# An Information Paradox and Its Resolution in de Sitter Holography

Hao Geng  Department of Physics, University of Washington, Seattle, WA 98195,
USA  Yasunori Nomura  Berkeley Center for Theoretical Physics, Department of
Physics, University of California, Berkeley, CA 94720, USA;
Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, CA
94720, USA;
Kavli Institute for the Physics and Mathematics of the Universe (WPI), UTIAS,
The University of Tokyo, Kashiwa, Chiba 277-8583, Japan  Hao-Yu Sun  Theory
Group, Department of Physics, University of Texas, Austin, TX 78712, USA.

###### Abstract

We formulate a version of the information paradox in de Sitter spacetime and
show that it is solved by the emergence of entanglement islands in the context
of the DS/dS correspondence; in particular, the entanglement entropy of a
subregion obeys a time-dependent Page curve. Our construction works in general
spacetime dimensions and keeps the graviton massless. We interpret the
resulting behavior of the entanglement entropy using double holography. It
suggests that the spatial distribution of microscopic degrees of freedom
depends on descriptions, as in the case of a black hole. In the static
(distant) description of de Sitter (black hole) spacetime, these degrees of
freedom represent microstates associated with the Gibbons-Hawking (Bekenstein-
Hawking) entropy and are localized toward the horizon. On the other hand, in a
global (effective two-sided) description, which is obtained by the quantum
analog of analytic continuation and is intrinsically semiclassical, they are
distributed uniformly and in a unique semiclassical de Sitter (black hole)
vacuum state.

###### pacs:

04.20.Cv, 04.60.Bc, 98.80.Qc

##  I Introduction

Recent development in AdS/CFT holography provides important new elements to
our understanding of the geometrization of the entanglement structure of the
boundary conformal field theory, called entanglement islands [1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25,
26, 27, 28, 29, 30, 31]. They play an important role in proposed resolutions
of various information paradoxes, including Hawking’s original one for an
evaporating black hole [32]. (For recent reviews, see Refs. [33, 34, 35, 36].)
A general lesson is that we can compute the Page curve of black hole radiation
[37] by coupling the black hole to a boundary bath system which absorbs
radiation. The Page curves calculated in various contexts are all consistent
with unitarity, a fundamental postulate of quantum mechanics.

The central technique behind these calculations is the so-called quantum
extremal surface (QES) prescription [38, 39]. This prescription is fully
quantum and requires calculation of entanglement entropies in bulk curved
spacetime, which is a hard task in general. However, if the quantum field
theory in the bulk is holographic itself, then this quantum prescription can
be realized as a classical prescription in a higher dimensional spacetime [7],
in which we only need to calculate a classical extremal area surface [40, 41,
42] under specific boundary conditions near the boundary of the higher
dimensional spacetime [43]. Models having this property are dubbed as doubly
holographic, and they can be naturally constructed in Karch-Randall braneworld
[44, 45, 46, 47] which uses subcritical branes in the original Randall-Sundrum
framework [48].

While it is possible to construct models in more than $2+1$ spacetime
dimensions in which a time-dependent Page curve is obtained by coupling a bulk
black hole spacetime to a non-gravitational boundary bath [1], such models
necessarily lead to a massive graviton [10]. This is most easily seen in the
context of the Karch-Randall braneworld, in which a $d$-dimensional AdS
spacetime is modelled as a subcritical brane in a $(d+1)$-dimensional AdS
spacetime and the bath is the conformal boundary of this AdSd+1. From the
higher dimensional ($(d+1)$-dimensional) point of view, the subcritical brane
cuts off the portion of the spacetime behind it, but the remaining part is
still non-compact. This makes the would-be massless graviton modes localized
on the subcritical brane non-normalizable, leaving only massive graviton modes
to be physical.

In an attempt to rectify this issue, Ref. [43] considered a setup in which
another subcritical brane cuts off the leftover conformal boundary. This makes
the extra dimension compact and massless graviton modes survive. In this case,
however, the bath is a portion of the second subcritical brane, which is
gravitating by itself. This motivated Ref. [43] to apply the same prescription
to both the first and second branes, resulting in a time-independent “Page
curve”.

In this paper, we take the view that to obtain the standard Page curve, we
must focus on fine-grained entropy of semiclassical Hawking radiation in a
weakly gravitating regime, and not the fine-grained entropy of the full
microscopic degrees of freedom. We postulate that this can be done by fixing
the boundary of the region on the second brane for which the entanglement
entropy is calculated, rather than determining it by extremization.111This has
also been suggested by Pratik Rath. For an earlier discussion on this issue
and the graviton mass, see Refs. [15, 17]. For a black hole on an AdS brane,
this indeed reproduces features of the time-dependent Page curve. We suspect
that with this prescription, the non-factorizable nature of the quantum
gravitational Hilbert space identified in Ref. [36] does not play a major
role, although more detailed studies are needed to obtain a definite
conclusion.

To further test the feasibility of the picture described above, in this paper
we perform an analogous calculation in de Sitter (dS) spacetime. We first
formulate a version of the information paradox about entanglement entropy in
dS spacetime, in analogy with that proposed for AdS black holes in Ref. [49].
The entanglement entropy and fast scrambling dynamics of dS quantum gravity
have been studied in various contexts [50, 51, 52, 53, 54, 55]. In this paper,
we work in the context of the DS/dS correspondence [56, 57], which is a
mathematically well-defined holographic framework for dS quantum gravity.

We embed the paradox into the DS/dS context and see if it is resolved by
entanglement islands. There are works on entanglement islands in dS spacetime
using lower dimensional models of quantum gravity [18, 19, 21, 22]. In this
paper, we analyze the problem in general dimensions. Using double holography
in the DS/dS framework, we find that the standard, time-dependent Page curve
is obtained while preserving the masslessness of the graviton. In addition to
increasing the feasibility of the postulated island prescription in a weakly
gravitating regime, we find it interesting itself that a time-dependent Page
curve is obtained in dS spacetime. We will also find that the resulting
picture is consonant with the interpretation of maximally extended spacetime
in quantum gravity developed in Refs. [58, 59, 60, 61].

This paper is organized as follows. In Section II, we briefly review the DS/dS
correspondence. We then formulate a version of the information paradox in dS
spacetime in Section III, which we embed into the context of the DS/dS
correspondence in Section IV. In Section V, we will see how the paradox is
resolved because of the emergence of entanglement islands. In Section VI, we
discuss deeper implications of our result for the structure of spacetime in
quantum gravity. Finally, conclusions are given in Section VII.

##  II The DS/dS Correspondence

The DS/dS correspondence is most easily understood as a deformation of the
AdS/CFT correspondence [62, 63, 64] as follows. Both a $(d+1)$-dimensional de
Sitter space (dSd+1) and a $(d+1)$-dimensional anti-de Sitter space (AdSd+1)
can be radially foliated by $d$-dimensional dS slices, but with slightly
different warp factor:

$ds^{2}_{\text{(A)dS}_{d+1}}=dr^{2}+\sin(\text{h})^{2}\\!\left(\frac{r}{\ell}\right)ds^{2}_{\text{dS}_{d}},$
(1)

where $\ell$ is the curvature length and we call $r$ the radial coordinate
(for AdSd+1, $r\in(-\infty,\infty)$, and for dSd+1, $r\in[0,\pi\ell]$). For dS
spacetime, the spacetime region covered by these coordinates is called a DS/dS
patch.

The ultraviolet (UV) conformal boundary of AdSd+1 is living at $r=\pm\infty$
and the infrared (IR) bulk point is near $r=0$. We can see that when $r\to
0,\pi\ell$, the metric of dSd+1 behaves in the same way as that of AdSd+1 at
$r\to 0$. The standard lessons of holography then tell us that the holographic
dual field theories of quantum gravity in dSd+1 and AdSd+1 have the same IR
fixed point. However, their UV behaviors are very different. For AdSd+1, it is
UV completed to a conformal field theory (CFT) living on its conformal
boundary. But for dSd+1, it is not UV-complete and its field theory dual
consists of two UV-cutoff CFTs, coupled to each other via dynamical gravity
living on the central slice $r=\frac{\pi}{2}\ell$ [56]. This central slice can
be thought of as a built-in Randall-Sundrum brane and the graviton localized
on it is massless [56].

Summarizing, quantum gravity in dSd+1 is dual to two cutoff CFTs living on dSd
coupled to each other via dynamical gravity [57]. In this context, questions
regarding entanglement entropy, how the duality works, and its implications
for the underlying fast scrambling dynamics have been studied in recent years,
where several important progresses have been made [65, 66, 67, 68, 69, 70,
54].

##  III The Paradox

We now present an apparent paradox involving dS spacetime, which we will
resolve later using DS/dS double holography.

Consider two dSd’s. Each of these is dSd in its extended static patch, which
is the union of two standard static patches associated with the north and
south poles (Fig. 1). We adopt the metric [54]

$ds^{2}=-\cos^{2}\beta\text{ }dt^{2}+d\beta^{2}+\sin^{2}\beta\text{
}d\Omega_{d-2}^{2}.$ (2)

Here, we have set the curvature length to be one, and $\beta\in[0,\pi]$ for
$d\geq 3$ and $\beta\in[0,2\pi)$ for $d=2$. (The difference of the range of
$\beta$ between $d=2$ and $d\geq 3$ comes from the fact that we have taken the
angular direction consisting of a point, not two points, in $d=2$.)

Figure 1: The Penrose diagram of dS spacetime in the extended static patch
(left) and its spatial section at $t=0$ (right). The blue and red dots
represent the north and south poles at $t=0$. The green arrows represent the
direction of time evolution.

Let us call the two dS spacetimes dS${}_{d}^{1}$ and dS${}_{d}^{2}$. We let
them interact through some intersecting defects. We impose transparent
boundary conditions there, so that radiation can flow from one dSd to the
other. We assume that dS${}_{d}^{1}$ is gravitating and potentially have some
matter fields, and that dS${}_{d}^{2}$ supports a fast scrambling field theory
system.222For the purpose of formulating the paradox, we need not consider
that dS${}_{d}^{2}$ is gravitating. Embedding it into DS/dS, however, requires
dS${}_{d}^{2}$ to be gravitating as dS${}_{d}^{1}$. We take the state of the
system in dS${}_{d}^{2}$ at $t=0$ to be in the thermofield double state
between the two hemispheres denoted by $L$ and $R$ in Fig. 1.

On the Penrose diagram of a dSd in Fig. 1, the parts $L$ and $R$ live on
different sides of the horizon, so that the timelike Killing vector
$\frac{\partial}{\partial t}$ in Eq. (2) goes opposite on the two sides. We
break this time translational symmetry, for both dS${}_{d}^{1}$ and
dS${}_{d}^{2}$, by letting the time to go up for both sides as indicated by
the green arrows in Fig. 1. With this, the dynamics of the fast scrambling
system on dS${}_{d}^{2}$ is nontrivial.

For the purpose of embedding the setup into DS/dS later, we take a
$\mathbb{Z}_{2}$ quotient of dS${}_{d}^{2}$ with respect to a surface that
goes through both the north and south poles (and thus is orthogonal to the
horizon separating $L$ and $R$ regions); see Fig. 2. The region of which we
calculate the entanglement (von Neumann) entropy, $A$, is an interior region
of this $\mathbb{Z}_{2}$ quotient, which we take to be symmetric with respect
to the horizon, as depicted by the green shaded region in Fig. 2. The
$\mathbb{Z}_{2}$ quotient dS${}_{d}^{2}$ space described here plays the role
of the bath in Refs. [1, 2, 3, 4, 5, 6, 7, 8, 9].

Figure 2: The $\mathbb{Z}_{2}$ orbifolding removes a half of dS${}_{d}^{2}$.
The region $A$, of which we calculate the entanglement entropy, at $t=0$ is
depicted as the green shaded region on the right panel.

We assume that the two spaces dS${}_{d}^{1}$ and dS${}_{d}^{2}/\mathbb{Z}_{2}$
have the same temperature, but they are not significantly entangled with each
other at $t=0$. We then time evolve both dS${}_{d}^{1}$ and
dS${}_{d}^{2}/\mathbb{Z}_{2}$ as described above; see the green arrows in Fig.
1. This makes the volume of the region $A$, specified by the coordinate
$\beta$, increase in time; it also increases entanglement between
dS${}_{d}^{1}$ and dS${}_{d}^{2}/\mathbb{Z}_{2}$ as well as that between $A$
and its complement $\bar{A}$ on dS${}_{d}^{2}/\mathbb{Z}_{2}$. Given that the
dynamics of the system on dS${}_{d}^{2}/\mathbb{Z}_{2}$ is fast scrambling,
the semiclassical expectation is that the entanglement entropy of $A$, which
we denote by $S_{A}$, grows indefinitely in time.

