Complexity and Emergence of Warped AdS 3 Space-time from Chiral Liouville Action
PPrepared for submission to JHEP
Complexity and Emergence of WarpedAdS Space-time from Chiral LiouvilleAction
Mahdis Ghodrati, a,ba
Center for Gravitation and Cosmology, College of Physical Science and Technology,Yangzhou University, Yangzhou 225009, China b School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240,China
E-mail: [email protected]
Abstract:
In this work we explore the complexity path integral optimization pro-cess for the case of warped AdS /warped CFT correspondence. We first present thespecific renormalization flow equations and analyze the differences with the case ofCFT. We discuss how the “chiral Liouville action" could replace the Liouville actionas the suitable cost function for this case. Starting from the other side of the story,we also show how the deformed Liouville actions could be derived from the spacelike,timelike and null warped metrics and how the behaviors of boundary topologicalterms creating these metrics, versus the deformation parameter are consistent withour expectations. As the main results of this work, we develop many holographictools for the case of warped AdS , which include the tensor network structure forthe chiral warped CFTs, entangler function, surface/state correspondence, quantumcircuits of Kac-Moody algebra and kinematic space of WAdS/WCFTs. In addition,we discuss how and why the path-integral complexity should be generalized and pro-pose several other examples such as Polyakov, p-adic strings and Zabrodin actionsas the more suitable cost functions to calculate the circuit complexity. a r X i v : . [ h e p - t h ] D ec ontents /WCFT geometry 194.2 Deriving chiral Liouville action from warped AdS and deriving warped AdS /WCFT Various tools for studying the connections between the information on a d dimensionboundary and the emergent d + 1 dimensional bulk geometry have been developed inthe holographic setups. Entanglement entropy would certainly play an essential role– 1 –s it describes how information would be organized to arise an additional dimension.Therefore, the first holographic tool constructed in [1] was for entanglement entropyand then its covariant versions were discussed in [2].To understand better the interplay between information and geometry, math-ematical tools such as kinematic space in the setup of integral geometry in holog-raphy was developed in [3–5]. Tensor network (TN) models such as multi-scale-renormalization ansatz (MERA) [6] have been implemented to depict the construc-tion of geometry out of organizing information which create a renormalization groupflow in real space for various scales. Using this picture then the surface/state corre-spondence [7] has been proposed which could act as another framework for hologra-phy.Then, holographic complexity which acts as another duality between geometry,(volume of a subregion, or action on a Wheeler-de Witt patch), and boundary infor-mation has been advanced in various works, [8–10]. Another proposal for holographiccomputational complexity using path integral optimization procedure has been de-veloped in [11–14]. In this proposal, Liouville action plays the central role as for thetwo-dimensional CFTs, it would be the measure of computational complexity and itwould be the functional needed to be minimized. Considering their idea, one wouldthink how this procedure could work for other non-CFT quantum theories and whichfunctional should replace the Liouville action. This question was the main motivationfor the this work.One of the best non-CFT theories one could consider to study such questionsand developing various tools that have already been established for CFTs, is actu-ally the warped conformal field theories (WCFTs) and its duality with warped AdSspace times, ( WAdS / WCFT ) [15–23]. One reason this duality is very useful is thatwarped CFTs contain enough symmetries that many techniques of normal CFTscould still be applied. Also, in this setup, it has been shown that the entropy of athermal state matches with the entropy of a warped BTZ black hole. In addition, thecalculations for entanglement entropy and even subregion complexity have alreadybeen done for such solutions [24–26]. Furthermore, interesting relationships betweenWCFTs which show semi-local properties and other models such as SYK with com-plex fermions which has maximal chaos have been found in [27]. Specifically, thisfact would make studying the complexity of WCFTs more compelling, where study-ing the tensor network models of WCFTs and the behaviors of its partition functionunder renormalization, would also be useful to study the SYK model. There are evenconnections between WCFTs and cold atoms models [15, 28, 29] which make theminteresting even from the applicational point of view.In spite of these advantages, many schemes which reveal the connections be-tween information and geometry have not yet been applied for any of these non-CFTs, including WCFTs. In this work we aim to develop such programs for theWAdS / WCFT scenario. – 2 –e start by reviewing the warped conformal field theory, its global and localsymmetry algebra and its holographic dual warped AdS geometry, in section 2.1.Also, the related chiral Liouville gravity will be reviewed in section 2.2.Then, in section 3, we postulate the framework of path integral optimization,RG flow equations and holographic computation complexity for warped CFT case.The idea comes from [11], where using the Liouville action and by optimization of thepath integrals in conformal field theories, the time-slice of anti-de Sitter spacetimehas been derived. Consequently, one of the aims of this work is to derive “warped AdSspace" using “chiral Liouville gravity" with a similar optimization procedure wherethe background picture of tensor network is also in mind.The important difference here is the existence of two fields for the chiral gaugewhich both could run in the RG flow and the interplay between them produces awarped geometry where the chiral symmetry is broken. We specifically discuss theseissues in section 3.1. We show how two different cutoffs are needed, one for eachfield, and then how this affects the form of the entangler function and consequentlythe path-integral complexity. Then, in sections 3.3, we consider a scalar and a Weyl-fermion model of WCFTs where their specific actions have been derived in [20, 22].For these models, we explain how each parameter of the theory plays a role in theoptimization procedure and therefore the path-integral computation complexity.In section 4, we start from the other side. First, in section 4.1 we show howthe Liouville action could be derived starting from an AdS geometry, similar to theprocedure shown in [30]. Then, in sections 4.2 and 4.3, we do the same thing for thespacelike, timelike and null warped AdS . We derive the deformed, chiral action foreach metric and then we compare the role of the boundary, Chern-Simons term inconstructing these geometries, which the results are shown in figure 2.For the main part of this work, in section 5, we extend various holographic toolswhich have been discussed only for the AdS geometry in the literature. In section5.1, using chiral Liouville action versus Liouville gravity, we discuss the form oftensor networks and the kinds of gates which are needed to construct a warped CFTalgebra and as a result the warped AdS geometry. In 5.2, we study the structure ofentanglement entropy in warped CFTs and its evolution and the way each parametersuch as the warping factor would play a role. In 5.3, we discuss the implications ofsurface/state correspondence of [7] for the case of WAdS / WCFT . Also, there wediscuss the form of Fisher information metric for the case of warped AdS. In part 5.4,similar to the work of [14], we build the quantum circuits of Kac-Moody algebra andthen we calculate its cost functions and then the circuit complexity. In part 5.5, wecalculate the Crofton form of WCFTs and discuss the geometry of kinematic spacefor this case.In section 6, we discuss the importance of generalizing Liouville action as thecost function for calculating complexity, to Polyakov action and then its variousgeneralizations such as p-adic strings and Zabrodin actions.– 3 –inally, we conclude with a discussion in section 7. In this section, we first review warped conformal field theories (CFTs) and the chiralLiouville gravity. We discuss the symmetries, group algebra, operators, the metricof warped AdS, the derivation of chiral Liouville theory from Polyakov action usingchiral gauge, the equations of motion and the meaning of various parameters in theholographic picture. /WCFT In [31], the idea of omitting one of the four global symmetries of the CFT whichthen could lead to a chiral, Lorentz-breaking d QFT has been studied. This in factwould lead to a theory which has only one, “left moving, global scaling symmetry".The authors there discussed how the left conformal symmetry is still intact, whiledepending on the assumptions, the right translations would be enhanced either toa right conformal or a left U (1) “Kac-Moody symmetry". Here, we always considerthe later case.So the global symmetries of the warped CFT that we consider are x − → x − + a, x + → x + + b, x − → λ − x − . (2.1)Therefore, these kinds of field theories only lack the the dilation symmetry of x + → λ + x + . Also, instead of Brown-Henneaux boundary condition [32], the correctboundary algebra of WCFTs would be the CSS boundary conditions discussed in[33, 34].The goal is to study how the lack of this dilation symmetry for the x + case, (orany other global chosen, omitted symmetry which could lead to other field theories)and also perhaps the choice of boundary conditions, would affect the emergence ofspacetimes out of the boundary information and any proposed tensor network forsuch theories.The connections between local and global symmetries in the boundary and bulkin the AdS/CFT setup, their exact definitions and the interplay between them havebeen recently studied in more details in [35, 36]. Their results would help to constructthe tensor network model for each of these deformed CFTs. We would also like tosee how lack of this symmetry would change the optimization procedure proposed in[11, 12] and how it would change the properties of computational complexity.Along the way, we use many known results for the partition function, propertiesof the dual holographic gravitational theories such as chiral Liouville action, theproposed actions for WCFTs such as those found in [22], the anomaly behaviors, andthe structure of entanglement entropy and holographic complexity of warped CFTsderived in previous works [24, 37]. – 4 –t worths to mention here that in fact there are various examples in real worldfor these field theories, such as the continuum limit of large N chiral Potts model[31, 38] which is a spin model on a planar lattice. In this model each spin couldhave n = 0 , ..., N − values and to each pair of nearest neighbor of spins n and n (cid:48) , a Boltzman weight W ( n − n (cid:48) ) would be assigned. For the chiral case though W ( n − n (cid:48) ) (cid:54) = W ( n (cid:48) − n ) , but as these weights satisfy the Yang-Baxter equation,the model would be integrable. So considering these real-world systems and theirapplications, would make studying various quantum information properties of thesemodels, such as holographic complexity or their models of tensor network much moreinteresting, even from a practical point of view.So as the reminder, we note that the operators of WCFTs satisfy the followingcommutation relations on the plane [17] i [ T ξ , T ζ ] = T ξ (cid:48) ζ − ζ (cid:48) ξ + c π (cid:90) dx − ( ξ ” ζ (cid:48) − ζ ” ξ (cid:48) ) ,i [ P χ , P ψ ] = k π (cid:90) dx − ( χ (cid:48) ψ − ψ (cid:48) ,i [ T ξ , P χ ] = P − χ (cid:48) ξ , (2.2)where the local operators T ξ = − π (cid:90) dx − ξ ( x − ) T ( x − ) , P χ = − π (cid:90) dx − χ ( x − ) P ( x − ) , (2.3)are the left moving stress tensor and the left moving U (1) current respectively. Theright moving is associated with x − and left moving with x + .Then, with the change of coordinate x − = e iφ to go from plane to cylinder, theKac-Moody algebra of warped CFTs could be written as [17] [ L n , L m ] = ( n − m ) L n + m + c n ( n − n + 1) δ n + m , [ P n , P m ] = k nδ n + m , [ L n , P m ] = − mP m + n , (2.4)where L n = iT ξ n +1 , P n = P χ n are the operators on the cylinder and ξ n = ( x − ) n = e inφ are the test functions. Also, c is the central charge of the Virasoro algebra and k isthe Kac-Moody level.Considering a chiral gauge, one could consider a general metric for the back-ground of warped CFTs as ds = − e ρ ( t + ,t − ) (cid:16) dt + dt − − h ( t + , t − )( dt + ) (cid:17) , ∂ − h = 0 . (2.5)The symmetries acting infinitesimally on the coordinates t ± , instead of being δt + = (cid:15) + ( t + ) and δt − = (cid:15) − ( t − ) as for the conformal case would be δt + = (cid:15) ( t + ) and– 5 – t − = σ ( t + ) for the chiral case, which at the end will lead to two non-zero Noetherconserved charge j − (cid:15) and j − σ .We will explain further about this metric in the next part.In the holographic setup, these field theories holographically are the dual to thewarped AdS geometries in the bulk. The metric of warped AdS could be writtenas g W ADS = L (cid:18) dr r − r dx − + α ( dx + + rdx − ) (cid:19) , (2.6)which is a fibration over AdS with a squashing parameter α . For α = 1 , this is anAdS metric.The global isometry of these geometries is SL (2; R ) × U (1) and with a suit-able boundary condition this asymptotic symmetry group would be enhanced to theVirasoro-Kac-Moody algebra.Also, note that the above geometries appear in the near-horizon limit of four-dimensional extremal Kerr black holes where at fixed polar angle the three-dimensionalwarped AdS would appear, leading to the Kerr/CFT correspondence. Here, we review the chiral Liouville action, the