PPrepared for submission to JHEP
Quenched free energy from spacetime D-branes
Kazumi Okuyama
Department of Physics, Shinshu University,3-1-1 Asahi, Matsumoto 390-8621, Japan
E-mail: [email protected]
Abstract:
We propose a useful integral representation of the quenched free energy whichis applicable to any random systems. Our formula involves the generating function of multi-boundary correlators, which can be interpreted on the bulk gravity side as spacetime D-branes introduced by Marolf and Maxfield in [arXiv:2002.08950]. As an example, we applyour formalism to the Airy limit of the random matrix model and compute its quenchedfree energy under certain approximations of the generating function of correlators. It turnsout that the resulting quenched free energy is a monotonically decreasing function of thetemperature, as expected. a r X i v : . [ h e p - t h ] J a n ontents Z g from constitutive relation 143.4 Low temperature resummation 16 Recent works suggest that the gravitational path integral generically includes the contri-bution of Euclidean wormholes and the result should be interpreted as a certain ensembleaverage, at least for lower dimensional gravities in spacetime dimensions d ≤ (cid:104) log Z (cid:105) and the annealed free energy log (cid:104) Z (cid:105) , where Z is the partitionfunction with the inverse temperature β = 1 /TZ ≡ Z ( β ) = Tr e − βH , (1.1)and the bracket (cid:104)·(cid:105) denotes the ensemble average. In a random system, thermodynamicquantities like the free energy F and the entropy S should be defined via the quenched freeenergy F = − T (cid:104) log Z (cid:105) , S = − ∂F∂T , (1.2) In this paper, we will loosely use the name “quenched free energy” for both (cid:104) log Z (cid:105) and F = − T (cid:104) log Z (cid:105) depending on the context. We hope this will not cause a confusion to the readers. – 1 –hich are expected to satisfy various thermodynamic inequalities. In particular, the posi-tivity of entropy S ≥ F ann = − T log (cid:104) Z (cid:105) is not necessarilya monotonic function of temperature.In a recent paper [9], the quenched free energy in JT gravity is studied by the replicamethod (cid:104) log Z (cid:105) = lim n → (cid:104) Z n (cid:105) − n , (1.3)and it is argued that the wormholes connecting different replicas, so-called replica worm-holes, play an important role (see also [10, 11]). However, the replica computation in [9]does not give rise to a well-behaved monotonic free energy at low temperature due to thelimitation of the approximation used in [9]. More importantly, as emphasized in [9] thereis a fundamental problem in the replica computation of the quenched free energy since theanalytic continuation of (cid:104) Z n (cid:105) from a positive integer n to n = 0 is highly non-unique.The replica method (1.3) is just a convenient trick to compute the quenched free energyand one can in principle compute it by directly taking the average of the quantity log Z .This is demonstrated in our previous paper [15] in a simple example of the Gaussian matrixmodel. We computed the average (cid:104) log Z (cid:105) directly by the matrix integral at finite N andfound that the resulting quenched free energy is a monotonic function of the temperature,as expected. It should be emphasized that if we take the average (cid:104) log Z (cid:105) directly withoutusing the replica method (1.3), the result is uniquely determined and there is no ambiguityin the computation. However, the direct computation in [15] relies on the property of thefinite N matrix model and it is not straightforward to generalize it to the double scaledmatrix model of JT gravity. Thus, it is desirable to find a general technique to computethe quenched free energy without resorting to the replica trick (1.3).In this paper, we propose a useful integral representation of the quenched free energy (cid:104) log Z (cid:105) = log (cid:104) Z (cid:105) − (cid:90) ∞ dxx (cid:104)(cid:10) e − Zx (cid:11) − e −(cid:104) Z (cid:105) x (cid:105) , (1.4)which can be applied to any random system. As far as we know, this expression of quenchedfree energy (1.4) has not appeared in the literature before. Our formula (1.4) follows fromthe simple identity (2.1) and the derivation of (1.4) does not rely on any approximation. Insection 2.2, we will argue that our formula (1.4) can be obtained from the replica method(1.3) under a certain prescription of the analytic continuation of (cid:104) Z n (cid:105) .Our formula (1.4) might have an interesting bulk gravity interpretation. As discussedin [3], the insertion of Z into the expectation value (cid:104)·(cid:105) corresponds to adding an extraboundary to the spacetime in the bulk gravity picture. The insertion of e − Zx creates manyboundaries in spacetime and it is identified in [3] as the “spacetime D-brane” (or SD-branefor short). Based on this picture, our formula (1.4) suggests that the bulk gravity dual of In a different context of the computation of von Neumann entropy of Hawking radiation, replica worm-holes also play an essential role to recover the unitary Page curve [12, 13]. See also [14] for a nice reviewon this subject. A similar, but slightly different expression of (cid:104) log Z (cid:105) is discussed in [16]. – 2 –he quenched free energy involves a superposition of SD-branes. Here, by “superposition”we mean that we have to integrate over the parameter x in the SD-brane operator e − Zx .Introducing the generating function Z ( x ) of the connected correlators (cid:104) Z n (cid:105) c Z ( x ) = − ∞ (cid:88) n =1 ( − x ) n n ! (cid:104) Z n (cid:105) c , (1.5)(1.4) is written as (cid:104) log Z (cid:105) = log (cid:104) Z (cid:105) − (cid:90) ∞ dxx (cid:104) e −Z ( x ) − e −(cid:104) Z (cid:105) x (cid:105) , (1.6)where we used the relation (cid:104) e − Zx (cid:105) = e −Z ( x ) . Thus the problem boils down to the compu-tation of the generating function Z ( x ) in (1.5). As a simple application of our formalism(1.6), we consider the quenched free energy in the Airy limit of Gaussian matrix model. We find that at genus-zero the generating function Z ( x ) can be written in a closed form interms of the Lambert function. If we plug this genus-zero approximation of Z ( x ) into (1.6),we find a monotonically decreasing behavior of the quenched free energy as a function ofthe temperature. However, if we include higher genus corrections to Z ( x ) in (1.6) it leadsto an unphysical diverging behavior of free energy as T →
0. This simply means thatthe genus expansion is not a good approximation at low temperature, and hence we willconsider a different approximation of Z ( x ) in (3.6) which is applicable at low temperature.Then, we find a monotonic behavior of the quenched free energy in this low temperatureapproximation (3.6).This paper is organized as follows. In section 2, we explain the derivation of theformula (1.4) for the quenched free energy. We also see that this expression (1.4) can beobtained from the replica method (1.3). Then we briefly comment on the possible bulkgravity interpretation of our formula (1.4). In section 3, we apply our formalism (1.6) tothe Airy limit of Gaussian matrix model. We find that the genus expansion of Z ( x ) in theAiry limit is written in terms of the Lambert function. The appearance of the Lambertfunction can be understood from the string equation in a shifted background due to theinsertion of the operator e − Zx . We find a monotonic behavior of the quenched free energyin a certain approximation of Z ( x ). Finally, we conclude in section 4 with some discussionfor the future problems. In appendix A, we compute the high temperature expansion ofthe quenched free energy in the Airy limit. (1.4)Let us prove our general formula (1.4) for the quenched free energy. (1.4) is based on thefollowing simple relation − (cid:90) ∞ dxx ( e − ax − e − bx ) = log ab . (2.1) See e.g. [17] for a review of the Airy limit of matrix model. – 3 –ote that, if we take the first term of (2.1) only, then the integral is logarithmicallydivergent − (cid:90) ∞ ε dxx e − ax = log( aε ) + γ + O ( ε ) , (2.2)where ε is a small regularization parameter and γ is the Euler-Mascheroni constant. How-ever, if we compute the difference of two such integrals as in (2.1) the divergence from x = 0 is canceled and we get the finite result (2.1). Setting a = Z and b = (cid:104) Z (cid:105) in (2.1) weobtain − (cid:90) ∞ dxx (cid:16) e − Zx − e −(cid:104) Z (cid:105) x (cid:17) = log Z (cid:104) Z (cid:105) . (2.3)Finally, taking the expectation value of the both sides of (2.3) and using the relation (cid:104) (cid:105) = 1, we find our desired result (1.4). We emphasize that our formula (1.4) is an exactrelation and we did not use any approximation in deriving (1.4).Let us consider an alternative derivation of (1.4) by expanding the quenched free energyaround the annealed free energy. (cid:104) log Z (cid:105) = (cid:10) log (cid:0) (cid:104) Z (cid:105) + Z − (cid:104) Z (cid:105) (cid:1)(cid:11) = log (cid:104) Z (cid:105) + (cid:42) log (cid:32) Z − (cid:104) Z (cid:105)(cid:104) Z (cid:105) (cid:33)(cid:43) = log (cid:104) Z (cid:105) − ∞ (cid:88) k =1 ( − k k (cid:10)(cid:0) Z − (cid:104) Z (cid:105) (cid:1) k (cid:11) (cid:104) Z (cid:105) k . (2.4)Then using the relation (cid:90) ∞ dyy e − y y k = ( k − , ( k ≥ , (2.5)we rewrite the summation in (2.4) as − ∞ (cid:88) k =1 ( − k k (cid:10)(cid:0) Z − (cid:104) Z (cid:105) (cid:1) k (cid:11) (cid:104) Z (cid:105) k = − (cid:90) ∞ dyy e − y ∞ (cid:88) k =1 ( − y ) k k ! (cid:10)(cid:0) Z − (cid:104) Z (cid:105) (cid:1) k (cid:11) (cid:104) Z (cid:105) k = − (cid:90) ∞ dyy e − y (cid:34)(cid:42) exp (cid:32) − y Z − (cid:104) Z (cid:105)(cid:104) Z (cid:105) (cid:33)(cid:43) − (cid:35) = − (cid:90) ∞ dyy (cid:34)(cid:42) exp (cid:32) − y Z (cid:104) Z (cid:105) (cid:33)(cid:43) − e − y (cid:35) (2.6)Finally, by rescaling the integration variable y = (cid:104) Z (cid:105) x we arrive at our formula (1.4). Let us consider a derivation of our formula (1.4) from the replica method (1.3). In thereplica method, we need to analytically continue the n -point function (cid:104) Z n (cid:105) from a positiveinteger n to n = 0. As pointed out in [9], the analytic continuation of (cid:104) Z n (cid:105) is not uniqueand there might be some ambiguity in the computation of the right hand side of (1.3). It– 4 –urns out that there is a natural definition of the analytic continuation of (cid:104) Z n (cid:105) which leadsto our formula (1.4).In general, the correlator (cid:104) Z n (cid:105) is expanded as a combination of the connected corre-lators (cid:104) Z k (cid:105) c . If we order this expansion by the number of connected components, the firstterm is the totally disconnected part (cid:104) Z (cid:105) n . The next term is (cid:104) Z (cid:105) n − (cid:104) Z (cid:105) c and the coeffi-cient of this term is (cid:0) n (cid:1) = n ( n −
