Normalization of ZZ instanton amplitudes in minimal string theory

We use insights from string field theory to analyze and cure the divergences in the cylinder diagram in minimal string theory with both boundaries lying on a ZZ brane. We focus on theories with worldsheet matter consisting of the $(2,p)$ minimal model plus Liouville theory, with total central charge 26, together with the usual $bc$-ghosts. The string field theory procedure gives a finite, purely imaginary normalization constant for non-perturbative effects in minimal string theory, or doubly non-perturbative effects in JT gravity. We find precise agreement with the prediction from the dual double-scaled one-matrix integral. We also make a few remarks about the extension of this result to the more general $(p',p)$ minimal string.

The quantity Z admits an expansion of the form Z = Z (0) + Z (1) + . . . = Z (0) 1 + N e −T + . . . . (1.1) Here Z (0) is the perturbative contribution to the matrix integral and Z (1) is the contribution to the matrix integral when one eigenvalue is in the classically forbidden region. One can also write (1.1) in terms of the free energy as log Z = log Z (0) + N e −T + . . ..
The object of interest to us in this paper is the normalization constant N. Roughly speaking, in string theory, N is the exponential of the worldsheet annulus with ZZ boundary conditions on both ends. This annulus amplitude has been computed using the worldsheet theory [7,8] and is divergent.
This state of affairs is very reminiscent of the recent computations in the c = 1 system, where the annulus amplitude between ZZ branes is also divergent, while the matrix side of the duality provides a finite unambiguous answer [15]. It has been shown by one of us [16] that string field theory techniques allow us to compute N in this case and the result matches with the matrix computation.
The purpose of this note is to apply these string field theory tools to the (2, p) minimal string theories and compute the value of N in these theories. We find perfect agreement with the matrix integral computations [2,[9][10][11][12][13]. We record the final result cot(π/p) Let us make a few comments about the form of this answer. First, the combination N T 1 2 is natural to consider since the dependence on g s cancels out in this combination. This is important since it is impossible to fix the multiplicative constant between the genus counting parameters on the two sides of the duality, since we can always add the Euler characteristic term to the worldsheet action with an arbitrary coefficient. So, when trying to match precise numerical constants, one should compute quantities that are independent of g s , like N T 1 2 rather than N or T separately. 1 On the matrix integral side, the gaussian integral around the one-eigenvalue instanton gives a multiplicative factor in N that is proportional to T − 1 2 . On the string theory side, this factor arises because the proper volume of the rigid U (1) gauge group on the instanton is proportional to T 1 2 [16]. Division by this gauge group volume in the path integral produces the factor of T − 1 2 . Second, the overall sign of the right hand side of (1.2) is ambiguous on both sides of the duality, as it depends on a two-fold choice of the contour of integration over one unstable mode. One should make this choice so that the result is the same for the matrix integral and the string theory. Third, the normalization constant N is purely imaginary and the instanton correction we are studying computes the leading imaginary part of the free energy. In this sense, this correction is similar to the case of "bounce" solutions in instanton physics [20] and the instanton correction is meaningful. Finally, note that the coefficient on the right hand side of (1.2) is finite in the JT gravity limit p → ∞.
The organization of this paper is as follows. In section 2, we present the computation of N in the double-scaled one-matrix integral, which is dual to the (2, p) minimal string. The results of this section are not new, and we are including them to illustrate the relevant tools in the simpler setting of the one-matrix integral. In section 3, we first present a general string field theory analysis of the divergences in the cylinder diagram with both boundaries lying on a D-instanton. We then apply these tools to the (2, p) minimal string and obtain a finite answer that agrees with the matrix integral result.
In section 4, we make a few remarks about the extension of these results to the more general (p , p) minimal string.

The matrix computation
In this section we will compute the normalization constant N for the one-matrix integrals that are dual to the (2, p) minimal string. The results in this section are not new and can be found in many papers, including [2,[9][10][11][12][13][14]21]. We choose to follow the streamlined presentation given in the recent work [2].
We start by explaining the setup. The starting point is an integral over all L × L hermitian matrices Z = dH e −L Tr V (H) . (2.1) Here V is a potential which can be taken to be an even polynomial of degree p + 1. The matrix integral Z is a function of the coefficients in this polynomial. In the large L limit, we can talk about a smooth density of eigenvalues and it is supported on a finite interval on the real axis. The double-scaling limit refers to a procedure where, in addition to taking L → ∞, we zoom in near the left edge of the spectrum and tune the coefficients of the potential such that the dominant double-line Feynman diagrams in the perturbation expansion of (2.1) resemble continuum surfaces [1] . In this limit, the density of states is non-normalizable and is supported on the entire positive real axis.
