Topological recursion for hyperbolic string field theory
Atakan Hilmi Fırat Nico Valdes-Meller



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        ./images/ MIT-CTP-5753September 9, 2024
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        		Topological recursion for hyperbolic string field theory 0.5cm
        		
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        		Atakan Hilmi Fırat^1,2 and Nico Valdes-Meller^1 0.5cm
        		
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        			^1
        			Center for Theoretical Physics 
        
        			Massachusetts Institute of Technology
        
        			Cambridge MA 02139, USA
        			
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        			^2
        			Center for Quantum Mathematics and Physics (QMAP) 
        
        			Department of Physics & Astronomy, 
        
        			University of California, Davis, CA 95616, USA
        			
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        			mailto:ahfirat@ucdavis.eduahfirat@ucdavis.edu, mailto:nvaldes@mit.edunvaldes@mit.edu 5pt  2.5cm
        		 Abstract
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        		We derive an analog of Mirzakhani's recursion relation for hyperbolic string vertices and investigate its implications for closed string field theory. Central to our construction are systolic volumes: the Weil-Petersson volumes of regions in moduli spaces of Riemann surfaces whose elements have systoles L ≥ 0. These volumes can be shown to satisfy a recursion relation through a modification of Mirzakhani's recursion as long as L ≤ 2 sinh^-1 1. Applying the pants decomposition of Riemann surfaces to off-shell string amplitudes, we promote this recursion to hyperbolic string field theory and demonstrate the higher order vertices are determined by the cubic vertex iteratively for any background. Such structure implies the solutions of closed string field theory obey a quadratic integral equation. We illustrate the utility of our approach in an example of a stubbed scalar theory.
        
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        § INTRODUCTION
        
        
        Despite its three decade history, covariant closed string field theory (CSFT) still remains notoriously impenetrable, refer to <cit.> for reviews. This can be attributed to the seemingly arbitrary structure of its elementary interactions that have been reverse-engineered from string amplitudes. Even though CSFT provides a complete and rigorous definition for closed string perturbation theory, its construction is also responsible for why almost all of its nonperturbative  features remain inaccessible—for example its nonperturbative vacua. Among them the most important one is  arguably the hypothetical tachyon vacuum of the critical bosonic CSFT <cit.>. Its construction is expected to provide insights on the nonperturbative nature of string theory and its background-independent formulation.
        
        Therefore obtaining any nonperturbative information from CSFT (or the theory and/or principles that govern CSFT perturbatively) appears to require seeking finer structures within its current formulation—beyond those demanded by perturbative quantum field theory, such as amplitude factorization, to tame the infinitely many, nonlocal interactions between infinitely many target space fields. The main purpose of this work is to begin unearthing some of these structures and investigate their implications.
        
        This objective immediately brings our attention to string vertices encoding the elementary interactions of CSFT, which is arguably the most nontrivial ingredient of CSFT. Roughly speaking, these are the subsets of appropriate moduli spaces of Riemann surfaces that satisfy the consistency condition known as the geometric master equation
            ∂𝒱  + 1  2{𝒱, 𝒱} + ħΔ𝒱 = 0   ,
        
        see section <ref> for details. This equation encodes the perturbative consistency of CSFT in the sense of Batalin-Vilkovisky (BV) formalism <cit.>. A particular solution to (<ref>) is equivalent to a particular choice of field parametrizations in CSFT <cit.>. Although any allowed field parametrization can be used for a given theory, having one that most efficiently parametrizes the interactions and contains additional structure would be the superior choice.
        
        Hyperbolic string vertices have the potential to provide such a canonical choice for all-order investigations in CSFT <cit.>. The overall objective of this paper is to demonstrate the existence of a topological recursion among the elementary interactions of hyperbolic CSFT and initiate the investigation of its consequences for the nonperturbative structure of CSFT and its solutions.  
        
        Briefly stated, we show there is an analog of Mirzakhani's recursion for the Weil-Petersson (WP) volumes of moduli spaces <cit.> for the hyperbolic vertices ⟨𝒱_g,n (L_i) |  = ⟨𝒱_g,n (L_1, ⋯, L_n) | of hyperbolic CSFT, which are defined using hyperbolic surfaces of genus g with n punctures whose local coordinates are determined up to a phase by grafting semi-infinite flat cylinders to geodesic borders of lengths |L_1|, ⋯, |L_n| <cit.>. It takes the following form (see (<ref>) and figure <ref>)
        
            |L_1| ·⟨𝒱_g,n (L_i) | = ∑_i=2^n  ∫_-∞^∞  d ℓ [ ⟨ℜ (L_1, L_i, ℓ) | ⊗ ⟨𝒱_g, n-1 (-ℓ, 𝐋∖{L_i }) |
            	] |ω^-1⟩
                + 1  2∫_-∞^∞  d ℓ_1   ∫_-∞^∞  d ℓ_2  [⟨𝔇 (L_1, ℓ_1, ℓ_2) | ⊗ (⟨𝒱_g-1,n+1 (-ℓ_1, -ℓ_2, 𝐋 )| 
            	
