Genus expansion of open free energy in 2d topological gravity
PPrepared for submission to JHEP
Genus expansion of open free energy in 2d topologicalgravity
Kazumi Okuyama a and Kazuhiro Sakai b a Department of Physics, Shinshu University,3-1-1 Asahi, Matsumoto 390-8621, Japan b Institute of Physics, Meiji Gakuin University,1518 Kamikurata-cho, Totsuka-ku, Yokohama 244-8539, Japan
E-mail: [email protected], [email protected]
Abstract:
We study open topological gravity in two dimensions, or, the intersection the-ory on the moduli space of open Riemann surfaces initiated by Pandharipande, Solomonand Tessler. The open free energy, the generating function for the open intersection num-bers, obeys the open KdV equations and Buryak’s differential equation and is related bya formal Fourier transformation to the Baker-Akhiezer wave function of the KdV hierar-chy. Using these properties we study the genus expansion of the free energy in detail. Weconstruct explicitly the genus zero part of the free energy. We then formulate a method ofcomputing higher genus corrections by solving Buryak’s equation and obtain them up tohigh order. This method is much more efficient than our previous approach based on thesaddle point calculation. Along the way we show that the higher genus corrections are poly-nomials in variables that are expressed in terms of genus zero quantities only, generalizingthe constitutive relation of closed topological gravity. a r X i v : . [ h e p - t h ] S e p ontents z ∗ ( s ) 16 Two-dimensional gravity is one of the simplest models of quantum gravity which has beenintensively studied for quite some time. In [1–4] it was found that two-dimensional gravityis described by a certain double-scaled random matrix model (see [5] for a review). Math-ematically, two-dimensional gravity corresponds to an intersection theory on the modulispace of closed Riemann surfaces, as first conjectured by Witten [6] and proved by Kont-sevich [7]. It is known that the free energy of two-dimensional gravity on closed Riemannsurfaces satisfies the KdV equations [6–8] and the Virasoro constraints [9, 10]. Recently, itis realized that this story holds for Jackiw-Teitelboim (JT) gravity as well; Saad, Shenkerand Stanford [11] showed that JT gravity is described by a doubled-scaled matrix modeland it corresponds to a particular background of Witten-Kontsevich topological gravity[12–14].Recently, Pandharipande, Solomon and Tessler [15] initiated the study of open topo-logical gravity, i.e. the intersection theory on the moduli space of Riemann surfaces withboundary. See also [16–21] for related works. It is conjectured in [15] and proved in [18]that the open free energy F o ( s ), or the generating function of the open intersection num-bers, satisfies the open version of the KdV equations and the Virasoro constraints. As The variable s is related to the ’t Hooft parameter λ in our previous papers [14, 22] by λ = √ s . – 1 –xplained in [13], open topological gravity is physically realized by adding vector degrees offreedom to the matrix model of two-dimensional gravity. After integrating out the vectordegrees of freedom, this amounts to the insertion of the determinant operator det( ξ − M ) tothe matrix integral, where ξ is a parameter and M is the random matrix. The expectationvalue of this determinant operator ψ ( ξ ) = e − g s V ( ξ ) (cid:104) det( ξ − M ) (cid:105) (1.1)corresponds to the wavefunction of the FZZT brane [23, 24]. Here V ( ξ ) is the matrix modelpotential and g s is the genus counting parameter (denoted as u in [15–18]). ψ ( ξ ) is alsoidentified as the Baker-Akhiezer (BA) function of the KdV hierarchy [25].It is known that the exponential of the open free energy e F o ( s ) and the BA function ψ ( ξ ) are related by the formal Fourier transformation [13, 18] e F o ( s ) = (cid:90) ∞−∞ dξe sξg s ψ ( ξ ) . (1.2)One can compute the small g s expansion of F o ( s ) F o ( s ) = ∞ (cid:88) ˜ g =0 g ˜ g − F o˜ g ( s ) (1.3)from the result of the WKB expansion of the BA function by evaluating the integral (1.2)by the saddle point method. For instance, the leading term of the small g s expansion of ψ ( ξ ) ≈ e − g s V eff ( ξ ) is given by the so-called effective potential V eff ( ξ ). In our previous paper[22], we obtained the explicit form of V eff ( ξ ) for arbitrary background couplings { t n } . Thenthe leading term F o0 ( s ) in (1.3) is given by the Legendre transform of V eff ( ξ ). One can inprinciple continue this saddle point computation for the higher order corrections in g s , butthe computation becomes very cumbersome as the order of g s increases.It turns out that the small g s expansion of F o ( s ) can be computed systematically byrecursively solving Buryak’s equation [17], which is understood as the Fourier transformof the Schr¨odinger equation of the KdV hierarchy. This is based on the fact that F o˜ g ≥ ( s )in (1.3) is written as a polynomial in variables which are expressed in terms of genus zeroquantities only. This is similar to the situation in original Witten-Kontsevich topologicalgravity, where the genus- g ( ≥
2) closed free energy F c g is written as a polynomial in a certainbasis [26, 27].This paper is organized as follows. In section 2 we briefly review closed and opentopological gravities. We also explain how the open KdV equations and Buryak’s equationare derived from the KdV hierarchy. In section 3 we study the genus expansion of theopen free energy. We first compute it from the genus expansion of the BA function bythe saddle point calculation. We next derive an explicit expression of the genus zero openfree energy. We then formulate a method of computing the genus expansion by solvingBuryak’s equation. We conclude in section 4 with discussions on the future directions.Some details of the calculations are relegated to the appendices A and B. The inverse transformation of (1.2) is considered in [18]. Our (1.2) is equivalent to eq.(4.76) in [13]. – 2 –
Brief review of topological gravity
In this section we will briefly review the basics and known results about closed and opentopological gravities. We will also explain how the open KdV equations and Buryak’sequation, which will be the main tools of our study of the open free energy, are derivedfrom the KdV hierarchy.