chiral gauge and its connection toWAdS/WCFT.In a conformal gauge, any metric of a gravitational theory should satisfy g −− = g ++ = 0 , (2.7)which is dual to a conformal field theory without gravity where the component g + − acts as one of the fields. The classical d Liouville gravity is written in this gaugeand has a residual left and right Virasoro symmetry algebra.However, one could alternatively choose the “chiral gauge" [39] g −− = 0 , ∂ − ( g + − g ++ ) = 0 , (2.8)which on the boundary side would lead to a warped conformal field theory withoutgravity. The residual symmetry is a right Virasoro Kac-Moody algebra with no leftVirasoro symmetry group. The general metric which satisfy such constraints wouldbe of the form 2.5, ds = − e ρ ( dt + dt − − h ( dt + ) ) . (2.9)Note that from the chiral gauge constraint, we have also the relation ∂ − h = 0 .The usual Liouville gravity, in fact, has been written starting from the nonlocalPolyakov action [39] S L = c π (cid:90) d x ( Z R − λ √− g ) , (2.10)– 6 –ith the following definitions for the parameters, R ≡ √− gR,Z ( x ) ≡ (cid:90) d x (cid:48) G ( x, x (cid:48) ) R ( x (cid:48) ) , √− gg ab ∇ a ∇ b G ( x, x (cid:48) ) = δ ( x, x (cid:48) ) . (2.11)Then, by gauge fixing it, using the conformal gauge 2.7, one gets the Liouvillegravity theory.For the metric 2.5, the Ricci density would be R = 4 ∂ − ∂ + ρ + 4 h∂ − ρ + [4 ∂ − h∂ − ρ + 2 ∂ − h ] = − √− g ∇ ρ + [2 ∂ − h ] . (2.12)The terms in the bracket will vanish only when the gauge condition ∂ − h = 0 isimposed. For such case, considering the chiral gauge, the scalar curvature would onlydepend on the gradient of the field ρ . This point would be important in constructingthe tensor network and understanding the emergent bulk spacetimes out of thesechiral theories.Additionally, the equations of motion for ρ and h from the action (2.17) wouldbe [39] ∂ + ∂ − ρ = 0 , ∂ − ∂ + ρ + 2 h ∂ − ρ + ∂ − h = 0 . (2.13)On a solution of (2.13), one could get the complexity of a quantum state whichis specified by the boundary conditions of ρ and h .To study the chiral Liouville theory in a fixed sector for the left moving zeromodes, an additional term would be added to S L in the following from [39] S L = S L + ∆4 π (cid:90) d x √− gg −− = S L − ∆2 π (cid:90) d xh, (2.14)and the equation of motion for g −− would be T −− = T −− + ∆2 = 0 , where ∆ / is theleft-moving energy density. Now the idea is to look for the effects of this additionalterm on the computational complexity and the tensor network structures. Setting t − as the time, the above action could then be written as the form of relation 2.16.In [12], one way of evaluating computational complexity is proposed to be derivedby minimizing the Liouville action, where each term, i.e, the kinetic and the potentialparts, would determine the number of quantum gates in the tensor network as [12, 40] S L = c π (cid:90) dx (cid:90) ∞ (cid:15) dz (cid:104) ( ∂ x φ ) + ( ∂ z φ ) (cid:124) (cid:123)(cid:122) (cid:125) of Isometries + δ − e φ (cid:124) (cid:123)(cid:122) (cid:125) of Unitaries (cid:105) . (2.15)Now we would like to check if one could also derive the time slice of warped AdS from warped CFTs, by using the chiral Liouville theory , and also by considering the– 7 –onnections between the different terms in that action with the number of quantumgates of the tensor network structure.Furthermore, in [40], the complexity of path integral C [ φ ] has been character-ized as the functional of φ ( z, x ) where varying that could determine the minimallycomplex preparation of the state, leading to the derivation of Einstein equations byvarying complexity. One would expect the same strategy could work here as the com-plexity of a path integral carried over the warped AdS would be the chiral Liouvilleaction. So the equations of motion 2.13 could be derived by varying the complexityhere. For demonstrating this idea, we could also use the methods of [11].In fact, in our previous work [24] we proposed the following connections, S L = c π (cid:90) d x ∂ + ρ∂ − ρ (cid:124) (cid:123)(cid:122) (cid:125) of Isometries − Λ8 e ρ (cid:124) (cid:123)(cid:122) (cid:125) of Unitaries + h ( ∂ − ρ ) + [ ∂ − h∂ − ρ ] − c h ∆ (cid:124) (cid:123)(cid:122) (cid:125) of WCFTs new gate, Chiraleons , (2.16)or one can also write this action in the following form S = S L + (cid:90) dt + dt − ∂ + φ∂ − φ (cid:124) (cid:123)(cid:122) (cid:125) of Isometries + h∂ − φ∂ − φ (cid:124) (cid:123)(cid:122) (cid:125) of WCFTs new gate − m e ρ φ (cid:124) (cid:123)(cid:122) (cid:125) of Unitaries . (2.17)Similar to the case of [3–5], we took the kinetic term of the chiral action to beproportional to the number of isometries, and the potential term to the number ofunitaries or disentangler [41]. As no combinations of these gates could lead to thedeforming term, then for creating this term, one would need an additional kind ofquantum gates which we dub “chiralons".So the chiral theory here could be considered as the usual CFT deformed by aterm, h∂ − φ∂ − φ . In this term h is proportional to a right moving current and hasdimension (1 , , the term ( ∂ − φ ) has dimension (0 , , therefore the middle term isa dimension (1 , operator. Such deforming operators which appear in WCFT andKerr/CFT are related to the IR limit of the “dipole deformed gauge theories" [42, 43].For constructing a tensor network structure for these theories, one could imagineseveral subtleties and complications. For instance the chirality of WCFTs could breakthe sense of locality of the network. However, it has been shown that, in fact thesetheories are semi-local, and therefore have enough locality to be constructed using a“semi-local entangler function". Also, the involuted way of coupling the two fields h and ρ would be another issue, though considering enough constraints such as ∂ − h = 0 would make many simplifications here, as it has been observed in the relation for Ricciscalar 2.12. Also, the fermion doubling issue when a fermion theory is being put ona lattice should be in mind too which we will discuss further later.– 8 –here are though beautiful studies of the fractal structure of gravity models as in[44, 45] in the setup of γ -Liouville quantum gravity (LQG) surfaces, where γ ∈ (0 , which could be applied here. One could check how this structure would be relatedto the emergence of spacetime, the structure of tensor network, and the complexity.For instance one of the Kac-Moody current j σ mentioned here is related to one of theKnizhnik-Polyakov-Zamolodchikov (KPZ) SL (2 , R ) currents in [45]. Further methodsof discretizations of these theories could be studied. Now in this section, similar to the studies of [11–14], we develop a path-integraloptimization formulation for the computation complexity of warped CFTs.
To understand the mechanism of AdS/CFT and the emergence of spacetimes out ofinformation, the idea of multi-scale renormalization ansatz (MERA) has been appliedin [6]. To further study such idea, specially for the continuum limit, the optimizationprocedure of Euclidean path-integral which can evaluate a CFT wave functional hasbeen introduced in [11, 12]. This optimization would be performed by minimizingthe Liouville action.We could then apply the same mechanism for the case of a chiral, warped CFT .Our conjecture is to replace the Liouville action with the chiral Liouville actionintroduced in previous section, and then check how different symmetries, anomaliesand chirality would affect the process of emergence of spacetime out of boundary fieldtheory. For that, one needs to understand the microscopic structure of complexityfor the case of warped CFTs where the left and right modes behave differently. Sowe would like to understand how a warped CFT state | Ψ Σ (cid:105) , and then the warpedgeometry, could be constructed using the tensor network description and from asimilar optimization procedure.A relation similar to [30] in the form of e C ( M Σ ) . Ψ Σ [ ϕ ( t + )] = (cid:90) (cid:34) (cid:89) r ∈ M Σ Dϕ ( r ) (cid:35) e − S WCFT M Σ [ ϕ ] (cid:89) x ∈ Σ δ ( ϕ ( t + ) − ϕ ( t + )) , (3.1)for the case of warped CFT states should be derived . In the above relation, S WCFT M Σ [ ϕ ] should be replaced by the action for WCFT theories, which would be the chiralLiouville action as discussed previously or other Lagrangians for warped CFTs whichwe will introduce later. The optimization procedure then would be performed byminimizing the functional C L ( M Σ ) with respect to the two fields of ρ and h .The plot of the manifold and its boundary is shown in figure 1. Note that inthe WCFT case we don’t have the complete conformal invariance, therefore, the– 9 – igure 1 . The relation between the codimension one surface M Σ and where it ends on Σ could be the same as those of [30] shown in the left side and the quantum circuit modeledby the path-integration on M Σ Σ connecting Σ to Σ is modeled in the right side. Weshow however later that a twist is needed to be considered for the WCFT/WAdS case. state | Ψ Σ (cid:105) after path integration could depend not only on Σ but the whole M Σ .However, in the warped CFT case, there is still a warped analogue of the conformaltransformation connecting the metrics of the two surfaces M (1)Σ and M (2)Σ which willbe used to simplify the process.So first, deriving the corresponding equation of motion of (2 . in [12], for thecase of warped CFT and chiral Liouville action would lead to ∂ + ∂ − ρ + 3 k c h + h ( ∂ − ρ ) + Λ8 e ρ = 0 , (3.2)where we also have ∂ − h = 0 . One could see from this equation how these two fieldsare coupled, behave differently in left versus right, and how central charge and Kac-Moody level simultaneously play a role. After choosing a boundary condition, oneneeds to solve these coupled equations for the field h and ρ which then lead to thetime-slice of warped AdS .Then, the renormalization group flow for the case of WCFTs could be consideredin order to understand further its path-integral optimization complexity. In fact,for this case, there should be two position dependent cut offs which would be thefunctions of coordinates ( t + , t − ) as Λ = e ρ ( t + ,t − ) , Λ = e h ( t + ,t − ) . (3.3)If we want to have a theory that would not change after optimization, both cutoffsshould satisfy the RG equation, Λ i dλ i (Λ i ) d Λ i = β ( λ i (Λ i )) , (3.4)where β is the beta function. – 10 –he Callan-Symmanzik equation for the partition function Z and also the cor-relation function would satisfy the following equations (cid:18) Λ ρ dd Λ ρ + β ( λ ρ ) ddλ ρ + ξ ( λ ρ ) (cid:19) Z [Λ ρ , λ ρ ] = 0 , (cid:18) Λ h dd Λ h + β ( λ h ) ddλ h + ξ ( λ h ) (cid:19) Z [Λ h , λ h ] = 0 . (3.5)The ξ ( λ ) functions here would break the conformal symmetry leading to theWCFT case.We then need to understand how to couple these two RG equations for these twofields. As for a general partition function Z for the whole system, we could have thefollowing relation (cid:18) Λ ρ dd Λ ρ + Λ h dd Λ h + β ( λ ρ ) ddλ ρ + β ( λ h ) ddλ h + ξ ( λ ρ ) + ξ ( λ h ) (cid:19) Z [Λ ρ , Λ h , λ ρ , λ h ] = 0 . (3.6)In [46, 47], the renormalization group flow equations for field theories with quenchdisorder as an example of non-relativistic, coupled RG flow have been studied wheretheir couplings could vary randomly in space. In the case they have studied, thecouplings were also mixed together, similar to the situation for WCFTs. So theirmethods could be applied here as well. So in the next parts, using various meth-ods, we try to understand these coupled renormalization group flows for the case ofWCFTs better. As mentioned, Liouville action is always written in the conformal gauge, it is localand has a left and right Virasoro algebra. However, in the chiral gauge, the chiralLiuoville theory has only the right-moving Virasoro Kac-Moody algebra and it is infact semi-local.Also, for the case of CFT we have the ground state and its Virasoro descendants.For the case of warped CFTs, however, we just have a left Virasoro and a right U (1) Kac-Moody algebra. As the right and left descendants behave differently, theEuclidean path integral optimization process that prepares the left and right movingstates would be different.For the ground state of CFT we have the relation