In quantum gravity, however, the Hilbert space dimension of dS spacetime is
expected to be finite [71, 72], so such an indefinite increase of entropy
would lead to an information paradox. The correct time dependence of $S_{A}$
must be the Page curve, i.e., its value must be saturated at late times by the
finite Hilbert space dimension of the union of dS${}_{d}^{1}$ and $\bar{A}$
(or of $A$).

We note that it is important for dS${}_{d}^{2}$ to be in the extended static
patch of Eq. (2), since it is what leads to the increase of the volume of $A$.
The same, however, is not true for dS${}^{1}_{d}$; here we put it in this
patch simply for convenience of demonstrating the paradox. Indeed, as we will
see in the next section, utilizing double holography requires us to put
dS${}^{1}_{d}$ in the DS/dS extended static patch, rather than the extended
static patch. The time evolution of dS${}^{1}_{d}$ will then be that of the
DS/dS patch.

##  IV Embedding the Paradox into DS/dS

The first step to embed the setup in the previous section into the DS/dS
context is to identify the fast scrambling system on dS${}_{d}^{2}$ as the
dual of quantum gravity in dSd+1. This implies that dS${}_{d}^{2}$ is on the
central slice of this dSd+1, i.e., $r=\frac{\pi}{2}\ell$ in Eq. (1), where
$ds^{2}_{\text{dS}_{d}}$ should now be described by the extended static patch
metric in Eq. (2). The full metric is now in the DS/dS${}_{\text{extended
static}}$ patch considered in Ref. [54], given by

$ds^{2}_{\text{dS}_{d+1}}=dr^{2}+\sin^{2}\\!r\left(-\cos^{2}\\!\beta\,dt^{2}+d\beta^{2}+\sin^{2}\\!\beta\,d\Omega_{d-2}^{2}\right).$
(3)

Here and below, we set $\ell=1$ for convenience.

We model dS${}_{d}^{1}$ as a Randall-Sundrum brane living on a slice
“orthogonal” to that on which dS${}_{d}^{2}$ lives. We call this slice $Q$.
For $d\geq 4$,

$d\Omega_{d-2}^{2}=d\chi^{2}+\sin^{2}\\!\chi\,d\Omega_{d-3}^{2}\quad(0\leq\chi\leq\pi),$
(4)

and $Q$ is the $\chi=\frac{\pi}{2}$ slice. For $d=3$,

$d\Omega_{1}^{2}=d\chi^{2}\quad(0\leq\chi<2\pi),$ (5)

and $Q$ is at $\chi=0$ and $\pi$; note that $Q$ as well as its intersection
with dS${}_{d}^{2}$ are still single connected surfaces—the $\chi=0$ and $\pi$
components touch at $\beta=0$ and $\pi$. Finally, for $d=2$, $Q$ lives at
$\beta=0$ and $\pi$; in this case, $Q$ is a single connected slice with the
$\beta=0$ and $\pi$ components touching at $r=0$ and $\pi$, but the
intersection of $Q$ and dS${}_{d}^{2}$ consists of two disconnected points.

It is easy to see that the geometry of $Q$ is dSd in the
DS/dS${}_{\text{extended static}}$ patch. This tells us that the bulk
gravitational action is given by the Einstein-Hilbert action and boundary
terms associated with $Q$:

$\begin{split}S=\frac{1}{16\pi
G_{d+1}}&\int\\!d^{d+1}x\,\sqrt{-g}(R-2\Lambda)\\\ &-\frac{1}{8\pi
G_{d+1}}\int_{Q}\\!d^{d}x\,\sqrt{-h}(K-T),\end{split}$ (6)

where $\Lambda$ is the cosmological constant, $G_{d+1}$ is Newton’s constant;
$h_{ab}$ is the induced metric on the $Q$ brane, $K$ is the trace of its
extrinsic curvature, and $T$ is the brane tension. The boundary condition for
the metric fluctuation near $Q$ is of Neumann type, and hence the equation of
motion splits into bulk and boundary parts. The bulk part is the vacuum
Einstein equation with a cosmological constant, and it is satisfied by the
dSd+1 in Eq. (3) for $\Lambda=\frac{d(d-1)}{2}$. The boundary equation of
motion determines the brane tension by [73, 74]

$K_{ab}=(K-T)h_{ab}.$ (7)

Since it is easily computed that $K_{ab}=0$ for $Q$, the brane $Q$ is
tensionless and not fluctuating [75, 76].

The fact that the slice $Q$ is a tensionless brane and non-fluctuating implies
that the bulk dSd+1 is actually a $\mathbb{Z}_{2}$ quotient with $Q$ as its
boundary, where we impose Neumann boundary conditions for bulk fields. This
$\mathbb{Z}_{2}$ orbifolding is the $\mathbb{Z}_{2}$ identification of
dS${}_{d}^{2}$ discussed in Section III, i.e.,
$\mbox{dS}_{d}^{2}\rightarrow\mbox{dS}_{d}^{2}/\mathbb{Z}_{2}$ depicted in
Fig. 2. The two dS systems dS${}_{d}^{1}$ and dS${}_{d}^{2}$ intersect at the
fixed point of this $\mathbb{Z}_{2}$ orbifolding of dS${}_{d}^{2}$.

We see that we have successfully embedded the paradox formulated in the
previous section into DS/dS. The entanglement entropy of a region $A$
($\subset\mbox{dS}_{d}^{2}/\mathbb{Z}_{2}$) is expected to increase in time
(beyond the Page transition point), which can now be seen as the Hartman-
Maldacena surface with increasing area [49] in the dSd+1 bulk. In order to
terminate this increase, some other element must come in. We stress that the
graviton in the current setup is massless, since the holographically generated
space is compact.

We finally note that this setup can be viewed as realizing double holography,
since the dS${}_{d}^{1}$ system on $Q$ itself can be dualized to a dSd-1 using
the DS/dS correspondence. This dual system then lives on the the boundary of
the $\mbox{dS}_{d}^{2}/\mathbb{Z}_{2}$ system (the horizontal great circle on
the right panel of Fig. 2). From this viewpoint, the picture of dSd+1 is
obtained by double holography, analogous to the setup for AdS black holes in
Ref. [10].

##  V Resolution of the Paradox and a Page Curve

To see the picture described above more explicitly, here we focus on the $d=2$
case. We will see how the paradox manifests itself as a growing bulk
entangling surface as well as how it is resolved. The same conclusions apply
to the higher dimensional case as well.

For $d=2$, the coordinate $\beta$ goes from 0 to $\pi$ (after the
$\mathbb{Z}_{2}$ quotient), and the horizon is located at
$\beta=\frac{\pi}{2}$. The field theory system of
$\mbox{dS}_{2}^{2}/\mathbb{Z}_{2}$ lives on the $r=\frac{\pi}{2}$ slice, with
$\beta=\frac{\pi}{2}$ being the bifurcation surface. The spatial section of
the geometry at $t=0$ is depicted in Fig. 3. The two dS systems dS${}_{2}^{1}$
and $\mbox{dS}_{2}^{2}/\mathbb{Z}_{2}$ are a circle and a semicircle,
respectively, and the bulk dS3 is a hemisphere bounded by dS${}_{2}^{1}$.

Figure 3: The spatial section at $t=0$ for $d=2$. The two intersecting dS
systems are depicted as the blue circle (dS${}_{2}^{1}$) and black semicircle
($\mbox{dS}_{2}^{2}/\mathbb{Z}_{2}$). The green segment represents region $A$,
of which we calculate the entanglement entropy, and the cross represents the
horizon at $\beta=\frac{\pi}{2}$. The holographic bulk space
($\mbox{dS}_{3}/\mathbb{Z}_{2}$) is the hemisphere bounded by dS${}_{2}^{1}$.

What about time evolution? As discussed in Section III, we evolve the left
($\beta\in[0,\frac{\pi}{2}]$) and right ($\beta\in[\frac{\pi}{2},\pi]$)
portions of $\mbox{dS}_{2}^{2}/\mathbb{Z}_{2}$ in the same and opposite
directions of the timelike Killing vector, respectively (or vice versa). We
consider the time evolution of the entanglement entropy $S_{A}$ of region $A$,
which we take to be $\beta\in[\beta_{*},\pi-\beta_{*}]$
($0<\beta_{*}<\frac{\pi}{2}$) on the $r=\frac{\pi}{2}$ slice. The region $A$
is depicted by the green line segment in Fig. 3.

For this purpose, let us consider the geodesic in dS3 between the two end
points of $A$ as a function of time. In terms of the embedding space
coordinate for the DS/dS${}_{\text{extended static}}$ patch, these two points
are located at

$\begin{split}X_{0}&=\cos\beta_{*}\sinh t\\\ X_{1}&=0\\\
X_{2}&=\cos\beta_{*}\cosh t\\\ X_{3}&=\sin\beta_{*}\end{split}$ (8)

and

$\begin{split}X_{0}^{\prime}&=\cos(\pi-\beta_{*})\sinh(-t)=\cos\beta_{*}\sinh
t\\\ X_{1}^{\prime}&=0\\\
X_{2}^{\prime}&=\cos(\pi-\beta_{*})\cosh(-t)=-\cos\beta_{*}\cosh t\\\
X_{3}^{\prime}&=\sin(\pi-\beta_{*})=\sin\beta_{*},\end{split}$ (9)

respectively (see Ref. [54]). Then the geodesic distance connecting these
points, which would give the entanglement entropy of $A$ through the
holographic formula, is given by

$\begin{split}D&=\arccos(-X_{0}X_{0}^{\prime}+X_{1}X_{1}^{\prime}+X_{2}X_{2}^{\prime}+X_{3}X_{3}^{\prime})\\\
&=\arccos(1-2\cos^{2}\\!\beta_{*}\cosh^{2}\\!t).\end{split}$ (10)

This monotonic growth in time is consistent with the fact that dS spacetime is
a fast scrambling system.

However, the entanglement entropy of $A$ given by Eq. (10) through the
holographic formula, $S_{A}=\frac{D}{4G_{3}}$, is problematic, since it
becomes imaginary at $t=t_{c}\equiv\big{|}{\rm
arccosh}\frac{1}{\cos\beta_{*}}\big{|}$. In fact, for $t>t_{c}$ the expression
in Eq. (10) is inapplicable, and the geodesic between the two end points of
$A$ stops existing.

The existence of another QES, however, solves this issue. Since the
dS${}_{2}^{1}$ brane (on the $\beta=\\{0,\pi\\}$ circle) can be viewed as
arising holographically from two dS1’s on $(r,\beta)=(\frac{\pi}{2},0)$ and
$(\frac{\pi}{2},\pi)$, a QES for calculating $S_{A}$ can end on it [3, 43].
After the extremization, we find that the relevant QES is the complement of
$A$ on $r=\frac{\pi}{2}$, i.e., $\bar{A}$ on the equal-time hypersurface of
$\mbox{dS}_{2}^{2}/\mathbb{Z}_{2}$, whose area (length) is simply
$2\beta_{*}$. We thus obtain

$S_{A}=\frac{1}{4G_{3}}\min(D,2\beta_{*}).$ (11)

This is plotted in Fig. 4 for a specific value of $\beta_{*}=\frac{\pi}{3}$.
An important point is that the value of $S_{A}$ given by $\frac{D}{4G_{3}}$ at
$t=t_{c}$ is $\frac{\pi}{4G_{3}}$, so that the second QES starts dominating
before $t=t_{c}$ for any choice of region $A$. We also note that if
$\beta_{*}\leq\frac{\pi}{4}$, $S_{A}$ stays to be a constant value
$\frac{\beta_{*}}{2G_{3}}$ ever since $t=0$, so to see the transition we need
to take region $A$ sufficiently small.

Figure 4: We plot $4G_{3}S_{A}$ for $\beta_{*}=\frac{\pi}{3}$ as an example.
The blue and red curves represent $D$ and $2\beta_{*}$, respectively; see Eq.
(11). The actual $4G_{3}S_{A}$ is given by the solid part of the curves.

