D-instantons, String Field Theory and Two Dimensional String Theory

In arXiv:1907.07688 Balthazar, Rodriguez and Yin (BRY) computed the one instanton contribution to the two point scattering amplitude in two dimensional string theory to first subleading order in the string coupling. Their analysis left undetermined two constants due to divergences in the integration over world-sheet variables, but they were fixed by numerically comparing the result with that of the dual matrix model. If we consider n-point scattering amplitudes to the same order, there are actually four undetermined constants in the world-sheet approach. We show that using string field theory we can get finite unambiguous values of all of these constants, and we explicitly compute three of these four constants. Two of the three constants determined this way agree with the numerical result of BRY within the accuracy of numerical analysis, but the third constant seems to differ by 1/2. We also discuss a shortcut to determining the fourth constant if we assume the equality of the quantum corrected D-instanton action and the action of the matrix model instanton. This also agrees with the numerical result of BRY. Note added: The apparent discrepancy in the third constant has been resolved in arXiv:2210.11473.

1 Introduction and summary D-instantons represent saddle points of the path integral in second quantized string theory and give non-perturbative contribution to the string amplitudes. The usual world-sheet approach to computing these corrections suffers from infra-red divergences from the boundaries of the moduli spaces of Riemann surfaces. Furthermore, unlike the perturbative amplitudes in string theory, these infra-red divergences cannot be removed by analytic continuation in the external momenta, since the open strings living on the D-instanton carry strictly zero momentum. Even if the divergences cancel at the end [1][2][3], the fact that they are present in the intermediate stages of the analysis and need to be tamed by suitable regulator, makes the final result ambiguous.
A concrete example of such ambiguities arose in a recent analysis of two dimensional string theory [4]. The authors of [4] computed the D-instanton contribution to the closed string scattering amplitude, but could determine the final result only up to two undetermined constants. These constants were then determined by comparison with the results in the dual matrix model [5][6][7][8][9]. This comparison was carried out numerically, leading to the best fit values −1.399 and .496 for these constants. 1 String field theory is well poised to address the issues related to infra-red divergences arising in string theory [11][12][13]. It does so by drawing insights from quantum field theory. Indeed, in an earlier analysis [14], string field theory was used to determine the first of these constants unambiguously, leading to the value − ln 4, which is within 1% of the numerical result −1.399.
In this paper, we shall use the systematic procedure for carrying out perturbation theory in the presence of D-instantons, as developed in [15], to rederive this result in a more general set-up.
We shall also see that to the order to which the analysis of [4,14] was carried out, there are actually four independent constants that are apparently divergent / ambiguous in the worldsheet description, but can be fixed in string field theory. The second constant .496 that appeared in the analysis of [4] is a particular linear combination of these four constants. We determine three of these four constants explicitly using string field theory. Determination of the fourth constant is possible within string field theory, but involves computations more elaborate than what will be performed in this paper. All of these constants can also be determined independently by (numerically) comparing the string theory results with the known results for higher point amplitudes in the matrix model. This has now been done [16]. We find that two of the three constants, determined from string field theory, agree with the results of the matrix model. However, the third constant differs by an additive constant close to 1/2.
Since most of the paper will be technical in nature, we shall now give an overview of the main results. We begin by describing our convention. A naive amplitude with D-instanton boundary condition does not satisfy energy conservation, since time translation invariance is broken by the Dirichlet boundary condition on the time coordinate. As will be discussed later, energy conservation is eventually restored after integration over the collective modes of the Dinstanton, but it will be convenient for us to work with the amplitude before integration over the collective coordinates. Furthermore, we focus on amplitudes on connected world-sheets, since the contribution from world-sheets with more than one connected component can be found by taking the product of the contributions from the connected components. We shall refer to these as the building blocks from which the final amplitude will be constructed. Furthermore, every D-instanton amplitude is eventually multiplied by a factor of N e −1/gs where g s is the string coupling constant and N is a numerical constant. We shall drop this factor from the building blocks. The final amplitude is to be constructed after taking sums of products of appropriate building blocks and then including an overall factor of N e −1/gs and the energy conserving delta function. Finally, we shall use a convention in which all energies are counted as outgoing, i.e. an ingoing state of energy ω will be described as a state of energy −ω. In this convention the final energy conserving delta function that comes from integration over the collective coordinate takes the form 2πδ(ω tot ) where ω tot is the total energy carried by all the external closed string states in an amplitude.
Ref. [4] computed three building blocks in two dimensional string theory. One is the one point disk amplitude of a closed string tachyon of energy ω. This is given by: 2 sinh(π|ω|) . (1.1) The powers of g s in this amplitude are explained as follows. The disk gives a factor of 1/g s , and the closed string vertex operator is normalized so that it carries a factor of g s . Therefore the powers of g s cancel from this amplitude. The second building block computed in [4] is the two point function of a pair of closed string tachyons of energy ω 1 and ω 2 on the disk. This takes the form: where f (ω 1 , ω 2 ) = 2 −1/4 π 1/2 2 (ω 2 Here V |ω 1 |/2 (i)V |ω 2 |/2 (i y) UHP denotes the two point function of a pair of primaries in the c=25 Liouville theory of momenta |ω 1 |/2 and |ω 2 |/2, inserted at i and iy respectively in the upper half plane. The third building block computed in [4] is the one point function of a closed string tachyon of energy ω on the annulus. This takes the form: 2 g s sinh(π|ω|) g(ω) , (1.4) where g(ω) = 2 π 2 1 sinh(π|ω|) Here θ 1 (z|τ ) is the odd Jacobi theta function and θ 1 (z|τ ) ≡ ∂ z θ 1 (z|τ ). V |ω|/2 (2πx) A denotes the one point function of the Liouville primary of momentum |ω| on an annulus described by 0 ≤ Re(w) ≤ π, w ≡ w + 2π i t, with the vertex operator inserted at Re(w) = 2πx. Explicit expressions for the correlation functions in the Liouville theory can be found in [4].
There is one more building block that was not analyzed in [4], but will be needed for our discussion below. This is the zero point function on surfaces of Euler number −1, that includes a disk with two holes and a torus with one hole. Since there are no external momenta, we can parametrize it as C g s for some constant C.
Ref. [4] used these building blocks to construct a D-instanton induced two point closed string tachyon amplitude to order g s , but we shall present the result in a more general form where we use the same building blocks to construct a general (n + 1)-point amplitude of closed string tachyons to this order. The leading order contribution comes from the product of n + 1 disk one point functions. Our interest will be in the subleading order g s contribution which comes from three sources: 1) the product of (n − 1) disk one point functions and one disk two point function, 2) the product of n disk one point functions and one annulus one point function, and 3) the product of (n + 1) disk one point functions and one zero point function on a surface of Euler number −1. A fourth contribution, given by the product of a sphere three point function and (n − 2) disk one point functions, has a separate energy conserving delta-function from the sphere three point function and does not contribute for generic energies of incoming and outgoing particles satisfying overall energy conservation. After restoring the factors of N e −1/gs and the 2π δ( i ω i ), the D-instanton induced (n + 1)-point amplitude up to order g s takes the form: (1.13) Using (1.6), (1.7) and (1.12) we now get, g finite (ω i ) + 1 2 n(n + 1)A f + (n + 1) (1.14) where in the last term we have used the energy conservation law to replace i<j ω i ω j by − i ω 2 i /2. This shows that the collection of the amplitudes for different values of n depends on the coefficients A f , A g and C individually, but only on the combination 2B g − B f . The matrix model result for the scattering amplitude of a closed string tachyon of energy −ω n+1 into n closed string tachyons of energy ω 1 , · · · , ω n is known explicitly, and is given by [4,17] 2 n+1 N e −1/gs 2πδ(ω 1 + ω 2 + · · · ω n + ω n+1 ) n+1 i=1 sinh(π|ω i |) 15) with N = −1/(8π 2 ). Ref. [4] compared (1.14) and (1.15) for n = 1 numerically and found that the results match well if we choose: (1. 16) We emphasize however that we can numerically compare (1.14) and (1.15) for different values of n to extract the coefficients A f , A g , C and 2B g − B f separately. This has now been done, and the comparison yields [16]: with an expected error of order .01 in each term.
In this paper we use string field theory to show that, where λ is an arbitrary parameter that enters the construction of string field theory. Different values of λ give apparently different string field theories, but they can be shown to be related by field redefinition [18]. Comparison with (1.12) now gives: (1. 19) We see that B f and B g are individually ambiguous but the combination 2B g − B f that enters the physical amplitudes is unambiguous and is given by: (1.20) Even though we have not computed C explicitly, we have argued in §7 that if we make the additional assumption that one can equate the renormalized D-instanton action with the renormalized action of the matrix model instanton, we get The values of A f , 2B g −B f and C given in (1.19), (1.20) and (1.21) agree with the numerical values (1.17) determined by comparison with the matrix model within an error of about .01.
However A g given in (1.19) seems to differ from the numerical value of A g given in (1.17) by an additive constant −1/2. While this discrepancy could be removed by rescaling the closed string tachyon field by (1 + 1 2 g s ), this option is not available since it would generate corrections of order g s in the perturbative amplitudes that are not present in the theory. The origin of this discrepancy is not clear to us. The most likely explanation is some missing contribution in the string field theory analysis, but we have not been able to identify such a contribution. Furthermore, as discussed in §8, our results pass various consistency tests. We discuss various other possible origins of this discrepancy in §8.
We shall now briefly mention the main subtleties that arise in our analysis.
1. The divergent parts of the amplitude are handled by interpreting these divergences in string field theory and following the standard procedure of quantum field theories to remove these divergences. This in particular requires us to not use the Schwinger parameter representation for the tachyon propagator that is commonly used in arriving at the world-sheet description of the amplitude. Instead we use the string field theory result for the tachyon propagator given by the inverse of the mass 2 of the tachyon. This removes all power law divergences.
2. We also need to remove the zero mode fields propagating in the internal lines, and treat them as we would do in a quantum field theory. Once the contribution due to the zero mode fields is removed, all logarithmic divergences also disappear. The effect of integration over the zero mode fields will be discussed separately below.
3. In the D-instanton background the Siegel gauge, that is normally used in string field theory, breaks down and we need a different gauge fixing [15]. In this new gauge there is an additional field whose contribution to the amplitude is not captured in the usual world-sheet formulation. We need to explicitly include the contribution from this field propagating in the internal propagators in all the amplitudes. 4. The open string field theory has a zero mode field that represents the freedom of translating the D-instanton along the (euclidean) time direction. This can be related to the collective coordinate of the D-instanton by a field redefinition, after which the integration over the collective coordinate produces the usual energy conserving delta function. However, the original string field theory integration measure produces an additional Jacobian factor during this change of variable. This gives additional contributions to the amplitudes that need to be accounted for. 5. In the Siegel gauge the ghost sector contains a pair of zero mode fields, one of which is removed in the new gauge that we use. However the other one remains, and can be interpreted as the ghost field associated with a particular gauge symmetry of string field theory. Integration over this ghost field can be interpreted as division by the volume of the corresponding gauge group. Now this particular gauge symmetry is related, by a field dependent multiplicative factor, to the rigid U (1) gauge symmetry under which an open string stretched from the D-instanton under consideration to any other D-brane picks up a phase. Since the rigid U (1) gauge group has a constant volume, the effect of division by this volume can be absorbed into a redefinition of the overall normalization constant N . However the field dependent multiplicative factor that takes us from the string field theory gauge transformation parameter to the rigid U (1) gauge transformation parameter gives rise to an additional Jacobian factor in the path integral measure. This also needs to be taken into account while computing amplitudes.
String field theory that we use has two ad hoc parameters λ and λ and a function f (β).
The contribution to the coefficients (1.19) from the different effects mentioned above -the two jacobians, the effect of integration over the out of Siegel gauge modes, and the original contribution after removal of the zero mode contribution and the string field theory treatment of the tachyonic modes, all depend on these ad hoc parameters and the function f (β) in a nontrivial way. An illustration of this can be found in (6.55), which lists contribution to g div (ω) from various sources. However after summing over all the contributions, the dependence on these parameters and the function f (β) disappears and we get the unambiguous results for the coefficients A f , A g and 2B g − B f as given in (1.19). This is consistent with the fact that different string field theories corresponding to different values of these ad hoc parameters and the function f (β) are related by field redefinition [18].
The rest of the paper is organized as follows. In §2 we review some aspects of string field theory and the main results of [15] that we shall be using to tame the infrared divergences in the computation of string amplitudes in a D-instanton background. One of the ingredients in this computation is the determination of the relationship between the standard coordinates of the moduli space of Riemann surfaces and the coordinates that arise from the string field theory description of the amplitudes. However it is often sufficient to determine approximate relations between these coordinates instead of the exact relations. In §3 we estimate the errors that we make in using these approximations, and determine how much error we are allowed to make so as not to affect the final results. In §4 we introduce the interaction vertices of string field theory by choosing appropriate local coordinates at the punctures of various Riemann surfaces. The latter are needed for describing off-shell amplitudes. Using these data, we find the (approximate) relations between the coordinates of the moduli space of Riemann surfaces and the parameters arising from string field theory description for various amplitudes. In §5 we use these results to compute the D-instanton contribution to the two point disk amplitude of closed string tachyons in the two dimensional string theory, avoiding the infrared divergences by using the procedure described in §2. This determines the function f div (ω 1 , ω 2 ) leading to the result given in the second line of (1.18). In §6 we use the results of §4 to compute the D-instanton contribution to the one point function of closed string tachyon on the annulus. This determines g div (ω), given in the first line of (1.18). In §7 we discuss the steps needed to compute the two loop correction C to the D-instanton action. However we stop short of actually doing this computation, which involves an open string two loop amplitude and a closed string one loop amplitude. Nevertheless we give an indirect determination of this constant C by assuming the equality of the quantum corrected actions of the D-instanton and the matrix model instanton. We end in §8 by listing some open questions. The appendices describe the details of some of the computations whose results were used in the text.

