FZZT branes in JT gravity and topological gravity

We study Fateev-Zamolodchikov-Zamolodchikov-Teschner (FZZT) branes in Witten-Kontsevich topological gravity, which includes Jackiw-Teitelboim (JT) gravity as a special case. Adding FZZT branes to topological gravity corresponds to inserting determinant operators in the dual matrix integral and amounts to a certain shift of the infinitely many couplings of topological gravity. We clarify the perturbative interpretation of adding FZZT branes in the genus expansion of topological gravity in terms of a simple boundary factor and the generalized Weil-Petersson volumes. As a concrete illustration we study JT gravity in the presence of FZZT branes and discuss its relation to the deformations of the dilaton potential that give rise to conical defects. We then construct a non-perturbative formulation of FZZT branes and derive a closed expression for the general correlation function of multiple FZZT branes and multiple macroscopic loops. As an application we study the FZZT-macroscopic loop correlators in the Airy case. We observe numerically a void in the eigenvalue density due to the eigenvalue repulsion induced by FZZT-branes and also the oscillatory behavior of the spectral form factor which is expected from the picture of eigenbranes.


Introduction
Two-dimensional Jackiw-Teitelboim (JT) gravity [1,2] is a useful toy model to study various aspects of quantum gravity and holography. In a remarkable paper [3] it was shown that JT gravity is holographically dual to a certain double-scaled matrix model. This is an example of the holography involving the ensemble average, where the Hamiltonian H of the boundary theory becomes the random matrix. This in particular implies that the correlator of two partition functions Z(β 1 )Z(β 2 ) includes the contribution of spacetime wormhole connecting the two asymptotic boundaries and thus the correlator is not factorized.
As argued in [4], one can fix some of the eigenvalues of the matrix integral by introducing the Fateev-Zamolodchikov-Zamolodchikov-Teschner (FZZT) branes [5,6], which are called "eigenbranes" in [4]. Adding branes in JT gravity is also considered in [7] to recover the Page curve for black hole evaporation. In a recent paper [8], the matrix model for the dynamical end of the world (EOW) brane in JT gravity is proposed. 1 In this paper, we consider FZZT branes in JT gravity. Our prescription for introducing the FZZT branes is a natural generalization of the JT gravity matrix model by Saad, Shenker and Stanford [3]. We obtain the amplitude in the presence of FZZT branes by gluing several building blocks. As shown in [3], the JT gravity amplitude is obtained by gluing the "trumpet" (2.9) and the Weil-Petersson (WP) volume (2.6). A new ingredient in the presence of a FZZT brane is the factor M(b) = −e −zb with z being a parameter, which is attached to the geodesic boundary of length b and we integrate over b (see Figure 1). This construction can be generalized to multiple FZZT branes by replacing the factor with M(b) = − i e −z i b . In particular, the trumpet can end on a FZZT brane as shown in Figure 1a. This implies that the two-boundary correlator Z(β 1 )Z(β 2 ) in the presence of FZZT brane receives a contribution depicted in Figure 2b, which reminds us of the "half-wormhole" introduced in [10].  It turns out that the above construction of FZZT branes in JT gravity can be generalized to encompass arbitrary background {t k } of 2d topological gravity. We can compute FZZT brane amplitude by gluing certain building blocks. In general background {t k }, we can use the same "trumpet" and M(b) as in JT gravity, but the WP volume should be replaced by the generalized WP volume defined in (2.37). This construction defines an FZZT brane amplitude only perturbatively in genus expansion. One can obtain the non-perturbative expression of the correlator of FZZT branes and macroscopic loop operators Z(β) = Tr e −βH by taking a double scaling limit of the correlator of determinants in the finite N matrix model. We find a compact expression of the generating function of the Z(β)-FZZT correlators in terms of the Baker-Akhiezer (BA) function and the Christoffel-Darboux (CD) kernel (see (5.38) and (5.39)). As an application of our formalism, we consider the spectral form factor in the presence of two FZZT branes (see Figure 6). This paper is organized as follows. In section 2, we find the prescription for the construction of FZZT brane amplitude by gluing the trumpet, (generalized) WP volume and M(b). We find that the half-wormhole amplitude is given by the complementary error function (2.57). We also comment on the relation between the trumpet and the Liouville wavefunction. In section 3, we compute the genus-zero density of states in JT gravity with K FZZT branes in the 't Hooft limit (3.1). We also comment on the relation to the deformation of the potential of dilaton gravity. In section 4, we review the known results about the BA function, the CD kernel and the multi-FZZT amplitude. In section 5, we find the generating functions (5.38) and (5.39) of the correlators of FZZT branes and macroscopic loop operators. In section 6, we apply our formalism to the Airy case, i.e. the trivial background t k = 0 (k ≥ 1) of topological gravity. We study the spectral form factor in the presence of two FZZT branes in the Airy case. Finally we conclude in section 7. In appendix A we summarize the result of (2, p) minimal string. In appendix B we provide an alternative derivation of (5.11) using the correlator of inverse determinant.

