(Half) Wormholes under Irrelevant Deformation

Recently it has been shown by Almheiri and Lin [1] that the reconstruction of black-hole interior is sensitive to knowing the exact coupling of the boundary theory even if the coupling is irrelevant. This motivates us to enlarge the set of the one-time and two-time toy models inspired from the SYK by deforming the same with \textit{irrelevant} coupling. We find that both half-wormholes as well as the wormholes persist in presence of the deformation, leading to a similar mechanism for curing the factorization problem. While for the one time case, the deformed partition function and its moments change by an overall factor, which can completely be absorbed into a renormalization of coupling, for the two time (or coupled one time) SYK we find non-trivial dynamics of the saddles as the couplings are varied. Curiously, the \textit{irrelevant} deformations that we consider can also be thought of as an ensemble average over an overall scaling of the original undeformed Hamiltionian with an appropriate probability distribution, this allows for the possibility that half-wormholes may also be present in suitably defined ensemble of theories.


Introduction and summary
In AdS/CFT, the factorization problem arises if we have a bulk manifold X with two (or multiple) disconnected conformal boundaries say L and R. From the boundary perspective, the partition function Z L∪R on L ∪ R should factorize as a product of two partition functions Z L and Z R , defined solely on the L and R component respectively. On the other hand, the holographic dictionary instructs us to perform a bulk computation summing up contribution from all possible choice of X. This allows for in particular a contribution from wormhole geometries connecting the L and R. Therefore the bulk computation seems to capture the correlation between two boundary quantities i.e Z L Z R − Z L Z R , leading to non-factorization. Following [2], let us call these kind of contribution as connected amplitudes with disconnected boundaries i.e CADB. They have recently gained lot of interest. In particular, the wormhole saddle contributions are responsible for the late time behaviours of the spectral form factor [3,4] and correlators [5], of holographic theories in the ramp regime, as well as the late time Hawking radiation entropy [6,7].
The CADB leads to factorization problem only if one insists on having a unitary dual conformal field theory with fixed Hamiltonian. The puzzle can be avoided if one instead assumes the Hamiltonian is drawn from a random matrix ensemble. In fact, in 2D JT gravity, this is precisely what happens, the dual theory of JT gravity is not a single fixed theory rather an ensemble of theories [4], hence provides a plausible explanation why CADB exists from the boundary perspective. Even though ensemble average saves a potential embarassment, this raises puzzle in cases where we expect and know the AdS/CFT duality to work without any averaging i.e when a theory with specific set of couplings in the boundary is dual to a bulk theory with specific set of parameters (as in the original examples of AdS/CFT based on maximally supersymmetric QFT models).
The dichotomy is that on one hand wormhole contributions are important for instance to find the ramp in the spectral form factor (SFF) while on the other hand they spoil factorization. Therefore it is desirable to search for wormholes even in the factorisable observables and naturally we are led to find other contributions which restore factorization in those scenarios. Moreover, the SFF is not self averaging [8], for a single fixed theory there are erratic fluctuations around the ramp, hence there should be some bulk contribution responsible for these fluctuations. As the problem for the full SYK model is a difficult one, the authors of [9,10] focussed on the SYK model at a single time instance and within this toy problem resolved factorization by discovering half -wormholes on top of the usual wormhole contributions. We emphasize that in these toy models, the wormholes survive even in the un-averaged theory and the halfwormholes added to it restore factorization. See also [11] and [12] on general mechanism of how half-wormholes can possibly cure the facorization puzzle. It worths mentioning that [13,14] discuss 2-dimensional gravity models where they can successfully address the factorization puzzle. While the former discusses an interpolating model between a random matrix ensemble and a fixed Hamiltonian, the latter utilizes non-local interactions in the action.
In this short note we enlarge the set of such toy examples by considering irrelevant deformations of the one (and two-) time SYK model. Since the toy models are zero dimensional, any deformation is marginal. Our nomenclatures are made with reference to the actual SYK model. Let us motivate why we care about irrelevant deformations. First of all, one might wonder how robust is the prediction of one time SYK model in terms of having wormholes and half-wormholes. More importantly, it has been pointed out recently that the reconstruction of black hole interior is highly theory sensitive [1], in particular depends on the marginal and irrelevant couplings non-trivially. Hence it is a natural step to deform the original one time SYK model by irrelevant deformations and ask about the presence of wormholes and half-wormholes, to which we answer affirmatively. On top of the one time case, we also consider the coupled (two time instances) case which is closer to the actual SYK model.
The relevant quantities that has been studied in [9,10] for the undeformed case are given by We note that ζ SYK (0) = z 2 SYK . Non-zero µ can be thought of as two-time or coupled one-time SYK model. One of the salient conclusions from [9,10] is and z 2 SYK is identified with the wormhole contribution. Later we will refer them as moments of partition function. Furthermore, it has been shown in [10] that the wormholes appear as the final term in a perturbation series around the half-wormholes. Finally, the half-wormholes are responsible for curing the factorization problem.
We study the above physical observables for irrelevant deformation of H 0 , denoted as f (H 0 ). In particular, we expound upon • Several interesting features come out of the analysis

