Hyperbolic Geometry and Closed Bosonic String Field Theory I: The String Vertices Via Hyperbolic Riemann Surfaces

The main geometric ingredient of the closed string field theory are the string vertices, the collections of string diagrams describing the elementary closed string interactions, satisfying the quantum Batalian-Vilkovisky master equation. They can be characterized using the Riemann surfaces endowed with the metric solving the generalized minimal area problem. However, an adequately developed theory of such Riemann surfaces is not available yet, and consequently description of the string vertices via Riemann surfaces with the minimal area metric fails to provide practical tools for performing calculations. We describe an alternate construction of the string vertices satisfying the Batalian-Vilkovisky master equation using Riemann surfaces endowed with the metric having constant curvature $-1$ all over the surface.


Introduction
String field theory is the quantum field theory that describes the dynamics of interacting strings. The perturbative amplitudes computed in this theory by evaluating the Feynman diagrams agree with the string amplitudes calculated using the standard formulation of string perturbation theory whenever the latter are finite [1][2][3]. They formally agree with the standard string amplitudes whenever the latter are infected by infrared divergences arising from the infrared effects such as mass renormalization and tadpoles. Compared with the standard covariant formulation of string theory, covariant string field theory has the following advantages. Unlike the standard formulation of string theory, string field theory provides standard quantum field theory techniques for taming the infrared divergences and compute unambiguously the S-matrix elements that are free from divergences [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. Furthermore, this S-matrix can be shown to be unitary [19][20][21][22]. Since string field theory is based on a Lagrangian, it also has the potential to open the door towards the non-perturbative regime of string theory [23], even though no one has succeeded in studying the non-perturbative behaviour of closed strings using closed string field theory yet [24,25].
Due to the complicated gauge structure, the quantization of closed string field theory needs the sophisticated machinery of Batalian-Vilkovisky (BV) formalism [26][27][28][29][30][31][32]. The BV formalism requires the introduction of an antifield for each field in the theory and finding the master action which is a functional of both the fields and the antifields that is a solution of the quantum BV master equation. The perturbative solution of quantum BV master equation for the closed bosonic string field theory in closed string coupling has been constructed [3]. This construction requires constructing the string vertices which satisfy the geometrical realization of BV master equation. Each string vertex contains a collection of string diagrams with specific genus and number of punctures, describing the elementary interactions of closed strings. The string vertices satisfying the BV master equation can be used to construct a cell decomposition of the moduli space of Riemann surfaces. The prominent feature of this cell decomposition is that all the string diagrams belong to a specific cell can be associated with a unique Feynman diagram. Such a decomposition for the moduli space can be achieved by using Riemann surfaces endowed with metric that solves the generalized minimal area metric problem [3].
The generalized minimal area problem asks for the metric of least possible area under the condition that all nontrivial closed curves on the surface be longer than or equal to some fixed length, conventionally chosen to be 2π. A Riemann surface endowed with minimal area metric has closed geodesics of length 2π that foliate the surface. These geodesics form a set of foliation bands. Foliation bands are the annuli foliated by the homotopic geodesics.
The shortest distance between the boundaries of the foliation band is defined as its height.
If the surface has no finite height foliation of height bigger than 2π, then the whole string diagram corresponds to an elementary interaction. Therefore, the set of all inequivalent genus g Riemann surfaces with n punctures endowed with minimal area and no closed curves having length less than 2π and no finite height foliation of height bigger than 2π is defined as the string vertex V g,n .
Unfortunately, a concrete description of the minimal area metric is available for genus zero Riemann surfaces. There, the minimal area metrics always arise from the Jenkins-Strebel quadratic differentials [34]. In the case of higher genus Riemann surfaces, in contrast with metrics that arise from Jenkins-Strebel quadratic differentials, where the geodesics (horizontal trajectories) intersect in zero measure sets (critical graphs), the minimal area metrics can have bands of geodesics that cross. Therefore, for higher genus, the minimal area metric is not the same as the metric that arises from the Jenkins-Strebel quadratic differentials. Moreover, beyond genus zero, a fairly concrete description is available only in terms of the structure of the foliations by geodesics that is expected to exist. Even at genus zero level, the explicit construction of Jenkins-Strebel quadratic differentials is a daunting task [35]. Moreover, a rigorous proof of the existence for such metrics is not yet available. Consequently, at present, the formulation of closed string field theory based on Riemann surfaces endowed with metric solving the generalized minimal area problem is not well suited for performing computations in closed string field theory.
In this paper, we describe an alternate construction of the string vertices using the Riemann surfaces with metric having constant curvature −1 all over the surface. As we will discuss in this paper and the follow up papers, in contrast with the theory of Riemann surfaces endowed with the minimal area metric, the theory of Riemann surfaces endowed with hyperbolic metric is sufficiently developed for providing a calculable formulation of the closed string field theory.
Every genus-g Riemann surface with n distinguished punctures subject to the constraint 2g + n ≥ 3 admits a hyperbolic metric. Such surfaces, known as hyperbolic Riemann surfaces, can be obtained by the proper discontinuous action of a Fuchsian group on the Poincaré upper half-plane [58]. The Fuchsian group is a subgroup of the automorphism group of the Poincare metric on the upper half-plane. Furthermore, the theory of the moduli space of the hyperbolic Riemann surfaces is well suited for performing integrations over the moduli space [43].
The string vertex that corresponds to an elementary vertex of the closed bosonic string field theory with g loops and n external legs can be naively defined as the set of n punctured genus g Riemann surfaces endowed with a metric having constant curvature −1 and having no simple closed geodesic of length less than an infinitesimal parameter c * . The surface obtained by the plumbing fixture of surfaces belong to the naive string vertices can be associated with a unique Feynman diagram. These naive string vertices together with the Feynman diagrams only provide an approximate cell decomposition of the moduli space, with a slight mismatch between the adjacent cells. The size of the mismatch is shown to be of the order c 2 * . A systematic algorithm for improving the naive string vertices perturbatively in c * is proposed.
Following this algorithm, the string vertices with leading order corrections are obtained. The leading order corrected string vertices together with the cells associated with different Feynman diagrams obtained by the plumbing fixture of surfaces belong to the naive string vertices provide a cell decomposition of the moduli space having no mismatch up to the order c 2 * . These improved string vertices can be used to build a consistent closed bosonic string field theory by keeping the parameter c * very small. This construction closely follows the construction of gluing compatible 1PI regions inside the moduli space needed for defining the off-shell amplitudes in string perturbation theory [36]. The essential difference is that the 1PI region inside the moduli space include degenerate Riemann surfaces with non-separating degenerations, unlike the string vertices which do not include any degenerate Riemann surface. Therefore, the string vertices has more boundaries compared to the gluing compatible 1PI regions, and consequently the string vertices are needed to satisfy more stringent conditions than the gluing compatible 1PI regions.
