts; The action (4.5) when κ = 0 has been studied extensively as the effective action of the complex SYK model. The goal in this note is to consider the consequences when κ = 0. Specially once k = 0 the shift (4.6) is no more valid.

Saddle points of the action
Upon variation of the the action (4.2) w.r.t. f andg (4.3), in general we have the following field equations; δf , c 12 In general when κ = 0 these two equations are coupled. If k = 0, one can solve the second equation forg ′ and insert it into the first equation to obtain the f -equation; c eff 12 With loss of generality we assume c eff = 0, otherwise the function f (τ ) remains arbitrary and the classical action become zero. Furthermore, f ′ > 0 and finite as a consequence of f (τ ) being an orientation-preserving diffeomorphism. Thus the equation (4.9) can be solved as tan π β f (τ ) = a/λ sin λτ + b cos λτ c/λ sin λτ + d cos λτ . (4.10) The numerator and the denominator in (4.10) has to be linearly independent so we should have ad − bc = 0. The solution to the second equation (4.8b) determines the fieldg(τ ); At finite temperature β and finite chemical potential α, the periodicity properties (2.16) and (4.7) fix integration constants as λ = πn β and B 0 = α eff . The zero temperature limit (β → ∞) of these solutions is taken by replacing tan π β f and exp 2πi β f in (4.10) and (4.11) by f while in this case λ is not quantized anymore.
The fields f and g subject to their boundary conditions are elements of the constant representative coadjint orbits of the twisted warped Virasoro group. The phase space which restricts the integration space in (5.1) is the infinite dimensional quotient space The coadjoint orbit (5.2) as a manifold is symplectic (and in this case also Kähler) and as a consequence the measure of integration in (5) is fixed using the Pfaffian of the symplectic matrix. The Duistermaat-Heckman (DH) theorem applied to symplectic manifolds endowed with a U(1) time translation generators L 0 playing the role of the action in the integral (5.1) states that the path integral is 1-loop exact; In order to evaluate the one-loop integral it is enough to expand around the classical solution to the equations of motion. Around the saddle point we can expand as f (τ ) ≃ τ + ǫ(τ ) and since g = σ • f , we have g(τ ) ≃ σ(τ ) + ǫ(τ )σ ′ (τ ). We can think of ǫ(τ ) and σ(τ ) as Goldstone modes for the broken SL(2,R) and U(1). The action (4.2) to quadratic order in ǫ(τ ) and σ(τ ) becomes S = S (0) + S (2) with; The expression for S (0) is the saddle point contribution S on-shell in (4.12) for λ = π/β as expected.The goal would be to compute the Euclidean one-loop path integral Hereσ = σ ′ + αǫ ′ . This change of variable leaves the measure of the integral invariant as σ →σ is a group action. In order to perform the path integral on the quotient space we should not include zero modes or the kernel of the symplectic from (3.12) in the measure of integration. The periodicity properties of f and g in (2.16) and (4.7) imply that ǫ(τ + β) = ǫ(τ ) and σ(τ + β) = σ(τ ). We employ the Fourier mode expansion of the fluctuations; and evaluate the quadratic effective action (5.5) to quadratic order in fluctuations around the SL(2,R)×U(1) vacuum, we find the individual contribution to each cocycle as follows; n≥2 n(n 2 − n)ǫ nσ−n + 2πiκ n≥1 n(n 2 + n)ǫ −nσn + kβ 2 n≥1 n 2 |σ n | 2 .
(5.8) whereσ n = σ n + αǫ n . Using the symplictic from (3.12) and the corresponding Pfaffian we can perform the 1-loop path integral (5.6). We find it useful to discuss cases where the twist term κ or the level k is zero separately.

Untwisted warped Schwarzian theory with non-zero level
The twistless case when κ = 0 is the same as in the complex SYK low energy carried out in [9,41,42]. The modes ǫ andσ are decoupled in both the action and the measure of the path integral and we can evaluate the 1-loop path integral by evaluating each contribution separately. Using the Pfaffian (3.18), the contribution of each piece is as follows; 5 The Gaussian integrals can easily be evaluated. In order to evaluate the infinite products we used the zeta function regularization The final answer is the product of contributions from the c-and the k-cocycles; This one-loop result matches with our expectation regarding the number of zero modes which is four in the case of warped Virasoro algebra. The full partition function in this case is

Twisted warped Schwarzian theory at level zero
Another interesting case would be to carry out the Partition function when the U(1) level is vanishing k = 0 but κ = 0. At the first glance to the action (4.2), in this case the g field plays the role of a Lagrange multiplier that can be integrated out; The path integral would then reduce to evaluating a Dirac delta functional in the integral. This is again naive as one should be cautious with the domain of the integration. In the path 5 We have d 2 ǫ n = dǫ R n dǫ I n = i 2 dǫ n dǫ −n and d 2 σ n = dσ R n dσ I n = − i 2 dσ n dσ −n .
integral (5.1), we should avoid integrating over zero modes that are globally quotiented in the integration space (5.2). Again using the mode expansion of the action (5.8) and the Pfaffian of the symplectic matrix (3.20) in this case we have where A n = cπ 2 3β n 2 (n 2 − 1) , J n = 4πκ in 2 σ R n − n 3 σ I n , K n = 4πκ n 3 σ R n + in 2 σ I n . (5.18) We evaluated the norm of the integral I 1,−1 by the doubling trick 6 ; We can also do a direct calculation by expanding ǫ −1 and σ 1 ; (5.20)

Twisted warped Schwarzian theory with non-zero level
Once κ = 0 and k = 0, it is possible to hide the κ-contribution to the quadratic action in cand k-terms where σ eff = σ + α eff ǫ − 2κ k ǫ ′ (5. 22) with the effective values c eff and α eff being defined in (3.10). A quick conclusion would be to map the problem to the previous case 5.1 where κ = 0 and k = 0 with physical quantities, c and α being modified to effective ones (3.10). However we note that the shift inσ in (5.22) is violating the holomorphicity of the σ field. In other words, the new σ eff is a complex field whose real and pure imaginary values are independent from each other. This means that the positive and negative Fourier modes (σ eff ) n are no more related to each other as they were before (3.13). So in order to keep track of all terms we preferably work with old variables ǫ andσ in the action; In this case we used; The full partition function of the warped Schwarzian theory is where α = −iµ with µ playing the role of a chemical potential to the U(1) charge.

Discussion
In order to compare the spectrum of the warped Schwarzian theory with its Schwarzian parent we separate the full partition function (6.1) as follows Z WSch (β, µ) = Z Sch (β) √ β exp 2πµκ + kµ 2 4 β . (6.1) The density of states can be determined as the inverse Laplace transform to the partition function. If we assume k < 0, and shift the zero point energy to − kµ 2 4 we have