In Witten-Kontsevich topological gravity [6, 7] (see also [13]) observables are made up ofthe intersection numbers (cid:104) τ d · · · τ d n (cid:105) g,n = (cid:90) M g,n ψ d · · · ψ d n n , d , . . . , d n ∈ Z ≥ . (2.1)They are defined on a closed Riemann surface Σ of genus g with n marked points p , . . . , p n .We let M g,n denote the moduli space of Σ and M g,n the Deligne-Mumford compactificationof M g,n . Here τ d i = ψ d i i and ψ i is the first Chern class of the complex line bundle whosefiber is the cotangent space to p i . The intersection numbers (2.1) obey the selection rule (cid:104) τ d · · · τ d n (cid:105) g,n = 0 unless d + · · · + d n = 3 g − n. (2.2)The generating function for the above intersection numbers is defined as F c ( s, { t k } ) := ∞ (cid:88) g =0 g g − F c g ( { t k } ) , F c g ( { t k } ) := (cid:68) e (cid:80) ∞ d =0 t d τ d (cid:69) g . (2.3)We will call F c the closed free energy.It was conjectured [6] and proved [7] that e F c is a tau function for the KdV hierarchy.This means that u := g ∂ F c (2.4)satisfies the KdV equations ∂ k u = ∂ R k +1 , (2.5)where R k are the Gelfand-Dikii differential polynomials of u R = 1 , R = u, R = u g ∂ u , · · · . (2.6)Here we have introduced the notation ∂ k := ∂∂t k . (2.7)For k = 1, (2.5) gives the traditional KdV equation ∂ u = u∂ u + g ∂ u. (2.8)– 3 –ntegrating (2.5) once in t we have g ∂ k ∂ F c = R k +1 . (2.9)It is well known (see e.g. [28]) that the KdV equations (2.5) are obtained as thecompatibility condition of the Schr¨odinger equation Qψ = ξψ (2.10)and the KdV flow equations ∂ k ψ = M k ψ, (2.11)where Q := g ∂ + u, M k := (2 Q ) k +1 / (2 k + 1)!! g s . (2.12)Here we have decomposed (2 Q ) k +1 / = (2 Q ) k +1 / + (2 Q ) k +1 / − and the subscript + meansthat (2 Q ) k +1 / contains only non-negative powers of ∂ . Indeed, (2.5) is recovered by usingthe relation [ M k , Q ] = ∂ R k +1 . (2.13)The wave function ψ that satisfies (2.10) and (2.11) is known as the Baker-Akhiezer func-tion.Another important constraint that the closed free energy F c obeys is the string equa-tion [8]. For Witten-Kontsevich gravity it is written as u − ∞ (cid:88) k =0 t k R k = 0 . (2.14)The genus zero part of this string equation is written as u − I ( u , { t k } ) = 0 , (2.15)where u is the genus-zero part of u u := ∂ F c0 (2.16)and we have introduced the Itzykson-Zuber variables [26] I n ( v, { t k } ) = ∞ (cid:88) (cid:96) =0 t n + (cid:96) v (cid:96) (cid:96) ! ( n ≥ . (2.17)Throughout this paper I n without specifying its arguments should always be understoodas I n = I n ( u , { t k } ) . (2.18)– 4 –t is also convenient to introduce the variable t := ( ∂ u ) − = 1 − I . (2.19)It was conjectured [26] and proved [27, 29] that F g ( { t k } ) ( g ≥
2) are polynomials in I n ≥ and t − . This fact significantly helps us to compute higher genus free energy F c g .An efficient way to compute F c g is as follows. (See [14] for a more detailed explanation.)Let us expand u as u = ∞ (cid:88) g =0 g g s u g , u g = ∂ F c g . (2.20) u g can be computed by recursively solving the KdV equation (2.8). To do this, let usregard t k ≥ as parameters and consider the change of variables from ( t , t ) to ( u , t ). Thedifferentials ∂ , are then written in the new variables as ∂ = 1 t ( ∂ u − I ∂ t ) , ∂ = u ∂ − ∂ t . (2.21)By expanding both sides of the equation in g s (2.8) is written as the recursion relation − t ∂ t ( tu g ) = g − (cid:88) h =1 u g − h ∂ u h + 112 ∂ u g − ( g ≥ . (2.22)This is easily solved with the help of (2.21). First few of u g are u = I t + I t ,u = 49 I t + 11 I I t + 84 I I + 109 I I t + 32 I I + 51 I I t + I t . (2.23)As explained in [14] one can easily integrate u g twice in t and obtain the well-knownresults [26] F c1 = −
