Entanglement and chaos in warped conformal field theories
Luis Apolo Song He Wei Song Jianfei Xu Junjie Zheng


























	
	
	
	
	
	
	
	
	C[1]>p#1<fpheaderPrepared for submission to JHEP

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[a][b]Max Planck Institute for Gravitational Physics, Albert Einstein Institute, Mühlenberg 1, 14476 Golm, Germany[c]Center for Theoretical Physics and College of Physics, Jilin University, Changchun 130012, China[d]Shing-Tung Yau Center and School of Mathematics, Southeast University, Nanjing 211189, China













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§ INTRODUCTION AND SUMMARY


Warped conformal field theories (WCFTs) are two-dimensional quantum field theories invariant under warped conformal transformations, spacetime symmetries where the left and right-moving coordinates x^+ and x^-  transform as <cit.>
  x^+' = f(x^+),     x^-' = x^- + g(x^+), 
and f(x^+) and g(x^+) are two arbitrary functions. Unlike standard conformal transformations, the warped conformal symmetries are generated by a left-moving Virasoro-Kac-Moody algebra. Notably, the SL(2,R)_L × U(1)_R global symmetries of the theory feature translations and chiral scale transformations, but do not include Lorentz transformations. Known examples of WCFT include chiral Liouville gravity <cit.>, free Weyl fermions <cit.>, and free scalars <cit.>. Furthermore, the Sachdev-Ye-Kitaev model with complex fermions <cit.> has been argued to be a broken-symmetry phase of WCFT <cit.>.

The infinite-dimensional algebra and the lack of Lorentz symmetry make WCFTs useful in the study of holography beyond AdS spacetimes. In particular, both warped AdS_3 and the near-horizon throat of extremal Kerr black holes feature an SL(2,R)× U(1) isometry group <cit.> that matches the global symmetries of WCFTs. By choosing appropriate boundary conditions, the asymptotic symmetries of a large class of gravitational theories are generated by a Virasoro-Kac-Moody algebra. This leads to a variety of holographic duals to WCFTs that include topologically massive gravity <cit.>, stringy truncations that include matter <cit.>, Einstein gravity with Dirichlet-Neumann boundary conditions <cit.>, and lower-spin gravity <cit.>. WCFTs also share common features with other Lorentz-violating theories, such as Galilean conformal field theories relevant to condensed matter physics and BMS field theories relevant to flat space holography <cit.>. 

Several aspects of WCFTs have been studied in the literature that uncover a rich structure reminiscent of their CFT cousins. For example, the modular properties of WCFT partition functions lead to a Cardy-like formula that matches the Bekenstein-Hawking entropy of black holes in the corresponding holographic duals <cit.>. Via the modular bootstrap <cit.>, it is furthermore known that WCFTs with a negative U(1) level require states with purely imaginary charge, in agreement with bulk studies <cit.>. Relatedly, correlation functions and the consequences of crossing symmetry are known and readily applicable to the bootstrap program <cit.>. 


One of the main features of WCFT studied in this paper is their entanglement entropy. The latter was computed for a single-interval on the vacuum state via the Rindler method in <cit.>. Interestingly, the entanglement entropy of WCFTs reveals hints of nonlocality along the U(1) direction, a feature that is also observed in their correlation functions <cit.>. The holographic entanglement entropy has also been derived in both locally warped and unwarped AdS_3 spacetimes <cit.> as well as lower spin gravity <cit.>. A novel feature of the holographic entanglement entropy is that it is determined by the minimal length between two null lines, the latter of which lie tangent to the modular flow emanating from the boundary end points <cit.>. This differs from the Ryu-Takayanagi prescription <cit.> and suggests a deep connection between geometry and quantum entanglement beyond the AdS/CFT correspondence. 

In this paper, we explore further aspects of entanglement in WCFTs, including the entanglement entropy on excited states and the time evolution of the Rényi entropy after a local quench. Using similar techniques and out-of-time-order (OTO) correlators we also probe the chaotic behavior of large-c WCFTs. It would be interesting to find a holographic description or our results, a task that is left for future work. 

Our results can be summarized as follows: (i) assuming a large central charge and dominance of the vacuum block in the conformal block expansion, we show that the entanglement entropy on a heavy state ψ on the cylinder is given by
  
  
  S^ψ_   = -i P_0^vacℓ̅ + μ q_ψℓ -2  ( i μ P_0^vac + 2 L_0^vac ) log ( _ψ/π sinhπℓ/_ψ ), 
where ℓ and ℓ̅ denote the lengths of the subsystem along the Virasoro and U(1) directions, μ is the spectral flow parameter defined in eq. (<ref>), and L_0^vac and P_0^vac are the vacuum expectation values of L_0 and P_0— the zero modes of the Virasoro-Kac-Moody algebra. In particular, we note that eq. (<ref>) takes the same form as the entanglement entropy of WCFTs at inverse temperatures $̱ andby means of the thermodynamical relations

  
  -i P_0^vac ( / - 2πμ/ ) = μ q_ψ,     =̱_ψ≡2π/√(24 h_ψ/c -1), 
whereh_ψ = h_ψ - q_ψ^2/kwithh_ψandq_ψdenoting the conformal weight and charge ofψ. We note that eq. (<ref>) is the transformation between the canonical and microcanonical ensembles in WCFT. Despite the additional ingredients and subtleties involved in this analysis, the result is similar to that found in CFT where the entanglement entropy on an excited state was also related to the entanglement entropy at finite temperature <cit.>. 


(ii) For a generic WCFT, we find that the change of the Rényi entropy S^(n)_after the insertion of a local operatorψvanishes at early and late times, as in two-dimensional CFTs <cit.>. On the other hand, at intermediate times of order the length of the subsystem, we show that S^(n)_is generically nonzero and universal, depending on the spectral flow parameterμof the WCFT and the chargeq_ψof the operatorψ,
  
   S^(n)_ = {[                    0, early and late times,;              2πμ q_ψ,   intermediate times. ]. 
A precise definition of early, late, and intermediate times will be given in eq. (<ref>). This is the so-called memory effect of the Rényi entropy <cit.> which, while nonvanishing, differs from the one found in rational CFTs <cit.> and irrational CFTs <cit.>. Relatedly, we note that S^(n)_vanishes at all times in a chiral CFT which features only one copy of the Virasoro algebra.
 
(iii) Assuming once again a large central charge and vacuum block dominance, we compute the OTO four-point function and show that WCFTs are maximally chaotic, like their CFT cousins <cit.>. This result is compatible with the existence of black holes in the holographic duals to this class of WCFTs, and it further distinguishes warped from chiral CFTs, the latter of which feature no chaos <cit.>. Interestingly, both S^(n)_and the OTO correlator exploit the analytic properties of correlation functions in different ways. We will see thatn-point correlators in WCFT factorize into two parts whose form is determined by the globalSL(2,R)andU(1)symmetries of the theory. In particular, we find that a nonvanishing S^(n)_relies on multivaluedness of theU(1)part of correlation functions, while the OTO correlator exploits multivaluedness of theSL(2,R)part instead.
 
The paper is organized as follows. In Section <ref> we set up our conventions and discuss relevant aspects of WCFT including the symmetries, spectral flow, and correlation functions. The entanglement entropy on excited states is derived in Section <ref>. Therein the semiclassical approximation of the Virasoro-Kac-Moody vacuum block is also discussed. The time evolution of the Rényi entropy after a local quench is considered in Section <ref> while the OTO correlator and its consequences are studied in Section <ref>. We derive the Virasoro-Kac-Moody block in Appendix <ref> and the warped conformal transformation between ann-sheeted Riemann surface and the complex plane in Appendix <ref>.




§ ASPECTS OF WARPED CFT

In this section we review the salient features of WCFT that are needed in our analysis following refs. <cit.>. We begin by considering WCFTs on the “reference plane” where the vacuum state is left invariant under the globalSL(2,R) × U(1)symmetries of the theory. We will pay special attention to the “tilt”μof the WCFT which is obtained via spectral flow of the Virasoro-Kac-Moody algebra. Finally, we describe the structure of the two, three, and four-point functions and the conformal block expansion of the latter.




 §.§ Warped CFT on the reference plane


Warped CFTs are characterized by two chiral currentsT(x^+)andP(x^+)that generate the warped conformal transformations (<ref>) <cit.>. In this subsection we consider WCFTs on the reference plane whereT(x^+) =  P(x^+) = 0.  On the reference plane the infinitesimal warped conformal transformations are generated by 
 
 
  l_n = - (x^+)^n+1_+,      p_n = i (x^+)^n _-,    
where_±≡/ x^±. In particular,l_0generates scale transformations ofx^+whilep_0is responsible for translations alongx^-. TheU(1)symmetry is not necessarily compact and, as a result, theU(1)charge is not necessarily quantized. 

The charges corresponding to the infinitesimal transformations (<ref>) are given by the modes of theT(x^+)andP(x^+)currents,

   
  L_n    = -i/2π∫ dx   (x^+)^n+1 T(x^+),      P_n = -1/2π∫ dx  (x^+)^n P(x^+),
  
which generate a chiral Virasoro-Kac-Moody algebra with central chargecand levelk,
 
 
 [L_n, L_m]    = (n-m) L_n+m + c/12 n(n^2 - 1) _̣n+m, 

 [L_n, P_m]     = -m P_n+m, 

 [P_n, P_m]     = k/2 n _̣n+m. 
 In principle, the level of the Kac-Moody algebra can be rescaled to one. However, different sectors in some holographic WCFTs may be interpreted as having different signs ofkso we leave theU(1)level intact <cit.>. 
 
On the reference plane the vacuum is invariant under the globalSL(2,R)× U(1)symmetries of the WCFT and is assumed to satisfy
 
  
 L_n0 = P_n+10 =0,      n ≥ -1.
 