Background
In this section we shall give a brief introduction to some aspects of string field theory [19][20][21] that we shall use, and also review the algorithm developed in [15] for carrying out systematic computation of closed string amplitudes in the presence of D-instantons. We shall focus on the contribution associated with a single D-instanton. Furthermore, we shall assume that the background does not have any other D-branes, so that all the boundaries of the world-sheet have D-instanton boundary conditions.

String field theory
The usual world-sheet description of string amplitudes involves integration over moduli spaces of Riemann surfaces with punctures, with the external closed string vertex operators inserted at the punctures in the bulk of the world-sheet and the external open string vertex operators inserted at the punctures on the boundary of the world-sheet. The integrand is computed from appropriate correlation functions of the vertex operators and world-sheet ghost fields on the Riemann surface. In the two dimensional string theory that we shall consider, the world-sheet theory consists of a free scalar field X describing the time coordinate, a Liouville field theory with central charge 25, and the usual b, c ghost system.
At a superficial level, string field theory may be regarded as a regular quantum field theory, whose perturbation expansion generates the same result at the one in the world-sheet approach.
Each Feynman diagram of string field theory generates an integral similar to what we would get in the world-sheet approach, but the integration over the moduli runs over a subspace of the full moduli space. The sum over all Feynman diagrams gives back integration over the full moduli space. We shall see however, that this correspondence is only formal, -often the world-sheet expression is divergent while the corresponding string field theory expression yields finite result. then {χ r } are the dynamical degrees of freedom on which the path integral is to be performed (after suitable gauge fixing). For string field theory on a Dp-brane the sum over r will also include integration over (p + 1) continuous variables labelling the energy and momenta of the state, but on the D-instanton the open strings do not carry any continuous momenta and the sum over r is a discrete sum. Therefore we can regard this as a field theory in zero space-time dimensions. The grassmann parity of χ r is opposite to that of the state |φ r . Therefore the string field Ψ is grassmann odd.
The string field theory action takes the form: where Q B is the world-sheet BRST operator and · · · denotes interaction terms. Note that we have not explicitly written down the kinetic term for the closed string fields since they will be treated as background fields and the kinetic term will not be needed for our analysis. The interaction term takes the form m≥0 n≥0 where the functions F (m,n) r 1 ···rn (ω 1 , P 1 , · · · , ω m , P m ) are constructed from appropriate correlation functions of m closed string tachyon vertex operators with energies ω 1 , · · · , ω m , n open string vertex operators φ r 1 , · · · , φ rn and appropriate set of b-ghost insertions, integrated over some appropriate subspaces of moduli space of Riemann surfaces with punctures. In general the functions F (m,n) r 1 ···rn depend on the choice of local coordinate systems at the punctures, but since we shall need these functions only for on-shell closed string states, the dependence on the local coordinates at the punctures in the bulk of the world-sheet, where closed string vertex operators are inserted, drops out. For on-shell open string states the correlation function is also independent of the choice of local coordinate system at the punctures on the boundaries of the world-sheet, but this is not the case for off-shell open string states. Different choices of local coordinates lead to apparently different string field theories, but they can be shown to be related to each other by field redefinition [18].
The action (2.2) is to be regarded as the quantum master action in the Batalin-Vilkovisky formalism [22,23] and expand the string field as, |Ψ = r (ψ r |ϕ r + ψ r |ϕ r ) . (2.5) Then we declare ψ r as a field and ψ r as the conjugate anti-field (up to a sign). String field theory constructed this way has gauge symmetries where the gauge transformation parameters are in also in one to one correspondence with states of the open string.
Therefore we can represent a gauge transformation parameter as: with θ r 's being the parameters of infinitesimal gauge transformation. Even though θ r and χ r appear in similar expansions (2.1) and (2.6), their grassmann parities are opposite of each other.
The significance of this will become clear later The gauge transformation of a field χ r under a gauge transformation parameter θ s is given (up to a sign) by the double derivative of the action with respect to χ * r and χ s , where χ * r is the anti-field conjugate to χ r , computed via (2.5) [24]. More general gauge transformations where the transformation parameters themselves are functions of fields have been discussed in [25], but we shall not discuss them here.
The actual path integral of string field theory is carried out over fields in a Lagrangian submanifold -this is the analog of gauge fixing in the Fadeev-Popov formalism. We shall not give a general definition of Lagrangian submanifolds, but it suffices to say that one set of choices correspond to setting, for each pair (ψ r , ψ r ), either ψ r to 0 or ψ r to 0. This can be done independently for each r. The result of the path integral for physical quantities can be shown to be formally independent of the choice of the Lagrangian submanifold.
A particular choice of Lagrangian submanifold corresponds to setting all the anti-fields to 0. Therefore |Ψ has ghost number ≤ 1. It then follows from ghost number conservation that the action depends on only the ghost number 1 sector fields. These are called classical fields since the physical open string states of the string belong to this sector. In this case the path integral decomposes into an integration over the classical fields on which the action depends and an integration over the fields of ghost number ≤ 0 on which the action does not depend.
Since the string fields χ r have opposite grassmann parity compared to the corresponding gauge transformation parameters θ r , the integration over the string fields of ghost number ≤ 0 can be interpreted as division by the volume of the gauge group generated by gauge transformation parameter Λ carrying ghost number ≤ 0. This connects the BV formalism to the usual path integral formulation of gauge theories, in which we start with the path integral over the classical fields weighted by the exponential of the classical action and divide this by the volume of the gauge group. While this is a sensible starting point when the gauge group has a finite volume, as in lattice gauge theories on a finite lattice and with a compact gauge group, this is usually not suitable for perturbation theory, since the kinetic operator of the classical action has flat directions associated with the gauge deformations. This makes the kinetic operator noninvertible, requiring us to gauge fix the theory. In the BV formalism this will correspond to choosing a different Lagrangian submanifold.
The Lagrangian submanifold commonly used for carrying out perturbation theory is called the Siegel gauge. Here we restrict the path integral to a subspace of string fields of the form: More precisely this means the following. We can choose the basis states |ϕ r and |ϕ r in (2.5) in such a way that one of the following holds: 1. b 0 |ϕ r = 0, c 0 |ϕ r = 0. In this case we choose ψ r = 0.
Since for each r we set to zero either the field or the anti-field, this is clearly a Lagrangian submanifold. The corresponding |Ψ given in (2.5) satisfies (2.7).
In the Siegel gauge, the kinetic operator Q B appearing in (2.2) reduces to c 0 L 0 and the open string propagator is proportional to where L n 's denote the total Virasoro generators of the world-sheet conformal field theory of matter and ghosts. As we shall discuss later, equality of the two sides of (2.8) is somewhat formal and breaks down for states with L 0 ≤ 0, but let us proceed with this for now. In this case there is a simple algorithm to determine which region of the moduli space is covered by a given Feynman diagram. The local coordinate system at the punctures, needed for defining the interaction term of string field theory for off-shell external states, play an important role in this analysis. If we contract the external state χ r from one interaction vertex with the external state χ s from another (or the same) interaction vertex 3 by a propagator, then geometrically this corresponds to sewing the Riemann surfaces associated with the two interaction vertices by identifying the corresponding local coordinates w and w via the relation For this reason, we shall call q the sewing parameter associated with the open string propagator. Following this procedure we can determine the regions in the moduli space of punctured Riemann surfaces covered by each Feynman diagram. This not only generates the family of Riemann surfaces corresponding to the Feynman diagram, but also generates the local coordinates at the punctures of these Riemann surfaces in terms of the local coordinates at the punctures of the Riemann surfaces associated with the interaction vertices. The interaction vertices are chosen such that the region corresponding to the sum of all Feynman diagrams covers the full moduli space. We shall discuss several examples of this in §4. This condition is needed for string field theory to reproduce the usual world-sheet results for the amplitudes, but at a more fundamental level, this condition is also necessary for the action to satisfy the BV master equation.
We shall use a convention in which the path integral measure is weighted by e S . In this convention the contribution to an amplitude from a Feynman diagram containing a single interaction vertex is given directly by the corresponding function F (m,n) r 1 ···rn (ω 1 , P 1 , · · · , ω m , P m ) without any additional sign or numerical factors. However we shall keep the overall normalization of S arbitrary, by choosing an arbitrary normalization of the vacuum in the world-sheet theory (see (6.20)). This will have the advantage that we can use our result both in the Euclidean and in the Lorentzian theory (in which the standard measure is e iS ) by changing the overall normalization of S.

Dealing with the open string tachyons
We shall now review the divergences that arise in the world-sheet description of the amplitudes in the presence of D-instantons. They are of two kinds -those associated with the open string tachyons and those associated with the open string zero mode fields. In this subsection we shall describe some general aspects of D-instanton perturbation theory, and review the procedure discussed in [15] for dealing with the open string tachyons. The effect of open string zero modes will be discussed in §2.5.
1. In the presence of D-instantons we need to sum over contributions to string amplitudes from Riemann surfaces with boundaries, with the world-sheet fields satisfying boundary condition associated with the D-instantons at the boundaries. We need to include disconnected world-sheets in the computation, with the rule that each connected component must have either at least one external closed string insertions or at least one boundary with D-instanton boundary condition, and at least one of the connected components must have both, an external closed string insertion and a boundary.
2. The integrals over the moduli spaces of Riemann surfaces suffer from infra-red divergences from the boundaries of the moduli space. These divergences are most problematic for open string degenerations since the open strings living on the D-instanton do not carry any momentum and therefore the usual approach of dealing with the divergences by analytic continuation in the external momenta does not work. However, as we shall review below, string field theory can be used to deal with these divergences.
3. The infrared divergences associated with open string degeneration arise from the q 0 region on the right hand side of (2.8). As is obvious from (2.8), these divergences arise from L 0 ≤ 0 states. The divergences associated with L 0 < 0 states can be easily tackled by using the left hand side of (2.8) instead of the right hand side. This leads to the replacement rule: For α > 0 this is an identity but for α < 0 this is a prescription that ensures that the world-sheet computations reproduce the string field theory results.
Since the L 0 < 0 states represent tachyonic modes, the reader may worry about treating them this way when they propagate in the loops. Indeed, this will not be a sensible way to treat tachyons in a quantum field theory. However, for D-instantons the tachyons do not carry any momentum and the path integral over the tachyons represent ordinary integrals, with the integrand having a local minimum in the direction of the tachyon field instead of a local maximum. This means that the steepest descent contour runs along the imaginary axis of the tachyon field. Once we take the integration contour over the tachyonic mode along this steepest descent contour, as is customary for evaluating contributions from saddle points of a path integral, there is no problem in carrying out systematic perturbation expansion to evaluate the contribution to the integral.
4. For treating the divergences associated with the L 0 = 0 states we need to introduce the notion of Wilsonian effective action that we shall discuss next.