(Half )-Wormholes persist after Deformation:
We find that both half-wormhole as well as the wormhole persist in presence of the deformation, leading to a similar mechanism for curing the factorization problem.
2. Wormholes as large fluctuations around Half-Wormhole even after Deformation: Just like the original undeformed model, the wormhole can be thought of as a large fluctuation around the half-wormhole saddle. One need not add it separately. The scenario mimics the case of tensionless string where one can prove the background independence of the partition function [15].

Deformation = Renormalization:
For the one time case, the deformed partition function changes by an overall factor, which can completely be absorbed into a renormalization of coupling J amongst fermions. Quantitatively, consider the following deformation f (H 0 ) (E 0 is the ground state energy of the undeformed model) and define the function B so that for the one time model, the renormalized coupling J renormalized is given by For more details, see the end of §2, in particular eq. (41). For the two time (or coupled one time) SYK we find non-trivial dynamics of the saddles as the couplings are varied. In particular one can understand the deformation for the two-time case as renormalization of the two-time coupling µ. In this case the wormholes lose their self-averaging characteristics if the renormalized µ renormalized becomes very large. On the other hand, in the regime, 1/N µ renormalized 1, the wormholes persist and dominate over half-wormhole just like in the undeformed case.
4. Deformation = Ensemble average over Overall Scale: The irrelevant deformations that we consider can also be thought of as an ensemble average over an overall scaling of the original undeformed Hamiltionian (along with possible shift in ground state energy) with an appropriate probability distribution. This allows for the possibility that the half wormhole may persist in a suitably defined ensemble theory as well. In particular, we can express eq. (3) for appropriate p( ), which depends on the deformation coupled with a possible shift in the ground state energy E 0 : Specifically, for TT deformation and Gaussian deformation, we have Here λ, s, * parametrize the deformation. For more details, see §2 and §4. It is worth mentioning that in [16], it has been shown using harmonic analysis on SL(2, Z) that the fixed N ensemble average of SL(2, Z) invariant physical observable over the N = 4 supersymmetric conformal manifold upon taking the large N limit reproduces the large N, large 't-Hooft coupling limit. This provides another example where the ensemble average is equivalent to a theory with particular coupling, albeit in some particular limit. See also [17] for the implication of harmonic analysis of SL(2, Z) in context of 2D CFT. In recent years, the ensemble average over some appropiate moduli space of 2D CFT has been considered in many papers, which includes [18][19][20][21][22][23][24][25][26][27][28], see also [29][30][31]. It deserves mention that the ensemble average over overall scaling induces a deformation is briefly discussed in the concluding section of [1].
Deformations of the SYK theory have previously been studied in [32,33]. The 1d analog of the TT deformation was shown to be equivalent to coupling an undeformed theory to the worldline quantum gravity. The generic deformation H 0 → f (H 0 ) is speculated to couple the undeformed theory to other 1d quantum gravities [32]. For the SYK model, [33] finds that the effect of the deformation H(J) → H(J) + λTT , can essentially be captured in the renormalization of the coupling of the theory : J → J renormalized = J(λ). It is also quite important to normalize the vacuum energy by adding a shift: H 0 → H 0 − E 0 . This constant shift though trivial for the undeformed theory, gives inequivalent deformed theories. In particular the renormalization becomes trivial when E 0 = 0. We show that similar features are also present in deformation of one and two-time SYK model. It deserves mentioning that the TT deformation in context of JT gravity is studied in [34] (see also [35]).
The organization of the rest of the note along with glimpses of some results are as follows. In §2 we review the one-time undeformed SYK model, then go on studying its TT deformation. The deformed version is investigated in two different ways. The first method involves using einbein to recast the model as original SYK model with an intergral over the einbein variable. The integral over einbein for the TT deformation can also be interpreted as ensemble averaging. From the exact answer we conclude that both wormholes as well as half-wormholes are present in the deformed theory. The other method involves explicitly evaluating the z 2 deformed as a finite perturbation series around the half-wormhole saddle, where the wormhole appears as the final term in the series. We further generalize this second method to any arbitrary deformation f (H 0 ). In §3, we look at two time SYK and its deformed version i.e study ζ deformed and dynamics of saddle points. As our work reveals interpretation of deformation in terms of ensemble averaging in §4 we study the Gaussian deformation. In particular, one chooses the overall scaling of the Hamiltonian from a Gaussian ensemble; this model is exactly equivalent to a quadratic deformation of the original Hamiltonian. Since this falls under the class of arbitrary deformation, the conclusion regarding the presence of (half)-wormholes remain true.