FZZT branes in the genus expansion
In this section, we will explain our formalism of computing the genus expansion of the correlator of partition functions Z(β) = Tr e −βH in the presence of FZZT branes.

JT gravity and Weil-Petersson volume
Let us first consider JT gravity. As shown by Saad, Shenker and Stanford [3], the genus expansion of the connected correlator of Z(β)'s can be obtained by gluing the "trumpet" and the Weil-Petersson (WP) volume i db iZtrumpet (β,b i )Ṽ g,n (b 1 , . . . ,b n ), (2.1) where the subscript c of · · · c refers to the connected part. The WP volumeṼ g,n (b) is given byṼ where M g,n denotes the Deligne-Mumford compactification of the moduli space of M g,n of Riemann surface Σ of genus g with n marked points p 1 , . . . , p n , κ is the first Miller-Morita-Mumford class and ψ i is the first Chern class of the complex line bundle whose fiber is the cotangent space to p i . Note that (2.2) is valid for 3g − 3 + n ≥ 0 andṼ 0,1 ,Ṽ 0,2 are undefined. Correspondingly, (2.1) makes sense except the contributions of the disk and annulus amplitudes, which are to be discussed separately. The trumpet partition functioñ Z trumpet (β,b) in (2.1) is given bỹ where γ is the asymptotic value of the dilaton field near the boundary of AdS 2 . It is convenient to set Then we findṼ where we defined the rescaled WP volume by In what follows we will call V g,n (b) the WP volume instead of the originalṼ g,n (b). As in our previous paper [11] we define the genus counting parameter g s by It is also convenient to rescale the trumpet so that Then we find In this new normalization, (2.1) is written as Now let us consider the effect of adding FZZT branes. In the matrix model the FZZT brane corresponds to the insertion of the determinant operator det(ξ + H) in the matrix integral, where H is the matrix variable and ξ is a formal parameter [12,13]. 2 The FZZT brane operator det(ξ + H) can be represented by means of the vector degrees of freedom as det(ξ + H) = dχdχ e χ(ξ+H)χ , (2.12) where χ and χ are Grassmann-odd vector variables. From the relation we see that the insertion of FZZT brane creates infinitely many boundaries corresponding to the single trace operator Tr log(ξ + H). This single trace operator has an integral representation where ε is a small regularization parameter. The logarithmically divergent term log ε can be absorbed into the overall normalization of the determinant operator and we will ignore such divergence unless otherwise stated. We can apply the integral transformation (2.14) to some of the Z(β)'s in (2.10). It turns out that the integral transformation of the trumpet partition function is free from the logarithmic divergence. We introduce M(b) by 3 Using the integral representation of the modified Bessel function K 1 2 (x) This implies that the FZZT brane operator (2.12) is written as det(ξ − M ). 3 In [8], this integral transformation is called the "inverse trumpet".
we find that M(b) in (2.15) is given by where we used the definition of M(b) in (2.15). This expression is valid except the terms of the order of g k s (k = −1, 0), which can be calculated separately. To summarize, we can introduce the FZZT branes in the correlator of Z(β)'s by gluing M(b) along the geodesic boundary and integrate over b with measure M(b)db (see Figure 1). Note that the factor of b in the integration measure bdb for the trumpet is canceled out by the 1/b in (2.15).
We can interpret our expression of (2.17) as follows. M(b) in (2.17) has the form where the particle action S particle is given by Namely, M(b) can be interpreted as the contribution of a particle with mass z running around the geodesic boundary with length b. The overall minus sign in (2.21) comes from the fact that the particle running around the loop is a fermion (see (2.12)).
If we change the sign of M(b) it corresponds to the anti-FZZT brane represented by the inverse determinant where φ and φ are the Grassmann-even (bosonic) vector degrees of freedom. We can generalize our formula (2.20) to include multiple FZZT branes. Using the relation we can use the same formula (2.20) for the correlator of Z(β)'s in the presence of multiple FZZT branes by simply replacing where z i is related to ξ i by ξ i = z 2 i /2. In [8], the matrix model description of the end of the world (EOW) brane in JT gravity is considered. The prescription of [8] is the same as our formula (2.20) with M(b) replaced by In our interpretation, this corresponds to infinitely many anti-FZZT branes with a particular set of parameters z n = µ + n + 1 2 (n ≥ 0). As discussed in [8], the integral of (2.27) has a divergence coming from the pole at b = 0 and a certain regularization is required to define the EOW brane. On the other hand, in our case of FZZT brane M(b) in (2.17) has no pole at b = 0 and the b-integral is well-defined.