TheSL(2,R)× U(1)invariance of the theory on the reference plane constrains the correlation functions of WCFTs in a similar way that theSL(2,R)× SL(2,R)group constrains correlation functions of two-dimensional CFTs <cit.>.




 §.§ Spectral flow and vacuum charges


Besidescandk, WCFTs are characterized by the tilt parameterμwhich has the following equivalent physical interpretations: (i) as a spectral flow parameter, (ii) as theL_0andP_0charges of the vacuum, (iii) as the tilt of the cylinder <cit.>. First of all, the tiltμcan be introduced via a warped conformal transformation
  
  
  x^- → y,      y = x^- + μlog x^+. 
This transformation is compatible with eq. (<ref>) and is equivalent to spectral flow. Indeed, under a finite warped conformal transformation, theT(x^+)andP(x^+)currents transform as <cit.>
  
  T'(x^+')     =  (  x^+/ x^+')^2  [ T(x^+) - c/12{ x^+', x^+ } ] +  x^+/ x^+' x^-/ x^+' P(x^+) - k/4 (  x^-/ x^+' )^2, 

  P'(x^+')     =  (  x^+/ x^+' )  [ P(x^+) + k/2 x^-'/ x^+ ], 
 
where{f,x}is the Schwarzian derivative. It is not difficult to check that the newL_n'andP'_nmodes satisfy
  
  L'_n     = L_n - i μ P_n - μ^2 k/4_̣n,     P'_n = P_n - iμ k/2_̣n, 
which is the spectral flow transformation that leaves the Virasoro-Kac-Moody algebra intact. Furthermore, if we map the plane to the cylinder via the exponential mapx^+ → e^i x^+, then the transformation (<ref>) tilts the cylinder. 
  
In what follows we consider WCFTs withμ 0. Hence the coordinates of the WCFT on the (spectrally-flowed) plane are given by
  
  
 (x^+, y) = (x^+,  x^- + μlog x^+). 
Although this is equivalent to spectral flow, a nonvanishing tiltμcarries physical consequences.[This is reminiscent of string theory on AdS_3 where spectral flow generates altogether new representations of the symmetry algebra <cit.>.] Indeed, spectral flow affects both the entropy <cit.> and the entanglement entropy of WCFTs <cit.>, and plays an important role in the holographic derivation of the latter <cit.>. Relatedly, we note that spectral flow leads to the following nonvanishing vacuum expectation values[The Sugawara-like combination of charges L_0 - P_0^2/k ≥ 0 is invariant under spectral flow. The inequality is required by unitarity and is saturated by the vacuum.]
  ⟨L'_0⟩= -μ^2 k/4,     ⟨P'_0 ⟩ = -i μ k /2. 
From this point of view, the spectral flow transformation is similar to the conformal transformation in CFT that maps the plane to the cylinder. The latter introduces a length scaleR— the size of the circle that is analogous toμ^-1— which leads to vacuum expectation values for the zero mode charges of the CFT. Note that if the spectral flow parameterμis real, then⟨P'_0 ⟩is imaginary. This is not a bug but a feature of WCFTs with holographic duals <cit.>. In this class of holographic WCFTs the dual theory is sensitive only toP_0'^2and the metric and conserved charges are always real.




 §.§ Correlation functions and conformal blocks


On the reference plane, the globalSL(2,R) × U(1)symmetries of a WCFT completely fix the coordinate dependence of its two and three-point functions. On the other hand, as in standard CFT, the four-point function is determined only up to a function of the cross-ratio,
  
  
  z = x^+_12 x^+_34/x^+_13 x^+_24, 
where we usea_ij≡ a_i - a_jfor any indexed variablea_i. In particular, the two, three, and four point functions are given by <cit.>
  ϕ_1 ϕ_2   =   (e^i ∑_j q_j x^-_j_̣ ß_i q_i ) _̣h_1 - h_2/(x^+_12)^2 h_1, 
ϕ_1 ϕ_2 ϕ_3   =  (  e^i ∑_j q_j x^-_j_̣ ß_i q_i ) C_123/(x^+_12)^h_1+h_2-h_3 (x^+_23)^h_2 + h_3 - h_1 (x^+_31)^h_3 + h_1 - h_2, 
⟨ϕ_1 ϕ_2 ϕ_3 ϕ_4⟩   = (  e^i ∑_j q_j x^-_j_̣ ß_i q_i )   ( x^+_24/x^+_14)^ h_12 ( x^+_14/x^+_13)^ h_34 G(z)  /(x^+_12)^h_1+h_2(x^+_34)^h_3+h_4 ,   
whereC_123is the OPE coefficient andϕ_i ≡ϕ_i(x^+_i,x^-_i)is a Virasoro-Kac-Moody primary field with conformal weighth_iandU(1)chargeq_i— the eigenvalues ofL_0andP_0, respectively. Note that then-point functions vanish unless the total momenta alongx^-is zero, i.e. unless the totalU(1)charge vanishes
  
  ∑_i q_i = 0.
  
Thex^+-dependence of then-point functions is the same as that of chiral CFTs with a left-moving Virasoro algebra. In contrast, the dependence onx^-suggests WCFTs are nonlocal along this direction. This interpretation is supported from the single-interval entanglement entropy of WCFTs, the latter of which features hints of nonlocality along thex^-coordinate <cit.>. A similar phenomenon is observed in the entanglement entropy on excited states derived in Section <ref>.

Under the warped conformal transformation (<ref>) a primary fieldϕ(x^+,x^-)transforms as anh-form under Virasoro transformations and as a scalar underU(1)transformations such that
  
  ϕ'(x^+',x^-') =  (x^+'/ x^+ )^-hϕ(x^+,x^-).
  
This implies that then-point functions transform as
  
  ⟨∏_i=1^n ϕ'_i(x^+_i',x^-_i') ⟩ = ∏_i=1^n  (x^+_i'/ x^+_i )^-h_i⟨∏_i=1^nϕ_i(x^+_i,x^-_i) ⟩ ,
without any additional factors ofx^-'on the right hand side <cit.>. Since eq. (<ref>) affects only thex^-coordinate we find that, after spectral flow, then-point functions still transform as a scalar. In particular, in terms of the coordinates(x^+,   x^-)on the reference plane, the two, three, and four-point-point functions are given by eqs. (<ref>)– (<ref>). One may choose to express the latter entirely in terms of the(x^+, y)coordinates of the spectrally-flowed theory. In that case one simply replaces thex^-_icoordinates featured in then-point functions by
  
  
  x^-_i→ y_i - μlogx^+_j. 

As in CFTs, the four-point function of WCFTs can be expressed in terms of Virasoro-Kac-Moody blocks such that, in thes-channel <cit.>,
  
  
  G(z) = ∑_p C_p 12 C^p_34_p(z), 
where the sum runs over primary states with weighth_pand chargeq_p = q_3 + q_4 = -q_1 - q_2, and the block_p(z)depends onc,k,h_i,q_i,h_p, andq_p. The contribution of theU(1)descendant states to_p(z)can be determined exactly using the Sugawara basis. In our conventions the Virasoro-Kac-Moody block is given by
  
  _p(z)= z^(q_3 + q_4)^2/k (1-z)^2q_2 q_3/k (c-1, h_i, h_p,z), 
whereh_i = h_i - q_i^2/kis the conformal weight in the Sugawara basis and(c, h_i, h_p,z)is the standard Virasoro conformal block. Up to conventions, eq. (<ref>) is the same expression found in CFTs with an internalU(1)symmetry in <cit.>. The derivation of eq. (<ref>) in WCFT is given in Appendix <ref>. The Virasoro conformal blockcan be expanded nearz = 0as

   (c-1, h_i, h_p,z) = z^h_p∑_K=0^∞_K (c-1, h_i, h_p)  z^K, 
where the_K (c-1, h_i, h_p)coefficients are determined by the Virasoro algebra with central chargec - 1. Finally, note that the weightsh_iandh_pare invariant under spectral flow. Hence the conformal block expansion is independent of the spectral flow parameter.




§  ENTANGLEMENT ENTROPY ON EXCITED STATES


In this section we derive the entanglement entropy on excited states in WCFTs defined on the cylinder. Our derivation is valid provided the following assumptions, which are natural from a holographic point of view, hold: (i) a large central chargec, (ii) dominance of the vacuum block in the conformal block expansion, and (iii) excited states whose conformal weights scale withc. The interpretation of our results is facilitated by comparison to the entanglement entropy of WCFTs at finite temperature to which we now turn.




 §.§ Entanglement entropy in warped CFT


Let us begin by considering a WCFT on the spectrally-flowed plane. We define the subsystemby the line connecting the points(x^+_1, y_1)and(x^+_2, y_2)which we denote by
  
  :    (x^+, y) ∈ [ (x^+_1, y_1),   (x^+_2, y_2) ]. 
The reduced density matrix onis obtained by tracing out the contribution of the complement^c, namelyρ_ = _^cρ, whereρ = 00and0is the vacuum. The moments of the reduced density matrixρ_are given by the Rényi entanglement entropies
  
  
  S^(n)_ = 1/1-nlog ( ρ^n_ ), 
whosen → 1limit yields the von Neumann entropy of subsystem, that is
  
  
  S_ = lim_n → 1 S^(n)_ = -ρ_logρ_. 