A similar behavior of $S_{A}$ is obtained in higher dimensions as well. In
this case, the saturation value corresponding to $\frac{\beta_{*}}{2G_{3}}$
for $d=2$ is given by

$S_{A,\infty}=\frac{V_{\bar{A}}}{4G_{d+1}}=\begin{cases}\frac{\pi(1-\cos\beta_{*})}{2G_{4}}&\mbox{for
}d=3\\\
\frac{\bigl{(}\frac{\pi}{2(d-2)}-\cos{\beta_{*}}f(d,\beta_{*})\bigr{)}\Sigma_{d-2}}{2G_{d+1}}&\mbox{for
}d\geq 4\end{cases},$ (12)

where $V_{\bar{A}}$ is the ($d-1$)-dimensional volume of $\bar{A}$ on
$\mbox{dS}_{d}^{2}/\mathbb{Z}_{2}$ at $t=0$, $\Sigma_{d-2}$ is the volume of
the ($d-2$)-dimensional unit sphare, and

$f(d,\beta_{*})=\frac{\sqrt{\pi}\Gamma(\frac{d}{2}-1)}{2\Gamma(\frac{d-1}{2})}\,{}_{2}F_{1}\biggl{[}\frac{1}{2},\frac{3-d}{2},\frac{3}{2},\cos^{2}\\!{\beta_{*}}\biggr{]}.$
(13)

The initial value at $t=0$ is given by $S_{A}=\frac{V_{A}}{4G_{d+1}}$, where
$V_{A}$ is the volume of $A$ on $\mbox{dS}_{d}^{2}/\mathbb{Z}_{2}$.

In the next section, we see that the time dependence of $S_{A}$ obtained here
can indeed be interpreted as a Page curve, reflecting the finite Hilbert space
of dS spacetime in quantum gravity.

##  VI Interpretation

Let us consider what the result obtained in the previous section implies for
the fundamental structure of spacetime. We discuss only the nontrivial case of
$\beta_{*}>\frac{\pi}{4}$. We first observe that the initial and final values
of $S_{A}$ can be written as

$\displaystyle S_{A}(t=0)$ $\displaystyle=\frac{V_{A}}{4G_{d+1}}$
$\displaystyle=\frac{\mathcal{A}}{4G_{d}}\biggl{(}\frac{V_{A}}{V}\biggr{)},$
(14) $\displaystyle S_{A}(t\rightarrow\infty)$
$\displaystyle=\frac{V_{\bar{A}}}{4G_{d+1}}$
$\displaystyle=\frac{\mathcal{A}}{4G_{d}}\biggl{(}\frac{V_{\bar{A}}}{V}\biggr{)},$
(15)

where

$G_{d}\equiv\frac{\cal A}{V}G_{d+1},$ (16)

and $V$ and ${\cal A}$ are the volume and horizon area of
$\mbox{dS}_{d}/\mathbb{Z}_{2}$ at $t=0$ (i.e., the area of the hemisphere and
the length of the half equator in the right panel of Fig. 2), respectively. It
is easy to see that $G_{d}$ is, indeed, Newton’s constant in the
$d$-dimensional theory on $\mbox{dS}_{d}^{2}/\mathbb{Z}_{2}$.

To decipher the meaning of Eqs. (14, 15), we take the “original” boundary
description of double holography discussed at the end of Section IV. In this
description, the total system is a gravitational system on ${\cal
M}\equiv\mbox{dS}_{d}^{2}/\mathbb{Z}_{2}$ with boundary $\partial{\cal M}$. In
the ($d+1$)-dimensional bulk picture, $\partial{\cal M}$ is the intersection
of ${\cal M}$ and $Q$, and the number of degrees of freedom associated with it
is related to the tension of the $Q$ brane [46, 74]. In our case, the $Q$
brane is tensionless, implying that $\partial{\cal M}$ does not carry its own
degrees of freedom at the leading order.

This allows us to interpret Eqs. (14, 15) that the initial and final values of
$S_{A}$ are given, respectively, by the fractions of the spatial volume which
$A$ and $\bar{A}$ occupy at $t=0$, multiplied by the quantity $\frac{\cal
A}{4G_{d}}$, the Gibbons-Hawking entropy of the spacetime
$\mbox{dS}_{d}/\mathbb{Z}_{2}$. Is this interpretation reasonable?

The interpretation just described, in fact, is consonant with the picture of
spacetime developed in Refs. [58, 59, 60, 61, 35]. Let us consider a spacetime
with a black hole. A “fundamental,” unitary description of this system is
obtained by viewing the black hole from a distance [77, 78], in which the
Bekenstein-Hawking entropy is interpreted as the logarithm of the number of
independent microstates associated with the black hole spacetime. The relevant
degrees of freedom, called soft modes [58, 59], are localized mostly on the
stretched horizon [78]. The picture of the black hole interior (or a two-sided
black hole, corresponding to the maximally extended spacetime in general
relativity) arises as a collective phenomenon, with the resulting vacuum state
being the unique semiclassical vacuum state because of the coarse graining
employed in obtaining the picture [60, 61, 35].

A similar construction also applies to dS spacetime [59], though it is more
speculative. In this case, the description based on the static coordinates
corresponds to the distant picture of a black hole. The Gibbons-Hawking
entropy represents the logarithm of the number of microstates of the dS
spacetime, and the degrees of freedom representing them, i.e., soft modes, are
located mostly on the (stretched) horizon. This localization of soft modes
occurs because of a large gravitational blueshift associated with the
existence of the horizon.

The extension of the dS spacetime covering the full spatial section at $t=0$
can be performed through coarse graining [59], analogous to the black hole
case. In the extended space, the vacuum state is the unique semiclassical dS
vacuum state, regardless of the microstate we started from. (In fact, the
state has a very special, antipodal entanglement structure at $t=0$ [79, 70,
54].) The concept of vacuum microstates does not exist in the (maximally)
extended theory because of the coarse graining involved in erecting such a
theory [35, 58, 59].

Nevertheless, there is a concept of microscopic vacuum degrees of freedom,
even though they are in the unique vacuum state. How are these degrees of
freedom distributed in space? Since the extended geometry at $t=0$ is a
homogeneous space with no special point selected, it is natural to expect that
the number of degrees of freedom indicated by the Gibbons-Hawking entropy is
distributed uniformly throughout the space. In this picture, the Gibbons-
Hawking entropy represents the entanglement entropy between two subregions
when the dS space at $t=0$ is divided into two hemispheres. The whole
discussion given here also goes through in our setup with the $\mathbb{Z}_{2}$
identification if the appropriate adjustments are made; for example, the
entanglement surface in the last sentence must be taken to be orthogonal to
the fixed surface of $\mathbb{Z}_{2}$ (as depicted in Fig. 2), and the value
of the entropy must be divided by a factor of $2$.

The picture described above is precisely what is suggested by our result.
Given that dS spacetime at $t=0$ is a homogeneous, scrambled system, the
entropy of subregion $A$ at $t=0$ is given by the number of degrees of freedom
there:

$n_{A,0}=\frac{S_{\rm GH}}{2}\biggl{(}\frac{V_{A}}{V}\biggr{)}=\frac{\cal
A}{4G_{d}}\biggl{(}\frac{V_{A}}{V}\biggr{)},$ (17)

where $S_{\rm GH}$ is the Gibbons-Hawking entropy, and the factor $2$ in the
denominator of the middle expression comes from the $\mathbb{Z}_{2}$
identification. (Note that we are taking subregion $A$ smaller than $\bar{A}$
at $t=0$.) We emphasize that the spatial distribution of the microscopic
degrees of freedom appears very different between the static and global
pictures (between the boundary and two-sided pictures in the case of a
collapse-formed black hole).

The initial increase of $S_{A}$ implies that the number of degrees of freedom
in $A$ increases in time. This is possible because the number of degrees of
freedom captured by the holographic entropy is, in fact, that of effective
degrees of freedom necessary to describe the emergent bulk spacetime, which is
determined by the entanglement structure of (presumably an infinite number of)
more fundamental degrees of freedom and can thus vary in time [79, 80, 81]. In
our context, the increase of $S_{A}$ arises from the emergence of the
“interior” region, i.e., the upper quadrant of the Penrose diagram in Fig. 1,
analogous to the Hartman-Maldacena setup [49].

This increase of $S_{A}$, however, does not last forever, since subregion
$\bar{A}$ does not “sense” the emergence of the interior, and hence the number
of degrees of freedom there,

$n_{\bar{A}}=\frac{S_{\rm
GH}}{2}\biggl{(}\frac{V_{\bar{A}}}{V}\biggr{)}=\frac{\cal
A}{4G_{d}}\biggl{(}\frac{V_{\bar{A}}}{V}\biggr{)},$ (18)

stays constant. If the state of the total system is pure, as we assume here,
then the entanglement entropy of $A$ cannot become larger than this number.
This is the origin of the saturation of $S_{A}$ we have seen, which is nothing
but the Page transition phenomenon representing unitarity of the underlying
dynamics.

##  VII Conclusions

In this paper, we have formulated a version of the information paradox in dS
spacetime and shown that it is solved by the emergence of entanglement islands
in the context of double DS/dS holography. In particular, we have obtained a
time-dependent Page curve in a setup in which the graviton stays massless.

Beyond the specific context of dS spacetime, our result suggests the
following:

  * •

The entanglement entropy of Hawking radiation appearing in the calculation of
the Page curve can be computed by the QES prescription by fixing a region in
which the radiation resides, even if it is in a (weakly) gravitating region.

  * •

While the Gibbons-Hawking/Bekenstein-Hawking entropy represents the logarithm
of the number of independent dS/black-hole microstates in a static/distant
description (i.e., in unitary gauge construction [35, 82]), the vacuum state
in an analytically extended spacetime—a global/two-sided description—is a
unique, semiclassical state.

  * •

In the static/distant description of a dS/black-hole spacetime, the degrees of
freedom associated with the microstates are mostly localized near the horizon,
due to a strong gravitational blueshift. On the other hand, in the
analytically extended spacetime, they are distributed uniformly in space (at
the time the extension is made).

  * •

The increase of volume in the “interior” region of dS spacetime (the top
quadrant of the Penrose diagram) has a physical meaning in the sense that the
entanglement entropy of a region surrounding it increases in time (until it is
saturated by the Page transition).

We hope that these findings shed further light on how spacetime and gravity
work at the fundamental level.

###### Acknowledgements.

We thank Andreas Karch, Suvrat Raju, Lisa Randall, and Pratik Rath for useful
discussions. The work of Y.N. was supported in part by the Department of
Energy, Office of Science, Office of High Energy Physics under contract DE-
AC02-05CH11231 and award DE-SC0019380 and in part by MEXT KAKENHI grant number
JP20H05850, JP20H05860. The work of H.-Y.S. was supported by the Simons
Collaborations on Ultra-Quantum Matter, grant 651440 (AK) from the Simons
Foundation. H.G. is very grateful to his parents and recommenders.