FZZT branes in general background of topological gravity
It turns out that the above construction can be generalized to encompass arbitrary background {t k } of 2d topological gravity. Recall that JT gravity is a special case of topological gravity with infinitely many couplings turned on as t k = γ k where [11,14,15] In Witten-Kontsevich topological gravity [16,17] observables are made up of the intersection numbers τ d 1 · · · τ dn g,n = Mg,n ψ d 1 1 · · · ψ dn n , d 1 , . . . , d n ∈ Z ≥0 . (2.29) The generating function for the intersection numbers is defined as (2. 30) We assume that the trumpet is background independent (i.e. independent of t k ) In section 2.5 we will consider the validity of this assumption. Then it is natural to define the generalized WP volume V g,n by Note that the correlator of Z(β)'s can be written as where B(β) is the boundary creation operator [18] As in the previous section (2.32) and (2.33) are valid except the (g, n) = (0, 1), (0, 2) parts and we leave V 0,1 , V 0,2 undefined. Here we are interested in the general structure and do not go into the details of these parts. See e.g. [19] for a precise treatment of them. Let us introduce the operator V (b) by Then we find From (2.32), (2.33) and (2.35) we find that the generalized WP volume has a simple expression From (2.30), this is also written as (2.38) One can see that this is a natural generalization of the WP volume (2.6). With our definition of the generalized WP volume in (2.32), the computation of the correlator of Z(β)'s in the presence of FZZT branes is completely parallel to that in the previous subsection for the JT gravity case (2.20). Thus we conclude that the FZZT brane amplitude of topological gravity in the general background {t k } is given by the same formula (2.20) with the WP volume replaced by the generalized WP volume in (2.32).
As an application of our formalism (2.20), let us consider the FZZT brane amplitude without Z(β)-insertions where M(b) is given by (2.26). Since V 0,1 , V 0,2 are undefined, the calculation here is valid for positive powers of g s . From the relation This shift of coupling t k due to the insertion of the FZZT branes agrees with the known result in the literature 4 [20] (see e.g. [21] for the expression in our convention). In the theory of soliton equations the shift is generated by the action of the vertex operator of [22], which is viewed as the infinitesimal Bäcklund transformation for the KdV equation [23]. When t k = 0, this expression of the couplings in terms of z i (2.42) has appeared in the Kontsevich matrix model [17] and it is known as the Miwa transform. In the context of minimal string theory [24,25], this shift is interpreted as the open-closed duality where the insertion of FZZT branes is replaced by the shift of closed string background {t k } [13,26]. 4 In the literature the convention x 2k+1 ≡ t k gs(2k + 1)!! is often used for the couplings. In terms of x the shift (2.42) is written as

Example of generalized WP volume
Let us compute a few examples of the generalized WP volume V g,n (b) for small g and n. One can compute V g,n (b) from the known result of the correlator of Z(β)'s using the relation (2.32). The generalized WP volume at genus-zero has been computed in [27]. From the result of the genus-zero n-point function of Z(β) [18] Z(β 1 ) · · · Z(β n ) g=0 one can read off V 0,n (b) using (2.32). Here u 0 = ∂ 2 0 F 0 . To this end, it is convenient to define the function One can easily show that I k (b) is given by where I k (x) denotes the modified Bessel function of the first kind. The following property (2.47) From (2.44) we find that the generalized WP volume at genus-zero is given by This agrees with the result of [27]. At genus-one, from our previous result of the correlators of Z(β)'s [19,28] (2.50) Here I k is the Itzykson-Zuber variable [29] Alternatively, we can compute V g,n (b) using (2.37). Let us compute V 1,1 (b) in this way. Applying the operator V (b) to the genus-one free energy F 1 = − 1 24 log t we find Using the relation we find This agrees with the result in (2.50) obtained from Z(β) g=1 . We should stress that the generalized WP volume V g,n (b) in not a polynomial in b i for general background {t k }. However, when u 0 = 0 V g,n (b) becomes a polynomial since . (2.56) In particular, in the JT gravity case t k = γ k (2.28) we have u 0 = 0. One can check that the generalized WP volume reduces to the WP volume when t k = γ k . A certain generalization of the WP volume is considered in [27] in the minimal model background. Our definition of V g,n (b) is different from [27].

Half-wormholes and factorization
In this section we consider the genus-zero amplitude between Z(β) and FZZT brane, which we call the "half-wormhole" following [10]. 5 The half-wormhole amplitude is easily found as
where Erfc(x) denotes the complementary error function On the other hand, the "wormhole" amplitude is obtained by gluing two trumpets where we assumed u 0 = 0. In the presence of FZZT brane, the spacetime of JT gravity can end on the FZZT brane along the geodesic boundary. For instance, let us consider the amplitude with two asymptotic boundaries Z(β 1 ) and Z(β 2 ) with one FZZT brane At the order of O(g 0 s ), there are two contributions as shown in Figure 2. The contribution of Figure 2b is factorized Of course, the total amplitude (2.60) is not factorized. But if we increase the number of FZZT branes as in (2.26) the trumpet can end on various branes labeled by z i and there are many contributions like Figure 2b with various choices of z i . This is a concrete realization of the idea of "eigenbrane" introduced in [4]. As discussed in [4], adding FZZT branes corresponds to fixing some of the eigenvalues of matrix. 6 This suggests that adding many FZZT branes amounts to pick a particular member of the ensemble of random matrices and we expect that the factorization is restored, at least partially.

