Is Action Complexity better for de Sitter space in Jackiw-Teitelboim gravity?

According to the CA conjecture, the complexity of a state with a holographic realization is proportional to the full gravitational action of the Wheeler-DeWitt (WDW) patch:

$$\mathcal{C}_{\rm A}=\frac{I_{\rm WDW}}{\pi}\,.$$

(7)

We compute action complexity in Jackiw-Teitelboim (JT) gravity, whose bulk action is

$$I_{\rm JT}=\frac{1}{8G}\int_{\mathcal{W}}d^{2}x\sqrt{-g}\,\left(\phi\,R+L^{-2}\,U\left(\phi\right)\right)+\frac{1}{4G}\int_{\partial\mathcal{W}}dy\sqrt{-h}\,\phi_{b}\,K\,,$$

(8)

with $\phi$ the dilaton and $\phi_{b}$ its value at the boundary $\partial\mathcal{W}$. The dilaton potential is $U(\phi)=2\phi$ for anti-de Sitter (AdS) spacetime and $U(\phi)=-2\phi$ for de Sitter (dS) spacetime. The boundary term represents the Gibbons-Hawking-York (GHY) action York:1972sj ; Gibbons:1976ue , where $K=h^{\mu\nu}K_{\mu\nu}$ is the trace of the extrinsic curvature. The latter is defined as $K_{\mu\nu}=h^{\rho}_{\mu}h^{\sigma}_{\nu}\nabla_{\rho}n_{\sigma}$, with $n_{\sigma}$ the outward-directed normal to $\partial\mathcal{W}$.

The equations of motion for the dilaton and the metric are

	$\displaystyle R$	$\displaystyle=-\frac{U^{\prime}(\phi)}{L^{2}},$		(9)
	$\displaystyle 0$	$\displaystyle=\nabla_{a}\nabla_{b}\phi-g_{ab}\nabla^{2}\phi+\frac{g_{ab}}{2L^{2}}U(\phi).$		(10)

In this paper, we work in dS spacetime, so we choose

$$U(\phi)=-2\phi\,.$$

(11)

With this choice, the equations of motion are solved by the following metric and dilaton

$$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}\,,\qquad\phi=\frac{r}{L}\,,$$

(12)

where the blackening factor $f(r)$ is

$$f(r)=1-\frac{r^{2}}{L^{2}}\,.$$

(13)

In the following, we focus on the portion of spacetime with $0\leq r<\infty$ (for the reason we shall explain at the beginning of subsection 2.2). Therefore, the position of the cosmological horizon is $r_{h}=L$. It is convenient to introduce the tortoise coordinate

$$r^{*}(r)=\int^{r}\frac{d\hat{r}}{f(\hat{r})}\,.$$

(14)

Taking as a boundary condition $r^{*}(0)=0$, we get

$$r^{*}(r)=\frac{L}{2}\log\left|\frac{r+L}{r-L}\right|\,.$$

(15)

The Eddington-Finkelstein (EF) coordinates are given by

$$v=t+r^{*}(r)\,,\qquad u=t-r^{*}(r)\,.$$

(16)

To describe the whole of spacetime, we need to consider two copies of EF coordinates, one for the right (R) and one for the left (L) side of the Penrose diagram, see figure 1. Note that $u_{R}$ is constant along null rays falling into the cosmological horizon, whereas $v_{R}$ is constant along outgoing null rays. The reverse behaviour applies to the EF coordinates for the left side of the Penrose diagram.

Action complexity is obtained by evaluating the full gravitational action of the WDW patch as in eq. (7). The question of which action should be considered is non-trivial since the full JT gravity action of the relevant spacetime region is not just given by $I_{\rm JT}$ in eq. (8). In fact, there are several additional boundary contributions. In order to determine the right action, we exploit the fact that the dS JT gravity action in two dimensions can be obtained by dimensional reduction from three-dimensional pure dS, starting from the Einstein-Hilbert (EH) action plus boundary terms Svesko:2022txo .²²2The dimensional reduction allows us to circumvent the study of the variational problem in JT gravity, leading to the same result for the full gravitational action. We employ this procedure to explicitly compute action complexity in dS₂.

Before we proceed, we point out that there are two ways to obtain dS₂ JT gravity from dimensional reduction. One is half reduction from pure dS₃ Svesko:2022txo and the other is full reduction from Schwarzschild-dS_d ($d\geq 4$) in the near-Nariai limit Nariai1950 ; Ginsparg:1982rs (see Anninos:2012qw ; Maldacena:2019cbz ; Svesko:2022txo as well). For our analysis, we use half reduction from pure dS₃, which automatically reduces to the $r\geq 0$ range. The dS₂ spacetime obtained from Schwarzschild-dS_d ($d\geq 4$) in the near-Nariai limit also includes $r<0$ and a horizon at negative $r$. In this limit, we are zooming in on the very tiny region near the black hole and cosmological horizons, and the $r$-negative horizon corresponds to the original black hole one. Since our main interest is holographic complexity beyond the cosmological horizon, rather than black hole complexity, we focus on $r>0$. For this purpose, half reduction from three-dimensional pure dS is enough. In three dimensions, the full gravitational action is Lehner:2016vdi