The one-time SYK under TT .
In this section after a short review of the undeformed single-time SYK model, we analyse the deformed case. We write down the exact deformed partition function and its moments using both the einbein approach as well as through a Taylor expansion in the deformation parameter. We find that the deformed partition function is proportional to the undeformed single-time z SYK . Recalling that z SYK has half-wormholes and wormholes, we establish the presence of half-wormholes and wormholes in the deformed theory. The method involving the einbein further reveals that one can view the deformation as an ensemble average. Using the Taylor expansion method, one can conclude the presence of (half)-wormholes for H → f (H) for any nice enough function f .

Undeformed theory
We start with the toy version of the undeformed SYK model wherein we focus on a single time instance. The analog of the partition function is built out of the following multi-Grassmann valued number : The couplings J i 1 ···iq are drawn randomly from a Gaussian distribution : We also assumed that both N, q are even integers and the ratio N/q = p is a positive integer. The partition function can be expressed as a Grassmannian integral: where in the last line the fermions were integrated out. The labels A i denote ordered non-intersecting subsets of {1, . . . , N } with cardinality q. We remark that on non intersecting subsets . . , N } with cardinality q and a im < a in if m < n, ordering is defined as follows: a i1 < a j1 ⇔ A i < A j for i = j. In the undeformed model with the explicit form of z SYK we may evaluate the averaged (over J) moments : z k SYK . Clearly, z SYK = 0 since there are no possible Wick contractions of J indices. The two lowest non-trivial moments comes from : In the large N limit, it follows that The correlations in the z 2 SYK = z L z R SYK contraction is the wormhole contribution. We have added replica indices L and R to distinguish the replicas. For z 4 SYK we therefore have : z L z R z L z R SYK . Among all the possible contractions in the large N limit the ones that dominate turn out to be: It also turns out that each of the averages are equal to each other and equal to the wormhole correlation, from whence eq. (12) follows.