Relation to Liouville wavefunction
In the traditional approach to 2d gravity, the correlator of macroscopic loop operators is commonly written in terms of the (mini-superspace) wavefunction of the Liouville theory [18,35] In (2.62) K ν (x) denotes the modified Bessel function of the second kind and we defined (2.65) 6 Fixing the eigenvalue of the random matrix model is also discussed in [34]. 7 Our normalization of ψE( ) is different from that in [18,35] by a factor of √ 2. To correctly normalize it as (2.65) we need the extra factor of √ 2.
In our formalism (2.20), the correlator of macroscopic loop operators is obtained by gluing the trumpet and the generalized WP volume. In the traditional approach to 2d gravity, the trumpet has not appeared in the literature before, as far as we know. The trumpet partition function naturally appears in JT gravity from the path integral of Schwarzian mode describing the wiggles near the asymptotic boundary of AdS 2 [3]. One might think that the trumpet is tightly connected to JT gravity and it cannot be generalized to the topological gravity with arbitrary background {t k }. However, as we will see below it turns out that the trumpet partition function is written in terms of the Liouville wavefunction ψ E (β), which connects our formalism (2.20) and the traditional approach. Thus we can use the trumpet in arbitrary background {t k } as we did in the previous section. In Multiplying both sides of (2.68) by cos(Eσ) and using the relation By using the orthogonality of ψ E (β) in (2.65), this is inverted as This is our desired result: the trumpet partition function can be expressed in terms of the Liouville wavefunction ψ E (β). Let us consider the wormhole amplitude In the last step we used the orthogonality of ψ 2E (2b/ √ E 0 ). The last line of (2.72) agrees with the known result of genus-zero two-point function [18] written in terms of the Liouville wavefunction ψ E (β). As discussed in [18,37], the factor G(E) = (2 cosh πE) −1 in (2.72) is interpreted as the propagator of 2d gravity. It is interesting to observe that the square root of the propagator 2G(E) = (cosh πE) − 1 2 appears in the trumpet (2.71). We can compute various amplitudes by gluing the trumpet (2.71) and M(b). Let us consider the Z(β)-FZZT amplitude (or half-wormhole) where z is related to σ by (2.67). This is written as the error function (2.57). Appearance of the error function in the Z(β)-ZZ brane amplitude is observed in [12]. From (2.75) we can compute the annulus amplitude between two FZZT branes with parameters z = 2 √ E 0 cosh σ 2 and z = 2 (2.76) The last integral has a divergence coming from E = 0. As discussed in [12], this divergence can be regularized by taking the principal value (2.77) After removing the divergent term, the annulus amplitude is written as This expression agrees with the result of (2, p) minimal string [12]. To summarize, we have shown that the trumpet can be written in terms of the Liouville wavefunction (2.71) and the known result of annulus amplitude in minimal string theory is reproduced by gluing the two trumpets. This justifies our use of trumpet in the general background {t k } away from the JT gravity point t k = γ k .
We note in passing that the annulus amplitude between two FZZT branes can be obtained directly by using (2.15) (2.79) Here we have ignored the logarithmic divergence coming from b = 0. This agrees with (2.78) as expected.

Deformation of JT gravity
In this section we focus on JT gravity and study the effect of adding FZZT branes. We also comment on how it is related to the deformation of the dilaton potential in JT gravity that corresponds to adding conical defects.

