Modular conjugations in 2D conformal field theory and holographic bit threads

We study the geometric action of some modular conjugations in two dimensional (2D) conformal field theories. We investigate the bipartition given by an interval when the system is in the ground state, either on the line or on the circle, and in the thermal Gibbs state on the line. We find that the restriction of the corresponding inversion maps to a spatial slice is obtained also in the gauge/gravity correspondence through the geodesic bit threads in a constant time slice of the dual static asymptotically AdS background. For a CFT in the thermal state on the line, the modular conjugation suggests the occurrence of a second world which can be related through the geodesic bit threads to the horizon of the BTZ black brane background. An inversion map is constructed also for the massless Dirac fermion in the ground state and on the line bipartite by the union of two disjoint intervals.


Introduction
The modular theory of Tomita and Takesaki [1][2][3][4][5], developed within the context of the algebraic quantum field theory, provides important rigorous tools to understand various aspects of the bipartite entanglement in quantum field theories.
Considering a von Neumann algebra A of observables localised in a spacetime region and in standard form (meaning that it acts on a Hilbert space H containing a cyclic and separating vector Ω), the conjugate linear operator S : AΩ → AΩ is such that S OΩ = O * Ω, for any operator O ∈ A. The unique polar decomposition of S reads S = J ∆ 1/2 , where ∆ is a self adjoint, positive (unbounded in general) operator called modular operator of (A, Ω) and J is an antiunitary operator called modular conjugation of (A, Ω).It turns out that the vector Ω is invariant, i.e. ∆Ω = Ω and JΩ = Ω, and also that J = J * = J −1 .The modular operator ∆ can be written as ∆ = e −K where K is the (full) modular Hamiltonian, a self adjoint operator whose real spectrum extends in general over R and such that Ω is one of its eigenvectors with eigenvalue equal to zero.A crucial role is played by the one-parameter group made by the unitary operators ∆ iτ with τ ∈ R. The theorem of Tomita and Takesaki tells us that ∆ −iτ A ∆ iτ = A and ∆ −iτ A ∆ iτ = A for A and its commutant A for any τ ∈ R, and that JAJ = A .This theorem leads to introduce the group of the modular automorphisms of the state Ω on A, whose generic element is the map σ τ O ≡ ∆ iτ O ∆ −iτ , for any O ∈ A, which determines the modular evolution (flow) of O generated by K.
For a local relativistic quantum field theory in its ground state whose space is bipartite by a region A and its complement B in a non trivial way, the theorem of Tomita and Takesaki can be applied and K = K A ⊗ 1 B − 1 A ⊗ K B , where K A and K B are the modular Hamiltonians of A and B respectively.Typically, σ τ and J do not have a geometric action; indeed, a direct geometric meaning for σ τ and J has been found only in very few special cases [2].The most important one corresponds to the theorem of Bisognano and Wichmann [6,7], which considers the bipartition associated to half space for a local relativistic quantum field theory in its ground state.In this case the geometric action of σ τ is given by the Lorentz boosts in the direction preserving the wedge and the geometric action of J by a reflection of the time coordinate and of the spatial coordinate orthogonal to the entangling hyperplane.Another important example is described in the theorem of Hislop and Longo [8,9], which considers a conformal field theory (CFT) in its ground state and the bipartition determined by a sphere, providing also the geometric actions for the corresponding σ τ and J. Instead, the action of σ τ and J is non geometric e.g. for a free massive field theory in its ground state when the space bipartite by a sphere [4,10,11].
In this manuscript we study the geometric action of some modular conjugations in 2D CFT by employing the results in the Euclidean spacetime discussed in [17].They include the case of a 2D CFT in a thermal state on the line bipartite by an interval and the massless Dirac fermion in the ground state on the line bipartite by the union of two disjoint intervals.
In the context of the gauge/gravity (holographic) correspondence, the prescription proposed by Ryu an Takayanagi (RT) [33,34] to evaluate the holographic entanglement entropy in asymptotically Anti de Sitter (AdS) gravitational backgrounds has provided a very important tool to explore quantum gravity through quantum information concepts in certain models.Focussing on static gravitational backgrounds, this formula claims that the entanglement entropy S A of a region A in the dual CFT on the boundary of the asymptotically AdS spacetime can be computed in such dual gravitational background as S A = Area(γ A )/(4G N ), where G N is the Newton constant and γ A (often called RT hypersurface) is the codimensiontwo minimal area hypersurface belonging to the same constant time slice of A, anchored to the boundary of A and homologous to A.
An insightful reformulation of the RT prescription has been suggested by Freedman and Headrick in terms of the holographic bit threads [35].Consider the bulk vector fields V which are divergenceless ∇ • V = 0 and norm bounded as |V | 1.By applying the Riemannian geometry version of the max-flow/min-cut theorem, it turns out that Area(γ A ) is equal to the maximum flux of one of these vector fields through the spatial region A in the boundary of the asymptotically AdS background.The integral curves (or flow lines) of this vector field are called (holographic) bit threads.In this setting, the RT hypersurface γ A is the bottleneck of the flow of a vector field satisfying this extremization condition.It is important to remark that, while such vector field is uniquely fixed on γ A , it is infinitely degenerate on A. Further properties of the holographic bit threads at some fixed time slice have been discussed in [36][37][38][39][40] and also a covariant generalisation has been recently proposed [41].Some holographic bit threads configurations have been explicitly constructed in [38] by imposing further reasonable requirements in order to reduce the above mentioned degeneracy.
Among the explicit holographic bit threads constructions proposed in [38], in this manuscript we consider the geodesic flows (where the integral lines of the holographic bit threads are also geodesics), showing that, for simple three dimensional gravitational backgrounds and for some bipartitions given by an interval, they provide the same inversion relation obtained from the geometric action of the modular conjugation for the corresponding setup in the dual CFT.
The outline of the paper is as follows.In Section 2 the main quantities considered throughout our analyses are introduced.The CFT analysis for the simplest case, which involves the bipartition of the line given by a single interval when the entire system is in the ground state, is carried out in Section 3. In Section 4 the results of the previous section are related to the geodesic bit threads in a constant time slice of Poincaré AdS 3 .In Section 5 we discuss the geometric action of the modular conjugation for a CFT in a thermal state and on the line bipartite by an interval.In Section 6, the resulting inversion map is related to the geodesic bit threads in the BTZ black brane geometry.In Section 7 we propose an inversion map for the free massless Dirac field in the ground state and on the line bipartite by the union of two disjoint intervals.Some conclusions are drawn in Section 8.In Appendix A we explore the inversion map for a CFT on a circle bipartite by an interval and in its ground state, and in Appendix B we discuss the corresponding holographic construction through geodesic bit threads in a constant time slice of global AdS 3 .

Single interval: Modular flow of a chiral field
In this section we summarise some basic results about the modular Hamiltonians in 2D CFT and their modular flows employed in the rest of the paper [8, 9, 12, 15-17, 21, 30].
In the 1 + 1 dimensional Minkowski spacetime M described by the coordinates (x, t), we consider the light ray coordinates u ± ≡ x ± t (2.1) and a right (+) and a left (−) primary chiral field φ ± (u ± ) of dimension h.Such fields satisfy the following commutation relation [42] T where T ± are the chiral components of the energy-momentum tensor.For the cases we are interested in, the right and left modular Hamiltonians of a single interval A = [a, b] can be written in terms of T ± as follows where the explicit form of the function w depends on the state of the entire system.For instance, in the ground state we have At finite temperature 1/β, we consider a thermal state for which the mean energy density satisfies the Stefan-Boltzmann law [43] T tt (t, x) ≡ 1 2 where c is the central charge of the 2D CFT.According to [28,29], for 0 < c < 1 this state is unique, while for c 1 the condition (2.6) selects a specific (geometric) thermal state.In this case The modular evolution (flow) of φ ± , generated by (2.3), reads (see e.g. the Appendix B of [21]), where τ ∈ R is the modular parameter and the function ξ(τ, u) is fully determined by the function w as follows where u ∈ A. Since ξ(0, u) = u for any u ∈ A by construction, the fields in (2.8) satisfy the initial condition φ ± (0, u ± ) = φ ± (u ± ).
In our analyses we also employ the modular evolution in Euclidean spacetimes described in [17], which is given by a complex function related to (2.9) as follows where x ∈ A and θ ∈ [0, 2π).
A fundamental characteristic feature of any modular flow is the Kubo-Martin-Schwinger (KMS) condition [2].In order to check that the flow (2.8) satisfies this condition, consider the two-point functions where x 1 and x 2 belong to A. The KMS condition implies which can be verified deriving the expectation values (2.11) in explicit form.To deal with the left and right movers simultaneously, let us adopt the notation From (2.8), for the two-point function along the modular flow one obtains where the expectation value in the r.h.s. is determined by the choice of representation for φ.
We observe also that, in addition to (2.12), the thermal two-point function (2.17) satisfies the KMS condition in the physical time which involves β.
In the 1 + 1 dimensional Minkowski spacetime in the coordinates (x, t), let us consider the domain of dependence D A of the interval A = [a, b] ∈ R (also known as diamond or double cone).A modular trajectory in D A is made by the points whose spacetime coordinates are in terms of (2.9), where τ ∈ R and u ±,0 are the light ray coordinates of the point at τ = 0.It is often convenient to describe a modular trajectory in terms of the point x ∈ A at t = 0. Setting τ = 0 at this point, this modular trajectory is given by (x(τ ), t(τ )) ∈ D A with in terms of (2.9), where τ ∈ R.