Ψ( ˜ ϕ ( x )) = (cid:90) e − S CFT ( ϕ ) (cid:89) x (cid:89) δ For a free scalar field ϕ , in [22], the Lagrangian has been constructed as S = (cid:90) dt + dt − √ γ (cid:110) 12 ( ω µ ∂ µ ϕ ) + 14 (cid:16) ω µ ∂ µ N − N (cid:17) ϕ − m ϕ (cid:111) , (3.22)where N = (cid:15) µν ∂ µ n ν . The first term is the kinetic term, the last term is the potentialterm and the middle one models the chiral coupling to the curvature. In the flatspace this action would become [22] S = (cid:90) dt + dt − (cid:110) 12 ( ∂ + ϕ ) − m ϕ (cid:111) . (3.23)In the above action the kinetic term which corresponds to the number of disentan-glers or isometries in the tensor network structure has only a right moving direction.Therefore, the disentanglers only would influence these right-moving modes leadingto such a mathematical relation.The mass here could be considered as a small, perturbative parameter whichis a function of both of the coordinates, in the form of m ( t + , t − ) . Then, similarprocedure to [13] could be applied to derive the normalization functional and thepath integral complexity for this theory. Note here though the operator ϕ is not aprimary operator.The solution for the the scalar field of this action is ϕ = ϕ + ( t − ) e im ( t + ,t − ) t + + ϕ − ( t − ) e − im ( t + ,t − ) t + , (3.24)where the parameter m is setting the Kac-Moody level. Also, these theories admitan infinite number of “exactly marginal non-local deformations".The stress tensor T and momentum P , as the most important physical parame-ters of the theory, have been calculated in [22], leading to the following relations, T = 14 π ( ∂ − ϕ∂ + ϕ − ϕ∂ − ∂ + ϕ ) ,P = 14 π (( ∂ + ϕ ) − m ϕ ) . (3.25)To search for the normalization functional in the path integral complexity ofwarped CFTs, one then should use the correlation functions, or specifically the re-tarded Green’s function for the primary operators of WCFTs calculated in [23], as G R ( ω ) ∼ β − e Q ¯ qβ sin (cid:0) π ∆ − iQ ¯ qβ (cid:1) Γ (cid:16) ∆ + i ω + Q ¯ q π/β (cid:17) Γ (cid:16) − ∆ + i ω + Q ¯ q π/β (cid:17) , (3.26)– 15 –here ∆ is the weight and Q is the charge of the primary operator which defines thestate O (0 , ∼ | ∆ , Q (cid:105) . Also, the two-point function for the twist operator Φ( X, Y ) which has dimension ∆ n and charge Q n has been found in [23] as (cid:104) Φ n ( X , Y )Φ † n ( X , Y ) (cid:105) C ∼ e iQ n (cid:16) Y − Y + ¯ β − αβ ( X − X ) (cid:17) (cid:18) βπ sinh π ( X − X ) β (cid:19) − n , (3.27)where here Y specifies the classically U (1) preferred axis, and X denotes the quantumanomaly selected axis with a scaling SL (2 , R ) symmetry. The thermal identificationhere is ( X, Y ) ∼ ( X + iβ, Y − i ¯ β ) . Also, ∆ n and Q n could be written as ∆ n = n (cid:18) c 24 + L vac n + i P vac α nπ − α k π (cid:19) , Q n = n (cid:18) − P vac n − i kα π (cid:19) . (3.28)Knowing the Green’s function, similar to equation ( A. of [13], the normaliza-tion functional for calculating the path-integral optimization could be found as N ∼ S CL − m (cid:90) dt + dt − √ gG R ( ω ) . (3.29)From this, one could see how each parameter such as the Kac-Moody parameter,central charge, the tilt angle α , or the mass term change the path integral complexity.Additionally, since the correlation functions of warped CFTs have already beenfound in [23], then using similar relations, (5 . and (5 . of [13], for the WCFTs,one could find the first and second order of the normalization functional in terms ofthe correlators of WCFTs on a half plane with the boundary at t + = (cid:15) as N ∼ S CL + N pt + N pt + .... (3.30)where N pt ∼ (cid:90) dt + dt − e ρ ( t + ,t − ) λ ρ ( t + ) λ h (cid:104) O (cid:105) . (3.31)For the second order, the following relation could be written in terms of the twopoint correlation function, as N pt ∼ (cid:32) (cid:89) i =1 (cid:90) dt + i dt − i √ g (cid:33) λ ρ ( t +1 ) λ ρ ( t +2 ) λ h ( t +1 ) λ h ( t +2 ) × (cid:16) (cid:104) O ( t +1 , t − ) O ( t +2 , t − ) (cid:105) − (cid:104) O ( t +1 , t − ) (cid:105)(cid:104) O ( t +2 , t − ) (cid:105) (cid:17) . (3.32)The exact normalization functional which has a more complicated form woulddepend on both fields φ and h , the coupling parameter for each of them, and alsothe Kac-Moody parameter, as well as central charge which could be calculated fromthe relation 3.29. In this process, further methods of [49], for calculating the heatkernel of such Lorentz-violating theories would be very helpful.– 16 – .3.2 The Hofman/Rollier theories Similar to [13], other forms of actions could be considered for the warped CFT case.For the first time, in [20], using warped geometry, Hoffman and Rollier wrote twoLagrangians for the WCFTs, which were a theory of “Weyl fermions" with a Lorentz-violating mass and also a warped “bc" system which then for the case of free scalarWCFTs were reviewed and extended further in [22]. For each of these theories, similarto [13], the optimization procedures and then, the derivation of the wave functionalbehaviour could be studied.Considering an anti-commutating, complex spinor field Ψ = (Ψ − , Ψ + ) , with itsconjugate ¯Ψ one could write the action in the following form, where the mass is afunction of the coordinates as in [13] S weyl = (cid:90) dt + dt − √ γ { i Ψ − ω µ ∂ µ Ψ − + m ( t + , t − )Ψ − Ψ − } . (3.33)In the above action, Ψ − , Ψ + are the two components of an anti-commuting field Ψ which under boosts transform as ¯Ψ + → ¯Ψ + + ψ − , ¯Ψ − → ¯Ψ − , (3.34)and also Ψ − → e − Ω2 Ψ − , ¯Ψ − → e − Ω2 ¯Ψ − . (3.35)The dimension of the fermion field here is / .It worths to refer here to another action called warped bc theory which has areal spinor Ψ written in the form S bc = (cid:90) d x √ γ { i Ψ − ν µ ∂ µ Ψ − − i Ψ + ω µ ∂ µ Ψ − − m Ψ + Ψ − } . (3.36)In [50], the partition function for a free warped Weyl fermion with a complexanti-commuting field Ψ and with the action I = (cid:90) dtdϕ ( i ¯Ψ ∂ + Ψ + m ¯ΨΨ) , (3.37)has been calculated as ˆ Z ( z (cid:12)(cid:12) τ ) = T r (cid:16) e πiz ˆ P e − πiτ (ˆ L − c ) (cid:17) , (3.38)where ˆ L and ˆ P are the plane generators. For this system, as the first approximation,the functional which should be minimized could be given by the log of this partitionfunction, S ∝ log ˆ Z , which would depend on both of the generators of the algebra.– 17 –n fact in [20], it has been shown that the partition function of WCFTs definedon a non-trivial background, up to a quantum anomaly, has a notion of warped Weylsymmetry as Z [ A µ , ¯ A µ ] ∼ Z [(1 + γ ) A µ , ¯ A µ + νA µ ] , (3.39)where γ and ν are arbitrary deformation parameters which only depend on the space-time coordinates.There would also be another interesting example of dual bulk geometry for thewarped CFTs, which is a SL (2) × U (1) Chern-Simons model, dubbed lower spingravity , written in the following form [20, 51] S = k (cid:90) bulk B ∧ dB + B ∧ dB − B ∧ dB + 2 B ∧ B ∧ B − ξ (cid:90) bulk ¯ B ∧ d ¯ B. (3.40)In the above action, we have k = κc , where κ is an overall normalization and ξ = +1 , , − is the sign of the last term which depends on the parameter of thetheory. So this bulk theory similar to the dual WCFT case, has a free continuousparameter k which is dual to the central charge and a discrete parameter ξ whichdetermines the level of U (1) Kac-Moody algebra. Therefore, this action could beconsidered as the minimal bulk gravity theory dual to WCFTs. The parameters ofthis action could be then connected to the parameters of warped AdS noted in [20].The main point here is that, as mentioned in [22], both of the actions 3.33 and3.36 could admit an infinite number of exactly marginal deformation operators and itmeans both are on the “brink" of being non-local along x + , so building an RG flow forthese specific theories using a “semi-local" entangler K would be rather challenging.However, still some constraints on WCFTs could be considered which tune awaysuch non-localities and therefore still building a physical tensor network would bepossible.As the Hofman/Rollier theories describe a fermion system and our objective is tocalculate the discretized path integral optimization and finding the tensor networkstructure for such systems, several subtleties involving issues about fermions on alattice should be considered. First, as has been shown in [52, 53], the chiral theoriesare renormalizable both within and outside of perturbation theory which lead to theconclusion that the theory in fact would be physical.Another issue is the fermion doubling problem [54] which arrises when one triesto put fermionic fields on a lattice leading to the appearance of spurious statessuch as d fermion particles for each previous fermion, where here d is the numberof discretized dimension. There is also the Nielsen-Ninomiya theorem [55], whichshows that a local, real free fermion lattice action which has chiral and translationalinvariance would necessarily has fermion doubling. However, there are various ways– 18 –o circumvent this problem, such as “perfect lattice fermions", “Wilson fermions",“twisted mass fermions", “domain wall fermions", “overlap fermions", “interactingfermions", " staggered fermions ", etc. Specifically the staggered fermion introducedby Susskind and Kogut [56], could be of interest as it introduces a novel "nonlocalaction" similar to the actions presented here.Also, note that for the construction of tensor networks, one could use additionaltensors such as “ smoothers " introduced in [57], or “ chiraleon " introduced here whichcould help to circumvent some of these problems.In [58], already the tensor network for fermionic topological quantum states hasbeen worked out where a Grassmann number tensor network ansatz for a fermionictwisted quantum model has been formulated. There, the fermionic projected entan-gled pair state fPEPS for a simple string-net model has been shown. One could alsoextend their procedure for the above chiral system. In the previous section, we generally tried to start from the boundary warped CFTsand using tools such as path-integral optimization complexity, the possible structureof RG flow and the form of Hamiltonian and different Lagrangian of the theory,understand the emergence of warped spacetimes out of a chiral theory. In this section,however, we start from the other side of the story. We will first take the metric ofwarped AdS and by using the form of the induced metric on a boundary profilederive a deformed chiral action.First, we review the procedure of deriving Liouville action from AdS similar tothe approach of [30] and then we repeat it for the warped AdS and chiral Liouvilleaction. geometry Considering the Euclidean Poincare AdS as ds = R AdS ( dz + dT + dX ) /z , (4.1)we could take the boundary on the cut off point at z ≥ (cid:15).e − ˜ φ ( T,X ) . We also have therelation R z = e φ (cid:15) . (4.2)The induced metric on the boundary z = (cid:15) . e − ˜ φ would be ds = e φ (cid:15) (cid:104)(cid:16) (cid:15) e − φ ( ∂ T ˜ φ ) (cid:17) dT + 2 (cid:15) e − φ ( ∂ T ˜ φ )( ∂ X ˜ φ ) dT dX + (cid:16) (cid:15) e − φ ( ∂ X ˜ φ ) (cid:17) dX (cid:105) , (4.3)– 19 –nd then the extrinsic curvature on this boundary could be found as K = R − AdS . (cid:16) − (cid:15) e − φ ( ∂ T + ∂ X ) ˜ φ (cid:17) . (4.4)Then, the bulk gravity action could be calculated as I G = 14 πG N R AdS (cid:90) N √ g − πG N (cid:90) M √ γK = − c π (cid:90) dT dX (cid:34) e φ (cid:15) + ( ∂ T ˜ φ ) + ( ∂ X ˜ φ ) (cid:35) . (4.5)The metric of CFT on the boundary would be ds CFT = e φ ( t,x ) ( dt + dx ) . (4.6)To go from the coordinates ( T, X ) to ( t, x ) we need to calculate the Jacobian J such that dT dX = J dtdx . This J could be found as e φ = J e φ (cid:15) (cid:114) (cid:15) e − φ (cid:16) ( ∂ T ˜ φ ) + ( ∂ X ˜ φ ) (cid:17) . (4.7)Till here we actually derived a relation which is very similar to the equation (22)of [59] D L (cid:39) (cid:90) dtdy (cid:15) (cid:113) a + (cid:15) η ( ∂a ) ( ∂ y a ) + (cid:15) η ( ∂b ) ( ∂ y b ) + ..., (4.8)where a and b are the parameters of the metric ds = (cid:0) a ( t, y ) + b ( t, y ) (cid:1) dt + 2 b ( t, y ) dtdy + dy , (4.9)where the constant Euclidean time t slices are flat lines.In fact, the connection with the DBI action D DBI (cid:39) (cid:90) d χ (cid:113) det ( g µν + (cid:15) ∂ µ χ∂ ν χ ) , (4.10)for the case of zero angle σ where t (cid:48) = − ˙ t tan σ has been proposed in [59]. Thedual action of DBI case for the warped case could be considered.So getting back to our calculations, using the Jacobian 4.7, one could write thefirst term of I G as (cid:90) dT dX e φ (cid:15) (cid:39) (cid:90) dtdx (cid:20) e φ − (cid:0) ( ∂ t φ ) + ( ∂ x φ ) (cid:1)(cid:21) . (4.11)Putting this into (4.1) would lead to I EG = − c π (cid:90) dtdx (cid:104) ( ∂ t φ ) + ( ∂ x φ ) + 2 e φ (cid:105) , (4.12)– 20 –s it was shown in the appendix section of [30]. Now we could repeat these calcula-tions for the warped AdS .Before doing that, we should remind that, from the perspective of [59], theLiouville action is a particular cost function. The chiral Liouville action then couldbe considered as a particular cost function for the non-local and Lorentz-breakingtheory of warped CFTs. By fine-tuning the cost function in the case of Liouville, onecould recover a complexity measure of the same form. Now, we could check how thisfine tuning would work for the case of chiral Liouville action. In this section, we repeat the previous calculations for the case of various warpedAdS to check what kind of geometries could be derived and therefore to get furtherinformation about the kind of quantum circuits which could be built for a chiral,warped geometry.The metric of warped AdS in the Poincare coordinate could be written as ds = L W AdS (cid:18) dr r − r dX − + α ( dX + + rdX − ) (cid:19) = L W AdS (cid:18) dr r + dX − (cid:0) α r − r (cid:1) + α dX +2 + 2 α rdX + dX − (cid:19) , where comparing with the notation of [15], one could write L W AdS = (cid:96) ν + 3 , α = 4 ν ν + 3 . (4.13)Again, considering a simple and naive cutoff surface as r = (cid:15) . e − ˜ φ ( X + ,X − ) , onecould find the induced metric on this surface as ds = e φ (cid:15) L W AdS (cid:104)(cid:16) r α + (cid:15) e − φ ( ∂ X + ˜ φ ) (cid:17) dX +2 + 2 (cid:15) e − φ ∂ X + ˜ φ ∂ X − ˜ φ dX + dX − + (cid:16) r ( α − 1) + (cid:15) e − φ ( ∂ X − ˜ φ ) (cid:17) dX − (cid:105) . It could be seen that the difference between this induced metric and the corre-sponding one for the AdS case, eq. 4.3 is much less than their actual metrics.The extrinsic curvature on this boundary would be K = L − W AdS . (cid:18) L W AdS r − (cid:15) e − φ (cid:18) r α ∂ X + ˜ φ + 1 r ( α − ∂ X − ˜ φ (cid:19)(cid:19) . (4.14)One could compare this with the corresponding term for the AdS metric in 4.4.The metric of warped CFT on the boundary is ds = − e ρ ( t + ,t − ) (cid:16) dt + dt − − h ( t + , t − )( dt + ) (cid:17) . (4.15)– 21 –o go from ( X + , X − ) to ( t + , t − ) coordinates, one could find the correspondingJacobian dX + dX − = J dt + dt − as i e ρ ( t + ,t − ) = J . e φ (cid:15) (cid:114) r α ( α − 1) + (cid:15) e − φ (cid:16) r ( α − ∂ X + ˜ φ ) + r α ( ∂ X − ˜ φ ) (cid:17) , (4.16)The above relation could then give a hint to derive a DBI action of WCFT case.The final action that could be derived from this calculation would be I G = − L πG N (cid:90) dt + dt − (cid:20) (1 + 3 Lr ) (cid:16) ( ∂ t + φ ) + ( ∂ t − φ ) (cid:17) + r α ( α − Lr ) e φ (cid:21) . (4.17)The first term is the deformed kinetic term and the second term is the deformedpotential term.However, this relation is not exactly correct, since we used the Einstein gravityalong the way, which does not have enough fields to produce the needed IR behaviorand the exact warped AdS geometry as we want. Another action that we couldconsider could be the one