Genus-zero density of states
In this section we consider K FZZT branes det(z 2 /2 + H) K in JT gravity background and take the 't Hooft limit 8 As shown in (2.42), adding FZZT branes amounts to shifting the couplings t k where γ k in (2.28) corresponds to the JT gravity background. The Itzykson-Zuber variable I 0 (u) in the shifted background (3. 2) reads Here J 1 (x) denotes the Bessel function. As discussed in [11], the genus-zero density of states is determined in terms of the function f (u) in (3.3) In this section t denotes the 't Hooft coupling, which should not be confused with t = 1 − I1 in (2.51).
Here I 0 (2 √ v) denotes the modified Bessel function of the first kind, which should not be confused with the Itzykson-Zuber variable I 0 (u) in (3.3). The threshold energy E 0 is determined by the condition When t = 0, the coupling (3.2) reduces to the JT gravity background t k = γ k . Indeed, from (3.5) one can see that E 0 = 0 when t = 0 and (3.4) reproduces the known result of JT gravity density of states [38] For t = 0, we do not have a closed form expression of the solution E 0 of (3.5). However, one can easily find the small t and large t behavior of E 0 . From (3.5), one can show that in the small t regime E 0 is expanded as On the other hand, in the large t regime E 0 is large and (3.5) is approximated as This is rewritten as The solution of this equation is given by the Lambert W -function obeying z = W (z)e W (z) (3.10) In the intermediate value of t, one can solve the equation (3.5) numerically. In Figure 3, we show the plot of E 0 as a function of t. Once we know E 0 , we can numerically evaluate the integral (3.4) to find ρ 0 (E). In Figure 4, we show the plot of ρ 0 (E) for t = 3, z = 1, g s = 1 as an example. As a comparison, we also plot the pure JT gravity density of states (3.6) (see orange dashed curve in Figure 4). When z 2 > 0, the eigenvalue corresponding to the FZZT brane is located at E = −z 2 /2 < 0. Since the eigenvalues behave as fermions and they repel each other, the other eigenvalues are pushed toward the positive E direction, as we can see from Figure 4.
On the other hand, if E = −z 2 /2 > 0 the corresponding eigenvalue is inserted on the original support of the pure JT gravity eigenvalue density (3.6). We expect this insertion creates a void in the support of ρ 0 (E). As we increase t, we expect that the support of eigenvalue density is split into two parts and the model exhibits a phase transition from a one-cut phase to a two-cut phase. However  in (2.17). In other words, our formalism (2.20) is valid only for the one-cut phase of the matrix model. To see the effect of FZZT branes with Re(z 2 ) < 0 we need a non-perturbative treatment of FZZT branes, which we will consider in section 5. Before closing this subsection, let us compute the order O(t 1 ) correction of the density of states (3.4). One can show that (3.11) Then the order O(t) correction to the one-point function Z(β) is given by where Z HW (β, z) is the "half-wormhole" amplitude in (2.57). In (3.12) Z HW (β, z) is multiplied by a factor of K since we are considering K FZZT branes. Thus our expression of the genus-zero density of states in (3.4) correctly reproduces the half-wormhole contribution in (2.57).

String equation via the Lagrange reversion
As discussed in [39], the modification of the genus-zero string equation f (u) = 0 with f (u) in (3.3) can also be seen by using the Lagrange reversion theorem. Following [39], let us consider the modification of the genus-zero part of the two-point function Z(β 1 )Z(β 2 ) g=0 c due to the insertion of K FZZT branes det(z 2 /2 + H) K . By using our formalism (2.20) in the general background {t k }, the genus-zero two-point function in the presence of K FZZT branes is written as where V 0,n+2 (b) is given by (2.48) and M(b) for K FZZT branes is given by (3.14) By using (2.45) and (2.47), (3.13) is written as where t 0 -dependence has been explicitly expressed for convenience of explanation. Let us now recall the Lagrange reversion theorem (see e.g. [40]): Suppose that ζ is related to a by the equation with φ(z) being some function. Then for any function g(z) and for small enough t, g(ζ) is expanded as Using this theorem one immediately sees that (3.15) is rewritten as Now, the genus zero string equation u 0 − I 0 (u 0 ) = 0 is written in the off-shell JT gravity background t k = γ k (k ≥ 1) as Writing this equation at t 0 = x with x in (3.19) and then setting t 0 = 0 to restrict ourselves to the on-shell JT gravity background, we have (3.21) We have thus seen that the genus zero two-point function in the presence of K FZZTbranes is written as the genuine genus zero two-point function (3.18) with u 0 (x) satisfying the equation (3.21). This is identical with the modified string equation f (u 0 (x)) = 0 with f (u) given in (3.3), which was obtained from the shift of couplings (3.2).

Comment on dilaton gravity
In [39,41], it is shown that we can add conical defects in JT gravity by modifying the dilaton potential For the defects with deficit angle 2πα i with fugacity ε i , U (φ) is given by As discussed in [39] the effect of defects is summarized as the modification of the string equation (see also [42][43][44] From this we can read off the shifted coupling as Let us consider the relation between this shifted coupling and the dilaton potential U (φ). To see this, it is convenient to define (3.28) For the dilaton potential in (3.23) we find From (3.27), this is also written as As argued in [41], the equivalence of the matrix model and dilaton gravity holds for a wide class of dilaton potential written as a superposition of the defect potential (3.23) Thus we expect that (3.30) can be applied to a wide class of background couplings {t k }.
For instance, if we add K FZZT branes the coupling is shifted as (3.2). Plugging (3.2) into (3.30), the corresponding dilaton potential becomes (3.32) 9 This is obtained as follows. The correction to the one-point function of Z(β) due to defects is given by This is formally equal to the trumpet partition function with the replacement b = √ 2iαi. Using the Lagrange reversion theorem one can show that the Itzykson-Zuber variable I0(u0) is shifted by where I0(b) is defined in (2.46). Thus we find (3.26).
In this approach we can only find the even part of U (φ) = U (φ)e 2πφ . We do not know how to recover the odd part of U (φ) from the data of {t k }. Also, it is not clear to us what is the condition for the applicability of the formula (3.30). We do not know how far we can deform the coupling from the JT gravity background t k = γ k when we use (3.30). Putting this problem aside, if we naively apply (3.30) to the (2, p) minimal model background (A.1) we find In the p → ∞ limit, U + (φ) in (3.33) vanishes, which is consistent with the statement that JT gravity is a p → ∞ limit of the (2, p) minimal string [3,46]. In [27,46,47], it is proposed that a dilaton gravity theory with sinh φ potential corresponds to the (2, p) minimal string. It would be interesting to understand the relation between our (3.33) and the proposal in [27,46,47], if any.