We focus on the so-called "conformal background" [22], where the leading density of states in the double-scaling limit reads 2 Here Θ(E) denotes the Heaviside theta function. This is the density of states that is dual to standard Liouville theory with only the cosmological constant term in the action turned on. See, for example, [17] for an explicit family of potentials that lead to the density of states (2.2) in the double scaling limit. 2 To get to the density of states in JT gravity, we need to take κ ∼ p 2 as p → ∞ [2].
Here, e S 0 is the genus counting parameter after taking the double-scaling limit and κ is an arbitrary energy scale.
Using the relationship between the form of the density of states and the spectral curve, we conclude that the spectral curve is given by [8,23] y(z) = sin p arcsin where T p denotes the p-th Chebyshev-T polynomial. One way to see this is to note that the leading density of states ρ(E) (0) is determined from the spectral curve as for E > 0. It is also a standard result in one-matrix integrals that the derivative V eff (E) of the one-eigenvalue effective potential V eff (E), that includes contributions from both the potential V that appears in (2.1) and the Vandermonde determinant, is proportional to y( √ −E) in the forbidden region E < 0 (see, for example, [2] for a recent exposition). The precise relationship is Integrating this using (2.3) and taking V eff (E = 0) = 0 we get, for E < 0 that Let us now look at the extrema of the one-eigenvalue effective action. From (2.4) and (2.3), we see that as we move towards negative energies starting at E = 0, the first zero of V eff (E) occurs at E = −2κ sin 2 π p . (2.7) We record the values of V eff (E ) and V eff (E ), which are obtained from (2.5) and (2.4) using (2.3): Now we organize various contributions to the integral (2.1) depending on how many eigenvalues are in the classically allowed region E > 0 and how many are in the classically forbidden region E < 0. The leading contribution Z (0) comes from the integration region where all eigenvalues are in the classically allowed region. The next important contribution Z (1) comes from the integration region when only one eigenvalue is in the forbidden region. Next, we borrow a couple of results from [2,10,11,13], which in the notations of [2] are as follows: with κ = 1 2 but it is qualitatively similar for all p. In string theory, the integration contour for the open string tachyon also looks like this.
Here the subscript F on the integral denotes integration over the classically forbidden region E < 0.
The formula (2.11) captures the small amount of quantum mechanical leakage of eigenvalues into the classically forbidden region. 3 We now plug in (2.11) into (2.10) and use the saddle point approximation about E to compute the integral (along a contour to be specified momentarily): It is important to note from (2.9) that V eff (E ) < 0 and thus the steepest descent contour is parallel to the imaginary-E axis. Furthermore, we only integrate over half of the steepest descent contour, since, in the perturbative region E κ, the defining contour must lie along the real axis [2]. Figure 1 shows this contour. On the string theory side, this "unstable mode" is the open string tachyon and one has a similar contour of integration over the tachyon mode [16]. 4 These facts give us the factor of i/2 in the gaussian integral. 5 Comparing (2.13) to (1.1) and using equations (2.7), (2.8) and (2.9), we get As explained in the introduction, it is natural to factor out T − 1 2 from the expression for N, and so we write the above result as Refs. [9][10][11][12] contain this result for p = 3, while the result for general p can be found in [13]. 6 Ref. [2] was interested in the limit p → ∞.
We would like to explain one subtlety in the above analysis. One can explicitly check that the effective potential given in equation (2.6) has (p − 1)/2 extrema on the negative-E axis. Roughly half of them are maxima and half are minima. The extremum at E in (2.7) is the one closest to the origin and is a local maximum. However, even among the local maxima, this is not the one with the smallest value of the effective potential, in general. This raises the question of why we have chosen the saddle point E in (2.7) as the relevant saddle. The point is that we want the perturbation series of the matrix integral to match with the vacuum string perturbation theory, and so we should not allow the integration contour for the matrix eigenvalues to pass through regions on the real axis with V eff < 0, since these regions will give real contributions to the matrix integral that are much larger than the terms in perturbation theory around the saddle point (2.2). This can be avoided by turning the integration contour along the steepest descent contour once it reaches E * .

The string theory computation
In this section we shall describe the string theory computation of the leading imaginary part of the partition function, arising from a single ZZ-instanton contribution.
