Fixing an ambiguity in two dimensional string theory using string field theory

In a recent paper, Balthazar, Rodriguez and Yin found some remarkable agreement between the results of c = 1 matrix model and D-instanton corrections in two dimensional string theory. Their analysis left undetermined two constants in the string theory computation which had to be fixed by comparing the results with the matrix model results. One of these constants is affected by possible renormalization of the D-instanton action that needs to be computed separately. In this paper we fix the other constant by reformulating the world-sheet analysis in the language of string field theory.


Introduction
In a recent paper [1], Balthazar, Rodriguez and Yin found some remarkable agreement between the results of c = 1 matrix model [2][3][4][5] and two dimensional string theory. In their analysis on the string theory side they left two constants undetermined. Our goal in this paper will be to fix one of these constants by carefully repeating their analysis in the language of string field theory. We find that the constant c in the analysis of [1], that was left undermined in the world-sheet analysis, can be determined using the framework of string field theory and takes value: c = − ln 4 −1.38629 .
(1.1) This is within 1% of the numerical value −1.399 that was required for the agreement between the results of world-sheet calculation and the matrix model results. The other constant, called S (1) ZZ in [1], is affected by possible renormalization of the D-instanton action that needs to be computed separately. In particular since the D-instanton contribution to any amplitude is proportional to exp[−c/g s ] for some positive constant c, a multiplicative renormalization of the D-instanton action c/g s by (1 − b g s − a g 2 s ) will generate an overall multiplicative factor of e c(b+a gs) . Therefore contributions proportional to g s multiplying the leading order term will be affected by this renormalization of the D-instanton action. 1 We shall begin in section 2 by describing the analysis of [1] in the language of string field theory, and the origin of the apparent ambiguity in their analysis. We shall then describe in section 3 why in a careful treatment using the framework of string field theory there is no ambiguity. Finally in section 4 we shall use the machinery of string field theory to fix the undetermined constant c .
Before we turn to the details, let us first make a general remark on why string field theory is useful for fixing the ambiguity described in [1]. The ambiguity arises from having to cancel the divergences between two diagrams, each of which is separately infrared JHEP03(2020)005 divergent. Therefore we need to regulate the diagrams before combining them, and since a priori the regulators can be chosen independently, we can be left with a finite term after the cancellation that depends on the finite parts of each regulated diagram. This is precisely the kind of situation where string field theory is useful. String field theory makes clear the physical origin of the divergences and therefore gives precise relation between the regulators in different diagrams, leaving behind no scope of an ambiguity except those induced by field redefinition.
2 Formulation of the problem in the language of string field theory In this section we shall interpret the analysis of [1] in the language of string field theory. The main goal of [1] was to compute the contribution of a single D-instanton to the scattering of massless closed strings. Since the fluctuations of D-instanton are described by open strings with ends on the D-instanton, the relevant string field theory is that of open and closed strings [6], with the open strings living on the D-instanton. Since our ultimate goal is to compute closed string scattering amplitudes, it would be natural to integrate out the open string degrees of freedom and write an effective action for closed string fields only. However on the D-instanton world-volume there is a zero mode, describing the freedom of translating the D-instanton along the time direction. This corresponds to a particular mode of the open string with vanishing propagator. For this reason, this mode cannot be integrated out at the beginning. The best we can hope is to integrate out all the open string modes other than the zero mode. The corresponding action still satisfies the BV master equation and was described as the Wilsonian effective action in [7].
Let us denote by φ the zero mode of the open string -called x 0 in [1]. It was argued in [1] that the φ dependence of the Wilsonian action has a particularly simple form. The sum of the terms in the Wilsonian action with a fixed set of external closed string fields but arbitrary number of φ modes is proportional to n (iωφ) n /n! = e iωφ were ω is the total energy of all the closed string fields. If we integrate over φ after computing the Green's function of a set of closed string fields in some background φ, we generate a factor proportional to δ(ω tot ), -ω tot being the total energy of all the closed string states in the Green's function. This restores energy conservation, which is otherwise broken for a fixed D-instanton configuration. From this we can compute the S-matrix of closed string states by restricting the external closed string states to be on-shell.
We are now in a position to describe the origin of the apparent ambiguity encountered in [1] in the language of string field theory. The stems from the fact that the string field theory action is not unique. The construction of the action requires the choice of local coordinates at the punctures where vertex operators are inserted, and different choices lead to apparently different string field theories. As we shall elaborate in section 3, the apparent ambiguity encountered in [1] can be traced to this ambiguity. However, it follows from a general analysis in [8], after suitable generalization to the field theory of open and closed strings, that the ambiguity described above can be regarded as a result of field redefinition. Therefore it is not expected to change any of the physical results. We shall see in section 3 how the ambiguity in the result of [1] is actually resolved.