The Rényi entropies (<ref>) can be computed in the path integral formalism by gluing togethernreplicas of the system along the interval<cit.> (see <cit.> for a review). This defines ann-sheeted two-dimensional manifold_nwith cuts alongsuch that
 
 ρ_^n = Z_n/Z_1,
 
whereZ_nis the partition function evaluated on_n. We will use this approach in the next section when we study the time evolution of the Rényi entropy after a local quench. Alternatively, the trace ofρ_^ncan be evaluated by consideringncopies of the original WCFT defined on the plane. In this case, the replica boundary conditions are enforced by the insertion of twist fields_nat the endpoints ofso that
  
  ρ_^n = ⟨_n^†(x^+_2, y_2) _n(x^+_1, y_1) ⟩, 
where the two-point function is evaluated on the cyclic orbifold WCFT^n/ℤ_non the plane.  

The conformal weight andU(1)charge of the twist field_nwere determined in <cit.> via a generalization of the Rindler method used in CFT <cit.>. The twist field conformal weighth_nand chargeq_nare given by[Note that in our conventions q_n has the opposite sign from the U(1) charge used in <cit.>.]
  
  h_n = n  ( c/24 + L_0^vac/n^2 + i  P_0^vac/2π n - ^2 k/16π^2),       q_n = ( P_0^vac + i n k /4π), 
whereL_0^vacandP_0^vacdenote the vacuum expectation values ofL_0andP_0on the cylinder. The variableαin eq. (<ref>) was first introduced as a free parameter in ref. <cit.> in the generalized Rindler transformation of WCFT. Its value was later determined by matching the entanglement entropy of WCFT holographically in BTZ black holes. A similar observation has also been made for warped AdS black holes in lower spin gravity in <cit.>.

We now argue thatαshould be identified with the tilt parameterμof the WCFT. Notice that whenn → 1the symmetric orbifold WCFT must reduce to the original theory which implies thath_1 = q_1= 0for the corresponding twist fields. This is possible if the following relation between the vacuum charges and the parameterαholds
   
  
  L_0^vac  = -c/24 -α^2  k/16π^2,      P_0^vac = -i α k /4π. 
These are precisely the spectrally-flowed charges (<ref>) defined on the cylinder provided that
  
  μ = /2π.
Equation (<ref>)  guarantees thath_nandq_nare proportional ton - 1as expected,


h_n = (n-1)/n [ (n+1) c/24 - (n - 1) μ^2 k/4 ] ,      q_n = (n-1) i k μ/2. 

We also note that eq. (<ref>) saturates the unitarity boundL_0^vac - (P_0^vac)^2 /k = -c/24, and that the following choice ofμreproduces the values ofL_0^vacandP_0^vacfound in the holographic entanglement entropy of WCFT (wherekis assumed to be negative) <cit.>
  μ =  √(-c/61/k)    ⇒     L_0^vac = 0,      P_0^vac = i/2√(-c  k/6).
  

The entanglement entropy of subsystemis thus given by <cit.>
  
    S_ = -i P_0^vac [ y_21 - μlog(x^+_2/x^+_1)  ]   -2  ( i μ P_0^vac + 2 L_0^vac ) log( x^+_21/ ), 
where we introduced a UV regulatorandx^+_21,y_21denote the lengths of the interval. Eq. (<ref>) is valid for WCFTs defined on the plane with nonvanishing tilt parameterμ. When the latter vanishes, the entanglement entropy reduces to the result derived in <cit.>. To compute the entanglement entropy at finite temperature we can exploit the symmetries of the WCFT and map the two-point function of twist fields (<ref>) to the thermal cylinder. This can be accomplished by the following warped conformal transformation <cit.>
  
  x^+ = e^2π/ w^+,      x^- = w^- +  ( / - 2πμ/ ) w^+, 
where the coordinates on the thermal cylinder satisfy(w^+,  w^-) ∼ (w^+ + i ,̱  w^- - i ). Using eq. (<ref>) we obtain
  
  
  S^,̱_ = -i P_0^vac [ ℓ̅ +  (/ - 2πμ/ ) ℓ ] -2  ( i μ P_0^vac + 2 L_0^vac ) log ( /π sinhπℓ/ ), 
whereℓ = |w^+_21|andℓ̅ = |w^-_21|denote the lengths of the intervalon the(w^+, w^-)system. 
  



 §.§ Excited states and four-point functions


Let us now consider the entanglement entropy on an excited state. The latter is obtained from the vacuum via the state/operator correspondence
  
  ψ = ψ(0,0) 0,
  
whereupon the density matrix becomesρ  = ψψ. The Rényi entropy can be computed in the symmetric orbifold theory using the twist fields to enforce the replica boundary conditions. In this case the trace ofρ_^nis given by a four-point function <cit.>
  
    S_n ≡ρ_^n     =  Ψ_n^†(x^+_2, y_2) _n(x^+_1, y_1) Ψ , 
    = lim_x^+ →∞ (x^+)^2 h_Ψ⟨Ψ^†(x^+,0)  _n^†(x^+_2, y_2) _n(x^+_1, y_1) Ψ(0,0) ⟩, 
whereΨdenotes the operator in the WCFT^n/ℤ_norbifold that is obtained from the insertion ofψin each of its copies.[In particular, this implies that h_Ψ = n h_ψ and q_Ψ = n q_ψ where h_ψ is the conformal weight and q_ψ the U(1) charge of ψ.] As reviewed in Section <ref>, the four-point function (<ref>) can be expressed in terms of Virasoro-Kac-Moody blocks via eq. (<ref>). In thet-channel the conformal block expansion is given by
  
  
  S_n = e^-i q_n x^-_21 z^2 q_Ψ q_n (1-z)^-2 h_n∑_p C_pΨΨ C^p_ (c-1, h_i, h_p, 1-z),    
where we have setx^+_2 = 1without loss of generality,z ≡ x^+_1is the cross-ratio (<ref>), and we recall thatdenotes the Virasoro conformal block. In particular, note that eq. (<ref>) is expected to converge forz ∼ 1and that the operators exchanged in the conformal block expansion are all neutral, i.e.q_p = 0. 




 §.§ Entanglement in the semiclassical limit


In this section we consider WCFTs in the semiclassical limit defined by 
  
  
  c →∞    with    h̃_i/c,  h̃_p/c   fixed. 
 There are two motivations for this definition of the semiclassical limit. One comes from bulk intuition. For instance, in Einstein gravity with Dirichlet-Neumann boundary conditions <cit.>, the spectral flow invariant weighth̃is identified with the sum of mass and angular momentum, i.e.h̃ = M + J +c/24. The other motivation is that, similar to the chiral sector of CFT, the Virasoro conformal block is expected to exponentiate <cit.>
  log (c-1, h_i, h_p, 1-z) ≈ -c/6 f ( h̃_i/c, h̃_p/c,1- z  ) + Ø(c^0),
  
wheref ≈ -(6/c) h_p log (1-z) + Ø(1-z). In the semiclassical limit the conformal block expansion is dominated by the contribution of light states with smallh̃_p. 
In what follows we assume that the vacuum block dominates the conformal block expansion and study its consequences. This is analogous to the assumptions made in the derivation of entanglement entropy on excited states in CFT in refs. <cit.>.[In sparse CFTs where the density ρ of light states h + h̅ < c/12 is bounded by ρ≲ e^π c/6 <cit.>, the vacuum block is expected to dominate the conformal block expansion at large c <cit.>. A precise definition of sparseness in WCFTs and its connection to vacuum block dominance have not been spelled out yet. We hope to further understand this assumption and its implications to WCFT holography in the future.]

Note that in WCFTs the vacuum block captures the contribution of all of the descendants of the vacuum including the conserved currentsT(x^+)andP(x^+), its derivatives, and its normal-ordered products. The holographic duals of WCFTs are gravitational theories (see Section <ref> for examples) that feature the same Virasoro-Kac-Moody symmetries and the fields dual to theT(x^+)andP(x^+)currents. Therefore, we expect the vacuum block to capture the dynamics of the gravitational sector of the dual theory in the semiclassical limit. For another application of vacuum-block dominance in large-cWCFTs see <cit.>.

We will henceforth approximate the trace ofρ_^nby
  
  
  S_n ≈ e^-i q_n x^-_21 z^2 q_Ψ q_n/k (1-z)^-2 h_n (c-1, h_i, 0, 1-z).   
The Virasoro vacuum block (c-1, h_i, 0,z)is not known in closed form (see e.g. <cit.> for a nice discussion and several series expansions). If we assume that the tiltμof the WCFT scales at most linearly withc, then the conformal weight and charge of the twist field satisfy
  
  h_n/c = h_n - q_n^2/k/c≪ 1,     when     n → 1.
  
In this case the semiclassical vacuum block is approximately given by <cit.>[We have implicitly assumed that the descendant states in the vacuum block are unitary. This property does not extend to all blocks in some WCFTs with known holographic duals <cit.>.]
   (c-1, h_i, 0, 1-z) ≈ [(1-z)z^ -1/2/1 - z^ ]^2h_n, 
whereis defined by
  
   = √(1 - 24 h_ψ/c).
  
Note that eq. (<ref>) is nonperturbative inh_ψ/cbut perturbative inh_n/c. Hence it is valid in the limitn → 1. We conclude that in the semiclassical limit the trace of the reduced density matrixρ_^nis given by
  
  
 S_n → 1   ≈ e^-i q_n x^-_21 z^2 q_ψ q_n /k (1-z)^-2 q_n^2/k [ z^ -1/2/1 - z^ ]^2h_n. 