Baker-Akhiezer function and FZZT branes
In this section we review the known results about the Baker-Akhiezer (BA) function, the Christoffel-Darboux (CD) kernel and the multi-FZZT amplitude.

Multi-FZZT amplitude from matrix integral
As discussed in [13], the BA function can be thought of as the wavefunction of FZZT brane, which is obtained as a double-scaling limit of the orthogonal polynomial. In the matrix model at finite it is convenient to define the polynomial P n (λ) = λ n + · · · of degree n which is orthogonal with respect to the weight e −V (λ) dλe −V (λ) P n (λ)P m (λ) = h n δ n,m .

(4.2)
One can show that Z N in (4.1) is written in terms of the norm h n of the orthogonal polynomial P n (λ) It is convenient to define the normalized orthogonal polynomial by which satisfies dλψ n (λ)ψ m (λ) = δ n,m . It is well known that ψ N (λ) is given by the expectation value of the determinant operator in the matrix model (see e.g. [48] for a review) (4.6) where Ψ N (λ) is given by Another important ingredient is the CD kernel Using the relation the CD kernel is written as One can show that the CD kernel (4.8) is given by the two-point function of determinant operators where the expectation value is taken in the (N − 1) × (N − 1) matrix integral. In the double-scaling limit the index n of ψ n (λ) becomes a continuous coordinate t 0 ψ n (λ) → ψ(λ; t 0 ), (4.12) and the double-scaling limit of (4.8) becomes Here is related to the genus counting parameter g s by = g s √ 2 . (4.14) From (4.13), one can see that K(ξ, η) satisfies where ψ(ξ) = ψ(ξ; t 0 ). One can also show that (4.9) reduces to the Schrödinger equation in the double-scaling limit (see [35] for a review) where u is the specific heat Note that the CD kernel of the form (4.10) in the double scaling limit becomes This is also derived from (4.13) by using (4.16). As discussed in [13] the multi-point function of FZZT branes can be obtained by taking the double scaling limit of the finite N expression [49,50] where ∆(ξ) = i<j (ξ i − ξ j ) is the Vandermonde determinant. In the double scaling limit one finds [13] k Here d j is the shorthand for the action of d = ∂ 0 on ψ(ξ j ).

Genus expansion of BA function
Let us consider the genus expansion of the BA function ψ(ξ) = ψ(ξ; t 0 ). It is known that the BA function is written as a ratio of tau-function τ (t) = e F (t) [20,21]: where ξ = z 2 /2 and This is consistent with our result of the shift of couplings (2.42). Let us consider the genus expansion of the BA function The A 0 term is known as the effective potential whose explicit form is obtained in [19] as On the other hand, from (4.24) we find We can show the equivalence of (4.26) and (4.27) as follows. It turns out that (4.26) has the following integral representation This can be shown by using the relations and repeating the partial integration: (4.30) Now, let us decompose (4.28) into two parts One can show that the first term of (4.31) agrees with the first term of (4.27) (4.32) Using the relation [29] F 0 = 1 2 33) one can show that the second term of (4.31) is equal to the second term of (4.27). Thus we find the equivalence of (4.26) and (4.27). We can compute the higher genus corrections of the genus expansion (4.24) by introducing the operator (4.34) Then we find For instance, the first few terms are given by (4.36) In this computation, the following relations are useful From (4.37) and F 1 = − 1 24 log t one can show that This reproduces the known result of the genus expansion of BA function [19] (up to the normalization), as expected.

Genus expansion of the multi-FZZT amplitude
The multi-FZZT amplitude is written as The genus expansion of A(z i ) can be obtained as follows. The Itzykson-Zuber variable for t k in (2.42) is written as The string equation for the shifted background t k becomes From (4.42) we find the genus expansion of u 0 Then the genus expansion of (4.44) Recall that F g (t) (g ≥ 2) are given by polynomials of I k (u 0 ) (k ≥ 2) and (1 − I 1 (u 0 )) −1 [29]. For g = 0 and g = 1 we have

Annulus amplitude between two FZZT branes
Let us consider the annulus amplitude between two FZZT branes. From (4.40) we have The prefactor agrees with the exponentiated annulus amplitude (2.78) up to an overall normalization constant. On the other hand, from (4.20) we find At the leading order in the genus expansion, the prefactor becomes where we used the relation obtained from (4.26) The difference between the prefactors of (4.46) and (4.48) is compensated by (4.50) Thus the two expressions (4.46) and (4.48) are consistent at this order.