	$\displaystyle I_{\rm tot}$	$\displaystyle=I_{\rm EHGHY}+I_{\rm null}+I_{\rm joint}+I_{\rm ct}\,,$		(17)
	$\displaystyle I_{\rm EHGHY}$	$\displaystyle\equiv\frac{1}{16\pi G_{(3)}}\int_{\tilde{\mathcal{W}}}d^{3}X\sqrt{-g_{(3)}}\left(R_{(3)}-2\Lambda_{(3)}\right)+\frac{1}{8\pi G_{(3)}}\int_{\partial\tilde{\mathcal{W}}}d^{2}Y\sqrt{-h_{(3)}}\,K_{(3)}\,,$
	$\displaystyle I_{\rm null}$	$\displaystyle\equiv\frac{1}{8\pi G_{(3)}}\int_{\tilde{\mathcal{B}}}dS\,d\lambda\,\sqrt{\gamma}\,k\,,\qquad I_{\rm joint}\equiv\frac{1}{8\pi G_{(3)}}\int_{\tilde{\mathcal{J}}}d\theta\,\sqrt{\gamma}\,\tilde{\mathfrak{a}}\,,$
	$\displaystyle I_{\rm ct}$	$\displaystyle\equiv\frac{1}{8\pi G_{(3)}}\int_{\tilde{\mathcal{B}}}dS\,d\lambda\,\sqrt{\gamma}\,\Theta\,\log\left|\tilde{L}\,\Theta\right|\,.$

The term $I_{\rm EHGHY}$ contains the EH and the GHY actions, the latter coming from spacelike and timelike codimension-one boundaries. Contributions from codimension-one null surfaces $\tilde{\mathcal{B}}$ are given by $I_{\rm null}$ Lehner:2016vdi ; Parattu:2015gga . In $I_{\rm joint}$, we have the contributions from codimension-two joints $\tilde{\mathcal{J}}$ Hayward:1993my ; Lehner:2016vdi at the intersection between codimension-one surfaces. Finally, $I_{\rm ct}$ is a counterterm for null boundaries $\tilde{\mathcal{B}}$ Lehner:2016vdi which ensures invariance of the full action under reparameterization of null normals. In what follows, we analyze each term separately and we apply dimensional reduction to determine the right action for holographic complexity in dS₂ JT gravity.

The EH-GHY action in three dimensions with positive cosmological constant reads

	$\displaystyle I_{\rm EHGHY}$	$\displaystyle=\frac{1}{16\pi G_{(3)}}\int_{\tilde{\mathcal{W}}}d^{3}X\sqrt{-g_{(3)}}\left(R_{(3)}-2\Lambda_{(3)}\right)$		(18)
		$\displaystyle+\frac{1}{8\pi G_{(3)}}\int_{\partial\tilde{\mathcal{W}}}d^{2}Y\sqrt{-h_{(3)}}\,K_{(3)}\,,$

where

$$\Lambda_{(3)}=\frac{1}{L_{(3)}^{2}}\,.$$

(19)

This admits a three-dimensional dS spacetime solution

$\displaystyle ds_{(3)}^{2}=g_{(3)MN}dX^{M}dX^{N}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\theta^{2}\,,\quad f(r)=1-\frac{r^{2}}{L_{(3)}^{2}}\,.$

(20)

In the GHY boundary term in eq. (18), $h_{(3)}$ is the determinant of the induced metric on $\partial\tilde{\mathcal{W}}$ and $K_{(3)}$ denotes the trace of the extrinsic curvature.
We now introduce the metric ansatz

$$ds_{(3)}^{2}=g_{(2)\mu\nu}dx^{\mu}dx^{\nu}+L_{(3)}^{2}\phi^{2}(x)d\theta^{2}\,,$$

(21)

where we have expressed $X^{M}=\left(t,r,\theta\right)=\left(x^{\mu},\theta\right)$ with $M=0,1,2$ and $\mu=0,1$. As in eq. (12), we consider a solution for the dilaton $\phi(x)$ depending just on the radial coordinate: $\phi=\phi(r)$. From the metric ansatz (21), we get Svesko:2022txo

	$\displaystyle R_{(3)}$	$\displaystyle=R_{(2)}-\frac{2}{\phi}\square_{(2)}\phi\,,$		(22)
	$\displaystyle K_{(3)}$	$\displaystyle=K_{(2)}+\frac{1}{\phi}n^{\mu}\nabla_{(2)\mu}\phi\,,$
	$\displaystyle\sqrt{-g_{(3)}}$	$\displaystyle=L_{(3)}\,\phi\,\sqrt{-g_{(2)}}\,,$

with $n^{\mu}$ the normal vector to the two-dimensional boundary $\partial\tilde{\mathcal{W}}$. Plugging into eq. (18), we obtain