shown in relation 3.40, or other higher derivative gravitytheories which have the warped AdS as their solutions such as Chern-Simons, Newor Topologically Massive Gravity (TMG).Here, we use the TMG for the gravity action. In the first order formalism, itsaction with a negative cosmological constant Λ = − /(cid:96) could be written as [60] I = − πG (cid:90) M (cid:15) ABC (cid:18) R AB + 13 (cid:96) e A e B (cid:19) e C + 132 πGµ (cid:90) M (cid:0) L CS ( ω ) + 2 λ A T A (cid:1) + (cid:90) ∂M B. (4.18)In the above action, M is a three-dimensional manifold where x µ are the localcoordinates, G is the gravitation constant, µ is a constant parameter with the dimen-sion of mass and L CS is the gravitational Chern-Simons 3-form with the followingrelation L CS ( ω ) = ω AB dω BA + 23 ω AB ω BC ω C A . (4.19)By defining the dreibein e A = e Aµ dx µ and the spin connections ω AB = ω ABµ dx µ one could write the curvature 2-form as R AB = 12 R ABµν dx µ dx ν = dω AB + ω AC ω CB . (4.20)For the TMG case, in [60], the boundary term which makes the variationalprinciple well-defined were introduced as B = 132 πG (cid:15) ABC ω AB e C . (4.21)– 22 –his action then could be used after finding the Jacobians.In topological and also Chern-Simons theories, the contribution of the boundaryterm is its most significant part as it is the case for the modes on the boundary oftopological matters. Therefore, the gates dual to the boundary part play the mostsignificant role.One could also consider a parity-preserving theory such as New Massive Gravity(NMG) which contains warped AdS as its solutions, and then compare the resultwith the previous cases. The path-integral optimization complexity and the kind ofquantum gates or circuits needed would take a different form in that case.To review that theory, one could note that the action of NMG is I = 116 πG (cid:90) M d x √− g (cid:20) R − − m (cid:18) R µν R µν − R (cid:19)(cid:21) , (4.22)where m is a dimensionful parameter.This theory could also be written in the following form [61] I = 116 πG (cid:90) M d x √− g (cid:20) R − 2Λ + f µν G µν + m (cid:0) f µν f µν − f (cid:1)(cid:21) , (4.23)where in the above term, G µν is the Einstein tensor and the auxiliary field f µν is f µν = − m (cid:18) R µν − d + 1) Rg µν (cid:19) . (4.24)The Gibbons-Hawking boundary term would be I GGH = 116 πG (cid:90) ∂ M d x √− γ (cid:16) − K − ˆ f ij K ij + ˆ f K (cid:17) , (4.25)where K ij is the extrinsic curvature of the boundary and K = γ ij K ij is the trace ofthe extrinsic curvature.The auxiliary filed ˆ f ij is also defined as ˆ f ij = f ij + 2 h ( i N j ) + sN i N j . (4.26)The functions in the above relation are defined from the following ADM metric ds = N dr + γ ij ( dx i + N i dr )( dx j + N j dr ) . (4.27)In the NMG action, the mass of graviton plays the essential role which affectsvarious quantum information quantities of the model, such as entanglement and itsevolutions during quench processes, entanglement and complexity of purification, etc,which were studied in more details in [24, 37, 62, 63].Since this action is parity-preserving, one would expect to see a final simplertensor network structure and quantum circuit model. This point has been validated– 23 –y constructing and comparing the Hawking-Page phase diagrams for both NMGand TMG warped black hole solutions in [64–67].The effect of corner terms in the form of [68] I c = 116 πG N (cid:90) Σ √ γ (2 α − π ) . (4.28)should also be considered at the end as it could affect the form of the circuit com-plexity and therefore the corresponding final tensor network model. and deriving warped AdS Before deriving a chiral, deformed Liouville action starting from a warped AdS andusing TMG, it is better to understand the components of warped AdS geometry, itsfibration form and its Killing vectors.The metric of AdS could be written as a spacelike (or timelike) fibration withfiber coordinate u or τ as ds = (cid:96) − cosh σdτ + dσ + ( du + sinh σdτ ) ] (4.29) = (cid:96) σdu + dσ − ( dτ + sinh σdu ) ] , (4.30)which has SL (2 , R ) L × SL (2 , R ) R isometries. By multiplying the fiber by a warpingfactor, one could break the isometry group to SL (2 , R ) × U (1) and derive severalsolutions of warped AdS as in [15].In addition, in [15], the Killing vectors for various warped AdS metrics havebeen derived which would be useful in studying the boundary CFTs and the inducedmetric on it. As it has been shown, the SL (2 , R L ) isometries are given by J = 2 sinh u cosh σ ∂ τ − u∂ σ + 2 tanh σ sinh u∂ u ,J = 2 ∂ u ,J = − u cosh σ ∂ τ + 2 sinh u∂ σ − σ cosh u∂ u . (4.31)Also, the isometries of SL (2 , R ) R are ˜ J = 2 sin τ tanh σ∂ τ − τ ∂ σ + 2 sin τ cosh σ ∂ u , ˜ J = − τ tanh σ∂ τ − τ ∂ σ − τ cosh σ ∂ u , ˜ J = 2 ∂ τ . (4.32)The point here is that, for the spacelike warped anti-de Sitter case, the Killingvectors are given by the SL (2 , R ) R and J , so for this case we should choose u as the– 24 –adial coordinate. For the timelike warped anti-de Sitter case, however, the Killingvectors are given by the SL (2 , R ) L and ˜ J , so for this case we choose τ as the radialcoordinate. For the null warped AdS, the Killing vectors would be N = ∂ − , N = x − ∂ − + u ∂ u , N − = ( x − ) ∂ − − u ∂ + + x − u∂ u , N = ∂ + . (4.33)For this case then u should be chosen as the radial coordinate. In this part, we start from a spacelike warped AdS metric and using TMG and theprocedure introduced in [30], we derive a form of a deformed chiral Liouville actionand then we study its properties.The spacelike or hyperbolic warped anti-de Sitter solution could be written as ds = (cid:96) ( ν + 3) (cid:20) − cosh σdτ + dσ + 4 ν ν + 3 ( du + sinh σdτ ) (cid:21) . (4.34)The case of ν > , is spacelike stretched AdS , and the case of ν < is spacelike squashed AdS .Note that near the boundary of spacelike warped AdS , where u → ∞ we couldget the following geometry ds ( νν +3 (cid:96) ) (cid:39) (cid:18) sinh σ − cosh σ (cid:0) ν + 34 ν (cid:1)(cid:19) dτ + ν + 34 ν dσ . (4.35)In this limit we could remove the non-diagonal term σdudτ as well asthe part du . This is the warped analogue of conformal transformation of the flatspacetime. This point to the fact that the warped CFT vacuum could be defined bythe path-integral, and the two surfaces of M (1)Σ and M (2)Σ in the bulk warped AdS could be related by the Weyl and boost transformations in the UV region.The lattice spacing then is determined by the coordinate u and the lattice sitecorresponds to the unit area measured by ds / ( νν +3 (cid:96) ) . In the warped AdS case,similar to the case of [30] for AdS, one could deform the shape of each surface inthe bulk using path-integral optimization and create a duality between the surfacesin warped AdS and the non-unitary quantum circuits of path-integration with anappropriate UV cut off which we will discuss further below.So, in the bulk side, we consider the action I TMG = 116 πG (cid:90) M dσ du dτ √− g (cid:18) R + 2 (cid:96) (cid:19) + (cid:96) πGν (cid:90) M dσ du dτ √− g(cid:15) λµν Γ rλσ (cid:18) ∂ µ Γ σrν + 23 Γ σµτ Γ τνr (cid:19) + 132 πG (cid:90) ∂ M du dτ (cid:15) ABC ω AB e C , – 25 –nd then calculating each term of this action for the spacelike warped AdS and thenby performing the coordinate transformation, we could get a new gravity theorywhich corresponds to the complexity functional of path-integral optimization for thewarped AdS.The contribution from the first term is πG (cid:90) M dσ du dτ √− g (cid:18) R + 2 (cid:96) (cid:19) = − ν(cid:96) (3 ν + 8)4 πG ( ν + 3) (cid:90) dudσdτ cosh σ, (4.36)as the even part of the first term is (cid:15) λµν Γ rλσ ∂ µ Γ σrν ( even ) = 4 ν (2 ν − ν + 3) cosh σ, (4.37)and the odd part of the first term would give (cid:15) λµν Γ rλσ ∂ µ Γ σrν ( odd ) = 4 ν cosh σ ( ν + 3) , (4.38)leading to (cid:15) λµν Γ rλσ ∂ µ Γ σrν = 12 ν ( ν − ν + 3) cosh σ. (4.39)The even part of the second term in above relation gives I CS , Even part = 2 ν ( ν + 3) sech σ (cid:0) ν − 3) cosh 2 σ (cid:1) , (4.40)and the odd part is just the negative of the above result. So for the second term wejust get zero, (cid:15) λµν Γ rλσ Γ σµτ Γ τνr = 0 . (4.41)The total integral of the second line would then be I CS = (cid:96) πGν (cid:90) M dσ du dτ √− g(cid:15) λµν Γ rλσ (cid:18) ∂ µ Γ σrν + 23 Γ σµτ Γ τνr (cid:19) = (cid:96) ν ( ν − πG ( ν + 3) (cid:90) dσdudτ cosh σ. (4.42)For calculating the last term, which is the boundary term, we choose the followingvielbein and spin connections as e = (cid:96) (cid:113) (7 ν − 3) sinh σ − ν − ν + 3 dτ ,e = (cid:96) √ ν + 3 dσ,e = 2 (cid:96)νν + 3 (sinh σ dτ + du ) , – 26 –nd ω = cosh σ (cid:16) (5 ν − 3) sinh σ dτ − ν du (cid:17) √ ν + 3 (cid:113) (7 ν − 3) sinh σ − ν − ,ω = − ν cosh σ (cid:113) (7 ν − 3) sinh σ − ν − d σ,ω = − ν cosh σ √ ν + 3 d τ . So we get (cid:15) ABC ω AB e C = − ν (7 ν − 3) cosh σ sinh σ(cid:96) √ ν + 3 (cid:113) (7 ν − 3) sinh σ − ν − du ∧ dτ. (4.43)The third term in the action then would give πG (cid:90) ∂ M du dτ (cid:15) ABC ω AB e C = (cid:90) dudτ − ν (7 ν − 3) cosh σ sinh σ πG(cid:96) (cid:112) ( ν + 3) (cid:113) (7 ν − 3) sinh σ − ν − , (4.44)So the total action would be I G = ν(cid:96) πG (cid:90) dudτ sinh σ (cid:16) (cid:96) ν ( ν − ν + 3) − ν + 8( ν + 3) − (7 ν − 3) cosh σ (cid:96) √ ν + 3 (cid:113) (7 ν − 3) sinh σ − ν − (cid:17) . (4.45)Note that σ here is a function of the two fields ˜ ρ and ˜ h which then could be writtenin terms of the actual ρ and h of the metric 2.5, after the coordinate transformation.Therefore, the above action could lead to a deformed version of Liouville. To findthe exact form in terms of t + , t − , h, ρ after the coordinate transformation, we need tochoose a suitable physical function for the “cut off surface" for the warped geometries.If we assume the following cutoff surface as σ = − (cid:15) . e − ˜ ρ ( τ,u ) (cid:16) − ˜ h ( τ, u ) − (cid:17) , (4.46)then after considering the chiral gauge relation, ∂ − h = 0 [39], which here would be ∂ u ˜ h = 0 , the induced metric on this boundary could written as ds = (cid:96) ( ν + 3) (cid:104)(cid:16) ν − ν + 3 sinh σ − (cid:15) e − ρ ( ∂ τ ˜ ρ − ∂ τ ˜ ρ ˜ h − − 12 ˜ h − ∂ τ ˜ h ) (cid:17) dτ + (cid:16) ν ν + 3 sinh σ + 2 (cid:15) e − ρ ( ∂ τ ˜ ρ − ∂ τ ˜ ρ ˜ h − − 12 ˜ h − ∂ τ ˜ h )( ∂ u ˜ ρ − ∂ u ˜ ρ ˜ h − − 12 ˜ h − ∂ u ˜ h ) (cid:17) dudτ + (cid:16) ν ν + 3 + (cid:15) e − ρ ( ∂ u ˜ ρ − ∂ u ˜ ρ ˜ h − − 12 ˜ h − ∂ u ˜ h ) (cid:17) du (cid:105) . (4.47)– 27 –e then need to find the two Jacobians J ρ and J h which do the following trans-formations dudτ = J ρ dt + dt − , dudτ = J h dt + dt + . (4.48)For doing so, one should compare the relation 4.47, with 3.8. Finding theseJacobians could then let us to write the action 4.45 in terms of t + , t − , h, ρ whichwould lead to the deformed Liouville action in the “actual coordinate". The timelike or elliptic warped anti-de Sitter solution could be found by warping thesecond part of 4.29 as ds = (cid:96) ( ν + 3) (cid:20) cosh σdu + dσ − ν ν + 3 ( dτ + sinh σdu ) (cid:21) . (4.49)The case of ν > is timelike stretched and the case of ν < is timelike squashedAdS . For the case of timelike stretched though we would have close timelike curves(CTCs).Near the boundary of time-like warped AdS , where τ → ∞ we have ds ( νν +3 (cid:96) ) (cid:39) (cid:18) − sinh σ + cosh σ (cid:0) ν + 34 ν (cid:1)(cid:19) dτ + ν + 34 ν dσ . (4.50)In this limit we also removed the term − dτ − σdτ du .The above metric is again the warped analogue of conformal transformation ofthe flat spacetime where timelike warped CFT vacuum is defined by the path-integral,so the two surfaces M (1)Σ and M (2)Σ in the bulk time like warped AdS could be relatedby Weyl and boost transformations in the UV region.For the case where M Σ is timelike, the state | Ψ Σ (cid:105) could be obtained by a“Lorentzian" path-integral on M Σ . So the timelike surfaces of warped AdS couldalso be interpreted as quantum circuits.For this case we should again calculate the action 4.36.The contribution for each term is as follows. First, πG (cid:90) M dσ du dτ √− g (cid:18) R + 2 (cid:96) (cid:19) = − ν(cid:96) (3 ν + 8)4 πG ( ν + 3) (cid:90) dudσdτ cosh σ. (4.51)The even part of the first term is (cid:15) λµν Γ rλσ ∂ µ Γ σrν ( symmetric ) = 4 ν (2 ν − ν + 3) cosh σ. (4.52)The odd part of the first term would be (cid:15) λµν Γ rλσ ∂ µ Γ σrν ( non-symmetric ) = 4 ν cosh σ ( ν + 3) . (4.53)– 28 –o we get (cid:15) λµν Γ rλσ ∂ µ Γ σrν = 12 ν ( ν − ν + 3) cosh σ. (4.54)Then, for the second term we get (cid:15) λµν Γ rλσ Γ σµτ Γ τνr = 0 . (4.55)The even part of the above relation actually iseven Part = 2 ν ( ν + 3) sech σ (cid:0) ν − 3) cosh 2 σ (cid:1) , (4.56)and the odd part is just the negative of this result.So the total integral of the second line would be (cid:96) πGν (cid:90) M dσ du dτ √− g(cid:15) λµν Γ rλσ (cid:18) ∂ µ Γ σrν + 23 Γ σµτ Γ τνr (cid:19) = (cid:96) ν ( ν − πG ( ν + 3) (cid:90) dσdudτ cosh σ. (4.57)For calculating the last boundary term, we choose the following vielbein and spinconnections as e = 2 (cid:96)ν cosh σ √ ν + 3 (cid:113) − ν ) sinh σ + ν + 3 dτ ,e = (cid:96) √ ν + 3 dσ,e = (cid:96) (cid:113) − ν ) sinh σ + ν + 3 ν + 3 du − (cid:96)ν sinh σ ( ν + 3) (cid:113) − ν ) sinh σ + ν + 3 dτ , and ω = ν ( ν + 3) (cid:113) − ν ) sinh σ + ν + 3 (cid:0) ν sinh σdτ + (3( ν − 1) cosh 2 σ − ν + 9) du (cid:1) ,ω = ν (3 (1 − ν ) cosh(2 σ ) + ν − √ ν + 3 (3 ( ν − 1) cosh(2 σ ) − ν − dσ,ω = cosh σ √ ν + 3 (cid:113) − ν ) sinh σ + ν + 3 (cid:16) ν dτ + 3( ν − 1) sinh σdu (cid:17) . – 29 –herefore, the third term in the action would become πG (cid:90) ∂ M dudτ (cid:15) ABC ω AB e C = ν πG(cid:96) ( ν + 3) (cid:90) (cid:16) sinh σ ( − ν + ( ν − ν ) − ν ( ν − 1) cosh(2 σ ) ν + 3 + 3(1 − ν ) sinh σ (cid:17) dudτ. So the total action could be written in a rather simplified form as I G = ν(cid:96) πG (cid:90) dudτ sinh σ (cid:16) (cid:96) ν ( ν − ν + 3) − ν + 8( ν + 3) − ν + 34 (cid:96) ( ν + 3 + 3(1 − ν ) sinh σ ) (cid:17) . (4.58)Note the difference between the last term in the above action and the