This paper is organized as follows. In section 2, we briefly review the Batalian-Vilkovisky quantization procedure. In section 3, we review the general construction of the quantum BV master action for the closed string field theory. In section 4, we discuss the geometrical identity satisfied by the string vertices. In section 5, we present a short discussion of the hyperbolic Riemann surfaces and the construction of the naive string vertices using them. In section 6, we check the consistency of these naive string vertices and find that together with the Feynman diagrams, they fail to provide the exact cell decomposition of the moduli space. In the last section 7, we describe a systematic procedure for correcting the naive string vertices defined using the hyperbolic Riemann surfaces and find explicitly the leading order correction to the naive string vertices. 2 Brief Review of the Batalian-Vilkovisky Formalism In this section, we present a brief review of the BV formalism. The construction of an arbitrary gauge theory based on a Lagrangian requires specifying the basic degrees of freedom and gauge symmetries. The next step is to construct the action having the specified gauge structure.
Finally, quantize the theory by gauge fixing the path integral. The gauge group of the theory chooses the minimal procedure that is required for the quantization. For simple gauge groups, like the unitary groups, we can quantize the theory using a relatively simple quantization procedure such as Fadeev-Popov quantization method. However, the gauge group associated with the closed string field theory, namely the homotopy Lie algebra L ∞ , endows it with all the features of the most general gauge theory with a Lagrangian description. Therefore, the quantization of such a gauge theory requires the sophisticated machinery of the BV formalism [26][27][28][29][30][31][32].
The most studied examples of gauge theories are the non-Abelian Yang-Mills theories with simple gauge groups. The gauge transformations of such theories form a simple Lie groups and have the following properties: • The commutators of the generators of the Lie group can be expressed as a linear combination of the generators of the Lie group.
• The coefficients of the resulting expression, called the structure constants of the algebra, are literally constants.
• The algebra of the Lie group is associative and satisfies the Jacobi Identities.
• All of the above statements are true irrespective of whether the field configuration satisfies the classical equations of motion or not.
A general gauge theory can have more flexible gauge group structure. We are free to allow the following generalizations: • The structure constants can be made to depend on the fields involved in the theory with appropriately modified Jacobi Identities.
• The gauge transformations itself may have further gauge invariance that make it a reducible system (see below for the definition of reducible systems).
• Two successive gauge transformations can be be allowed to produce another gauge transformation plus a term that vanishes only on-shell.
Consider an arbitrary gauge theory with m 0 number of gauge invariances whose gauge transformations are not invariant under any other gauge transformation. At the classical level, we need to introduce a ghost field for each of the m 0 gauge invariances. Assume that the gauge theory also has m 1 gauge transformations that keep the m 0 gauge transformations invariant. Suppose that these m 1 gauge transformations are not invariant under any further transformations. We call such a gauge theory a first-stage reducible gauge theory. In such theories we need to add m 1 ghost for the ghost fields. Therefore, a general L th -stage reducible gauge theory with N gauge fields φ i has the following set of fields Φ i , i = 1, ..., N where C αs s denotes a ghost field in the theory. With each of these fields let us assign a conserved charge, which we call the ghost number, as follows. The gauge field φ i has the ghost number zero and the ghost field C αs s has the ghost number Similarly, we can assign a statistics for each of the ghost fields. The statistics (ǫ) of the ghost field C αs s is given by where ǫ αs is the statistics of the level-s gauge parameter. To quantize a general L th -stage reducible theory, one has to use the BV quantization procedure. The first step in the BV formalism is the introduction of a set of antifields Φ * i for each set of the fields Φ i . The assignment of the ghost numbers and the statistics of the antifields are as follows Note that a field and its corresponding antifield have opposite statistics. The second step is the construction of the classical master action S[Φ, Φ * ]. The classical master action is a functional of the fields and the antifields. The ghost number of the classical action must be zero and its Grassmanality must be even. The classical master action is required to satisfy the following equation known as the classical BV master equation: where {, } denotes the antibracket, the subscript r denotes the right derivative and l denotes the left derivative. The left and right derivatives are defined as follows Assume that X and Y are two functionals of the fields Φ i and the antifields Φ * i with the statistics ǫ X and ǫ Y . Then the anti-bracket {·, ·} is defined as The action of the left and the right derivatives on the functional X are related to each other as follows However, only those solutions of the classical master equation (2.5) that satisfy the following set of regularity conditions can be considered as the classical master action S[Φ, Φ * ]: • The classical master action should reduce to the classical action of the gauge theory upon setting all the antifields to zero. This condition is needed to ensure that we will get back the correct classical limit.
• The classical master action should allow the consistent elimination of all the antifields Φ * . This is needed because antifields are auxiliary fields and they should not be able to make any contribution to the physical observables in the theory. The usual BRST formalism allows the gauge fixed action to have a residual gauge symmetry (the BRST symmetry), whose action is a graded derivation that is nilpotent. Similarly, the BV formalism also allows the gauge fixed action to have a residual gauge symmetry (the generalized BRST symmetry), whose action is a graded derivation that is nilpotent. The proper solution of the classical BV master equation has a generalized BRST symmetry even after gauge-fixing.
The generalized BRST transformation, δ B , of a functional X of fields and antifields generated by a proper solution S is given by The classical master action S is invariant under this transformation due to the classical BV master equation. It is straightforward to check that δ 2 B = 0. Therefore, all the classical observables belong to the cohomology of δ B .
Consider the classical master action S of a gauge theory. For any function of Υ of fields, it is straightforward to verify that, the deformed action also satisfies the classical BV-master equation, where ǫ is an arbitrary parameter and Υ is a fermionic functional only of the fields. Using this freedom, we can gauge fix the antifields to ∂Φ , and get rid of the antifields altogether. Finally, we quantize the classical gauge theory by considering the partition function It is important to make sure that physical quantities of the theory do not depend on the choice of the gauge fixing function Υ. This is true only if we demand that S, the quantum master action, satisfies the the quantum BV-master equation given by

The Quantum BV Master Action
In this section, we review the construction of the quantum master action for the closed string field theory following the Zwiebach's seminal work [3]. This master action contains a kinetic term for the string field and infinite number of interaction vertices.