Wilsonian effective action
In string field theory, it is possible to integrate out the set of fields corresponding to a given set of L 0 eigenvalues and write an effective action for the other fields. The resulting effective action shares all the properties of the original master action, both in the classical theory [26][27][28][29] and in the quantum theory [30]. We can use this to integrate out all the open string fields with The {1 − δ α,0 } reflects that for α = 0, we need to remove the contribution from the region 0 ≤ q ≤ 1 altogether. Since this removes all divergent terms, computation of the Wilsonian effective action does not suffer from any divergences.

An alternative replacement rule
We shall now describe an alternative to the replacement rule (2.11). We shall consider the cases α = 0 and α = −1 only, since these are the relevant cases for our analysis. If we replace the lower limit of q integration by some small number δ, then (2.11) can also be stated as, On the other hand if we explicitly evaluate the left hand side, we get Therefore the replacement rule simply corresponds to dropping the boundary terms proportional to δ −1 and ln δ. This procedure is particularly useful if the integrand is a total derivative -which is indeed the case in many of the expressions that we shall encounter. In this case we can express this as an integral over the various boundary segments labeled by q i = δ i following the procedure described in [12], and then drop terms proportional to δ −1 i and ln δ i in the final expression.
Although we shall not describe this analysis in this paper, we have checked that following this procedure we get the same results as the ones reported here.

Dealing with the zero mode fields
At the level of the Wilsonian effective action, the open string fields associated with the L 0 = 0 states, which we shall collectively call the zero modes, remain unintegrated. However in computing physical amplitudes we have to eventually integrate over them. Doing this requires physical understanding of the origin of these fields.
1. First let us consider the zero modes arising from the pure ghost sector. We begin by writing down the full expansion of the string field in the L 0 = 0 sector, involving pure ghost excitations: 14) The analog of the field ψ 1 in heterotic string theory has been identified in [31] with the zero mode of the Nakanishi-Lautrup field [32,33], while the fields ψ 2 and ψ 1 can be identified with the zero mode of the ghost and the anti-ghost fields associated with the U (1) gauge symmetry. Siegel gauge condition would set ψ 1 and ψ 2 to zero, but the resulting kinetic operator c 0 L 0 will vanish while acting on the remaining L 0 = 0 states, leading to divergent propagator of the fields ψ 2 and ψ 1 . For this reason, in the L 0 = 0 sector we choose a new gauge [15] in which we set all the anti-fields to zero, and express the string field involving pure ghost excitations as: We are supposed to integrate over the fields ψ 1 and ψ 2 . First consider the effect of ψ 1 . One can show that the field ψ 1 associated with the state c 0 |0 has finite kinetic term and propagator, and can be treated perturbatively. This means however that besides the usual world-sheet contribution, with the replacement rule (2.11), we must also separately include the contribution from the propagator of this new field ψ 1 in every propagator of a Feynman diagram. A classical version of a similar result can be found in [31].
3. The field ψ 2 associated with the state |0 multiplies a state of ghost number zero. Using the same argument as for the full string field theory, one can show that it decouples from the action due to ghost number conservation. The interpretation of integration over ψ 2 follows from the same analysis that was used for the full string field theory -since the string field ψ 2 associated with the state |0 is grassmann odd, the integration over ψ 4. Finally, let us turn to the zero modes coming from the matter sector in the ghost number 1 sector. Typically for single instanton, these arise from the collective excitations of the D-instanton that translate the D-instanton along transverse directions along which we have translational symmetry. For the two dimensional string theory that we shall analyze, there is only one such zero mode described by the vertex operator c∂X in the world-sheet theory. We shall denote this field by φ. The integration over this mode is carried out at the end of the computation. In general, φ itself is not the collective coordinate, but it is expected to be related to the collective coordinate φ describing time translation of the Dinstanton by a field redefinition [13]. We can find this explicit field redefinition by noting that the dependence of the amplitude on the collective mode φ must be proportional to e −i φω where ω denotes the total outgoing energy of all the external closed string states in the amplitude. Therefore we first make the change of variable of integration in the path integral from φ to φ so that the amplitude has the correct dependence on φ, and then perform the φ integration to pick up the energy conserving delta function. Any Jacobian that arises from this change of variable must be taken into account in the computation of the amplitudes.
This discussion can be summarized by saying that after evaluating the amplitudes using the replacement rule (2.11) that renders the world-sheet integrals finite, we need to include the following extra contributions: 1. We need to take into account the contribution from the ψ 1 exchange in each internal propagator of the Feynman diagrams.

Approximations
In order to implement the procedure described in §2, we need to know the precise relation between the integration parameters entering the expression for a world-sheet amplitude and the variables q i that are used to express the propagators via (2.8). We can then express the integral as integration over the variables q i 's near the q i = 0 region, expand the integrand in a power series expansion in the q i 's and make the replacement (2.11). Our goal in this section will be to analyze how much error we are allowed to make in identifying the variables q i without actually changing the value of the integral.
In order to convince the reader that this is not an empty exercise, we shall first demonstrate, with an example, that identifying the correct variables q i is important before applying the replacement rule (2.11). Let us consider an integral where we have used the replacement rule (2.11). Now suppose that instead of analyzing (3.1) directly, we first make a change of variable: with c < 1, so that the change of variable is well defined in the range 0 ≤ q ≤ 1. It is easy to check that dq q −2 = dq q −2 . Using this we can express (3.1) as where in the last but one step we have used the replacement rule (2.11) for the q integral from 0 to 1, and have explicitly evaluated the finite integral from 1 to 1/(1 − c). We now see that the final result in (3.3) differs from (3.1). This demonstrates that it is important to identify the variable q correctly using string field theory before applying (2.11).
Before proceeding with the analysis, we shall make a change in the notation. For the type of string field theory we shall be considering, the integration parameters entering the expression for a world-sheet amplitude will have natural relation not with the q i 's but with u i = i q i 's where i are small parameters. In other words, the integrand, when expanded in powers of the u i 's, will have finite coefficients of expansion. In this case it is more convenient to trade in the variables q i 's for the u i 's, and express the replacement rule (2.11) by If we can determine the exact relation between the moduli parameters of the Riemann surface in terms of which we compute the integrand, and the u i 's, then we can carry out the procedure described above. However this is not strictly necessary -we shall show that even if we make some error in finding the relation between the moduli parameters and the u i 's, as long as the error goes to zero sufficiently fast in the u i → 0 limit, we can still get the result for the integral exactly. Our goal in this section will be to study how fast the error should vanish as u i → 0 so as to not make an error in the final value of the integral. This question is important since during our analysis we often will need to make approximations in relating the natural moduli space variables to the string field theory variables q i , or equivalently, u i .
We shall first consider the case where we have one integration variable u, and the integrand is u −2 . In this case the rule (3.4) can be stated more explicitly as: for any positive constant a. Note that I is a formal integral since the integral diverges from u = 0, but I is well defined. It can be easily verified that I defined in (3.5) is independent of . Now suppose that instead of using the variable u, we erroneously use the variable v, related and use the replacement rule (3.4) with the variable u replaced by v. Then the formal integral I, when converted into an integration over the variable v, reduces to another formal integral: we can express I as Note that only the first integral on the right hand side is divergent. If we now use the replacement rule (3.4) with u replaced by v, we can replace the formal integral I by: It can be verified that the finite integral I is independent of . Using we get We can now use the independence of I and independence of I to evaluate both I and I in the limit , → 0. Since (3.6) gives we get, from (3.5), (3.12), This shows that if instead of applying the replacement rule (3.4) on the original integral I expressed as integration over u, we first make a change of variables from u to v as given in (3.6), and then use analogous replacement rule on the integral, the result differs by −a 1 /a 2 0 . The example described at the beginning of this section is a special case of this general result.
Next we consider the case α = 0. In this case the replacement rule can be expressed as: This time J does depend on the cut-off since the effect of this cut-off is to leave unintegrated the zero mode fields whose definition implicitly depends on the cut-off . Under a change of variable of the form (3.6), the formal integral J may be expressed as, Only the first term is divergent. Using the replacement rule (3.4) with u replaced by v, we can now replace J by: where we have used the fact that in a given variable, the rule for subtraction in a logarithmically divergent integral is to replace the lower limit of integration by instead of 0. Using we get It is worth noting that the origins of the errors (3.14) and (3.19) are somewhat different. The error (3.14) associated with linearly divergent integral arises due to the fact that under a change of variable from u to f (u), the replacement rule (3.4) does not remain invariant but gets shifted by a constant. However, since the right hand sides of (3.5) and (3.10) are independent of the cut-offs and , the error is not related to a change in the cut-off. On the other hand the error (3.19) arises from the fact that for logarithmically divergent integral the integration region u ≤ is removed by hand, making the integral dependent on the cut-off . Therefore if we make a change of variable from u to v and use the same cut-off for the v variable, then it corresponds to a different cut-off in the original variable u and result changes.
Eqs.(3.14) and (3.19) show that if we do not want to make an error in the evaluation of an amplitude, then for a linearly divergent integral we are allowed to make a fractional error of order u 2 , while for a logarithmically divergent integral we are allowed to make a fractional error of order u. Intuitively these results may be understood as follows. When there is a tachyon in the intermediate state, it can produce a contribution of order 1/ in the integral according to (3.4). The 1/ term is expected to cancel at the end, leaving behind a finite contribution. However a fractional error of order can combine with the 1/ term to give a finite contribution that does not cancel. In contrast, divergences associated with massless particle poles are logarithmic. Therefore a fractional error of order is not important, but the leading order identification of variables must be correct.
Even though we have not carried out a detailed analysis for the many variable case, we shall adopt this guideline for determining what are allowed approximations. Regarding the i 's associated with different degenerations as independent, we can see that in general we cannot ignore fractional errors of order i , but can ignore higher order corrections. Thus, for example, for two variables we shall not ignore terms of order 1 , 2 or 1 2 but will ignore terms of order 2 1 and 2 2 . One might worry that 1/ 1 2 contribution from the tachyon may combine with the 2 1 term to give a finite contribution 1 / 2 . However since we know that the final results are independent of 1 and 2 , such terms must cancel at the end. If in some cases 1 and 2 cannot be taken to be independent -since they are determined by the underlying string field theory -then we need to be a bit more careful and keep terms up to sufficient order in order not to make an error.
In our analysis, the i 's will be determined in terms of two string field theory parameters λ and α, which we shall take to be large. The contributions from different Feynman diagrams will have the structure of Laurent series expansion in α −1 and λ −1 . We shall drop terms with positive powers of α −1 and / or λ −1 by taking α and λ to be large. Since the final result is expected to be independent of α and λ, we could in principle also drop terms with positive powers of α and / or λ, keeping only the terms that are independent of α and λ. However we keep these terms, since their cancellation in the final result provides a consistency test of our results. Finally, we note that the results of this section are also valid if we use the alternative procedure described in §2. 4. To see this, recall that the procedure of §2.4 involves dropping terms proportional to δ −1 and ln δ from the boundary at q = δ before taking the δ → 0 limit.
. Therefore, if we were using the variableq as the sewing parameter, then we would drop terms proportional toδ −1 and lnδ. It . Therefore as long as b 0 = 1 and c 0 = 0, we shall not make any error if we interpretq as the sewing parameter instead of q. This argument easily generalizes to the case of multiple sewing parameters.

