End of the world brane in double scaled SYK

We study the end of the world (EOW) brane in double scaled SYK (DSSYK) model. We find that the boundary state of EOW brane is a coherent state of the $q$-deformed oscillators and the associated orthogonal polynomial is the continuous big $q$-Hermite polynomial. In a certain scaling limit, the big $q$-Hermite polynomial reduces to the Whittaker function, which reproduces the wavefunction of JT gravity with an EOW brane. We also compute the half-wormhole amplitude in DSSYK and show that the amplitude is decomposed into the trumpet and the factor coming from the EOW brane.


Introduction
The double scaled SYK (DSSYK) model [1,2] is a useful toy model for the study of quantum gravity and holography.As shown in [2], DSSYK is exactly solvable by using the technique of the chord diagrams and the transfer matrix.DSSYK allows us to explore the high energy regime of holography and quantum gravity, beyond the realm of JT gravity and the Schwarzian modes at low energy.The bulk dual of DSSYK has a peculiar feature: the length scale is quantized in units of the coupling λ and the geodesic length is replaced by the discrete chord number n [3].In [4], it is also suggested that the length of geodesic loop b is replaced by a discrete length b as well.
To understand the bulk dual of DSSYK better, in this paper we consider the end of the world (EOW) brane in DSSYK.EOW branes in JT gravity were very useful for the study of black hole information problem and the Page curve of Hawking radiation [5].If we quantize JT gravity in the presence of an EOW brane, the problem boils down to a quantum mechanical problem of a particle moving in the Morse potential, and the wavefunction is given by the Whittaker function [6].We will show that the DSSYK analogue of the Whittaker function is the continuous big q-Hermite polynomial H n (x, a|q), where a is related to the tension µ of EOW brane by a = q µ+ 1 2 . (1.1) We also find that the boundary state |B a ⟩ of the EOW brane is a coherent state of the q-deformed oscillator where A ± acts on the chord number state |n⟩ as a creation and annihilation operator of chords.We find that the JT gravity amplitude of the half-wormhole ending on the EOW brane [6] has a complete parallel in DSSYK.We find that the half-wormhole amplitude in DSSYK is decomposed into the trumpet and the factor a b 1−q b coming from the EOW brane where the trumpet partition function is given by dθ 2π e −βE(θ) 2 cos(bθ). (1.4) Note that the integral over the length b of geodesic loop is replaced by the sum over the discrete length b in (1.3).This paper is organized as follows.In section 2, we review the transfer matrix formalism of DSSYK.In section 3, we review the EOW brane in JT gravity and the computation of the half-wormhole amplitude in [6].In section 4, we consider the EOW brane in DSSYK.We find that the wavefunction in the presence of EOW brane is given by the big q-Hermite polynomial.We also find that the boundary state of EOW brane is a coherent state of the q-deformed oscillators.In section 5, we compute the half-wormhole amplitude in DSSYK.Along the way, we find the trumpet partition function in DSSYK.In section 6, we study the cylinder amplitude in DSSYK, which is obtained by gluing two trumpets.Finally we conclude in section 7.In appendix A we summarize useful formulae of the q-Pochhammer symbols.In appendix B we summarize useful properties of the continuous big q-Hermite polynomial.