corre-sponding one for the space-like case, in relation 4.44. For the undeformed case where ν = 1 , the two actions would match. Also, still here σ ( τ, u ) is a function of u and τ which then needs to be integrated out.Let’s compare this differentiating term more closely. Assuming an arbitrary valuefor the σ such as σ = π , and also (cid:96) = 1 , we could compare this two terms in figure2. There, one could see that, first, for a big enough value of ν , the contribution fromthe boundary term for the time-like case is positive, and for space-like is negativeand then when ν becomes very big, for both cases, they become constant.Also, one could see that for a specific value of the deformation, for the case oftime-like or space-like geometries, the contributions to the action from the boundaryterm would blow up, indicating that at this value of ν , the geometry would not bewell-defined. This irregularity could also be modeled as the cosmological singularitiessimilar to [69].This then even could be studied further from the point of view of [69]. Forinstance, a quench system in the warped AdS/CFT case could be considered andthe fluctuations for this particular value of ν and the effects on the emergent space-time could be examined. For those special values of ν , again one would expectthat if the geometry is not well-defined, the couplings for fluctuations would diverge.Furthermore, one could look for the quench systems where a well-defined emergentwarped AdS could be constructed.Note that the case of ν < leads to the squashed geometry and ν > to thestretched case, while the case of ν = 1 corresponds to AdS background. One couldsee that by increasing ν , i.e. for the case of stretched AdS, the contribution to theaction from the boundary term would be a decreasing function of ν , while for thesquashed case, the boundary action term for a specific value of ν < would diverge.For the case of null AdS, the boundary term would be zero. As a matter of fact,in Chern-Simons theory, most of the contributions to various physical parameterscome from the boundary term. The consistency of these two pictures then once– 30 – ime - LikeSpace - LikeNull ν - - - - I B Figure 2 . In the left part, the differentiating, boundary term in the chiral Liouvilleaction versus deforming parameter ν for the case of time-like and space-like warped AdS is shown. The right part is similar to figure 3 of [30] which is for the case of AdS. Thewarped AdS would have deformed constructions depending on the value of ν . These plotsshow the respective relations between these three manifolds again shows that the “complexity=action" could be used for the WAdS/WCFT case[24, 30].Then, to complete the calculations, we find the induced metric on the boundary σ = − (cid:15) . e − ˜ ρ ( τ,u ) (cid:16) − ˜ h ( τ, u ) − (cid:17) , (4.59)as ds = (cid:96) ( ν + 3) (cid:104) cosh σ + (cid:15) e − ρ ( ∂ u ˜ ρ − ∂ u ˜ ρ ˜ h − ) (cid:17) du + (cid:16) − ν ν + 3 sinh σ + 2 (cid:15) e − ρ ( ∂ τ ˜ ρ − ∂ τ ˜ ρ ˜ h − − 12 ˜ h − ∂ τ ˜ h )( ∂ u ˜ ρ − ∂ u ˜ ρ ˜ h − ) (cid:17) dudτ + (cid:16) − ν ν + 3 + (cid:15) e − ρ ( ∂ τ ˜ ρ − ∂ τ ˜ ρ ˜ h − − 12 ˜ h − ∂ τ ˜ h ) (cid:17) dτ (cid:105) , (4.60)where again the chiral gauge relation, ∂ − h = 0 [39], or ∂ u ˜ h = 0 has been used. The case of null or parabolic warped AdS would be ds = (cid:96) (cid:34) du u + dx + dx − u ± (cid:18) dx − u (cid:19) (cid:35) . (4.61)This metric is also a solution of TMG, but only for the case of ν = 1 . The pureAdS could then be derived by removing the last term.It worths to mention that the null warped AdS could be used to study coldatom systems [15, 28, 29]. – 31 –gain, for this case we calculate the action 4.36, I TMG = 116 πG (cid:90) M dudx + dx − √− g (cid:18) R + 2 (cid:96) (cid:19) + (cid:96) πGν (cid:90) M dudx + dx − √− g(cid:15) λµν Γ rλσ (cid:18) ∂ µ Γ σrν + 23 Γ σµτ Γ τνr (cid:19) + 132 πG (cid:90) ∂ M dx + dx − (cid:15) ABC ω AB e C . For this metric, considering the boundary being located at u ≥ (cid:15).e − ˜ ρ ( x + , x − ) would lead us to the effective boundary field theory.The first part of the action would give πG (cid:90) dudx + dx − √− g (cid:18) R + 2 (cid:96) (cid:19) = 116 πG (cid:90) dudx + dx − × (cid:96) u = (cid:96) πG (cid:90) dx + dx − ( − u − ) (cid:12)(cid:12)(cid:12) ∞ u = (cid:15).e − ˜ ρ ( x + , x − ) = (cid:96) πG(cid:15) (cid:90) dx + dx − e ρ ( x + , x − ) . (4.62)The Chern-Simons part would give (cid:96) πGν (cid:90) M dudx + dx − √− g(cid:15) λµν Γ rλσ ( ∂ µ Γ σrν + 23 Γ σµτ Γ τνr ) = 0 . (4.63)The even part has a factor of u and the odd part has a factor of − u , whichfor the null warped AdS will sum up to zero.For the null warped AdS, the tetrad and spin connections could be chosen as e = (cid:96) dx + , e = (cid:96)u du, e = (cid:96) dx + + (cid:96)u dx − ,ω = − (cid:96) dx + − (cid:96)u dx − , ω = − (cid:96)u du, ω = (cid:96) dx + + 2 u dx − , (4.64)or for another example, we could take e = i(cid:96) dx + , e = (cid:96)u du, e = − i(cid:96) dx + + i(cid:96)u dx − ,ω = i dx + − iu dx − , ω = − duu , ω = − i dx + + 2 iu dx − , (4.65)which using any of the two, would give a zero boundary term, i.e, (cid:15) ABC ω AB e C = 0 . (4.66)The total action would be just I G = (cid:96) πG(cid:15) (cid:90) dx + dx − e ρ ( x + ,x − ) . (4.67)– 32 –ssuming u = − (cid:15) . e − ˜ ρ ( x + ,x − ) (cid:16) − ˜ h ( x + , x − ) − (cid:17) , and then taking ∂ x − h = 0 , theinduced metric on the boundary would be ds = (cid:96) u (cid:40) (cid:15) e − ρ (cid:16) ∂ x + ˜ ρ (cid:0) − ˜ h − (cid:1) − 12 ˜ h − ∂ x + ˜ h (cid:17) dx +2 + (cid:32) ± u + (cid:15) e − ρ (cid:16) ∂ x − ˜ ρ (cid:0) − ˜ h − (cid:1)(cid:17) (cid:33) dx − + (cid:32) (cid:15) e − ρ (cid:16) ∂ x − ˜ ρ (cid:0) − ˜ h − (cid:1)(cid:17)(cid:16) ∂ x + ˜ ρ (cid:0) − ˜ h − (cid:1) − ∂ x + ˜ h ˜ h − (cid:17) dx + dx − (cid:41) . (4.68)The null surfaces M Σ in the warped AdS could be understood as the degeneratelimit of the time-like metrics. Also, the final result for complexity, similar to the AdScase, would be minimized for the null warped AdS geometries.In this setup it would be possible to understand the way any component of thewarped and deformed AdS metric would emerge. It would also be possible to checkhow the density of unitary quantum gates, and the behavior of the scrambling of thequantum states of the dual deformed CFT would be different from the usual CFT.The same procedure could later be done for other geometries such as Lifshitz orhyperscaling violating [70], starting from the Newton-Cartan group theory, or thecorresponding boundary algebra for flat or dS geometries or even by using othermassive gravity theories; check [71–73] for more details of the dual algebra for suchbackgrounds. In this section we construct and extend various holographic tools for WAdS / WCFT that have already been implemented in the setup of AdS / CFT for studying theconnections between geometry and encoded boundary information. The idea of tensor network and renormalization of entanglement entropy to under-stand the mechanism of AdS/CFT and emergence of spacetime out of geometry hasfirst started from the work of [6].In [30], the holographic spacetimes have been modeled as quantum circuits ofpath-integrations. There, it has been argued that the holographic spacetimes areactually some collections of quantum circuits. Using path-integration method andan appropriate UV cutoff a codimension one surface in gravity has been connected toa quantum circuit model (extension of surface/state duality) and therefore the dualitybetween tensor networks and AdS/CFT has been generalized. There, the relationsbetween the numbers of quantum gates to surface areas have been discussed and theholographic entanglement entropy formula has been generalized. In addition, the– 33 –mergence of “time" in AdS from the density of unitary quantum gates of CFT andalso the gravitational force from quantum circuits has been shown.One could speculate about the indications of these results for the case of thewarped or chiral geometries, for instance, how each deformation parameter or com-ponents of the metric, or even the presence CTCs of geometry are hidden in theboundary information. One could think that for these systems, other more “exotic"quantum gates would be needed that one should search for.In [57], the tensor networks as conformal transformation has been studied. There,in addition to disentanglers u and isometries w which construct the MERA, otherforms of tensors have been studied. Examples are, euclideons e which are the tensorsin the euclidean time evolution and smoothers which would be placed at the end ofa truncated network.Then, in [74], eucliedons , e and lorentzions l have been defined. The authorsassigned a path integral geometry to the Null, Euclidean and Lorentzian MERA.They found that the local rescaling in the CFT makes the path integral map V ,while the tensor network map W remains unchanged. This rule which was explainedfurther in [75] could be applied for the warped CFT case as well.Here we would like to point out to several other features that a proposed tensornetwork for warped CFTs should poses. First, we noted that, for constructing tensornetwork for warped CFTs we actually need chiraleons which produce only rightmoving dilation symmetry. This point has been considered to show a proposedidea for the structure of warped MERA as in figure 3. In fact, the WCFTs have aright moving stress tensor T where ∂ + T = 0 , and a right moving current P where ∂ + P = 0 . However, the right moving current generates left translations. So thetransformations of the left-moving direction x + are generated by the right-movingcurrent P ( x − ) . Therefore, the insertion of chiraleons would be necessary.Also, unlike the usual AdS/CFT systems, here instead of coupling to the Rie-mannian geometry which is the case of d CFT, the WCFTs should be coupled to the“warped geometry" [20] discussed before, which is a variant of the Newton-Cartangeometry.In [22] it was argued that WCFTs have a dynamical critical exponent z = ∞ , asunder dilatations, time rescales but space does not. However, since for constructingthe tensor network, we fix time, this factor would not affect our analysis.In addition, insights from the constructions of Hawking-Page phase diagramsfor the vacuum warped AdS and warped AdS black holes which have already beenestablished in [24, 64, 67] could point out to the fact that the construction of geometryfrom the tensor network for WCFT should be “deformed" in one direction as it hasbeen shown in the schematic figure 4.To understand better the difference between the tensor network system for theCFTs, versus those of WCFTs, we could examine further the difference between theLiouville action versus chiral Liouville gravity.– 34 –s we saw, the usual Liouville gravity theory deriving from the nonlocal Polyakovaction written in the conformal gauge would be [39] S L = c π (cid:90) d x ( Z R − λ √− g ) , R ≡ √− gR,Z ( x ) ≡ (cid:90) d x (cid:48) G ( x, x (cid:48) ) R ( x (cid:48) ) , √− gg ab ∇ a ∇ b G ( x, x (cid:48) ) = δ ( x, x (cid:48) ) . (5.1)On the other hand, in the case of the chiral Liouville gravity (Polyakov actionin a chiral gauge), for studying the theory in a fixed sector for the left moving zeromodes, an additional term in the following form should be added to the S L term, as S L = S L + ∆4 π (cid:90) d x √− gg −− = S L − ∆2 π (cid:90) d xh, (5.2)which makes all the difference. Later, the effects of this additional term on thecomplexity from the view of complexity = action should also be considered further.One could then see that the action becomes asymmetric and the deformationat each point of the tensor network becomes a function of the parameters h ( t + , t − ) where the left moving energy density is ∆ / .In the canonical formalism of chiral Liouville gravity, t − could then be consideredas time. Then, from the relation ∂ − h = 0 , one could see that h becomes independentof time which then is a necessary point in constructing the tensor network of WCFTs. Figure 3 . The tensor chiraleon for constructing WCFTs. Note that the yellow trianglesare gates that are combination of unitary disentaglers and chiraleons. So one could considerthe two layers are combined together. To get further information, similar to [76], using the wavelet decompositions, thequantum circuits for Majorana fermions could also be constructed. Similar to theSchrödinger holography, for the case of WAdS/WCFT, the Fourier modes aroundthe spatial circle of WAdS with momentum k are dual to the boundary operatorswhose conformal dimension depends on k .Other ideas such as quantum error correction could also help to get a betterpicture. In [59], where the connections between path integral complexity and circuitcomplexity has been outlined, the authors showed that the Liouville action which in– 35 – paceDepth (increasing course grained) Figure 4 . The chiral structure of TN for warped CFT creates a deformed version of warpedAdS space. The blue triangles could be a combination of chiraleons and disentanglers. Also,yellow triangles here are isometric coarse graining transformations and circles are the latticesites. So this structure due to the presence of chiraleons does not have a dilation invariancein the left direction. general is expressed in terms of Weyl parameter, could be considered as a generalcost function in the gate counting setup. In terms of quantum error correcting codes(QECCs), the dual of this physical CFT anomaly, have been conjectured to be alogical bulk gate. The same result still could be applied for WCFTs where we possesa “warped Weyl" symmetry. Specifically, the level of the energy of the dual of theseoperators in terms of the eigenstate thermalization hypothesis (ETH) [77] have beenstudied, which could be repeated for WCFTs. One then could check whether forWCFTs the gates would satisfy the ETH ansatz. In addition, using this setup, onecould estimate the energy difference for a layer of chiraleon as (cid:12)(cid:12) E i +1 − E i (cid:12)(cid:12) ∝ e − cN ,where c is an independent parameter related to the deformation parameter ν in themetric or α in the action.Another point is that the complexity of Hamiltonian circuit C ( e − itH ) would beproportional to √− g tt . For the warped AdS case it is proportional to the deformingparameter α . So for a fixed time period, the number of quantum gates in onedirection relative to the number of quantum gates in the other direction would exactlydetermine the deforming parameter of the bulk geometry. It could in fact determinewether the geometry is squashed or stretched and also determine the extent of thisdeformation with respect to other parameters of the theory. From the Hamiltonianformalism of chiral Liouville theory, we deduced that this deformation is actuallyrelated to the “energy density parameter ∆ ". This argument could then point out toa relation between the deformation parameter ν and the emergence of “time" fromthe density structure of the deformed circuit in the WCFTs tensor network structure.