124 log t,F c2 = I t + 29 I I t + 7 I t . (2.24) Pandharipande, Solomon and Tessler proposed an open analog of Witten-Kontsevich topo-logical gravity [15]. They introduced the open intersection numbers (cid:104) τ d · · · τ d n σ k (cid:105) o˜ g,n = 2 − ˜ g + k − (cid:90) M ˜ g,k,n e ( E, s ) , d , . . . , d n ∈ Z ≥ . (2.25)The new insertion σ corresponds to the addition of a boundary marking and the power k of σ specifies the number of boundary markings. In [15] a natural lift e ( E, s ) of the This change of variables was originally introduced by Zograf (see e.g. [30]). – 5 –uler class e ( E ) = ψ d · · · ψ d n n is defined. M ˜ g,k,n denotes a suitable compactification ofthe moduli space M ˜ g,k,n of Riemann surfaces with boundary of doubled genus ˜ g with k boundary markings and n interior markings. The open intersection numbers (2.25) obeythe selection rule (cid:104) τ d · · · τ d n σ k (cid:105) o˜ g,n = 0 unless 2 n (cid:88) j =1 d j = 3˜ g − k + 2 n. (2.26)The generating function for the open intersection numbers is defined as F o ( s, { t k } , g s ) := ∞ (cid:88) ˜ g =0 g ˜ g − F o˜ g ( s, { t k } ) , F o˜ g ( s, { t k } ) := (cid:68) e sσ + (cid:80) ∞ d =0 t d τ d (cid:69) o˜ g . (2.27)We will call F o the open free energy.It was conjectured [15] and then proved [18] that F o satisfies the open KdV equations2 n + 12 ∂ n F o = g s ∂ s F o ∂ n − F o + g s ∂ s ∂ n − F o + g ∂ F o ∂ ∂ n − F c − g ∂ ∂ n − F c ( n ≥ . (2.28)In fact it is known [17] that F o is fully determined by the above system of equations withthe initial condition F o (cid:12)(cid:12)(cid:12) t k ≥ =0 = 1 g s (cid:18) s t s (cid:19) , (2.29)given the closed free energy F c . Buryak proved that F o further satisfies another differentialequation [17] ∂ s F o = g s (cid:20)
12 ( ∂ F o ) + 12 ∂ F o + ∂ F c (cid:21) . (2.30)These equations play a crucial role in the study of F o in this paper.In [16] Buryak constructed an explicit expression for e F o in terms of F c . In this sensean explicit form of F o is known. For many purposes, however, it is still useful to express F o in the form of genus expansion (2.27) and construct an explicit, closed expression of F o˜ g ( s, { t k } ) for fixed ˜ g . This is our primary goal in this paper. It is known [13, 18] that the open partition function exp F o is related to the BA function ψ by the formal Fourier transformation (1.2). Using this relation one can show [18] that theopen KdV equations (2.28) and Buryak’s equation (2.30) are in fact derived from the KdVflow equations (2.11) and the Schr¨odinger equation (2.10) respectively. In what follows wewill present a derivation in a manner slightly different from [18].Let (cid:98) ψ denote the formal Fourier transform of the BA function ψ (cid:98) ψ ( s ) := (cid:90) ∞−∞ dξe sξg s ψ ( ξ ) . (2.31)– 6 –n terms of (cid:98) ψ the Schr¨odinger equation (2.10) is written as Q (cid:98) ψ = g s ∂ s (cid:98) ψ. (2.32)On the other hand, (1.2) is written as (cid:98) ψ = e F o . (2.33)Substituting (2.33) into (2.32) one obtains g ∂ e F o + ue F o = g s ∂ s e F o . (2.34)Rewriting u by (2.4) we immediately see that this is equivalent to (2.30). We have thusseen that Buryak’s equation (2.30) is nothing but the formal Fourier transform of theSchr¨odinger equation (2.10).Similarly, let us consider the Fourier transform of the KdV flow equations (2.11). It isclear that the same equations hold for (cid:98) ψ as well ∂ n (cid:98) ψ = M n (cid:98) ψ = (2 Q ) n + + (2 n + 1)!! g s (cid:98) ψ. (2.35)It is shown [16] that these equations are satisfied by (cid:98) ψ in (2.33) with F o obeying theopen KdV equations (2.11). Conversely, we can directly derive the open KdV equations(2.11) from (2.35) using (2.33). Since the derivation is rather technical, we relegate itto Appendix A. We stress that the open KdV equations are equivalent to the KdV flowequations (2.11) under the identification (2.33). In this section we will study the genus expansion of the open free energy. We will firstcompute it from the genus expansion of the BA function by the saddle point calculation.We will next derive a fully explicit expression of the genus zero open free energy. Finally, wewill formulate a method of computing the genus expansion by solving Buryak’s equation,which turns out to be much more efficient than the saddle point calculation.