	$\displaystyle I_{\rm JT}$	$\displaystyle=\frac{L_{(3)}}{8G_{(3)}}\int_{\mathcal{W}}d^{2}x\sqrt{-g_{(2)}}\left[\phi\left(R_{(2)}-2\Lambda_{(3)}\right)-2\,\square_{(2)}\phi\right]$		(23)
		$\displaystyle+\frac{L_{(3)}}{4G_{(3)}}\int_{\partial\mathcal{W}}dy\sqrt{-h_{(2)}}\,\left(\phi\,K_{(2)}+n^{\mu}\nabla_{(2)\mu}\phi\right)\,,$

where $\mathcal{W}$ and $\partial\mathcal{W}$ are the manifolds endowed with metric $g_{(2)\mu\nu}$ and $h_{(2)\mu\nu}$, respectively. Defining $G\equiv G_{(2)}=G_{(3)}/L_{(3)}$ and $L\equiv L_{(3)}$, we recognize the JT gravity action $I_{\rm JT}$ in eq. (8) with some additional terms. Three comments regarding the EH-GHY contributions are in order:

• •

  According to the CA conjecture, we must evaluate the full gravitational action of the WDW patch. Such a spacetime region, which we denote by $\mathcal{W}$, is bounded by null codimension-one surfaces. Since there are neither timelike nor spacelike boundaries, the GHY term in eq. (23) vanishes.

• •

  With the metric ansatz in eq. (21), the three-dimensional dS spacetime solution in eq. (20) comes down to the two-dimensional dS solution with linear dilaton given in eqs. (12) and (13). On-shell we have

  $$R_{(2)}=2\Lambda_{(3)}=\frac{2}{L_{(3)}^{2}}\,,$$

  (24)

  so the bulk contribution $I_{\rm JT}$ in eq. (23) reduces to

  $$I_{\rm bulk}=-\frac{1}{4G_{(2)}}\int_{\mathcal{W}}d^{2}x\sqrt{-g_{(2)}}\,\square_{(2)}\phi\,.$$

  (25)

  Even though we refer to this expression as two-dimensional “bulk” action, $I_{\rm bulk}$ is in fact a boundary term since it comprises a total divergence.³³3As it will be clear later, at late times all contributions to the full action have the same divergence structure. Therefore, by neglecting $I_{\rm bulk}$ the qualitative behavior of action complexity at late times does not change.

• •

  We can integrate the term $\square_{(2)}\phi$ getting a codimension-one contribution for the null boundaries of the WDW patch distinct from $I_{\rm null}$ in eq. (17):

  $$I_{\rm bulk}=-\frac{1}{4G_{(2)}}\int_{\tilde{\mathcal{B}}}d\lambda\,k^{\mu}\nabla_{(2)\mu}\phi=-\frac{1}{4G_{(2)}}\int_{\tilde{\mathcal{B}}}d\lambda\,\partial_{\lambda}\phi\,,$$

  (26)

  where $k^{\mu}$ is the null normal to $\tilde{\mathcal{B}}$, see Goto:2018iay for an analog analysis in AdS₂ JT gravity. As we will see, this term yields nonzero contribution and is important to get consistency with the three-dimensional CA result.

Besides the bulk term, the full gravitational action of the WDW patch contains boundary terms for null codimension-one surfaces and joint terms for codimension-two surfaces at the intersection of null bounding surfaces, which we consider next.

The contribution from null boundaries has been originally studied in Lehner:2016vdi ; Parattu:2015gga . In three-dimensional spacetime it is

$$I_{\rm null}=\frac{1}{8\pi G_{(3)}}\int_{\tilde{\mathcal{B}}}dS\,d\lambda\,\sqrt{\gamma}\,k\,,$$

(27)

where $\lambda$ is a parameter running along the null geodesics generating the surface $\tilde{\mathcal{B}}$, $S$ is the transverse spatial direction to such generators, and $\gamma$ is the determinant of the induced metric in the $S$ direction. The constant $\kappa$ is defined by the geodesic equation

$$\tilde{k}^{M}\nabla_{(3)M}\tilde{k}^{N}=\kappa\,\tilde{k}^{N}\,,$$

(28)

with $\tilde{k}^{M}=\frac{dX^{M}(\lambda)}{d\lambda}$ the null generator. In other words, $\kappa$ measures the failure of $\lambda$ to be an affine parameter. Consequently, we can set $\kappa=0$ by a wise parameterization choice, getting rid of the $I_{\rm null}$ contribution.

At the intersection between bounding codimension-one surfaces, where the boundary is non-smooth, the joint term comes into play. The joint contributions involving just timelike and spacelike intersecting surfaces have been investigated in Hayward:1993my , while joints involving at least one null boundary have been studied in Lehner:2016vdi . In our computation we will meet only joints involving two null boundaries, which in three-dimensional spacetime are given by