Single interval: Ground state
In this section we focus on a CFT in the ground state and on the line bipartite by an interval A = (a, b) ⊂ R and its complement.

Internal modular evolution
The function w to consider in this case is given in (2.5) [8,12,[15][16][17].Specifying (2.9) to this function, for each chirality we obtain the following geometric action of the modular operator [2,8,9] which is also known as the geometric action of the modular automorphism group of the diamond D A induced by the vacuum state.The expression (3.1) must be employed for both the chiralities and it provides the modular trajectories in the diamond D A through (2.19), or (2.20) equivalently.
< l a t e x i t s h a 1 _ b a s e 6 4 = " f a + 8 y t B 5 c g s z / 2 h 3 I 5 q E K D T l R K m W 6 8 T a S 4 n U j H I c 5 d q J w p j Q A e l h y 6 A g I S o v n Q w 6 s g + N 0 7 W D S J o n t D 1 x f 3 a k J F R q G P q m M i S 6 r + a z s f l f 1 k p 0 c O a l T M S J R k G n H w U J t 3 V k j 7 e 2 u 0 w i 1 X x o g F D J z K w 2 7 R N J q D a 3 y Z k j u P M r / 4 X 6 c d E 9 K T p V p 1 A 6 h 6 m y s A 8 H c A Q u n E I J y l C B G l B A u I d H e L J u r A f r 2 X q Z l m a s W c 8 u / J L 1 + g 2 z q p C L < / l a t e x i t > b < l a t e x i t s h a 1 _ b a s e 6 4 = " 0 H < l a t e x i t s h a 1 _ b a s e 6 4 = " m 2 d F n d + v A S e v y I P p u e P t P 8 6 / f t k = " > A A A B 5 H i c b V B N S 8 N A E J 3 U r x q / q l c v i 0 X w V B J B 1 J M F L x 4 r 2 A 9 o Q 9 l s J + 3 a z S b s b o Q S + g u 8 e F C 8 + l P 8 D d 7 8 N 2 7 T H r T 1 w c D j v R l m 5 o W p 4 N p 4 3 r d T W l v f 2 N w q b 7 s 7 u 3 e X H e 5 4 0 l Z z F x B H / g f P w A Y 2 m M h A = = < / l a t e x i t > 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " 8 5 u j u R V 4 0 t E b q P l F Y s j L E e B P c g 0 |⌧m| < l a t e x i t s h a 1 _ b a s e 6 4 = " j L j 0 0 D j D u i J K J 5 2 Y r g D a 5 B i 9 l y X p 2 3 a e u S M 5 v Z h z 9 w 3 n 8 A q y G T V Q = = < / l a t e x i t > |⌧m| < l a t e x i t s h a 1 _ b a s e 6 4 = " The important case considered by Bisognano and Wichmann [6,7] corresponds to the limiting regime where the subsystem A becomes the semi-infinite line x > 0. Setting a = 0 first and then taking b → +∞ in (3.1), one finds ξ(τ, x) = x e 2πτ with x > 0, i.e. the dilations of the semi-infinite line parameterised by τ .
The action of the modular operator and of the modular conjugation in the Minkowski space can be found by combining the corresponding actions for the two chiralities.In the right panel of Fig. 1, the grey region is the domain of dependence D A of the interval A = [a, b] ∈ R, which is also called diamond or double cone.The null rays departing from its central point ( a+b 2 , 0) (dot-dashed segments in D A ) provide the following partition of D A into four smaller diamonds where in the r.h.s. the letter in the subindex stands for either right (R) or left (L) or future (F) or past (P), depending on the position of the corresponding diamond with respect to the central point ( a+b 2 , 0) of D A .The latter point is the only one which provides a partition of D A in four diamonds; indeed, the partitions of D A induced by the other points include at least two rectangular regions.The solid line inside D A is the modular trajectory obtained from (2.20), with ξ(τ, x) and ξ(−τ, x) given by the solid lines shown in left panel of the same figure (an arc having certain colour in the left panel provides the arc with the same colour in the right panel).The different arcs of the modular trajectory in D A can be identified through their intersections with the light rays departing from (c, 0) (dotted segments in D A ) and from the mid-point of the interval at t = 0 (dot-dashed segments in D A ).The modular parameter τ grows along the modular trajectory as indicated by the arrows.

Geometric action in Minkowski spacetime
For a CFT in the ground state and on the line bipartite through an interval, the geometric action of the modular conjugation is the following inversion map (x, t) 3) which is idempotent, as expected.
In terms of the light ray coordinates (2.1), the geometric action (3.3) leads to define the inversion map z inv as follows which allows to write (3.3) as Another insightful form of the geometric action of the modular conjugation can be written in terms of the distance from the mid-point of A. Indeed, by introducing where ≡ b − a is the length of the interval A, whose endpoints are the entangling points.
When t = 0, the inversion map (3.3) simplifies to which sends A into its complement on the line, and viceversa.In particular, z inv (x) < a when x ∈ (a, a+b 2 ) and z inv (x) > b when x ∈ ( a+b 2 , b).We remark that z inv (x) in (3.8) sends a point in A close to an entangling point into a point in B close to the same entangling point, and viceversa.
In the left panel of Fig. 1 the dot-dashed curves correspond to ξ(τ, z inv (x)) and ξ(−τ, z inv (x)), with τ ∈ R, which intersect at τ = 0 and z inv (x) ∈ B. These curves and the corresponding ones for ξ(τ, x) and ξ(−τ, x) in the grey region are partitioned into five arcs (indicated with different colours) at the values τ = ∓|τ c | and τ = ∓|τ m | of the modular parameter.Notice that, while |ξ(τ, x)| remains finite, we have that ξ(τ, In the right panel of Fig. 1, the light blue region W A can be partitioned into four infinite triangular domains as follows (the notation introduced in (3.2) has been adopted also here) where the subindex indicates the position of the vertex of the corresponding infinite triangle with respect to the center of D A .The vertices of the left spacelike wedge W L and the right spacelike wedge W R are (a, 0) and (b, 0) respectively, while the forward light cone V F and backward light cone respectively.The geometric action of the modular conjugation given by (3.3), or equivalently by (3.4) in the light ray coordinates, maps D A into the light blue region and viceversa (see also Fig. ).The mid-point ( a+b 2 , 0) is sent to infinity by this map.Considering the geometric action (3.3) separately on each element of the partitions of D A and W A introduced in (3.2) and (3.9), one finds that it relates domains that share a vertex (hence with the same subindex).
The modular trajectory corresponding to the solid curve in D A passes through (x, 0) when τ = 0 and is mapped into the dot-dashed curve in the light blue region.The modular parameter τ grows along these curves as indicated by the arrows.Hence, both the modular trajectory in D A and its dot-dashed image in W A originate in the lower vertex of D A as τ → −∞ and end in the upper vertex of D A as τ → +∞.Each arc in the modular trajectory having a certain colour is sent by (3.3) to the arc on the dot-dashed curve having the same colour and they both correspond to the same range of the modular parameter τ .As already remarked, the partition of the modular trajectory into the five arcs with different colours is determined by its intersections with the light rays departing from the mid-point of the interval P m = ( a+b 2 , 0) (black dot-dashed segments in D A ) and from the generic point P c = (c, 0) in the interval (black dotted segments in D A ) at t = 0.The same holds for the image of the modular trajectory under (3.3) in the light blue region, with the obvious difference that its partition is determined by the light rays departing from the images of P m and P c under the same map.Since this map sends P m to infinity, the red, purple and magenta dot-dashed arcs in the light blue region reach infinity in W R , V P and V F respectively.In particular, the red dot-dashed arc in W A entirely belongs either to W R or to W L , depending on whether x > a+b 2 or x < a+b It is worth considering two special bipartitions of the line.When a = − b, the interval A is centered in the origin of the x-axis and the inversion map which further simplifies for t = 0 to the following suggestive form When b → +∞, the interval A and its diamond D A become respectively the half line x a and the right Rindler wedge with vertex in (a, 0).The inversion map (3.3) simplifies to the modular conjugation for such Rindler wedge, which reads [2] In terms of the light ray coordinates (2.1), taking b → +∞ in (3.4), for this map one obtains which is consistent with (3.12), as expected.
The above results can be extended in a straightforward way to the case in higher dimensions where the subsystem is a spatial sphere in the Minkowski spacetime [2,8], by employing the light cone coordinates r ± t, being r the radial coordinate when the origin coincides with the center of the sphere.