We saw in [22] that the BA function ψ admits the following expansion ψ = e A , A = ∞ (cid:88) ˜ g =0 g ˜ g − A ˜ g . (3.1)– 7 –irst few of A ˜ g are A = − tz ∞ (cid:88) n =1 I n +1 (2 n + 3)!! z n +3 ,A = −
12 log z + const. ,A = − tz − I t z ,A = 516 t z + 1 t (cid:18) I z + I z (cid:19) + I t z , (3.2)where we have introduced z := (cid:112) ξ − u ) . (3.3) A ˜ g can be computed up to any order by solving the recursion relation for v ˜ g := ∂ A ˜ g v ˜ g = − v (cid:32) ∂ v ˜ g − + ˜ g − (cid:88) k =1 v k v ˜ g − k + (cid:40) u ˜ g (˜ g even)0 (˜ g odd) (cid:33) , ( n ≥ ,v = z, v = 12 tz . (3.4)In [22] we performed this computation with special values of t k corresponding to the caseof JT gravity, but as advertised in [31] it can be generalized without any effort to the caseof general values of t k , as we have seen above. The Fourier transformation (1.2) enables us to calculate the genus expansion (1.3) of F o from that of A = log ψ just obtained above. (1.2) is written as e F o = (cid:90) ∞−∞ dξe sξg s + A = (cid:90) ∞−∞ dξe [ sξ + A ( ξ )] g − + A ( ξ )+ A ( ξ ) g s + O ( g ) . (3.5)As in [14, 22] one can calculate F o˜ g by the saddle point method.The saddle point ξ ∗ is given by the condition ∂ ξ [ sξ + A ( ξ )] (cid:12)(cid:12)(cid:12) ξ = ξ ∗ = 0 . (3.6) In [22] the constant part of A is fixed so that it fits well with the convention of closed topologicalgravity. In this paper we will use this degree of freedom later for compensating the difference of thenormalizations of e F o and ψ , so that we can avoid putting an inessential normalization factor in (1.2). A ˜ g , v ˜ g and z in this paper are related to those in our previous paper [22] by A here˜ g = (cid:0) √ (cid:1) − ˜ g A there˜ g , v here˜ g = (cid:0) √ (cid:1) − ˜ g v there˜ g , z here = √ z there . – 8 –his is equivalent to s = − ∂ ξ A (cid:12)(cid:12)(cid:12) ξ = ξ ∗ = tz ∗ − ∞ (cid:88) n =1 I n +1 (2 n + 1)!! z n +1 ∗ , (3.7)where z ∗ := (cid:112) ξ ∗ − u ) ⇔ ξ ∗ = z ∗ u . (3.8)By using the Lagrange inversion theorem this is inverted as (see Appendix B) z ∗ = (cid:88) j a ≥ (cid:80) a j a = k (cid:80) a aj a = n (2 n + k )!(2 n + 1)! s n +1 t n + k +1 ∞ (cid:89) a =1 I j a a +1 j a !(2 a + 1)!! j a . (3.9)As in [14] let us introduce a new variable φ as ξ = ξ ∗ + √ g s φ. (3.10)The integral (3.5) is then written as e F o = e [ sξ ∗ + A ( ξ ∗ )] g − + A ( ξ ∗ ) (cid:90) ∞−∞ √ g s dφe ∂ ξ ∗ A ( ξ ∗ ) φ + O ( g / ) . (3.11)By expanding the integrand in g s , the integral in φ can be performed order by order as aGaussian integral. In fact, we did essentially the same calculation in [14] up to the orderof g . We thus immediately obtain F o0 = sξ ∗ + A ( ξ ∗ ) = sξ ∗ − (cid:90) ξ ∗ u dξ (cid:48)∗ s ( ξ (cid:48)∗ )= (cid:90) s ds (cid:48) ξ ∗ ( s (cid:48) ) ,F o1 = A ( ξ ∗ ) + 12 log 2 πg s − ∂ ξ ∗ A ( ξ ∗ )= 12 log ∂ s ξ ∗ z ∗ = 12 log ∂ s z ∗ ,F o2 = A ( ξ ∗ ) + 12 (cid:32) A (3)0 ∗ (cid:33) (cid:104) φ (cid:105) + (cid:32) A (4)0 ∗
4! + A (3)0 ∗ A (1)1 ∗ (cid:33) (cid:104) φ (cid:105) + (cid:0) A (1)1 ∗ (cid:1) + A (2)1 ∗ (cid:104) φ (cid:105) = − tz ∗ − I t z ∗ + ξ (3) ∗ (cid:0) ξ (1) ∗ (cid:1) − (cid:0) ξ (2) ∗ (cid:1) (cid:0) ξ (1) ∗ (cid:1) − ξ (2) ∗ z ∗ ξ (1) ∗ + 5 ξ (1) ∗ z ∗ , (3.12)where we have introduced the notation A ( n )˜ g ∗ := ∂ nξ A ˜ g (cid:12)(cid:12)(cid:12) ξ = ξ ∗ , ξ ( n ) ∗ := ∂ ns ξ ∗ . (3.13)– 9 –n the last equality in (3.12) we have used A ( n )0 ∗ = − (cid:32) ξ (1) ∗ ∂∂s (cid:33) n − s ( n ≥ , (cid:104) φ m (cid:105) = (cid:82) ∞−∞ dφe A (2)0 ∗ φ φ m (cid:82) ∞−∞ dφe A (2)0 ∗ φ = (2 m − (cid:0) − A (2)0 ∗ (cid:1) m = (2 m − (cid:0) ξ (1) ∗ (cid:1) m ( m ≥ . (3.14)We have fixed the constant part of A in (3.2) in such a way that the initial condition (2.29)is satisfied. Using this method one can in principle calculate F o˜ g up to any order. However,this calculation gets quickly involved as ˜ g increases. We will propose an alternative, muchmore efficient method of computing F o˜ g in the following subsections.An advantage of the above calculation is that we