The entanglement entropy obtained from eq. (<ref>) has a nice interpretation when the WCFT is placed on the cylinder. This can be accomplished by the warped conformal transformation
  
  
  x^+ = e^i w^+,      x^- = w^-,
  
where(w^+, w^-) ∼ (w^+ + 2π,  w^-)are the coordinates on the cylinder. In these coordinates the cross-ratio is given byz ≡ x^+_1 = e^i w^+_1and we useℓ = |w^+_21|andℓ̅ = |w^-_21|to denote the lengths of the interval. Thus, the entanglement entropy on the excited stateψis given by

  
  
  S^ψ_   = lim_n→11/1-nlog S_n 
   = -i P_0^vacℓ̅ + μ q_ψℓ -2  ( i μ P_0^vac + 2 L_0^vac ) log ( _ψ/π sinhπℓ/_ψ ), 
where we used eqs. (<ref>) and (<ref>), and_ψis defined by
  
  _ψ = 2π/i  = 2π/√(24 h_ψ/c -1).
  

This equation is reminiscent of the entanglement entropy of WCFT at finite temperature (<ref>). This is not a coincidence, as the entanglement entropy on the stateψcan be obtained from eq. (<ref>) by changing ensembles (canonical→microcanonical), an observation first made in the context of CFT in <cit.>. In order to see this let us consider the partition function of a WCFT in the canonical ensemble[The WCFT partition function is sensitive to the orientation of the spatial circle where the theory is quantized, a consequence of the absence of Lorentz invariance. However, as shown in <cit.>, the expectation values given above turn out to be insensitive to the orientation of the spatial circle.]
  
  Z(,̱) = ( e^- Ḻ^cyl_0-  P^cyl_0). 
whereL^cyl_0andP^cyl_0are the zero modes defined on the cylinder. In the high temperature limit→̱0, the expectation values ofL^cyl_0andP^cyl_0were computed  in <cit.> and read
  
  ⟨ L_0^cyl⟩   = -/ Z(,̱) = k/4^2/^̱2 - 2π i P_0^vac/^̱2 - 4π^2 L_0^vac1/^̱2, 
⟨ P_0^cyl⟩   = -/ Z(,̱) = -k/2/ + 2π i P_0^vac1/. 

Although these expressions were derived in the→̱0limit we expect them to be valid for any<̱ 2πwhencis large. This result has been shown to hold in CFTs with a sparse spectrum in <cit.> which strongly suggests that a similar result is possible in WCFTs. Using eqs. (<ref>) and (<ref>) we find

  
  -i P_0^vac ( / - 2πμ/ ) = μ q  ,     =̱2π/√(24 h/c -1), 
where we used eq. (<ref>) and defined⟨ L_0^cyl⟩≡ h - c/24and⟨ P_0^cyl⟩≡ q. This is the transformation between the canonical and microcanonical ensembles in WCFT. It is not difficult to see that eq. (<ref>) withh = h_ψandq = q_ψrelates the entanglement entropy at finite temperature (<ref>) to the entanglement entropy on the stateψ(<ref>). 




§ LOCAL QUENCH


In this section we study the time evolution of S^(n)_, the change in the Rényi entropy after a local quench. We find that S^(n)_is topological and independent of the size and shape of the subsystem. While S^(n)_vanishes at early and late times for any WCFT, we show that S^(n)_ 0at “intermediate” times, but only for charged states in spectrally-flowed WCFTs.




 §.§ Rényi entropy via replica


We begin by specifying what time and space are in the WCFT. As warped CFTs are not relativistic, different choices may lead to different results. In this paper, we align the Virasoro andU(1)directions with the lightcone coordinatesx^+andx^-on the reference plane such that
  
  
  x^+ = x + i τ,      x^ - = x - i τ, 
whereτis the Euclidean time andx^±transform as in eq. (<ref>). This choice is motivated by the identification of coordinates made in known WCFT models <cit.> as well as in holographic duals to WCFTs <cit.>. An alternative choice corresponds to identifying the Virasoro directionx^+with space and theU(1)directionx^-with time <cit.>. In what follows we find that the dependence on theU(1)coordinate drops out. Hence, in order to probe the real time dynamics of WCFTs the Virasoro direction must depend on time. Another set of coordinates compatible with this requirement identifies the Virasoro direction with time and theU(1)direction with space. This choice is relevant in AdS_2gravity <cit.> and the near horizon region of extremal Kerr black holes in four dimensions <cit.>. One problem with this identification is the lack of well-defined Fourier expansion and charges. We leave further exploration of these coordinates to future work. 

Let us define the subsystemby the interval 
  
  :   { (x,τ)  |  x ∈ [0, ℓ], τ = 0 }.  
The local quench is modeled by the insertion of an operatorψ(x,τ)at 
  
  ψ (t) = ψ(-x_0, -i t ) 0 = e^-i t Hψ(-x_0,0) 0, 
whereH = - L_-1 + i P_0is the Hamiltonian of the system in the reference plane responsible for the time translation_t. In a different frame, say(x^+,y),His given by a different linear combination of the spectrally-flowedSL(2,R)× U(1)generators, cf. eq. (<ref>). By constructionψ(x,τ)lies outside the subsystemat early (t=0) and late times (t →∞). As time evolves we expect the operatorψ(x,τ)to cross the interval.

We are interested in the change of the Rényi entropy of subsystembefore and after the local quench. This quantity is defined as, cf. eq. (<ref>),
  
   S^(n)_ = 1/1-n{log [( ρ^ψ_)^n ] - log [ (ρ_^1)^n  ] },
  
whereρ_^Ø = _^cρ^Øis the reduced density matrix of subsystemon the stateØ. More concretely,ρ^1 = 00denotes the density matrix of the vacuum whileρ^ψis the density matrix of the stateψ(t)defined by
  
  ρ^ψ = ψ(-x_0,τ_2) 00ψ^†(-x_0,τ_1), 
whereτ_1andτ_2read
  
  τ_1 =  - it,     τ_2 = - - it. 
We have introduced a regulatorin eq. (<ref>) that is necessary to control the UV divergences in the correlation functions ofψ(x,τ). In particular, when→ 0we recover the expected expression for the density matrix on the stateψ(t), namelyρ = ψ(t)ψ(t). 

In this section we evaluate the traces of(ρ_^ψ)^nvia the replica method,
 
 ρ_^n = Z_n^ψ/(Z_1^ψ)^n, 
whereZ^ψ_nis the partition function on_nwith2ninsertions of the operatorψand_1denotes the complex plane. Using eq. (<ref>) we obtain
  
   S^(n)_ = 1/1-n{log(Z^ψ_n/Z^1_n ) - n log(Z^ψ_1/Z^1_1) },
  
so the change in then-th Rényi entropy requires the computation of2n-point correlation functions. More concretely, let us denote the coordinates on_nby   
  
  (x^+, y) = (x^+, x^- + μlogx^+),
  
wherex^± = x± i τ. One then finds
  
  Z^ψ_n/Z^1_n    = ⟨ψ^†(x_1, τ_1) ψ(x_2, τ_2)…ψ^†(x_2n-1, τ_2n-1) ψ(x_2n, τ_2n)⟩__n, 
where the pair of operatorsψ^†(x_2k-1, τ_2k-1) ψ(x_2k, τ_2k)is located on thek^thsheet of_nsuch that
  
  
arg ( x^±_2k-1) =  arg ( x^±_1 ) ± 2π i (k-1) ,        arg ( x^±_2k) =  arg (  x^±_2  ) ± 2π i (k-1). 
Thus, after an appropriate normalization, S^(n)_reduces to then-point function of the bilocal operatorψ^†ψ, namely
  
   S_^(n) = 1/1-nlog⟨∏_k=1^nψ^†(x_2k-1, τ_2k-1) ψ(x_2k, τ_2k) ⟩__n/ ( ⟨ψ^†(x_1, τ_1) ψ(x_2, τ_2)⟩__1)^n. 
 
We can evaluate the correlation functions on_nby exploiting the symmetries of WCFTs. Indeed, eq. (<ref>) allows us to express correlators in_nin terms of correlators on the complex planeprovided that a warped conformal transformation exists that maps_nto. This transformation takes the form (<ref>) and is given by
 
  
  z^+ = (x^+/x^+ - ℓ)^1/n,    z^- = x^- + μ( n-1/n)log( x^+/x^+ - ℓ) , 
where(z^+, z^-)denote the coordinates on the reference plane whereT(z^+) = P(z^+) = 0. A derivation of eq. (<ref>) is given in Appendix <ref>. Here we note that eq. (<ref>)
  (i) features the desired branch points at x^+ = 0 and x^+ = ℓ,
  (ii) maps the two coordinates x^+_2k-1 and x^+_2k on the k^th sheet of _n to arg(z^+_2k-1) ∈2π/n(k-3/2,k-1) and arg(z^+_2k) ∈2π/n(k-1,k-1/2),
  (iii) generalizes the analogous transformation in CFT <cit.>.


 §.§ Second Rényi entropy after the quench


We now consider the change in the second Rényi entropy S^(2)_in detail and generalize our results to S^(n)_in the next subsection. Forn = 2, eq. (<ref>) maps thex^+_1andx^+_2coordinates to the right half of the complex plane and thex^+_3andx^+_4coordinates to the left half such that
  
  
  z^+_3 = e^π i z^+_1,      z^+_4 = e^π i z^+_2.
  