1) which is the number of ways to choose two boundariesout of n boundaries. Then (cid:104) Z n (cid:105) is expanded as (cid:104) Z n (cid:105) = (cid:104) Z (cid:105) n + 12 n ( n − (cid:104) Z (cid:105) n − (cid:104) Z (cid:105) c + · · · = (cid:104) Z (cid:105) n (cid:20) n ( n − (cid:104) Z (cid:105) c (cid:104) Z (cid:105) + · · · (cid:21) . (2.7)Taking the n → (cid:104) log Z (cid:105) = log (cid:104) Z (cid:105) − (cid:104) Z (cid:105) c (cid:104) Z (cid:105) + · · · . (2.8)In this computation, up to the overall factor (cid:104) Z (cid:105) n the n -dependence in (2.7) is essentiallya polynomial in n and hence the analytic continuation can be defined unambiguously.To see the general structure, it is convenient to define w n = (cid:104) Z n (cid:105) c (cid:104) Z (cid:105) n . (2.9)Then the expansion (2.7) is generalized as (see e.g. [18]) (cid:104) Z n (cid:105) = (cid:104) Z (cid:105) n (cid:88) j i ≥ i =2 , , ··· ) n !( n − (cid:80) l ≥ lj l )! (cid:89) k ≥ j k ! (cid:16) w k k ! (cid:17) j k . (2.10)Now using the relation lim n → n n !( n − m )! = ( − m − ( m − , (2.11)the quenched free energy becomes (cid:104) log Z (cid:105) = log (cid:104) Z (cid:105) + (cid:88) j i ≥ i =2 , ··· )( j ,j , ··· ) (cid:54) =(0 , , ··· ) ( − (cid:80) l ≥ lj l − (cid:16)(cid:88) l ≥ lj l − (cid:17) ! (cid:89) k ≥ j k ! (cid:16) w k k ! (cid:17) j k . (2.12)Again using the trick (2.5) this is rewritten as (cid:104) log Z (cid:105) = log (cid:104) Z (cid:105) − (cid:88) j i ≥ i =2 , , ··· )( j ,j , ··· ) (cid:54) =(0 , , ··· ) (cid:90) ∞ dyy e − y ( − y ) (cid:80) l ≥ lj l (cid:89) k ≥ j k ! (cid:16) w k k ! (cid:17) j k = log (cid:104) Z (cid:105) − (cid:90) ∞ dyy e − y (cid:34) exp (cid:32) ∞ (cid:88) k =2 ( − y ) k w k k ! (cid:33) − (cid:35) = log (cid:104) Z (cid:105) − (cid:90) ∞ dxx (cid:34) exp (cid:32) ∞ (cid:88) k =1 ( − x ) k k ! (cid:104) Z k (cid:105) c (cid:33) − e −(cid:104) Z (cid:105) x (cid:35) , (2.13)– 5 –here in the last step we changed the integration variable y = (cid:104) Z (cid:105) x . One can see that thefirst term of the integrand in (2.13) is e −Z ( x ) defined in (1.5), and hence (2.13) agrees withour formula (1.6).To summarize, we have derived our formula (1.4) from the replica method by usingthe prescription (2.11) for the analytic continuation. For instance, the example in (2.8) ofthis computation corresponds to the m = 2 case of (2.11)lim n → n n !( n − n → n n ( n −
1) = − . (2.14)At fixed m , the factor n !( n − m )! in (2.11) is a polynomial in nn !( n − m )! = n ( n − · · · ( n − m + 1) , (2.15)and hence we believe that our prescription (2.11) of the analytic continuation is a naturalchoice. (1.4)Our formula (1.4) is very suggestive for a possible interpretation of the bulk spacetimepicture of the quenched free energy. In [3], it is discussed that the operator e − Zx corre-sponds to the so-called “spacetime D-brane” (SD-brane). Our formula (1.4) suggests thatthe bulk spacetime picture of quenched free energy involves a superposition of SD-branes.As mentioned in section 1, in order to obtain the quenched free energy, we have to integrateover the parameter x in the SD-brane operator e − Zx .Recently, in [12, 13] the von Neumann entropy of Hawking radiation is computed bythe replica method and it is argued that the so-called replica wormholes are essential forrecovering the unitary Page curve. In the saddle point approximation of gravitational pathintegral, the dominant contribution comes from either maximally connected or maximallydisconnected spacetimes and these contributions exchange dominance around the Pagetime. On the other hand, in our formula (1.4) no such saddle point approximation ismade. In our derivation of (1.4) from the replica method, all terms in the decompositionof (cid:104) Z n (cid:105) in (2.10) contribute to the quenched free energy and they add up to the generatingfunction (cid:104) e − Zx (cid:105) . The first term log (cid:104) Z (cid:105) in (1.4) comes from the maximally disconnectedpart (cid:104) Z (cid:105) n , while the remaining term in (1.4) is a contribution of SD-brane e − Zx which doesnot have a simple interpretation as the contribution of maximally connected part. As wewill see in section 3.3, the insertion of the operator e − Zx can be interpreted as a shift ofthe background couplings, or a deformation of the matrix model potential.It is argued in [9] that the contribution of the so-called replica wormholes should beincluded in the replica computation of the quenched free energy. The second term of (1.4)represents the deviation from the annealed free energy, which might be interpreted as thecontribution of replica wormholes. It is interesting that even after taking the limit n → (cid:104) e − Zx (cid:105) = e −Z ( x ) in (1.5). In other words,we need to include a coherent superposition Z ( x ) of the infinite number of boundaries forthe computation of quenched free energy. – 6 –n general, the large N limit serves as a semi-classical approximation in the randommatrix model. One might think that we can see an exchange of dominance between dis-connected and connected contributions in the large N limit. However, if we naively takethe large N limit at fixed temperature, the disconnected correlator always dominates overthe connected correlators. Around the temperature T ∼ N − / , the disconnected and con-nected contributions become comparable [18], but this temperature is pushed to zero as N → ∞ which implies that the disconnected part is always dominant at finite temperature.As discussed in [9, 10], to focus on this temperature scale T ∼ N − / one can take a certaindouble scaling limit, called the Airy limit of random matrix model, which we will considerin the next section. In this section we apply our general formula (1.4) to the Airy limit of Gaussian randommatrix model. In this case, the connected part of the n -point correlator (cid:104) Z n (cid:105) c is known inthe integral representation [19]. For instance, (cid:104) Z n (cid:105) c for n = 1 , , (cid:104) Z ( β ) (cid:105) = e ξ √ πξ , (cid:104) Z ( β ) (cid:105) c = (cid:104) Z (2 β ) (cid:105) Erf (cid:32)(cid:114) ξ (cid:33) , (cid:104) Z ( β ) (cid:105) c = (cid:104) Z (3 β ) (cid:105) (cid:34) − T (cid:16)(cid:112) ξ, √ (cid:17)(cid:35) , (3.1)where Erf( z ) is the error function and T ( z, a ) denotes the Owen’s T -function defined by T ( z, a ) = 12 π (cid:90) a dt e − z (1+ t ) t . (3.2)The parameter ξ in (3.1) is given by ξ = (cid:126) β , (3.3)where (cid:126) is the genus-counting parameter. In the Airy limit, (cid:126) and β appear in (cid:104) Z n (cid:105) c onlythrough the combination ξ .In principle one can compute (cid:104) Z n (cid:105) c using the integral representation of [19], but it isnot straightforward to find the generating function Z ( x ) of the connected correlators sincethe integral in [19] is difficult to compute in a closed form for general n . In fact, the closedform expression of (cid:104) Z n (cid:105) c is not known for n ≥
4. Instead, here we will consider the genusexpansion of the generating function Z ( x ) Z ( x ) = ∞ (cid:88) g =0 ξ g − Z g ( x ) , (3.4)– 7 –here Z g ( x ) is the generating function of the genus- g part of the connected correlators. Itturns out that Z g ( x ) is written as a combination of the Lambert W -function. If we use thegenus-zero approximation Z ( x ) ≈ ξ − Z ( x ) in (1.6), we find that the quenched free energy F = − T (cid:104) log Z (cid:105) decreases monotonically as a function of temperature. However, the genusexpansion is not a good approximation at low temperature since the expansion parameter ξ in (3.4) becomes large when T (cid:28) (cid:126) / .At low temperature we can use the result of [18] that the connected part of the n -pointcorrelator is approximated by the one-point function (cid:104) Z ( β ) n (cid:105) c ≈ (cid:104) Z ( nβ ) (cid:105) , ( T (cid:46) (cid:126) / ) . (3.5)One can see this behavior explicitly from the exact result of (cid:104) Z ( β ) n (cid:105) c for n = 2 , n .Then Z ( x ) at low temperature is approximated as Z ( x ) ≈ − ∞ (cid:88) n =1 ( − x ) n n ! (cid:104) Z ( nβ ) (cid:105) . (3.6)The right hand side of this equation can be computed explicitly since the exact form of theone-point function (cid:104) Z ( nβ ) (cid:105) is known in the Airy limit. Plugging (3.6) into (1.6), we findthe monotonic behavior of the quenched free energy even at low temperature T (cid:46) (cid:126) / . Let us consider the genus-zero part of the expansion of Z ( x ) in (3.4). The connectedcorrelator (cid:104) Z n (cid:105) c is obtained by acing the “boundary creation operator” (cid:98) Z to the freeenergy F ( { t k } ) of 2d topological gravity [21] (cid:104) Z n (cid:105) c = (cid:0) (cid:98) Z (cid:1) n F , (3.7)where (cid:98) Z is given by (cid:98) Z = g s (cid:114) β π ∞ (cid:88) k =0 β k ∂ k (3.8)with ∂ k ≡ ∂∂t k . Then the generating function Z ( x ) is written as Z ( x ) = − ∞ (cid:88) n =1 ( − x ) n n ! (cid:0) (cid:98) Z (cid:1) n F = F − e − (cid:98) Zx F . (3.9) g s in (3.8) is the natural genus-counting parameter in the topological gravity F = ∞ (cid:88) g =0 g g − s F g , (3.10) Strictly speaking, this result (3.5) is shown in the Gaussian matrix model [18], but we expect that (3.5)holds in the Airy limit as well. – 8 –nd g s is related to (cid:126) in (3.3) by g s = √ (cid:126) . (3.11)The Airy limit corresponds to the trivial background t n = 0 ( n ≥ u = g s ∂ F [25, 26]. In the KdV approach of thecomputation of (cid:104) Z n (cid:105) c , it is convenient to turn on t , t t , t (cid:54) = 0 , t n = 0 ( n ≥ , (3.12)and set t = t = 0 at the end of the calculation. One can show that in this subspace(3.12) u is given by u = t − t , (3.13)without any higher genus corrections.At genus-zero, the general expression of the connected correlator (3.7) reduces to [21] (cid:104) Z n (cid:105) g =0 c = (cid:18) β π (cid:19) n ( g s ∂ ) n − e nβu nβ , (3.14)where u is determined by the genus-zero string equation [27] u = ∞ (cid:88) k =0 t k u k k ! . (3.17)In our case (3.12), u is equal to u in (3.13) u = ∂ F = t − t . (3.18)Plugging this u into (3.14) the genus-zero part of the n -point correlator (cid:104) Z n (cid:105) g =0 c becomes (cid:104) Z n (cid:105) g =0 c = (1 − t ) − n ξ n n − (cid:18) ξπ (cid:19) n e nβt − t , (3.19)where ξ is defined in (3.3). Then, by setting t = t = 0 we find (cid:104) Z n (cid:105) g =0 c in the Airy limit (cid:104) Z n (cid:105) g =0 c = 12 ξ n n − (cid:18) ξπ (cid:19) n . (3.20) The all-genus string equation is given by u = ∞ (cid:88) k =0 t k R k , (3.15)where R k = u k k ! + O ( ∂ ) is the Gelfand-Dikii differential polynomial determined by the recursion relation(2 k + 1) ∂ R k +1 = (cid:126) ∂ R k + 2 u∂ R k + ( ∂ u ) R k . (3.16) – 9 –ow let us consider the generating function of the genus-zero correlators in (3.20) ξ − Z ( x ) = − ∞ (cid:88) n =1 ( − x ) n n ! (cid:104) Z n (cid:105) g =0 c = 12 ξ ∞ (cid:88) n =1 ( − n − n n − n ! (cid:32) x (cid:114) ξπ (cid:33) n . (3.21)It turns out that this summation is closely related to the Lambert function W ( z ) W ( z ) = ∞ (cid:88) n =1 ( − n − n n − n ! z n , (3.22)which satisfies the relation z = W ( z ) e W ( z ) . (3.23)From (3.21) and (3.22), one can see that Z obeys( z∂ z ) Z = 12 W ( z ) , (3.24)where z is related to x by z = x (cid:114) ξπ . (3.25)Using the property of the Lambert function z∂ z W ( z ) = W ( z )1 + W ( z ) , (3.26)(3.24) is integrated as Z = 12 (cid:32) W + 34 W + W (cid:33) . (3.27)Here we have suppressed the argument of W ( z ) for brevity.Then at genus-zero the quenched free energy (1.6) is written as (cid:104) log Z (cid:105) g =0 = log (cid:104) Z (cid:105) g =0 − (cid:90) ∞ dxx (cid:104) e − ξ − Z ( x ) − e −(cid:104) Z (cid:105) g =0 x (cid:105) , (3.28)where (cid:104) Z (cid:105) g =0 = √ πξ is the genus-zero one-point function. Using the fact that z ∈ [0 , ∞ ] ismapped to W ∈ [0 , ∞ ], we can take W as the integration variable in (3.28). From (3.26),(3.28) is rewritten as (cid:104) log Z (cid:105) g =0 = −