	$\displaystyle I_{\rm joint}$	$\displaystyle=\frac{1}{8\pi G_{(3)}}\int_{\tilde{\mathcal{J}}}d\theta\,\sqrt{\gamma}\,\tilde{\mathfrak{a}}\,,$		(29)
	$\displaystyle\sqrt{\gamma}=\sqrt{g_{(3)\theta\theta}}$	$\displaystyle\,,\qquad\tilde{\mathfrak{a}}={\rm sign}({\rm joint})\times\log\left|\frac{\mathbf{\tilde{k}_{1}}\cdot\mathbf{\tilde{k}_{2}}}{2}\right|_{\tilde{\mathcal{J}}}\,,$

where $\mathbf{\tilde{k}_{1}},\mathbf{\tilde{k}_{2}}$ denote the one-forms normal to null boundaries, which are taken to be outward-directed from the spacetime region of interest. The sign of the joint term is fixed by the following rule. Choosing either of the two null boundaries intersecting at the joint, the sign of the corresponding contribution (29) is positive if the bulk region $\mathcal{W}$ is at the future (past) of the segment and the joint itself is located at the past (future) edge of the selected segment. In the remaining configurations, the joint term has a negative sign Lehner:2016vdi .

Applying the dimensional reduction, we obtain

$$I_{\rm joint}=\frac{1}{4G_{(2)}}\sum_{\mathcal{J}}\phi(r_{\mathcal{J}})\,\mathfrak{a}_{\mathcal{J}}\,,\qquad\mathfrak{a}_{\mathcal{J}}={\rm sign}({\rm joint})\times\log\left|\frac{\mathbf{k_{1}}\cdot\mathbf{k_{2}}}{2}\right|_{J}\,.$$

(30)

Here $r=r_{\mathcal{J}}$ is the radial position of the joint, and we consider one-forms $\mathbf{k_{1}}$, $\mathbf{k_{2}}$ normal to one-dimensional boundaries. By spherical symmetry of the dS₃ geometry, we trivially have $\mathbf{\tilde{k}_{i}}=\mathbf{k_{i}}$ and $\tilde{k}_{i}^{M}=\left(k_{i}^{\mu},0\right)$, with $i=1,2$.

The joint contributions in eq. (30) are affected by the arbitrariness of choosing the normalization of null normals $\mathbf{k_{1}},\mathbf{k_{2}}$. Such ambiguity can be partially removed by requiring that $\mathbf{k_{i}}\cdot\partial_{t}=\pm\alpha$ at the spacetime boundary Carmi:2016wjl ; Chapman:2016hwi ; Carmi:2017jqz , with $\partial_{t}$ the timelike Killing vector in the boundary theory and $\alpha$ a positive constant. Still, the constant $\alpha$ can be arbitrarily chosen. For the gravitational action to be invariant under the reparameterization of null generators, a counterterm must be added for each null boundary Lehner:2016vdi . In three-dimensional spacetime, the counterterm has the following form

$$I_{\rm ct}=\frac{1}{8\pi G_{(3)}}\int_{\tilde{\mathcal{B}}}dS\,d\lambda\,\sqrt{\gamma}\,\Theta\,\log\left|\tilde{L}\,\Theta\right|\,,$$

(31)

where $\tilde{L}$ is an arbitrary length scale. In the above expression, $\Theta$ is the expansion of null geodesics given by $\Theta=\partial_{\lambda}\log\sqrt{\gamma}$. An explicit calculation with the metric ansatz (21) leads to

$$\Theta=\partial_{\lambda}\log\sqrt{g_{(3)\theta\theta}}=\partial_{\lambda}\log\phi=\frac{\partial_{\lambda}\phi}{\phi}\,.$$

(32)

Therefore, in JT gravity the counterterm reads

$$I_{\rm ct}=\frac{1}{4G_{(2)}}\int_{\mathcal{B}}d\lambda\,\partial_{\lambda}\phi\,\log\left|\tilde{L}\,\partial_{\lambda}\log\phi\right|\,.$$

(33)

Summarizing, the full gravitational action for holographic complexity in dS₂ JT is given by

	$\displaystyle I_{\rm JTdS_{2}}$	$\displaystyle=I_{\rm bulk}+I_{\rm joint}+I_{\rm ct}$		(34)
		$\displaystyle=-\frac{1}{4G_{(2)}}\int_{\tilde{\mathcal{B}}}d\lambda\,\partial_{\lambda}\phi+\frac{1}{4G_{(2)}}\sum_{\mathcal{J}}\phi(r_{\mathcal{J}})\,\mathfrak{a}_{\mathcal{J}}$
		$\displaystyle\,\,\,\,\,\,\,+\frac{1}{4G_{(2)}}\int_{\mathcal{B}}d\lambda\,\partial_{\lambda}\phi\,\log\left|\tilde{L}\,\partial_{\lambda}\log\phi\right|\,.$

The lesson we learn from the dimensional reduction analysis is that the effects of the dilaton, which explicitly appears in all terms of the full action (34), are important for the evaluation of holographic complexity. This is consistent with the discussion in the introduction. Note that in JT gravity the value of dilaton varies, and as a result, one has an effective Newton “constant”

$$\frac{1}{G_{(2)\,{\rm eff}}}\equiv\frac{\phi}{G_{(2)}}\,.$$

(35)

We recall that in two dimensions there is no “area” for a black hole horizon, but instead, there is a dilaton. The dilaton plays the role of area. In fact, in $d(\geq 3)$-dimensional dS spacetime, the orthogonal $S^{d-2}$ area grows to infinity near the future spacelike infinity, and by dimensional reduction the size factor of the $S^{d-2}$ area becomes the dilaton. Since the divergence of complexity in higher-dimensional dS is associated with the growth of the orthogonal $S^{d-2}$ area Jorstad:2022mls , without taking into account the dilaton one cannot see the divergence of complexity in two-dimensional dS spacetime in JT gravity.