## References

  * Penington [2020] G. Penington, JHEP 09, 002 (2020), arXiv:1905.08255 [hep-th] .
  * Almheiri  _et al._ [2019a] A. Almheiri, N. Engelhardt, D. Marolf, and H. Maxfield, JHEP 12, 063 (2019a), arXiv:1905.08762 [hep-th] .
  * Almheiri  _et al._ [2020a] A. Almheiri, R. Mahajan, J. Maldacena, and Y. Zhao, JHEP 03, 149 (2020a), arXiv:1908.10996 [hep-th] .
  * Almheiri  _et al._ [2019b] A. Almheiri, R. Mahajan, and J. Maldacena, (2019b), arXiv:1910.11077 [hep-th] .
  * Rozali  _et al._ [2020] M. Rozali, J. Sully, M. Van Raamsdonk, C. Waddell, and D. Wakeham, JHEP 05, 004 (2020), arXiv:1910.12836 [hep-th] .
  * Chen  _et al._ [2020a] H. Z. Chen, Z. Fisher, J. Hernandez, R. C. Myers, and S.-M. Ruan, JHEP 03, 152 (2020a), arXiv:1911.03402 [hep-th] .
  * Almheiri  _et al._ [2020b] A. Almheiri, R. Mahajan, and J. E. Santos, SciPost Phys. 9, 001 (2020b), arXiv:1911.09666 [hep-th] .
  * Penington  _et al._ [2019] G. Penington, S. H. Shenker, D. Stanford, and Z. Yang, (2019), arXiv:1911.11977 [hep-th] .
  * Almheiri  _et al._ [2020c] A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, and A. Tajdini, JHEP 05, 013 (2020c), arXiv:1911.12333 [hep-th] .
  * Geng and Karch [2020] H. Geng and A. Karch, JHEP 09, 121 (2020), arXiv:2006.02438 [hep-th] .
  * Gautason  _et al._ [2020] F. F. Gautason, L. Schneiderbauer, W. Sybesma, and L. Thorlacius, JHEP 05, 091 (2020), arXiv:2004.00598 [hep-th] .
  * Hashimoto  _et al._ [2020] K. Hashimoto, N. Iizuka, and Y. Matsuo, JHEP 06, 085 (2020), arXiv:2004.05863 [hep-th] .
  * Hartman  _et al._ [2020a] T. Hartman, E. Shaghoulian, and A. Strominger, JHEP 07, 022 (2020a), arXiv:2004.13857 [hep-th] .
  * Hollowood and Kumar [2020] T. J. Hollowood and S. P. Kumar, JHEP 08, 094 (2020), arXiv:2004.14944 [hep-th] .
  * Krishnan  _et al._ [2020] C. Krishnan, V. Patil, and J. Pereira, (2020), arXiv:2005.02993 [hep-th] .
  * Chen  _et al._ [2020b] H. Z. Chen, R. C. Myers, D. Neuenfeld, I. A. Reyes, and J. Sandor, JHEP 10, 166 (2020b), arXiv:2006.04851 [hep-th] .
  * Krishnan [2020] C. Krishnan, (2020), arXiv:2007.06551 [hep-th] .
  * Chen  _et al._ [2020c] Y. Chen, V. Gorbenko, and J. Maldacena, (2020c), arXiv:2007.16091 [hep-th] .
  * Hartman  _et al._ [2020b] T. Hartman, Y. Jiang, and E. Shaghoulian, JHEP 11, 111 (2020b), arXiv:2008.01022 [hep-th] .
  * Balasubramanian  _et al._ [2020a] V. Balasubramanian, A. Kar, and T. Ugajin, (2020a), arXiv:2008.05274 [hep-th] .
  * Balasubramanian  _et al._ [2020b] V. Balasubramanian, A. Kar, and T. Ugajin, (2020b), arXiv:2008.05275 [hep-th] .
  * Sybesma [2020] W. Sybesma, (2020), arXiv:2008.07994 [hep-th] .
  * Ling  _et al._ [2020] Y. Ling, Y. Liu, and Z.-Y. Xian, (2020), arXiv:2010.00037 [hep-th] .
  * Matsuo [2020] Y. Matsuo, (2020), arXiv:2011.08814 [hep-th] .
  * Goto  _et al._ [2020] K. Goto, T. Hartman, and A. Tajdini, (2020), arXiv:2011.09043 [hep-th] .
  * Caceres  _et al._ [2020] E. Caceres, A. Kundu, A. K. Patra, and S. Shashi, (2020), arXiv:2012.05425 [hep-th] .
  * Deng  _et al._ [2020] F. Deng, J. Chu, and Y. Zhou, (2020), arXiv:2012.07612 [hep-th] .
  * Karananas  _et al._ [2020] G. K. Karananas, A. Kehagias, and J. Taskas, (2020), arXiv:2101.00024 [hep-th] .
  * Wang  _et al._ [2021] X. Wang, R. Li, and J. Wang, (2021), arXiv:2101.06867 [hep-th] .
  * Verheijden and Verlinde [2021] E. Verheijden and E. Verlinde, (2021), arXiv:2102.00922 [hep-th] .
  * Kawabata  _et al._ [2021] K. Kawabata, T. Nishioka, Y. Okuyama, and K. Watanabe, (2021), arXiv:2102.02425 [hep-th] .
  * Hawking [1976] S. W. Hawking, Phys. Rev. D14, 2460 (1976).
  * Almheiri  _et al._ [2020d] A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, and A. Tajdini, (2020d), arXiv:2006.06872 [hep-th] .
  * Liu and Sonner [2020] H. Liu and J. Sonner, Nature Rev. Phys. 2, 615 (2020), arXiv:2004.06159 [hep-th] .
  * Nomura [2020a] Y. Nomura, (2020a), arXiv:2011.08707 [hep-th] .
  * Raju [2020] S. Raju, (2020), arXiv:2012.05770 [hep-th] .
  * Page [1993] D. N. Page, Phys. Rev. Lett. 71, 3743 (1993), arXiv:hep-th/9306083 .
  * Engelhardt and Wall [2015] N. Engelhardt and A. C. Wall, JHEP 01, 073 (2015), arXiv:1408.3203 [hep-th] .
  * Faulkner  _et al._ [2013] T. Faulkner, A. Lewkowycz, and J. Maldacena, JHEP 11, 074 (2013), arXiv:1307.2892 [hep-th] .
  * Ryu and Takayanagi [2006] S. Ryu and T. Takayanagi, Phys. Rev. Lett. 96, 181602 (2006), arXiv:hep-th/0603001 [hep-th] .
  * Hubeny  _et al._ [2007] V. E. Hubeny, M. Rangamani, and T. Takayanagi, JHEP 07, 062 (2007), arXiv:0705.0016 [hep-th] .
  * Lewkowycz and Maldacena [2013] A. Lewkowycz and J. Maldacena, JHEP 08, 090 (2013), arXiv:1304.4926 [hep-th] .
  * Geng  _et al._ [2020] H. Geng, A. Karch, C. Perez-Pardavila, S. Raju, L. Randall, M. Riojas, and S. Shashi, (2020), arXiv:2012.04671 [hep-th] .
  * Karch and Randall [2001a] A. Karch and L. Randall, JHEP 05, 008 (2001a), arXiv:hep-th/0011156 .
  * Karch and Randall [2001b] A. Karch and L. Randall, JHEP 06, 063 (2001b), arXiv:hep-th/0105132 .
  * Takayanagi [2011] T. Takayanagi, Phys. Rev. Lett. 107, 101602 (2011), arXiv:1105.5165 [hep-th] .
  * Karch and Randall [2020] A. Karch and L. Randall, JHEP 09, 166 (2020), arXiv:2006.10061 [hep-th] .
  * Randall and Sundrum [1999a] L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 4690 (1999a), arXiv:hep-th/9906064 .
  * Hartman and Maldacena [2013] T. Hartman and J. Maldacena, JHEP 05, 014 (2013), arXiv:1303.1080 [hep-th] .
  * Nomura [2011] Y. Nomura, JHEP 11, 063 (2011), arXiv:1104.2324 [hep-th] .
  * Maldacena and Pimentel [2013] J. Maldacena and G. L. Pimentel, JHEP 02, 038 (2013), arXiv:1210.7244 [hep-th] .
  * Nomura  _et al._ [2018a] Y. Nomura, P. Rath, and N. Salzetta, Phys. Rev. D 97, 106010 (2018a), arXiv:1711.05263 [hep-th] .
  * Aalsma and Shiu [2020] L. Aalsma and G. Shiu, JHEP 05, 152 (2020), arXiv:2002.01326 [hep-th] .
  * Geng [2020a] H. Geng, (2020a), arXiv:2005.00021 [hep-th] .
  * Haque and Underwood [2020] S. S. Haque and B. Underwood, (2020), arXiv:2010.08629 [hep-th] .
  * Karch [2003] A. Karch, JHEP 07, 050 (2003), arXiv:hep-th/0305192 [hep-th] .
  * Alishahiha  _et al._ [2004] M. Alishahiha, A. Karch, E. Silverstein, and D. Tong, AIP Conf. Proc. 743, 393 (2004), arXiv:hep-th/0407125 .
  * Nomura [2019] Y. Nomura, Phys. Rev. D 99, 086004 (2019), arXiv:1810.09453 [hep-th] .
  * Nomura [2020b] Y. Nomura, Phys. Rev. D 101, 066024 (2020b), arXiv:1908.05728 [hep-th] .
  * Nomura [2020c] Y. Nomura, Phys. Rev. D 102, 026001 (2020c), arXiv:1911.13120 [hep-th] .
  * Nomura [2020d] Y. Nomura, (2020d), arXiv:2010.15827 [hep-th] .
  * Maldacena [1999] J. M. Maldacena, Int. J. Theor. Phys. 38, 1113 (1999), [Adv. Theor. Math. Phys.2,231(1998)], arXiv:hep-th/9711200 [hep-th] .
  * Gubser  _et al._ [1998] S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Phys. Lett. B428, 105 (1998), arXiv:hep-th/9802109 [hep-th] .
  * Witten [1998] E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998), arXiv:hep-th/9802150 [hep-th] .
  * Dong  _et al._ [2018] X. Dong, E. Silverstein, and G. Torroba, JHEP 07, 050 (2018), arXiv:1804.08623 [hep-th] .
  * Gorbenko  _et al._ [2019] V. Gorbenko, E. Silverstein, and G. Torroba, JHEP 03, 085 (2019), arXiv:1811.07965 [hep-th] .
  * Geng  _et al._ [2019] H. Geng, S. Grieninger, and A. Karch, JHEP 06, 105 (2019), arXiv:1904.02170 [hep-th] .
  * Lewkowycz  _et al._ [2020] A. Lewkowycz, J. Liu, E. Silverstein, and G. Torroba, JHEP 04, 152 (2020), arXiv:1909.13808 [hep-th] .
  * Geng [2020b] H. Geng, Phys. Lett. B 803, 135327 (2020b), arXiv:1910.03594 [hep-th] .
  * Geng [2020c] H. Geng, JHEP 02, 005 (2020c), arXiv:1911.02644 [hep-th] .
  * Banks [2001] T. Banks, Int. J. Mod. Phys. A 16, 910 (2001), arXiv:hep-th/0007146 .
  * Witten [2001] E. Witten, in  _Strings 2001: International Conference_ (2001) arXiv:hep-th/0106109 .
  * Kraus [1999] P. Kraus, JHEP 12, 011 (1999), arXiv:hep-th/9910149 .
  * Fujita  _et al._ [2011] M. Fujita, T. Takayanagi, and E. Tonni, JHEP 11, 043 (2011), arXiv:1108.5152 [hep-th] .
  * Randall and Sundrum [1999b] L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999b), arXiv:hep-ph/9905221 .
  * Karch  _et al._ [2020] A. Karch, Z.-X. Luo, and H.-Y. Sun, (2020), arXiv:2012.02067 [hep-th] .
  * ’t Hooft [1990] G. ’t Hooft, Nucl. Phys. B 335, 138 (1990).
  * Susskind  _et al._ [1993] L. Susskind, L. Thorlacius, and J. Uglum, Phys. Rev. D 48, 3743 (1993), arXiv:hep-th/9306069 .
  * Miyaji and Takayanagi [2015] M. Miyaji and T. Takayanagi, PTEP 2015, 073B03 (2015), arXiv:1503.03542 [hep-th] .
  * Nomura  _et al._ [2018b] Y. Nomura, P. Rath, and N. Salzetta, Phys. Rev. D 98, 026010 (2018b), arXiv:1805.00523 [hep-th] .
  * Murdia  _et al._ [2020] C. Murdia, Y. Nomura, and P. Rath, Phys. Rev. D 102, 086001 (2020), arXiv:2008.01755 [hep-th] .
  * Langhoff and Nomura [2020] K. Langhoff and Y. Nomura, Phys. Rev. D 102, 086021 (2020), arXiv:2008.04202 [hep-th] .


its coordinate ring $\Gamma(X, \mathcal O_X)$.  Swan extended Serre's result
replacing $X$ by a paracompact topological space of bounded topological
dimension and $\mathcal O_X$ by its sheaf of continuous functions <cit.>.
Several other variations exist.  See Morye <cit.> for a general
theorem from which many classical results follow.

§.§ Differential refinements of the smooth Serre–Swan theorem

Let $X$ be a smooth manifold. A connection on a sheaf of $C^\infty_X$-modules $\mathcal M$ is a $\C$-linear map

[The notation $\mathcal{D}$ is used in this appendix only to refer to a connection on a sheaf. This notation is not to be conflated with that of a polynomial path of connections, which only appears in the body of this work.]