String field theory as regulator
We shall begin with some simple examples that illustrate how string field theory can be used to systematically analyze the infrared divergences in string theory, arising as ultraviolet divergences in the world-sheet theory [9,10]. Let us consider a unit disk with a bulk puncture at the centre and a boundary puncture at the point z 1 on the boundary of the disk. Using rotational symmetry we can fix z 1 to 1, but we shall keep it arbitrary for later use. For defining off-shell amplitudes we need to choose local coordinates at both punctures, with the property that the local coordinate vanishes at the puncture, and that the local coordinate around a boundary puncture must take real values on the boundary. Let us for definiteness choose the local coordinate w at the bulk puncture to be βz and the local coordinate w 1 at the boundary puncture to be for some real positive constants β and λ. Note that (3.1) gives real w 1 as long as z 1 and z are on the boundary of the unit disk. It is convenient to take λ and β to be large -in the language of string field theory this is described as adding long stubs to the vertices [8].
Once a choice of local coordinates is made, we can compute a disk amplitude with an offshell closed string vertex operator inserted at the center and an off-shell open string vertex operator inserted at the boundary at the point z 1 . This corresponds to an interaction vertex of string theory with one open string and one closed string. Now consider a Feynman diagram where we take two such vertices and join the open string states by a propagator. This has been shown in figure 1 with closed strings represented by thick lines and open strings represented by thin lines. This describes a disk amplitude with two external closed strings. However this does not cover the full moduli space of the disk with two bulk punctures. Instead, it covers part of the moduli space that corresponds to the surface Figure 2. This diagram represents two other contributions, besides the one shown in figure 1, to the disk two point function of a pair of closed strings. The first diagram represents the two point closed string interaction vertex described by a disk amplitude with two closed strings, and the second diagram represents a Feynman diagram that joins the closed string three point vertex to a closed string one point vertex by a closed string propagator. Since in our analysis the second diagram will not give a divergent contribution, we shall include its contribution in the definition of the interaction vertex of two closed strings shown in the first diagram.
where z andz are the coordinates on two unit disks and are the local coordinates around the boundary punctures on the two disks. The identification (3.2) joins the two disks into a single disk. If we parametrize this by the coordinate z, then the bulk punctures at z = 0 andz = 0 are situated at In string field theory the parameter q introduced in (3.2) appears as follows. An open string propagator contains a 1/L 0 factor, which is expressed as where u and are the variables introduced in (3.2). Upon substituting this into the expressions for the Feynman diagram we can recover the geometric picture where connecting a pair of interaction vertices by a propagator corresponds to sewing the surfaces via the relation (3.2). Note however that (3.4) is valid only for states with positive L 0 eigenvalue. For L 0 < 0, the right hand side diverges but the left hand side is finite. Therefore while using the right hand side to find a geometric interpretation of the amplitude is problematic, we can always use the left hand side. Equivalently, whenever we encounter an expression of the form given in the right hand side of (3.4), we use the replacement rule: Note that this does not work for α = 0. We shall return to this issue shortly.