can prove the polynomial structure ofthe higher genus free energies F o˜ g ≥ . The expansion (3.11) implies that F o˜ g ≥ are polynomialsin A ( n ≥ ∗ , A ( n ≥ ∗ , A ( n ≥ g ≥ ∗ and (cid:104) φ m (cid:105) ( m ≥ A ˜ g ≥ arepolynomials in t − , I k ≥ and z − . Combining these two lemmas we arrive at the conclusionthat F o˜ g ≥ are polynomials in the variables t − , I k ≥ , z − ∗ , (cid:0) ξ (1) ∗ (cid:1) − and ξ ( n ≥ ∗ .It is well known that closed topological gravity exhibits the constitutive relation [32],i.e. higher genus quantities are expressed in terms of genus zero quantities only. In thecase of Witten-Kontsevich gravity F c1 is given as in (2.24) and F c g ≥ are expressed aspolynomials in the variables t − and I k ≥ , as we saw in section 2.1. These variables areexpressed explicitly in terms of genus zero quantities ∂ n u ( n ≥
1) [33]. Since z ∗ and ξ ( n ) ∗ = ∂ n +1 s F o0 are also genus zero quantities, the form of F o1 in (3.12) and the abovepolynomial structure of F o˜ g ≥ ensure that Pandharipande-Solomon-Tessler open topologicalgravity exhibits a generalized constitutive relation. In the last subsection we have obtained an explicit expression of F o0 : By plugging (3.8)into the second line of (3.12) we have F o0 = u s + 12 (cid:90) s ds (cid:48) z ∗ ( s (cid:48) ) (3.15)with z ∗ ( s ) given in (3.9). As we will see below, we can write down a more direct expressionfor F o0 by using the relations among F o0 , ξ ∗ and z ∗ which follow from the system of equations(2.30) and (2.28).Buryak’s equation (2.30) at the order of g − reads ∂ s F o0 = 12 ( ∂ F o0 ) + u . (3.16)Note also that the second line of (3.12) gives ∂ s F o0 = ξ ∗ . (3.17) See (3.18). It is also possible to express ξ ( n ) ∗ in terms of t -derivatives only. This is done by repeatedlyusing (3.29). – 10 –omparing these with (3.8) one finds ∂ F o0 = z ∗ . (3.18)On the other hand, the open KdV equation (2.28) for n = 1 at the order of g − reads32 ∂ F o0 = ∂ s F o0 ∂ F o0 + 12 ∂ F o0 ∂ F c0 . (3.19)By using (2.21), (3.16), (3.18) and (2.16) this becomes − ∂ t F o0 = z ∗ . (3.20)Applying ∂ s to both sides of the equation and using again (3.16) and (3.18) one obtains − ∂ t (cid:18) z ∗ u (cid:19) = z ∗ ∂ s z ∗ , (3.21)which gives − ∂ t z ∗ = z ∗ ∂ s z ∗ = ∂ s ξ ∗ = ∂ s F o0 . (3.22)Hence, differentiating (3.9) once in t and then integrating it twice in s , one obtains F o0 = u s + (cid:88) j a ≥ (cid:80) a j a = k (cid:80) a aj a = n (2 n + k + 1)!(2 n + 3)! s n +3 t n + k +2 ∞ (cid:89) a =1 I j a a +1 j a !(2 a + 1)!! j a . (3.23)The integration constants have been fixed accordingly so that (3.23) matches with (3.15).We verified by series expansion in s that (3.23) and (3.15) are indeed in perfect agreement.Note that when t k ≥ = 0, we have t = u , t = 1 , I k ≥ = 0 and thus the above F o0 becomes F o0 = t s + s . (3.24)This is consistent with the initial condition (2.29). By substituting the genus expansions (1.3) and (2.20) Buryak’s equation (2.30) at the orderof g ˜ g − (˜ g ≥
1) is written as D F o˜ g = 12 ˜ g − (cid:88) k =1 ∂ F o˜ g − k ∂ F o k + 12 ∂ F o˜ g − + (cid:40) u ˜ g (˜ g even)0 (˜ g odd) , (3.25) Note that z ∗ is the uniformization coordinate on the spectral curve. In the context of minimal stringtheory, it is known that z ∗ is given by the t -derivative of the disk amplitude F o0 [25]. – 11 –here we have introduced the differential operator D := ∂ s − z ∗ ∂ . (3.26)(3.25) can be viewed as a recursion relation: one can recursively compute F o˜ g if one is ableto perform the integration D − on the l.h.s. of (3.25). This is indeed feasible, as we willsee below.To do this, let us first study the operator D , which has in fact several interestingproperties. For instance, one can show that D z ∗ = 1 t , D ξ ∗ = 0 , D ξ ( n − ∗ = D ∂ n − s ξ ∗ = 12 n − (cid:88) k =1 (cid:32) nk (cid:33) ∂ n − ks z ∗ ∂ ks z ∗ ( n ≥ . (3.27)The first line of (3.27) follows from z ∗ ∂ z ∗ = ∂ ( ξ ∗ − u ) = ∂ ξ ∗ − t (3.28)and ∂ ξ ∗ = ∂ s z ∗ , (3.29)which follows from (3.18) by differentiating