Using eqs. (<ref>) and (<ref>) the four-point function on_2reads
  
  ⟨∏_k=1^2ψ^†(x_2k-1, τ_2k-1) ψ(x_2k, τ_2k) ⟩__2   =  [ ( (z^+_1)^2-1)( (z^+_2)^2-1)/2 ℓ z^+_21 ]^4h_ψ e^iq_ψ (z^-_21 + z^-_43) G(z)/(z^+_1)^2h_ψ (z^+_2)^2h_ψ, 
 whereh_ψandq_ψdenote the conformal weight andU(1)charge ofψ, and the cross-ratiozis now given by
   
   
   z = z_12 z_34/z_13z_24. 
Eq. (<ref>) has been written in terms of thez^-coordinate, reason why the spectral flow parameterμis not manifest, cf. the discussion above eq. (<ref>). The two-point function on_1is given by
   
   ⟨ψ^†(x_1, τ_1) ψ(x_2, τ_2)⟩__1    = e^i q_ψx^-_21/(x_12^+)^2h_ψ,
and is also independent ofμin these coordinates. Theμ-dependence of the two-point function (<ref>) is manifest when expressed in the same(z^+, z^-)coordinates used in (<ref>), namely
     
   ⟨ψ^†(x_1, τ_1) ψ(x_2, τ_2)⟩__1    = e^iq_ψ [z^-_21-μlog (z^+_2/z^+_1)]/ℓ^2h_ψ [ ( (z^+_1)^2-1)( (z^+_2)^2-1)/(z^+_2)^2 - (z^+_1)^2  ]^2h_ψ. 
Thus, the change in the second Rényi entropy after the local quench is given by
  
   S^(2)_ = - log [ e^iq_ψμlog (z^+_2/z^+_1)^2 (1 - z)^2h_ψ G(z)  ],
where we have usedz = -(z^+_12)^2/(4z^+_1 z^+_2)andz^-_43 = z^-_21, equations that follow from the map (<ref>). In particular, in terms ofx_0,t,, andℓ, the cross-ratio satisfies
    
  
  z =  1/2( 1 - (x_0 - t) (x_0 + ℓ - t) + ^2/√([(x_0-t)^2 + ^2] [(x_0 + ℓ - t)^2 + ^2]) ).
  
Note that eq. (<ref>) is independent of theU(1)direction, i.e. independent of thez^-_icoordinates. Hence, provided that we set the spectral flow factor to zero, our derivation is also valid for chiral CFTs featuring only one left-moving copy of the Virasoro algebra.[Although eq. (<ref>) with μ = 0 matches the corresponding expression in left-moving chiral CFTs, we expect the chiral CFT function G(z) to differ from that of warped CFT.]

Using the conformal block expansion (<ref>) we can recast eq. (<ref>) as
  
   S^(2)_ =     -log [ e^iμ q_ψlog (z^+_2/z^+_1)^2 (1 - z)^2h_ψ∑_p (C_pψψ)^2  (c-1,h_ψ,h_p,z) ], 
where we recall thath_ψandh_pdenote the conformal weights in the Sugawara basis. This expression is valid in thes-channel where we expand aroundz=0. Using crossing symmetry, i.e. by computing the four-point function in thet-channel, we obtain an expression that is instead valid aroundz=1. Thus, S^(2)_is alternatively given by
  
   S^(2)_, t- ch =    -log [ e^i μ q_ψlog (z^+_2/z^+_1)^2 z^2h_ψ∑_p (C_pψψ)^2  (c-1,h_ψ,h_p, 1-z) ].

The expansions around thes(<ref>) andt-channels (<ref>) capture different regimes in the time evolution of S^(2)_.  This follows from the fact thatzis a function of time for any value of the regulator. In particular, for small but non-vanishingwe havez ∼Ø(^2)at early and late times whereasz ∼ 1-  Ø(^2)at intermediate times (see below for a precise definition). When the regulatoris set strictly to zero the cross-ratio becomes
  
  lim_→ 0 z = 1/2{ 1 - sign[(x_0 -t )(x_0 + ℓ -t)] },
  
and we find more precisely that
  
  
  z  =  {[                                                0, 0  ≤ t < x_0   (early)  and  x_0 + ℓ < t   (late);                                                1,                x_0 < t < x_0 + ℓ  (intermediate). ]. 
In particular we note thatzis real, even for finite, and that it does not circle the pointz = 1as the system is evolved in time (instead approachingz → 1from below). Hence the cross-ratio does not probe the multivaluedness of the four-point function, in contrast to the out-of-time-order correlator considered in the next section. 

Forz = 0orz = 1, the leading contribution to each of the conformal blocksin eqs. (<ref>) and (<ref>) is given by the primary state sans descendants, cf. eq. (<ref>)
   (c-1, h_ψ, h_p, 0) = lim_z → 0 z^h_p[1 + Ø(z)].
  
Since the conformal weighth_pis invariant under spectral flow, its minimum value is saturated by the vacuum, namely byh_p  = 0. Hence, the conformal block expansion of (<ref>) is dominated by the vacuum state in both thesandt-channels, and S^(2)_is given by[When 0, z varies smoothly as a function of time between z=0 and z=1. As z grows away from zero the contribution of other primary states to the conformal block expansion becomes important and cannot be neglected. Indeed, as z approaches 1 these states are summed into the conformal block expansion in the t-channel which is dominated by the vacuum state at z = 1.]
   S^(2)_ =     -  iμ q_ψlog (z^+_2/z^+_1)^2 .
Thus, the time-dependence of S^(2)_comes entirely from thez^+_2/z^+_1ratio. Furthermore note that eq. (<ref>) implies S^(2)_ = 0at all times in a left-moving chiral CFT.
  
The ratioξ≡ (z^+_2/z^+_1)^2featured in eq. (<ref>) is a pure phase given by
  
  ξ = e^i  = (x_0 - t + i)(x_0 + ℓ - t - i)/(x_0 - t - i)(x_0 + ℓ - t + i ). 
 When = 0, it seems thatξ = 1and S^(2)_vanishes. However, care must be exercised as the limit→ 0is taken. Note that asϵ→ 0^+we have the following expansion  
   
   ξ∼ 1+  2iϵℓ (x_0-t)(x_0+ℓ-t), 
  
For small but finite,evolves counter-clockwise in the complex plane until it reaches = 2πatt = x_0. As time evolves further,reverses its path reaching = 0after a timet = x_0 + ℓ.
 Hence, asϵ→ 0^+, the phase takes different values in different regions,
   
  lim_→0^+ = {[                                0, 0  ≤ t < x_0    and  x_0 + ℓ < t,;                              2π ,               x_0 < t < x_0 + ℓ . ]. 
 As a consequence, the change of the second Rènyi entropy picks up a phase in the intermediate region such that
  
   S^(2)_ = {[                                0, 0  ≤ t < x_0    and  x_0 + ℓ < t,;                          2πμ q_ψ,               x_0 < t < x_0 + ℓ . ]. 
The second Rènyi entropy is independent of the lengthℓof the subsystem. It is in this sense that S^(2)_is topological. Notably, the change in the second Rényi entropy after a local quench is nontrivial at intermediate times but only for charged statesψin spectrally-flowed WCFTs. This is a feature that extends to higher-order Rényi entropies as we show next. 
  



 §.§ n-th Rényi entropy after the quench


We now generalize eq. (<ref>) and comment on its interpretation. For arbitrarynthe change in the Rényi entropy S^(n)_is given, up to a normalization, by a2n-point function on_n(<ref>). In particular, this means that we havenpairs of operatorsψ^†andψdepending respectively onnpairs of coordinatesx^±_2k-1andx^±_2kwherek = 1, …, n. Thex^±_2k-1andx^±_2kcoordinates live on thek-th sheet of_nand are obtained from thex^±_1andx^±_2coordinates of the original manifold via eq. (<ref>). As in the previous subsection we will evaluate the2n-point function on the complex plane, a feat that is possible due to the properties of the uniformizing map (<ref>) that we discuss next.

We begin by noting that to linear order in, thez_2k-1^+andz^+_2kcoordinates obtained via eqs. (<ref>) and (<ref>) satisfy
  
    (z^+_2k-1,z^+_2k) =  {[            e^2π i (k-1)/nζ( e^-i /n, e^i /n) ,              0  ≤ t < x_0    and  x_0 + ℓ < t,; e^2π i (k-1)/nζ( e^-iπ/n - i/n, e^iπ/n + i/n),                            x_0  ≤ t < x_0 + ℓ, ]. 
whereζand$̣ are defined as
  
  ζ≡ | x_0 - t/x_0 + ℓ - t |^1/n ,     =̣ℓ/(x_0 - t)(x_0 +ℓ - t).
  
In particular, eq.(<ref>) implies that the coordinates pair up at early and late times as
  
      z^+_1 z^+_2 … z^+_2n-1 z^+_2n→ (z^+_1 z^+_2)… (z^+_2k-1 z^+_2k) … ( z^+_2n-1 z^+_2n), 
where the notation (z^+_i z^+_j) means that z^+_i - z^+_j ∝. As → 0 there is a sharp transition at intermediate times where we instead find
 
     z^+_1 z^+_2 … z^+_2n-1 z^+_2n→ (z^+_2 z^+_3)… (z^+_2k z^+_2k+1) … ( z^+_2n z^+_1). 
When n =2 eqs. (<ref>) and (<ref>) are compatible with the behavior of the cross-ratio found in the previous subsection. There, z → 0 at early and late times while z → 1 at intermediate times (<ref>).
  
The 2n-point function featured in S^(n)_ is given via the uniformizing map (<ref>) by
    
  ⟨∏_k=1^nψ^†(x_2k-1, τ_2k-1) ψ(x_2k, τ_2k) ⟩__n    =  ( ζ/n  )^2 n h_ψ⟨∏_k=1^nψ^†(z^+_2k-1, z^-_2k-1) ψ(z^+_2k,z^-_2k) ⟩.
   
As a consequence of eqs. (<ref>) and (<ref>) the n-point function of the bilocal operator ψ^†(z^+_2k-1,z^-_2k-1)ψ(z^+_2k,z^-_2k) on the complex plane factorizes accordingly into the product of n two-point functions. Interestingly, we find that
  
  ⟨∏_k=1^nψ^†(x_2k-1, τ_2k-1) ψ(x_2k, τ_2k) ⟩__n    = ∏_k=1^n e^i q_ψ ( z^-_2k - z^-_2k-1 )/(2i )^2 n h_ψ, 
is valid at all times. To obtain the change in the Rényi entropy S^(n)_ we must first normalize eq. (<ref>) by n copies of the two-point function
    
  ⟨∏_k=1^2ψ^†(x_2k-1, τ_2k-1) ψ(x_2k, τ_2k) ⟩__1    = e^i q_ψx^-_21/(2ix)^2h_ψ.
  