12 log(4 πξ ) − (cid:90) ∞ dWW (1 + W ) (cid:104) e − ξ ( W + W + W ) − e − ξ W e W (cid:105) . (3.29)In Fig. 1 we show the plot of rescaled free energy F (cid:48) = (cid:126) − / F of the Airy case atgenus-zero. In other words, F (cid:48) is defined by F (cid:48) = − T (cid:48) (cid:104) log Z (cid:105) , T (cid:48) = (cid:126) − / T, ξ = T (cid:48)− . (3.30)From Fig. 1 one can see that the quenched free energy at genus-zero (blue solid curve)is a monotonic function of the temperature. As a comparison, we have also plotted theannealed free energy at genus-zero (orange dashed curve in Fig. 1) F (cid:48) ann = − T (cid:48) log (cid:104) Z (cid:105) g =0 = 12 T (cid:48) log(4 πT (cid:48)− ) , (3.31)which is not a monotonic function of the temperature.– 10 – .5 1.0 1.5 2.0 2.5 3.0 T ′ - - F ′ Figure 1 : Plot of the genus-zero free energy for the Airy case. The vertical and thehorizontal axes are the rescaled free energy F (cid:48) = (cid:126) − / F and the rescaled temperature T (cid:48) = (cid:126) − / T , respectively. The blue solid curve represents the quenched free energy in(3.29) while the orange dashed curve is the annealed free energy in (3.31). Next let us consider the higher genus corrections Z g in (3.4). As we will see below, Z g can be systematically obtained from the KdV equation. Our starting point is the KdVequation for u ∂ u = u∂ u + (cid:126) ∂ u. (3.32)Introducing the potential φ by u = ∂ φ, (3.33)(3.32) is integrated once as ∂ φ = 12 ( ∂ φ ) + (cid:126) ∂ φ. (3.34)Note that φ is written as φ = g s ∂ F = 2 (cid:126) ∂ F . (3.35)We can derive an equation obeyed by Z ( x ) by acting the operator e − (cid:98) Zx on both sidesof (3.34). From (3.9) and (3.35), e − (cid:98) Zx φ is written as e − (cid:98) Zx φ = 2 (cid:126) ∂ e − (cid:98) Zx F = 2 (cid:126) ∂ ( F − Z ) = φ − (cid:126) W , (3.36)where we defined W = (cid:126) ∂ Z . (3.37)Acting e − (cid:98) Zx on both sides of (3.34), we find ∂ ( φ − (cid:126) W ) = 12 ( ∂ φ − (cid:126) ∂ W ) + (cid:126) ∂ ( φ − (cid:126) W ) . (3.38)From (3.34) and (3.38), we arrive at the equation for W ∂ W = u∂ W − (cid:126) ( ∂ W ) + (cid:126) ∂ W . (3.39) This equation was originally found by Kazuhiro Sakai. The author would like to thank him for sharinghis unpublished note. – 11 –et us consider the genus expansion of WW = β − ∞ (cid:88) g =0 ξ g − W g . (3.40)It turns out that W g is a rational function of W ≡ W ( z ) with z given by (3.25). Tocompute W g recursively from (3.39), it is useful to work on the subspace (3.12). On thisspace (3.12), one can easily show that z in (3.25) is generalized as (see (3.19)) z = 11 − t xe βt − t (cid:114) ξπ . (3.41)Then the derivative ∂ , appearing in (3.39) is written in terms of D = z∂ z . We also notethat (cid:126) and t appear only via the combinationˆ (cid:126) ≡ (cid:126) − t . (3.42)With these remarks in mind, ∂ W and ∂ W at t = t = 0 are given by ∂ W (cid:12)(cid:12)(cid:12) t = t =0 = βD W ,∂ W (cid:12)(cid:12)(cid:12) t = t =0 = ( D + (cid:126) ∂ (cid:126) ) W . (3.43)Then after setting t = t = 0 (3.39) reduces to( D + 2 g − W g = 16 D W g − − g (cid:88) h =0 D W h D W g − h . (3.44)From the definition of W (3.37), one can easily show that Z g and W g are related by W g = D Z g . (3.45)To solve (3.44) recursively, we rewrite (3.44) as (cid:104) (1 + 2 D W ) D + 2 g − (cid:105) W g = 16 D W g − − g − (cid:88) h =1 D W h D W g − h . (3.46)Note that, from (3.26) D = z∂ z is written as D = W W ∂ W , (3.47)and from (3.24) D W is given by D W = D Z = 12 W. (3.48)Finally we arrive at the recursion relation for W g ( W ∂ W + 2 g − W g = 16 D W g − − g − (cid:88) h =1 D W h D W g − h . (3.49)– 12 –ne can easily solve this relation recursively starting from D W in (3.48). More explicitly,(3.49) is solved as Z g = (cid:90) dW (1 + W ) W − g (cid:90) dW W g − (cid:34) D W g − − g − (cid:88) h =1 D W h D W g − h (cid:35) . (3.50)Using this formalism, we can compute Z g up to any desired order. For instance, the firstfew terms are given by Z = 124 log(1 + W ) , Z = 5 W − W + 4 W W ) , Z = 35 W − W + 12393 W − W + 3960 W − W W ) . (3.51)Using this result (3.51), we can study the behavior of free energy including the highergenus corrections up to genus- g (cid:104) log Z (cid:105) [ g ] = log (cid:104) Z (cid:105) [ g ] − (cid:90) ∞ dWW (1 + W ) (cid:20) e −Z [ g ] − e −(cid:104) Z (cid:105) [ g ] (cid:113) πξ W e W (cid:21) , (3.52)where we defined (cid:104) Z (cid:105) [ g ] and Z [ g ] by (cid:104) Z (cid:105) [ g ] = 1 √ πξ g (cid:88) n =0 ξ n n n ! , Z [ g ] = g (cid:88) h =0 ξ h − Z h . (3.53)In Fig. 2a, 2b and 2c, we show the plot of quenched free energy including the highergenus corrections up to g = 1, g = 2 and g = 3, respectively (see blue solid curves inFig. 2). For g = 2 and g = 3, we see that the quenched free energy diverges as T → Z and Z in (3.51)become negative at some value of W . This implies Z [ g ] in (3.53) diverges in the negativedirection at some value of W as T →