(Half) wormholes of the undeformed theory
The half-wormholes are the contributions to the unaveraged z 2 SYK in addition to z 2 SYK which restore factorization. The regime where the half-wormholes dominate is best seen by introducing collective fields, g ∝ i ψ L i ψ R i and its conjugate σ. It turns out that the factorized answer can be expressed as: where Here we have and is independent of the disorder. Therefore disorder average, Φ 0 (σ) is well approximated by the contributions of the wormhole saddles. On the complex σ plane these lie symmetrically on the unit circle separated out by angle 2π/q. In order to compute the RMS we need Φ 2 0 (σ) = d 4 g ab d 4 σ ab I(σ, σ ab , g ab ) , where we have .
(17) where (a, b) ∈ {(L, L ), (L, R ), (R, R ), (R, L )} and σ LR = σ L R fixed to σ and not integrated over. σ ab = 0 is a trivial saddle for any value of σ and responsible for Φ 0 (σ) 2 contribution to Φ 2 0 (σ) . When σ = 0, the nontrivial saddles lie on unit circle and there are q × q × 2 of them. One of the two kinds corresponds to (L, R ), (L , R) contraction while the other represents (L, L ), (R, R ) contraction. As we make σ = 0, these non trivial saddles take the form σ LR σ RL = s 2 e 2πim/q , σ LL σ RR = s 2 e 2πim/q . Without loss of generality, let us focus on the case σ LR σ RL = s 2 e 2πim/q and σ LL σ RR = 0. Let us name the saddles as σ ab * , g ab * . So we have two contributions : Now as we vary σ on the complex plane, Φ 2 0 (σ) gets most of the contribution either from the trivial saddle or from the nontrivial ones. The region dominated by the trivial saddle is self averaging one while the other region is non self averaging. Thus we can write It turns out that for the one time SYK model, the wormhole saddles (which corresponds to some specific value of σ) always lie in the self-averaging region . The non-self averaging region takes the shape of a scallop (see Fig 1 in [9] for q = 4). For future reference, we define and use it as diagnostic of the self-averaging (or non self-averaging) region. The above exercise shows that the wormholes persist for the one-point SYK model before averaging since they live in the self-averaging region. Now we turn our attention to the half-wormholes. While the wormhole saddles are in self averaging regime, we know that z 2 is factorizable and can not be self-averaging exactly. This can be seen by setting σ = 0 and noticing that the dominant contribution to Φ 2 0 come from non-trivial saddles, hence σ = 0 is in non self-averaging region. It turns out that the following is a good approximation: where Φ 0 (0) is dubbed as half wormhole [9,10] and responsible for factorization. The RMS value of the fluctuation is Φ 2 0 (0) , which is dominated by nontrivial saddles is of the same order as the trivial contribution: Furthermore, one can show that other contributions to z 2 SYK are typically suppressed [9, 10].

TT Deformed theory
In this subsection we focus on the λTT deformation. The deformation of the spectrum indexed by energy levels, E, can fully be solved from the non-perturbative flow equation: The above can be integrated to find the full deformed Hamiltonian in terms of the original one as: Where, H 0 is the undeformed SYK hamiltonian. We have also included a constant, E 0 shift to the ground state energy of the undeformed theory. The solution with negative sign is perturbatively connected from the undeformed spectrum, hence we consider, H(λ) with only the negative sign in front of the square root.
In what follows we will compute the moments of this model using two methods. The first one involves introducing an einbein and then doing an ensemble average over the einbein to implement the deformation. The second method proceeds without the einbein by doing simple brute force Taylor expansion of deformed Hamiltonian around the original undeformed one.

Moments from exact computation-I via Einbeins
A convenient way to implement the TT deformation given by eq. (24) is to use an intergral over einbeins. The zeroth moment i.e partition function now involves also an integral over the einbein as follows 1 : We note that 1 can be thought of as a probability distribution of , the overall scaling of shifted Hamiltonian H 0 − E 0 . The distribution has its support on [0, ∞) with unit mean and 4λ variance. This is only when the deformation coupling λ is positive. 2 As promised in the introduction, we see the ensemble average over overall scaling implements a deformation of the original Hamiltonian.
We can rescale ψ in H 0 to separate out the integral so that we obtain with, S ( ) = ( − 1) 2 8λ Note that we have used the Grassmannian identity d(aψ) = dψ/a above. The integral can be now be done explicitly to obtain Since the effect of the deformation appears as an overall factor to z SYK the existence of wormholes and half-wormholes follow naturally. One can also check this by looking at the variance and moments, e.g., the k th moment of the partition function upon performing the SYK like average over the coupling J evaluates to: 1 We thank Baur Mukhametzhanov for pointing out the one loop exactness of the deformation. 2 When λ < 0, the analog process with the probability distribution 1 √ and noting the integral only affects the z k SYk factor of z k deformed . Thus Note that in the limit λ → 0 with λE 0 held fixed, we can use the asymptotics of Bessel K to obtain Similarly, we have The same conclusion will be reached using the exact computation without using einbein.