The Worldsheet CFT
The closed bosonic string theory is formulated in terms of a conformal field theory (CFT) defined on a Riemann surface. The worldsheet CFT, describing the propagation of a closed bosonic string, can be divided into two sectors. They are the matter sector and the ghost sector.
The matter CFT has the central charge (26,26)   The non vanishing anti-commutation relations of these modes are the following {b n , c m } = {b n ,c m } = δ m+n,0 (3.14) The closed bosonic string theory has a family of SL(2, C)-invariant vacua |1, p labeled by the eigenvalue of the momentum operatorp. The family of vacua |1, p is annihilated by b n ,b n for n ≥ −1 and c n ,c n for n ≥ 2. The dual vacua 1, p| is annihilated by b n ,b n for n ≤ 1 and c n ,c n for n ≤ −2. The vacua and the dual vacua are related each other by the inner product where c ± 0 and b ± 0 are defined as follows The family of vacua |1, p is also annihilated by the first quantized ghost number operator G given by Note that the vacuum |1, p is not annihilated by the modes c 0 , c 1 , c −1 . As a result, the state space breaks into four sectors that are built on the top of the following four different vacua: The BRST operator for the worldsheet CFT has the following expression where T m (z) and T m (z) denote the stress tensors of the holomorphic and anti-holomorphic sectors of the matter CFT and T g (z) and T g (z) denote the stress tensors of the holomorphic and anti-holomorphic sectors of the ghost CFT. The stress tensors of the holomorphic and the anti-holomorphic sectors of the ghost fields are given by: Using the operator product expansion we can verify that These equations imply that where the dots indicate the terms that do not involve zero modes of the ghost fields and Here L n and L n denote the total Virasoro generators in the left and right moving sectors of the worldsheet theory. The Virasoro generators are the modes of the mode expansion of the total energy-momentum tensor (i.e. matter plus ghost) given by

The Fields and the Antifields
The basic degrees of freedom in the closed string field theory are the closed string fields. An arbitrary closed string field is an arbitrary vector in the Hilbert space H of the closed string worldsheet CFT and can be expressed as an arbitrary linear superposition of the basis states: where the set of states {|Φ s } forms a basis for the Hilbert space H. Each target space field ψ s is a function of the target space coordinates. It is the component of the vector |Ψ along the basis vector |Φ s . The ghost number of a component of the string field is declared to be the ghost number of the corresponding first quantized state. Each target space field ψ s entering into the string field as |Φ s ψ s are assigned a target space ghost number defined by where G s is the ghost number of the state |Φ s . The Grasmmanality of the string field |Ψ is declared to be even. Since the Grassmanality of the vacua |1, p is also declared to be even, the Grassmanality of the state |Ψ s is same as the Grassmanality of the CFT operator Ψ s that creates the state by acting on the vacumm.
The string fields that enter into the BV master action of the closed string field theory are called the dynamical string fields and are required to satisfy the following conditions: • They must be annihilated by both b − 0 and L − 0 : This is necessary to make the closed string field theory action invariant under the local Lorentz transformations on the worldsheet. This imply that the dynamical string fields can be expanded as where |φ s , ↓↓ denotes the subset of the basis states |Φ s which are built on the vacuum | ↓↓ and |φ s , ↑↓ ψ s denotes the subset of the basis states |Φ s which are built on the vacuum | ↑↓ .
• They must satisfy the following reality condition: Here superscript dagger denotes the Hermitian conjugation and Ψ| denotes the BPZ conjugate state.
Given a state |Φ = Φ(0)|1 , one defines the associated BPZ conjugate state to be where I denotes the conformal mapping I(z) = − 1 z . The first step in the BV formalism is the specification of the fields and the antifields in the theory. The fields and antifields are specified by splitting the dynamical string field Ψ as The string field |Ψ − contains all the fields and the string field |Ψ + contains all the antifields.
Both |Ψ − and |Ψ + are annihilated by b − 0 and L − 0 . They have the following decomposition The sum in (3.33) extends over the basis states |Φ s with ghost number less than or equal to two. The prime over the summation sign reminds us that the sum is only over those states that are annihilated by L − 0 .
The target space field ψ * s is the antifield that corresponds to the field ψ s . The target space ghost number of the fields g t (ψ s ) takes all possible non-negative values and that of antifields g t (ψ * s ) takes all possible negative values. The target space ghost numbers of a field and its antifield are related via the following relation (see 2.4) Therefore, the statistics of the antifield is opposite to that of the field, as it should be. Since the state |Ψ − and the state |Ψ + are annihilated by the b − 0 mode, half of the fields appear along the states built on the | ↓↓ vacuum and the other half along the states built on | ↑↓ vacuum. It is straightforward to verify that the field corresponding to a state that is built on | ↓↓ vacuum is always paired with the antifield corresponding to a state built on the | ↑↓ vacuum, and vice versa.

The Master Kinetic Term
The kinetic term for the classical closed bosonic string theory is given by [3]: where g s denotes the closed string coupling. The string fields appearing in classical kinetic term are allowed to have only ghost number 2. Due to (3.28), this action is Hermitian. The master kinetic term satisfying the classical master equation is given by the same expression for classical kinetic term (3.35). The only difference is that the string fields appearing in the master kinetic term can have any ghost number. It is straightforward to check that by simply setting all the antifields to zero, we recover the classical kinetic term from the master kinetic term.