$\cD\colon \mathcal{M}\ra \mathcal{M}\tsr_{C^\infty_X} \Om^1_X$ satisfying $\cD(vf)=\cD(v)f+v\tsr df$ for every $U\subset X$, $v\in \mathcal{M}(U)$, and $f\in C^\infty(U)$.  If $\mathcal M$ and $\mathcal N$ are $C^\infty_X$-modules with connection, denoted by $\mathcal D_{\cM}$ and $\mathcal D_{\cN}$, respectively, a morphism $F\colon \mathcal M \to \mathcal N$ is compatible with the connections if the diagram

\[
\xy0;/r.25pc/:
\endxy
\]

commutes.  In a formula, this means that

\begin{equation*}
\mathcal D_\cN F(v) = (F\tsr_{C^{\infty}_{X}} \text{id}_{\Om_X^1})(\mathcal D_{\cM} v)
\end{equation*}

holds for all local sections $v$ of $\mathcal M$.

Let $\LfbD(X)$ denote the category of locally free sheaves of $C^\infty_X$-modules of bounded rank endowed with a connection. Morphisms are $C^\infty_X$-linear homomorphisms compatible with the given connections. Recall, $\FgpD^{\text{iso}}(X)$ denotes the category of finitely-generated projective modules with connection over $C^\infty(X;\C)\equiv C^\infty(X)$ with respect to the DGA $\Om_{\dR}^\bl(X;\C)$ (cf. Notation <ref>,  Definition <ref>, and Example <ref>).

The tensor product of vector bundles and tensor product of modules over $C^\infty(X)$ make $\Bun^{\text{iso}}(X)$ and $\Fgp^{\text{iso}}(C^\infty(X))$ into symmetric monoidal categories. Recall that if $\mathcal M$ and $\mathcal N$ are sheaves of $\mathcal A$-modules, we define $\mathcal M \otimes_{\mathcal A} \mathcal N$ as the sheafification of the presheaf $U \mapsto \mathcal M(U) \otimes_{\mathcal A(U)} \mathcal N(U)$.
In general, the sheafification step in the definition of tensor product of sheaves is essential, and the global sections functor is not monoidal. In the context of $C^\infty_X$-modules, however, the situation is simplified.

Given sheaves of $C^\infty_X$-modules $\mathcal M$ and $\mathcal N$, there is a natural isomorphism of $C^\infty(X)$-modules

\begin{equation*}
\mathcal M(X) \otimes_{C^\infty(X)} \mathcal N(X)
\xrightarrow{\cong}
(\mathcal M \otimes_{C^\infty_X} \mathcal N)(X).
\end{equation*}

Before proving the lemma, we recall some notation and fundamental facts.  Let $\mathcal M$, $\mathcal N$ and $\mathcal Z$ be sheaves of $C^\infty_X$-modules.

* The set of homomorphisms $\mathcal M \to \mathcal N$ is denoted $\Hom_{C^\infty_X}(\mathcal M, \mathcal N)$.  It has a natural structure of a $C^\infty(X)$-module: for any $f\in C^\infty(X)$, $F\in \Hom_{C^\infty_X}(\mathcal M, \mathcal N)$, and an open set $U$ of $X$, set $(f\cdot F)(U):=f|_{U}F(U)$.

* The sheaf of homomorphisms from $\mathcal M$ to $\mathcal N$ assigns to an open set $U \subseteq X$ the $C^\infty(U)$-module $\Hom_{C^\infty_U}(\mathcal M\vert_U, \mathcal N\vert_U)$.  This sheaf is denoted $\underline\Hom_{C^\infty_X}(\mathcal M, \mathcal N)$, and it has a natural structure of a $C^\infty_X$-module. Note that $\Hom_{C^\infty_X}(\mathcal M, \mathcal N)$ is the set of global sections of the sheaf $\underline\Hom_{C^\infty_X}(\mathcal M, \mathcal N)$.

* The previous construction is the internal hom in the category of sheaves of $C^\infty_X$-modules, in the sense that there is a natural isomorphism (hom-tensor adjunction)

\begin{equation*}
\Hom_{C^\infty_X}(\mathcal M \otimes_{C^\infty_X} \mathcal N, \mathcal Z)
\cong
\Hom_{C^\infty_X}(\mathcal M,
\underline\Hom_{C^\infty_X}(\mathcal N, \mathcal Z)).
\end{equation*}

Consider the map

\begin{equation}
\label{eq:Hom-vs-Gamma}
\Hom_{C^\infty_X}(\mathcal M, \mathcal N)\
\to \Hom_{C^\infty(X)}(\mathcal M(X), \mathcal N(X)),
\end{equation}

which sends a sheaf homomorphism to the associated map on
global sections. We claim that, because all sheaves involved are fine,
(<ref>) is an isomorphism of $C^\infty(X)$-modules. To
show that (<ref>) is injective, suppose that
$F\colon\mathcal M \to \mathcal N$ is such that $F_X\colon \mathcal
M(X) \to \mathcal N(X)$ vanishes. We need to show that $F_U\colon
\mathcal M(U) \to \mathcal N(U)$ vanishes for any open $U \subseteq X$.
Fix $\sigma \in \mathcal M(U)$, let $U' \subseteq X$ be open with
closure contained in $U$, and let $\rho$ be a function with support in
$U$ and $\rho\vert_{U'} = 1$

Then $\rho\sigma$ defines a global section of $\mathcal M$, and

\begin{equation*}
= F_{U'}(\rho\sigma\vert_{U'})
= F_X(\rho\sigma)\vert_{U'} = 0.
\end{equation*}

Since, by the sheaf property of $\mathcal N$, any section over $U$ can be reconstructed from its restrictions to each open $U'$ with closure in $U$, we conclude that $F\vert_U = 0$ and therefore $F = 0$.

To show that (<ref>) is surjective, fix $F_X\colon \mathcal M(X) \to \mathcal N(X)$. We want to find a sheaf homomorphism $F\colon \mathcal M \to \mathcal N$ that acts as $F_X$ on global sections. Fix $\sigma \in \mathcal M(U)$ and consider, for any open $U'$ with closure in $U$, a function $\rho$ as above. Then we claim that the section $F_X(\rho\sigma)\vert_{U'} \in \mathcal N(U')$
is independent of $\rho$.  In fact, let $\rho'$ be another admissible choice, and set $\eta = \rho - \rho'$.  Then we want to show that $F_X(\eta\sigma)\vert_{U'} = 0$.  Choosing a function $\xi$ that is identically $1$ in $U'$ and has support in the set $\{x\in X \mid \eta(x) = 0 \}$, we can write

\begin{equation*}
= \xi\vert_{U'} \cdot F_X(\eta\sigma)\vert_{U'}
= F_X(\xi\eta\sigma)\vert_{U'}
= 0,
\end{equation*}

since the section $\xi\eta\si$ vanishes globally by definition of $\xi$. Now, using the sheaf property of $\mathcal N$ to glue these coherent values $F_X(\rho\sigma)\vert_{U'} \in \mathcal N(U')$ as $U' \subseteq U$ varies over all open subsets with closure in $U$, we obtain an element $F_U(\sigma) \in \mathcal N(U)$. This recipe, applied to the open $X$, gives us the original $F_X$, and the naturality condition $F_U(\sigma)\vert_V = F_V(\sigma\vert_V)$ is satisfied for all $V \subseteq U$. This proves (<ref>) is surjective.

Finally, let $\mathcal Z$ be an arbitrary sheaf of $C^\infty_X$-modules.
Then the lemma follows from the Yoneda lemma and the following natural

\begin{equation*}
\def\iHom{\underline\Hom_{C^\infty_X}}
\def\sHom{\Hom_{C^\infty_X}}
\def\mHom{\Hom_{C^\infty(X)}}
\def\sotimes{\otimes_{C^\infty_X}}
\def\motimes{\otimes_{C^\infty(X)}}
\begin{aligned}
\mHom((\mathcal M \sotimes \mathcal N)(X), \mathcal Z(X))
& \cong \sHom(\mathcal M \sotimes \mathcal N, \mathcal Z)
\\ & \cong \sHom(\mathcal M, \iHom(\mathcal N, \mathcal Z))
\\ & \cong \mHom(\mathcal M(X), \sHom(\mathcal N, \mathcal Z))
\\ & \cong \mHom(\mathcal M(X), \mHom(\mathcal N(X), \mathcal Z(X)))
\\ & \cong \mHom(\mathcal M(X) \motimes \mathcal N(X), \mathcal Z(X)).
\end{aligned}
\end{equation*}

Above, $\underline\Hom_{C^\infty_X}$ denotes the sheaf of homomorphisms between $C^\infty_X$-modules (which is again a $C^\infty_X$-module), while $\Hom_{C^\infty_X}$ denotes its global sections, that is, the $C^\infty(X)$-module of sheaf homomorphisms.  Each of the above identifications is justified by the isomorphism (<ref>), the adjunction between $\Hom_{C^\infty(X)}$ and $\otimes_{C^\infty(X)}$, or its analogue for $\underline\Hom_{C^\infty_X}$ and $\otimes_{C^\infty_X}$.

Suppose that the sheaves of $C^\infty_X$-modules $\mathcal M$ and $\mathcal N$ are locally free of bounded rank.  Then so is their tensor product $\mathcal M \otimes_{C^\infty_X} \mathcal N$.  Thus, $\Lfb(X)$ is a symmetric monoidal category.  Moreover, $\Gamma_X$ is a symmetric monoidal functor.

The functor $\Gamma_\loc\colon \Bun^{\text{iso}}(X) \to \Lfb^{\text{iso}}(X)$ from (<ref>) is symmetric monoidal.

Let $E, F \in \Bun^{\text{iso}}(X)$. The natural bilinear map $E \times F \to E \otimes F$ induces, on local sections, a map $\Gamma_\loc(E) \otimes_{C^\infty_X} \Gamma_\loc(F) \to \Gamma_\loc(E \otimes F)$. We need to show that this is an isomorphism.  Taking stalks, this gives a map

\begin{equation}
\label{eq:1}
\Gamma_\loc(E)_x \otimes_{C^\infty_{X, x}} \Gamma_\loc(F)_x
\cong
(\Gamma_\loc(E) \otimes_{C^\infty_X} \Gamma_\loc(F))_x
\to
\Gamma_\loc(E \otimes F)_x.
\end{equation}

The isomorphism on the left-hand side follows from the fact that the tensor product preserves colimts. Next, by taking stalks at $x \in X$, we get $\Gamma_\loc(E)_x \cong C^\infty_{X, x} \otimes E_x$, where $E_x$ is the fiber of $E$ at $x$ and all other subscripts $x$ denote stalks.  Using this, and the corresponding fact for $F$ and $E \otimes F$, we see that (<ref>) is an isomorphism.  A map of sheaves inducing isomorphisms on stalks is an isomorphism, as desired.

The functor $\Gamma\colon \Bun^{\text{iso}}(X) \to \Fgp^{\text{iso}}(X)$ is symmetric monoidal. In particular, given vector bundles $E$ and $F$ over $X$, there is an isomorphism

\begin{equation*}
\Gamma(E\otimes F) \cong \Gamma(E) \otimes_{C^\infty(X)} \Gamma(F)
\end{equation*}

of $C^\infty(X)$-modules.

The natural isomorphism (<ref>) between $\Gamma$ and the monoidal functor $\Gamma_X \circ \Gamma_{\loc}$ gives the former the structure of a monoidal functor. For a more direct proof, not involving sheaf tensor products, see Nestruev <cit.>.

Recall that a connection on a vector bundle $E \to X$ is a $\mathbb
C$-linear map

\begin{equation*}
\nabla\colon \Gamma(E) \to \Gamma(E \otimes T^*X)
\end{equation*}

satisfying the familiar version of the Leibniz rule. By Corollary <ref> (with $F$ being the cotangent bundle $T^*X$ of $X$), $\nabla$ determines a connection on the $C^\infty(X)$-module $\Gamma(E)$. This assignment extends to a functor

\begin{equation}\label{EQN.GammaCon}
\Gamma^\con\colon \BunD^{\text{iso}}(X) \to \FgpD^{\text{iso}}(X),
\end{equation}

which on morphisms does the same as the functor $\Gamma$ of

Similarly, applying Lemma <ref> to the case $\mathcal N = \Omega^1_X$, we see that a connection on a $C^\infty_X$-module $\mathcal M$ determines a connection on its global sections $\Gamma_X(\mathcal M)$. Thus, we get a functor $\Gamma_X^\con\colon \LfbD(X) \to \FgpD^{\text{iso}}\big(C^\infty(X)\big)$.