Thus, since x_21 = z^-_21 + μ (1-n) log (z^+_2/z^+_1) and z^-_2k - z^-_2k-1 = z^-_21 for all k, the change in the n-th Rényi entropy is given by
    
   S^(n)_ =  - i μ q_ψlog (z^+_2/z^+_1)^n. 

When n=2 in eq. (<ref>) we recover the results found in the previous subsection. One motivation behind the more detailed analysis of the n =2 case concerns the analytic properties of the four-point function. Indeed, when n =2 we observed that the cross-ratio does not circle the branch point at z = 1. Instead, a nontrivial value of S^(2)_ comes solely from the multivaluedness of the U(1) phases in the two and four-point functions of WCFTs. Motivated by the n=2 result, we have implicitly assumed that the multivaluedness of S^(n)_ derived in eq. (<ref>) comes only from the U(1) phases. Thus, provided this assumption holds, the change in the n-th Rényi entropy is given by
  
   S^(n)_ = {[                                0, 0  ≤ t < x_0    and  x_0 + ℓ < t,;                          2πμ q_ψ,               x_0 < t < x_0 + ℓ . ]. 
where we used the properties of the ratio (z^+_2/z^+_1)^n = ξ = e^i studied in the previous subsection, namely the fact that = 2π at intermediate times and vanishes otherwise (<ref>).

Eq. (<ref>) highlights the importance of spectral flow in the study of WCFTs, the latter of which is necessary to obtain nontrivial values of S^(n)_. Furthermore, eq. (<ref>) vanishes unless the operator ψ is charged, i.e. unless it carries momentum along x^-. In fact, the only non-vanishing contribution to S^(n)_ comes from the U(1) sector of the WCFT. Thus, eq. (<ref>) further distinguishes warped from chiral CFTs, as internal U(1) symmetries do not lead to the spacetime-dependent phases in correlation functions that are needed to obtain a nonvanishing S^(n)_.

In refs. <cit.>, eq. (<ref>) was argued to be compatible with the causal structure of relativistic field theories. While WCFTs are inherently nonrelativistic and nonlocal, our results suggest a similar interpretation. Namely, that the operator ψ(-x_0,0) creates an entangled pair of particles at x=-x_0 that propagate oppositely at the speed of light. This is compatible with the fact that S^(n)_ 0 at x_0 < t < x_0 + ℓ, that is, when one of the particles lies in subsystem . It would be interesting to relate our results to the observation that the causal domain of WCFT is a strip <cit.>, as opposed to a diamond, and to understand the consequences to the causal structure of the holographic dual theories (see <cit.> for a related discussion). It would also be interesting to understand the physical interpretation behind the μ-dependence of S^(n)_.




 §.§ Comments on different approaches



We conclude this section with a few comments on different approaches to the calculation of the entanglement entropy after a local quench, specifically refs. <cit.> and <cit.>. In these references the local quench is also modeled by the insertion of a local operator ψ. However, while ref. <cit.> uses the four-point function with twist operators (<ref>) to evaluate the entanglement entropy, ref. <cit.> uses the replica four-point function (<ref>) instead. For this reason, the entanglement entropy computed in these approaches exploits the nonanalytic structure of CFT n-point functions in different ways. 

In more detail, ref. <cit.> computed the entanglement entropy after a local quench in large-c CFTs in the semiclassical limit. Therein the vacuum block is assumed to dominate in each channel of the conformal block expansion and different OPE channels dominate at different times. In this case a nontrivial value for the entanglement entropy is obtained at intermediate times when the cross-ratio in the right-moving sector, namely z̅, moves onto the second sheet of the vacuum block. On the other hand, ref. <cit.> considered the change in the n-th Rényi entropy of rational CFTs. There, instead of exploiting the analytic properties of 2n-point functions, the authors use fusion rules to relate (the chiral half of) the vacuum block in one OPE channel to the sum over blocks in different OPE channels. This procedure yields nontrivial values for S^(n)_ at intermediate times only. 




The change in the Rényi entropy of WCFTs computed in  this paper differs from the previous approaches in a novel way. First of all, due to the chirality of the symmetry algebra, the correlation functions can be readily evaluated at all times when → 0, and the resulting 2n-point functions are multivalued. This feature is used at intermediate times when the argument of the correlation function moves onto a different branch. However, unlike ref. <cit.>, the nonanalyticity of the four-point function originates from the U(1) sector and is present only in spectrally-flowed WCFTs. As a consequence, S^(n)_ 0 at intermediate times provided that the tilt μ of the theory and the U(1) charge q_ψ of the state ψ do not vanish.




§ BUTTERFLIES IN WCFT


In this section we show that large-c WCFTs are maximally chaotic provided that the vacuum block dominates the conformal block expansion of the four-point function. As in Section <ref> we also consider operators whose conformal weights scale with the central charge. These set of assumptions are holographic in nature and the maximal Lyapunov exponent obtained herein suggests the corresponding WCFTs are dual to theories of gravity that admit black hole solutions.




 §.§ Semiclassical limit and the vacuum block


In order to probe the chaotic behavior of WCFTs we turn our attention to the out-of-time order four-point function of two generic operators W(t) and V(0),
  
  
  R(t) = W^†(t) V^†(0) W(t) V(0)_/W^†(t) W(t)_V^†(0) V(0)_, 
where t is the Lorentzian time. In chaotic systems the OTO correlator (<ref>) decays exponentially at late times, a phenomenon that is absent in time-ordered correlators <cit.>. The late-time behavior of (<ref>) is preceded by a rapid decrease, R(t) ≈ 1 - f_0 e^ł_L t with f_0 ≪ 1, that is characterized by the Lyapunov exponent ł_L. The latter is bounded by ł_L ≤ 2π/$̱ in systems with a parametrically large number of degrees of freedom (largec) <cit.>. In particular, the Lyapunov exponent vanishes in theories without chaos, e.g. integrable models, chiral CFTs <cit.>, and permutation orbifolds <cit.>, while its upper bound is saturated by black holes in Einstein gravity. For this reason, we expect theories withł_L = 2π/$̱ to admit gravitational duals in the large-c limit <cit.>.

We begin by considering the four-point function (<ref>) in Euclidean time. This finite-temperature correlator can be obtained from the four-point function in the complex plane via the change of coordinates (<ref>), reproduced here for convenience
    
  
  x^+ = e^2π/ w^+,      x^- = w^- +  ( / - 2πμ/ ) w^+,
  
where w^± = x± i τ denote the coordinates on the “thermal" cylinder. The equation above features a chemical potential β̅ for the U(1) charge and a nonvanishing tilt μ. However, unlike Sections <ref> and <ref>, neither β̅ nor μ play a role in the OTO correlator  in our setup. Using eq. (<ref>) we find
  
  
  R(z)    ≡⟨ W^†(x_1,τ_1) W (x_2,τ_2) V^†(x_3, τ_3) V(x_4, τ_4) ⟩_/⟨W^†(x_1, τ_1) W(x_2, τ_2)⟩_⟨V^†(x_3, τ_3) V(x_4, τ_4)⟩_ = G(z), 
where we have assumed the operators W(x_i,τ_i) and V(x_j,τ_j) have unit norm and z is the cross-ratio
  
  
  z = x^+_12 x^+_34/x^+_13 x^+_24.
  

We note that eq. (<ref>) does not depend on w^-_i and is a multivalued function of z. In contrast to correlation functions in CFT, the warped CFT four-point function is multivalued even before analytic continuation to Lorentzian time. This is a consequence of the chiral nature of the Virasoro-Kac-Moody algebra. Aside from the U(1) phases, the correlation functions in WCFT correspond to the chiral half of CFT. This partly motivates using the Euclidean theory as our starting point, since several results, including the conformal block expansion and the approximate expressions for individual blocks and their properties are already known from Euclidean CFT.

If we assume the Virasoro-Kac-Moody vacuum block dominates the conformal block expansion of eq. (<ref>) we obtain, cf. eq. (<ref>),
  
  
  R(z) ≈ (1-z)^-2q_W q_V/k (c-1, h_i, 0,z), 
where h_i = {h_W, h_V} are the conformal weights of W and V in the Sugawara basis, and q_W, q_V are their U(1) charges. Furthermore, in the semiclassical limit[Note that a small h_W/c is a further assumption beyond the semiclassical limit considered in Section <ref>.]
  
  c →∞,     h_W/c, h_V/c≪ 1,
  
the expression for the vacuum block given in eq. (<ref>) yields
  
  
   R(z) ≈ (1-z)^ [z/1 - (1-z)^1 - 12 h_W/c ]^2h_V, 
where  is given by
  
   = - 12h_W h_V/c -  2q_W q_V/k. 
The branch cut extending along z = [1, ∞) is a crucial feature of eq. (<ref>).
It is ultimately responsible for the chaotic behavior of the OTO correlator to be derived in eq. (<ref>). 
The same nonanalytic behavior (<ref>) appears in both the holomorphic and antiholomorphic sectors of CFT <cit.> but the full Euclidean correlator remains analytic. In this case the analytic continuation to Lorentzian time breaks the symmetry between the two chiral halves of the four-point function and different time orderings explore different branches of one of the chiral sectors. In contrast, WCFTs are already chiral but an appropriate time ordering prescription is required to probe the nonanalytic behavior of (<ref>), as we will show in the next subsection.