The String Field Interaction Vertices
The conventional formulation of the perturbative string theory computes the g loop contribution to the scattering amplitude of n closed string states by integrating the string measure 6g−6+2n over M g,n , the moduli space of genus g Riemann surfaces with n punctures. The basic intuition behind closed string field theory is that perturbative expansion of any amplitude in the closed string theory can be constructed by joining the elementary interaction vertices in string field theory and the propagators using the usual Feynman rules, just like in any quantum field theory. Therefore, we need to construct the propagators and the elementary interaction vertices for string field theory.
We identify the integration of the string measure over a set of cylinders as the string propagator. Therefore, it is natural to identify the integration of the string measure over the region inside the moduli space M g,n in which one can not find any Riemann surface having regions that can be identified with the cylinders used for constructing the string propagator as the g loop elementary interaction vertex with n external string states. Let us denote this region inside the moduli space M g,n as V g,n . Hence, the g-loop elementary interaction vertex {Ψ 1 , · · · , Ψ n } g for n closed string fields can be defined as the integral of the off-shell string measure Ω (g,n) 6g−6+2n (|Ψ 1 , · · · , |Ψ n ) over the string vertex V g,n : where Ψ 1 , · · · , Ψ n denotes the off-shell closed string states |Ψ 1 , · · · , |Ψ n .
The off-shell string measure Ω (g,n) 6g−6+2n (|Ψ 1 , · · · , |Ψ n ) can be constructed using the vertex operators of arbitrary conformal dimension. Remember that the integrated vertex operator having conformal dimension zero represents a state satisfying classical on-shell condition. Hence, the off-shell string measure depends on the choice of local coordinates around the punctures on the Riemann surface. As a result, the integration measure of an off-shell amplitude is not a genuine differential form on the moduli space M g,n , because this space does not know about the various choices of local coordinates around the punctures. Instead, we need to consider it as a differential form defined on a section of a larger space P g,n . This space is defined as a fiber bundle over M g,n . The fiber direction of the fiber bundle π : P g,n → M g,n contains the information about all possible choices of local coordinates around the n punctures on a genus g Riemann surface. If we restrict ourselves to the dynamical string fields (see (3.28)), then we can consider the differential form of our interest as a form defined on a section of the space P g,n . This space is smaller compared to the space P g,n . We can understand P g,n as a base space of the fiber bundle π : P g,n → P g,n with the fiber direction contains the information about different choices of local coordinates around each of the n punctures that differ by only a phase factor.
Let us discuss the construction of a p-form on a specific section of the space P g,n . The section of our interest corresponds to the choice of a specific set of local coordinates around the punctures for each point R ∈ M g,n . Therefore, we need to only worry about the tangent vectors of P g,n that corresponds to the tangent vectors of the moduli space. They are given by the Beltrami differentials spanning the tangent space of the moduli space of Riemann surfaces ( [57]). Consider p tangent vectors V 1 , · · · , V p of the section of the space P g,n and an operatorvalued p-form B p , whose contraction with the tangent vectors V 1 , · · · , V p is given by Here µ k denotes the Beltrami differential associated with the moduli t (k) of the Riemann surfaces belong to the section of the fiber space P g,n in which we are interested. The p-form on the section can be obtained by sandwiching the operator valued p-form, B p , constructed using (3.37), between the surface state R| and the state |Φ built by taking the tensor product of external off-shell states |Ψ i , i = 1, · · · , n inserted at the punctures: The state R| is the surface state associated with the surface R. It describes the state that is created on the boundaries of the discs D i , i = 1, · · · , n by performing a functional integral over the fields of CFT on R− i D i . The inner product between R| and a state |Ψ 1 ⊗· · ·⊗|Ψ n ∈ H ⊗n is given by the n-point correlation function on R with the vertex operator for |Ψ i inserted at the i th puncture using the local coordinate system w i around that puncture.

The Quantum Master Action
The quantum BV master equation for the closed bosonic string field theory is given by where the target space field ψ * s is the antifield corresponding to the field ψ s . The perturbative solution of this equation in the closed string coupling g s is given by [3]: where Ψ denotes the dynamical string field (3.32) having arbitrary integer ghost number.
This can be considered as a solution to the quantum master equation only if the string field interaction vertices satisfy the following equation Here the sum is over all states in a complete basis of the Hilbert space of the world-sheet CFT that are annihilated by This equation imposes a very stringent condition on the string vertices V g,n , the region inside the moduli space over which we integrate the off-shell string measure to obtain the elementary string field theory interaction vertex with n dynamical string fields and g loops. The precise definition of the string vertices and the geometric equation satisfied by them is discussed in detail in the next section.
The quantum BV master action (3.42) is invariant under the master transformation given by where µ is an anti-commuting parameter. These gauge redundancies can be fixed by specifying the anti-fields by a relation of the form where Υ is a fermionic functional of the fields and antifields. Then, the gauge fixed path integral for the closed string field theory is obtained by integrating only over fields, and substituting the gauge fixing condition (3.45) for the antifields in the master action S(ψ s , ψ * s ) given in (3.42): With the help of master equation (3.43), one can verify that this gauge fixed action is independent of the choice of the gauge fermion Υ.

The Cell Decomposition of the Moduli Space
In this section, we discuss the precise definition of the string vertex, its properties and the geometric identity satisfied by them. The string vertex V g,n for the closed strings can be understood as a collection of genus g Riemann surfaces with n punctures that belong to a specific region inside the compactified moduli space M g,n . This region inside the moduli space has the following properties [3]: • The surfaces that are arbitrarily close to the degeneration are not included in it.
• The surfaces that belong to it are equipped with a specific choice of local coordinates around each of its punctures. The coordinates around the punctures are only defined up to a constant phase and these coordinates are defined continuously over the set V g,n .
• The assignment of the local coordinates around the punctures on the Riemann surfaces that belong to a string vertex are independent of the labeling of the punctures. Moreover, if a Riemann surface R with labeled punctures is in V g,n then copies of R with all other inequivalent labelings of the punctures also must be included in V g,n .
• If a Riemann surface belongs to the string vertex, then its complex conjugate also must be included in the string vertex. A complex conjugate Riemann surface of a Riemann surface R with coordinate z can be obtained by using the anti-conformal map z → −z.
The string vertices must satisfy the following geometric identity. This identity can be understood as the geometric realization of the quantum BV master equation (3.41): Here ∂V g,n denotes the boundary of the string vertex V g,n and S denotes the operation of summation over inequivalent permutations of the external punctures. {V g 1 ,n 1 , V g 2 ,n 2 } denotes the set of Riemann surfaces with the choice of local coordinates that can be glued at one of the puncture from each via the special plumbing fixture relation: where z and w denote the local coordinates around the punctures that are being glued. The special plumbing fixture corresponds to the locus |t| = 1 of the plumbing fixture relation The resulting surface has genus g = g 1 + g 2 and n = n 1 + n 2 − 2. ∆ denotes the operation of taking a pair of punctures on a Riemann surface corresponding to a point in V g−1,n+2 ⊂ M g−1,n+2 and gluing them via special plumbing fixture relation. Therefore, the first term of where ∂ p denotes the operation that take us to the boundary obtained by propagator collapse (|t| = 1). Since we assumed that the string vertices V g,n together with the Feynman diagrams V g,n;I , I = 1, · · · , 3g − 3 + n provide a single cover of the compactified moduli space M g,n , we have the identity M g,n = V g,n V g,n;1 · · · V g,n;3g−3+n