Interaction vertices and covering of the moduli space
The string field theory that we shall use falls in the general framework discussed in [19][20][21], but the actual interaction vertices that we use in our analysis will be based on the choice of local coordinates that are related to the global upper half plane coordinates by SL(2, R) There are more elegant choice of interaction vertices based on half string overlap [34], the minimal area metric [20,21] or the hyperbolic metric [35][36][37][38][39], but for practical calculations the vertices based on SL(2, R) transformations seem to be more useful. 4 Since string field theories associated with different choices of interaction vertices are related to each other by field redefinition [18], all physical results will be independent of this choice.

C-O amplitude on the disk
The C-O amplitude on the disk has zero dimensional moduli space, and therefore the entire where λ is a large positive number. We can map the unit disk to the upper half plane labelled by z via so that (4.1) takes the form In the z-plane the closed string puncture is located at i.
In the z coordinate system this translates to Figure 2: This figure shows two Feynman diagrams contributing to the disk amplitude with one external closed string and two external open strings.

O-O-O amplitude on the disk
The O-O-O amplitude on the disk also has zero dimensional moduli space, and therefore the entire contribution to the amplitude comes from the elementary O-O-O interaction vertex shown in Fig. 1(b). In the coordinate z, labelling the upper half plane, the punctures are taken to be located at 0,1 and ∞ with local coordinates: where α is a large positive number. These choices of local coordinates get cyclically permuted under the transformation z → 1/(1 − z) that cyclically permutes 0, 1 and ∞.

C-O-O amplitude on the disk
This gives: This gives an upper half plane labelled by z, with a closed string puncture located at z = i and a pair of open string punctures located atz = 1, ∞. In the z plane, these translate to: Therefore this Feynman diagram covers part of the moduli space corresponding to β ≤ (2 λ) −1 .
Furthermore, the local coordinates at the open string punctures are given by w 2 and w 3 in (4.6), which we shall rename as w 1 and w 2 respectively: These can be written in a compact form as   We restrict β to lie in the range β ≤ 1 since a z → −1/z transformation leaves the closed string puncture at i∞ fixed and exchanges the range 0 < β < 1 with 1 < β < ∞. This Note that we could have multiplied the choice of w a given in (4.12) by an arbitrary function g(β) of β that takes value 1 at β = (2 λ) −1 and satisfies the condition but is otherwise arbitrary. This will make the later analysis a bit more complicated. For this reason, we shall proceed with the choice g(β) = 1. We shall check in appendix E that inclusion of such multiplicative function does not change the final results.

C-C amplitude on the disk
The moduli space of the C-C amplitude on the disk is one dimensional, and can be represented by an upper half plane with the first closed string puncture at i and the second closed string puncture at i y with 0 < y < 1. This amplitude receives contribution from two sources, the  This mapsẑ = i to z = iy with The condition q ≤ 1 corresponds to the following region in the y-space: Therefore the C-C interaction vertex shown in Fig. 3(b) must cover the region Note that the y ≥ 1 region can be brought back to the y ≤ 1 region by z → −1/z transformation.
The region B actually contains a boundary of the moduli space associated with closed string degeneration, since as y → 1 the two closed string punctures approach each other. In the full open-closed string field theory, we need to represent the contribution from this region as coming from a new Feynman diagram in which a C-C-C vertex is connected to a C vertex (one point function of the closed string on the disk) by an internal closed string propagator.
This will be necessary for evaluating the full contribution to f (ω 1 , ω 2 ) given in (1.3) that has a singular contribution from this region [17]. However, since our goal is to evaluate f div given in (1.9) which does not have any divergence from the y → 1 region, we do not need this special treatment of the y 1 region.

C-C-O amplitude on the disk
The moduli space of the C-C-O amplitude on the disk is two dimensional, and can be rep- Using (4.3) and (4.6) we get the sewing relations to be, will describe a continuous curve. This will play no role in the analysis presented in this paper, but is useful when we follow the alternative procedure described in §2.4 and [12].
In the z plane, the location of the first closed string puncture at z = i, the second closed string puncture atẑ = i and the open string puncture atz = 1 are mapped to: We shall now make a coordinate transformation and adjust the real constant a so that in the z plane the closed string punctures are located at i and iy for some real number y. This gives The vanishing of the real part of the right hand side fixes a, which can then be substituted into the same equation to determine y. We shall carry out the analysis for small u 1 , u 2 . This gives: and Also in the z plane the open string puncture is located at x with The region of the moduli space covered by this diagram corresponds to The range (4.26) translates to a region in the x-y plane bounded by the following curves: I : where we have neglected fractional errors of order higher than λ −2 . This can be justified as follows. We shall see in §6.3 that the maximally divergent part of the integrand goes as In the z plane, the locations of the first closed string puncture at z = i, the second closed string puncture atẑ = i and the open string puncture atz = 1 are given by We now make a change of coordinate z = (z − a)/(a z + 1) so that in the z plane the closed string punctures are located at i and i y for some real number y. This gives and Also in the z plane the open string puncture is located at x with The region of the moduli space covered by this diagrams corresponds to This translates to the region in the x-y plane bounded by the curves (4.34) Next we consider the diagram shown in Fig. 4(b). If we denote by z the coordinate on the upper half plane associated with the C-O interaction vertex, and byẑ the upper half plane coordinate associated with the C-O-O interaction vertex, then, using (4.12) and (4.3), the sewing relation takes the form: We also need to separately consider First consider the case of sewing via (4.35). In this case the second closed string puncture, located atẑ = i, is mapped in the z-plane to where we have used λ = α λ. We need to bring this on the imaginary axis via the change of coordinate z = (z − a)/(az + 1) that maps the location of the first closed string puncture at z = i to z = i and the second closed string puncture at z = z c to z c = (z c − a)/(1 + a z c ).
a is determined as before by demanding that the real part of z c vanishes. This gives, using and, We also need to find the location of the external open string puncture in the z coordinate.
In theẑ coordinate it is located atẑ = β. Using (4.35), we get In the z coordinate this is located at Note that due to the presence of the λ −1 factor in the third term inside the curly bracket, this term can contribute only for β ∼ 1/ λ. Therefore we can replace the β dependent factor multiplying this by any function of β that approaches 1/(2β) for small β. We shall choose this to be (1 − β 2 )/(2β) for convenience, but we have confirmed using Mathematica that the final result remains unchanged even if we use the original form given in (4.41). Labelling z c by iy and z o by x, and ignoring fractional corrections of order higher than λ −2 , we can write the relations as, We now see from (4.42) that the range corresponds to the region bounded by the curves: where F (x) is a function of x defined as the result of elimination of β from the equations: The case of sewing via (4.36) can be analyzed similarly, and yields the result: The range of x and y covered by this diagram is bounded by: where, F (x) is now defined via: Next we consider the diagram shown in Fig. 4(c). This is related to the contribution from and The range of u and β given in (4.43) correspond to the regions in the x-y plane bounded by the curves: and  where F (x) is defined via the equation: A property of the function F (x) that will be useful later is that to leading order in the expansion in powers of λ −1 , F (x) satisfies the equation: for all values of x, to prove (4.54) we need to verify that the leading order relation between x and β remains unchanged under x → λ −2 /x. Now, ignoring fractional corrections of order λ −2 , we see that for |x| > λ −1 , the relation between x and β given in (4.53) takes the from |x| = 2β/(1 − β 2 ), whereas for |x| ≤ λ −1 , the relation between x and β given in (4.45), (4.48) takes the from |x| = λ −2 (1 − β 2 )/(2β). Since these are related by x → λ −2 /x transformation, this establishes (4.54). The rest of the moduli space must be covered by the C-C-O interaction vertex shown in Fig. 4(d). This corresponds to the region:

O amplitude on the annulus
Next we consider the one point amplitude of an open string on the annulus. We shall label the annulus by the complex coordinate w with the identification This can be expressed as where We can 'diagonalize' this matrix by defining new coordinateẑ via Then the identification (4.58) takes the form Comparison with (4.56) gives This agrees with the result of [40]. Since according to the result of §3, fractional errors of order u 2 can be absorbed into a redefinition of u without changing the final result, we shall use the relation Therefore the range of v spanned by Fig.6(a) corresponds to In this coordinate system the open string puncture located at z = 0 is mapped to, The local coordinate at the open string puncture is given by Since translational invariance along the imaginary w axis allows us to change this to any arbitrary imaginary value, the location of the open string puncture is not a modulus. Nevertheless, for definiteness, we shall stick to (4.65), (4.66).
The contribution from Fig. 6(b), corresponding to the open string one point interaction vertex on the annulus, will have to span the range We can choose the local coordinate at the open string puncture to be the same as the one given in (4.66) for u = α −2 , i.e.
We could make more general choice of w o , e.g. include in the definition of w o an arbitrary multiplicative function h(v) that takes value 1 at v = 1/(α 2 − 1 2 ) but is otherwise arbitrary. In our analysis below, this will have the effect of replacing u by u/h(v) in (4.86), and multiplying the upper limit of x in (4.88) and the lower limit of x in (4.102) by 1/h(v). This in turn will affect the integrals I (b) and I (d) in (6.11) and (6.5) respectively. It is easy to verify that these changes cancel each other and do not affect the final result.

C amplitude on the annulus
Next we consider the one point amplitude of a closed string on the annulus. The moduli space for this amplitude is two dimensional. We shall label the annulus by the coordinate w with the identification and the closed string puncture will be taken to be located at w c with Re w c = 2πx . The pair (v, x) label the moduli on the Riemann surface, with the full range given by: The range x > 1 4 is related to the region described above by w → π − w transformation. The imaginary part of w c can be shifted using the translation invariance of the annulus along the periodically identified imaginary w axis, and therefore does not represent a modulus. Fig. 7 shows the different Feynman diagrams contributing to this amplitude. Our goal will be to find out the relation between the sewing parameters {q i } associated with these diagrams and the variables v, x introduced above. Since we shall derive approximate relations between these quantities instead of exact relations, we need to make sure that we do not throw away terms that could give finite contribution to the amplitude. For this we note the following facts: Figure 7: This figure shows the Feynman diagrams contributing to the annulus amplitude with one external closed string. The interaction vertex with a × represents part of a disk amplitude, while the interaction vertex with ⊗ represents annulus amplitude. As in Fig. 4, the q i 's represent the sewing parameters of the corresponding propagators.
1. The relations we are after depend on two string field theory parameters λ and α which are both taken to be large, and the amplitude, to be evaluated in §6.1, will be given as a Laurent series expansion in α −2 and λ −1 . In this expansion we shall keep all terms that contain non-negative power of α 2 and λ, but will throw away terms containing negative powers of α 2 or λ, even if they contain positive power of the other variable. The reason for this is that the α and λ dependent terms are expected to cancel at the end anyway, and once a term has got negative power of one large variable, there is no way it can give a α and λ independent term at higher order in the expansion.
2. The most divergent term in the integrand that we shall encounter during our analysis in §6.1, where these results will be used, goes as dv dx x −2 v −2 for small v, x.
We begin by analyzing the contribution from Fig. 7 We shall define: The second equation of (4.72) now givesz then the identification (4.74) takes the form In thez plane, the closed string puncture at z = i is at In the w coordinate system this maps to up to addition of imaginary terms which can be removed using translational invariance along the imaginary w direction. Comparison of (4.77), (4.79) with (4.69), (4.70) gives (4.80) shows that for small v, x the most divergent part dx dv x −2 v −2 of the integrand goes as It then follows from the analysis of §3 that it will be sufficient to keep terms up to fractional errors of order u 1 ∼ λ −1 and u 2 ∼ α −2 but we can ignore fractional errors of order u 2 1 ∼ λ −2 and / or u 2 2 ∼ α −4 . With this understanding, we can replace (4.80) by: Therefore Fig. 7(a) covers the range in the moduli space. This has been shown as region (a) in Fig. 8. We now turn to the contribution from Fig. 7(b). We denote byẑ = e iw the coordinate Also, using (4.3), (4.68), the sewing relation takes the form: Using this we find that the closed string puncture at z = i is located in the w = 1 i lnẑ plane at Comparison with (4.70) gives: Since dx dv x −2 v −2 grows as du dv u −2 v −2 for small u, v, and since v is bounded from below by α −2 , it follows from the analysis of §3 that we need to keep terms up to fractional errors of order u ∼ λ −1 and α −2 . Therefore we can approximate (4.86) as: (4.87) Therefore Fig. 7(b) covers the region: This has been shown as region (b) in Fig. 8.
Next we turn to the contribution from Fig. 7(c). We denote by z the coordinate on the C-O-O interaction vertex with the closed string puncture located at z = i. Then the sewing relation gives the identification where w 1 and w 2 have been defined in (4.12). Denoting f (β) by f for notational simplicity and using the relation f (−β) = −f (β), we can express (4.89) as and (4.92) The transformation (4.90) can be 'diagonalized' using new coordinatez, defined via We also define In these variables (4.90) takes the form: Also, in this variable the closed string puncture at z = i is located at Comparison of (4.95), (4.96) with (4.69), (4.70), and use of the x → 1 2 − x symmetry, gives Since dv dx v −2 x −2 ∼ du dβ u −2 λ 2 for small u, we can ignore fractional errors of order u 2 , since they will generate corrections containing inverse powers of α 2 . Therefore we can write (4.97) The range 1 translates to the region bounded by the curves: where now f ≡ f (tan(πx)) and we have used The region (4.100) has been shown as region (c) in Fig. 8.
The contribution from Fig. 7(d) should cover the left-over region in the x-v space: In the full open-closed string field theory, we need to represent the contribution from this region as coming from new Feynman diagrams containing internal closed string propagators. This will be necessary for evaluating the full contribution to g(ω) given in (1.5) that has a singular contribution from this region [17]. However, since our goal is to evaluate g div (ω) given in (1.11) which does not have any divergence from the v → 1 region, we do not need this special treatment of the v 1 region.