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As discussed in [10], as long as α is not an integer, the rule (3.5) is invariant under a change of variables u → v = a 1 u + a 2 u 2 + · · · that is regular at u = 0. Therefore given a divergent integral, we can apply this rule blindly without knowing if the integration variable is actually the sewing parameter u introduced in (3.2). However for negative integer α, application of this rule using a different variable v could yield an expression that differs from the original expression by a constant. Therefore in this case we need to identify the variable u first before applying the rule.
In Next let us consider an interaction vertex with two external open strings and one external closed string, again associated with the disk amplitude. Using appropriate SL(2, R) transformation we take the bulk puncture at the origin of the disk. The boundary punctures then lie at points z 1 and z 2 on the disk. We now need to choose local coordinates at the punctures. Let w =βz describe the local coordinate around the bulk puncture, and be the local coordinates around the boundary punctures. Hereβ andλ are some large positive constants that are a priori independent of the constants β and λ appearing in (3.1). The choice (3.6), although not unique, is consistent with various symmetry requirements, e.g. symmetry under the cyclic permutation of the two open strings (which in this case is simply implemented by overall rotation of the z-plane). The choice ofβ will not affect any of our results since in our analysis the closed string inserted at the vertex will always be on-shell. The ambiguity mentioned in [1] is related to the freedom of choosingλ to be independent of λ. As we shall now explain however, the choiceλ = λ is in conflict with another hidden assumption that was used in the analysis of [1]. Using the fact that the vertex operator of the state φ is proportional to ∂X 0 where X 0 is the world-sheet scalar field associated with the time coordinate, it is easy to see that after integration over z 2 , the contribution to the amplitude with an external open string zero mode field φ is given by iωφ times the amplitude where the external open string state is removed. 2 Here ω is the total energy carried by the external closed strings. However over certain subspace of the moduli space, where the second bulk puncture is located at z 1 (1−u)/(1+u) with 0 ≤ u ≤ (λλ) −1 , the amplitude will still have missing internal φ state. On the other hand the contribution to the term in the Wilsonian effective action without any external φ insertion is given by a similar term, except that the subspace of the moduli space, over which the internal φ contribution is removed, corresponds to the second bulk puncture being located at z 1 (1 − u)/(1 + u) with 0 ≤ u ≤ λ −2 (see (3.2), (3.3)). Ifλ = λ, then these two subspaces are identical, and we can express the sum of the terms with and without external φ as (1 + iωφ) times the term without external φ. This gives the first two terms in the expansion of e iωφ . Requiring that the terms with higher powers of φ add up to the result exp[iωφ] would demand using the same parameter λ in defining the local coordinates at the boundary punctures in the vertices with one closed and multiple open strings.
Since [1] used exp[iωφ] as the φ dependence of the Wilsonian effective action, we see that we do not have the freedom of choosingλ independently of λ in (3.6). Instead we must choose:λ = λ .  figure 4. Geometrically this describes an annulus amplitude with one external closed string. Naively, in order to evaluate this contribution, the u parameter associated with the sewing has to be integrated from 0 to . However intermediate tachyons need special treatment, and we need to remove the contribution from the state φ in the internal open string propagator. Following our earlier discussion, this leads to the following prescription for computing the full Wilsonian effective action. In the expression for the full annulus amplitude, expressed as integrals over the independent moduli parameters u and z 2 /z 1 , we need to impose a lower cut-off = λ −2 on the u integral and add a compensating term using the replacement rule (3.5) for α = 1. This would correspond to integrating out the tachyonic mode in figure 4, but leaving the zero mode unintegrated. Integration over the zero mode will have to be taken care of at the end.
To summarize, we need to augment the world-sheet analysis of [1] using the following procedure for regulating divergences: 1. When we encounter a divergence, we need to identify the degeneration responsible for the divergence, and change the integration variables to the analog of the parameter u appearing in sewing relations of type described in (3.2) and the moduli of the Riemann surfaces that are being sewed.
2. If near u = 0 the integrand has a term of the form A du u −2 for some u independent A, we replace the lower limit of integration by , and subtract A −1 from the integral. The subtraction of A −1 represents the effect of 'integrating out' the tachyon field in the Wilsonian effective action.
3. If near u = 0 the integrand has a term of the form B du u −1 for some u independent B, we replace the lower limit of u integration by without adding any compensating term. A term proportional to du u −1 in the integrand represent the effect of zero mode exchange, and we cannot integrate out these modes.

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4. Integration over the open string zero modes is performed at the end after computing the desired Green's function/S-matrix. This produces the energy conserving delta function.