The Naive String Vertices Via the Hyperbolic Metric
The foremost difficulty in constructing string field theory is to find a suitable cell decomposition of the moduli spaces of Riemann surfaces. Any naive set of Feynman rules led to multiple or infinite overcounting of surfaces. Given a Riemann surface, we must be able to associate to it a unique Feynman diagram. In principle, the string vertices satisfying the conditions listed in the section (4) can be constructed using the Riemann surfaces endowed with the metric solving the generalized minimal area problem [3]. The generalized minimal area problem asks for the metric of least possible area under the condition that all nontrivial closed curves on the surface be longer than or equal to some fixed length, conventionally chosen to be 2π. However, as we already discussed in the introduction, the description of the string vertices using surfaces endowed with minimal area metric, at present, fails to provide a calculable framework for closed string field theory. In this section, we shall provide an alternate construction of the string vertices using Riemann surfaces endowed with a metric having constant curvature −1.
The description of the string vertex with genus g and n punctures in the closed bosonic string field theory requires : 1. Specifying the region inside the moduli space that corresponds to the elementary string interactions.

Specifying the choice of the local coordinates around the punctures.
For this, consider a Riemann surface endowed with metric having constant curvature −1 all over the surface. The uniformization theorem promises that every genus-g Riemann surface Then, the natural local coordinate for the puncture that corresponds to a parabolic element, whose fixed point is at z = i∞ on the upper half-plane H, is given by As it is required, this choice of local coordinate is invariant under the translation, z → z + 1, which represents the action of the generator of the corresponding parabolic element. In terms of the local coordinate w, the metric around the puncture takes the form If the fixed point of the parabolic element is at z = x on the upper half-plane H, then it is given by [37] It generates the following transformation on H: Then, the natural local coordinate for the puncture that corresponds to a parabolic element, whose fixed point is at infinity z = x on the upper half-plane H, is given by As it is required, this choice of local coordinate is invariant under the translation, 1 z−x → 1 z−x +1, which represents the action of the generator of the corresponding parabolic element. In terms of the local coordinate w, the metric around the puncture takes the form Therefore, for a puncture p on a hyperbolic Riemann surface, there is a natural local conformal coordinate w with w(p) = 0, induced from the hyperbolic metric and its local expression is given by We define the string vertices using the Riemann surfaces endowed with the hyperbolic metric as below: The naive string vertex V 0 g,n : Consider R g,n , a genus-g hyperbolic Riemann surface with n punctures, which has no simple closed geodesics with geodesic length l ≤ c * . Here c * is positive real number that is much less than one, c * ≪ 1. The local coordinates around the punctures on R g,n are chosen to be w = e π c * w, where w is the natural local coordinate induced from the hyperbolic metric on R g,n . The set of all such inequivalent hyperbolic Riemann surfaces with the above-mentioned local coordinates around the punctues form the naive string vertex V 0 g,n .

Inconsistency of the Naive String Vertices
In this section, we check the consistency of the the string vertices V 0 g,n . In subsection (4), we discussed that the string vertices satisfying the geometrical equation For this, we should study the hyperbolic metric on the Riemann surfaces obtained via the plumbing fixture of the string vertices.

The Plumbing Fixture of the Hyperbolic Riemann Surfaces
A degenerate Riemann surface is obtained by pinching a non-trivial simple closed curve on the surface. There are two ways of pinching a surface of genus g. One way is to pinch a curve along which if we cut, we get two separate Riemann surfaces. Such a degeneration is called a separating degeneration. Another way is to pinch a curve along which if we cut, we get a Riemann surface with lower genus and two more boundaries. Such a degeneration is called a non-separating degeneration. The local model around the pinch for both type of degenerations is the same, i.e. a hyperboloid with thin waist (see figure 3). This limiting case, where the loop degenerates to a point, can be described in terms of the Deligne-Mumford stable curve compactification of the moduli space of Riemann surface [54]. An alternate description for the degenerating families of the hyperbolic Riemann surfaces can be obtained using the cut and paste construction in the hyperbolic geometry following Fenchel and Nielsen [38,39]. In this subsection, we discuss the relation between these two approaches. surfaces to the moduli space produce the compactified moduli space M g,n of the the genus g Riemann surface [54]. By definition, a neighbourhood of a node p of R is complex isomorphic where w (1) and w (2) are the local coordinates around the two sides of the node p. We can obtain a family of non-degenerate Riemann surfaces from the degenerate Riemann surface R by identify U with the 0-fibre of the following family (see figure (4)) {w (1) w (2) = t| |w (1) |, |w (2) | < ǫ, |t| < ǫ} (6.66) A deformation of R ∈ M g which opens the node is given by varying the parameter t (see figure (5)).
Let us discuss a more general construction. Consider an arbitrary Riemann surface Here w (1) and w is a quasiconformal deformation of R 0 corresponds to this Beltrami differential. Then, we parametrize the opening of the nodes as follows. Given the m-tuple we construct the non-degenerate Riemann surface R t,s as follows. Remove the discs {0 < |w (1) i | ≤ |t i |} around the puncture a i and {0 < |w (2) i | ≤ |t i |} around the puncture b i from the Riemann surface R s (see figure 4). Then, attach the annular region {|t i | < |w (1) i | < 1} to the annular region {|t i | < |w (2) i | < 1} by identifying w This construction is complex: the tuple (t, s) parametrizing R t,s provides a local complex coordinate chart near the degeneration locus of the compactified moduli space.