We now move to the explicit computation of the full gravitational action, mainly following Jorstad:2022mls . To ease the notation, we set $L_{(3)}\equiv L$. First, we introduce the stretched horizons for both the left and the right sides of the Penrose diagram. These are $r$-constant surfaces described by

$$r_{st}=\rho L\,,\qquad 0<\rho<1\,,$$

(36)

where the limit $\rho\sim 1$ is intended. We then attach the WDW patch to the two stretched horizons, and we define the anchoring times as $t_{L}$ and $t_{R}$ for the left and the right horizons, respectively. It is not restrictive to focus on the symmetric case $t_{R}=-t_{L}$. For convenience, we define a dimensionless time as $t_{R}=L\,\tau$. We focus on the case where the tip of the WDW patch does not meet the future spacelike infinity at $r=\infty$.

For the right side of the geometry, the future boundary of the WDW patch is at constant $u_{R}$, whereas the past boundary is at constant $v_{R}$. From eq. (16), the defining equations are

	$\displaystyle u_{FR}$	$\displaystyle=t_{R}-r^{*}(r_{st})=L\,\tau-\frac{L}{2}\log\left(\frac{1+\rho}{1-\rho}\right)\,,$		(37)
	$\displaystyle v_{PR}$	$\displaystyle=t_{R}+r^{*}(r_{st})=L\,\tau+\frac{L}{2}\log\left(\frac{1+\rho}{1-\rho}\right)\,.$

Let us denote by $r=r_{\pm}$ the position of the future and past tip of the WDW patch, respectively. By symmetry, both tips are located on the vertical axis in the middle of the Penrose diagram, which is described by $t=0$. So, evaluating $u_{FR}$ defined in eq. (37) at the future tip, from eq. (16) we get

$$L\,\tau-\frac{L}{2}\log\left(\frac{1+\rho}{1-\rho}\right)=-\frac{L}{2}\log\left(\frac{r_{+}+L}{r_{+}-L}\right)\,,$$

(38)

which leads to

$$\frac{r_{+}}{L}=\frac{\cosh\tau-\rho\sinh\tau}{\rho\cosh\tau-\sinh\tau}\,.$$

(39)

Similarly, computing $v_{PR}$ defined in eq. (37) at the past tip we obtain

$$L\,\tau+\frac{L}{2}\log\left(\frac{1+\rho}{1-\rho}\right)=\frac{L}{2}\log\frac{r_{-}+L}{r_{-}-L}\,,$$

(40)

from which

$$\frac{r_{-}}{L}=\frac{\cosh\tau+\rho\sinh\tau}{\rho\cosh\tau+\sinh\tau}\,.$$

(41)

The time $\tau=\tau_{\infty}$ at which the future tip meets the future spacelike infinity $r\to\infty$ can be obtained by eq. (39) as

$$\tau_{\infty}=\tanh^{-1}\rho\,.$$

(42)

We can thus rewrite eqs. (38) and (40) as

$$\tau_{\infty}-\tau=\tanh^{-1}\left(\frac{L}{r_{+}}\right)\,,\qquad\tau+\tau_{\infty}=\tanh^{-1}\left(\frac{L}{r_{-}}\right)\,,$$

(43)

or equivalently

$$r_{\pm}=L\coth\left(\tau_{\infty}\mp\tau\right)\,.$$

(44)

Since we are interested in how complexity grows as $\tau$ approaches $\tau_{\infty}$, we focus on $\tau\leq\tau_{\infty}$. We now evaluate the gravitational action in eq. (34) term by term.

Explicit computation of the boundary term in eq. (26) gives

	$\displaystyle I_{\rm bulk}$	$\displaystyle=\left.\frac{\phi}{2G_{(2)}}\right|^{r_{+}}_{\rho L}+\left.\frac{\phi}{2G_{(2)}}\right|^{r_{-}}_{\rho L}=\frac{1}{G_{(2)}}\left(\frac{r_{+}+r_{-}-2\rho L}{2L}\right)$		(45)
		$\displaystyle=\frac{1}{G_{(2)}}\,\frac{\rho\left(1-\rho^{2}\right)\cosh^{2}\tau}{1-\left(1-\rho^{2}\right)\cosh^{2}\tau}\,,$

where in the last step eqs. (39) and (41) has been used.