Geometric action in the Euclidean spacetime
In [17] the modular evolution in the Euclidean spacetime for a 2D CFT in its ground state and on a line bipartite by an interval A = (a, b) has been described in terms of the function w(x) given in (2.5) through the following complex function which can be obtained also by replacing 2πτ with iθ in (3.1).This complex function can be equivalently written in the form where (3.16) Hence, in the complex plane described by the complex coordinate z, the curve z(θ, x) corresponding to an assigned value x ∈ A and parameterised by θ ∈ (0, 2π) is a circle with radius R centered in the point (C, 0) on the real axis.From (3.14), notice that z(θ, a) = a and In Fig. 2, considering the complex plane described by the complex variable z (grey plane), we show the half circles obtained from (3.14) (or (3.15) equivalently) for various assigned x ∈ A and parameterised either by θ ∈ (0, π) (solid arcs) or by θ ∈ (π, 2π) (dot-dashed arcs).In the same complex plane, it is useful to explore also the curves given by (3.14) (or (3.15) equivalently) parameterised by x ∈ A for some fixed θ ∈ (0, 2π).For these curves we have that ∂ x z(θ, x) → e iθ as x → a and ∂ x z(θ, x) → e −iθ as x → b; hence, the curve that intersects orthogonally the real axis at x = a and x = b corresponds to θ = π/2.From (3.14) one finds that such curve is the half circle centered in the mid-point of A (see also Sec. 4 and Fig. 3).
The inversion map (3.4) naturally provides the complex map z inv : C → C given by which is a rational function with real parameters.By introducing the real variable t E ∈ R and exploiting (3.5), we can relate (3.17) to the inversion map (3.3) by replacing t with it E , finding The inversion on the real line given in (3.8) can be obtained by restricting (3.17) to the real line or by evaluating (3.14) at θ = π, which is equivalent to (2.9) evaluated at τ = i/2, namely This tells us that, in Fig. 2, the endpoints of a black solid half circle in the upper half plane (or of the corresponding black dotted half circle in the lower half plane) belong to the real axis and are related through the inversion map (3.8).Hence, any point in the interval A (red segment) is uniquely mapped into a point in its complement on the line (union of the blue half lines), and viceversa.From (3.20), we also observe that (3.14) e (3.17) are related as follows Finally, by combining (3.20) with (3.5) and (3.14) (or (2.9)), one observes that the geometric action of the modular conjugation (3.3) can be expressed through w(x) in (2.5).
4 Geodesic bit threads in Poincaré AdS 3 In this section we relate the geodesic bit threads in Poincaré AdS 3 for the RT curve corresponding to an interval on the line to the results discussed in Sec.3.2 and Sec.3.3.
The Poincaré AdS 3 can be described by the coordinates (t, x, ζ), where ζ > 0 parameterises the holographic direction, hence its boundary corresponds to ζ = 0.A t = const slice of Poincaré AdS 3 is the upper half plane described by the coordinates (x, ζ) and equipped with the following metric (yellow region in Fig. 3) In this Euclidean manifold, the set of geodesics having both the endpoints on the boundary is the two-parameter family of half circles centered on the real axis, namely The vertical lines orthogonal to the boundary provide the remaining set of geodesics.According to the AdS/CFT correspondence, on the boundary of the Poincaré AdS 3 we have a dual 2D CFT on the real line and in its ground state.
In order to recover the same situation investigated in Sec. 3, the spatial direction on the boundary is bipartite by an interval of length ≡ 2b (red segment in Fig. 3) and its complement (union of the blue half lines in Fig. 3).The interval can be placed symmetrically with respect to the origin (i.e.A = {x ; |x| b}) without loss of generality.The RT formula [33,34] provides the holographic prescription to evaluate the holographic entanglement entropy through a bulk computation.For static three dimensional gravitational spacetimes and a single interval A in the dual CFT, the RT prescription requires the minimal length curve anchored to the endpoints of A (the RT curve, denoted by γ A in Sec. 1) because its regularised length multiplied by 1/(4G N ) provides the holographic entanglement entropy.In a fixed time slice of Poincaré AdS 3 and for the interval A = (−b, b), the RT curve is the half circle of radius b centered in Figure 3: Geodesic bit threads (green dashed curves) in a fixed time slice of Poincaré AdS 3 (yellow region) associated to an interval on the line (red segment), whose RT curve is the red solid half circle anchored to its endpoints.the origin (red solid curve in Fig. 3), whose generic point P m (that must not be confused with the point indicated in the same way in Sec.3.2) have coordinates (x m , ζ m ) satisfying It is worth reminding that also the curve z(θ = π/2, x) for x ∈ A obtained from (3.14) is the half circle centered in the mid-point of A in the complex plane, as already noticed in Sec.3.3.
Among the holographic bit threads introduced in [35], let us consider the specific configuration given by the geodesic bit threads discussed in [38].A geodesic bit thread for a RT curve is a geodesic such that it connects a point x ∈ A to a point that does not belong to A and intersects the RT curve orthogonally.At such intersection point, its unit norm tangent vector coincides with the unit norm outward-pointing vector normal to the RT curve.
For the RT curve (4.3), the generic geodesic bit thread is (4.2) with [38] where (x m , ζ m ) are the coordinates of the intersection point between the geodesic bit thread and the RT curve, hence they are constrained by (4.3).Some geodesic bit threads obtained by combining (4.2), (4.3) and (4.4) are shown in Fig. 3 (green dashed half circles).
We find it more convenient to parameterise a geodesic bit thread in terms of the coordinate x A ∈ A of its endpoint in A. The coordinate x m of the intersection point between the geodesic bit thread and the RT curve is related to x A as follows5 which leads to write (4.4) as implying that (R s , x s ) = (0, ±b) when x A = ±b, as expected.From (4.6), one obtains Plugging (4.6) into (4.2),we find that the geodesic bit thread anchored to x A ∈ A reads From (4.7) or (4.8), we have that the geodesic bit thread intersects the AdS boundary in A at x = x A , as expected by construction, and in the complement of A on the line at x = x B , where This relation between the endpoints of a geodesic bit thread coincides with the inversion relation on the real axis for t = 0 provided by the geometric action of the modular conjugation for the interval A = (−b, b), which has been reported in (3.11).Furthermore, let us observe that a geodesic bit thread anchored at x A ∈ A (see (4.8) and the green dashed curves in Fig. 3) and the trajectory corresponding to the Euclidean modular evolution for θ ∈ (0, π) that starts at the same x A when θ = 0 (see (3.15) and the black solid curves in Fig. 2) are described by the same curve.Indeed, they are both half circles whose endpoints are related in the same way.However, these two curves belong to different Euclidean manifolds.
Finally, comparing (4.4) and (4.9), we find it worth remarking that the same inversion relation occurs between x A and x B and between x m and x s .
We remind that the contour function for the entanglement entropies [44,45] can be associated holographically to the restriction on the boundary of the norm bounded and divergenceless vector field providing the holographic bit threads [46].In particular, from the vector field for the geodesic bit threads in Poincaré AdS 3 found in [38], the function β 0 (x) for a CFT in the ground state (see (2.4) and (2.5)) is obtained [47].
A straightforward extension of the above analysis allows to address the case of a CFT in the ground state and in the higher dimensional Minkowski spacetime bipartite by a sphere of radius b.For this bipartition it is convenient to adopt the polar coordinates in the Minkowski spacetime with the origin in the center of the sphere.In this case, the geodesic bit threads are obtained from the expressions discussed above restricted to x > 0, which corresponds to the radial coordinate.

Single interval: Thermal state
In this section we consider a CFT in a thermal state at temperature 1/β and on the line bipartite by an interval A = (a, b) and its complement.This thermal state is the geometric one within the analysis performed in [28,29], which satisfies the Stefan-Boltzmann law (2.6).In this case the function w(u) to employ is reported in (2.7) [16,17].The KMS condition is fulfilled, both in the modular time and in the physical time (see (2.12) and (2.18) respectively).