The Fenchel-Nielsen Cut and Paste Construction
The Teichmüller space of the hyperbolic Riemann surfaces can be parametrized using the Fenchel-Nielsen coordinates [38]. The Fenchel-Nielsen parametrization is based on the observation that every hyperbolic metric on an arbitrary Riemann surface can be obtained by piecing together the metric from simple subdomains. A compact genus g Riemann surface with n boundary components can be obtained by taking the geometric sum of 2g − 2 + n pairs of pants (see figure (6)). The boundary components are the curves with lengths L i , i = 1, · · · , n.
When all L i = 0, i = 1, · · · , n, we have a genus g Riemann surface with n punctures.
Every hyperbolic metric on genus g Riemann surface with n borders can be obtained by varying the parameters of this construction. There are two parameters at each attaching site. For the pair of pants P and the pair of pants Q, these parameters are the length ℓ(β P 1 ) = ℓ(β Q 1 ) ≡ ℓ of the boundaries β P 1 , β Q 1 and the twist parameter τ . The twist parameter measures the amount of relative twist performed before glued between the boundaries of the pairs of pants that are being glued. The precise definition of the twist parameter is as follows. Let p 1 on the boundary β P 1 and q 1 on the boundary β Q 1 be two points with the following property. The point p 1 is the intersection of β P 1 and the unique orthogonal geodesic connecting β P 1 and β P 2 . Similarly, the point q 1 is the intersection of β Q 1 and the unique orthogonal geodesic connecting β Q 1 and β Q 2 .
The twist parameter τ is the distance between p 1 and q 1 along β P 1 ∼ β Q 1 . Then the parameters for a fixed pairs of pants decomposition P endows the Teichmüller space T g,n of the genus-g Riemann surfaces with n boundary components with a global real-analytic coordinates. In this coordinate system, the Weil-Petersson (WP) symplectic form takes the following very simple form [53]: dℓ j ∧ dτ j (6.69)

The Plumbing Fixture Vs the Fenchel and Nielsen Construction
Let us discuss the the relation between the plumbing fixture construction and the cut paste construction of Fenchel and Nielsen for the hyperbolic Riemann surfaces when the simple closed geodesic along which we are performing the cut and paste has infinitesimal length. Here, we follow [45].
We start by discussing the notion of a collar. For a simple closed geodesic α on the hyperbolic surface R of length l α , the collar around the geodesic α is a neighbourhood around the curve α having area 2l α cot l α 2 (6.70) The standard collar around the geodesic α is the collection of points p whose hyperbolic distance from the geodesic α is less than w(α) given by sinh w(α) · sinh l α 2 = 1 (6.71) The standard collar can be described as a quotient of the upper half-plane H. To describe this quotient space, consider the deck transformation z → e lα z (6.72) It generates a cyclic subgroup of PSL(2, R). We denote this cyclic subgroup by Γ lα . A fundamental domain for the action of Γ lα is given by a strip in H. If we quotient H with z → e lα z relation, we identify the two sides of the strip. This gives a hyperbolic annulus with a hyperbolic structure induced from H. The core geodesic of this hyperbolic annulus has hyperbolic length l α . Then, the standard collar can be approximated by the quotient of the following wedge with the cyclic group Γ l : The hyperbolic annulus can also be constructed via the plumbing fixture of two discs D 1 =  the discs D 1 and D 2 joined at the origin (see figure 7). We need to remove the origin from D 1 and D 2 to obtain a hyperbolic metric. Then, each punctured disc has the following complete hyperbolic metric, The space M D = {|t| < 1} can be considered as the moduli space for the hyperbolic annuli.
The Weil-Petersson metric on the moduli space of the hyperbolic annuli M D is given by [45]: We can identify the Fenchel-Nielsen coordinates (ℓ, τ ) for the moduli space of hyperbolic annulus as follows: If we use this form, then (6.78) can be written as dℓ ∧ dτ . This perfectly agrees with the Wolpert's formula for the WP metric [53].
Let us return to our discussion of degenerate Riemann surfaces. It is known that there exists a positive constant c * such that if the length l γ of any simple closed geodesic γ on a hyperbolic Riemann surface R is less than or equal to c * , then the standard collar having finite width embeds isometrically about γ [55,56]. This constant c * is known as the collar constant. We call the geodesics having length at most c * as short geodesics. Therefore, whenever the length of the simple geodesic along which we perform the cut and paste construction of Fenchel and Nielsen becomes less than the collar constant c * , we can replace the collar around this short geodesic with a hyperbolic annulus having finite width and interpret it as a plumbing collar.
This observation provides as the needed bridge between plumbing fixture construction and the Fenchel and Nielsen construction and persuades to claim that the hyperbolic Riemann surface near the boundary of the moduli space can be obtained by the plumbing fixture of another hyperbolic Riemann surfaces. Before asserting this claim, we must determine the metric on the plumbing family of the Riemann surfaces obtained via the plumbing fixture procedure. We explain in the next section that the plumbing fixture of hyperbolic Riemann surfaces generates a new Riemann surface with a metric that has constant curvature −1 over almost all of the resulting Riemann surface, but not over all of the surface.