Before we consider the OTO correlator let us comment on chiral CFTs which also feature a chiral symmetry algebra.[Chiral CFTs could have different meanings in different contexts. See <cit.> for a partial classification.] In chiral CFTs with c=24n and n ∈ℕ <cit.>, all primary states are conserved currents with integer conformal weights. Consequently, eq. (<ref>) is single-valued and features no chaos <cit.>. This is not expected to be the case in WCFT where the conformal weights are not assumed to be integers except for the states corresponding to the vacuum, the U(1) current P(x^+), and their Virasoro descendants. In particular, the analyticity of the four-point function in chiral CFTs implies that the vacuum block cannot be assumed to dominate the conformal block expansion at large c. In this case, additional blocks must be included which restore single-valuedness of the four-point function. From the point of view of the conformal block expansion this is possible due to fine-tuning of the OPE coefficients and the conformal weights. Thus, in the derivation of eq. (<ref>) we have assumed that WCFTs feature no such fine tuning. 
  



 §.§ Out-of-time order correlator and chaos


Although eq. (<ref>) is multivalued, not all orderings of the operators W(x_i,τ_i) and V(x_j,τ_j) probe different sheets in real time. When eq. (<ref>) is confined to the first sheet, the correlator remains of Ø(1) at late times. Only out-of-time-order correlators probe the second sheet and decay exponentially at late times <cit.>. This motivates the following choice of coordinates for the operators W(x_i,τ_i) and V(x_j,τ_j) in eq. (<ref>)
  w^+_k = {[ -x_0 + i (_k - i t),            k = 1, 2,;                i _k,            k = 3, 4, ]. 
where x_0 > 0 and the _i terms are small but finite and chosen to satisfy[The coordinates w^-_i do not play a role in our story and can be obtained from conjugation of w^+_i if _k - it is assumed to be real. ]
  _1 > _3 > _2 > _4. 
The Euclidean-time ordering of the operators W(x_i,τ_i) and V(x_j,τ_j) given by eq. (<ref>)  corresponds to the desired out-of-time ordering of the correlator R(t) in eq. (<ref>). This choice of coordinates implies that the cross-ratio z evolves from z=0 at early times to
  
  
  z ≈ - e^2π/ (x_0 - t) _12^* _34,     at  t ≫ x_0, 
where we have defined _jk as in <cit.> by
  
 _jk = i  ( e^2π/ i _j - e^2π/ i _k ).
 

The ordering (<ref>) guarantees that z evolves counterclockwise in the complex plane circling once the branch point at z = 1. As a result, the late time behavior of the OTO correlator is obtained by taking the limit z → 0 on the second sheet of eq. (<ref>). Using the late time expression of the cross-ratio (<ref>) we find
  
  
  R(t) ≈ e^-2π i  [ 1/1 + 24π i h_W/^*_12_34e^2π/ (t - t_⋆ -x_0)  ]^2 h_V, 
where we defined the scrambling time t_⋆ by <cit.> (see also <cit.>)
  
  
  t_⋆ = /2πlog c.
  
The OTO correlator (<ref>) decreases by a term proportional to - e^2π/ (t - t_⋆ - x_0 ) at times /̱2π < t < t_⋆ + x_0. This suggests that large-c WCFTs where the vacuum block dominates the conformal block expansion are maximally chaotic and saturate the bound on the Lyapunov exponent
  
  ł_L = 2π/.
We therefore expect WCFTs satisfying the aforementioned assumptions to admit gravitational duals featuring black holes. On the other hand, the OTO correlator decays as e^-4πh_V/ (t - t_⋆ - x_0) at late times t > t_⋆ + x_0, which is the expected behavior in chaotic systems.

Up to the phase factor, eq. (<ref>) agrees with the CFT out-of-time-order correlator after the identification h_i → h_i <cit.>. From a technical point of view, the main difference between the warped and CFT calculations comes from the expansion of the four-point function in terms of Virasoro-Kac-Moody blocks. This leads to the conformal weights in the Sugawara basis featured in eq. (<ref>). Physically, the similarities between the warped and CFT calculations are to be expected from a generalization of the arguments given in ref. <cit.>. Therein it was argued that the leading contribution to the OTO correlator in a holographic CFT comes from the current of highest spin, namely the stress energy tensor. In contrast to CFTs with Virasoro or _N algebras, warped CFTs feature an additional spin-1 current P(x^+) which, as we have seen, does not spoil the leading order contribution of the spin-2 current T(x^+).

Holographic duals for WCFTs come in a variety of flavors, including topologically massive gravity <cit.>, stringy truncations that include matter <cit.>, Einstein gravity with modified boundary conditions <cit.>, and lower-spin gravity <cit.>. It is not clear which of these theories is compatible with vacuum-dominance in the dual large-c WCFT. It would be interesting to use the shock wave methods of refs. <cit.> to answer this question and reproduce eq. (<ref>) directly from gravity.

Finally note that the OTO correlator in CFTs featuring an internal U(1) symmetry, semiclassical limit (<ref>), and vacuum block dominance, is also given by eq. (<ref>). This follows from the fact that (i) a U(1) current enhances the Virasoro algebra of CFT to the Virasoro-Kac-Moody algebra of WCFT, and (ii) features characteristic of WCFTs such as spectral flow do not play a role in the OTO four-point function (<ref>).






§ ACKNOWLEDGMENTS

We are grateful to Alex Belin, Pankaj Chaturvedi, Bin Chen, Peng-xiang Hao, Max Riegler, Boyang Yu, and Yuan Zhong for helpful discussions. This work was supported by the National Thousand-Young-Talents Program of China and NFSC Grant No. 11735001. The work of L. A. was also supported by the International Postdoc Program at Tsinghua University. S. H. is supported by a Max-Planck Fellowship and the German-Israeli Foundation for Scientific Research and Development.




§ THE VIRASORO-KAC-MOODY BLOCK


In this appendix we derive the Virasoro-Kac-Moody block given in eq. (<ref>). A convenient way to compute conformal blocks is to insert the identity operator ∑_p_p_p in the four-point function (<ref>), where the sum runs over primary states with conformal weight h_p and charge q_p = q_1 + q_2 = - (q_3 + q_4) (the q_i charges, where i = 1, …, 4, are defined below). Using eqs. (<ref>) and (<ref>) we have
 
 _p(z) =  lim_x_1^+ ∞ (x^+_1)^2 h_1 z^h_3 + h_4e^-i∑_jq_j x^-_jϕ_1(x_1^+) ϕ_2(1) _p_pϕ_3(z) ϕ_4(0)/C_p12C^p_34,   
 where z is the cross-ratio (<ref>), the operators ϕ_i(x^+_i) have weight h_i and charge q_i, and we have suppressed their x^-_i-dependence for convenience. The _p_p term in eq. (<ref>) sums the contributions of all the descendant states of the primary state h_p, q_p. In the Sugawara basis it reads
   
  _p_p = ∑_N,M∑_{n_i, ρ_i}∑_{m_j, ρ_j} | _p^N_i,M_j⟩⟨_p^N_i,Mj|,
  
where N_i and M_i are shorthand for N_i ≡{N, {n_i, ρ_i}}, M_j ≡{M, {m_j, _j}} and | _p^N_i,M_j⟩ is given by
  
      | _p^N_i,M_j⟩ =  [ _{n_i, ρ_i }^-1/2∏_i=0^N (L̃_-n_i)^ρ_i ] [ _{m_j, _j }^-1/2∏_j=0^MP_-m_j^_j ] h_p, q_p,   
In eq. (<ref>)|_p^N_i,M_j⟩ denotes the state with ∑_i ρ_i = N Virasoro and ∑_j _j = MU(1) descendants, _{n_i,ρ_i}×_{m_j,_j} is the norm of the state, and L̃_n denotes the Virasoro modes in the Sugawara basis, namely
  
  L̃_n = L_n - 1/k∑_m ≤ -1 P_m P_n-m + ∑_m ≥ 0 P_n-mP_m. 
In the Sugawara basis the symmetry algebra simplifies to the direct product of a Virasoro and U(1) algebras with c  c - 1. As a consequence, the Virasoro and U(1) descendant states are orthogonal to each other — reason why the norm factorizes in eq. (<ref>)— and the conformal block _p(z) splits into the product of Virasoro and U(1) Kac-Moody blocks,
  
  _p(z) = ^Vir_p(z)^KM_p(z).
  
The derivation of eq. (<ref>) proceeds as follows <cit.>. We first consider the contribution of one Virasoro descendant to _p_p, i.e. the contribution of the state |_p^N_1,0⟩  = _{n,1}^-1/2L̃_-nh_p, q_p where N_1 = {1, {n,1}}. Let ϕ_i ≡ϕ_i(x^+_i, x^-_i). Using
  
  
  [L_n, ϕ_i ]    =  [ h_i (n+1) (x^+_i)^n + (x^+_i)^n+1_x^+_i ] ϕ_i,      [P_n, ϕ_i] = q_i (x^+_i)^n ϕ_i, 
and eq. (<ref>), it is straightforward to show that
  
  ⟨_p^N_1,0 | ϕ_1 ϕ_2⟩ =       _{n,1}^-1/2{∑_i=1^2  [(n+1) h̃_i   (x^+_i)^n  + (x^+_i)^n+1_x^+_i ]  
   -2 q_1 q_2/k·(x_1^+)^n+1 - (x^+_2)^n+1/x^+_1 - x^+_2}h_p, q_pϕ_1 ϕ_2, 
where h_i = h_i - q_i^2/k (the eigenvalue of L_0) is the conformal weight in the Sugawara basis. Crucially, the three-point function in eq. (<ref>) may be written as
  
  h_p, q_pϕ_3 ϕ_4 = C_p34e^i/3 (q_3 x^-_3 + q_4 x^-_4)/(x^+_34)^h_3 + h_4 - h_p = (x^+_34)^2 q_3 q_4/kh_p, q_pϕ_3 ϕ_4_h_i h̃_i,  h_p h̃_p, 
where we used q_p = q_3 + q_4. When the factor of (x^+_34)^2 q_3 q_4/k is moved through the derivatives in eq. (<ref>), the three-point function can be expressed in terms of h̃_i and h̃_p such that
  
  ⟨_p^N_1,0 | ϕ_3ϕ_4⟩    = (x^+_34)^2 q_3 q_4/k⟨_p^N_1 | ϕ_3ϕ_4⟩_h_i h̃_i,  h_p h̃_p,  c  c -1, 
where | _p^N_i⟩ denotes the normalized state with ∑_i ρ_i = N Virasoro descendants, namely[The shift in the central charge in  eq. (<ref>) is necessary to reproduce the correct normalization, that is _{n_i,ρ_i} = _{n_i,ρ_i}|_h_p h̃_p,  c  c -1.]
   | _p^N_i⟩  = _{n_i, ρ_i }^-1/2∏_i=1^N (L_-n_i)^ρ_ih_p, q_p. 