We now move to the boundary terms. The null normals $\mathbf{k}=k_{\mu}dx^{\mu}$ to the four codimension-one boundaries of the WDW patch are

	$\displaystyle\mathbf{k}_{FR}$	$\displaystyle=\alpha\,du_{R}=\alpha\left(dt-\frac{dr}{f(r)}\right)\,,$		(46)
	$\displaystyle\mathbf{k}_{PR}$	$\displaystyle=-\beta\,dv_{R}=\beta\left(-dt-\frac{dr}{f(r)}\right)\,,$
	$\displaystyle\mathbf{k}_{FL}$	$\displaystyle=-\alpha^{\prime}\,dv_{L}=\alpha^{\prime}\left(-dt-\frac{dr}{f(r)}\right)\,,$
	$\displaystyle\mathbf{k}_{PL}$	$\displaystyle=\beta^{\prime}\,du_{L}=\beta^{\prime}\left(dt-\frac{dr}{f(r)}\right)\,,$

where $\alpha,\beta,\alpha^{\prime},\beta^{\prime}$ are positive arbitrary constants and F (P) stands for future (past), see figure 2. This choice corresponds to an affine parameterization of null normals, thus the boundary term (27) for null surfaces vanishes. To compute the four joint terms in eq. (30) we need

	$\displaystyle\mathfrak{a}_{+}$	$\displaystyle=+\log\left|\frac{\mathbf{k}_{FR}\cdot\mathbf{k}_{FL}}{2}\right|_{r=r_{+}}=\log\left|\frac{\alpha\alpha^{\prime}}{f(r_{+})}\right|\,,$		(47)
	$\displaystyle\mathfrak{a}_{-}$	$\displaystyle=+\log\left|\frac{\mathbf{k}_{PR}\cdot\mathbf{k}_{PL}}{2}\right|_{r=r_{-}}=\log\left|\frac{\beta\beta^{\prime}}{f(r_{-})}\right|\,,$
	$\displaystyle\mathfrak{a}_{st,R}$	$\displaystyle=-\log\left|\frac{\mathbf{k}_{FR}\cdot\mathbf{k}_{PR}}{2}\right|_{r=r_{st}}=-\log\left|\frac{\alpha\beta}{f(\rho L)}\right|\,,$
	$\displaystyle\mathfrak{a}_{st,L}$	$\displaystyle=-\log\left|\frac{\mathbf{k}_{FL}\cdot\mathbf{k}_{PL}}{2}\right|_{r=r_{st}}=-\log\left|\frac{\alpha^{\prime}\beta^{\prime}}{f(\rho L)}\right|\,.$

Then, the total joint contribution is

	$\displaystyle I_{\rm joint}$	$\displaystyle=I_{+}+I_{-}+I_{st,R}+I_{st,L}$		(48)
		$\displaystyle=\frac{1}{4G_{(2)}L}\left(r_{+}\,\log\frac{\alpha\alpha^{\prime}L^{2}}{r_{+}^{2}-L^{2}}+r_{-}\,\log\frac{\beta\beta^{\prime}L^{2}}{r_{-}^{2}-L^{2}}-\rho L\,\log\frac{\alpha\beta\alpha^{\prime}\beta^{\prime}}{\left(1-\rho^{2}\right)^{2}}\right)\,.$

Finally, we have to calculate the counterterm for null boundaries given by eq. (33). We start from the FR boundary. The null vector orthogonal to this surface is

$$k_{FR}^{\mu}=g^{\mu\nu}k_{FR\,\nu}=-\alpha\left(\frac{1}{f(r)}\,,1\right)\,.$$

(49)

By definition, the null vector is $k^{\mu}=\partial_{\lambda}x^{\mu}=\left(\dot{t},\dot{r}\right)$, where the dot denotes a derivative with respect to the affine parameter $\lambda$. We thus conclude that $\dot{r}=-\alpha$, implying that $r=-\alpha\,\lambda$ is an affine parameter too which increases when $\lambda$ decreases. Putting all together, we get

	$\displaystyle I_{ct,FR}$	$\displaystyle=\frac{1}{4G_{(2)}L}\int_{r_{+}}^{\rho L}dr\,\log\frac{\tilde{L}\,\alpha}{r}$		(50)
		$\displaystyle=\frac{1}{4G_{(2)}L}\left(\rho L-r_{+}+\rho L\log\left(\frac{\tilde{L}\,\alpha}{\rho L}\right)-r_{+}\log\left(\frac{\tilde{L}\,\alpha}{r_{+}}\right)\right)\,.$

By symmetry, the contribution on the future left boundary is simply $I_{ct,FL}=I_{ct,FR}\left(\alpha\rightarrow\alpha^{\prime}\right)$. Similarly, the counterterm for the right past boundary is

	$\displaystyle I_{ct,PR}$	$\displaystyle=\frac{1}{4G_{(2)}L}\int_{r_{-}}^{\rho L}dr\,\log\frac{\tilde{L}\,\beta}{r}$		(51)
		$\displaystyle=\frac{1}{4G_{(2)}L}\left(\rho L-r_{-}+\rho L\log\left(\frac{\tilde{L}\,\beta}{\rho L}\right)-r_{-}\log\left(\frac{\tilde{L}\,\beta}{r_{-}}\right)\right)\,,$

and the contribution from the past left boundary is $I_{ct,PL}=I_{ct,PR}\left(\beta\rightarrow\beta^{\prime}\right)$. So, the total counterterm reads