– 36 –he coupling of the WCFTs to the background fields should also be considered,as this point could manifest itself in the causality wedge during the emergence ofspace from the tensor network. Here, instead of the light-cone in the CFT case,for the WCFT case we have another geometric structure called “scaling structure"which essentially plays the same role. There, instead of the Weyl invariance we geta corresponding “warped Weyl invariance" which should be used similarly. In this section, we mention several observations about the structure of entanglemententropy (EE) of warped CFTs in several separate points, as for understanding betterthe renormalization group flow, structures of tensor network and the connectionsbetween chirality of the boundary theory and emergent geometry, EE and extremalsurfaces still are very important tools.For the case of WCFTs, the entanglement entropy has been studied in variousworks, see [21, 78, 79] for instance.The spacelike warped AdS metric in global coordinates with warp factor a ∈ [0 , would be written as [19] ds = (cid:96) (cid:16) − (1 + r ) dτ + dτ r + a ( du + rdτ ) (cid:17) . (5.3)For this metric, similar to [40], starting from the derived entanglement entropyand the proposed model of tensor network for warped CFTs, by varying the circuitpath integral-optimization complexity using the chiral Liouvile action, one couldderive the equations of motion 2.13 or 3.2.In fact, the entanglement entropy of warped CFTs has been found in [19, 21]and then modified in [78]. Later, we will see that we really need to use the modifiedversion of it in order to get a physically well-behaved kinematic space.The simpler relation which first we use to explain the structure of entanglementhere is S EE = − L vac log (cid:18) Lπ(cid:15) sin π(cid:96)L (cid:19) + iP vac (cid:96) (cid:18) ¯ LL − ¯ (cid:96)(cid:96) (cid:19) , (5.4)where (cid:96) and ¯ (cid:96) are the separations of the endpoints of an assumed interval in spaceand time respectively, and L and ¯ L are related to the identification pattern of thecircle which defines the vacuum of the theory. The second term would actually act asthe twist term. The interval is considered to be on the plane, and then one assumes L, ¯ L → ∞ . The angle α ≡ ¯ LL could also be defined, see figure 7.In [80], the emergence of gravitation from entanglement in holographic CFTswhich have a gravitational anomaly (with unbalanced left and right moving centralcharges) has been studied. The Wald-Tachikawa covariant phase space formalism– 37 –as been implemented there in order to obtain the linearized equations of motion ofTMG from the entanglement.For the gravity action with Chern-Simons term, the extremization prescription,corresponding to Ryu-Takayanagi formula should then be modified in the followingway S HEE = ext γ G N (cid:90) γ ds (cid:18)(cid:113) g µν ˙ X µ ˙ X ν + 1 µ g µν ˜ n µ ν ρ ∇ ρ n ν (cid:19) . (5.5)This relation, for the locally warped AdS , could be considered as the length ofthe geodesic and the twist of the normal frame along the geodesic, as S HEE = Length G N + Twist G N µ . (5.6)In the case of complexity of such theories also, a corresponding twist operatorcould be imagined which produces the specific warped anomaly, and then the totalnumber of these operators create the additional term in the chiral Liouville action.So for the warped AdS case, one could also have a corresponding relation of [30]in the form dS A ¯ A = dA (Γ P ¯ P )4 G N + Twist . (5.7) Σ Σ A BB 'A' Γ PP' P ' P Figure 5 . The codimension surface Γ P ¯ P which connects P and ˜ P in the warped AdS . For the warped CFTs, the evolution of entanglement entropy corresponds to thesum of the area elements and the twist. The deformation of the manifold by theWeyl anomaly of warped CFTs will then come into play which the schematic formhas been shown in Fig. 5.The twist fields of WCFT which would be the local operators in C q are in thefollowing form (cid:104)O (Σ ( i ) ) (cid:105) C , R q = (cid:104)O i (Σ)Φ q (Σ )Φ † q (Σ ) (cid:105) C q , C (cid:104) Φ q (Σ )Φ † q (Σ ) (cid:105) C q , C , (5.8)– 38 –here the conformal dimension and charge of these twist fields Φ q have been foundas ∆ q = q (cid:18) c 24 + L vac q (cid:19) , Q q = P vac . (5.9)Then, the effect of any non-trivial topology could be modeled by considering q decoupled copies of field theories and then adding local twist fields Φ q (Σ) at theendpoints of the interval. This would then couple together the replica copies.Moreover, the complete partition function of warped CFTs has been found in[21] as Z ¯ a | a (¯ τ | τ ) = exp (cid:18) iπk τ aτ ) + 2 πiP vac ( a ¯ ττ − ¯ a ) + 2 πiaτ L (cid:19) (5.10)The complexity path integration and the replica manifold could then becomeconnected to each other using the above relations. As mentioned in [21], ttheOPE of the twist field in the correlation function of WCFTs with the U (1) cur-rent J ( x )Φ q ( y ) ∼ iQ q O x − y , could also detect a singularity which shows its signature inthe dual geometrical space.The main point here then could be seen from taking the limit of q → in relation5.9. The result would be non-zero due to the fact that the vacuum of the theory isnot invariant under SL (2 , R ) × U (1) . Changing from cylinder to plane, the vacuumwould be mapped to a non-trivial operator. So the vacuum operator would be relatedto Φ , which is being inserted locally at the end points of the interval, shown in figure5. Also, similar to the case of entanglement entropy, the path-integral complexityis a function of vacuum charges L vac and P vac , and so only the central charge of thetheory would not be enough. As most realistic field theories, such as warped CFTs ornon-unitary CFTs, are not invariant under conformal group, this statement is actuallyimportant and it should be applied while studying their circuit complexities. /WCFT The surface/state correspondence was introduced in [7]. If this indeed is a frameworkthat could work for any quantum gravity theory; i.e, saying that tensor networks arereally equivalent to gravitational theories, then one should consider how this frame-work could be applied for the chiral theories such as warped CFTs, and then checkhow a deformed AdS spacetime could be emerged by considering some particular dis-tributions of quantum states in the chiral gauge. Then, one would like to understandhow the quantum circuits of path integration in the setup of warped CFTs make theevolution of quantum entanglement, and how the area of a particular surface and thenumber of quantum gates creating entanglement could be connected. In this section– 39 –e consider this problem. We also study the specific form of the information metricfor WAdS/WCFT case.As for the tensor network structure of warped CFTs, one could imagine that thenumber of intersections between any convex surface Σ and the links between chira-leons and other elements of the tensor network which intersect Σ should determinethe chirality of the theory. In addition all kinds of links between various gates on thenetwork which intersects Σ could give an estimation of the effective entropy S (Σ) of the particular surface. However, as mentioned in [7], this would not be a precisematch as it just could act as a first estimation, since the links would not necessarilybe maximally entangled and efficient. However, we could still take this first orderestimation in what follows.For the warped AdS metric in the Poincare coordinate, ds = L W AdS (cid:18) dr r − r dX − + α ( dX + + rdX − ) (cid:19) ,L W AdS = (cid:96) ν + 3 , α = 4 ν ν + 3 , (5.11)one could compare the results we already got with those of [7].First, as the deformation parameter α depends on the chirality and thereforethe number of chiraleons intersecting the surface, one could check its effect on thegeometry as well. The figure of α versus deforming parameter ν is shown in figure6. By warping the geometry further, some more chiraleon gates would be needed.Considering a positive ν , it could be seen that the behavior is a monotonically in-creasing function of the deforming parameter, and then it becomes a constant valuefor larger ν s, i.e, here it happens around ν ∼ . - - ν α - - ν S eff Figure 6 . The parameter α and the effective entropy S eff versus deforming factor ν . Theleft plot could give a first estimation of the number of chiraleon gates versus the deformationof the geometry and the right one could show a c-like theorem for WCFTs as the effectingaction is a monotonically increasing function of the physical positive values of ν . In this case, the surface M Σ , similar to the Lorentzian AdS [30], could be time-like, null or space-like. The null warped AdS (G ¨ odel geometries) could be consideredas a degenerate limit of time-like surfaces. The space-like surfaces would then simi-larly correspond to the path-integrals on the space-like surfaces M Σ .– 40 –ath-integration then would change the normalization of the wave functional.The corner contribution in the gravity dual would actually point to the (non-Lorentzian)evolutions in these models. Also, one should note that similar to the AdS case, inWAdS geometries, the propagation of local excitations could break the causality inthe bulk and therefore the circuits would be non-unitary. In fact, the quantum circuiton M Σ includes both unitary and non-unitary quantum gates.If one considers the state dual to the vacuum state of warped CFT, i.e, | (cid:105) as | Ψ Σ (cid:105) , then due to the warped Weyl invariance here and similar to the conformalscenarios, a time-like path-integration which starts from another quantum state | Ψ ˜Σ (cid:105) would not affect modes that their wave lengths are larger than the ones in Σ . Thetime-like path integration would only create vacuum states for the modes whoselengths are between the ones in ˜Σ and Σ .The Weyl invariance of path-integrations then points out to the fact that, similarto the AdS case, for any codimention one “time-like" surface M Σ in warped AdS which would connect the surface Σ to ˜Σ , the dual quantum circuit would also maps | Ψ ˜Σ (cid:105) to | Ψ Σ (cid:105) . So the codimension one surface M Σ could be regarded as a path-integration with a suitable cut off and then by a discretization with the cut off scale,this path integration could be regarded as a quantum circuit.To complete our discussion here, similar to [7], for the WAdS geometries, theFisher information metric G ( B ) uu could be calculated. The definition of this metric isas follows − (cid:12)(cid:12) (cid:104) Φ(Σ u ) (cid:12)(cid:12) Φ(Σ u + du ) (cid:105) (cid:12)(cid:12) = G ( B ) uu du . (5.12)In [7], in their eq. (3.19), a conjectured from for G ( B ) uu for the AdS case and ina gravity theory in M d +2 has been proposed. For the warped AdS metric of 5.3, asthe determinant of the non-radial parts X + and X − is √ g = αr , the above relationwould result in the following form of Fisher metric G ( B ) uu = 1 G N (cid:90) dX + dX − αr (cid:18) rα ( α − 1) ( P −− + − + P −−− + ) + α ( P + − + − + P − ++ − + P + −− + + P − + − + )+4 r ( α − P −−−− + 2 rα ( α − P + −−− + P − + −− ) (cid:19) . (5.13)Note that the components of the tensor P µνξη are non-negative functions of thedegrees of the freedom such as central charge and the Kac-Moody level parameter.From the relation 5.13, one could note that the information metric for the warpedAdS is an order of α . As the effective entropy S eff (Σ u ) is proportional to G ( B ) uu , thenwe deduce that for the WAdS geometry also we have the relation S eff (Σ u ) ∝ α .Therefore, it is a monotonically increasing function of ν , but it becomes saturated atvery big deformation parameters ν . Its plot is shown in the right part of figure 5.3.