Review of DSSYK
We first review the result of DSSYK [2].SYK model is defined by the Hamiltonian for N Majorana fermions ψ i (i = 1, • • • , N ) with all-to-all p-body interaction where J i 1 •••ip is a random coupling drawn from the Gaussian distribution.DSSYK is defined by the scaling limit SYK model is exactly solvable in this double scaling limit.As shown in [2], the ensemble average of the moment tr H k reduces to a counting problem of the intersection number of chord diagrams ⟨tr where q is given by q = e −λ . (2.4) One can introduce a matter field in the bulk which is dual to an operator in DSSYK.One simple example is the length s strings of Majorana fermions with Gaussian random coefficients K i As shown in [2], the counting problem of the chord diagram can be solved by introducing the transfer matrix T where A ± are the q-deformed oscillators (2.9) A ± acts on the states {|n⟩} n=0,1,••• as where n labels the number of chords and A ± creates/annihilates the chord.Then the disk partition function of DSSYK is written as One can diagonalize T by using the q-Hermite polynomial ⟨θ|n⟩ = H n (cos θ|q) which is orthogonal with respect to the measure µ(θ) defined by1 µ(θ) = (q, e ±2iθ ; q) ∞ , (2.13) and the orthogonality relation reads Note that |θ⟩ is normalized as and the resolution of the identity is written as The transfer matrix T is diagonal in the |θ⟩-basis where the eigenvalue E(θ) is given by One can see that the energy spectrum is supported in the range E ∈ [−u 0 , u 0 ] and u 0 specifies the edge of the spectrum.Now the disk partition function (2.11) becomes One can also compute the correlator of matter operators in DSSYK using the technique of the chord diagrams and the transfer matrix T .After the Wick contraction of where N is the number operator As shown in [2], B β,∆ is invariant under the time translation and commutes with T T, B β,∆ = 0.
(2. 22) This implies that T and B β,∆ are simultaneously diagonalized in the |θ⟩-basis and the eigenvalue of B β,∆ is given by where the factor ⟨θ|q ∆ N |θ ′ ⟩ in (2.23) is given by This factor also appears in the two-point function of matter operators See also [7][8][9] for the semi-classical limit of matter correlators in DSSYK.

EOW brane in JT gravity
In this section, we review the wavefunction of JT gravity in the presence of an EOW brane [6].As shown in [6], quantization of JT gravity with an EOW brane leads to the Hamiltonian with Morse potential and the eigenvalue equation for this Hamiltonian reads where L denotes the renormalized geodesic length between the EOW brane and the AdS 2 boundary where SYK lives, and µ is the tension of the brane.Note that L can be negative since L is a renormalized length.One can show that (3.1) is solved by the Whittaker function which is normalized as where ρ(k) is the density of states of the Schwarzian theory [10] As discussed in [6], one can extract the coupling between the trumpet and the EOW brane by considering a half-wormhole ending on the EOW brane.The half-wormhole amplitude is defined by which is schematically depicted as The integral over L represents a trace on the Hilbert space spanned by the states {Ψ k (L)} and the top and the bottom of the left hand side of (3.6) are periodically identified.Thus the corresponding spacetime is a half-wormhole as shown on the right hand side of (3.6).This trace is obviously divergent due to the first relation in (3.3).However, as demonstrated in [6], one can rewrite this amplitude as an integral over the length b of geodesic loop and interpret this divergence as a UV divergence coming from b = 0.
Here we review the computation in [6], which serves as a warm-up for a similar computation of DSSYK in section 5.As we will see in section 5, the computation below has a complete parallel in DSSYK.First, we rewrite Ψ k (L) 2 appearing in (3.5) by using the formula 6.647-1 in [11] where z = e −L and K ν (x) denotes the modified Bessel function of the second kind.If we further make a change of variable then (3.7) becomes with Then the half-wormhole amplitude (3.5) is written as where K β (b, z) is the so-called boundary particle propagator [12][13][14] 2 The integral over L can be performed by using the integration formula of the modified Bessel function which was heavily utilized in the study of minimal string theory [16].Using this formula, we find Using (3.14), the L integral of the boundary particle propagator in (3.12) becomes3 where the trumpet partition function is given by This agrees with the known result of trumpet in JT gravity [17], as expected.Note that ρ(k) is canceled in this computation and the trumpet is independent of ρ(k).Finally, we arrive at the desired expression of the half-wormhole amplitude (3.17) This amplitude has a divergence coming from b = 0, which is interpreted as a UV divergence in the bulk gravitational theory.