The string theory that is dual to the double-scaled one-matrix integral described in section 2 is Liouville theory coupled to the (2, p) minimal model and the bc-ghost system. The b parameter that appears in the Liouville lagrangian is determined by p and is such that the total central charge of Liouville, the matter CFT and ghosts adds up to zero. One finds b = 2/p. 5 Since we are only interested in computing the imaginary part, we don't need to worry about the part of the contour along the real axis, which contributes something real. 6 Note that some of these references are computing an integral over the full steepest contour through the saddle point, and others are including contributions from both ends of the eigenvalue cut, and thus the pre-factors quoted there are a multiple of the value in (2.16).

The cylinder diagram and its divergences
We shall begin by describing some general issues that arise in the analysis of the cylinder diagram with boundaries lying on a D-instanton (whose analog in non-critical string theory is the ZZ instanton). We can express the cylinder partition function in the open string channel as: where F (t) has the structure In theories of interest to us in this paper, the integral (3.1) has no divergence in the t → 0 limit, indicating that the (regulated) number of fermionic and bosonic states are equal. In the hypothetical situation where h b andĥ f are all positive, there are no divergences in the t → ∞ limit either, and A is given by For positive h b ,ĥ f this can be used to express the normalization factor N accompanying the instanton amplitude as an integral, to make sense of this using insights from string field theory. We shall now describe this procedure.
First we note that, for h b , h f > 0, we can pick any non-negative integer n and write hybrid expressions for A and N as Now, when some of the h b 's orĥ f 's are negative or zero, we shall choose n to be such that for b, f > 2n all the h b 's andĥ f 's are positive. Then the term in the first line of (3.7) is finite since we have subtracted the 'bad' contributions involving h b ,ĥ f ≤ 0 terms from F (t). Furthermore, since the subtraction term vanishes as t → 0, the integral is free of divergences from the t → 0 end as well. Thus, we are left with the goal of making sense of the integral over the modes φ b for b ≤ 2n and p f , q f for f ≤ n.
For the D-instantons that we shall discuss, the bad modes consist of one bosonic mode -the tachyon mode φ 1 corresponding to the state c 1 |0 with h b = −1, and a pair of fermionic modes p 1 , q 1 corresponding to the states i|0 and ic 1 c −1 |0 withĥ f = 0. The coefficients i in these states have been chosen to ensure that the modes multiplying these states are real. Since there is only one bad bosonic mode and two bad fermionic modes, we can choose n = 1 in (3.7). See Table 1 for a list of the states that are relevant for the discussion and their basic properties.
First we shall discuss the integration over the bosonic modes φ 1 and φ 2 . Since h 2 > 0, the integration over φ 2 gives a standard gaussian integral. The integration over φ 1 is problematic since the exponent takes the form exp(φ 2 1 /2). We shall carry out this integral by regarding this as a contour integral in the complex φ 1 plane as follows [16]. Since the vacuum without any D-instanton is represented by a State L 0 eigenvalue Ghost number In Siegel gauge? Field name Grassmann parity of field  Choosing the contour to be from −i∞ to 0 for definiteness, we can write (the leading imaginary part of) the bosonic part of the integral as: (3.8) Next we turn to integration over the fermion zero modes p 1 , q 1 . We can get physical insight into the origin of these modes if, instead of a D-instanton, we consider a Dp-brane extending along some directions in space-time in any bosonic string theory. In that case the gauge field a µ (k) living on the brane appears in the expansion of the open string field as a term proportional to d p+1 k a µ (k)α µ −1 c 1 |k where α µ n are the oscillators associated with the scalars X µ describing coordinates tangential to the brane and |k = e ik·X(0) |0 are momentum carrying states. In string field theory, gauge transformations appear as BRST exact states Q B |Λ (plus higher order terms), and |Λ is referred to as the "gauge transformation parameter". For instance, usual spacetime gauge transformations of the gauge field δa µ (k) ∝ ik µ θ(k) appear via the term i d p+1 k θ(k)|k in |Λ . Note that this term in |Λ has ghost number zero. Then the linearized gauge transformation Q B |Λ produces a term proportional to i d p+1 k θ(k)k µ α µ −1 c 1 |k . Comparing to the state representing the gauge field, we see that this generates the usual gauge transformation law δa µ (k) ∝ i k µ θ(k).