Let $\mathcal M$ be a sheaf of $C^\infty_X$-modules and $M = \mathcal M(X)$ the $C^\infty(X)$-module of global sections. Then a connection $D$ on $M$ uniquely determines a connection $\mathcal D$ on $\mathcal M$ characterized by the property that $\mathcal{D}_{X}=D$.

Set $\mathcal D_X\colon\mathcal M(X) \xrightarrow D \mathcal M(X) \otimes_{C^{\infty}(X)} \Omega^{1}(X) \cong (\mathcal M \otimes_{C^\infty_X} \Omega_X^1)(X)$ to be $D$ followed by the isomorphism from Lemma <ref>. Our goal is to show this extends to a unique connection $\mathcal D_U\colon \mathcal M(U) \to  \mathcal M(U) \otimes_{C^{\infty}(U)} \Omega^{1}(U) \cong (\mathcal M \otimes_{C^\infty} \Omega^1)(U)$ on each open $U\subseteq X$.

Let $U \subseteq X$ be open and $\sigma \in \mathcal M(U)$.  Assume for the moment that $\sigma$ admits an extension $\sigma'\in \mathcal M(X)$ to a global section.  Then we set
$\mathcal D_U \sigma := (\mathcal D_X \sigma')\vert_U.$ Note that this is independent of the choice of $\sigma'$. To see this, pick a different extension $\sigma'' \in \mathcal M(X)$ of $\sigma$, so that $\sigma' - \sigma'' = 0$ on $U$. For each $x \in U$, we can choose a function $\rho \in C^\infty(X)$ with support in $U$ and such that $\rho = 1$ in a neighborhood of $x$, so that $\rho (\sigma' - \sigma'') = 0$ on all of $X$. Then, in a neighborhood of $x$, we have

\begin{equation*}
0 = \mathcal D_X \rho (\sigma' - \sigma'')
= d\rho \otimes (\sigma' - \sigma'')
+ \rho \mathcal D_X (\sigma' - \sigma'')
= \mathcal D_X (\sigma' - \sigma'').
\end{equation*}

By the sheaf property (or, more specifically, the fact that sections agree if they agree locally), we see that $\mathcal D_U\sigma$ is well-defined.

Now, in general, a section $\sigma \in \mathcal E(U)$ may not admit a global extension. Nevertheless, for each $x\in U$, there exists an open neighborhood $U'\subseteq U$ of $x$ such that $\sigma|_{U'}$ has a global extension to $X$.
Thus, using global extensions of restrictions $\sigma\vert_{U'}$ for small open disks $U' \subset U$ and using the sheaf condition, we get a unique $\mathcal D_U$ with the property that $\mathcal D_U (\sigma\vert_U) = (\mathcal D_X \sigma)\vert_{U}$ for all global sections $\sigma \in \mathcal M(X)$.

It remains to check that $\mathcal D_U$ satisfies the Leibniz rule.  This holds if and only if it holds for all sections with compact support in $U$.  Now, the condition $\mathcal D_U (\sigma\vert_U) = (\mathcal D_X \sigma)\vert_{U}$, and the fact that $\mathcal D_X$ satisfies the Leibniz rule, shows that the same is true of $\mathcal D_U$.

We now define a functor $\Gamma_\loc^\con\colon \BunD^{\text{iso}}(X) \to \LfbD(X)$. Given a vector bundle with connection $(E, \nabla)$, we set $\Gamma_\loc^\con(E, \nabla) := (\mathcal E, \mathcal D)$, where $\mathcal E := \Gamma_\loc(E)$ is the sheaf of sections of $E$, and $\mathcal D$ is the connection obtained via Lemma <ref> from the connection on $\Gamma^\con(E, \nabla)$. At the level of morphism, $\Gamma_\loc^\con$ performs the same assignments as its counterpart in diagram <ref>. One can check that this is compatible with the connections constructed in Lemma <ref>.

To sum up, we obtained a diagram of groupoids and functors

\begin{equation}
\label{eq:bun-lfb-fgp-conn}
\vcenter{\hbox{
\xymatrix{
\BunD^{\text{iso}}(X) \ar[d]_{\Gamma_{\mathrm{loc}}^\con} \ar[r]^{\Gamma^\con} & \FgpD^{\text{iso}}(X)\\
\LfbD(X) \ar[ur]_{\Gamma_X^\con} &
\end{equation}

which commutes up to isomorphism.
The reason this diagram does not commute on the nose is because the definition of a connection involves the tensor product and these functors are only monoidal up to isomorphism.

[Differential Serre–Swan theorem]  Each functor in (<ref>) is an equivalence of categories.

The forgetful functors mapping the triangle (<ref>) to its version (<ref>) without connection are faithful, since morphisms in the former are morphisms in the latter satisfying additional conditions (to preserve connections, in each of the three different senses). It follows that all functors in (<ref>) are faithful.

By a similar argument, $\Gamma^\con$ is full.  We now verify that $\Gamma^\con$ is essentially surjective. Take any $(M,D)\in\FgpD^{\text{iso}}(X)$. The essential surjectivity of the horizontal map in (<ref>) gives us an isomorphism $\al:\Ga(E)\srl{\isom}\ra M$ for some $E\in \Bun^{\text{iso}}(X)$. We define a connection $\na$ on $E$ by the composite

\begin{equation*}
\na:\Ga(E)\srl{\al}\ra M\srl{D}\ra M\tsr_{\Cinf(X)}
\Om^1(X)\xrightarrow{\al^{-1}\tsr\text{id}} \Ga(E)
\tsr_{\Cinf(X)}\Om^1(X)\srl{\isom}\ra\Ga(E\tsr T^*X).
\end{equation*}

Since $D$ satisfies the Leibniz rule, so does $\nabla$. Furthermore, $\alpha$ defines an isomorphism of modules with connection $\alpha:(\Gamma(E),\nabla)\to(M,D)$ by construction of $\na$. This proves that $\Gamma^\con$ is essentially surjective.

We conclude that $\Gamma_X^\con$ is essentially surjective and full since $\Gamma^\con$ is essentially surjective and full and the former factors through the latter. We already know they are faithful, and it follows that $\Gamma^\con$ and $\Gamma_X^\con$ are equivalences. Therefore, $\Gamma_\loc^\con$ is an equivalence as well.

Let $M\in\Fgp(A)$ and $\Omega_{\bullet}(A)$ a DGA on top of $A$. A Grassmann connection on a $M$ is a connection $D$ on $M$ for which there exists an $n\in\N$, an embedding $i:M\hookrightarrow A^{n}$, and a projection $p:A^{n}\to M$ such that $D:=(p\otimes\mathrm{id})\circ d\circ i$, where $d$ is the standard connection on $A^{n}$.

A Grassmann connection is also called a Berry connection in the physics literature <cit.> and a Levi-Civita connection by Karoubi <cit.>.

Let $X$ be a manifold and $A=\Cinf(X;\C)$. Then any connection on a finitely generated projective $A$-module is a Grassmann connection.

Suppose $(M,D)\in\FgpD^{\text{iso}}(\Cinf(X;\C))$. From the proof of Proposition <ref> on essential surjectivity of $\Ga^\con$, there exists $(E,\na)\in \BunD^{\text{iso}}(X)$ together with an isomorphism $\alpha:\Gamma(E)\srl{\isom}\ra M$ such that $\na=(\al^{-1} \tsr \text{id})\circ D \circ \al$. By the theorem of Narasimhan and Ramanan NR1, NR2,Qu, $\na= (p \tsr \text{id})\circ D \circ i$, where $p:X\times \C^N\ra E$ is a projection such that $\text{Im}(p)=E$ and $i:E\ra X\times \C^N$ is an embedding. Therefore $D= (\al\tsr\text{id})\circ p \circ d \circ i\circ \al^{-1}$, which is a Grassmann connection.

§ ON GROTHENDIECK FIBRATIONS AND THE EXTENSION OF SCALARS

Naturality of the noncommutative Chern character necessitates the need to introduce the extension of scalars. Since naturality is a categorical notion, it is important to describe the extension of scalars from a functorial viewpoint, and, for this reason, it is helpful to review Grothendieck fibrations and opfibrations. This justifies some of the terminology we have been using, such as cocartesian morphisms. We also make everything explicit since taking opposite categories can be quite confusing due to the many ways in which it can occur. Our references include St18,MoVa18,Ma19.

Let $\mathcal{E}$ and $\mathcal{X}$ be two categories and let $\mathcal{E}\xrightarrow{\pi}\mathcal{X}$ be a functor. A morphism $u\xrightarrow{\beta}v$ in $\mathcal{E}$ is cartesian iff for any morphism $x\xrightarrow{f}\pi(u)$ in $\mathcal{X}$ and any morphism $w\xrightarrow{\gamma}v$ in $\mathcal{E}$ such that $\pi(\beta)\circ f=\pi(\gamma)$, there exists a unique morphism $w\xrightarrow{\alpha}u$ in $\mathcal{E}$ such that

\[
\pi(\alpha)=f
\qquad\text{ and }\qquad
\beta\circ\alpha=\gamma.
\]

A visualization of the data in Definition <ref> is often helpful:

[The morphism $\beta$ has been bolded to emphasize that it is the morphism that is cartesian.]

\[
\xy0;/r.25pc/:
\endxy
\qquad\qquad\qquad
\xy0;/r.25pc/:
\endxy
\]

Note that the conditions necessitate $\pi(w)=x$ so that one can think of the morphisms in $\mathcal{E}$ as lying above the morphisms in $\mathcal{X}$ via the functor $\pi$. In Example <ref>, we will see that a cartesian morphism is a fibrewise isomorphism in the setting of vector bundles over smooth manifolds.

A closely related definition is that of a cocartesian morphism.

Let $\mathcal{E}$ and $\mathcal{X}$ be two categories and let $\mathcal{E}\xrightarrow{\pi}\mathcal{X}$ be a functor. A morphism $u\xleftarrow{\beta}v$ in $\mathcal{E}$ is cocartesian iff for any morphism $x\xleftarrow{f}\pi(u)$ in $\mathcal{X}$ and any morphism $w\xleftarrow{\gamma}v$ in $\mathcal{E}$ such that $f\circ\pi(\beta)=\pi(\gamma)$, there exists a unique morphism $w\xleftarrow{\alpha}u$ in $\mathcal{E}$ such that

\[
\pi(\alpha)=f
\qquad\text{ and }\qquad
\alpha\circ\beta=\gamma.
\]

Cocartesian morphisms are visualized as:

\[
\xy0;/r.25pc/:
\endxy
\qquad\qquad\qquad
\xy0;/r.25pc/:
\endxy
\]

In other words, a cocartesian morphism in $\mathcal{E}\xrightarrow{\pi}\mathcal{X}$ is a cartesian morphism in $\mathcal{E}^{\op}\xrightarrow{\pi^{\op}}\mathcal{X}^{\op}$.