0. As Z [ g ] → −∞ , the factor e −Z [ g ] in (3.52) growsexponentially and it leads to the diverging behavior of the quenched free energy in Fig. 2band 2c.However, this is not the expected behavior for the generating function Z ( x ). In fact,since Z and x are positive quantities, the generating function (cid:104) e − Zx (cid:105) satisfies the inequality e −Z ( x ) = (cid:104) e − Zx (cid:105) < , (3.54)which implies that Z ( x ) is always positive. The appearance of the negative coefficients in(3.51) simply means that the genus expansion is not a good approximation of Z ( x ) at lowtemperature since the expansion parameter ξ is not small at low temperature T (cid:28) (cid:126) / .In other words, the genus expansion of Z ( x ) breaks down at low temperature.To study the low temperature behavior of Z ( x ) we need a different approximation of Z ( x ), which we will consider in section 3.4.– 13 – .5 1.0 1.5 2.0 2.5 3.0 T ′ - - F ′ (a) Free energy up to g = 1 T ′ - F ′ (b) Free energy up to g = 2 T ′ - F ′ (c) Free energy up to g = 3 Figure 2 : Plot of free energy including the higher genus corrections up to (a) g = 1, (b) g = 2 and (c) g = 3. The blue solid curves are the quenched free energy in (3.52) while theorange dashed curves represent the annealed free energy − T (cid:48) log (cid:104) Z (cid:105) [ g ] . Z g from constitutive relation Before discussing the low temperature behavior of Z ( x ) in section 3.4, let us consider analternative method for the computation of Z g . This method clarifies the appearance of theLambert function in this problem. Using the boundary creation operator (cid:98) Z in (3.8), thegenerating function Z ( x ) is obtained by acting e − (cid:98) Zx to the topological free energy (3.9) Z ( x ) = F ( { t k } ) − e − (cid:98) Zx F ( { t k } ) = F ( { t k } ) − F ( { t (cid:48) k } ) , (3.55)where t (cid:48) k is given by t (cid:48) k = t k − x (cid:126) (cid:114) βπ β k . (3.56)In the Airy case t k = 0 ( k ≥ Z ( x ) = −F (cid:16)(cid:110) − x (cid:126) (cid:114) βπ β k (cid:111)(cid:17) . (3.57)To see the structure of the derivatives of Z g with respect to t , t , it is convenient towork in the subspace (3.12) with non-zero t , t . On this subspace (3.12), the genus-zerostring equation (3.17) for the shifted coupling t (cid:48) k (3.56) becomes v = ∞ (cid:88) k =0 t (cid:48) k v k k ! = t + t v − x (cid:126) (cid:114) βπ e βv . (3.58)Here we denote u for the background t (cid:48) k (3.56) as v in order to distinguish it from u in(3.13) for the original background (3.12). Now we see that the string equation (3.58) forthe background (3.56) can be written in the form of the Lambert function z = W e W , (3.59)where z is given by (3.41) and W is defined by W = βt − t − βv. (3.60)Namely, the Lambert function naturally appears from the genus-zero string equation forthe shifted background (3.56). – 14 –s shown in [28], the genus- g free energy F g in (3.10) is written as a combination of v m = ∂ m v , which is known as the constitutive relation [29]. From (3.59) and (3.60), onecan show that v m at t = t = 0 is given by v m = δ m, − β m − D m W, (3.61)where D is defined in (3.47). In particular v is given by v = 1 − DW = 11 + W . (3.62)As discussed in [28], if we assign the weight m − v m , the genus- g free energy F g hasthe weight 3 g −
3. From (3.61), this implies that F g is written as F g = β g − (cid:98) F g , (3.63)where (cid:98) F g is a function of W only. Then, in the Airy limit t = t = 0 (3.57) becomes Z ( x ) = − ∞ (cid:88) g =0 ( √ (cid:126) ) g − β g − (cid:98) F g = ∞ (cid:88) g =0 ξ g − Z g , (3.64)which implies Z g = − g − (cid:98) F g . (3.65)For instance, using the expression of F g in terms of v m [28] and plugging v m in (3.61) into F g , we find Z = − (cid:98) F = −
124 log v = 124 log(1 + W ) , Z = − (cid:98) F = − β − (cid:18) v v − v v v + v v (cid:19) = 5 W − W + 4 W W ) . (3.66)As expected, this agrees with the result (3.51) obtained from the recursion relation. Wehave also checked that Z in (3.51) is correctly reproduced from F in [28].We note in passing that our Z g is related to the generating function of the Hodgeintegrals studied in [30]. For instance, Z is expanded as Z = − W ) + 11576(1 + W ) − W ) + 1720(1 + W ) . (3.67)This agrees with − H Hodge2 in eq.(87) of [30] after setting x i = 0 an T = − W . From theresult in [30], we find that Z g is expanded as Z g = 12 (1 + W ) − g g − (cid:88) n =1 κ g,n (cid:18) W W (cid:19) n , (3.68) It is well-known that the Lambert function also appears in the problem of Hurwitz numbers [31–33]. Aspointed out in [30], the generating function of Hurwitz numbers is a special case of the generating functionof the Hodge integrals (see the last page of [30] for details). – 15 –here κ g, = 112 g g ! , κ g, g − = ( − g c g g (5 g − g − . (3.69)As discussed in [30], c g is given by the coefficient of the formal power series solution of thePainlev´e I equation U = ∞ (cid:88) g =0 c g X (1 − g ) , d UdX + 116 U − X = 0 . (3.70) As shown in [18], at low temperature (cid:104) Z ( β ) n (cid:105) c is approximated by the one-point function (cid:104) Z ( nβ ) (cid:105) (3.5) and hence the generating function Z ( x ) is approximated by (3.6). Thesummation on the right hand side of (3.6) can be performed by rewriting the one-pointfunction (cid:104) Z ( nβ ) (cid:105) in terms of the eigenvalue density ρ ( E ) (cid:104) Z ( nβ ) (cid:105) = (cid:90) ∞−∞ dEρ ( E ) e − nβE . (3.71)For the Airy case, ρ ( E ) is given by ρ ( E ) = (cid:126) − / (cid:104) Ai (cid:48) ( − (cid:126) − / E ) − Ai( − (cid:126) − / E )Ai (cid:48)(cid:48) ( − (cid:126) − / E ) (cid:105) , (3.72)where Ai( z ) denotes the Airy function. We emphasize that this ρ ( E ) is the