Inversion on the real line
By adapting the observation (3.20), made for the ground state, to a CFT at finite temperature and on the bipartite line, from (5.1) or (5.2) we introduce the following inversion map e π(b−a)/β − e w(x) . (5.5) In this expression the argument of the logarithm becomes negative for some values of x ∈ A. The range of these values can be found by observing that the denominator and the numerator of the ratio in the argument of the logarithm in (5.5) vanish when x is equal respectively to where The argument of the logarithm in (5.5) is negative when x ∈ A β , where A β ≡ (a β , b β ), which is a proper subinterval of A of length β .From (5.6) and (5.7), one observes that A β and A have the same mid-point.Thus, when a = −b, also a β = −b β .
From (5.8), we have that β → 0 as β → +∞ and β → as β → 0; namely A β disappears at zero temperature while it coincides with A in the large temperature regime.
In the limit where A becomes the half-line, which is obtained by setting b = a + first and then taking → +∞, for the endpoints of A β in (5.6) and (5.7) we find that a β → a+ β 2π log(2) and b β → +∞ respectively.Now, taking a → −∞ in this result, A β becomes the whole line.This is found also by taking b → +∞ and a → −∞ simultaneously in (5.6) and (5.7).
Let us stress that A β has been introduced through the inversion map (5.5).It would be insightful to find a footprint of A β without invoking the modular conjugation.
Since the argument of the logarithm in (5.5) is negative when x ∈ A β , the inversion (5.5) is a complex map that can be written as follows Combining the two chiralities, in Sec.5.3 we will see that the imaginary part occurring in the complex map z inv (x) : A β → R cancels and therefore naturally provides a second Minkowski space M, which has been called second world in [24] or shadow (virgin) world in [25].We remark that this second world M, differently from the real one M, is not bipartite.
In the special case of A = (−b, b), the inversion map (5.10) simplifies to (5.11)This result will be re-obtained holographically also through the geodesic bit threads in the BTZ black brane background in Sec. 6.
In the zero temperature limit β → +∞, we have that A β → ∅, as remarked above; hence only the first expression in (5.10) must be considered, which becomes (3.8) in this limit, as expected.
< l a t e x i t s h a 1 _ b a s e 6 4 = " H Y T Q H e I J n 6 8 Z 6 t F 6 s 1 2 l p z p r e s A t / Z L 1 9 A + V s k 1 k = < / l a t e x i t > x Figure 4: The real part of the inversion map (5.10).The green segment corresponds to A β , the red segments to A \ A β and the blue half lines to B.
In the limiting regime b → +∞, the interval A becomes the half-line (a, +∞) and, by using also the above results a β → a + β 2π log(2) and b β → +∞, one finds that the map (5.10) simplifies to (5.12) which is real in A \ A β = (a, a + β 2π log(2)) and sends this limited interval onto the half line B = (−∞, a) complementary to A in a bijective way.By employing the exponential map characterising the thermal state, we can write (5.12) in the following suggestive form e 2π[z inv (x)−a]/β = 2 − e 2π(x−a)/β (5.13) which becomes (3.13) in the zero temperature limit, as expected.In the non negative variable y ≡ e 2π(x−a)/β , this map becomes to the reflection y → 2 − y with respect to 1.The reflection (5.13) suggests to continue this real positive variable y by including also the semi-axis made by the negative real numbers y < 0, which corresponds to the second world [24,25].Indeed, such reflection maps the points y > 2 into the negative real semi-axis in a bijective way.
In order to explore the meaning of the sub-interval A β , we find it worth exploiting the contour function for the entanglement entropies [44,45].For a CFT at finite temperature on the line bipartite by an interval, this function reads [45] where w(x) is given by (2.7) and C is a non universal constant such that s (n) A (x) > 0 for every x ∈ A. The main feature of the positive function (5.14) is that the entanglement entropies of A found in [48] are obtained by integrating it on A ≡ (a + , b − ), where is the UV cutoff, i.e.

S (n)
up to O(1) non universal contributions as → 0. In [48] it has been remarked that, when β, the entanglement entropy obtained from (5.15) becomes extensive and provides the Gibbs entropy S A π c /(3β) of an isolated system of length coming from the Stefan-Boltzmann law for a 2D CFT (see (2.6)) [43,49,50].
It is sometimes insightful to integrate the contour function for the entanglement entropies in (5.14) over a subsystem properly included in A (see e.g.[51]).In the case that we are exploring, the integration of (5.14 where β has been introduced in (5.8).When n = 1 and C = 0, the expression (5.16) provides the above mentioned Gibbs entropy π c /(3β) for any value of and β.
Considering the sub-interval A β , from (2.7) and its endpoints (5.6) and (5.7), one finds which can be interpreted through the modular parameter τ .Indeed, by considering (2.9) for x = a β , the expression (5.17) corresponds to the value τ β of the modular parameter such that ξ(τ β , a β ) = b β .As a consequence, also (5.16) can be interpreted through the modular parameter τ because it is proportional to (5.17) when C = 0.

Geometric action in Euclidean spacetime
In the finite temperature case that we are investigating, the geometric action of the modular conjugation can be studied by adapting the observations made in Sec.3.2 and Sec.3.3 for the ground state.It is more instructive to discuss first the Euclidean spacetime.
For a CFT at finite temperature 1/β on the line, the Euclidean spacetime to consider is the infinite cylinder whose section along the compactified direction is a circle having length equal to β.It can be equivalently represented by the horizontal strip {z ∈ C; |Im(z)| β/2} in the complex plane (parameterised by z) with vertical width equal to β (grey region in Fig. 5), where the horizontal lines at Im(z) = ±β/2 delimiting the strip (horizontal black dashed lines in Fig. 5) are identified.The space where the CFT is defined corresponds to the real axis Im(z) = 0, which is bipartite by the interval A and its complement.This bipartition is characterised by the entangling points x = a and x = b on the real axis, which correspond to the points where a red segment and a blue segment join in Fig. 5; hence A is the union of the two red segments and the green segment, while B corresponds to the union of the blue half lines.
The curves shown in Fig. 5 are obtained from (5.2) for some values of x ∈ A: the solid ones have θ ∈ (0, π), while θ ∈ (π, 2π) for the dotted ones.Any black solid arc (or equivalently the Figure 5: Euclidean modular evolution for a CFT at finite temperature 1/β on the bipartite line, obtained from (5.2), with θ ∈ (0, π) (solid arcs) or θ ∈ (π, 2π) (dotted arcs), for some x ∈ A. The green segment corresponds to A β .The solid black curves (or the dotted black ones equivalently) map A \ A β into B in a bijective way, while the solid yellow curves (or the dotted yellow ones equivalently) map A β into the horizontal dashed black line at Im(z) = +β/2 (which is identified with the horizontal dashed black line at Im(z) = −β/2) in a bijective way.dotted one having the same endpoints) associates a point in the red segments to a point in B in a unique way.Instead, any yellow arc (or equivalently the dotted one having the same endpoints) associates a point in the green segment to a point on the horizontal black dashed lines at Im(z) = ±β/2 (which are identified).This map is (5.10) and the green segment in Fig. 5 denotes the sub-interval A β .The two orange curves in Fig. 5 intersect the real axis at x = a β or x = b β and connect these points to −∞ or +∞ respectively; hence they provide the extremal lines separating the set of the black curves and the set of the yellow curves, whose behaviours are qualitatively different.In this Euclidean setup, the horizontal black dashed lines at Im(z) = ±β/2 corresponds to the space of the second world [24,25] introduced in Sec.5.1.
In this Euclidean strip, it is useful to consider also the curves obtained from (5.2) at fixed θ ∈ (0, 2π), parameterised by x ∈ A. For these curves, we find that ∂ x z(θ, x) → e iθ as x → a and ∂ x z(θ, x) → e −iθ as x → b; hence the curve corresponding to θ = π/2 intersects orthogonally the real axis at x = a and x = b.This result is independent of β and for the ground state it has been already remarked in Sec.3.3.