	$\displaystyle I_{ct}$	$\displaystyle=I_{ct,FR}+I_{ct,FL}+I_{ct,PR}+I_{ct,PL}$		(52)
		$\displaystyle=\frac{1}{4G_{(2)}L}\left(4\rho L-2r_{+}-2r_{-}+\rho L\log\left(\frac{\tilde{L}^{4}\alpha\beta\alpha^{\prime}\beta^{\prime}}{\rho^{4}L^{4}}\right)-r_{+}\log\left(\frac{\tilde{L}^{2}\alpha\alpha^{\prime}}{r_{+}^{2}}\right)-r_{-}\log\left(\frac{\tilde{L}^{2}\beta\beta^{\prime}}{r_{-}^{2}}\right)\right)\,.$

Note that the terms containing the arbitrary constants $\alpha,\beta,\alpha^{\prime},\beta^{\prime}$ cancel the corresponding terms in the joint contribution (48). In details,

	$\displaystyle I_{\rm joint}+I_{ct}$	$\displaystyle=\frac{1}{2G_{(2)}L}\left(2\rho L-r_{+}-r_{-}+r_{+}\,\log\frac{r_{+}L}{\tilde{L}\sqrt{r_{+}^{2}-L^{2}}}+r_{-}\,\log\frac{r_{-}L}{\tilde{L}\sqrt{r_{-}^{2}-L^{2}}}-\rho L\,\log\frac{\rho^{2}L^{2}}{\tilde{L}^{2}\left(1-\rho^{2}\right)}\right)$		(53)
		$\displaystyle=\frac{1}{2G_{(2)}L}\left(L\coth\left(\tau_{\infty}-\tau\right)\left(\log\left(\frac{L\cosh\left(\tau_{\infty}-\tau\right)}{\tilde{L}}\right)-1\right)+\left(\tau\rightarrow-\tau\right)\right.$
		$\displaystyle\left.-2L\tanh\left(\tau_{\infty}\right)\left(\log\left(\frac{L\sinh\left(\tau_{\infty}\right)}{\tilde{L}}\right)-1\right)\right)\,,$

where in the second equality eqs. (42) and (44) have been used.

Summing up eqs. (45) and (53), we finally get the action complexity

	$\displaystyle\mathcal{C}_{\rm A}$	$\displaystyle=\frac{I_{\rm JTdS_{2}}}{\pi}=\frac{I_{\rm bulk}+I_{\rm joint}+I_{ct}}{\pi}$		(54)
		$\displaystyle=\frac{1}{2\pi G_{(2)}}\left(\frac{2\rho\left(1-\rho^{2}\right)\cosh^{2}\tau}{1-\left(1-\rho^{2}\right)\cosh^{2}\tau}-2\tanh\left(\tau_{\infty}\right)\left(\log\left(\frac{L\sinh\left(\tau_{\infty}\right)}{\tilde{L}}\right)-1\right)\right.$
		$\displaystyle\left.+\coth\left(\tau_{\infty}-\tau\right)\left(\log\left(\frac{L\cosh\left(\tau_{\infty}-\tau\right)}{\tilde{L}}\right)-1\right)+\left(\tau\rightarrow-\tau\right)\right)\,,$

which matches the dS₃ result found in Jorstad:2022mls , with $G_{(2)}=G_{(3)}/L$.

Some comments are in order:

• •

  The $\mathcal{C}_{\rm A}$ given by eq. (54) diverges at the critical time

  $$\cosh^{2}\tau_{\infty}=\frac{1}{1-\rho^{2}}\,,$$

  (55)

  which is exactly eq. (42). In the limit $\tau\to\tau_{\infty}$, both $r_{+}$ and $\mathcal{C}_{\rm A}$ diverge at the leading order. In particular,

  $$r_{+}=\frac{L}{\left(\tau_{\infty}-\tau\right)}+\mbox{(subleading)}\qquad\mbox{as $\tau\to\tau_{\infty}$}\,,$$

  (56)

  where we have used eq. (39), and

  $$\mathcal{C}_{\rm A}=\frac{1}{\pi G_{(2)}}\frac{1}{\sqrt{2-\rho^{2}}}\frac{1}{\left(\tau-\tau_{\infty}\right)}\propto\frac{r_{+}}{G_{(2)}}+\mbox{(subleading)}\qquad\mbox{as $\tau\to\tau_{\infty}$}\,.$$

  (57)

  Therefore, the late-time behavior of $\mathcal{C}_{\rm A}$ is

  $$\mathcal{C}_{\rm A}\sim\frac{\phi}{G_{(2)}}\equiv\frac{1}{G_{(2)\,{\rm eff}}}\,.$$

  (58)

  As discussed in the introduction, this is exactly what we expect in JT gravity.

• •

  As first pointed out in Susskind:2021esx , in $d\geq 3$ dimensions volume complexity diverges at the critical time. This can be understood from the fact that maximal slices anchored at the two stretched horizons bend upwards to the future spacelike infinity, where local $(d-2)$-spheres exponentially expand. In JT gravity, the effect of such an exponential expansion appears in the linear dilaton. Note that the static coordinate $r$ behaves as time beyond the cosmological horizon $r\geq L$ and it diverges at the future spacelike infinity.