EOW brane in DSSYK
In this section, we consider the EOW brane in DSSYK.It turns out that the big q-Hermite polynomial H n (x, a|q) plays the role of the Whittaker function in (3.2).

Triple scaling limit
The continuous big q-Hermite polynomial H n (x, a|q) is a one parameter generalization of the q-Hermite polynomial H n (x|q).H n (x, a|q) reduces to H n (x|q) when a = 0. H n (x, a|q) is defined recursively by the three-term recursion relation with the initial condition H −1 (x, a|q) = 0, H 0 (x, a|q) = 1.An important property of H n (x, a|q) is the orthogonality relation It is convenient to introduce the normalized wavefunction which satisfies the orthogonality relation We will show that ψ n (θ) is the DSSYK analogue of the wavefunction Ψ k (L) (3.2) of JT gravity with an EOW brane.In terms of the wavefunction ψ n (θ) in (4.3), the three-term recursion relation (4.1) is written as To see the semi-classical picture of (4.5), we consider the triple scaling limit with µ, L, and k fixed.If we denote ψ n (θ) = Ψ k (L) 4 in this limit, the right hand side of (4.5) becomes Similarly, the left hand side of (4.5) is expanded as Finally, equating the O(λ 2 ) term of the both sides of (4.5) we find which is exactly the equation (3.1) for JT gravity with an EOW brane!This strongly suggests that ψ n (θ) in (4.3) is the DSSYK analogue of the EOW wavefunction Ψ k (L).
For the small a expansion to be well-defined, from (4.6) we see that µ should satisfy the inequality µ > − 1 2 .This inequality also appears in JT gravity with an EOW brane [6].

Boundary state of EOW brane
Let us consider a physical interpretation of the extra measure factor (4.11).We propose that the extra factor (4.11) is the θ-representation of the boundary state |B a ⟩ of the EOW brane 1 (ae ±iθ ; q) ∞ = ⟨θ|B a ⟩.(4.14) Using the formula (B.7), the left hand side of (4.14) is expanded as where we used (2.12) in the last equality.From (4.14) and (4.15), |B a ⟩ is given by Then the partition function in the presence of the extra factor ⟨θ|B a ⟩ is written as This expression suggests that |B a ⟩ is the boundary state of EOW brane and (4.17) is the partition function of half-disk ending on the EOW brane where the blue thick line represents the worldline of the EOW brane.
The boundary state |B a ⟩ in (4.16) has an interesting characterization: it is a coherent state of the q-deformed oscillator which can be shown by using the first relation of (2.10).|B a ⟩ is also written as where we used the relation and the formula (A.5).In the a → 0 limit, |B a ⟩ reduces to the Hartle-Hawking vacuum |0⟩ [18] lim and the half-disk amplitude (4.17) reduces to the disk amplitude (2.11) Thus, a = 0 corresponds to the absence of EOW brane.In fact, the big q-Hermite polynomial H n (x, a|q) reduces to the q-Hermite polynomial H n (x|q) when a = 0. From the relation a = q µ+ 1 2 , the absence of EOW brane corresponds to µ = ∞, not µ = 0.This is similar to the situation considered in [19]: a heavy object in the bulk corresponds to a projection to the vacuum |0⟩.For instance, the large dimension limit of the two-point function (2.25) of matter operator One might naively think that the heavy operator corresponds to a black hole in the bulk, but this is not the case in DSSYK: if we try to create a black hole by putting two heavy operators on the boundary, then the bulk spacetime is pinched and splits into two pieces and the black hole never forms!