Moments from exact computation-II
In this subsection we evaluate the Grassmanian integrals exactly to compute one-time z and its higher moments. We find that in the leading order in N , and more specifically when N λ q, the answers from the einbeins are reproduced. Our methods allow for a easy generalization to arbitrary deformations of the type H 0 → f (H 0 ) of which TT is only a special case. For the latter we can write: where h(E 0 , α) is given by Now let us consider ( ∞ α=0 h(E 0 , α) H α 0 ) r and separate out the α = 0 term from the rest, so that we have Now the idea is that we need exactly N Grassmannian variables to saturate the integral, thus we need to figure out the coefficient of H p , where N/q = p. From eq. (34) we find that: The sum runs over all possible sets {α i : i = 1, 2, · · · r 1 } such that i α i = p and α i > 0.
This happens because α i 1 and hence p = i α i r 1 . Finally performing the Grassmanian integral, we find that One can check numerically that the above equation is identical to eq. (27), for more details see Appendix §B. B(r 1 ) is maximized when r 1 = p and the most contributing term for small λ is coming from the set α i = 1. The next leading term comes from B(p − 1), where the most contributing term is when one of the α i = 2 and rest are set to 1: Hence in the approximation pλ 1, we have (38) If we now consider z 2 deformed at the leading order it will precisely give us eq. (30). Going back to eq. (38) we can use the replacements : and E 0 from eq. (30) to rewrite: In leading order, this is therefore the one time SYK model with renormalized coupling J(λ) and ground state energy shifted to E 0 . This is consistent with the results obtained in the full SYK model using diagrammatics and einbeins [33].

Arbitrary deformations
For arbitrary Hamiltonian deformations H 0 → f (H 0 ), we therefore see that once we know the Taylor coefficients f (H 0 ) = α h(E 0 , α)H α 0 , the partition function reduces to z SYK with renormalized couplings, and ground state shifted E 0 → h(E 0 , 0), where B(r 1 ) is defined in eq. (35).

Two time SYK and deformation
In this section we consider the two time version of the SYK model and its deformation. We begin with undefomred theory, which can be thought of as coupling two one time models with coupling denoted as µ. As µ increases while one saddle goes to zero, the other three merge at µe i3π/4 . Note, ζ(0) = z 2 SYK and we have where we repeat Now averaging leads to At large N the above integrals over σ and g maybe computed by saddle point. If we solve the saddle point equations simultaneously we can get the saddle points as specific values of σ. These are the analogs of the wormhole contributions since a non-zero saddle point value of σ implies a non-vanishing correlation between the two replicas. Unlike the one time case, for a non-zero µ the wormhole saddles are no longer present on the unit circle in the complex σ plane, though there are still q of them. We plot in Fig.1 for the q = 4 case how the saddles behave as a function of µ. When µ = 0 the saddles appear symmetrically on the unit circle as it should since ζ(0) = z 2 SYK . As µ gets turned on, the saddles start to move asymmetrically. One of the saddles always approach the origin σ = 0. As µ increases, the rest q − 1 saddles collapse and start moving at an angle π 2 + π q .
(Half) wormholes of the undeformed (two-time) theory We repeat the exercise done in the end of 2.1, but now with Φ 0 (σ−ie iπ/q µ). When µ = 0 the self averaging region takes the form of a scallop centered at origin. The wormhole saddles live outside the scallop and hence also exist in the non-averaged model. We contrast this with the large µ where the scallop is now centered at σ = ie iπ/q µ i.e. the center moves along a line with an angle π/2 + π/q. For large enough µ the σ = 0 saddles comes within the scallop. Thus for large enough µ, the non averaged model does not have the wormhole saddles. This is expected since increasing µ effectively makes the model a q = 2 non-averaged model, which is a Gaussian integrable theory. Effects of disorder in the spectral form factor for the q = 2 SYK model have been analyzed in [36,37], wherein the ramp is exponential. On the other hand we know that wormholes give rise to a linear ramp.
In Fig.3 we illustrate the dissolution process of the wormhole saddles with increasing |µ|. The plots take into account both the movement of the wormhole saddles as well as the motion of the scallop. A simple dimension analysis shows that the threshold value µ * scales with variance in following manner : Also as we see in the appendix §A, there is a q dependence on the size of the scallop. With increasing q the scallop gets smaller, and therefore the saddles survive further out in the complex σ plane as compared with a lower q result. Intuitively as increasing q increases; chaoticity this is also what one would expect.