– 41 –ater, the implications of quantum estimation theory such as Cramer-Rao boundfor the WAdS/WCFT could be studied. In this section, similar to the work of [14], we can build the quantum circuits for theKac-Moody algebra.First, taking T ( x − ) and P ( x − ) as the usual local operators on the plane, onecould define the following operators [17] T ζ = − π (cid:90) dx − ζ ( x − ) T ( x − ) , P χ = − π (cid:90) dx − χ ( x − ) P ( x − ) . (5.14)Here the right moving modes are associated with x − and left moving with x + .Then, taking the coordinate transformation from x − to φ , or from the Lorentzianplane to Lorentzian cylinder, in the following form x − = e iφ , x + = t + 2 αφ, (5.15)one would get P α ( φ ) = ix − P ( x − ) − kα,T α ( φ ) = − x − T ( x − ) + c 24 + i αx − P ( x − ) − kα . (5.16)Then, the modes of the algebra on the cylinder could be written as P αn = − π (cid:90) dφP α ( φ ) e inφ , L αn = − π (cid:90) dφT α ( φ ) e inφ , (5.17)which in terms of the original modes of Kac-Moody would be written as P αn = P n + kαδ n , L αn = L n + 2 αP n + ( kα − c 24 ) δ n . (5.18)Here α is an arbitrary tilt .The operators L n and P n of the Virasoro-Kac-Moody algebra would satisfy thefollowing commutator relations [ L n , L m ] = ( n − m ) L n + m + c n ( n − n + 1) δ n + m [ P n , P m ] = k nδ n + m [ L n , P m ] = − mP m + n (5.19)Now similar to [14], we could build the Virasoro + Kac-Moody symmetry gatesas U ( τ ) = (cid:126) P exp (cid:104) (cid:90) ζ Q [ ζ (cid:48) ] dζ (cid:48) + (cid:90) χ Q [ χ (cid:48) ] dχ (cid:48) (cid:105) , (5.20)– 42 –here L n = Q [ ζ n ] , P n = Q [ χ n ] , (5.21)are the associated charges of Virasoro-Kac-Moody U (1) algebra, where the centralextensions are [17] c = 5 ν + 3 ν ( ν + 3) , k = − ν + 36 ν . (5.22)Here the instantaneous gate could be defined as Q T ( τ ) = (cid:90) π dφ π (cid:15) ( τ, φ ) T ( φ ) ,Q P ( τ ) = (cid:90) π dφ π (cid:15) ( τ, φ ) P ( φ ) , (5.23)where T ( φ ) = (cid:88) n ∈ Z (cid:16) L n + 2 αP n + ( kα − c 24 ) δ n (cid:17) e − inφ ,P ( φ ) = (cid:88) n ∈ Z ( P n + kαδ n ) e − inφ , (5.24)and (cid:15) ( τ, φ ) = (cid:88) n ∈ Z (cid:15) n ( τ ) e − inφ . (5.25)As mentioned in [17], the transformation P αn = P n + kαδ n , L αn = L n + 2 αP n + ( kα − c 24 ) δ n , (5.26)is the usual shift proportional to the central charge for an exponential mappingwhere its cost function F has been calculated in [14], combined with a spectral flowtransformation which would be given by that tilt parameter α which will be modeledby our chiraleon gate.This gate could also be interpreted as generating anomaly in the system. Thiscould be understood by considering the following coordinate transformation on thecylinder, as in [17], φ = φ (cid:48) λ , t = t (cid:48) + 2 γλ φ (cid:48) . (5.27)As mentioned in [17], T ( x − ) is the generator for the coordinate transformationsin x − and P ( x − ) is the generator for the gauge transformation in the gauge bundle– 43 –hich is parametrized by x + . If similarly we assume the following functions for thecoordinate transformations x − = f ( w − ) , x + = w + + g ( w − ) , (5.28)then the general relations for the infinitesimal transformations, [17], P (cid:48) ( w − ) = ∂x − ∂w − (cid:104) P ( x − ) + k ∂w + ∂x − (cid:105) ,T (cid:48) ( w − ) = (cid:16) ∂x − ∂w − (cid:17) (cid:34) T ( x − ) − c (cid:40) ∂ w − ∂x − ∂w − ∂x − − (cid:32) ∂ w − ∂x − ∂w − ∂x − (cid:33) (cid:41) + ∂x − ∂w − ∂x + ∂w − P ( x − ) − k (cid:32) ∂x + ∂w − (cid:33) , (5.29)could be simplified as T (cid:48) ( w − ) = f (cid:48) (cid:104) T ( x (cid:48) ) − c (cid:110) f (cid:48)(cid:48) − f (cid:48) f (cid:48)(cid:48)(cid:48) f (cid:48) + 32 f (cid:48)(cid:48) (cid:111)(cid:105) + f (cid:48) g (cid:48) P ( x − ) − k g (cid:48) ,P (cid:48) ( w − ) = f (cid:48) P ( x − ) − k g (cid:48) , (5.30)where f (cid:48) = ∂f ( w − ) ∂w − and g (cid:48) = ∂g ( w (cid:48) ) ∂x − .This is the corresponding equation in [14] where instead of the case of usualCFT, it works now for the warped CFT case.Also, there would be the following relations for the generator of translations, Q [ ∂ t (cid:48) ] = Q [ ∂ t ] + kγ, Q [ ∂ φ (cid:48) ] = 1 λ , (5.31)which have been interpreted as constructing an anomalous term being generated bythe new chiraleon gate.Similar to [14], the cost functions for the gate generating the ordinary transfor-mation of the partial derivate and also the gate creating the anomaly, our chiraloenor "anomalon" could be calculated as in [14]. First, we could find the cost functionsin the following way F ≡ (cid:12)(cid:12) Tr (cid:16) ρ (cid:0) Q T + Q P (cid:1)(cid:17)(cid:12)(cid:12) = (cid:12)(cid:12) Tr (cid:16) ρ ( ˜ Q T + ˜ Q P ) (cid:17)(cid:12)(cid:12)(cid:12) , F ≡ (cid:113) − Tr (cid:0) ρ ( Q T + Q P ) (cid:1) = (cid:113) − Tr (cid:0) ρ ( ˜ Q T + ˜ Q P ) (cid:1) , (5.32)where ρ ≡ U ρ U † , ˜ Q T = U † Q T U, ˜ Q P = U † Q P U. (5.33)– 44 –o we have ˜ Q T = (cid:90) π dx + π (cid:15) ( x + , x − ) (cid:32) f (cid:48) (cid:104) T ( x (cid:48) ) − c (cid:110) f (cid:48)(cid:48) − f (cid:48) f (cid:48)(cid:48)(cid:48) f (cid:48) + 32 f (cid:48)(cid:48) (cid:111)(cid:105) + f (cid:48) g (cid:48) P ( x (cid:48) ) − k g (cid:48) (cid:33) , ˜ Q P = (cid:90) π dσ π (cid:15) ( x + , x − ) (cid:16) f (cid:48) P ( x − ) − k g (cid:48) (cid:17) . (5.34)We can insert the expectation values of T ( x (cid:48) ) and P ( x (cid:48) ) in the above relationsas T ( x (cid:48) ) → (cid:12)(cid:12) kα − c (cid:12)(cid:12) , P ( x (cid:48) ) → (cid:12)(cid:12) kα (cid:12)(cid:12) . (5.35)The last thing to do is to relate the velocities (cid:15) ( x + , x − ) to the path in the groupmanifold using the symmetries of Kac-Moody algebra, which would lead to the result, (cid:15) ( x + , x − ) = f (cid:48) g (cid:48) .At the end, similar to [14], the complexity action could be computed using thecost functions F or F , which for the warped CFT becomes C ( τ ) = (cid:90) τ F ( H ( τ (cid:48) )) dτ = 12 π (cid:90) x − d ˜ x − (cid:90) π d ˜ x + (cid:32) f (cid:48) g (cid:48) (cid:2) kα − c − c (cid:8) f (cid:48)(cid:48) − f (cid:48) f (cid:48)(cid:48)(cid:48) f (cid:48) + 32 f (cid:48)(cid:48) (cid:9)(cid:3) + kα − k g (cid:48) f (cid:48) (cid:33) + k π (cid:90) x − d ˜ x − (cid:90) π d ˜ x + (cid:16) αg (cid:48) − f (cid:48) (cid:17) . (5.36)In the gravity side, this result could also be derived using Polyakov action in thechiral gauge. So as we pointed out before, for calculating complexity, Polyakov actionshould be considered as the fundamental theory rather than the Liouville gravity.The equivalent connection between Polyakov action and the coadjoint orbit ac-tion of the Virasoro group has been shown in [81], and the connection betweenquantum complexity and coadjoint orbit action in [14]. So, here, we have empha-sized that complexity is not necessarily equivalent to coadjoint orbit action of theVirasoro group. This is only true if one chooses the conformal gauge fixing for thePolyakov action. If one chooses the chiral gauge, then complexity would be relatedto coadjoint orbit action of the Virasoro + Kac-Moody group [82].Then, for the case of warped geometries, the results of path-integral complexitycould be compared with those of holographic calculations using the complexity = action or complexity = volume [24–26]. Kinematic space is an auxiliary Lorentzian geometry where the components of itsmetric is being defined using conditional mutual information [83]. This concept– 45 –ould help to understand the interplay between information on the boundary CFTand the bulk geometry better. For holographic CFTs where Ryu-Takayanagi (RT)prescription works, kinematic space would be just the space of bulk geodesics. Thelengths of the bulk curves could then be calculated by the volumes in the kinematicspace.The kinematic space in general is related to the number of isometry gates in thetensor network and at least in the first order of approximation, it could be constructedby calculating the Crofton form [3, 4, 84] ω ( θ, ϑ ) = ∂ S ( u, v ) ∂u∂v du ∧ dv, (5.37)where u = θ − ϑ, v = θ + ϑ. (5.38)Now we would like to study the properties of this space for the warped CFTs andcheck how the warping factors and also the symmetry breaking change its properties.In this case, the geodesics are located in a space were they asymptote to a warpedAdS background.For the case of the static slice of AdS, the kinematic space would be a two-dimensional de-Sitter space. For warped CFTs, one could imagine that its kine-matic space which is the space of intervals on a constant time slice of a given two-dimensional “warped CFT", would then be a two-dimensional “deformed de-Sitter"space.If one considers an AdS background which is expressed as a real-line or a circlefibration over a Lorenzian AdS base space, then these geometries could certainlyget deformed with a warped factor into the warped AdS which then affects thekinematic space.In addition, the RT-like prescription for the warped AdS also has a modifiedversion.These structures have been discussed in [19, 78, 79], which we will use in ourcalculations later.As mentioned in [4], there would be a relation between the sign of the Croftonform and the strong subadditivity leading to the result I ( A, C, B ) = S ( AB ) + S ( BC ) − S ( B ) − S ( ABC ) ≥ , ≈ ∂ S ( u, v ) ∂u∂v dudv ≥ . (5.39)The volume of the kinematic space for the case of MERA is determined by thenumber of these isometries. This of course still could be true for the case of deformedMERA since we could write ds MERA = I (∆ u, ∆ v (cid:12)(cid:12) B ) . (5.40)– 46 – igure 7 . The interval setup D considered in [21] which shows the domain covered by thecoordinates ( t, x ) relative to ( T, X ) . We implement this plot to construct the structure ofkinematic space for WCFTs. Also, similar to the work of [85], we conjecture that the dynamics of the kinematicspace for warped CFTs would be the dynamics of the two-dimensional chiral Liouvillegravity, which would be the corresponding deformed version of Jackiw-Teitelboim(JT) gravity. In this case however, as shown in [22], the throats of WAdS with afinite volume of spatial circle would not completely decouple from the rest of thegeometry. Similar to the CFT case though, the chiral Liouville stress tensor forthe entanglement would be given by the vacuum expectation value of WCFT stresstensors evaluated at the interval endpoints.Now, for calculating the entanglement entropy and then the Crofton forms, wecould use the procedures done in [21] and their coordinate transformation shown infigure 7 and then we could construct the kinematic space for warped CFTs.First, we consider an interval D on a background which has the cylinder geometrydescribed by coordinates ( T, X ) where T is related to U (1) axis and X to the SL (2 , R ) symmetry. The identification of the cylinder would be ( T, X ) ∼ ( T + ¯ L, X − L ) . Theinterval inside this cylinder would be taken similar to the one in [21] as D : ( T, X ) ∈ (cid:104) ( ¯ (cid:96) , − (cid:96) , ( − ¯ (cid:96) , (cid:96) (cid:105) . (5.41)After mapping to the appropriate coordinate ( t, x ) , the interval would become ( t, x ) ∈ (cid:104) ( ¯ κ π ζ − (cid:96) LL + ¯ (cid:96) , − κ π ζ ) , ( − ¯ κ π ζ + (cid:96) LL − ¯ (cid:96) , κ π ζ ) (cid:105) , (5.42)where ζ = log (cid:16) Lπ(cid:15) sin π(cid:96)L (cid:17) + O ( (cid:15) ) . (5.43)– 47 –hen, the relation for the entanglement entropy of warped CFTs mentionedbefore, [21], in the form S EE = iP vac (cid:96) (cid:16) ¯ LL − ¯ (cid:96)(cid:96) (cid:17) − L vac log (cid:16) Lπ(cid:15) sin π(cid:96)L (cid:17) , (5.44)could be appliedSimilar to the CFT case, we take L = 2 π(cid:96) WCFT , (cid:96) = (cid:96) WCFT ( v − u ) and also u = θ − ϑ , v = θ + ϑ . All the angles for the warped AdS case are shown in figure 8.Taking the corresponding relations for the conjugate ones, the above relationcould be written as S EE = iP vac (cid:16) ¯ (cid:96) WCFT ( v − u ) − (cid:96) WCFT (¯ v − ¯ u ) (cid:17) − L vac log (cid:16) (cid:96) WCFT (cid:15) sin v − u (cid:17) . (5.45)Using above relation to derive the Crofton forms, one would get the CFT corre-sponding relations as ω = ∂ S EE ∂u∂v du ∧ dv = − ∂ ϑ Sdθ ∧ dϑ = 4 L sin ( ϑ ) dθ ∧ dϑ, ¯ ω = 0 , (5.46)which does not seem to be a plausible result.However, if we use the corrected, “modified" result for the entanglement entropyof WCFTs found in [78, 86] as S EE = iP vac (cid:96) (cid:18) ¯ L − αL − ¯ (cid:96)(cid:96) (cid:19) + (cid:16) − i απ P vac − L vac (cid:17) log (cid:18) Lπ(cid:15) sin π(cid:96)L (cid:19) , (5.47)which for the case of zero tilt angle α = 0 leads to the previous result 5.44, then forthe Crofton form, we could get the correct and reasonable result. Changing variablesas before simply leads to ω = ∂ S EE ∂u∂v du ∧ dv = − ∂ ϑ Sdθ ∧ dϑ = iαπ P vac + 4 L sin ( ϑ ) dθ ∧ dϑ, ¯ ω = 0 . (5.48)The difference in the volume form now is the term iαP vac π sin ϑ which is the gap in the“density of geodesics" between CFTs and warped CFTs.As mentioned in [78], α is not an arbitrary parameter and actually is being fixedby the theory. This statement corresponds to the statement that the entropy is alsoinvariant under the “warped conformal transformations".Our results could then help to reveal the nature of gates in the tensor networkstructures of warped CFTs. The density of “chiraleon gates" in such tensor networkswould be proportional to this additional term which depends on the U (1) chargeand the tilt angle α . This tilt parameter α then will produce the tilt angle in thetensor network structure shown in figures 4 and 8. This result could also give further– 48 – igure 8 . The geodesic and the corresponding point in the kinematic space are shown inblue color. Also, the angles ϑ , θ and α in relation 5.48 for the warped AdS case are shown.However, the tilt angle α introduced in relation 5.47 which should be fixed by the theoryand is “not" an arbitrary parameter of warped AdS backgrounds, would not necessarilyhave the exact geometrical meaning shown here. evidences that the “modified" relation for the entanglement entropy of WCFTs isindeed the correct one. The modified term actually shows its signature in the volumeform, geodesic density and tensor network structure.In [23], it was mentioned that α is proportional to the slope of the thermal identi-fication after the modular transformation and taking a finite value for α correspondsto the slow rotating limit of [17] c (cid:29) β Ω , ∆ gap β Ω (cid:29) . (5.49)In the above relations, ∆ gap is the dimension of L where the theory starts toget a large number of operators. From these statements then, one could deduce thatinserting the “chiraleon" gates in the tensor network would lead to such slow-rotatingregimes. The exact relations between circuit complexity, quantum gates, tensor network mod-els and the actual properties of string theory deserve further studies. By comparingvarious results from Polyakov action in conformal versus chiral gauge, we could gainseveral results here in this direction.Interesting factors such as choosing gauge, boundary conditions and its symme-try algebra would then affect the choice of cost functions and quantum gates forcalculating circuit complexity. For instance the behavior of the boundary modes, i.e,symmetry of the left and right-moving