Disk two point function in two dimensional string theory
Our analysis so far applies to any string theory D-instanton. Now we shall apply the results to the specific case of D-instantons in two dimensional string theory, whose world-sheet matter theory contains a free scalar describing the time coordinate and the Liouville theory with c = 25. As discussed in §1, we shall first analyze the building blocks of the amplitude, leaving out the energy conserving delta function and the N e −1/gs factor that are common to all amplitudes.
In this section we shall analyze the building block corresponding to the disk two point function of a pair of closed string tachyons of energies ω 1 and ω 2 . As described in §1, this has the form: (5.1) where f (ω 1 , ω 2 ) is a sum of a finite piece and an apparently divergent piece: f finite has been given in (1.8), but since we shall not need it, we do not reproduce it here.
f div (ω 1 , ω 2 ) is given by: We shall follow the general strategy introduced in §2 to evaluate f div (and also g div described in §6). Let us suppose that we have some on-shell amplitude, expressed as an integration over the moduli m ≡ {m 1 , · · · , m n } of the world-sheet with punctures. The issue that we have to deal with is that the integrand diverges at various corners of the moduli space and we have to make sense of these divergences. For this we express the amplitude as sum over Feynman diagrams of string field theory. Let us suppose that we have a Feynman diagram with k propagators and that q 1 , · · · q k are the sewing parameters associated with these propagators, each taking value in the range 0 ≤ q i ≤ 1. The analysis of the type performed in §4 determines the relation between the moduli m and the sewing parameters q 1 , · · · , q k and (n − k) other variables {m 1 , · · ·m n−k } arising from the interaction vertices of the Feynman diagram. We now express the integrand as a function of the q i 's andm j 's and expand it in a power series in q i 's, with the coefficients of the power series being functions of them j 's. Next we apply the replacement rule (2.11) to extract a finite answer. We must also include, for each propagator, the contribution from the propagation of the ψ 1 field introduced in (2.14) separately. The effect of the Jacobian factors discussed in §2.5 contribute at the same order as one loop amplitudes, and will affect the analysis of g div in §6. However they will not affect the analysis of f div that will be carried out in this section, since this, being a disk two point function, represents a tree amplitude.
To implement the strategy described above for the integral (5.3), we first need the worldsheet interpretation of y appearing in (5.3). There is a simple answer to this: i and iy are the positions of the vertex operators carrying energies ω 1 and ω 2 in the upper half plane [4]. Therefore we can use the relation between y and q given in (4.16) for analyzing (5.3). We first divide the range of integration over y into the regions A and B described in (4.17), (4.18). The contribution from the region B does not suffer from any divergences and simply gives back the same integral with integration range replaced by λ −2 ≤ y ≤ 1. The contribution from the region A is evaluated by replacing y by q/λ 2 according to (4.16) and expressing the integral as where in the second step we have used the replacement rule (2.11). Therefore the net result is We also have to add the contribution where in Fig. 3(a) Therefore (5.5) represents the full contribution to f div : 5 Comparing this with (1.12) we get Note that the vanishing of the ψ 1 exchange amplitude is a specific property of the choice of local coordinate system that we have made in (4.3). If we had chosen a different coordinate system, then the result would not have vanished. However the final result for any physical amplitude will be independent of the choice of local coordinates. In particular A f will be independent of the choice of local coordinate system since it can be extracted from the scattering amplitude of n closed string tachyons. We have illustrated this in appendix A by working with a different choice of local coordinate system for the C-O interaction vertex. B f by itself need not be independent of the choice of local coordinate system, but the combination 2B g − B f will be independent of the choice of local coordinates.

Annulus one point function in two dimensional string theory
We now turn to the second building block for the amplitudes in two dimensional string theory -the one point function of closed string tachyon of energy ω on the annulus. As discussed in §1, this has the form: 2 g s sinh(π|ω|) g(ω) , (6.1) where g(ω) is given by the sum of a finite piece and a divergent piece: The expression for g finite (ω) has been given in (1.10) and will not be repeated here. g div (ω) has the form: As in §5, we can analyze this by first relating the variables v and x to the parameters {q i } associated with the propagators of string field theory, and then using the replacement rule (2.11). We shall call this the world-sheet contribution to g div . However there are several other contributions to the annulus one point function that are not captured by the world-sheet expression for g finite . We shall include these also in the definition of g div . This includes the contribution due to the ψ 1 field propagating in the internal lines of the Feynman diagrams of string field theory. We also have two possible field redefinitions whose associated Jacobians give additional contributions to the one point function of the closed string tachyon. The first one is associated with the redefinition that relates the zero mode in open string field theory to the collective coordinate associated with the translation of the D-instanton along the time direction. The second redefinition involves the ghost field ψ 2 introduced in (2.14). This can also be interpreted as the redefinition that relates the gauge transformation parameter in open string field theory associated with the vacuum state |0 to the rigid U (1) symmetry transformation parameter that gives a phase to any open string stretching from the D-instanton to another D-brane. We shall analyze each of these contributions in turn.

World-sheet contribution
We shall first analyze the contribution to (6.3) that we obtain by replacing the variables x and/or v near the singular points by appropriate variables {q i } associated with the propagators of string field theory, and then using the replacement rule (2.11). For this we need to know the world-sheet interpretation of the variables v and x. As in (4.69), (4.70), −i ln v is the periodicity of the w coordinate on the annulus, with the real part of w taking value in the interval (0, π), and 2πx denotes the real part of the location of the closed string puncture in the w plane [4]. We shall divide the integration region over v and x in (6.3) into four regions shown in Fig. 8, representing the contributions from different Feynman diagrams in Fig. 7, and carry out the integration over each region separately, using the replacement rule (2.11) or (3.4). In this analysis we shall ignore terms that vanish in the large α and large λ limit, including terms proportional to α 2 / λ and λ 2 /α 2 , since these terms must cancel at the end. The reader may be somewhat uncomfortable in ignoring terms of order λ 2 /α 2 , since this ratio, being given by λ 2 , is large. Nevertheless, it follows from our general arguments that such terms must cancel at the end. As discussed at the end of this subsection, we have explicitly checked that such terms do cancel.
We begin with the contribution from the region (d) given in (4.102): where f stands for f (tan(πx)) The integral takes the form: The integrals in the third line on the right hand side are elementary. For the terms in the last line, the integral can be simplified by writing dx/ sin 2 (2πx) as −d cot(2πx)/2π and then doing an integration by parts. This gives, after dropping terms that have either powers of λ or of α in the denominator: An alternative procedure for evaluating the ω independent part of the integral is to express the integrand as a total derivative, e.g. as, and integrate the one form A along the boundary of region (d) in the anti-clockwise direction. This has the advantage that we do not need to explicitly find the shape of the boundary given in (4.102). For example, while evaluating the contribution from the boundary separating regions (c) and (d), we can simply change variables from (v, x) to (u, β) using (4.98) that is appropriate in region (c), set u = α −2 (4 λ 2 ) 2 (1 + 4 λ 2 ) −2 to parametrize the boundary, and then integrate over the variable β in the range (2 λ) −1 ≤ β ≤ 1. This give the same result, but this procedure is easier to use if we want to extend the analysis to higher orders in the expansion in powers of λ −1 and α −2 and check that these terms indeed cancel, as implied by our general arguments.
Next we turn to the contribution from region (c). In order to evaluate the contribution from this region, we need to change variables from (v, x) to (u, β) according to (4.98), and then integrate over the range 0 ≤ u ≤ α −2 {1 + (4 λ 2 ) −1 } −2 and (2 λ) −1 ≤ β ≤ 1 using the replacement rule (3.4). One finds: In region (b), v is restricted to the range α 2 − 1 2 −1 ≤ v ≤ 1, and we use (4.87) to trade in x for the variable u via This reduces the integral to ignoring fractional errors of order u 2 1 or u 2 2 . This gives and expresses the contribution to the integral from this region as (6.14) We now use the replacement rule (3.4) to get: Adding (6.6), (6.9), (6.11) and (6.15), we get the total world-sheet contribution to g div : where in the last step we have changed variable from x to β = tan(πx) and dropped terms containing inverse powers of λ.
We have carried out two further checks on the result: 1. Terms of order λ 2 /α 2 have been dropped in our analysis, despite the ratio being large, since we have argued that such terms must cancel at the end. To confirm this, we have checked that such terms appear in the expressions for I (c) and I (d) , and are given respectively by, This shows that these terms do cancel at the end.
2. The ω independent part of the integral in (6.3) can also be computed using the alternative procedure described in §2.4. For this we note that this part of the integrand may be expressed as a total derivative, e.g. as, Therefore the integral can be expressed as an anti-clockwise boundary integral of A in

ψ 1 exchange contribution
We shall now evaluate the ψ 1 exchange contributions. For this we need the ψ 1 propagator.
In order to demonstrate that the final results are independent of the choice of normalization conventions, we shall normalize the vacuum as where K 0 is some non-zero constant. This corresponds to normalizing the ghost correlation function on the upper half plane as: If we expand the string field as then with the convention (6.20) the quadratic term in the string field theory action takes the form: Therefore the tachyon propagator is given by K −1 0 while the ψ 1 propagator is given by K −1 0 /2. Although the propagators are convention dependent, their ratio is convention independent.
Our final results will depend only on the convention independent ratios of this type.
The C-O amplitude, where the closed string is an on-shell tachyon state and the open string represents ψ 1 , vanishes by (5.6). Therefore the relevant diagrams are the ones shown in Fig. 9. Both these diagrams involve computing a disk amplitude with one external C and two external O's, with the O's corresponding to the ψ 1 state, and then multiplying it by the ψ 1 propagator.
One subtlety in this computation is that the ψ 1 vertex operator ∂c is not a dimension zero primary of the form cW where W is a dimension one matter primary. Therefore we need to be careful in evaluating the integrand that we shall eventually integrate over the modulus of this diagram -the variable q appearing in (4.9) for Fig.9(a) and the variable β appearing in (4.12) for Fig. 9(b). Defining β = q/(2 λ) in (4.9), we can analyze the two diagrams together.
Let z be the coordinate on the upper half plane in which the closed string puncture is situated at i and the open string punctures are situated at z 1 = −β and z 2 = β. Let w 1 and w 2 be the local coordinates at the open string punctures, related to z via the relations z = F a (w a , β) , (6.24) so that z a = F a (0, β). Then the integrand multiplying dβ is given by the correlation function of the vertex operators of the closed string tachyon at i and of the two ψ 1 's at z 1 , z 2 in the w 1 , w 2 coordinate system, and the operator [11,20,21] where a is the anti-clockwise contour around z a = F a (0; β) and K 1 is another normalization constant. Note that ∂F a /∂β is computed at fixed w a , but after taking the derivative we need to express this back in the z coordinate system using (6.24). Therefore the amplitude takes the form: where ccV T denotes the vertex operator of the closed string tachyon of energy ω and F a • ∂c(0) is the conformal transform of ∂c(0) by the map F a . After computing the amplitude we can divide the result by 2 g s sinh(π|ω|) to compute its contribution to g(ω). We describe this computation in detail in appendix B, quoting here only the final results for the contribution g ψ 1 to g div : The first term inside the square bracket is the contribution from Fig. 9(b) and the second term is the contribution from Fig. 9(a). The constants K 0 and K 1 drop out of the final computation since we compare the result with the contributions where the ψ 1 in Fig. 9 is replaced by the open string tachyon. The latter contributions can be read out from (6.14) and (6.9) respectively.