Geometric action in Minkowski spacetime
In the Minkowski spacetime M parameterised by the coordinates (x, t), let us consider the diamond D A (see Sec. 3.1).The occurrence of the sub-interval A β introduced in Sec.5.1, whose endpoints on the spatial line are (5.6) and (5.7), naturally leads to the following partition identified by the null rays departing from the points (a β , 0) and (b β , 0) (see Fig. 6).In (5.18), D A β is the domain of dependence (diamond) of A β (green region in Fig. 6), while D A and R A are defined as follows The domains D k with k ∈ {R, L, F, P} are the dark grey diamonds in Fig. 6 and the letters for the subindex, which stand for right, left, future and past respectively (see also (3.2)), indicate their position with respect to D A β .Each of these diamonds shares a vertex with D A β .The domains R rs with r, s ∈ {+, −} are the light grey rectangles in Fig. 6 and the subindices indicate their position with respect to D A β (e.g.++ and +− correspond to the top right and top left light grey rectangle respectively).
In the limit where A becomes the half line, D A and D A β become the right wedges with vertex in (a, 0) and (a + β 2π log(2), 0) respectively.Furthermore, only D L remains a finite diamond in this limit, while the other dark grey regions D k with k = L disappear at infinity.
In the zero temperature limit β → +∞, we have that D A β and R A become zero measure regions in Minkowski space and D A → D A .Moreover, considering the partitions of D A introduced in (3.2) and (5.19), in this limit we have that D k → D k for each k ∈ {R, L, F, P}, up to zero measure domains.
A modular trajectory in Minkowski space is obtained by combining (5.1) and (2.20).It depends on the assigned value x ∈ A at τ = 0 and it entirely belongs to D A .The partition (5.18) of D A induces a partition of a modular trajectory into (at most) five arcs corresponding to its intersections with D A β , D A and R A (arcs denoted with different colours that partition each solid curve in Fig. 6).The only exception is the modular trajectory corresponding to x = a+b 2 , which is the vertical segment connecting the top and bottom vertices of D A ; indeed, its partition is made only by three vertical segments because it does not intersect R A .The five arcs partitioning the modular trajectory can be found also through the corresponding values of the modular parameter.The ranges of τ that provide the different arcs are identified by ±|τ a β | and ±|τ b β |, where A modular trajectory is characterised by its point (x, 0) with x ∈ A at τ = 0.At finite temperature, we have either x ∈ A \ A β (left panel in Fig. 6) or x ∈ A β (right panel in Fig. 6).Considering the partition (5.18), notice that, while all the modular trajectories have non vanishing intersections with D A ad R A , only those ones corresponding to x ∈ A β intersect also D A β .Figure 6: Modular trajectories (solid lines) in D A passing at some x ∈ A when t = 0 and their images (dot-dashed and dashed lines) under (5.23).Left: two values of x ∈ A \ A β .Right: three values of x ∈ A β .The arcs denoted through different kind of lines but having the same colour are related through (5.23).While the dot-dashed lines span W A (light blue region, see (3.9)) as x ∈ A \ A β , the dashed lines span the entire Minkowski spacetime (the second world M) as x ∈ A β .
In order to adapt to the thermal case the results discussed in Sec.3.2 for the ground state (see (3.3) and (3.5)), we introduce the geometric action of the modular conjugation as follows where the function z inv (x) is given by (5.9).In terms of the light ray coordinates (2.1), it is convenient to introduce u ±,inv ≡ x inv ± t inv , finding that From (5.21) we obtain t inv (x, t) = β 4π log e π(b+a)/β − e 2πb/β e w(x+t) e π(b−a)/β − e w(x−t) e π(b−a)/β − e w(x+t) e π(b+a)/β − e 2πb/β e w(x−t) . ( This map satisfies some consistency relations.For t = 0 we have t inv (x, t = 0) = 0, as expected, and, interestingly, x inv (x, t = 0) = Re[z inv (x)], which is shown in Fig. 4. In the zero temperature limit β → +∞, the inversion map (3.3) is recovered, as expected.We remark that (5.23) has been obtained by merging the two logarithmic functions coming from z inv (x + t) and z inv (x − t) into a single logarithmic function and that such seemingly innocent simplification leads to interesting consequences, as discussed below.
Consider the partitions in (5.18) and (5.19), together with the partition (3.9) for the light blue region in both panels of Fig. 6.When (x, t) ∈ D A , the inversion map (5.21) is equivalent to (5.23) and it defines the four bijective maps The action of the map (5.23) in D A β and R A is very interesting.When (x, t) ∈ D A β , the arguments of the logarithm in z inv (x + t) and z inv (x − t) are both negative; hence (5.21) gives Im(t inv ) = 0 and |Im(x inv )| = β/2.Instead, in (5.23) the arguments of the logarithms remains positive; hence (5.23) is a real map from D A β to the entire Minkowski spacetime M, which is identified with the second world.This is an interesting consequence of the combination of the two different chiralities; indeed, they separately provide results with non vanishing imaginary parts.Notice that, differently from M, the space direction of M is not bipartite.In the right panel of Fig. 6, a dashed curve of certain colour is the image under (5.23) of the solid arc in D A β having the same colour.As x ∈ A β , these dashed curves span the entire Minkowski spacetime M (the second world).Hence, the structure of the "real world" is transported with the help of the conjugation J to the "second world", as Borchers observed in [24].
The above discussion tells us that the full modular Hamiltonian in M ∪ M reads (5.24) In the zero temperature limit β → +∞, we have that the second world M disappears and (5.24) tends to K = K A ⊗1 B −1 A ⊗K B , where K B = JK A J, being J the modular conjugation whose geometric action has been described in Sec. 3.
Instead, the image of a point (x, t) ∈ R A under (5.21) or (5.23) has |Im(t inv )| = |Im(x inv )| = β/4; hence the reality condition suggest to consider this result unphysical.In Fig. 6, this implies that the cyan arcs do not have a dot-dashed or dashed counterpart.
When A becomes the half line, i.e. in the limit b → +∞, the inversion (5.23) simplifies to whose zero temperature limit gives (3.12), as expected.
In terms of the light ray coordinates u ± and of u ±,inv ≡ x inv ± t inv , the map (5.25) can be written in the form (5.12) or (5.13), as expected.