both sides of the equation in s . The secondline of (3.27) also follows from (3.29). The third line of (3.27) can easily be shown byinduction.It is also useful to note that ξ ( n ) ∗ = ∂ ns ξ ∗ = 12 n (cid:88) k =0 (cid:32) nk (cid:33) ∂ n − ks z ∗ ∂ ks z ∗ ( n ≥ , (3.30)which immediately follows from (3.8). This relation is important because it enables us toexpress ξ ( n ≥ ∗ in terms of z ( n ) ∗ := ∂ ns z ∗ ( n ≥ , (3.31)and vice versa. For instance, z ( n ≥ ∗ with small n are expressed in terms of ξ ( n ≥ ∗ as z (1) ∗ = ξ (1) ∗ z ∗ ,z (2) ∗ = ξ (2) ∗ z ∗ − (cid:0) ξ (1) ∗ (cid:1) z ∗ ,z (3) ∗ = ξ (3) ∗ z ∗ − ξ (1) ∗ ξ (2) ∗ z ∗ + 3 (cid:0) ξ (1) ∗ (cid:1) z ∗ . (3.32)– 12 –oreover, comparing (3.30) with (3.27) one finds that ξ ( n ) ∗ = D ξ ( n − ∗ + z ∗ z ( n ) ∗ ( n ≥ . (3.33)Therefore, by using (3.32) and (3.33) one can express D ξ ( n ≥ ∗ as polynomials in ξ ( k ≥ ∗ and z − ∗ : D ξ (1) ∗ = (cid:0) ξ (1) ∗ (cid:1) z ∗ , D ξ (2) ∗ = 3 ξ (1) ∗ ξ (2) ∗ z ∗ − (cid:0) ξ (1) ∗ (cid:1) z ∗ , D ξ (3) ∗ = 4 ξ (1) ∗ ξ (3) ∗ z ∗ + 3 (cid:0) ξ (2) ∗ (cid:1) z ∗ − (cid:0) ξ (1) ∗ (cid:1) ξ (2) ∗ z ∗ + 15 (cid:0) ξ (1) ∗ (cid:1) z ∗ . (3.34)On the other hand, to evaluate the r.h.s. of (3.25) it is convenient to use ∂ z ∗ = z (1) ∗ z ∗ − tz ∗ ,∂ ξ ( n ) ∗ = z ( n +1) ∗ ( n ≥ . (3.35)Again using (3.32) one can express these quantities as polynomials in t − , z − ∗ and ξ ( n ≥ ∗ .Hence, by using the low genus results (3.12) and the polynomial structure of F o˜ g ≥ derivedin section 3.2, it is easy to see that all quantities appearing in (3.25) are expressed aspolynomials in the variables t − , I k ≥ , z − ∗ , (cid:0) ξ (1) ∗ (cid:1) − and ξ ( n ≥ ∗ . We are now in a position to solve the recursion relation (3.25) and compute the highergenus free energy F o˜ g . To begin with, we verified that F o˜ g with ˜ g = 0 , , g = 1 ,
2. This is easily done by usingvarious identities derived in the last two subsections. Moreover, based on the polynomialstructure discussed above, one can perform the integration D − completely and determine F o˜ g unambiguously for ˜ g ≥
2. The algorithm to solve (3.25) and obtain F o˜ g from the dataof { F o˜ g (cid:48) } ˜ g (cid:48) < ˜ g is as follows:(i) Compute the r.h.s. of (3.25) using (3.35) and express it as a polynomial in the vari-ables t − , I k ≥ , z − ∗ , (cid:0) ξ (1) ∗ (cid:1) − and ξ ( n ≥ ∗ .(ii) Let t − m f ( I k , z ∗ , ξ ( n ) ∗ ) denote the highest order part in t − of the obtained expression.This part can arise only from D (cid:32) − f ( I k , z ∗ , ξ ( n ) ∗ )( m − t m − z ∗ I (cid:33) . (3.36)Therefore subtract this from the obtained expression.– 13 –iii) Repeat the procedure (ii) down to m = 3. Then all the terms of order t − automat-ically disappear and the remaining terms are of order t − or t . Note also that theexpression does not contain any I k .(iv) In the result of (iii), collect all the terms of order t − and let t − ∂ z ∗ g ( z ∗ , ξ ( n ) ∗ ) denotethe sum of them. This part arises from D g ( z ∗ , ξ ( n ) ∗ ) . (3.37)Therefore subtract this from the result of (iii). The remainder turns out to be inde-pendent of t .(v) In the obtained expression, let h (cid:0) ξ ( n ≥ ∗ (cid:1) z ∗ (cid:0) ξ (1) ∗ (cid:1) m (3.38)denote the part which is of order z − ∗ as well as of the lowest order in (cid:0) ξ (1) ∗ (cid:1) − . Thispart arises from D (cid:32) h (cid:0) ξ ( n ≥ ∗ (cid:1) ( m + 1) (cid:0) ξ (1) ∗ (cid:1) m +1 (cid:33) . (3.39)Therefore subtract this from the obtained expression.(vi) Repeat the procedure (v) until the resulting expression vanishes.