Let $\pi:\mathcal{E}\to\mathcal{X}$ be as in Definition <ref>. Let $\mathcal{E}_{x}$ be the subcategory of $\mathcal{E}$ consisting of the objects $u$ in $\mathcal{E}$ such that $\pi(u)=x$ and $\pi(\beta)=\mathrm{id}_{x}$ for all morphisms $u\xrightarrow{\beta}v$ with $\pi(u)=x=\pi(v)$. The category $\mathcal{E}_{x}$ is called the fibre of $\pi$ over $x$ and the morphisms in $\mathcal{E}_{x}$ are called vertical morphisms of $\pi$ over $x$.
Given a morphism $x\xrightarrow{f}y$ in $\mathcal{X}$ and an object $v$ in $\mathcal{E}_{y}$, a cartesian lifting of $f$ with target $v$ is a cartesian morphism $u\xrightarrow{\beta}v$ such that $\pi(\beta)=f$.
A functor $\pi:\mathcal{E}\to\mathcal{X}$ is a fibration iff for any morphism  $x\xrightarrow{f}y$ in $\mathcal{X}$ and an object $v$ in $\mathcal{E}_{y}$, a cartesian lifting exists.
Dually, given a morphism $x\xleftarrow{f}y$ in $\mathcal{X}$ and an object $v$ in $\mathcal{E}_{y}$, a cocartesian lifting of $f$ with source $v$ is a cocartesian morphism $u\xleftarrow{\beta}v$ such that $\pi(\beta)=f$.
A functor $\pi:\mathcal{E}\to\mathcal{X}$ is a opfibration iff for any morphism  $x\xleftarrow{f}y$ in $\mathcal{X}$ and an object $v$ in $\mathcal{E}_{y}$, a cocartesian lifting exists.
When $\pi$ is either a fibration or an opfibration, $\mathcal{E}$ is called the total category and $\mathcal{X}$ is called the base.
Given a fibration (opfibration) $\mathcal{E}\xrightarrow{\pi}\mathcal{X}$, the collection of cartesian (cocartesian) morphisms form a fibration (opfibration) $\mathbf{cart}(\mathcal{E})\xrightarrow{\pi}\mathcal{X}$ ($\mathbf{cocart}(\mathcal{E})\xrightarrow{\pi}\mathcal{X}$).

Let $\Man$ denote the category of smooth manifolds and smooth maps. Let $\Bun$ denote the category of smooth vector bundles over manifolds. Namely, an object of $\Bun$ is a triple $(X,E,\pi)$, with $X$ a smooth manifold and $E\xrightarrow{\pi}X$ a smooth vector bundle. A morphism $(X,E,\pi)\to(Y,F,\rho)$ consists of a pair of smooth maps $(X\xrightarrow{f}Y,E\xrightarrow{\varphi}F)$ such that $\varphi$ is a fibrewise linear map and such that the diagram

\[
\xy0;/r.25pc/:
\endxy
\]

commutes. Then the projection map $\Bun\to\Man$, sending an object $(X,E,\pi)$ to $X$ and a morphism $(X,E,\pi)\xrightarrow{(f,\varphi)}(Y,F,\rho)$ to $f$, is a Grothendieck fibration. Indeed, to every morphism $X\xrightarrow{f}Y$ in $\Man$, and to every vector bundle $(Y,F,\rho)$ over $Y$, the pullback bundle, defined by the categorical pullback

\[
\xy0;/r.25pc/:
\endxy
\]

in the category $\Man$, is a cartesian morphism over $f$. This follows immediately from the universal property of the pullback—see the visualization after Definition <ref>. This same universal property shows that all cartesian morphisms are of this form. Thus, $\mathbf{cart}(\Bun)$ can be identified with the category of vector bundles as objects and smooth maps on the base with fibrewise isomorphisms on the vector bundles as morphisms.

Let $\Ring$ denote the category of unital rings and unital ring homomorphisms (henceforth, all rings will be assumed unital in this appendix). Let $\Mod$ denote the category of modules. Namely, an object of $\Mod$ is a pair $(R,M)$ consisting of a ring $R$ and a right $R$-module $M$. A morphism $(S,N)\to(R,M)$ is a pair $(\psi,\Psi)$ with $S\xrightarrow{\psi}R$ a ring homomorphism and $N\xrightarrow{\Psi}M$ a $\Z$-linear map such that $\Psi(ns)=\Psi(n)\psi(s)$ for all $n\in N$ and $s\in S$. Then the forgetful functor $\Mod\to\Ring$ sending a pair $(R,M)$ to $R$ and $(\psi,\Psi)$ to $\psi$ is an opfibration. Indeed, to every ring homomorphism $S\xrightarrow{\psi}R$ and to every right $S$-module $N$, a cocartesian lift is given by the extension of scalars right $R$-module $N\otimes_{S}R:=N\otimes_{\Z}R/_{\sim}$, together with the $\Z$-linear map $N\xrightarrow{\Psi}N\otimes_{S}R$ defined by sending $n\in N$ to $n\otimes1_{R}$. This is cocartesian because to any other ring $Q$ and right $Q$-module $L$ together with morphisms as in

\[
\xy0;/r.25pc/:
\endxy
\]

the unique morphism $N\otimes_{S}R\xrightarrow{\Xi}L$ in $\Mod$ is uniquely determined by the assignment

\[
N\otimes_{\K}R\ni n\otimes r\mapsto \Phi(n)\xi(r).
\]

A quick calculation shows that this descends to a well-defined morphism $N\otimes_{S}R\xrightarrow{\Xi}L$ in $\Mod$ satisfying the required commutativity conditions. Its uniqueness follows from the fact that it is actually completely determined by its action on elements of the form $n\otimes 1_{R}$, which must be given by $\Phi(n)$ in order for the top diagram in $\Mod$ to commute.
Furthermore, all cocartesian morphisms are of this form. More precisely, if $N\xrightarrow{\Sigma}M$ is another cocartesian lift of $\psi$, then $M$ is canonically isomorphic to $N\otimes_{S}R$ in the category $\Mod$.
This follows from the fact that any two target objects of cocartesian lifts are canonically isomorphic (this follows from the usual arguments of canonical uniqueness of objects defined via universal properties).

Thus, the category of cocartesian morphisms $\mathbf{cocart}(\Mod)$ is equivalent to the category of modules and morphisms where the morphisms always map into a module that is canonically isomorphic to an extension of scalars of the first module along the ring homomorphism in the base.

Fibrations and opfibrations are occasionally constructed from functors via the Grothendieck construction, though elements of this construction were known earlier by Yoneda and Mac Lane (cf. <cit.>).

Let $\mathcal{X}$ be a category and let $\mathcal{X}^{\op}\xrightarrow{\mathfrak{E}}\mathbf{Cat}$ be a pseudofunctor, also called a weak 2-functor (for definitions, see Pa19,KeSt74). Then the following structure defines a category $\int\mathfrak{E}$.

* An object of $\int\mathfrak{E}$ is a pair $(X,E)$ with $X$ an object of $\mathcal{X}$ and $E$ an object of $\mathfrak{E}_{X}$.

* A morphism from $(X,E)$ to $(Y,F)$ is a pair $(f,\alpha)$, with $X\xrightarrow{f}Y$ a morphism in $\mathcal{X}$ and $E\xrightarrow{\alpha}\mathfrak{E}_{f}(F)$ a morphism in $\mathfrak{E}_{X}$.

* The composite of a composable pair of morphisms $(X,E)\xrightarrow{(f,\alpha)}(Y,F)\xrightarrow{(g,\beta)}(Z,G)$ is given by the composite $X\xrightarrow{g\circ f}Z$ in $\mathcal{X}$ and

\[
E\xrightarrow{\alpha}\mathfrak{E}_{f}(F)\xrightarrow{\mathfrak{E}_{f}(\beta)}\mathfrak{E}_{f}\big(\mathfrak{E}_{g}(G)\big)\xrightarrow{c_{(f,g)}}\mathfrak{E}_{g\circ f}(G)
\]

in $\mathfrak{E}_{X}$,
where $c_{(f,g)}$ is the compositor natural isomorphism from the data of a pseudofunctor.

* The identity morphism associated to a pair $(X,E)$ in $\int\mathfrak{E}$ is given by $E\xrightarrow{\epsilon_{X}}\mathfrak{E}_{\mathrm{id}_{X}}(E)$, where $\epsilon$ is the unitor natural isomorphism from the data of the pseudofunctor $\mathfrak{E}$.

Furthermore, the assignment

\[
\begin{split}
\int\mathfrak{E}&\xrightarrow{\pi_{\mathfrak{E}}}\mathcal{X}\\
(X,E)&\mapsto X\\
\Big((X,E)\xrightarrow{(f,\alpha)}(Y,F)\Big)&\mapsto\Big(X\xrightarrow{f}Y\Big)
\end{split}
\]

defines a functor. This functor is a fibration in which a cartesian lift of $X\xrightarrow{f}Y$ with target $F$ in $\mathfrak{E}_{Y}$ is the pair $(f,\mathrm{id}_{\mathfrak{E}_{f}(F)})$. Finally, a morphism $(X,E)\xrightarrow{(f,\alpha)}(Y,F)$ is cartesian if and only if $E\xrightarrow{\alpha}\mathfrak{E}_{f}(F)$ is an isomorphism in $\mathfrak{E}_{X}$.

Given a pseudofunctor $\mathcal{X}^{\op}\xrightarrow{\mathfrak{E}}\mathbf{Cat}$, the fibration $\int\mathfrak{E}\xrightarrow{\pi_{\mathfrak{E}}}\mathcal{X}$ from Lemma <ref> is called the Grothendieck construction/fibration associated to $\mathfrak{E}$.

The extension of scalars briefly described in Example <ref> can be described as a pseudofunctor $\mathfrak{E}:\Ring\to\mathbf{Cat}$ sending a ring $R$ to the category of right $R$-modules $\mathfrak{E}(R):=\Mod(R)$ and sending a ring homomorphism $S\xrightarrow{\psi}R$ to a functor $\Mod(S)\xrightarrow{\mathfrak{E}_{\psi}}\Mod(R)$ defined as follows. To each $S$-module $N$, the functor assigns the $R$-module $N\otimes_{S}R$. To each morphism of $S$-modules $N\xrightarrow{\kappa}N'$, it assigns the canonical extension $N\otimes_{S}R\xrightarrow{\wtl{\kappa}}N'\otimes_{S}R$ determined uniquely by the $R$-action on elements of the form $n\otimes1_{R}$. The reason $\mathfrak{E}$ is a pseudofunctor, as opposed to a (strict) functor, is that to a pair of composable ring homomorphisms $S\xrightarrow{\psi}R\xrightarrow{\xi}Q$, there is a natural isomorphism

\[
\xy0;/r.25pc/:
\endxy
\]

that is not necessarily the identity natural transformation. Indeed, to an $S$-module $N$, it provides the isomorphism

\[
\begin{split}
(N\otimes_{S}R)\otimes_{R}Q&\to N\otimes_{S}Q\\
n\otimes r\otimes q&\mapsto n\otimes\xi(r)q
\end{split}
\]

in $\Mod(Q)$.
The inverse is given by sending $n\otimes q$ to $n\otimes1_{R}\otimes q$. Secondly, given a ring $R$, there is a natural isomorphism

\[
\xy0;/r.25pc/:
\endxy
\]

given on a right $R$-module $M$ by the isomorphism $M\to M\otimes_{R}R$ sending $m\in M$ to $m\otimes1_{R}$.
We leave the other verifications of the definition of a pseudofunctor to the reader.

We now apply the Grothendieck construction to $\mathfrak{E}:(\Ring^{\op})^{\op}\to\mathbf{Cat}$, where the double “op” is taken to more directly apply Definition <ref>. This gives a fibration $\int\mathfrak{E}\xrightarrow{\pi_{\mathfrak{E}}}\Ring^{\op}$, where the objects of $\int\mathfrak{E}$ are given by pairs $(R,M)$ with $R$ a ring and $M$ a right $R$-module. A morphism from $(R,M)$ to $(S,N)$ in $\int\mathfrak{E}$ is a pair $(\psi^{\op},\alpha)$ with $R\xrightarrow{\psi^{\op}}S$ a morphism in $\Ring^{\op}$, which corresponds to a ring homomorphism $S\xrightarrow{\psi}R$, and a right $R$-module morphism $M\xrightarrow{\alpha}N\otimes_{S}R$.
The projection $\int\mathfrak{E}\xrightarrow{\pi_{\mathfrak{E}}}\Ring^{\op}$ takes an object $(R,M)$ to $R$ and a morphism $(\psi^{\op},\alpha)$ to $\psi^{\op}$.

In what follows, we will compare the three fibrations from Example <ref>, Example <ref>, and Example <ref>. To do this, we need to make sense of morphisms of fibrations.

Let $\mathcal{E}\xrightarrow{\pi}\mathcal{X}$ and $\mathcal{F}\xrightarrow{\rho}\mathcal{Y}$ be two fibrations. A fibred functor from $\pi$ to $\rho$ consists of a pair of functors $\big(\mathcal{E}\xrightarrow{\Gamma}\mathcal{F},\mathcal{X}\xrightarrow{\gamma}\mathcal{Y}\big)$ such that the diagram

\[
\xy0;/r.25pc/:
\endxy
\]

commutes and such that every cartesian lifting in $\pi$ gets sent to a cartesian lifting in $\rho$.