exact eigenvaluedensity including all-genus contributions. Then the low temperature approximation of Z ( x )in (3.6) becomes Z ( x ) ≈ − ∞ (cid:88) n =1 ( − x ) n n ! (cid:90) ∞−∞ dEρ ( E ) e − nβE = (cid:90) ∞−∞ dEρ ( E ) (cid:16) − e − xe − βE (cid:17) . (3.73)This expression of Z ( x ) is positive for x > ρ ( E ) is positive definite. Thus thisapproximation of Z ( x ) in (3.73) satisfies the necessary condition (3.54).In Fig. 3 we show the plot of the quenched free energy using the low temperatureapproximation (3.73). One can see that the quenched free energy is a monotonic functionof the temperature even in the low temperature regime. In this paper we have studied the quenched free energy in an ensemble average of randomsystems. We found the general formula (1.4) for the quenched free energy and argued thatit can be derived by the replica method (1.3) under a certain prescription of the analyticcontinuation of (cid:104) Z n (cid:105) . We also discussed that the bulk gravity picture of our formula (1.4)involves the so-called spacetime D-branes e − Zx introduced in [3]. Then we have applied ourgeneral formula (1.4) to the Airy limit of the Gaussian matrix model. We found that thegenus-zero approximation of the generating function Z ( x ) leads to a monotonic behaviorof the quenched free energy. The inclusion of higher genus corrections to Z ( x ) leads to a– 16 – .5 1.0 1.5 2.0 2.5 3.0 T ′ - F ′ Figure 3 : Plot of the free energy in the low temperature approximation. The blue solidcurve is the quenched free energy in the low temperature approximation while the orangedashed curve is the all-genus result of the annealed free energy.diverging free energy at low temperature, but this is not a problem of our formula (1.4); itsimply means that the genus expansion is not a good approximation at low temperature.Instead, we found that the alternative low temperature approximation (3.6) gives rise to awell-behaved monotonic free energy even at low temperature.There are many interesting open problems. It would be interesting to generalize ouranalysis of the Airy case to the matrix model of JT gravity [2]. It is speculated in [2]that the quenched free energy of JT gravity exhibits a spin glass phase with the replicasymmetry breaking at low temperature. Based on the result in [23, 24], it is argued in [9]that in the low temperature scaling limit β → ∞ , S → ∞ with βe S / : fixed , (4.1)the quenched free energy of JT gravity reduces to that of the Airy case. Thus, accordingto the argument in [9], we would be able to see the replica symmetry breaking and spinglass phase in the quenched free energy of the Airy case studied in our paper. However,in our computation in section 3, it is not obvious whether the replica symmetry breakingoccurs in the Airy case or not. Instead, our analysis is based on the spacetime D-branes e − Zx . It would be interesting to clarify the relation between these two pictures.Our analysis of the Airy case is limited to the genus expansion of Z ( x ) in (3.4) and thelow temperature approximation of Z ( x ) in (3.6). It would be nice if we can find a closedform expression of the generating function of the integral representation of (cid:104) Z n (cid:105) c given in[19]. We leave this as an interesting future problem.In section 3.3, we have seen that Z ( x ) in the Airy limit is closely related to the gener-ating function of the Hodge integrals studied in [30]. However, the quenched free energy isnot the generating function of the Hodge integrals itself, but some integral transformationof it. By introducing the chemical potential µ = log x , (1.6) is written as (cid:104) log Z (cid:105) = log (cid:104) Z (cid:105) − (cid:90) ∞−∞ dµ (cid:104) e −Z ( e µ ) − e −(cid:104) Z (cid:105) e µ (cid:105) . (4.2)This reminds us of the frame change in the topological string/spectral theory (TS/ST)correspondence (see e.g. [34] for a review). It would be interesting to understand themathematical meaning of this integral transformation.– 17 –n our previous paper [15], we find that the quenched free energy of Gaussian matrixmodel has a finite limit at zero temperaturelim T → F = E , (4.3)and E is given by the average of the minimal eigenvalue of the random matrix. Ourresult in the Airy case suggests that the quenched free energy in the Airy limit also has afinite limit at zero temperature (see Fig. 3). It would be interesting to compute this zerotemperature value of the free energy and clarify its physical meaning.Finally, let us comment on the status of our formula (1.6). We emphasize that (1.6)is not just a formal expression, but it is useful in practice for the actual computation ofthe quenched free energy in random matrix models. We have explicitly demonstrated thiscomputation for the Airy case in the genus expansion and also in the low temperatureapproximation (3.6). For the JT gravity case, as shown in [23, 24] the deviation from theAiry limit (4.1) is organized as a low temperature expansion and it would be possible tocompute Z ( x ) of JT gravity in this expansion. In general, the summation (1.5) over thenumber of boundaries might be an asymptotic series and one might worry that Z ( x ) is onlydefined perturbatively in the small x expansion. However, in principle Z ( x ) can be definednon-perturbatively as the free energy of random matrix model with a shifted background(3.55), as demonstrated in section 3.3 in the Airy case. In other words, Z ( x ) is the freeenergy of random matrix model with a deformed matrix potential due to the insertion of theSD-brane operator e − Zx . This definition of Z ( x ) does