(Half) wormholes of the deformed (two-time) theory
At this point, it is easy to figure out the effect of deformation. We will be brief and mention the salient features only.
where Φ deformed (σ − ie iπ/q µ) = Φ 0 (σ − ie iπ/q µ renormalized ), which can be obtained by carrying out the einbein integrals via saddle. Thus we can also express ζ deformed (µ) = ζ(µ renormalized ). Of course note, ζ(0) = z 2 . For the TT , at leading order in pλ 1 we find that the dissolution of the wormhole saddles now happen for a different value of µ : We therefore see for λ < 0 (recall E 0 is negative), if we increase |λ|, the renormalised varianceJ renormalized decreases and therefore µ * renormalized < µ * , hence the wormholes dissolve faster. Since the 1/N expansion in this toy model is finite it is possible to expand ζ deformed also around the half-wormhole i.e. σ = 0 saddle [10] : where,Φ (51) * * * * In terms of this expansion the wormhole saddles appear as the final terms in the series. Also note that in this expansion the effect of the deformation is encapsulated via µ → µ renormalized . In the next section we shall look into this expansion for the exactly solvable case of Gaussian ensemble of einbeins.

The Gaussian deformation
One of the key objects that we have investigated so far is: The purpose of the einbein integral is to implement the irrelevant deformation H → f (H). One can view the above as an ensemble average with a weight factor of e −S( L , R ) . Since S({ i }) = i S( i ) this averaging does not induce correlations between the replicas for µ = 0, rather just renormalizes J A which is now a coupling of the irrelevant Hamiltonian, f (H). On the other hand for µ = 0 , things are non trivial. One can not simply scale out , and cast the deformation as an overall factor.
The choice of S( L , R ) provides a class of toy models where one can replace e −S( L , R ) with a probability measure. An exactly solvable choice is the Gaussian ensemble for the einbeins i with means * i and variances s i . For simplicity, we will assume L * = R * = * . It results in the exact deformation: H 0 → 1 2 (s 2 H 2 0 + 2 * H 0 ). In general for the deformation of the type : H 0 → #H γ 0 , the einbein ensemble action that implements this (in the saddle-point approximation) is S( ) = # γ/(γ−1) . Coming back to the Gaussian deformation, the one-time partition function is given by: This integral can be computed exactly and leads to a result in terms of Hypergeometric function. We can arrive at the final answer also by using eq. (35) and (36) with appropriate h for the Gaussian case, which now reads for even p: Note that the sum runs from p/2 because B(r 1 ) = 0 for r 1 < p/2. This happens because α i can be 1 or 2 for the Gaussian deformation, leading to 2r ≥ r i=1 α i = p. Thus r i=1 α i = p can not have solution if r < p/2. A similar calculation can be done for odd p as well, where the sum runs from (p + 1)/2, leading to a slightly different final expression. For simplicity, in what follows, we will assume p is even without loosing any physical content.
In the two-time case, when µ = 0, one has: ζ(0) = z 2 If we set λ = 1 and parametrize x = √ 1+8E 0 4 then the above match follows from the following series representation for modified Bessel K with negative half integer index: which can be simplied into We have checked explicitly the expressions on both sides of the above equality for specific integer values of p and found a match. In what follows, we sketch a proof of the identity.
• Proof: The identity given by eq. (60) can be proven in following manner. Let us focus on the r.h.s of (60), call it I(x). Now we set s = p − r to rewrite which can further be rewritten as Here C β i is the β i th Catalan number. The generating function for the Catalan numbers is Using the generating function, we see that A standard way to derive the above coefficient is to do a contour integral. Now we plug the sum of product of Catalan numbers in eq. (62) to obtain This completes the proof. Note, here we have used K ν (x) = K −ν (x) for ν / ∈ Z and the following identity for Bessel function with half-integer index [38] : and set n = p − 1.