In Section 5 we explore a very interesting special point in the parameter space of the couplings, namely when λ=+ = 0 (or λ=−= 0), which is dual to the null-distorted AdS3 geometry (with non-vanishing dilaton and NS- NS B Mark). In the following discussion, we focus exclusively on the long strings of the R sector.
JHEP10(2022)143
The holographic 2 + 1-d background
In the dual sense, this geometry represents an integrable RG flow connecting a Lorentz-invariant local CFT (fixed point) in the IR with a Lorentz violating non-local theory in the UV, namely a deformed small string theory ( LST). Here we work in the string frame with a non-trivial dilaton background and the volume complexity proposal must be generalized.
Volume complexity in stationary coordinates (x, t)
In anticipation of the fact that the double boundary field theory violates Lorentz, we calculate the volume complexity in two different Lorentz frames and the comparison is drawn between the results. The divergence structure as evident from this expression does not correspond to that of the Lorentz covariant local field theory.
Volume complexity in static (X, T ) coordinates
- A comment on the nonlocality and Lorentz violation
Obviously, the divergence structure of the volume complexity (3.26) does not look like that of a local quantum field theory. Next, consider the coefficient of the log term (which is universal) in the volume complexity expression (3.26) in the deep UV (ie βH0 ), which is.
Thus, we have successfully reproduced the features of the RT curves for the special cases of the pure AdS, skewed AdS and M3 from our general formula relating the inflection point of the RT curve and the subregion length (4.7). We will use this relation to obtain the expression for subregion volume complexity next. Since we are not interested in perturbative responses, we will do this accurately but numerically.
Just as we did for the RT curve, before presenting the final results for the general case, we first perform sanity checks by studying several special cases where the effects of locality and Lorentz violation are removed and comparing those expressions with existing results in the literature obtained in the contexts where the boundary dual is a local CFT2, instead of an LST2.
The volume corresponding to this subregion (slices of maximum volume) as a function of the breakpoint U0 is given by. In order not to compromise accuracy, we instead perform the calculations numerically to illustrate the quantitative features of subregion complexity. In Fig. 2, we present numerical diagrams showing the dependence of the complexity (the modulus of the factor 8πGN√ . kls) on the length of the subregion L for three different values (different sizes) λ, T T of the deformation coupling parameter.
The volume complexity of a subregion undergoes a sharp (phase) transition when the size of the subregion increases beyond a certain critical size, as shown by the presence of a kink in each of the faces. The parabolic part of the curve, for the subregion size (length) is less than the critical length, refers to the linear dilaton region because that is where the subregion size is. The kink implies a break in the linear dilaton geometry and the mass is then taken over by the AdS geometry.
For fixed λ (i.e., degree of nonlocality held fixed), the critical size of the subregion at the phase transition point in the plots increases (shifts to the right) as the Lorentz+ (−) violating pairing increases. Interestingly, the critical size of the subregion changes (increases) even if only one of the + couplings becomes nonzero. We will keep this in mind when looking at the complexity of the static frame where it will turn out that the critical size of the subregion is a product function.
From the graphs, one can directly assess the appearance of the transition point Lc where the complexity characteristics switch sharply from linear parabolic dependence to AdS. A phase transition in holographic entanglement entropy as a function of subregion size for the same system is shown in [2]. What is interesting, however, is that not only does complexity undergo an analogous phase transition, but that the subregion complexity phase transition occurs at the same critical subregion length as the entanglement entropy phase transition, as seen from Table 1 showing the numerical value of the critical length derived from the graphs and the theoretical expression for the critical length of the subregion for the entropy of entanglement (refer to Eq.
Subregion volume complexity in static frame
This limiting value of L in the static frame, called L0c, gives the critical length of the subregion at the point of the Hagedorn phase transition. This is the critical subregion size where the entanglement entropy (RT curve) undergoes the Hagedorn phase transition in the static frame. Subregion volume complexity is a monotonically increasing function of subregion size, and it undergoes a phase transition as the subregion size is varied (like subregion complexity in the stationary frame) beyond some critical length, which turns out to be L0c of eq. .
The physics of this is the same as in the stationary frame - for small subregion sizes, the RT curve is confined to the near-boundary linear dilaton region, i.e. unlike in the stationary frame, in the static the critical subregion size at the transition point , extracted from the location of kinks in the parcels, does not change as it. This is potentially due to the fact that the complexity of the static frame subregion effectively becomes the function of λ0 =λ−4+−, so it is insensitive to distinguish between the different values of λ0 for vanishing value of the product+−.
Volume complexity
In this section we consider a special limit on the parameter space of trivial LST couplings, for which the largest dual is the Warped AdS null geometry, which is smoothly realized by sending λ→0 and one of the Lorentz violating couplings (say −) to zero. The limit theory in this case is adopted CFT [88, 89], a highly nonlocal field-breaking Lorentz theory with the CFT symmetry algebra now reduced to a semidirect product of Virasoro (left) and a U(1 ) Kac-Moody (right ). In particular, for null-deformed WAdS3, the double-deformed CFT is not complete UV, beyond a certain critical energy (deep UV), the theory is non-unitary since the energy spectrum is complex [61].