C-C-O disk amplitude, zero mode field redefinition and the Jacobian
In this subsection we shall explore whether we need a field redefinition to relate the zero mode field φ of the open string field theory associated with the vertex operator c∂X and the collective coordinate φ that translates the D-instanton along the time direction. This will be done by studying the C-C-O disk amplitude with a pair of closed string tachyon vertex operators carrying total energy ω and the open string vertex operator c∂X. If the open string zero mode had been the same as the collective coordinate, then this amplitude would be given by −iω times the C-C disk amplitude, since the dependence of the amplitude on the collective coordinate φ is expected to be of the form e −iω φ . Therefore any deviation from this form would have to be removed by a field redefinition and the associated Jacobian would have to be taken into account in computing amplitudes.
Therefore we consider the C-C-O amplitude, with the closed string states corresponding to a pair of tachyons as in §5 and the open string state corresponding to the zero mode φ associated with the vertex operator c∂X. As in the previous cases, here by amplitude we shall refer to the building block, obtained before getting the energy conserving delta function or the N e −1/gs factor. We shall insert the closed string vertex operators at i and i y as in §5, and the open string vertex operator at a point x on the real axis, and express the amplitude as an integral over x and y. The integrand is given by the integrand of the C-C amplitude, multiplied by 6 (6.28) Therefore if we integrate over x for fixed y, we get the result: multiplying the integrand of the C-C amplitude. Since the coupling of the collective coordinate φ to this order is also supposed to be given by −i(ω 1 + ω 2 ) times the C-C amplitude, this would be consistent with the hypothesis that φ and φ can be identified. This analysis, however is only formal, since the C-C-O amplitude has divergences from different regions of integration and they need to be treated following the replacement rule (2.11). Therefore we need to carefully evaluate the contributions from the different regions shown in Fig. 5, associated with the Feynman diagrams of Fig. 4. Possible divergences are associated with the q parameters of one or more internal propagators in Fig. 4 approaching zero. To analyze this, we shall split the integrand into the sum of two parts -one where we multiply (6.28) by the integrand of f finite defined in (1.8), and the other where we multiply (6.28) by the integrand of f div defined in (1.9). Now from Fig. 5 one can easily see that all the  potentially divergent regions I, I , II, II , III and III , associated with Feynman diagrams with at least one propagator, correspond to the small y region. On the other hand, f finite defined in (1.8) has the property that it has no divergence from the small y region. We also see from (6.28) that integration over x for fixed y does not produce any additional singularity as y → 0. Therefore if we multiply the integrand of f finite by (6.28) and carry out the integration over x and y, we can exclude the region y ≤ from the y integral at the cost of making an error that will be suppressed by a power of and will vanish in the → 0 limit. On the other hand, by the argument outlined in the previous paragraph, the contribution to the x integral for y ≥ will just produce a multiplicative factor of −i(ω 1 + ω 2 ), reproducing the expected coupling of the collective coordinate. Therefore the problematic terms come from the second integral where we multiply the integrand of f div by (6.28) and integrate over x and y. This contribution to the C-C-O amplitude is given by: A reader unconvinced by this argument can verify this explicitly by replacing the second line of (6.30) by y β for β ≥ 0, and then carrying out the integration over x and y from different regions of Fig. 5 following the same procedure as described in appendix C. For β ≥ 0, the result will always be given by −i(ω 1 + ω 2 ) times the integral of y β . In appendix C we have analyzed the contributions to (6.30) from the different regions shown in Fig. 5, associated with the Feynman diagrams of Fig. 4. The result is: The desired result for this is −i(ω 1 + ω 2 ) times the divergent part of the C-C amplitude given in (5.7): Note that we have put back the overall normalization factor 4 g s sinh(π|ω 1 |) sinh(π|ω 2 |) that multiplies f (ω 1 , ω 2 ) in the disk two point amplitude. We now see that (6.31) differs from (6.32) by the term: This means that the zero mode dependence of the effective action is not of the form e −i(ω 1 +ω 2 )φ times the amplitude without the zero mode. Let us denote by Φ C (ω, P ) the closed string tachyon field of energy ω and Liouville momentum P , with positive ω describing outgoing states and negative ω describing incoming states. Then the extra term in the effective action to order φ is of the form 7 Note that the integrand could be multiplied by some function of {ω i } and {P i } that reduces to 1 for on-shell closed string fields. But since eventually we shall use this to compute on-shell amplitudes, this factor will not be important and we shall drop it. We shall now look for an appropriate relation between the open string field φ and the collective coordinate φ so that the effective action has φ dependence proportional to e iω φ . This will allow us to recover the standard energy conserving delta function after integration over φ.
For closed string tachyon of energy ω, the one point function on the disk is given by, 2 sinh(π|ω|) . Since there is no divergence in this amplitude, this result is not modified. We now make a field for some function a(ω) that will be determined shortly. Upon substitution into (6.36), this produces (6.36) with φ replaced by φ, plus an extra term: −4 i g s φ dω dP dω dP ω sinh(π|ω|)Φ C (ω, P ) sinh(π|ω |) a(ω ) Φ C (ω , P ) 7 We are using the convention that if S is the action then the weight factor in the path integral is e S . In that case a term in the action contributes to the amplitude without any additional minus sign or factors of i. Also note that when we compute the amplitude from (6.34), we shall get an extra factor of 2 due to two possible ways of contracting Φ C with the external closed string states. This would account for the factor of 2 in (6.33).
When we express the path integral over φ in terms of path integral over φ using the field redefinition (6.37), we get a Jacobian This can be interpreted as a new term in the effective action, given by: This will give an additional contribution to the one point function of closed string tachyon besides the ones given by Feynman diagrams: Comparing this with (6.1) we see that (6.42) can be interpreted as an additional contribution to g(ω), given by: We end this section with two remarks: 1. The field redefinition described here is only one of the many field redefinitions that φ undergoes. In general (6.37) will contain terms involving other closed string fields and higher powers of closed string fields. These will affect higher order results but will not affect the closed string tachyon one point function to the order that we are doing our computation. There will also be terms that involve redefinition of φ involving higher powers of φ [13]. Such terms will lead to Jacobian factor involving non-linear terms in φ, which can be regarded as a contribution to the effective action for φ. However all such terms must cancel with explicit loop contribution to the effective action for φ, so that when the dependence of the amplitude on φ takes the form e −iω φ , the integration measure becomes φ independent. In this case the integration over φ will produce the δ(ω) factor.
2. The notion of effective action used here is different from the notion of closed string effective action discussed in [41]. Here the effective action refers to the sum over contributions from connected world-sheet before integrating over the collective modes, while in the analysis of [41] the effective action includes disconnected world-sheets and also include the final integration over the collective modes. Therefore in the analysis of [41] the Jacobian given in (6.40) will appear as a multiplicative contribution to the integrand that, after integration over the collective coordinates, gives the effective action. The difference between the two papers can be traced to the fact that here we are trying to analyze the building blocks of the amplitude instead of the full amplitude.

C-O-O-O disk amplitude, ghost field redefinition and the Jacobian
In this subsection we shall find the relation between the infinitesimal gauge transformation where ψ 2 is the string field multiplying |0 ⊗ 1 0 0 0 . The lowest order contribution to this gauge transformation law comes from the ξ-ξ * -ψ 2 coupling in the action, whose coefficient is given by: where we have normalized the ghost correlator as in (6.21) in keeping track of the overall signs, which depend on the precise arrangement of the different fields in the action and the order in which we take the derivatives, since some of the fields are grassmann odd. However such sign errors will be eventually cancelled since the important quantity is the ratio of the coupling described in (6.46) and the coupling in the presence of closed string tachyon, as given in (6.47) below.
When we take into account other terms in the action containing more fields, the rigid U (1) gauge transformation law of ξ remains unchanged, but the string field theory gauge transformation law given by the second derivative of the action with respect to ψ 2 and ξ * , gets modified. This causes θ and θ to be related by a field dependent transformation. We shall now study a particular example of this. Let us denote by A(ω) the disk amplitude (with contribution from massless internal open string states removed) with one closed string tachyon of energy ω and three open string states described by the vertex operators I, c∂c∂ 2 c/2 and I, carrying the same Chan-Paton factors as in (6.44). This translates to a term in the effective action of the form: where Φ C (ω, P ) is the closed string tachyon field of energy ω and Liouville momentum P . As in (6.34), the integrand in (6.47) can contain an extra factor that reduces to 1 when the closed string field is on-shell, but we shall not include this since we shall use this formula for on-shell closed string states. Adding this to (6.46) we get the following contribution to the effective action: Taking derivative with respect to ψ 2 and ξ * , we get the modified gauge transformation law of ξ under the gauge transformation parameter θ: Comparing this with δξ = i θξ we see that θ and θ are related by: where · · · denote higher order corrections that we shall ignore. Therefore we have Hence dividing the path integral over the open string fields by dθ can be regarded as a division by d θ and a multiplication by up to higher order terms. The effect of division by d θ generates a constant factor since d θ gives the volume of the rigid U(1) gauge group. This can be absorbed into a redefinition of the overall normalization constant N that accompanies all amplitudes. Therefore we can ignore this term. This leaves us with a correction to the action of the form:

52)
A(ω) has been evaluated in appendix D, leading to the result: This leads to an additional contribution to the closed string tachyon one point function due to the extra term (6.52) in the effective action. Comparing this with (6.1) we get the contribution to g(ω) from the ghost jacobian: We note in passing that ξ is the ghost field associated with the gauge transformation parameter I ⊗ 0 1 0 0 . Therefore, our calculation can also be interpreted as computing the algebra of the gauge transformations generated by θ and ξ, and finding appropriate redefinition of parameters that maps the algebra to the standard SU(2) algebra expected on a pair of Dinstantons.
Before concluding this section, let us discuss one subtlety that could have affected our analysis. Usually when one tries to relate the standard gauge transformation laws of the form δξ = i θξ to the string field theory gauge transformation laws as given in (6.49), we could add to (6.49) trivial gauge transformation proportional to the equations of motion [42]. Therefore one could wonder if it is possible to change the coefficient of the second term inside the square bracket in (6.49) by adding such trivial gauge transformation, and modify (6.50). For this to happen, there must be a field χ whose equation of motion is proportional to ξΦ C (ω, P ). Using various conservation laws one can see that the only candidate for χ is the coefficient of the open string state c 1 c −1 |0 ⊗ 0 0 1 0 . However explicit computation along the line of appendix B shows that the χ-ξ-Φ C coupling on the disk vanishes, and therefore we cannot modify (6.49) by adding to it trivial gauge transformation. The vanishing of the χ-ξ-Φ C coupling can in turn be traced to the fact that ξ and on-shell Φ C represent BRST invariant states, whereas χ represents a BRST trivial state.