The gauge transformation Q B |Λ also produces a state proportional to i d p+1 k θ(k)k 2 c 0 |k . This translates to a transformation δψ(k) ∝ k 2 θ(k) where ψ(k) is the field multiplying the state ic 0 |k . The Siegel gauge choice corresponds to setting ψ(k) = 0. This produces a Jacobian proportional to k 2 , which is represented by a pair of Fadeev-Popov ghosts p 1 (k), q 1 (k) multiplying the states ic 1 c −1 |k and i|k . Since these states have conformal weightĥ 1 = k 2 , integration over p 1 and q 1 will precisely produce the required Fadeev-Popov determinant k 2 , for k = 0. Now, the issue is that, on a D-instanton we have k = 0. Thus, neither the "gauge field" nor the field ψ multiplying ic 0 |0 transforms, showing that the Siegel gauge choice breaks down. This is a reflection of the fact that the usual local U(1) symmetry on the Dp-brane becomes a rigid symmetry on the D-instanton. The remedy is to go back to the "original" form of the path integral where we carry out integration over all the "classical" modes of the theory and explicitly divide by the volume of the gauge group. In string field theory language, fields that multiply states of ghost number one are referred to as classical since the physical open string states belong to this sector. Concretely, among the states in table 1, this means that instead of integrating over {φ 1 , p 1 , q 1 }, we integrate over {φ 1 , ψ} and divide by the volume of gauge group. The precise normalization of the integration measure can be fixed by carefully following the line of argument described above and gives the replacement rule [16]: The since θ has period 2π.
Substituting (3.8), (3.9) and (3.10) into (3.7) we get: where h = h 2 . One can easily check that the expression is independent of h by taking derivative with respect to h. Therefore we do not need to choose h = h 2 , any choice of h > 0 will give the same result.

Specialization to minimal string theory
We shall now use (3.11) to compute the normalization of the instanton amplitude in the (2, p) minimal string theory. The form of the integrand F (t) for the cylinder diagram in minimal string theory is well-known [7]. Since we are studying the cylinder diagram, we need to specify boundary conditions for the worldsheet fields. For the Liouville CFT, we pick the "(m, n) = (1, 1)" ZZ boundary condition [5], as this is the one that corresponds to the saddle point E in equation (2.7) in the matrix integral [8,18].
The partition function of the matter CFT with the given boundary conditions equals the identity character in the minimal models, which is given by [28,29] . (3.13) Multiplying the contribution η(it) 2 from the ghosts (see, for example, [24]), we find (3.14) It is important to note that the leading terms in F (t) as t → ∞ are the ones with k = 0: As already discussed in section 3.1, the e 2πt term arises from the open string tachyon, while the −2 arises from the two ghost zero modes. 11 10 For the (2, p) minimal string, there are (p − 1)/2 possible ZZ brane boundary conditions [23]. By comparing the relative tensions of these branes (given in, for example, [8]), to the relative heights of the extrema of the matrix effective potential (2.6), one can establish that it is the (m, n) = (1, 1) ZZ brane, with identity character from the matter CFT, that corresponds to the matrix saddle point at E in (2.7) with V eff (E ) as in (2.14). 11 In the c = 1 case, we have F (t) = e 2πt − 1 exactly. The change of coefficient in the L0 = 0 sector comes from an additional bosonic zero mode that corresponds to time translations of the D-instanton [15,16].
If we substitute (3.14) into (3.11) and choose h = 1, we can see that the k = 0 term in the sum exactly cancels the subtraction term e 2πt + e −2πt − 2 . The rest of the terms may be analyzed using the general result: Using this we can rewrite (3.11) as (3.17) We now use we pick the (m, n) = (1, 1) ZZ state for Liouville, and for the matter CFT, we pick the Cardy state on both ends so that the open string channel only contains the identity character [27]. This gives the partition functions [5,28,29] Combining the Liouville, matter and ghost contributions to F (t), using (3.11), and following the steps in section 3.2, we get (pk + 1)(p k + 1) − 1 k(pp k + p − p ) − 1 k(pp k + p − p ) (pk + 1)(p k + 1)   sin π p + π p sin π p − π p sin 2 (π/p) sin 2 (π/p ) (p 2 − p 2 ) This agrees with (3.21) when p = 2.
For p > p ≥ 3, these string theories are dual to the double-scaled limit of a two-matrix integral [30,31]. The two-matrix integral is more complicated, so we won't go into the full analysis of the eigenvalue instanton in this case [14,19], and just note that the result (4.3) agrees with the m = n = 1 expression given in [14]. 12 We leave a fuller investigation of the two-matrix case to future work. The BRST charge Q B is given by where T m is the matter stress tensor. There is also a cubic term in the action [32]. See for example, [25] for a detailed form of this coupling. If we normalize the string field so that the kinetic term is independent of the coupling as in (A.3), then the cubic term has an explicit factor of the open string coupling g o .
From the above equations, one can see, for example, that the contribution of the tachyon field φ 1 to the quadratic action is and thus the weight in the path integral is exp(φ 2 1 /2). The action for ψ is similarly seen to be ψ 2 .