The Approximate Cell Decomposition of the Moduli Space
In this subsection, we check the validity of the naive string vertices, by comparing the Riemann surfaces belong to the boundary of the naive string vertices with the Riemann surfaces obtained by the special plumbing fixture (4.48) of the hyperbolic Riemann surfaces belong to the appropriate naive string vertices.
For this, consider the Riemann surface R t , for t = (t 1 , · · · , t m ) obtained via plumbing fixture around m nodes of a hyperbolic surface R t=0 ≡ R 0 with m nodes. We denote the set of Riemann surfaces obtained by removing the nodes from R 0 by R 0 : Let us choose the local coordinates w  the j th such that in terms of these local coordinates, the hyperbolic metric around the punctures has the local expression These local coordinates are assumed to be zero at the punctures.
As the first approximation to the hyperbolic metric on R t , for t = 0 take the t j -fiber metric of the hyperbolic annulus in the j th collar of R t . Away from the collars, the hyperbolic metric can be taken to be the hyperbolic metric on R 0 . These two choices do not join smoothly at the two boundaries of the plumbing collar. Since there is a mismatch, we interpolate between the two choices at the boundaries of the collar. The resulting metric is the grafted metric ds 2 graft for R t (see figure 8).
In order to state the precise definition of the grafted metric, let us introduce a positive constant b * less than one and a negative constant a 0 for specifying the overlap of coordinate charts and to define a collar in each R t . Then the grafted metric on R t is given as below [46]: • The hyperbolic metric on R 0 restricted to R * b * is the grafted metric in the region com-plement to the region in R t described by the fiberation i |, |w (2) i | < b * ; i = 1, · · · , m} (6.82) The surface R * b * is obtained from R 0 by removing the punctured discs {0 < |w • The metric on the hyperbolic annulus given by is the grafted metric in the region in the plumbing collar F b * that is complement to the collar bands described by the fiberation • The geometric interpolation of the two metrics ds 2 and ds 2 t given by for j = 1, 2, is the grafted metric in the collar bands {e a 0 b * ≤ |w Here, η(a) is a smooth function and is given by: The Gaussian curvature 1 of the grafted metric is identically −1 on the regions complement to the collar bands on the surface. The first two leading order contribution to the Gaussian curvature of the grafted metric in the collar bands is given by [46]: for a ≡ ln |w (1) | or ln |w (2) | Therefore, the interpolation changes the constant curvature from −1 by a term having magnitude of (ln |t|) −2 . This deviation makes the resulting surface almost hyperbolic except at the boundaries of the plumbing collar. However, the Riemann surface that is obtained by gluing the hyperbolic Riemann surfaces can be considered as a hyperbolic Riemann surface only if the metric induced on the surface has −1 constant curvature all over the surface. Therefore, the glued Riemann surface is not a hyperbolic Riemann surface. Only in the t → 0 limit, we obtain a hyperbolic Riemann surface as a result of the plumbing fixture of hyperbolic Riemann surfaces. This can be seen from (6.87). Hence, the string vertices V 0 g,n defined as a set of Riemann surfaces with natural local coordinates induced from the hyperbolic metric around the punctures do not satisfy the geometrical identity (4.47) that is arising from the quantum BV master equation except in the c * → 0 limit.
Let us elaborate this. Assume that we obtained a Riemann surface R g,n by gluing two hyperbolic Riemann surfaces R g i ,n i and R g j ,n j belong to the string vertices V 0 g i ,n i and V 0 g j ,n j respectively via the special plumbing fixture construction (6.63). The length C * of the geodesic on the plumbing collar of R g,n computed using the hyperbolic metric on the glued surface, that we express in next section, is given by where c * is the length of the geodesic calculated using the grafted metric. Therefore, for vertex s-channel t-channel u-channel the finite values of c * , the geodesics length on the plumbing gets finite corrections. This in particular means that the Fenchel-Nielsen length parameters on the surfaces lying at the boundary of the string vertices and that on the glued surfaces obtained via the special plumbing fixture construction do not match. There is a mismatch of the order c 3 * (see figure 9). We can also match the local coordinates on the surfaces belong to the boundary of the string vertices and that on the glued surfaces obtained via the special plumbing fixture construction.
From the equation (7.101), it is clear that the hyperbolic metric on the surface obtained by gluing R g i ,n i and R g j ,n j do not match with the hyperbolic metric on the relevant regions of R g,n . Their ratios are different from unity by a term of order c 2 * . This suggests that the local coordinates on R g,n deviates from that on the surface obtained by gluing R g i ,n i and R g j ,n j by a term of order c 2 * . Therefore, we conclude that the naive string vertex V 0 g,n together with the Feynman diagrams won't be able to provide a single cover of the moduli space of hyperbolic Riemann surfaces with continuous choice of local coordinates on them. We are left with a mismatch of order c 2 * . As a result, the string vertices V 0 provide only an approximate cell decomposition of the moduli space which becomes more and more accurate as we take the parameter c * → 0.

7
Procedure for Correcting the Naive String Vertices In this section, we discuss an systematic procedure for improving the approximate cell decomposition of the moduli space by correcting the definition of the string vertices perturbatively in c * . We discussed in the previous section that the reason for the mismatch between the faces of the adjacent cells in the cell decomposition of the moduli space using the string vertices V 0 is that when we glue two hyperbolic surfaces using plumbing fixture, we get a surface which fails to be hyperbolic everywhere. In this section, we argue that the deviation of the induced metric from the hyperbolic metric is of order c 2 * . Therefore, the approximate cell decomposition of the moduli space can be improved by correcting the string vertices by modifying the definition of the boundary of the string vertices and the choice of local coordinates around the punctures on the surfaces belong to the boundary region of the string vertices perturbatively in c * , in a way that compensate for the deviation from the hyperbolic metric.