modes in the conformal case, versus the onlyleft moving modes of chiral gauge, would affect the nature of quantum gates to be– 49 –hosen for the TN models of these cases. Therefore, the extension of Liouville actionas the cost function is inevitable.To see this point better, we review the geometric definition of complexity intro-duced in [87, 88]. There, the circuit is modeled by an operator of the form V = P exp (cid:32) − (cid:90) λ dκ (cid:88) I Y I ( κ ) O I (cid:33) , (6.1)where O I are the elementary gates coming along each other in a sequence and Y I controls the insertion of these gates at each layer, (which could be considered as avelocity along the path κ ). The connection between the nature of these elementarygates O I and the properties of strings, such as tension T or mass could be studied.The gates are related to the components of the energy momentum tensor, and if oneconsiders more generalized form of Liouville, for instance the Polyakov action, onecould find also connections between the string tension T , or the Regge slope α (cid:48) andnature of these quantum gates.In fact, the cost function D [59], would be related to the dynamics of the string,and the way the end-points of these strings would be coupled to any boundary orD-brane. For instance, taking the L -norm D = (cid:82) λ dκ (cid:112)(cid:80) IJ η IJ Y I Y J for calculat-ing complexity would correspond to taking the Nambu-Goto action describing thedynamics of the strings. In [59], the Liouville action has been proposed to be a goodcost function which also match with result of [89]. Then, here we conjectured that,as we have an analogue warped Weyl invariance for WCFTs, for these theories, thechiral Liouville action would be a suitable cost function. This argument could thenbe extended to the Polyakov action and its various generalization discussed below.Let’s look at the Polyakov action more closely, S = T (cid:90) d σ √− hh ab g µν ( X ) ∂ a X µ ( σ ) ∂ b X ν ( σ ) . (6.2)Here T is the string tension, g µν is the metric of the target manifold and h ab isthe worldsheet metric.Remember the connection between the kinematic and potential terms of Liouvilleaction with the number of isometries and disentanglers of MERA network. Theunitaries of the system which are proportional to the potential term could execute aforce on the boundary modes of the system which would be related to the tension ofthe string T s , or other quantities of the model such as µ (the mass of graviton) forthe case of massive gravity. The disentanglers then, by removing the entanglemententropy between the virtual particles would create a kinetic energy, and consequentlythe number of isometries or disentanglers would be proportional to the kinetic termin the Liouville action. So for all these cases, an important parameter would be thetension of the open strings. – 50 –o consider a more generalized form, one could imagine that each of these cou-pling parameters in the action corresponds to a different tension, see figures 9, whichthen leads to the generalized form of Polyakov action where there are various T i pa-rameters. This action then could offer a more inclusive cost function for calculatingthe circuit complexity and for constructing various tensor network models.It worths to note here that the non-linear sigma models, such as O ( n ) modelas another example, have many interesting features which could give a better pic-ture. For instance the non-trivial renormalization-group flow fixed point of the O ( n ) symmetric model contains a critical point separating the ordered phase from thedisordered one. This phase transition could also be captured by the tensor networkmodels. The specific nature of the quantum gates and the string tension T , branetension T , and the tension between string and the brane T would affect this phasetransition then. This model could even be experimentally closer to the results of O ( n ) models describing the Heisenberg ferromagnets. Figure 9 . The different tensions in the system, T is the tension of the string, T is thetension between the string and the worldsheet and T is the brane tension itself. One could even go beyond that and think of more generalized actions in stringtheory, in order to find the suitable cost function for the circuit complexity. Forinstance one action, could be written in the form of sigma model action for “ p -adicstrings" which has been proposed by Zabrodin [90–92] as S ∼ (cid:88) (cid:104) ij (cid:105)∈ E ( T p ) η ab ( X ai − X aj )( X bi − X bj ) , (6.3)where η ab is target space Minkowski metric, the worldsheet is Bruhat-Tits tree T p ,and (cid:104) ij (cid:105) is the edge between vertices i and j . The generalized version of AdS/CFTusing this Bruhat-Tits tree would be more suitable for constructing tensor networkand so it could be more compatible with the results of quantum error correctionwhich was introduced in [93, 94]. The tensor network for the case of AdS / CFT ,using p-adic strings was constructed in [95].Then, in [90], the authors generalized the Zabrodin action which is equivalent toPolyakov action, to the case where the target space is changed from a flat geometry– 51 –o a curved one. The proposed action would be in the form S = (cid:88) (cid:104) ij (cid:105)∈ E ( T p ) d ( X i , X j ) V a (cid:104) ij (cid:105) , (6.4)where d ( X i , X j ) is the target space distance between the two points X i and X j and a (cid:104) ij (cid:105) is the length of edge (cid:104) ij (cid:105) in the tree, and V is the degree of any vertex. Thisaction could be one of the most generalized form to be considered as the cost functionfor circuit complexity.Adding nonlinear interactions such as flavor-chiral anomalies (which would leadto Wess-Zumino-Witten model) and extending geometry to include torsion or con-sidering the cobordism classes and its relations to complexity and swampland wouldbe other non-trivial directions to generalize these arguments, [96, 97]. In this work we studied the optimization path integral complexity for the case ofwarped conformal field theory in the setup of WAdS / WCFT . For this computationwe proposed the chiral Liouville action as the suitable cost function which then couldlead to the slice of a warped geometry. The RG equations where two cut offs wouldbe needed, one for each field in the metric, have been presented. Then, the RGequations for two special cases where the Lagrangian of WCFTs have been directlyobtained where discussed. Those Lagrangians have been written for a scalar fieldaction and a Weyl fermion action called Hofman/Rollier theory.We then looked at the problem of emergence of warped AdS geometry fromthe chiral warped CFT from the other side of the story. So, we started from thewarped AdS and by calculating the action of TMG as an example which has thesolution of warped AdS (in addition to other solutions such warped BTZ blackholes), we derived three warped Liouville actions from the spacelike, timelike and null warped AdS metric. We showed that the specific boundary parts which arisefrom the topological terms are consistent with the final emergent spacetimes onewould expect. It would be interesting then to repeat these calculations for othergravity theories which contain warped AdS as their solutions such as NMG, andthen derive new forms of deformed Liouville actions. Also, the specific forms ofJacobians and coordinate transformations using various boundary cut off profilesshould be studied further.Then, we implemented various holographic tools that have already been con-structed for the case of AdS / CFT , in the case of WAdS / WCFT . These includea deformed version of MERA for the tensor network of warped CFTs. Specifically,based on the symmetry group of the theory, we introduced a new kind of gate onewould need to construct the desired TN model, where we dubbed “chiraleon" . Fur-– 52 –hermore, we discussed some subtleties of putting fermions on a lattice which arerelevant to our discussions.We also discussed the entanglement structure of WCFTs which include a twistand discussed how it affects the emergent warped AdS geometry.The surface/state correspondence for the case of warped AdS and chiral theorieshas also been discussed where we showed the relationship between the deformationparameter of the geometry and the entanglement entropy. Also, a form of Fisherinformation metric for the warped cases has been proposed there.Next, similar to the approach of [14] done for the Virasoro group, the quantumcircuits model for the Kac-Moody group has been proposed and then using them andalso a specific cost function, the complexity for WCFTs in terms of the parameters ofthe algebra such as central charge, Kac-Moody level and symmetry transformationfunctions, has been derived. Finally, the structure of kinematic space for the casewarped AdS/warped CFTs has been analyzed.In the final section, we discussed the various extensions of Liouville action (in theconformal gauge) as the cost function for complexity to the case of chiral Liouvillefor the chiral gauge and then to Polyakov action and even its generalizations such asZabrodin action or p-adic strings which could be used to construct tensor networkmodels. The exact connection between each of these actions, the equations of motion,calculations of entanglement entropy and complexity and the connections with thetensor network structures deserve further investigations.So in this paper, we emphasized the role of gauge fixing in constructing tensornetwork and calculating complexity. Recently, the question of gauge fixing for tensornetworks has also been studied further in [98]. There, it has been mentioned that thecorrect gauge fixing could in fact help to optimize the iteration process of algorithms.In light of many recent developments, there would be various directions to gen-eralize and extend this work which here we mention to a few ideas.In [63], the connection between entanglement of purification (EoP) and com-plexity of purification (CoP) has been studied. These studies could also be done forthe case warped conformal field theories and check how the lack of dilation in onedirection could change the EoP and CoP and their evolutions [62].In [59], it was proposed that by including non-unitary transformations [99, 100] inthe Fubini-Study metric, one could define mixed state complexity. It would be inter-esting to see if such non-unitary transformations could also create a chiral state. Also,one could connect the “Fubini-Study metric" to chiral Liouville gravity, Polyakov ac-tion or its generalizations.In [85], the gravitational dynamics of kinematic space have been discussed andthere it has been argued that it could be described by the “Jackiw-Teitelboim" gravitytheory. The relationship between the modular Hamiltonian and the dilaton in thegravity model which underlies the kinematic space construction has been discussed.The complete set of equations for the connections between entanglement entropy S – 53 –nd modular Hamiltonian H mod have been presented where it has been called the kinematic space on-shell identities . These dynamical relations and similar equationsfor the case of warped AdS could also be derived.The construction of [101] could specifically be useful in studying the Hamiltoniandynamics of WCFTs. The symplectic structure of warped CFT could be constructedwhich would help to understand the holographic dictionary for WAdS/WCFT better.The construction for TMG could be a nice example for applying their algorithm, sincein their method one specifically could treat the important boundary terms of TMG ina more complete and efficient way which then could help to understand our derivedcomplexity relations better.In [48, 102], the construction of cMERA for gauge theories has been discussed,specially both a massless and massive U (1) gauge theory have been studied. So ithas been shown there that gauge invariance of a theory and “quasi-local" characterof the entangler which generates the cMERA wave functional are indeed compatiblewith each other. This could be a good evidence that cMERA would also work forthe case of warped conformal field theories. Note that this compatibility could alsowork for the interacting or non-Abelian gauge theories.Then, all of these studies could be repeated for other Lorentz-violating back-grounds such as Lifshtiz or hyperscaling violating geometries [70], and specificallyusing the already calculated results for the entanglement entropy of these back-grounds, in various works, one could check the effects of Lifshitz parameter z orhyperscaling exponent θ on the structure of the tensor network, kinematic space,quantum gates of symmetries, cost functions, Fisher information metric, etc.These studies could also be performed again for the case of T ¯ T and J ¯ T deforma-tions. For instance, in [103], it has been found that the integrable T ¯ T deformationwould decrease the degrees of freedom of the subsystem leading to a renormalization-like flow. So one could consider this deformation as the rescaling of the energy scale.The phase transitions in these systems have been studied as well. Similar calculationsthen for the case of warped CFTs or J ¯ T deformations could be performed. Recentprogress on path-integral optimization for T ¯ T has been reported in [104].The specific structure of casual cone of warped AdS could also be constructedand then the theory of minimal update proposal (MUP) and rayed MERA [105]could be applied to the warped CFTs. The existence of the anomalous symmetryand chirality could have interesting consequences in this setup.Also, in [106, 107], some new aspects of entanglement entropy in “gauge theories"have been studied. Using path integral complexity, the implications of these aspectsof gauge theories for the complexity could also be analyzed. One, for instance, couldcheck how the notion of path-integral complexity would be different for the Abelianversus non-Abelian gauge theories. 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