FZZT amplitude and CD kernel
As shown in [52], the correlator of 2k determinants in the finite N matrix model is written in terms of the CD kernel as . . . · · · . . . .
(4.52) In the double scaling limit, we find where ψ(ξ) and K(ξ, η) are the BA function and the CD kernel (4.13), respectively. In other words, the correlator of the even number of FZZT branes can be written as a product of two-point functions K(ξ, η). Although the permutation symmetry of the parameters ξ i and η i is not manifest in (4.53), the result is symmetric as proved in [52]. Of course, one can use (4.20) for the multi-point function of FZZT branes. However, ∂ n 0 ψ(ξ) with n ≥ 2 can be reduced to a linear combination of ψ(ξ) and ∂ 0 ψ(ξ) using the Schrödinger equation (4.16), and after this rewriting one finds that (4.20) is identical with (4.53).

Correlator of FZZT branes and macroscopic loops
In this section, we consider the correlator of FZZT branes Ψ(ξ i ) and macroscopic loop operators Z(β j ).

Bra-ket notation
To describe our results, it is convenient to introduce the bra-ket notation as follows.
Letx be the coordinate operator and∂ x be its conjugate "momentum operator" satisfying We then introduce the operator where u is the specific heat (4.17). We decided not to put a hatˆon Q just for notational simplicity. For a Roman letter x, we let |x denote the coordinate eigenstate, e.g.
For a Greek letter ξ, on the other hand, we let |ξ denote an eigenstate of Q with eigenvalue ξ: These states are normalized such that where ψ(ξ; x) is the BA function. Another important element is the projection operator As discussed in [53], Π can be interpreted as the projection below the Fermi level t 0 . In terms of Π the CD kernel (4.13) is expressed as Note that Π satisfies Note also that t 0 | and |t 0 satisfy so that the Schrödinger equation (4.16) holds.

Correlator of one FZZT brane and macroscopic loops
As a first step, we consider the correlator of one FZZT brane and macroscopic loops. Following [4], let us consider the correlator of det(ξ − M ) and Tr e βM at finite N : where Z N is given by (4.3) and M is related to H by M = −H (see footnote 2). It follows from (4.53) that in the double scaling limit this becomes = dλe βλ K(λ, λ)ψ(ξ) − K(ξ, λ)ψ(λ) .

(5.11)
In appendix B, we will give an alternative derivation of this result. Note that ρ(λ) = K(λ, λ) is the eigenvalue density. Thus the first term of (5.11) is the disconnected part Z(β) Ψ(ξ) and the second term is the connected part In the last step we have used (5.4) and (5.5).
Next consider the correlator of one FZZT brane with two macroscopic loops Tr e β 1 M Tr e β 2 M Ψ N (ξ) (5.13) From (4.53), in the double scaling limit this becomes (5.14) One can see that the connected part is given by In the same way, one can calculate the correlator of one FZZT brane and three macroscopic loops. The result is As a non-trivial check of this conjecture, in what follows we will prove that ψ(ξ) given above obeys the Schrödinger equation. We first notice that the connected correlator in (5.17) is obtained by applying the boundary creation operators to the BA function ψ(ξ) Z(β 1 ) · · · Z(β n )Ψ(ξ) c = B(β 1 ) · · · B(β n )ψ(ξ) =: ψ n (β 1 , . . . , β n ; ξ). (5.20) Applying the boundary creation operator to the both sides of the Schrödinger equation (4.16), we obtain Here . . , n} and the sum is taken for all possible proper subsets I of S including the empty set. In terms of the generating function, (5.21) is simply written as Here Z is the generating function of the connected correlator of macroscopic loops such that Z(β 1 ) · · · Z(β n ) c = Z O(w 1 ···wn) .
where we have introduced the notation Next consider the first term of (5.23). It is rewritten as 29) The first term of (5.29) is The second term of (5.29) is The third term of (5.29) is Finally we find

General correlator of FZZT branes and macroscopic loops
We found that the generating function (5.18) satisfies (5.23). The key observation is that (5.23) is itself a Schrödinger equation with modified potential where u = 2 2 ∂ 2 0 F with F being F = F + Z = F + Tr log G. (5.35) This suggests that ψ(ξ) is a BA function in a certain modified background. To see this, let us consider the correlator of Z(β)'s and FZZT branes at finite N where the deformed potential V (M ) is given by Since (5.36) is just the correlator of determinants in the deformed matrix model integral, it also takes the form (4.51). In the double scaling limit we find The overall factor det G = e F −F accounts for the different normalization of the matrix integrals. For odd number of FZZT branes we have The CD kernel in the modified potential can be found from the condition (4.15) We find that K(ξ, η) is given by Let us see that (5.41) indeed satisfies (5.40): The combination 1 − ΠG −1 A in the first factor is written as (5.43) Thus (5.42) becomes This confirms (5.41).
To summarize, the generating function of the correlator of FZZT branes and macroscopic loops is given by (5.38) for even number of FZZT branes and (5.39) for odd number of FZZT branes. Our formulae (5.38) and (5.39) do not rely on the genus expansion and in principle they can be defined non-perturbatively.