The Hyperbolic Metric on the Plumbing Family
For small |t| the hyperbolic metric (6.76) on the hyperbolic annulus that corresponds to a point t in the moduli space M of the hyperbolic annuli can be expanded as follows: with Θ = π ln |z| ln |t| . Our goal in this subsection is to find the analogous expansion for the hyperbolic metric on the plumbing family R t [46][47][48].
Consider an arbitrary compact Riemann surface having metric ds 2 with Gaussian curvature C. Then, another metric e 2f ds 2 on this surface has constant curvature −1 provided where D is the Laplace-Beltrami operator on the surface. This equation is known as the curvature correction equation [46,47]. Therefore, in order to get the hyperbolic metric on R t as an expansion in the plumbing fixture parameter, we need to solve the curvature correction equation perturbatively around the grafted metric by adding a compensating factor.
It is straightforward to check that the following metric on the plumbing family R t has constant curvature −1 if we neglect the O(ǫ 4 ) terms [46]: In which · is an appropriate norm and C graft is the Gaussian curvature of the grafted metric given by (6.87) and D graft is the Laplace-Beltrami operator on R t written using the grafted metric.
Assume that we have a smooth function f defined on the plumbing family satisfying the following equation Then, the hyperbolic metric on the plumbing family is given by This reduces our problem of finding the hyperbolic metric on the plumbing family to finding the function f satisfying the equation ( Given a choice of the function η, the parameters b * , a 0 and the plumbing fixture parameter t, let us consider the special truncation E # of the Eisenstein series, associated with the puncture p represented at infinity, that is given by the following modification in the cusp regions: • For Im(z) > 1, in the cusp region associated with the puncture that is used for the plumbing fixture, define • For Im(z) > 1, in the cusp region associated with other punctures, define We extend the special truncation E # of the Eisenstein series by zero on the components of R 0 not containing the puncture p. For the cusp coordinates w (1) around the puncture p and the cusp coordinate w (2) around another puncture of R 0 , say q, and the constant b * < 1, the punctured discs {0 < |w (1) | ≤ |t|/b * }, {0 < |w (2) | ≤ |t|/b * } are removed and the annuli form a collar. The extended E # is nonvanishing in the w (1) , w (2) cusp regions contained in In the complement of the collar, the grafted metric is the hyperbolic metric. So E † = E and (D graft − 2)E † becomes (D hyp − 2)E, where D hyp is the Laplace-Beltrami differential operator written using the hyperbolic metric on R 0 . By definition of the Eisenstein series E, given in (7.94), this quantity is zero. However, on the collar [47]: Using this result, we can obtain the degenerate expansion for the hyperbolic metric of R t in terms of the grafted metric (theorem 4 of [47]): where the functions E † i,1 and E † i,2 are the melding of the Eisenstein series E(·; 2) associated to the pair of cusps plumbed to form the i th collar. This expansion for the hyperbolic metric on R t can be written in terms of the i th collar geodesic computed using the ds 2 t metric on the hyperbolic annulus as follows: Then, the length of the geodesic in the i th plumbing collar is given by and the length of a simple closed geodesic α, disjoint from the plumbing collars is given by In this formula, l α ({l i }) is the length of α when the value of the core geodesic of the i th collar is l i computed in the ds 2 t metric which is given by l i = − 2π 2 ln |t| and l α ({0}) means the length of α when the lengths of all plumbing collars are zero.

The Second Order Corrections to the String Vertices
Now, we have enough prowess to describe to correct the naive string vertices to second order in c * . Consider the Riemann surface obtained by gluing m pairs of punctures on a set of hyperbolic Riemann surfaces via the special plumbing fixture construction. To the second order in c * , we see from the equation (7.101) that the grafted metric ds 2 graft on this Riemann surface is related to the hyperbolic metric ds 2 hyp on the Riemann surfaces belong to the boundary of the string vertex V 0 g,n corresponds to m nodes as follows: Therefore, to second order in c * , we use the definition of the region corresponding to the modified string vertex inside the moduli space as the same region defining the naive string vertex.
Correction to the choice of the local coordinates: Since there is a modification to the metric, we need to modify the choice of local coordinates around the punctures to make it gluing compatible to second order in c * . For an infinitesimal parameter δ, we modify the local coordinates on the surfaces belong to the naive string vertex as follows: • Consider the region inside the naive string vertex region of the moduli space where all the geodesics have length greater than (1 + δ)c * . On the Riemann surfaces belong to this region, choose e π 2 c * w i as the local coordinate around the i th punctures, where w i is the local coordinate induced from the hyperbolic metric around the i th puncture which vanishes at the puncture.
• On the Riemann surfaces belong to the rest of the regions inside the naive string vertex region, change the local coordinates around the punctures to e π c * w i , where w i is the local coordinates induced from the metric around the i th puncture.
Let us denote the string vertices corrected in this way by V 2 g,n . They provide an improved approximate cell decomposition of the moduli space that has no mismatch up to the order c 2 * . Therefore, to the order c 2 * , the corrected string vertices V 2 g,n together with the Feynman diagrams provide an exact cell decomposition of the moduli space. In other words, to second order in c * , the modified string vertices V 2 together with the Feynman diagrams defined using the original string vertices V 0 provide a single cover of the compactified moduli space. = M g,n (7.107) Therefore, the string vertices V 2 , corrected perturbatively to the second order in c * , provide a consistent closed string field theory to the order c 2 * . In other words, the corrected string vertices V 2 can be used to construct a consistent closed string field theory by keeping c * very small.
The parameter c * is related to the length of the stubs used for defining the string vertices.
Using the equation for the standard collar width (6.71), we can compute the length of the c * l stub Figure 10: The length of the stub l(stub) increases very fast as the length c * of the core geodesic on the special plumbing collar becomes small. stub. It is given by l stub = arcsinh 1 sinh (c * /2) (7.108) From figure (10), it is clear that the length of the stub l stub increases very fast as the parameter c * becomes small. Therefore, keeping the parameter c * very small corresponds to adding very long stubs to the string vertices.

Discussions
In this paper, we constructed the string vertices using Riemann surfaces endowed with metric having constant curvature −1 all over the surface. For this we introduced an infinitesimal parameter c * . The parameter c * is related to the lengths of the stubs used for defining the string vertices. The string vertices that we obtained together with the Feynman diagrams provide a single cover of the moduli space to the order c 2 * . Therefore, by keeping the parameter c * very small and using the string vertices constructed in this paper, we can obtain a consistent closed string field theory.
Adding stubs to the string vertex refers to the enlargement of the size of the region inside the moduli space that corresponds the string vertex. Taking c * very small corresponds to using very long stubs. For constructing a string field theory we are allowed to use stubs having arbitrary length. However, if we choose to add stubs having small length, then we need to find the higher order corrections to the string vertices. We can correct the string vertices up to an arbitrary order by solving the curvature correction equation (7.90) up to that order. We can then find the corrected string vertex by the procedure introduced in the previous section.
Interestingly, the length of stubs determines the energy scale of the Wilsonian effective action of the string field theory [24].
In the follow up papers, we develop these ideas further to provide a calculable framework for the covariant quantum closed bosonic string field theory. In particular, we explain: • The rules for the explicit evaluation of the closed bosonic string field theory action [59].
• An explicit construction of the gauge invariant 1PI and the Wilsonian effective action for the closed bosonic string field theory using the hyperbolic Riemann surfaces [60,61].
In a separate series of papers, we generalize the above constructions to the covariant closed superstring field theory [62][63][64][65]. To complete this program, it is essential to compute the correlation functions of conformal field theories on hyperbolic Riemann surfaces. We hope to report on this soon [66].