Representation of the q-oscillator algebra
Let us consider the action of A ± on the continuous big q-Hermite polynomials.To this end, it is convenient to introduce the notation The action of A ± on the basis |H n ⟩ is given by From (B.3), the big q-Hermite state |H n , a⟩ is written as a linear combination of the q-Hermite state From (4.27), one can deduce the action of A ± on the big q-Hermite basis from which one can easily check that the q-oscillator algebra (2.9) is realized on the big q-Hermite basis We should stress that the parameter a does not deform the algebra itself.Instead, a labels a representation of the q-oscillator algebra (2.9).From (4.29), one can see that A + + A − acts on the big q-Hermite basis as which is exactly the same combination appearing in the three-term recursion of the big q-Hermite polynomial (4.1).Thus the transfer matrix T = A + +A − √ 1−q is diagonal on the basis of H n (cos θ, a|q) as well, with the same eigenvalue 2 cos θ √ 1−q as the original one without EOW brane.Namely, we can use the common transfer matrix T = A + +A − √ 1−q for both with and without the EOW brane.The only difference is the presence of the extra factor ⟨θ|B a ⟩ for the case with the EOW brane pure DSSYK : DSSYK with EOW brane : (4.32)

Half-wormhole in DSSYK
Let us consider the DSSYK analogue of the computation in section 3. Using the dictionary The sum over n in (5.2) is divergent due to the relation n ψ n (θ) 2 ∝ δ(θ − θ).However, we can rewrite the amplitude (5.2) as a sum over the discrete length b of geodesic loop, in a similar manner as the computation in the JT gravity case in section 3.In analogy with (3.9), we define G(b, n, θ) by Using the small a expansion of ⟨θ|B a ⟩ in (4.15) and the expansion of big q-Hermite polynomial in (B.3), together with the linearization formula of the q-Hermite polynomials (B.9), after some algebra we find (q; q) n (q; q) 2 n−k (q; q) k q kb (q; q) b H 2n−2k+b (cos θ|q). (5.5) Using the DSSYK analogue of the relation (3.13) 5   2 cos(bθ) H 2k+b (cos θ|q) (q; q) k (q; q) k+b , (5.8) we find µ(θ) .
(5.9)This is the analogue of (3.14).It is also natural to define the DSSYK analogue of the boundary particle propagator In analogy with (3.15), the sum over n of K β (b, n) is written as where we introduced the DSSYK analogue of the trumpet Note that in this computation µ(θ) is canceled and Finally, the half-wormhole amplitude (5.2) is written as (5.13) See (1.3) for a pictorial representation of (5.13).If we set a = q µ+ 1 2 , (5.13) becomes which is the DSSYK analogue of (3.17) where the integral over b is replaced by the sum over b.Note that the b = 0 term in (5.14) is divergent: this is an analogy of the fact that the integral in (3.17) has a UV divergence coming from b = 0.