Total contribution
We now collect the various contributions to g div (ω) as given in (6.16), (6.27), (6.43) and (6.54): Adding these, we get the following total contribution to g div (ω): Note that g div stands for any contribution to the annulus one point function that is not already included in g finite defined in (1.10). Comparing this with (1.12) we get

Strategy for computing the correction to the D-instanton action
In this section we shall outline the strategy for computing the remaining building block that contributes to the one instanton correction to closed string tachyon scattering to the first subleading order. This involves a zero point function on a surface of Euler number −1. There are two different surfaces of the latter kind -a disk with two holes and a torus with a hole. If we denote the total contribution to this building block by C g s , then the net contribution to the (n + 1)-point tachyon amplitude from the product of (n + 1) disk one point functions and a zero point function on surfaces of Euler number −1 will be given by: {2 sinh(π|ω i |)} . (7.1) Adding (7.1) to the leading order contribution N e −1/gs i {2 sinh(π|ω i |)}, we see that the effect of (7.1) is to make the replacement, Therefore the instanton action is modified to Furthermore, this modification is independent of which amplitude we consider -this is the overall factor that multiplies the one instanton contribution to any amplitude. Now (7.3) shows that the value of C can be changed by a redefinition of g s of the form g s → g s (1 + ag 2 s + O(g 4 s )) for some constant a. In a critical string theory, g s is the expectation value of the dilaton field. Since different versions of open-closed string field theory are related by field redefinition, the string coupling g s used in different versions of string field theory are also related by redefinition of the kind described above [43,44]. If the same story holds in two dimensional string theory, then it would seem that C takes different values in different versions of string field theory. On the other hand, the matrix model yields a definite result for the tachyon scattering amplitudes, and determines C via (1.13) without any ambiguity. To resolve this apparent contradiction, let us recall that while comparing the matrix model results with the string theory results, [4] made a specific choice of the relation between the matrix model parameter µ and the string coupling g s : Let us take (7.4) as the definition of g s , and denote byg s the string coupling in some version of string field theory. Now in the matrix model one can compute perturbative contribution to the closed string tachyon scattering amplitude with three or more tachyons in the external state.
This will have an expansion in powers of g 2 s . In the dual string theory the same amplitude will have an expansion in powers ofg 2 s . These expansion coefficients may or may not depend on the version of string field theory we use, depending on whether or notg s in different versions of string field theory are different. In either case, in any given version of string field theory, we can determine the relation betweeng s and g s by comparing any particular perturbative amplitude in the matrix model and string theory. On general ground we expect this relation to take the form 8g Once this is done, we can use the same string field theory to compute the zero point function on surfaces of Euler number −1, use this to determine the quantum corrected instanton action as a function ofg s and then express this in terms of g s using the relation between g s andg s determined from perturbative amplitudes. This will generate an unambiguous expansion of the quantum corrected D-instanton action in powers of g s , and determine C unambiguously. Note also that the difference between g s andg s will not affect the functions f and g appearing in (1.13) since they are corrections of order g s to the leading order instanton induced amplitudes.
We have not computed C using this procedure. However, we shall now argue that if we assume the equality of the quantum corrected D-instanton action in two dimensional string theory and the quantum corrected instanton action in the matrix model, then C must vanish.
In the matrix model, the dynamics is that of free fermions moving in an inverted harmonic oscillator potential. Therefore the fluctuations around the instanton solution have purely quadratic action and the instanton action does not get renormalized (except for the one loop determinant that is related to the overall normalization constant N ). Therefore the overall µ dependent multiplicative factor associated with the instanton will be given by the leading order result: Comparing this with the right hand side of (7.2), we get On the other hand, g s is defined by (7.4) without any correction. By comparing this with (7.7) we get It should be noted however that the quantum corrected instanton action, by itself, is not an observable, and therefore this argument is not watertight. Nevertheless, agreement of (7.8) with the numerical results based on the comparison of the S-matrix elements in the matrix model and the world-sheet theory [16] provides a verification of the argument given above.

Some open problems
Clearly, the most urgent issue is to understand the origin of the discrepancy between the value of A g obtained from the string field theory computation and the result obtained by comparison with the matrix model. We discuss various possible sources of this discrepancy, in order of their likelihood.
1. The most likely explanation is that there is an error in the string field theory computation of A g . However we have not been able to identify the origin of this error. Furthermore the string field theory results pass various internal consistency checks -the final result is independent of the string field theory parameters λ, α and the function f (β). Some further consistency checks have been described in appendix E. Although not reported here, we have also performed the computation with a different choice of local coordinates for the O-O-O vertex, where we take the local coordinates at the punctures at 0, 1 and ∞ to be αz, α(z − 1) and −α/z respectively and then average over cyclic permutations generated by z → 1/(1 − z). Consequently the C-O-O vertex will also have to be defined by averaging over permutations. However, the final result for A g does not change. We have also used the procedure of §2.4 to reproduce our results. In fact this method can also be used to get the result without assuming α and / or λ to be large, and the final result remains unchanged.
2. Another possibility is that for the D-instanton, the world-sheet theory has an additional contribution that is not related to divergences from the boundary of the moduli space.
This could give an additive constant of 1/2 in the computation of g finite and cancel the value −1/2 of A g that we find, restoring the equality between the string field theory results and the matrix model results. If this is the case, then it will be important to understand what the missing contribution is, since it is likely to affect the D-instanton contribution to other string theories as well.
3. A third possibility is that the matrix model result for the instanton induced scattering amplitude may have an additional term that makes the result agree with the choice (1.14). This would restore the agreement between the results of the world-sheet theory and the matrix model. 4. The final (unlikely) possibility is that the duality between the bosonic string theory and the matrix model does not hold in the form suggested in [4].
We end this paper by listing some other open problems.
2. Matrix model dual to two dimensional string theory is described by free fermions in inverted harmonic oscillator potential. The analysis of [4,10] identified the two dimensional bosonic string theory with the particular vacuum where the exact energy eigenstates representing the scattering states incident from the left are filled up to the fermi level, but the scattering states incident from the right are empty. Refs. [63,64] proposed a different correspondence where they identified two dimensional type 0B theory to the system where both sides are filled up to the Fermi level. It will be interesting to compute Dinstanton induced amplitudes in type 0B theory to verify this. This would also establish that the bosonic string theory and type 0B string theory are different vacua of the same underlying theory, at least in two dimensions.
3. In our analysis we have not fixed the overall normalization constant N that appears in for some constant γ. The sewing relation ww = −q now gives In the z plane the closed string punctures at z = i and z = i are located at, respectively, We now introduce a new coordinateẑ = (z − a)/(1 + a z) for some real constant a. This maps the closed string punctures toẑ Our goal is to adjust a so thatẑ 2 is purely imaginary and then identifyẑ 2 as i y. Since it follows from the analysis of §3 that we need the relation between u and y to the first subleading order in the expansion in powers of u, we can work only to this order. We get Therefore the range 0 ≤ u ≤ λ −2 translates to 0 ≤ y ≤ λ −2 − 2 γ 2 λ −4 . In this range we change variable from y to q = uλ 2 in (5.3) and apply the replacement rule (2.11), and in the rest of the range of integration we carry out the y integration explicitly. This gives the following result for (5.3): where in all the steps we have ignored terms containing negative powers of λ. This differs from (5.5) by the additive term γ 2 . We shall now show that this is cancelled by the ψ 1 exchange contribution to Fig. 3(a).
As usual, in order to avoid normalization issues, it will be simplest to compare the ψ 1 exchange contribution with the tachyon exchange contribution, given by the first term in the second line of (A.6). Since it follows from (6.23) that the ψ 1 propagator is 1/2 times the tachyon propagator, and we see from the first term in the second line of (A.6) that the tachyon exchange contribution is −λ 2 /2, we can express the ψ 1 exchange contribution as: where C Φψ According to (6.26), the amplitude is given by: where ccV T denotes the vertex operator of the closed string tachyon of energy ω, a is the anti-clockwise contour around z a = F a (0; β) and K 1 is a normalization constant. We shall assume that includes a 1/(2πi) factor. F a • ∂c(0) denotes the conformal transform of ∂c(0) by F a .
Let F a (w a , β) have an expansion of the form: Then the locations of the punctures are given by, Furthermore, F a • c(0) and F a • ∂c(0) can be found from (A.9) and (B.3): Let us introduce another constant K 2 via the relation: Then (B.2) takes the form, In arriving at the right hand side of (B.7) we have used the operator product expansion where ∼ denotes equality up to non-singular terms. Now we have, from (4.12), Inverting these relations and expanding them in powers of w a , we get (B.9) Comparing these with (B.3) we get, (B.10) Substituting these into (B.7) and multiplying the result by the ψ 1 propagator K −1 0 /2 obtained from (6.23), we get the contribution to Fig. 9(b): (B.11) We shall now determine the constant K 1 K 2 K −1 0 . For this let us consider the effect of replacing ψ 1 by the open string tachyon in Fig. 9(b). The analysis proceeds as above, except that instead of the vertex operator of ψ 1 , we insert the tachyon vertex operators as given in the first equation of (B.5). The analog of (B.7) now takes the form: Multiplying this by the tachyon propagator K −1 0 and using (B.10), we get the contribution to Fig. 9(b) with ψ 1 replaced by the open string tachyon: On the other hand, the tachyon contribution, computed from the first term in the last line of (6.9), is given by: (B.14) Note that we have put back a factor of 2 g s sinh(π|ω|) that was factored out from (6.1) in defining g(ω). Comparison of (B.13) and (B.14) now gives Substituting this into (B.11) we get the ψ 1 exchange contribution to Fig. 9(b): The contribution from Fig. 9(a) may be analyzed similarly. The main difference is that the local coordinates w a now have a different relation to the coordinate z in the upper half plane.
We have, from (4.9), (4.10), Inverting these relations we can identify the functions F 1 and F 2 : Comparison with (B.3) now gives: Substituting this into (B.7), multiplying this by the ψ 1 propagator K −1 0 /2 and integrating over β, we get the contribution to Fig. (9)(a): where in the last step we have used (B.15), replaced β by q/(2 λ) according to (4.9) and used the replacement rule (2.11), throwing away terms of order 1/ λ.