In Example <ref>, we showed that the extension of scalars $\Ring\xrightarrow{\mathfrak{E}}\mathbf{Cat}$ defines a fibration $\int\mathfrak{E}\xrightarrow{\pi_{\mathfrak{E}}}\Ring^{\op}$. Let $\Man\xrightarrow{C^{\infty}}\Ring^{\op}$ be the functor that takes the ring of smooth functions. More precisely, it takes a smooth manifold $X$ to the ring of smooth functions $C^{\infty}(X)$ on $X$. It sends a smooth map $X\xrightarrow{f}Y$ to the pullback ring homomorphism $C^{\infty}(Y)\xrightarrow{f^*}C^{\infty}(X)$. Second, let $\Bun\xrightarrow{\Gamma}\int\mathfrak{E}$ be the functor that sends a smooth vector bundle $(X,E,\pi)$ to the pair $\big(C^{\infty}(X),\Gamma(\pi)\big)$, where $\Gamma(\pi)$ is the $C^{\infty}(X)$-module of smooth sections of the vector bundle $E\xrightarrow{\pi}X$. To describe what this functor assigns to a morphism $(X,E,\pi)\xrightarrow{(f,\varphi)}(Y,F,\rho)$, we first note that associated to any such morphism in $\Bun$ is a uniquely determined morphism of bundles $(X,E,\pi)\xrightarrow{(\mathrm{id}_{X},\xi)}(X,f^*F,f^*\rho)$ coming from the universal property of the pullback

\[
\xy0;/r.25pc/:
\endxy
\]

This map induces a map of $C^{\infty}(X)$-modules $\Gamma(\pi)\xrightarrow{\Xi}\Gamma(f^*\rho)$ given by sending a section $s$ to $\xi\circ s$. This does not yet define a morphism in $\int\mathfrak{E}$. To obtain such a morphism, note that there is a natural $C^{\infty}(X)$-module map

\[
\begin{split}
\Gamma(\rho)\otimes_{C^{\infty}(Y)}C^{\infty}(X)&\xrightarrow{\Upsilon}\Gamma(f^*\rho)\\
t\otimes\eta&\mapsto\bigg(X\ni x\mapsto \psi^{-1}\Big(t\big(f(x)\big)\Big)\eta(x)\bigg),
\end{split}
\]

which is obtained by taking a section $t$ of $\rho$ to the section of $f^*\rho$ by composing the appropriate morphisms in the diagram

\[
\xy0;/r.25pc/:
\endxy
\]

and using the fact that $\psi$ is a fibrewise isomorphism (and then just multiplying by the function $\eta$). This module map is an isomorphism satisfying many convenient properties, which are described in <cit.>.
For example, by sending the morphism $(X,E,\pi)\xrightarrow{(f,\varphi)}(Y,F,\rho)$ in $\Bun$ to the pair

\[
\Gamma(\pi)\xrightarrow{\Xi}\Gamma(f^*\rho)\xrightarrow{\Upsilon^{-1}}\Gamma(\rho)\otimes_{C^{\infty}(Y)}C^{\infty}(X)
\]

in $\int\mathfrak{E}$, the assignment $\Bun\xrightarrow{\Gamma}\int\mathfrak{E}$ defines a functor for which the diagram

\[
\xy0;/r.25pc/:
\endxy
\]

commutes. Furthermore, a cartesian lift in $\Bun\to\Man$, which corresponds to the pullback bundle, gets sent to the extension of scalars module by <cit.>. Thus, $\Gamma$ defines a fibred functor.

A similar situation happens when working with (unital) algebras and right modules over them. In particular, the analogue of this fibred functor is fully faithful when one takes the subcategory of finitely generated projective modules.

In Example <ref>, we showed that $\Mod\to\Ring$ is an opfibration. Hence, $\Mod^{\op}\to\Ring^{\op}$ is a fibration. The subcategory of cartesian morphisms in this fibration is equivalent to the subcategory of cartesian morphisms of $\int\mathfrak{E}\xrightarrow{\pi_{\mathfrak{E}}}\Ring^{\op}$ from Example <ref> by the comments in Example <ref>. More precisely, if $N\xrightarrow{\Psi}M$ is a cocartesian lift of $S\xrightarrow{\psi}R$, then there exists a canonical $R$-module isomorphism $M\xrightarrow{\Xi}N\otimes_{S}R$. This module isomorphism defines a cartesian morphism in $\int\mathfrak{E}$.

author=Antonini, Paolo,
author=Azzali, Sara,
author=Skandalis, Georges,
title=Flat bundles, von Neumann algebras and $K$-theory with
journal=J. K-Theory,

title=Green-hyperbolic operators on globally hyperbolic spacetimes,
author=Bär, Christian,
journal=Commun. Math. Phys.,

author=Basu, Devraj,
title=$K$-theory with $R/Z$ coefficients and von Neumann algebras,
pages=327–343 (2006),

author=Bunke, Ulrich,
author=Gepner, David,
title=Differential function spectra, the differential Becker-Gottlieb transfer, and applications to differential algebraic K-theory,
journal=arXiv preprint arXiv:1306.0247v2 [math.KT],

author=Bunke, Ulrich,
author=Nikolaus, Thomas,
author=Völkl, Michael,
title=Differential cohomology theories as sheaves of spectra,
journal=J. Homotopy Relat. Struct.,

author=Bunke, Ulrich,
author=Schick, Thomas,
title=Smooth $K$-theory,
language=English, with English and French summaries,
pages=45–135 (2010),

author=Bunke, Ulrich,
author=Schick, Thomas,
title=Uniqueness of smooth extensions of generalized cohomology
journal=J. Topol.,

author=Connes, Alain,
title=Noncommutative differential geometry,
journal=Inst. Hautes Études Sci. Publ. Math.,

author=Cuntz, Joachim,
author=Quillen, Daniel,
title=Algebra extensions and nonsingularity,
journal=J. Amer. Math. Soc.,

author=Freed, Daniel S.,
title=Dirac charge quantization and generalized differential cohomology,
title=Surveys in differential geometry,
series=Surv. Differ. Geom., VII,
publisher=Int. Press, Somerville, MA,

author=Freed, Daniel S.,
author=Lott, John,
title=An index theorem in differential $K$-theory,
journal=Geom. Topol.,

author=Grady, Daniel,
author=Sati, Hisham,
title=Spectral sequences in smooth generalized cohomology,
journal=Algebr. Geom. Topol.,

author=Grady, Daniel,
author=Sati, Hisham,
title=Twisted differential generalized cohomology theories and their Atiyah–Hirzebruch spectral sequence,
journal=Algebraic & Geometric Topology,
publisher=Mathematical Sciences Publishers,

author=Hopkins, M. J.,
author=Singer, I. M.,
title=Quadratic functions in geometry, topology, and M-theory,
journal=J. Differential Geom.,

author=Karoubi, Max,
title=Homologie cyclique et $K$-théorie,
language=French, with English summary,

author=Karoubi, Max,
title=Théorie générale des classes caractéristiques secondaires,
language=French, with English summary,

author=Kelly, G. Max,
author=Street, Ross,
booktitle=Category seminar,
title=Review of the elements of 2-categories,
series=Lecture Notes in Mathematics,
publisher=Springer, Berlin, Heidelberg,

author=Klonoff, Kevin Robert,
title=An index theorem in differential K-theory,
note=Thesis (Ph.D.)–The University of Texas at Austin,
publisher=ProQuest LLC, Ann Arbor, MI,

title=An introduction to noncommutative spaces and their geometries,
author=Landi, Giovanni,
publisher=Springer Science & Business Media,

author=Loday, Jean-Louis,
title=Cyclic homology,
series=Grundlehren der Mathematischen Wissenschaften [Fundamental
Principles of Mathematical Sciences],
note=Appendix E by María O. Ronco,
publisher=Springer-Verlag, Berlin,

author=Lott, John,
title=Secondary analytic indices,
title=Regulators in analysis, geometry and number theory,
series=Progr. Math.,
publisher=Birkhäuser Boston, Boston, MA,

author=Mac Lane, Saunders,
author=Moerdijk, Ieke,
title=Sheaves in geometry and logic: A first introduction to topos theory,
publisher=Springer Science & Business Media,

author=Mathai, Varghese,
author=Stevenson, Danny,
title=On a generalized Connes–Hochschild–Kostant–Rosenberg theorem,
journal=Adv. Math.,

author=Mazel-Gee, Aaron,
title=A user's guide to co/cartesian fibrations,
journal=Graduate J. Math,

author=Minasian, Ruben,
author=Moore, Gregory,
title=$K$-theory and Ramond–Ramond charge,
journal=J. High Energy Phys.,
doi = 10.1088/1126-6708/1997/11/002,

title=Monoidal Grothendieck construction,
author=Moeller, Joe,
author=Vasilakopoulou, Christina,
JOURNAL = Theory Appl. Categ.,

author=Morye, Archana S.,
title=Note on the Serre–Swan theorem,
journal=Math. Nachr.,

title=Geometry of differential forms,
author=Morita, Shigeyuki,
series=Translations of Mathematical Monographs,
publisher=American Mathematical Soc.

author=Narasimhan, M. S.,
author=Ramanan, S.,
title=Existence of universal connections,
journal=Amer. J. Math.,

author=Narasimhan, M. S.,
author=Ramanan, S.,
title=Existence of universal connections. II,
journal=Amer. J. Math.,

author=Nestruev, Jet,
title=Smooth manifolds and observables,
series=Graduate Texts in Mathematics,

publisher=Springer-Verlag, New York,

author=Byungdo Park,
title=A smooth variant of Hopkins–Singer differential $K$-theory,
journal=New York J. Math.,

title=Two-dimensional algebra in lattice gauge theory,
author=Parzygnat, Arthur J.,
journal=J. Math. Phys.,

author=Quillen, Daniel,
title=Superconnection character forms and the Cayley transform,

author=Rosenberg, Jonathan,
title=Algebraic $K$-theory and its applications,
series=Graduate Texts in Mathematics,
publisher=Springer-Verlag, New York,

author=Schreiber, Urs,
title=Differential cohomology in a cohesive infinity-topos,
journal=arXiv preprint arXiv:1310.7930v1 [math-ph],

author=Serre, Jean-Pierre,
title=Faisceaux algébriques cohérents,
journal=Ann. Math.,
YEAR = 1955,
PAGES = 197–278,
ISSN = 0003-486X,
DOI = 10.2307/1969915,

title = Holonomy, the Quantum Adiabatic Theorem, and Berry's Phase,
author = Simon, Barry,
journal = Phys. Rev. Lett.,
volume = 51,
issue = 24,
pages = 2167–2170,
year = 1983,
publisher = American Physical Society,
doi = 10.1103/PhysRevLett.51.2167,

author=Simons, James,
author=Sullivan, Dennis,
title=Structured vector bundles define differential $K$-theory,
title=Quanta of maths,
series=Clay Math. Proc.,
publisher=Amer. Math. Soc., Providence, RI,

author = Streicher, Thomas,
title = Fibred Categories a la Jean Benabou,
journal=arXiv preprint arXiv:1801.02927v11 [math.CT],
year = 2018,

AUTHOR = Swan, Richard G.,
TITLE = Vector bundles and projective modules,
JOURNAL = Trans. Amer. Math. Soc.,
VOLUME = 105,
YEAR = 1962,
PAGES = 264–277,
ISSN = 0002-9947,
DOI = 10.2307/1993627,

author=Szabo, Richard J.,
title=D-branes and bivariant K-theory,
title=Noncommutative geometry and physics. 3,
series=Keio COE Lect. Ser. Math. Sci.,
publisher=World Sci. Publ., Hackensack, NJ,

author=Tradler, Thomas,
author=Wilson, Scott O.,
author=Zeinalian, Mahmoud,
title=An elementary differential extension of odd K-theory,
journal=J. K-Theory,

author=Tradler, Thomas,
author=Wilson, Scott O.,
author=Zeinalian, Mahmoud,
title=Differential K-theory as equivalence classes of maps to Grassmannians and unitary groups,
journal=New York J. Math.,

title=Topological Vector Spaces, Distributions and Kernels,
author=Treves, Francois,
series=Pure and Applied Mathematics,

author=Witten, Edward,
title=D-branes and $K$-theory,
journal=J. High Energy Phys.,