(vii) By summing up all the above obtained primitive functions we obtain F o˜ g .Using this algorithm we computed F o˜ g for ˜ g ≤ We verified that F o2 computed by thisalgorithm reproduces the result (3.12) of our saddle point calculation. For ˜ g = 3 we obtain F o3 = I t z ∗ + I t z ∗ + I t z ∗ + 516 t z ∗ + I ξ (2) ∗ t z ∗ ξ (1) ∗ − I ξ (1) ∗ t z ∗ + 5 ξ (2) ∗ tz ∗ ξ (1) ∗ − ξ (1) ∗ tz ∗ − ξ (2) ∗ z ∗ + 35 (cid:0) ξ (1) ∗ (cid:1) z ∗ − ξ (4) ∗ z ∗ (cid:0) ξ (1) ∗ (cid:1) + 5 ξ (3) ∗ z ∗ ξ (1) ∗ − (cid:0) ξ (2) ∗ (cid:1) z ∗ (cid:0) ξ (1) ∗ (cid:1) − (cid:0) ξ (2) ∗ (cid:1) z ∗ (cid:0) ξ (1) ∗ (cid:1) + 7 ξ (3) ∗ ξ (2) ∗ z ∗ (cid:0) ξ (1) ∗ (cid:1) + ξ (5) ∗ (cid:0) ξ (1) ∗ (cid:1) − (cid:0) ξ (3) ∗ (cid:1) (cid:0) ξ (1) ∗ (cid:1) − ξ (4) ∗ ξ (2) ∗ (cid:0) ξ (1) ∗ (cid:1) + 3 ξ (3) ∗ (cid:0) ξ (2) ∗ (cid:1) (cid:0) ξ (1) ∗ (cid:1) − (cid:0) ξ (2) ∗ (cid:1) (cid:0) ξ (1) ∗ (cid:1) . (3.40) In this paper we have studied the small g s expansion (1.3) of the open free energy F o ( s )of topological gravity. We have obtained the explicit form (3.23) of the genus zero part The data of F o˜ g are available upon request to the authors. – 14 – o0 of the free energy. We have then argued that the higher order corrections F o˜ g can becomputed systematically by solving Buryak’s equation recursively. We have demonstratedthis computation explicitly for the first few orders. We have also elucidated the polynomialstructure of F o˜ g ≥ . We emphasize that our result of F o˜ g holds for arbitrary value of thecouplings { t n } . It is interesting that F o˜ g is written as a combination of genus-zero quantitiesonly, which can be thought of as an open analog of the constitutive relation for closedtopological gravity [26, 27].There are several interesting open questions. In general, the small g s expansion of F o in (1.3) is an asymptotic series and we expect that F o receives non-perturbative correctionsin g s . Such corrections are physically interpreted as the effect of the so-called ZZ-branes[34]. It would be interesting to find the general structure of the effect of ZZ-branes forthe arbitrary background { t n } . It is known that [35] some of the background { t n } exhibitsa non-perturbative instability and it does not lead to a well-defined theory. It wouldbe interesting to find the map of the “swampland” in the space of all two-dimensionaltopological gravities { t n } . In particular, it is argued that the JT gravity matrix modelsuffers from such a non-perturbative instability [11]. It is important to see if JT gravity isnon-perturbatively well-defined or not. Acknowledgments
This work was supported in part by JSPS KAKENHI Grant Nos. 19K03845 and 19K03856,and JSPS Japan-Russia Research Cooperative Program.
A Derivation of open KdV equations
In this section we will derive the open KdV equations (2.28) from the KdV flow equations(2.35).Let n be a positive integer. Since Q n + = Q · Q n − = Q (cid:18) Q n − + + Q n − − (cid:19) , (A.1)we have Q n + + = (cid:18) Q · Q n − + (cid:19) + + (cid:18) Q · Q n − − (cid:19) + = Q · Q n − + + (cid:18) Q · Q n − − (cid:19) + . (A.2)Therefore (2.35) is rewritten as2 n + 12 ∂ n (cid:98) ψ = Q (2 Q ) n − + (2 n − g s (cid:98) ψ + Q (2 Q ) n − − (2 n − g s + (cid:98) ψ. (A.3)Using (2.35) again one finds that the first term on the r.h.s. of (A.3) gives Q∂ n − (cid:98) ψ . Onthe other hand, it is known that (see e.g. [28]) Q n − − has the structure Q n − − = (2 n − n + g s {R n , ∂ − } + O ( ∂ − ) , (A.4)– 15 –rom which (2.13) follows. By using this, the second term on the r.h.s. of (A.3) becomes Q (2 Q ) n − − (2 n − g s + (cid:98) ψ = 14 (cid:0) ∂ {R n , ∂ − } (cid:1) + (cid:98) ψ = 14 (cid:0) ( ∂ R n ) ∂ − + ( ∂ R n ) + 2 ∂ R n (cid:1) + (cid:98) ψ = 14 (cid:16) ( ∂ R n ) (cid:98) ψ + 2 ∂ ( R n (cid:98) ψ ) (cid:17) = 34 ( ∂ R n ) (cid:98) ψ + 12 R n ∂ (cid:98) ψ. (A.5)Hence (A.3) becomes2 n + 12 ∂ n (cid:98) ψ = Q∂ n − (cid:98) ψ + 34 ( ∂ R n ) (cid:98) ψ + 12 R n ∂ (cid:98) ψ = (cid:18) g ∂ + u (cid:19) ∂ n − (cid:98) ψ + 34 ( ∂ n − u ) (cid:98) ψ + g ∂ ∂ n − F c ) ∂ (cid:98) ψ = ∂ n − (cid:18) g ∂ + u (cid:19) (cid:98) ψ −
14 ( ∂ n − u ) (cid:98) ψ + g ∂ ∂ n − F c ) ∂ (cid:98) ψ = ∂ n − Q (cid:98) ψ + g ∂ ∂ n − F c ) ∂ (cid:98) ψ −