Adding (B.16) and (B.20)
, and dividing the result by 2 g s sinh(π|ω|), we get the total ψ 1 exchange contribution to g(ω): C Computation related to collective mode redefinition As discussed in §6.3, in order to find the field redefinition that relates the open string zero mode associated with the vertex operator c∂X to the collective mode associated with time translation of the D-instanton, we need to evaluate the 'divergent' part of the contribution to the disk amplitude with a pair of closed string tachyons of energies ω 1 and ω 2 and an open string zero mode field described by the vertex operator c∂X. According to (6.30), the relevant integral is: and we have to evaluate the contribution to this integral from the different regions shown in First we evaluate the contribution to (C.1) from region IV in Fig. 5. There are no singularities from this region and we need to evaluate the integral explicitly. We see from (4.55) that the x integration is unrestricted and the y integration range may be expressed as: In the above we have ignored fractional corrections of order λ −4 . Our strategy for evaluating the integral is to divide region IV into two parts: y ≥ λ −2 , and y in the range ( λ −2 − λ −4 ∆(x), λ −2 ).
Since the latter range is small, we can evaluate the y integral by replacing y by λ −2 in the integrand and then multiplying the integrand by λ −4 ∆(x). If we denote these two integrals by I 1 and I 2 , then we have: On the other hand, we get, Making a change of variable x → λ −2 /x in the second term, and using (4.54), we can express this as where we have ignored fractional errors of order λ −2 . This can be justified as follows.
Next we consider regions III and III . In these regions, we need to change variable from (x, y) to (u, β) according to (4.49), (4.50), and then integrate over the range (4.43). It will be more convenient to trade in the variable β for ξ, and introduce the function φ(ξ) via the relations: so that (4.49), (4.50) may be combined into the equation: In these variables the range (4.43) translates to: We also have ignoring fractional corrections of order higher than λ −2 . This may be justified by noting that after using the replacement rule (3.4), the leading divergent term in the integral (C.1) is of order λ 2 . Therefore fractional corrections of order higher than λ −2 will be suppressed in the large λ limit. Using (C.9)-(C.11) we may now express the contribution to the integral (C.1) from regions III and III as: (C.12) Note that the ξ integration range includes both positive and negative ξ. We now expand the integrand in a Laurent series expansion in u and use the replacement rules: Integrals with non-negative powers of u in the integrand may be carried out explicitly. This gives (C.14) where we have thrown away the terms that would vanish in the large λ limit. Using (C.8) this may be written as: The change of variables in region II can be found in (4.42) and (4.46). Let us first definẽ x = −y/x , (C. 16) and express (4.42) and (4.46) as: where ξ and φ(ξ) have been defined in (C.8). As already discussed in §4, under (C.16) the regions II and II get mapped to III and III . Therefore the original integral (C.1) over regions II and II can be rewritten as follows: (C.18) (C.17), (C.18) have the same form as (C.9), (C.12), with ω 1 ↔ ω 2 and x replaced byx. Therefore this integral can be analyzed in the same way, leading to the analog of (C.15): This gives: Using this, the contribution to the integral from this region may be expressed as: We evaluate this by using the replacement rules: This leads to Finally we note that the ψ 1 exchange contribution to any of the diagrams in Fig. 4(a) This concludes our derivation of (6.31) which was used to find the field redefinition that relates the open string zero mode φ to the collective coordinate φ.
We end this appendix by noting that we can also use the procedure described in §2.4 to evaluate the integral given in (C.1). For example, the integrand of (C.1) may be expressed as: − dx ∧ dy i π ω 1 x 2 + 1 + ω 2 y x 2 + y 2 4 g s sinh(π|ω 1 |) sinh(π|ω 2 |) where, (C.28) Therefore, (C.1) may be represented as a boundary integral of B in Fig. 5, with the boundaries including the segments x = ∞, y = 1, x = −∞, and the regularized boundaries at y = 0, consisting of the segments q 1 = δ 1 in Fig. 4(a) and (b) and q 2 = δ 2 in Fig. 4(a) and (c). String field theory instructs us to drop all terms proportional to δ −1 1 , δ −1 2 , ln δ 1 and ln δ 2 in the final expression. We have checked that this gives the same result as (C.26).

D Computation related to gauge parameter redefinition
As described in §6.4, in order to find the relation between the string field theory gauge transformation parameter accompanying the state |0 , and the rigid U (1) gauge transformation parameter, we need to compute the disk amplitude A(ω) of the closed string tachyon vertex operator of energy ω, and the open string vertex operators I, c∂c∂ 2 c/2 and I. Furthermore, the open string vertex operators carry Chan-Paton factors so that only one cyclic ordering contributes. In this appendix we shall compute A(ω).
We shall use the upper half plane representation of the disk with the closed string tachyon vertex operator ccV T inserted at i and the vertex operators I, c∂c∂ 2 c/2 and I inserted at three points z 1 , z 2 and z 3 respectively on the real axis. The upper half plane with three punctures on the boundary and one puncture in the bulk has two real moduli. Let us denote them by β = (β 1 , β 2 ). Then we have z 1 = f 1 ( β), z 2 = f 2 ( β) and z 3 = f 3 ( β) where f 1 , f 2 , f 3 are three functions of these moduli. As in (B.3), we describe the choice of local coordinates w a at the punctures via the relations: We can now use the generalization of (B.2) to express the C-O-O-O amplitude A(ω) as: where K 3 is a normalization constant and a is the anti-clockwise contour surrounding z a . In writing (D.2) we have made use of the fact that all the vertex operators are dimension zero primary operators, so that conformal transformations by the F a 's leave them unchanged. Using (D.1), (B.6), and evaluating the contour integrals using the b-c operator product expansion, we can express (D.2) as: Next we shall discuss how to fix the orientation of the integral, i.e. given an integral of the form R dβ 1 ∧ dβ 2 over some finite region R, how to decide whether it gives positive or negative result. If we were using the same coordinate system everywhere in the moduli space then this would not be necessary, since an overall sign can always be absorbed inside K 3 . However we shall use different coordinate systems in different regions, and relative signs will be important. 9 For this we shall first make an SL(2, R) transformation to bring the third puncture to 0, keeping the closed string puncture fixed at i. This gives: and Since dz 1 ∧ dz 2 fixes the global orientation of the moduli space of a disk with one closed string puncture and three open string punctures, we need to ensure that the orientations we use in different segments coming from different Feynman diagrams agree with this orientation. We shall take dz 1 ∧dz 2 to be the positive integration measure, i.e. assume that R dz 1 ∧dz 2 for some finite region R gives positive result. The other convention where R dz 2 ∧ dz 1 gives positive result is related to the one used here by a change in the sign of K 3 introduced in (D.2). Once we decide that R dz 1 ∧ dz 2 gives positive result, (D.7) can be used to fix the orientation of the moduli space integration coming from different Feynman diagrams. This will be illustrated below.
We shall now analyze the contribution to A(ω) by expressing this as a sum of four Feynman diagrams shown in Fig. 10. First we shall argue that the contributions from Fig. 10  for some functions φ i (τ ). Furthermore, the local coordinates w a at the open string punctures must be functions of z/u and τ . Therefore we have the expansion: Comparing this with (D.1) we see that we have Substituting this into the first expression in (D.3) we get the form of the integral inside the region covered by Fig. 10(d) to be: where denotes derivative with respect to τ . Let us define ε such that it can take values 1 or −1 and that εdu ∧ dτ gives the integration measure with the correct sign, i.e. ε du ∧ dτ gives positive result when integrated over a finite region in u-τ space. ε can be determined from (D.7), but we shall not need this information.
We now use the replacement rule (2.11) to replace 1/λ 0 duu −2 by −λ. Terms in the integrand with non-negative powers of u can be made zero by taking the large λ limit. This gives the contribution to A(ω) from Fig. 10(d) to be where B 2 is the boundary at u = λ, with its normal along the direction of increasing u.
The contribution from Fig. 10(c) may be analyzed by using the fact that the integrand in (D.3) is a total derivative and therefore the integral may be expressed as boundary integrals. Now the region of the moduli space covered by Fig. 10(c) has two boundaries: B 1 where the moduli space connects to the q 1 = 1 boundary of the region covered by Fig. 10(b), and B 2 where the moduli space connects to the q 2 = 1 boundary of the region covered by Fig. 10(d). Therefore we have (D.14) where First let us evaluate the contribution from the boundary B 2 . For this we can use the u, τ coordinate system introduced in (D.9), and, describing the boundary as the curve u = 1/λ, express the boundary integral as: where we have dropped terms with negative powers of λ. An extra negative sign in (D. 16) comes from the fact that u = 1/λ is the lower limit of integration in the region covered by the diagram 10(c). Adding this to (D.13) we get, It remains to evaluate the contribution from the boundary B 1 . Since this boundary is shared with the boundary of the region associated with Fig. 10(b), we can use the moduli space parameters inherited from this diagram and describe the boundary to be at q 1 = 1, where q 1 and (D. 26) We shall now fix the orientation of the integral. For all three cases, the integration measure (D.7) takes the form, for small u, where F (β) is a positive function of β. Therefore, dβ ∧ du is the positive integration measure.
The results are given, respectively, by: where we have used f ((2 λ) −1 ) = 1/2 + O( λ −2 ) and f (1) = 0, and ignored terms containing inverse power of λ. Using (6.52) we now see that this gives an additional contribution to the closed string tachyon one point function of the from: We shall now determine the ratio and following the same steps as outlined below (D.2), we arrive at the analog of (D.3): For Fig. 10(b), we can use (D.24) to express this as: where we have kept the leading term in the expansion in a power series in u. Note that this is the contribution from one Feynman diagram and we have not summed over cyclic permutations. Furthermore, using (D.27) we see that dβ ∧ du is the positive integration measure. Now using u = {1 + (4 λ 2 ) −1 } −1 q/(2α 2 λ) and the replacement rule (2.11), we can carry out integration over q and express the leading contribution from Fig. 10(b) to the amplitude with one closed string tachyon and three open string tachyons to be: .
(D. 36) On the other hand, it follows from Fig. 10 This concludes the derivation of (6.53).
Finally we would like to note that the integral (D.3), having a total derivative as the integrand, can also be evaluated using the method described in §2.4. This is given as an integral of J over the boundaries of the moduli space, corresponding to the segments q 1 = δ 1 in Fig. 10 (a) and (b) and q 2 = δ 2 in Fig. 10 (a) and (c). Each of these boundary segments actually represent three separate terms, obtained by cyclic permutation of the three external lines. We have checked that the integral, evaluated this way, gives the same final result (D. 31) after dropping all terms proportional to δ −1 1 , δ −1 2 , ln δ 1 and ln δ 2 in the final expression.

E Closed string tachyon one point function on the annulus with a more general C-O-O vertex
In the analysis of the C-O-O amplitude in §4.3, we could include on the right hand side of (4.12) a multiplicative function of β that takes value 1 at β = (2 λ) −1 , but is otherwise arbitrary. In this appendix we shall show that the final result is independent of the choice of this function.
(E.1) The analysis may be easily generalized to the case of finite multiplicative function.
We shall first outline the general strategy for dealing with this problem. Let us suppose where we have labelled the sewing parameter by q to distinguish this from the original sewing parameter q that appeared for ℘(β) = 0. When we express this in terms of the upper half plane coordinates associated with the two vertices, then the ℘(β) dependence appears in the form of a multiplicative factor (1 + ℘) on the left hand side of (E.2). Therefore the sewing relations with the modified vertex may be read out from the original sewing relations by making the identification: The effect of this can be analyzed from the results of §3. As shown there, a scaling of this form does not affect integrals of the form dq q −2 -if we replace q in terms of q using (E. 3) and then use the replacement rule (2.11), we get the same result. However a rescaling of this type does affect integrals of the form dq q −1 . This is because according to (2.11), we are instructed to drop terms of the form Therefore our task is to identify all diagrams in our analysis that contain a C-O-O vertex.
We list them below: 1. Contributions to the C-C-O amplitude from Fig. 4 (b) and (c) involve the C-O-O vertex. This will modify the contributions given in (C.12) and (C.18).
2. Contribution to the annulus one point function given in Fig. 7(c) involves the C-O-O vertex. This, in turn, will modify the contribution given in (6.9).
Besides these, there are two further effects that need to be taken into account: 1. The modification of the C-O-O vertex also affects the ψ 1 exchange diagrams where the internal ψ 1 connects to the O of the C-O-O vertex. This has to be analyzed separately. In this case, besides the effect mentioned above, there will be additional dependence on ℘(β) since one of the external vertex operators involve c∂c∂ 2 c/2 that can connect to one of the O's of the C-O-O vertex. Since this vertex operator is not of the form cW where W is a dimension one matter primary, the integrand will have a complicated dependence on the local coordinates of the C-O-O vertex, and hence on ℘(β). This needs to be taken into account separately.

E.1 Modification of the annulus one point function
The off-shell C-O-O vertex appears in Fig. 7(c). Therefore it modifies the contribution from this diagram as given in (6.9). Since the integration variable u in (6.9) is proportional to the sewing parameter q, the additional term is obtained by replacing duu −1 term by 2℘ ( Here the integration variable u is proportional to q, so according to the algorithm given earlier we simply need to replace du u −1 by ℘(β) to get the extra term.

E.3 Modification of the ψ 1 exchange contribution
We shall now study the effect of modifying the C-O-O vertex on the ψ 1 exchange contribution shown in Fig. 9 1. In (D.24), u is replaced by u(1 − ℘(β)), the expression for g 1 is multiplied by (1 − ℘(β)) and the expression for h 1 is multiplied by (1 − 2℘(β)).
The effect of these changes is to modify (D. 28 We now see that the sum of (E.5), (E.8), (E.12) and (E. 19) vanishes. Therefore the modification of the C-O-O vertex has no effect on g(ω).