Trumpet and cylinder in DSSYK
Plugging E(θ) = −u 0 cos θ (2.18) into (5.12), the trumpet of DSSYK becomes 6 where I ν (x) denotes the modified Bessel function of the first kind.As discussed in [4], the cylinder amplitude is obtained as a "double trumpet" by gluing two trumpets 7 This is schematically depicted as we can perform the summation over b in (6.2) in a closed form Note that (6.5) is equal to the cylinder amplitude in the Gaussian matrix model [21][22][23].This is expected since the cylinder amplitude is independent of the details of the matrix model potential and it depends only on the endpoints ±u 0 of the spectrum [24,25]. 8 By using the asymptotic form of the modified Bessel function the low temperature limit of Z cylinder (β 1 , β 2 ) becomes e (β 1 +β 2 )u 0 .(6.7) 6 From the usual large N counting, the genus-g with n-boundary amplitude comes with the factor of e (2−2g−n)S 0 , where e S 0 = 2 N/2 is the dimension of the Hilbert space of N Majorana fermions.For instance, the trumpet with (g, n) = (0, 2) scales as e 0S 0 which is suppressed relative to the disk with (g, n) = (0, 1) of the order e S 0 .In this paper, we have suppressed this factor e (2−2g−n)S 0 for simplicity. 7See also [20] for the study of non-planar corrections in DSSYK. 8As argued in [26], trumpet in topological gravity is also independent of the background {t k }.
This agrees with the known result of cylinder amplitude in JT gravity [17], up to the factor e (β 1 +β 2 )u 0 coming from the threshold energy E = −u 0 .In the semi-classical limit λ → 0 with the trumpet in (6.1) reduces to up to an overall factor.This reproduces the known result of trumpet in JT gravity, as expected.In the semi-classical limit (6.8), the sum over b in (6.2) is replaced by the integral over b which is also consistent with the JT gravity result.One advantage of DSSYK over the continuum approach of JT gravity is that one can easily incorporate the effect of matter field within the framework of chord diagrams.According to [4], if we include the loop correction of the matter field with dimension ∆, the cylinder amplitude is modified as In the semi-classical limit (6.8), the denominator is replaced by 1 − e −b − e −∆b which vanishes at a certain value of b.This leads to a divergence which was interpreted as a Hagedorn divergence in [4].On the other hand, the denominator in (6.11) does not vanish for a generic value of q, ∆ with b ∈ Z ≥0 .Thus we expect that the Hagedorn divergence from the matter loop is regularized in DSSYK.We do not know how to perform the sum over b in (6.11) in a closed form, but we can easily evaluate (6.11) numerically by truncating the summation up to some cut-off b ≤ b cut .In figure 1, we show the plot of spectral form factor (SFF) Z cylinder (β + it, β − it, ∆) as a function of t.When ∆ is large, the effect of matter loop is negligible and the SFF exhibits a linear growth called "ramp" (see figure 1a).When ∆ becomes small, the effect of matter loop drastically changes the behavior of SFF; as we can see from figure 1b, SFF exhibits an oscillatory behavior as a function of t.It would be interesting to understand the bulk gravitational interpretation of this oscillation.

Conclusions and outlook
In this paper, we have studied the EOW brane in DSSYK.We found that the boundary state of EOW brane is a coherent state of the q-deformed oscillator and the associated We set u 0 = 1, β = 1, q = 1/2 in (6.11) and evaluate it numerically by truncating the summation up to b cut = 100.
orthogonal polynomial is the continuous big q-Hermite polynomial H n (x, a|q).We showed that ψ n (θ) in (4.3) reduces to the Whittaker function in the triple scaling limit (4.6).We also computed the half-wormhole amplitude in DSSYK and found the decomposition (5.13) into the trumpet and the factor a b 1−q b , which is a complete parallel of the decomposition of half-wormhole amplitude (3.17 Thus the bulk geodesic lengths L and b are quantized in units of λ and there is a minimal length λ in the bulk spacetime.It would be interesting to understand the physical meaning of this minimal length. 9e can generalize our setup by introducing two EOW branes and consider the amplitude Then the measure factor becomes and the associated orthogonal polynomial is the Al-Salam-Chihara polynomial Q n (x, a, b|q).This type of boundary condition has been considered in [28] in the context of ASEP (asymmetric simple exclusion process) under the identification It would be interesting to pursue this analogy with ASEP further.One missing ingredient in this analogy is the relation to the spin chain.In the context of ASEP, the problem can be mapped to the matrix product state of open XXZ spin chain and the quantum group naturally arises in the spin chain picture (see e.g.[29] for a review).It would be interesting to understand the role of spin chain in DSSYK, if any.As discussed in [4], one can construct various amplitudes of DSSYK by gluing basic building blocks such as the trumpet, in a similar manner as the JT gravity matrix model [17].However, we still do not have a complete set of the "Lego block" of DSSYK.For instance, we do not know the DSSYK analogue of the Weil-Petersson volume, which played an important role for the construction of higher genus amplitudes of JT gravity.Also, we have only briefly studied the effect of matter loop in section 6.Clearly, more work needs to be done to better understand the structure of DSSYK.

. 24 )
Note that the right hand side is the product of two disk amplitudes.(4.24) is schematically depicted