Geodesic bit threads in BTZ black brane
In this section we show that the inversion map on the real line discussed in Sec.5.1 can be found through the geodesic bit threads in the BTZ black brane geometry described in [38].
The BTZ black brane is a three dimensional gravitational background solving the Einstein equations with negative cosmological constant and without matter term.A horizon occurs and it is a two dimensional plane.This geometry can be described in terms of the coordinates (t, x, ζ), where t is the time coordinate and ζ > 0 is the holographic coordinate.The boundary at ζ = 0 is a 1 + 1 dimensional Minkowski spacetime M in the coordinates (x, t).A constant time slice of the BTZ black brane geometry is equipped with the following induced metric where ζ h denotes the position of the horizon.This two dimensional geometry is considered in Fig. 7, where the yellow region corresponds to 0 < ζ < ζ h and the grey region to ζ > ζ h .The AdS/CFT correspondence relates ζ h to the temperature 1/β of the dual 2D CFT in the Minkowski spacetime at the boundary of the BTZ black brane as follows Consider the bipartition of the spatial line of the dual CFT given by an interval, which can be set to A = (−b, b) ⊂ R without loss of generality (see the union of the red segments and of the green segment in Fig. 7, while the union of the blue half lines is the complement of A on the line), as already done in Sec. 4. In the geometry (6.1), the minimal length curve anchored to the endpoints of A (the RT curve) is made by the points P m (that must not be confused with the point indicated in the same way in Sec.3.2) whose coordinates (x m , ζ m ) satisfy the following constraint where ζ * denotes the maximum depth of this geodesic, which can be expressed in terms of ζ h and b as follows In Fig. 7, the RT curve (6.3) corresponds to the red solid curve in the yellow region.This RT curve does not coincide with the one discussed in the last paragraph of Sec.5.2.These two curves overlap in the zero temperature limit, as remarked in Sec. 4.
The geodesic bit threads for the BTZ black brane geometry (6.1) and the RT curve (6.3) have been found in [38].In the following we report some of their results, trying to keep their notation wherever it is possible.Some geodesic bit threads (dashed lines) are shown in Fig. 7 (see also the left panel of Fig. 4 in [38]).A crucial feature of these geodesic bit threads is that those ones whose intersection with the RT curve (6.3) Figure 7: Geodesic bit threads in BTZ black brane geometry for an interval on the line.The red solid curve is the RT curve (6.3).The green segment corresponds to A β (see Sec. 5.1).
have both the endpoints on the boundary at ζ = 0; hence they connect a point A to a point in its complement B on the line (green dashed curves in Fig. 7).Instead, the geodesic bit threads intersecting the RT curve (6.3) in a point whose coordinate x m satisfies |x m | x m,max reach the horizon; hence they connect a point in A to a point on the horizon (grey dashed curves in Fig. 7).
A convenient parameterisation for the geodesic bit threads connecting A to its complement on the real axis is where the constants C 1 , C 2 and C 3 are defined respectively as ) being s m the slope of the outward-pointing normal vector at a given point (x m , ζ m ) of the RT curve (6.3), that reads As for the geodesic bit threads connecting a point in A to a point of the horizon, they can be written also as being the constants D 1 and σ defined as follows where σ and s m must have opposite signs.Both the parameterisations (6.6) and (6.9) can be adopted to describe each one of the two different types of geodesic bit threads, by employing two branches if necessary.
In Fig. 7 we show some geodesic bit threads (dashed lines in the yellow region): the green dashed lines connect a point in A to a point in B, hence both these points are on the boundary; instead, the grey ones connect a point in A to a point on the horizon.The magenta dashed curves denote the geodesic bit threads corresponding to the transition between these two different behaviours, hence they intersect the horizon (or the boundary) at infinity.The thermal sub-interval A β ⊂ A introduced in Sec.5.1, whose endpoints are (5.6) and (5.7), can be found also through the geodesic bit threads in the BTZ black brane geometry described above.Consider the transition geodesic bit thread given by (6.9) with σ = −1 (hence s m > 0) and having x m = x m,max in (6.5), which corresponds to the magenta dashed curve on the right in Fig. 7. Setting ζ = 0 in the resulting expression, we recover b β in (5.7) in the special case of a = − b.Because of the symmetry of A, the magenta dashed curve on the left in Fig. 7 provides a β in (5.6) when a = − b.Thus, the green segment on the boundary in Fig. 7, obtained holographically through the geodesic bit threads corresponding to the magenta dashed curves, can be identified with A β , which is also the green segment on the real line in Fig. 5 and the segment on the real axis providing the green diamond in Fig. 6.
Plugging the expression for ζ * in (6.4) into (6.5) and comparing the result with b β in (5.7) specialised to the case a = −b, we observe that .11)This interesting holographic relation deserves some interpretation.
Also the inversion map (5.11) can be interpret through the geodesic bit threads.
Consider first a point on the boundary at x A ∈ A\A β , which is therefore connected through a geodesic bit thread described by (6.6) to another point on the boundary at some x B ∈ B. In order to find x B in terms of x A , notice that (6.6) vanishes for x = x ± given by where, from (6.7), we have Since ζ h > ζ m , the constant C 1 and both the expressions in (6.13) are positive quantities; hence one concludes that x + > x − , from (6.12).The symmetric choice of A with respect to the origin allows us to assume x A > b β > 0 without loss of generality; hence x A = x − and x B = x + (see Fig. 8).
To express x m in terms of x A = x − , first employ (6.8) and then (6.3) into x − given by (6.12); and finally invert the resulting expression.This gives different solutions, but the one providing (4.5) when ζ h → +∞ reads Now x B can be written in terms of x A by first using (6.8) and (6.3) into x + = x B given by (6.12), and then employing (6.14) in the resulting formula.After some algebra, we obtain Figure 8: Comparison between geodesic bit threads (green dashed curves) connecting a point x A ∈ A \ A β to a point x B ∈ B (see also Fig. 7) and the corresponding lines (starting at the same x A 's) for the modular evolution in the Euclidean spacetime described in Sec.5.2 (black solid curves, see also Fig. 5).
where the ratio within the round brackets in the r.h.s. is positive when x A ∈ A \ A β .Thus, the CFT result (5.11) in A \ A β has been obtained holographically through the geodesic bit threads.As consistency check, notice that the limit ζ h → +∞ of the expression for x B obtained from (6.15) when x A ∈ A \ A β leads to (4.9), as expected.
In Fig. 8, considering only the positive half real axis without loss of generality, we show some geodesic bit threads having both the endpoints on the boundary (green dashed curves).These bit threads are compared with the corresponding curves (black solid lines) starting at the same values of x A ∈ A \ A β obtained from the CFT expression (5.2) for the same value of β, which have been shown also in Fig. 5.We remark that these two sets of curves belong to two different Euclidean manifolds.These geodesic bit threads share the same endpoints with the corresponding CFT lines, as expected from the analytic result discussed above, but they do not coincide for the other points in general.There is no reason to expect such overlap.However, the overlap approximatively occurs for the geodesics bit threads close to the boundary, which connect points close to the entangling point b.Indeed, since these geodesics bit threads are weakly influenced by the occurrence of the horizon, they become approximatively half circles, i.e. geodesics in Poincaré AdS 3 .
The inversion map (5.11) obtained from CFT can be recovered holographically through the geodesic bit threads also when x A ∈ A β .The geodesic bit thread starting at this point on the boundary reaches the horizon at some point (x h , ζ h ) (grey dashed curves in Fig. 7 and Fig. 9).We observe that this bit thread is tangent to the horizon; indeed, from (6.9) one obtains , which diverges as ζ → ζ h .Such observation suggests us to merge this geodesic bit thread with the geodesic starting at (x h , ζ h ) tangentially to the horizon and ending at some point (x, 0) on the boundary (cyan dashed curves in Fig. 9).The latter geodesic can be obtained by reflecting the geodesic bit thread starting at (x A , 0) with respect to the vertical axis at x = x h .This construction implies that x = x A + 2(x h − x A ).
The same procedure described above to obtain x B from x A ∈ A \ A β can be employed to find x in terms of x A ∈ A β (in particular, (6.14) holds also in this case), obtaining (6.15) with Figure 9: Comparison between geodesic bit threads (grey dashed curves) connecting a point x A ∈ A β to a point on the horizon (see also Fig. 7) and the corresponding lines (starting at the same x A 's) of the modular evolution in the Euclidean spacetime (yellow solid curves, see also Fig. 5).
x B replaced by x.The crucial difference is that the ratio in the r.h.s. of (6.15) is negative when x A ∈ A β ; hence the proper sign must be introduced taking the square root of (6.15).This leads to x = Re[z inv (x A )], where z inv (x A ) is given in (5.11) with x A ∈ A β .We remind that x(x A ) is a bijective function from A β onto R.
In Fig. 9, considering only the positive half real axis without loss of generality, we display some geodesic bit threads that start at various x A ∈ A β , and therefore reach the horizon (grey dashed curves).Each geodesic bit thread of this kind is merged smoothly at the horizon with the corresponding geodesic (cyan dashed curves) that connects the horizon to the point (x, 0) on the boundary, where x = Re[z inv (x A )], as discussed above.The curves obtained from these geodesics are compared with the corresponding CFT curves described in Sec.5.2 (yellow solid curves, shown also in Fig. 5), that start at the same values of x A ∈ A β , grow in the strip {z ∈ C ; |Im(z)| β/2} according to (5.2) with θ ∈ (0, π) and end at Re[z inv (x A )] + i β 2 when θ = π.The relation x = Re[z inv (x A )] is highlighted by the vertical dotted lines in Fig. 9.

Two disjoint intervals: Massless Dirac field, ground state
In this section we consider the massless Dirac fermion in its ground state on the line bipartite by the union A = A 1 ∪ A 2 of two disjoint intervals A 1 = (a 1 , b 1 ) and A 2 = (a 2 , b 2 ).

Internal modular evolution
The modular Hamiltonian of two disjoint intervals for a CFT in its ground state and on the line depends on the model.For the massless Dirac field, this modular Hamiltonian and the corresponding modular flow for the Dirac field have been described in [12,13].Casini and Huerta [12] found that this operator can be written as