Cancellation of the extra terms
In this section we shall implement the general principles described in section 3 and determine the constant that was left unfixed in the world-sheet analysis of [1]. We shall begin by describing the strategy that we shall follow. The analysis of [1] involved addition of some extra terms to the result of the worldsheet computation on the string theory side. These terms were used to regulate the infrared divergences, and are given in the second, fourth and fifth lines of eq. (2.36) of [1]. They take the following form: where N is a normalization constant, and S (1) ZZ and c are constants which were eventually adjusted to make the amplitude agree with the matrix model result. Since our goal will be to show that string theory results agree with the matrix model results without any ad hoc adjustment, we need to show that (4.1) vanishes after we use the regulator implied by string field theory. 3 The terms inside the square bracket in (4.1) can be divided into two classes: the ones proportional to sinh 2 (πω) and the ones proportional to ω 2 sinh 2 (πω). The leading term (not displayed here) is also proportional to sinh 2 (πω). Therefore, renormalization of the D-instanton action, mentioned in the first paragraph of the introduction, will change the coefficient of the term proportional to sinh 2 (πω). The coefficient of the ω 2 sinh 2 (πω) term cannot be changed this way, and so we focus on that term. Changing the integration we can express the terms proportional to ω 2 sinh 2 (πω) inside the square bracket as:  3 The logic goes as follows. Since the counterterms (4.1) have been chosen to cancel the divergences in the world-sheet result, if we regulate both the original world-sheet results and the counterterms following the prescription given by string field theory, the result remains unchanged, But now the regulated world-sheet integrals give the correct result. Therefore the regulated counterterms must add up to zero.

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The first integral is divergent at y = 0 while the second integral is divergent at v = 0. Our goal now will be to identify the source of these divergences to appropriate degenerations of Riemann surfaces, find the relation between the integration variables in (4.4) and the sewing parameter u in (3.2) or (3.8) and possibly other moduli of the Riemann surfaces, and then translate the cut-off procedure on u prescribed by string field theory to a cut-off on y and v.
We begin with the y integral in the first term in (4.4). This term was chosen to remove the divergences in the two point function on the disk, with the closed string vertex operators inserted at i and i y in the upper half plane [1], which we shall label by w. If we denote by z the coordinate on the unit disk, related to w via z = i i−w i+w , then in the z coordinate the punctures are at z = 0 and z = i(1 − y)/(1 + y). Therefore the y → 0 limit corresponds to one of the closed string vertex operators coming close to the boundary point z = i. Comparing this with (3.3) we see that this corresponds to the degeneration of an open string propagator joining two disks associated with the Feynman diagrams shown in figure 1, with the identification: (4.5) Therefore the cut-off u > translates to y > . Next we turn to the divergence of the second term in (4.4) in the v → 0 limit. In the analysis of [1] this comes from the one point function of a closed string state on the annulus, parametrized by coordinate w, satisfying, with the bulk puncture located at w = 2πx. Here ∼ denotes identification of points under the given transformation. Our goal will be to compare this configuration with the one associated with the diagram shown in figure 4. The latter is parametrized by the sewing parameter u appearing in (3.8), and the ratio z 2 /z 1 , with z 1 , z 2 labelling the locations of the boundary punctures on the disk in the open-open-closed interaction vertex. Once we have determined the relation between (x, v) and (z 2 /z 1 , u), we can translate the cut-off u > into a cut-off on the (x, v) integration and carry out the integration over x and v. We begin by recalling the choice of local coordinates around the two boundary punctures in the open-open-closed interaction vertex: 4 and the sewing relation We now introduce the coordinatew = i z 1 − z z 1 + z , (4.9)

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taking value in the upper half plane, and define e 2iθ ≡ z 2 z 1 . (4.10) In thew coordinate the sewing relation (4.8) takes the form w ∼ u 1/2 sin θw + u 1/2 cos θ −u −1/2 cos θw + u −1/2 sin θ . (4.11) For small u, the right hand side of (4.11) is an SL(2, R) transformation with a hyperbolic element. We can diagonalize this by an SL(2, R) transformation a b c d such that the identification (4.11) takes the formŵ ∼ βŵ . (4.14) We shall not give the explicit form of β and a b c d for general u, since we only need this for small u. In the small u limit, Furthermore the identification (4.14) with β defined in (4.15) gives w ∼ w + i ln β = w + i ln(u −1 sin 2 θ) . Comparing this with (4.6) we get v = u/ sin 2 θ . (4.20)

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It remains to locate the position of the bulk puncture in the w plane since its real part gives the value of 2πx, -due to translational invariance of the annulas, Im w does not carry any physical information. Since in the z-plane the bulk puncture is located at z = 0, we see from (4.9) that in thew plane it is atw = i. (4.16) now gives its location at w = cot θ + i = e iθ / sin θ and therefore at w = θ + i ln sin θ. Since 2πx is the real part of w, we get 2 π x = θ .
(4.21) (4.20) and (4.21) now shows that the cut-off u ≥ can be translated to 5 v ≥ / sin 2 (2πx) .  This is within 1% of the numerical value −1.399 determined in [1] by comparing the string theory results with the matrix model results.