14 ( ∂ n − u ) (cid:98) ψ. (A.6)In the second equality we have used (2.5) and (2.9). Substituting (2.32) we have2 n + 12 ∂ n (cid:98) ψ = g s ∂ s ∂ n − (cid:98) ψ + g ∂ ∂ n − F c ) ∂ (cid:98) ψ − g ∂ ∂ n − F c ) (cid:98) ψ. (A.7)Under the identification (2.33) one sees that this is equivalent to the open KdV equations(2.28). B Derivation of z ∗ ( s ) In this section we will derive (3.9) from (3.7).Suppose that w is expressed as a function of z given by the formal power series w = f ( z ) = ∞ (cid:88) n =1 f n z n n ! (B.1)with f (cid:54) = 0. According to the Lagrange inversion theorem, the inverse function is given by z = g ( w ) = ∞ (cid:88) n =1 g n w n n ! (B.2)with g = 1 f , g n = 1 f n n − (cid:88) k =1 ( − k ( n + k − (cid:88) { j (cid:96) } n − k (cid:89) (cid:96) =1 j (cid:96) (cid:18) f (cid:96) +1 ( (cid:96) + 1)! f (cid:19) j (cid:96) ( n ≥ , (B.3)– 16 –here the second sum is taken over all sequences j , j , . . . , j n − k of non-negative integerssuch that j + j + · · · + j n − k = k,j + 2 j + · · · + ( n − k ) j n − k = n − . (B.4)In the present case we have w = s, z = z ∗ ,f = t, f n n ! = − n !! I n +12 n = 3 , , , . . . , n = 2 , , , . . . . (B.5)Since f (cid:96) +1 with odd (cid:96) are absent, the conditions (B.4) reduce to j + j + · · · + j (cid:98) n − k (cid:99) = k, j + 4 j + · · · + 2 (cid:98) n − k (cid:99) j (cid:98) n − k (cid:99) = n − . (B.6)It is clear that the second condition is satisfied only if n is odd. This means that all g n with even n vanish. For odd n ( ≥
3) we have g n = 1 t n n − (cid:88) k =1 ( − k ( n + k − (cid:88) { j a } (cid:98) n − k (cid:99) (cid:89) a =1 j a ! (cid:18) − I a +1 (2 a + 1)!! t (cid:19) j a = n − (cid:88) k =1 ( n + k − t n + k (cid:88) { j a } (cid:98) n − k (cid:99) (cid:89) a =1 j a ! (cid:18) I a +1 (2 a + 1)!! (cid:19) j a . (B.7)Therefore z ∗ = st + ∞ (cid:88) m =1 s m +1 (2 m + 1)! g m +1 = st + ∞ (cid:88) m =1 s m +1 (2 m + 1)! m (cid:88) k =1 (2 m + k )! t m + k +1 (cid:88) { j a } (cid:98) m − k +12 (cid:99) (cid:89) a =1 j a ! (cid:18) I a +1 (2 a + 1)!! (cid:19) j a = st + (cid:88) j a ≥ (cid:80) a j a = k ≥ (cid:80) a aj a = m (2 m + k )!(2 m + 1)! s m +1 t m + k +1 ∞ (cid:89) a =1 j a ! (cid:18) I a +1 (2 a + 1)!! (cid:19) j a = (cid:88) j a ≥ (cid:80) a j a = k (cid:80) a aj a = m (2 m + k )!(2 m + 1)! s m +1 t m + k +1 ∞ (cid:89) a =1 j a ! (cid:18) I a +1 (2 a + 1)!! (cid:19) j a . (B.8)By rewriting j a as j a this gives (3.9). – 17 – eferences [1] D. J. Gross and A. A. Migdal, “Nonperturbative Two-Dimensional Quantum Gravity,”Phys. Rev. Lett. (1990) 127.[2] D. J. Gross and A. A. Migdal, “A Nonperturbative Treatment of Two-dimensional QuantumGravity,” Nucl. Phys. B (1990) 333–365.[3] M. R. Douglas and S. H. Shenker, “Strings in Less Than One-Dimension,” Nucl. Phys. B (1990) 635.[4] E. Brezin and V. Kazakov, “Exactly Solvable Field Theories of Closed Strings,” Phys. Lett.B (1990) 144–150.[5] P. H. Ginsparg and G. W. Moore, “Lectures on 2-D gravity and 2-D string theory,” inProceedings, Theoretical Advanced Study Institute (TASI 92): From Black Holes and Stringsto Particles: Boulder, USA, June 1-26, 1992. arXiv:hep-th/9304011 [hep-th] .[6] E. Witten, “Two-dimensional gravity and intersection theory on moduli space,” SurveysDiff. Geom. (1991) 243–310.[7] M. Kontsevich, “Intersection theory on the moduli space of curves and the matrix Airyfunction,” Commun. Math. Phys. (1992) 1–23.[8] M. R. Douglas, “Strings in Less Than One-dimension and the Generalized KdV Hierarchies,”Phys. Lett. B (1990) 176.[9] M. Fukuma, H. Kawai, and R. Nakayama, “Continuum Schwinger-dyson Equations andUniversal Structures in Two-dimensional Quantum Gravity,” Int. J. Mod. Phys. A6 (1991)1385–1406.[10] R. Dijkgraaf, H. L. Verlinde, and E. P. Verlinde, “Loop equations and Virasoro constraints innonperturbative 2-D quantum gravity,” Nucl. Phys. B348 (1991) 435–456.[11] P. Saad, S. H. Shenker, and D. Stanford, “JT gravity as a matrix integral,” arXiv:1903.11115 [hep-th] .[12] M. Mulase and B. Safnuk, “Mirzakhani’s recursion relations, Virasoro constraints and theKdV hierarchy,” arXiv:math/0601194 [math] .[13] R. Dijkgraaf and E. Witten, “Developments in Topological Gravity,” Int. J. Mod. Phys.
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