A
, where is a local operator determined by the energy density T tt (t, x).For the massless Dirac field, the energy density reads9 in terms of the chiral components ψ 1 and ψ 2 of the massless Dirac field.The bi-local term where the bi-local operator T bi-loc (t, x, y) is the following quadratic operator T bi-loc (t, x, y) ≡ i 2 : ψ * 1 (x + t) ψ 1 (y + t) − ψ * 1 (y + t) ψ 1 (x + t) : The weight functions in (7.1) and (7.3) can be written as where which is the crucial function of our analysis throughout this section.This function naturally leads to introduce the point x c (x) conjugate to x ∈ A through the condition w(x c (x)) = w(x).
The explicit expression for x c (x) is [12] x where Notice that a point x and its conjugate point x c (x) belong to different intervals in A. Furthermore, x c (x) in (7.7) is invariant under a cyclic permutation of the sequence of endpoints (a 1 , b 1 , a 2 , b 2 ).It is sometimes convenient to write these expressions in terms of the lengths of the intervals j ≡ b j − a j with j ∈ {1, 2} and of the distance d ≡ a 2 − b 1 separating them.
The function providing the geometric action of the modular group of automorphisms induced by the vacuum reads [12,13,21] where Θ A j (x) is the characteristic function of A j with j ∈ {1, 2}, and satisfying ξ(0, x) = x for any x ∈ A. The functions in (7.10) are written in terms of the inverse functions w −1 ± of (7.6), which are 2(1 + e x ) (7.11) where the expression under the square root is a quadratic polynomial in the variable e x , whose discriminant is The map sending a generic configuration A of two intervals on the line into the symmetric configuration ) .
(7.13) Considering a point x ∈ A and its conjugate x c (x) given by (7.7), we observe that (7.12) satisfies Let us introduce the points c j ∈ A j with j ∈ {1, 2} such that Combining this definition with (7.14), one finds that f (c j ) 2 = a b for these two points.From (7.15), their explicit expressions are The curves ξ(τ, x) and ξ(−τ, x) (solid lines) and their images under the inversion map (dot-dashed lines) in the plane (ξ, τ ), obtained from (7.9) and (7.26).The vertical dotted lines in the grey strips correspond to (7.16) and (7.18).The bottom panel illustrates the limiting regime of adjacent intervals.colour intersect A in x and x c (x)), as done in Fig. 9 of [17] for the special case of A sym .
In order to write z inv,A (x), let us consider the points (7.16) and (7.18) (green points in Fig. 12).As observed in [17], these points can be found from the points in the complex plane where w (z) = 0, which are The points z 0,± (black dots in Fig. 12) are the intersections of the circles C a and C b whose centers are on the real axis, in (a 1 + a 2 )/2 and (b 1 + b 2 )/2 respectively.In z 0,± , the straight lines tangent to C a and C b provide two supplementary angles whose bisectors allow to identify in a unique way two circles among the ones passing through z 0,± .One of these two circles (red dashed circle in Fig. 12) intersects the real axis at c 1 and c 2 given by (7.16) and (7.18) (green points), while the other one (blue dashed circle) intersects the real axis at x c (c 1 ) and x c (c 2 ) (cyan points).
In [17] it has been argued that the points (7.24) induce the occurrence of two critical values for θ ∈ (0, 2π) whose sum is 2π.One of these angles is θ 0 ≡ Im[w(z 0,± )] ; hence tan θ 0 = Im e w(z 0,± ) Re e w(z 0 From (7.23), (7.16) and (7.18) we construct the following inversion map on the line bipartite by the union of two disjoint intervals which is one-to-one from A to B and maps points close to an entangling point into points close to the same entangling point.
At this stage let us introduce ξ(τ, x) ≡ Θ B 1 (x) ξ − (−τ, x) + Θ B 2 (x) ξ + (−τ, x) x ∈ B 1 ∪ B 2 (7.27) which has been employed to draw the dot-dashed curves in Fig. 10, obtained from (7.27) with x given by the image under (7.26) of the initial point in A providing the solid curves in the grey strips through (7.9).
In order to construct an inversion map z inv (x) on the entire line, we have to describe also the one-to-one map inversion map z inv,B (x) : B → A as the inverse of (7.26).This can be done by employing again the curves defined through (7.23), but now in the range θ ∈ (π, 2π).This leads to the curves in the lower half plane shown in Fig. 12  which is idempotent, as expected.

Conclusions
In this manuscript we have studied a holographic relation between the geometric action of some modular conjugations in 2D CFT and the geodesic bit threads considered in [35,38] in a constant time slice of the corresponding dual gravitational backgrounds.This comparison has been performed for three cases and in all of them the bipartition of the space is given by a single interval: for a CFT in the ground state, either on the line (Sec.3 and Sec.4) or on the circle (Appendices A and B), and in the thermal state on the line (Sec.5 and Sec. 6).
When the CFT is in the ground state and on the line bipartite by an interval, the geometric actions of the modular operator and of the modular conjugation are described by the theorem of Hislop and Longo [2,8,9] and are given by (3.1) and (3.3) respectively.Considering for simplicity and without loss of generality the interval A = (−b, b) on the line, the geometric action for the modular conjugation at t = 0 simplifies to (3.11).In the context of the gauge/gravity correspondence, this inversion relation is obtained between the endpoints of a geodesic bit thread on a constant time slice of Poincaré AdS 3 (see Fig. 3 and (4.9)).The case of a CFT in its ground state and on the circle bipartite by an interval is a straightforward generalisation whose geometric actions for the modular operator and the modular conjugation are (A.3) and (A.5) respectively.The corresponding holographic dual description of the geometric action of the modular conjugation is obtained through the geodesic bit threads in a constant time slice of global AdS 3 (see Fig. 15).
For a CFT in a thermal state at temperature 1/β and on the line bipartite by an interval, the geometric action of the modular conjugation is more insightful.In each chiral sector, the inversion map (5.5) on the bipartite line (see also (5.10)) obtained from the modular evolution in the Euclidean spacetime [17] naturally leads to introduce the subinterval A β ⊂ A, whose endpoints are (5.6) and (5.7).Interestingly, denoting by the size of the interval A, we find that the integration over A β of the corresponding contour function for the entanglement entropy [45] (see (5.16)) provides the Gibbs entropy π c /(3β) of an isolated system of length coming from the Stefan-Boltzmann law for a 2D CFT [43,49,50] for any value of and β.The inversion map (5.5) for a given chirality is a complex function which acquires a constant non vanishing imaginary part on A β and whose real part maps A β into the second world (see Fig. 4).The occurrence of such spacetime is induced by the non triviality of the commutant for a thermal state, as discussed in [24-26, 28, 29].The combination of the two different chiralities allows to extend this picture to the Minkowski spacetime M (see Fig. 6), where the geometric action of the modular conjugation is (5.23).Interestingly, this map is real also on the domain of dependence of A β (green region in Fig. 6) and sends this diamond onto the entire Minkowski spacetime M (which is interpreted as the second world) in a bijective way.
In the gauge/gravity correspondence, the restriction to the real line at t = 0 of this inversion map obtained for a 2D CFT in the (geometric) thermal state coincides with the relation between the endpoints of the geodesic bit threads on a constant time slice of the BTZ black brane geometry (see Fig. 8 and Fig. 9 for the geodesic bit threads anchored in A \ A β and in A β respectively).The geodesic bit threads anchored in A β reach the horizon.
It is instructive to compare the modular conjugation for the thermal case derived in Sec. 5 with the thermo field dynamic (TFD) setting of [52].Both constructions are based on the Gibbs state generated by the Hamiltonian H of the system and adopt [53] the Tomita-Takesaki modular theory.In the TFD case one applies this theory to the whole system and the modular Hamiltonian is H − JHJ.Instead, in our case the modular Hamiltonian (5.24) is induced by the bipartition of the system in two parts A and B. For this reason the modular evolution and the modular conjugation in the two settings are different.Heuristically, our construction approaches the TFD setting when the interval A becomes the whole system.
Finally, for the massless Dirac fermion in the ground state and on the line bipartite by the union of two disjoint intervals, in Sec.7.2 we have proposed the inversion map (7.26), obtained by following [17] (see Fig. 12) and adapting the procedure established in the previous cases.This inversion map can be extended to the Minkowski spacetime through (7.30) (see Fig. 13).It is not difficult to realise that a holographic description of this inversion map on the real line cannot be found only through the geodesic bit threads in Poincaré AdS 3 (see the corresponding discussion in [38]).However, it would be interesting to reproduce it holographically by employing other holographic bit threads in Poincaré AdS 3 .
The analyses discussed in this manuscript can be extended in various directions.In 1 + 1 spacetime dimensions, more complicated configurations can be investigated by considering e.g. a CFT at finite temperature and finite volume when the subsystem is still an interval [30-32, 54, 55] or a CFT in the ground state on the line when the bipartition is given by the union of an arbitrary number of disjoint intervals [12,13,[56][57][58].In higher dimensional CFT, it would be interesting to explore the action of the modular operators and of the modular conjugations for subregions having different shapes, comparing the results with the corresponding configurations of geodesic bit threads (see e.g.[59][60][61][62][63][64][65] for the shape dependence of the holographic entanglement entropy).Finally, we find it worth mentioning that various modular Hamiltonians in free models have been studied by taking the continuum limit of the corresponding operators (often called entanglement Hamiltonians) on the lattice [18,[66][67][68][69][70][71][72][73][74][75][76][77].It would be insightful to study also other operators occurring in the Tomita-Takesaki modular theory in quantum many-body systems.which tells us that the inversion maps at t = 0 discussed in Sec.3.2 and in this Appendix can be related through conformal mappings.