Airy case
In this section we consider the Airy case corresponding to the trivial background t k = 0 (k ≥ 1). In this case u = t 0 and the Schrödinger equation for the BA function is The solution to this equation is given by the Airy function Let us apply our formula (5.12) to the Airy case. For simplicity we set t 0 = 0. Then (5.12) becomes where 0| = x = 0|. As shown in [55] the matrix element of e βQ is given by Thus we find In the → 0 limit, the µ integral can be evaluated by the saddle point approximation.

Spectral form factor in FZZT brane background
Let us consider the two-point function of macroscopic loops in the presence of two FZZT branes As in the previous subsection, we put two FZZT branes since single FZZT brane is not positive definite. From our general formula (5.38) we find g(β 1 , β 2 , ξ) = Z(β 1 )Z(β 2 ) We can show that this is rewritten as g(β 1 , β 2 , ξ) = dλe (β 1 +β 2 )λ 1 K(ξ, ξ) K(λ, λ) K(λ, ξ) K(ξ, λ) K(ξ, ξ) . (6.20) We consider the spectral form factor (SFF) in the presence of two FZZT branes Ψ(ξ) 2 at ξ = −E g(t, β, E ) = g(β + it, β − it, −E ). (6.21) Using the CD kernel for the Airy case (6.12) and the integral representation (6.20), we can evaluate this SFF (6.21) numerically. In Figure 6 we show the plot of SFF in the Airy case with two FZZT branes for β = 1/100 with several different values of E . For E = −3 and E = 3, we do not see a notable deviation from the original SFF without the FZZT branes (orange dashed curve in Figure 6), but the E = 20 case in Figure 6c has some oscillatory behavior. As discussed in [4], we expect that the erratic behavior arises if we put many FZZT branes. It is tempting to speculate that the oscillatory behavior in Figure 6c might be an indication towards the erratic behavior as we increase the number of FZZT branes. It would be interesting to study the SFF with many FZZT brane insertions explicitly in the Airy case. However, this is a challenging problem since the generalization of (6.20) for many FZZT brane insertions is a multi-variable integral which is not easy to evaluate numerically with high precision. We leave this as an interesting future problem.

Conclusions and outlook
In this paper we have studied FZZT branes in JT gravity and topological gravity. We found that FZZT branes can be introduced in the matrix model of JT gravity [3] by attaching M(b) = −e −zb to the geodesic boundary of length b and integrating over b (see Figure 1). Our construction can be generalized to arbitrary background {t k } of topological gravity by introducing the generalized WP volume (2.37). We argued that the trumpet ending on a FZZT brane can be thought of as a "half-wormhole" introduced in [10] (see Figure 2). We found that the half-wormhole amplitude is given by the complementary error function (2.57). FZZT branes induce the shift of couplings (2.42) which is consistent with the known property of BA function in the literature. However, the expression such as (4.21) only gives an asymptotic expansion of the BA function in a certain sector of the complex z-plane (2.19). In section 5 we found the general formulae (5.38) and (5.39) for the generating function of the correlator of FZZT branes and macroscopic loops. Our formula expresses the correlator of FZZT branes and macroscopic loops in terms of the BA function and the CD kernel, which in principle can be defined non-perturbatively. As an example, in section 6 we studied the eigenvalue density and the spectral form factor (SFF) deformed by the insertion of two FZZT branes in the Airy case. In this explicit computation, we confirmed the picture of "eigenbranes" put forward in [4].
There are many interesting open questions. It would be interesting to generalize our study of SFF by inserting more FZZT branes to see if we recover the erratic behavior of SFF as anticipated in [4]. Another interesting direction is the derivation of the Page curve in our formalism of FZZT branes. In [7], it is argued that the Page curve is reproduced by adding EOW brane degrees of freedom to JT gravity. As we have seen in (2.27), the EOW brane in [8] is a special case of our formalism, corresponding to an infinite collection of anti-FZZT branes. It would be interesting to apply our formalism of FZZT branes to this problem. We should also mention that as argued in [13] the "meson" field S constructed out of the fermions in (2.12) S ij = χ ia χ aj (7.1) can be identified as the matrix appearing in the Kontsevich model e F (t) = dSe i 2gs Tr S 3 3 +ZS 2 , t k = −g s (2k − 1)!! Tr Z −2k−1 .

(7.2)
It would be interesting to study, say, the second Rényi entropy e −S 2 = Tr S 2 Tr S 2 (7.3) in Kontsevich model in the JT gravity background with FZZT branes (3.2) to see if the Page curve is recovered. We leave this as an interesting future problem.
where P m (x) denotes the Legendre polynomial. The genus-zero eigenvalue density is given by and the disk amplitude is given by One can easily see that in the p → ∞ limit the eigenvalue density of the (2, p) minimal string (A.3) reduces to the eigenvalue density of JT gravity in (3.6).
(B.11) In the double scaling limit this becomes This reproduces the deformed eigenvalue density due to the insertion of one FZZT brane in (5.11). One can in principle derive the multi-point correlator of Z(β)'s and FZZT branes in this approach using the result in [52,56,57]. We leave this as an interesting future problem.