Although correlation functions are difficult to compute in this warped CFT, we demonstrate that this feature (UV imperfection) is easily captured by complexity.
Subregion volume complexity for null WAdS 3
Unlike in the case of generic non-vanishing λ, + the subregion complexity does not undergo any phase transition. In this special point of the parameter space we will have to be satisfied exclusively with the volume complexity, subregion volume complexity and action complexity in the stationary coordinate system. Conventionally in the literature, these total derivative terms are abandoned as they do not contribute to the classical equations of motion.
For the action complexity calculation, the first order of business is to determine the WdW patch, that is to say, the solution of the WdW patch is however very complicated in the stationary frame coordinates (2.17) where constant t surfaces are not orthogonal to the vector∂/∂t . Life is much simpler in the static frame coordinates (3.14) since the constant T surfaces are orthogonal to the time direction vector ∂/∂T.
Volume (EH) pieces of the onshell action
- Action contributions from the null boundaries of the WdW patch The WdW patch action receives surface contributions (GHY terms) from the boundaries of
This will also introduce an additional contribution to the surface terms (GHY terms) in the string frame, in addition to the usual GHY surface term arising from the metric. The kinetic term of the dilaton in action: we refer the reader to section B.2.2 for details. The cosmological constant term in action: details are worked out in section B.2.3.
The Kalb-Ramond term in action: the final space-time volume type contribution from the Kalb-Ramond field in 4 dimensions or the full Kalb-Ramond derivative fields in 3 dimensions after dimensionality reduction (a Kalb-Ramond two-form field and a Kalb-Ramond one-form gauge field). The details, including the matching before and after dimension reduction, are prepared in appendix section B.2.1. Using (6.21), the induced metric on this time-like surface can be written as ds2 =−εkls2dU2. 6.22) The negative sign in the first term clearly indicates that this is a time-like surface.
Action complexity
We conclude this work by presenting results for the complexity of the zero-distorted AdS3 action defined by the limitλ=+= 0. Instead, work from zero in a stationary Lorentzian frame on the limit for which the dual metric is also stationary . In this case the dilaton field is simply a constant2Φ.=gs2, which points to the fact that the double limit theory, WCFT, has scale (Weyl) symmetry.
To construct the WdW patch bounds for this stationary metric, we first rearrange the WAdS3 null metric to form. The boundary of the WdW patch at time T0 is then described by the void surface obtained by summing the null radii obtained by varying U and x0. As a result for the space-time volume terms in action (Ricci, cc, the Kalb-Ramond part etc.) the integration sequences are
Bulk action terms
GHY surface terms from null boundaries of WdW patch
Let's first start working with the upper part of the time-like (near zero) surface (regulating surface). For the lower part of the deformed time-like surface the outward unit normal is. The normalization constant and the extrinsic curvature are the same as those for the upper part of the control surface.
We identify this leading quadratic divergence as a hallmark of the nonlocal nature of the LST. This was one of the main reasons for including the complexity of the subregion in this work. Qualitatively, the complexity of operation11 at the final temperature showed the same behavior as in the case of zero temperature.
KK reduction of the Kalb-Ramond field
With this choice of α, β one must of course change the Dilaton, so that the action has the same normalization for the Ricci and the c.c. Now, the first step in the KK reduction for the H2 term according to the recipe of [103] is to divide the 4 dimensional NS-NS B-field potential (A.10) by eq. B.6) We previously noted that from the Ricci sector, .
Matching the 4d action terms with the 3d action terms The volume terms in the bulk action (6.2) are
- Matching the contributions of the Kalb-Ramond field sector before and after KK reduction
- Matching the contributions of the dilaton sector before and after KK reduction
- Matching the contributions of the cosmological constant term before and after KK reduction
- Matching the contributions of the Ricci scalar sector before and after KK reduction
By considering separately the similarity of term blocks in the actions (B.17), (B.20), (B.22) and (B.32), in the following subsections and by summing over both sides of these terms, we demonstrate the equality of the two actions (B.12) and (B.13). Here we again used the value of the twist factor as −= 10−6 for the same reason, as mentioned at the end of section 5.2. A curious fact to note is that the holographic entanglement entropy shows no locality with respect to the appearing UV divergences - for any value of the twist.
This is the exact opposite of the pattern of UV divergences that occur in the volume complexity of the subregion (all orders of UV divergences occur there). The fact that the entanglement entropy of WCFT2 is a highly nonlocal theory that violates Lorentz augmentation but has exactly the same UV divergence structure as the entanglement entropy of local CFT2 has been established in previous works [92,93]. Efimov, Nonlocal quantum field theory, Lagrangian nonlinear interactions and series convergence of perturbation theory, Theor.