B Geodesic bit threads in global AdS 3
In this appendix we show that the inversion map discussed in the Appendix A can be obtained also through the geodesic bit threads in a constant time slice of global AdS where ρ 0 and θ ∈ [0, 2π).The last expression in (B.1) is written in terms of the polar coordinates (r, θ), with 0 r < 1, and corresponds to the Poincaré disk (yellow unit disk in Fig. 15).The Cayley map (A.7) relates the upper half plane, in the complex coordinate µ = x+iζ and equipped with the hyperbolic metric (4.1), to the Poincaré disk with the metric (B.1) in the complex coordinate ν = r e iθ .
The geodesics in the Poincaré disk having their endpoints on the boundary are circular arcs that intersect the boundary orthogonally.In a disk of radius R in polar coordinates (r, θ), where 0 r R, the orthogonality condition for the intersection of two circles provides the following second order equation O 7 8 y 4 u k d l p 0 z 4 r O t U m j B F N k 4 B C O 4 A R c O I c y X E E F q k B B w C M 8 w 4 t 1 a z 1 Z r 9 b b t H X J m s 3 s w x 9 Y 7 z + b 7 5 N L < / l a t e x i t > |⌧c| < l a t e x i t s h a 1 _ b a s e 6 4 = " K 5 b P P N Y T P i N w k 1 r A f o 0 / 5 e u 2 b w Q = " > A A A B 8 H i c b V D L S g N B E O y N r y S + o l 4 E L 4 N B 8 G L Y F c Q c g 1 4 8 R j A P S Z Y w O 5 k k Q 2 Z m l 5 l Z I W 4 C / o M X D 4 p 4 9 e i n e B P 8 G C e P g y Y 5 T 8 6 r 8 z Z t T T m z m T 3 4 A + f 9 B w a M k 4 I = < / l a t e x i t > |⌧c| < l a t e x i t s h a 1 _ b a s e 6 4 = " C z a + o A K T E D n R G J u M + / M 5 r F 7 i D k 8 = " > A A A B 8 H i c b V D L S g N B E O y N r y S + o l 4 E L 4 N B 8 G L Y F c Q c g 1 4 8 R j A P S Z Y w O 5 k k Q 2 Z m l 5 l Z I W 4 C / o M X D 4 p 4 9 e i n e B P 8 G C e P g y Y

2
respectively, and (3.8) tells us that it passes through the point (z inv (x), 0) when τ = 0. Any arc in the right panel of Fig.1is obtained through (2.20), by using the arcs denoted by the same kind of line and the same colour in the left panel of the same figure.

Figure 2 :
Figure 2: Euclidean modular evolution in the complex plane: The curves are half circles obtained from (3.14) (or (3.15) equivalently) with θ ∈ (0, π) (solid arcs) or θ ∈ (π, 2π) (dotted arcs), for some x ∈ A. They map a point in A (red segment) to a point in its complement (blue half lines) and viceversa, which are related by (3.20).

(5. 10 )
where Θ A β (x) it the characteristic function 7 of A β .It is worth remarking that the real functions z inv (x) : A \ A β → B and Re[z inv (x)] : A β → R are both bijective.They are shown in Fig.4through the solid curves and the dashed curve respectively.The latter one tells us that Re[z inv (x)] : A β → R is strictly increasing.
3 J P H s m T E z s P z r P z M m 1 d c m Y z + + Q P n L c f K w 2 T L w = = < / l a t e x i t > a < l a t e x i t s h a 1 _ b a s e 6 4 = " N h J h d f r o h 1 e t 0 w b + j b + x d q i P r I Y = " > A A A B 6 H i c b Z C 7 S g N B F I b P x l u M t 6 i l I I N B s A q 7 I m p n I I 1 l I 5 m J a x P J W X a 3 C Z n j u D M r / w X 6 s d F 5 7 R 4 U r U L p T J M l Y U 9 O I A j c O A M S n A J F a g B A 4 R 7 e I Q n 6 8 Z 6 s J 6 t l 2 l p x p r 1 7 M I v W a / f E o + Q 2 w = = < / l a t e x i t > a < l a t e x i t s h a 1 _ b a s e 6 4 = " N h J h d f r o h 1 e t 0 w b + j b + x d q i P r I Y = " > A A A B 6 H i c b Z C 7 S g N B F I b P x l u M t 6 i l I I N B s A q 7 I m p n I I 1 l I 5 m J a x P J W X a 3 C Z n j u D M r / w X 6 s d F 5 7 R 4 U r U L p T J M l Y U 9 O I A j c O A M S n A J F a g B A 4 R 7 e I Q n 6 8 Z 6 s J 6 t l 2 l p x p r 1 7 M I v W a / f E o + Q 2 w = = < / l a t e x i t > a < l a t e x i t s h a 1 _ b a s e 6 4 = " U U 5 z 8 2 8 X 4 8 F X a D m U h U s s 0 R + F M X w = " > A A A B 6 H i c b Z C 7 S g N B F I b P x l u M t 6 i l I I N B s A q 7 I m p n I I 1 l I u Y C y R J m J 2 e T M b M X Z m a F s K S 0 s r F Q x N a n 8 B G s 7 X w G X 8 L J p d D o D w M f / 3 8 O c 8 7 x Y s G V t u 1 P K 7 O w u L S 8 k l 3 N r a 1 v b G 7 l t 3 f q K k o k w x q L R C S b H l U o e I g 1 z b X A Z i y R B p 7 A h j c o j / P G L U r F o / B a D 2 N 0 A 9 o L u c 8 Z 1 c a q e p 1 8 w S 7 a E 5 G / 4 M y g c P F e / b r b f 7 u q d 1 z b X A Z i y R B p 7 A h j c o j / P G L U r F o / B a D 2 N 0 A 9 o L u c 8 Z 1 c a q e p 1 8 w S 7 a E 5 G / 4 M y g c P F e / b r b f 7 u q d h z o b g / T 7 5 L 6 l s F 7 3 d 4 s 6 J T e O Q D D F F V s k a 2 S Q e 2 S M H 5 J i U S J l w c k 3 u y A N 5 d G 6 c e + f J e R 6 2 j j i f M 8 v k B 5 y X D / T M o N s = < / l a t e x i t > Re[z inv (x)] < l a t e x i t s h a 1 _ b a s e 6 4 = " y e j k d V K 1 x U G d 7 l a 0 X f 3 X 9 1 o x s x w = " > A A A B 6 H i c b Z B L S w M x F I X v 1 F e t r 6 p L N 8 E i u C o z I u r O g i A u W 7 A P a I e S S e + 0 s Z k H S U Y s Q 8 G 9 G x e K u P X 3 6 M a d / 8 Z 0 6 k J b D 4 R 8 n H N D 7 r 1 e L r j 9 r 3 / d H 7 T b V b + O z 0 I p p I p o A K Y k z b c 2P w U 6 K B U 8 H G + U 5 i W E z o k P R Z 2 6 I i k h k / n c w 6 x i f W 6 e E w 0 v Y p w B P 3 d 0 d K p D E j G d h K S W B g 5 r P M / C 9 r J x B e + i l X c Q J M 0 e l H Y S I w R D h b H P e 4 Z h T E y A K h m t t Z M R 0 Q T S j Y 8 + T t E b z 5 l R e h c V b y z k t u z S 2 W K 2 i q H D p E x + g U e e g C l d E 1 q q I 6 o m i A H t A T e n a k 8 + i 8 O K / T 0 i V n 1 n O A /s h 5 + w F u t J I 5 < / l a t e x i t > ⌧ < l a t e x i t s h a 1 _ b a s e 6 4 = " f a + 8 y t B 5 c g s z I 4 D U u p G q 8 + H B U Q 4 = " > A A A B 6 n i c b V D L S g N B E O z 1 G e M r 6 l G Q w S B 4 C r u C 6 D G Q g x 4 j m g c k I c x O Z p M h s 7 P L T K 8 Y l v y A 4 M W D I l 7 9 G M / e / A f / Q S e P g y Y

Figure 11 :Figure 12 :
Figure 11: Spacetime domain D A for the union of two disjoint intervals A 1 and A 2 on the line, whose lengths are 1 = and 2 = 2 respectively.The distance d between the intervals changes in the different panels, while is kept fixed: d/ = 1 (top left), d/ = 1/5 (top right), d/ = 1/20 (bottom left) and d/ = 1/1000 (bottom right).The modular trajectories corresponding to following two choices of the initial point P are displayed: the black filled circle in D 2 and the blue filled circle in D + 12 .The empty symbols having the same colour denote the conjugate point P c (empty circle) and the auxiliary points P c,± (empty square and empty rhombus) obtained from P .

Figure 13 :
Figure 13: Modular trajectories in D A 1 ∪A 2 and their images under the inversion map (7.30).The bottom panel shows the limiting regime of adjacent intervals (see the right panel of Fig. 1).

Figure 15 :
Figure 15: Geodesic bit threads (green dashed circular arcs) in a fixed time slice of global AdS 3 (yellow unit disk, see (B.1)) for an arc of the boundary circle (red arc), whose RT curve is the red solid arc anchored to its endpoints.