• •

  Even though $\tau_{\infty}$ diverges in the limit $\rho\to 1$, it does as

  $$\rho\equiv 1-\epsilon\,,\quad\tau_{\infty}=\frac{1}{2}\log\frac{2}{\epsilon}+O(\epsilon)\,.$$

  (59)

  So, the $\tau_{\infty}$ dependence on the stretched horizon parameter $\epsilon$ is very mild Susskind:2021esx ; Jorstad:2022mls .

• •

  Without taking into account the dilaton, volume complexity in dS₂ remains finite due to the lack of a local exponentially expanding $(d-2)$-sphere in two dimensions Chapman:2021eyy ; Jorstad:2022mls .

In this section, we discuss volume complexity and its subtleties. Before we proceed, we review the basic point.

The most general two-dimensional dilaton gravity bulk action up to two derivatives can be written in the form

$\displaystyle S=\frac{1}{8G}\int_{\mathcal{W}}d^{2}x\sqrt{-\tilde{g}}\left(U_{1}(\Phi)\tilde{R}+U_{2}(\Phi)\tilde{g}^{\mu\nu}\tilde{\nabla}_{\mu}\Phi\tilde{\nabla}_{\nu}\Phi+{{L^{-2}}}U_{3}(\Phi)\right)\,.$

(60)

We can perform a Weyl field redefinition

$$\tilde{g}_{\mu\nu}=e^{2\omega}\,g_{\mu\nu}\,,\qquad\nabla_{\mu}\omega=-\frac{U_{2}(\Phi)}{2U^{\prime}_{1}(\Phi)}\nabla_{\mu}\Phi\,,$$

(61)

to get rid of the kinetic term. Then, by doing a simple field redefinition $\phi=U_{1}(\Phi)$, we can remove $U_{1}$ as well. Note that this Weyl field redefinition does not change the coefficient of the $\sqrt{-\tilde{g}}\tilde{R}$ terms and we are left with just one function of $\phi$ which is related to $U_{3}$ as $U(\phi)=U_{3}(\Phi)$. Therefore, a general dilation gravity system can be brought to the form

$\displaystyle S=\frac{1}{8G}\int d^{2}x\sqrt{-g}\left(\phi R+L^{-2}U(\phi)\right)\,,$

(62)

which admits the two-dimensional dS solution with linear dilaton in eq. (12).

Note that even though the intermediate Weyl transformation done above is just a field redefinition, and hence should not affect physical quantities, it has significant influence in the context of the CV conjecture, since the volume changes by Weyl transformation. In this section, we study the effects of Weyl transformations on the CV conjecture. To evaluate volume complexity, it is convenient to use the metric ansatz

$$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}\,.$$

(63)

In fact, by starting with the dS₂ solution in eq. (12) and by applying the Weyl transformation-like field redefinition

$$g_{\mu\nu}\equiv\Omega(\phi)\hat{g}_{\mu\nu}\,,$$

(64)

one can always put the metric in the form of eq. (63) by a coordinate change, as we will show soon. Here we always assume $\Omega(\phi)$ is regular. In what follows, we set $L=1$ for convenience. The $L$ scale can always be recovered by dimensional analysis.

Since $\phi=r$, we have $\Omega(\phi)=\Omega(r)$. Starting from $f(r)=1-r^{2}$, the Weyl-transformed metric is given by

$\displaystyle ds^{2}=\Omega(r)\left(-f(r)dt^{2}+\frac{dr^{2}}{f(r)}\right)=-\Omega(r)f(r)dt^{2}+\frac{\Omega^{2}(r)dr^{2}}{\Omega(r)f(r)}\,.$

(65)

Defining the coordinate $\tilde{r}$ in terms of $r$ as

$\displaystyle\tilde{r}(r)=\int^{r}\Omega(\hat{r})d\hat{r}\,,$

(66)

the line element can be written as

$\displaystyle ds^{2}=-\tilde{f}(\tilde{r})dt^{2}+\frac{d\tilde{r}^{2}}{\tilde{f}(\tilde{r})},\qquad\tilde{f}(\tilde{r})=\Omega(r(\tilde{r}))f(r(\tilde{r}))\,.$

(67)

One can check that the temperature of the solution remains unchanged upon Weyl transformation-like field redefinition because $t$ is unmodified. In this way, we can always make the metric in the form eq. (63). However, note that the dilaton is no more linear in terms of the new radial coordinate $\tilde{r}$

$$\phi=r(\tilde{r})\,.$$

(68)

In the end, this is just a field-redefinition, so physics should not be modified. Since the vacuum of JT gravity is characterized by both the metric and the dilaton, reasonable physical quantities should be determined by taking into account both. While the on-shell action is invariant under such field-redefinitions, the volume is not, because it is determined by the metric only. In other words, action complexity is invariant, but the volume (precisely, geodesic length) changes, as we will see explicitly.