The commutativity of the L̃_n and P_m generators allows us to generalize eq. (<ref>) to states with an arbitrary number of L_n descendants. There, the role of the (x^+_34)^2 q_3 q_4/k term in (<ref>) is to give, upon differentiation, the appropriate powers of the charges q_i such that h_i h̃_i. Thus, more generally we find that
  
  ⟨_p^N_i,0 | ϕ_3 ϕ_4⟩    =  (x^+_34)^2 q_3 q_4/k⟨_p^N_i | ϕ_3 ϕ_4⟩_h_i h̃_i,  h_p h̃_p,  c  c -1. 
Similarly, by first considering the case with one Virasoro descendant state and then exploiting the virtues of the Sugawara basis we obtain

  ⟨ϕ_1 ϕ_2 | _p^N_i,0⟩    = (x^+_12)^2q_1 q_2/k/ (x^+_1)^2 q_1(q_1 + q_2)/k (x^+_2)^2 q_2 (q_1 + q_2)/k⟨ϕ_1ϕ_2 | _p^N_i⟩_h_i h̃_i,  h_p h̃_p,   c  c-1. 
Finally, using eqs. (<ref>) and (<ref>) we find that ^Vir_p(z) in (<ref>) is given by
  
  ^Vir_p(z)    = lim_x^+_1 ∞ (x^+_1)^2 h_1 z^h_3 + h_4e^-i∑_jq_j x^-_j∑_N∑_{n_i, ρ_i}⟨ϕ_1 ϕ_2  | _p^N_i,0⟩⟨_p^N_i,0 | ϕ_3 ϕ_4 ⟩/C_p12C^p_34, 
    =  z^(q_3 + q_4)^2/klim_x^+_1 ∞ [ (x^+_1)^2 h_1 z^h_3 + h_4e^-i∑_jq_j x^-_j∑_N∑_{n_i, ρ_i}⟨ϕ_1 ϕ_2  | _p^N_i⟩⟨_p^N_i | ϕ_3 ϕ_4 ⟩/C_p12C^p_34 ]_h_i h̃_i,  h h̃, 
 c  c -1
    = z^(q_3 + q_4)^2/k_p (c-1, h_i , h_p , z), 
where, for convenience, we suppressed the coordinate dependence of ϕ_i with respect to eq. (<ref>) and _p (c-1, h_i , h_p , z) is the Virasoro conformal block in CFT, which in our conventions satisfies _p (c-1, h_i , h_p , z) = z^h_p (1 + …).


Let us now consider the contribution of the U(1) descendant sates to the Virasoro-Kac-Moody block. In what follows we will assume the level k of the U(1) algebra is positive so that all descendant states have positive norm. Recall from eq. (<ref>) that | _p^0,M_i⟩ denotes a descendant state with a total number M of U(1) generators P_-m. We thus have
  
  ^0,M_i_p = _{m_j, _j}^-1/2 P_-m_1^_1… P_-m_M^_Mh_p,q_p, 
where we assume m_1 > m_2 > … > m_N in order to avoid overcounting states.

Thanks to the abelianness of the U(1) algebra, the normalization _{m_j, _j} is given by
  
  _{m_j, n_j} = ∏_j=1^Mm_j  k/2^_j_j! .
This is a consequence of the anomaly and originates from terms of the form P^_1_m_1 P^_1_-m_1 in the inner product for which [P_n, P_m] = n k/2 _̣n+m, cf. eq. (<ref>). The factor of _j! in eq. (<ref>) accounts for the fact that we must commute each P_m_j with P_-m_j a total _j! number of times when computing the norm of P_-m_j^_jh_p,q_p. Using eq. (<ref>) we find
  
  ⟨^0,M_i_p | ϕ_3 ϕ_4⟩ =  _{m_j, _j}^-1/2∏_j=1^M  [q_3 (x^+_3)^m_j + q_4 (x^+_4)^m_j]^_jh_p,q_pϕ_3 ϕ_4,
  
and, likewise,
  
  ⟨ϕ_1 ϕ_2 | ^0,M_i_p⟩ = (-1)^M_{m_j, _j}^-1/2∏_j=1^M  [q_1 (x^+_1)^-m_j + q_2 (x^+_2)^-m_j]^_jϕ_1 ϕ_2h_p,q_p,
  
where we used the fact that ∑_j _j = M. Finally, from the definition of the Virasoro-Kac-Moody block in eqs. (<ref>) and (<ref>), it follows that

  ^KM_p(z)    = ∑_M∑_{m_j, _j}lim_x^+_1 ∞⟨ϕ_1(x_1^+) ϕ_2(1) | ^0,M_i_p⟩⟨^0,M_i_p | ϕ_3(z)ϕ_4(0)⟩/ϕ_1(x^+_1)ϕ_2(1)h_p,q_ph_p,q_pϕ_3(z)ϕ_4(0), 
   =  ∑_M∑_{m_j, _j}  (-1)^M ∏_j=1^M1/_j!2q_2 q_3/kz^m_j/m_j^_j.
  
The condition m_1 > m_2 > … > m_N and the multinomial theorem guarantee that
  
  ^KM_p(z)    = ∑_M=0^∞1/M!log (1-z)^2 q_2 q_3/k^M = (1-z)^2q_2 q_3/k. 
Thus,  the Virasoro-Kac-Moody block (<ref>) is given by
  
  _p(z) =  z^(q_3 + q_4)^2/k (1-z)^2q_2 q_3/k (c-1, h_i, h_p,z).
  


§ MAPPING _N TO THE COMPLEX PLANE IN WCFT


In this appendix we derive the map (<ref>) between the n-sheeted Riemann surface _n and the complex plane . Our strategy is to consider the one-point functions of T(x^+) and P(x^+) on _n obtained via the generalized Rindler method in <cit.>. We then find the warped conformal transformation that reproduces these one-point functions starting from the ones on the plane. This is the inverse of the approach used in <cit.> to derive the conformal weight of the twist fields _n in CFT.

Let (x^+, y) = (x^+, x^- + μlogx^+) denote the spectrally-flowed coordinates on the n-sheeted Riemann surface R_n glued along the interval . The latter is defined as the line connecting the points (0,0) and (ℓ, ℓ̅) which we denote by
  
  :   (x^+, x^-) ∈ [ (0,0),   (ℓ,ℓ̅) ].  
The one-point functions of T(x^+) and P(x^+) on _n are given by <cit.>,
  
  T(x^+)_ R_n    = ℓ^2/(x^+)^2 ( x^+ - ℓ)^2h_n/n, 
      P(x^+)_ R_n = ℓ/x^+( x^+ - ℓ)i q_n/n,  
  
where h_n and q_n are the conformal weight and charge of the twist operator in the WCFT
  
  
  h_n = n  ( c/24 + L_0^vac/n^2 + i  P_0^vac/2π n - ^2 k/16π^2),       q_n = ( P_0^vac + i n k /4π). 

Following the discussion in Section <ref>, we identify the parameter  in eq. (<ref>) with the spectral flow parameter μ of the WCFT via
  
  μ = /2π.
  
It follows that L_0^vac and P_0^vac, the expectation values of L_0 and P_0 on the cylinder, satisfy
   
  
  L_0^vac  = -c/24 -μ^2  k/4,      P_0^vac = -i μ k /2,
  
which implies that the expectation values of the T(x^+) and P(x^+) currents on the spectrally-flowed plane with coordinates (x^+, y) = (x^+, x^- + μlog x^+) are given by
  
  T(x^+)    = -k μ^2/4(x^+)^2,     P(x^+) = k μ/2 x^+. 

Using the transformation of the currents given in eqs. (<ref>) and (<ref>) we find that the following warped conformal transformation

  
  x^+ = (x^+/x^+ - ℓ)^1/n,      y = x^- + μlog(x^+/x^+ - ℓ), 
maps the expectation values of T(x^+) and P(x^+) on the spectrally-flowed plane (<ref>) to the corresponding expressions on _n (<ref>). Alternatively, the transformation

  
  x^+ = (x^+/x^+ - ℓ)^1/n,      x^- = x^- + μ(n-1/n) log(x^+/x^+ - ℓ) , 
maps the expectation values  on the reference plane, eq. (<ref>) with μ = 0, to the expectation values on _n given in (<ref>). 

Note that the warped conformal transformations (<ref>) and (<ref>) are independent of the length of the interval  along the x^- coordinate. This is a feature of the Rindler map used to derive the entanglement entropy in WCFT and is also reflected in eq. (<ref>).



apsrev4-1jhep