not rely on the small x expansion in(1.5) and it may serve as a non-perturbative definition of Z ( x ), which in turn leads to a non-perturbative definition of (cid:104) log Z (cid:105) via (1.6). Instead of computing the connected correlators (cid:104) Z n (cid:105) c term by term in (1.5), we should consider the expectation value of the SD-braneoperator (cid:104) e − Zx (cid:105) as a whole. We stress that this viewpoint is not obvious from the replicamethod (1.3) and it is one of the advantages of our formalism (1.6), at least conceptually.One might think that the expansion (2.10) used in the replica method breaks down when w n ’s in (2.9) become of order one. This is indeed the case at low temperature (see (A.1)).However, the computation in section 2.2 is just an illustration that our formula (1.4) can beobtained from the replica method and the final result (1.4) makes sense without referringto the replica method. Our formula (1.4) does not require (cid:104) log Z (cid:105) to be computed viathe expansion (2.10); (1.4) is well-defined even at low temperature once (cid:104) e − Zx (cid:105) is definednon-perturbatively by a shift of background (3.55). We hope that our expression of thequenched free energy (1.4) in terms of the SD-brane operator e − Zx provides us with a newperspective on our understanding of random systems and gravitational path integrals. Acknowledgments
The author would like to thank Kazuhiro Sakai for useful discussions. This work wassupported in part by JSPS KAKENHI Grant No. 19K03845.– 18 –
High temperature expansion in the Airy limit
In this appendix we consider the high temperature expansion of the quenched free energyin the Airy limit. The natural expansion parameter at high temperature is ξ = (cid:126) β . Forsmall ξ , w n defined in (2.9) behaves as w n = O ( ξ n − ) . (A.1)Up to the order O ( ξ ), the expansion (2.10) reads (cid:104) Z n (cid:105)(cid:104) Z (cid:105) n = 1 + 12 n ( n − w + 16 n ( n − n − w + 18 n ( n − n − n − w + O ( ξ ) . (A.2)For the Airy case, w and w are known in the closed form [19, 20]. From (3.1) we find w = (cid:114) πξ e ξ Erf (cid:32)(cid:114) ξ (cid:33) ,w = 4 πξ √ e ξ (cid:34) − T (cid:16)(cid:112) ξ, √ (cid:17)(cid:35) . (A.3)Plugging (A.3) into (A.2), we find the small- ξ expansion of (cid:104) Z n (cid:105) / (cid:104) Z (cid:105) n in the Airy limit (cid:104) Z n (cid:105)(cid:104) Z (cid:105) n = 1 + n ( n − ξ + n ( n − n − n + 2)24 ξ + O ( ξ ) . (A.4)This agrees with the result eq.(1.17) in [20]. Taking the n → ξ expansion (cid:104) log Z (cid:105) = log (cid:104) Z (cid:105) − ξ + 512 ξ + O ( ξ )= −
12 log(4 πξ ) − ξ + 512 ξ + O ( ξ ) , (A.5)where we used the result of one-point function (cid:104) Z (cid:105) in (3.1). For the higher w n ≥ we do notknow the closed form expression, but the small ξ expansion of w n is easily obtained fromthe KdV equation (3.39), as we will see below.Let us briefly explain the computation of w n in the small ξ expansion. To do this, it isconvenient to introduce the generating function of the connected correlators (cid:101) Z ( x ), whichis related to Z ( x ) in (1.5) by (cid:101) Z ( x ) = −Z ( − x ) = ∞ (cid:88) n =1 x n n ! (cid:104) Z n (cid:105) c . (A.6)Then one can show that (cid:102) W = (cid:126) ∂ (cid:101) Z satisfies a similar relation as (3.39) ∂ (cid:102) W = u∂ (cid:102) W + (cid:126) ∂ (cid:102) W + (cid:126) ( ∂ (cid:102) W ) . (A.7)– 19 –o solve this relation recursively, we work on the subspace (3.12). On this subspace (3.12),the one-point function is given by (cid:104) Z ( β ) (cid:105) = 1 (cid:113) π ˆ ξ exp (cid:32) ˆ ξ
12 + βt − t (cid:33) . (A.8)Here ˆ ξ = ˆ (cid:126) β and ˆ (cid:126) is defined in (3.42). From the result of (cid:104) Z n (cid:105) c for small n in (3.1), itis natural to make an ansatz (cid:104) Z ( β ) n (cid:105) c = (cid:104) Z ( nβ ) (cid:105) C n ( ˆ ξ ) . (A.9)The important point is that C n is independent of t and it depends on t only through thecombination ˆ (cid:126) in (3.42). From (A.7), W n = (cid:126) ∂ (cid:104) Z ( β ) n (cid:105) c satisfies ∂ W n = u∂ W n + (cid:126) ∂ W n + (cid:126) n − (cid:88) m =1 (cid:18) nm (cid:19) ∂ W m ∂ W n − m . (A.10)Using the fact that the one-point part (cid:104) Z ( nβ ) (cid:105) of the ansatz (A.9) satisfies the first line of(A.10), we find the equation for C n ( ˆ ξ )(1 − t ) ∂ C n = (cid:115) ˆ ξ π n − (cid:88) m =1 (cid:18) nm (cid:19)(cid:112) nm ( n − m ) e − nm ( n − m )ˆ ξ C m C n − m . (A.11)Since C n depends on t only through the combination ˆ ξ , the left hand side of (A.11) isequal to 2 ˆ ξ∂ ˆ ξ C n . Finally, after setting t = t = 0, we find the recursion relation for C n ( ξ )2 ξ∂ ξ C n = (cid:114) ξ π n − (cid:88) m =1 (cid:18) nm (cid:19)(cid:112) nm ( n − m ) e − nm ( n − m ) ξ C m C n − m . (A.12)One can easily solve this equation recursively starting from C = 1. For instance, C satisfies ∂ ξ C = e − ξ √ πξ . (A.13)This is solved as C = Erf (cid:16)(cid:112) ξ/ (cid:17) , (A.14)which agrees with the known result (3.1) of the two-point function in the Airy limit. In asimilar manner, for n = 3 we can show that C = 1 − T ( (cid:112) ξ, / √
3) (A.15)is the solution for the recursion relation (A.12).– 20 –or n ≥ C n , but it is not difficult to solvethe recursion relation (A.12) in the small ξ regime and find C n as a power series in ξ . Then w n = (cid:104) Z n (cid:105) c / (cid:104) Z (cid:105) n is also obtained as a power series in ξw n = n − (4 πξ ) n − e n − n ξ C n ( ξ ) . (A.16)For instance, w and w are expanded as w = 32 ξ + 84 ξ + 1936 ξ
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