                +
            	∑_stable⟨𝒱_g_1,n_1 (-ℓ_1, 𝐋_1 ) | ⊗ ⟨𝒱_g_2,n_2 (-ℓ_2, 𝐋_2 ) |
            	)
            	] |  ω^-1⟩_1  |  ω^-1⟩_2   ,
        
        where
        
            𝐋 = {L_2, ⋯, L_n }  ,     𝐋_1∪𝐋_2 = 𝐋   ,     𝐋_1∩𝐋_2 = ∅  ,
        
        and the “stable” in (<ref>), and henceforth, denotes the sum over all non-negative integers g_1,g_2,n_1,n_2 and partitions 𝐋_1, 𝐋_2 that satisfy
        
            g_1 + g_2 = g  ,    n_1 + n_2 = n + 1   ,    2g_1 - 2 + n_1 > 0   ,    2g_2 - 2 + n_2 > 0    ,
        
        for 2g -2 + n > 1 and n ≥ 1.[We comment on the cases g=1, n=1 and g≥ 2, n = 0 in section <ref>.] Here | ω^-1⟩ is the Poisson bivector <cit.> that twist-sews the entries of states whose length is integrated. Its subscript denotes the sewed entry.  The relation (<ref>) is a recursion in the negative Euler characteristic -χ_g,n = 2g-2+n of the underlying surfaces, hence we refer it topological recursion in this particular sense.
         
        
        The amplitudes ⟨𝒱_g,n (L_i) | are defined using the hyperbolic surfaces directly when L_i >0. However, forming the recursion also requires incorporating the cases when L_i are negative in the sense that they differ by the application of the 𝔅 ghost operators that result from changing the length of the border
        
            ⟨𝒱_g,n (L_1, ⋯, L_n) | =
            	⟨𝒱_g,n (|L_1|, ⋯, |L_n|)  |  (𝔅^⊗ θ(-L_1)⊗⋯⊗𝔅^⊗ θ(-L_n))   ,
        
        where θ(x) is the Heaviside step function: it is 1 whenever x ≥ 0, 0 otherwise. These cases are recursively determined by (<ref>) as well. We particularly highlight that the base case for the recursion is given by the cubic vertex
        
            ⟨𝒱_0,3 (L_1, L_2, L_3) |  =  ⟨Σ_0,3( |L_1|, |L_2| , |L_3| )|  
            	 (  𝔅^⊗ θ(-L_1)⊗𝔅^⊗ θ(-L_2)⊗𝔅^⊗ θ(-L_3))   ,
        
        where ⟨Σ_0,3( L_1, L_2 ,L_3)| is the generalized hyperbolic three-vertex of <cit.>, see appendix <ref>. As an example, a single application of 𝔅 to ⟨Σ_0,3( L_1, L_2 ,L_3)| is given by (see (<ref>))
        
            ⟨Σ_0,3( L_1, L_2 , L_3 )| (   𝔅⊗𝕀⊗𝕀)
            	
               = 
            	⟨Σ_0,3( L_1, L_2 , L_3 )|
            	[ 1  2 π(
            	1 ρ_1∂ρ_1 ∂λ_1 ( b_0 + b_0 )
            	+λ_1    (λ_2^2  - λ_3^2)  (1+ λ_1^2)^2 ρ_1 
            	( b_1+ b_1 )
            	+ ⋯) ⊗𝕀⊗𝕀]
            	  .
        
        Here L_i = 2 πλ_i > 0 and ρ_1 = ρ_1(L_1, L_2, L_3) is the mapping radius associated with the first puncture (<ref>). Observe that the 𝔅 insertion produces a dependence on all the border lengths L_1, L_2, L_3. One can similarly consider when there are multiple 𝔅 insertions.
        
        The recursion (<ref>) also contains the string kernels
            ⟨ℜ(L_1,L_2,L_3) |    = R_|L_1| |L_2| |L_3| ⟨𝒱_0,3(L_1,L_2,L_3)|   ,
            ⟨𝔇(L_1,L_2,L_3) |    = D_|L_1| |L_2| |L_3| ⟨𝒱_0,3(L_1,L_2,L_3)|   ,
        
         where R_L_1, L_2, L_3 , D_L_1, L_2, L_3 are the functions
        
            R_L_1 L_2 L_3   = R_L_1 L_2 L_3 - L_1  θ(L - L_3)   ,
            D_L_1 L_2 L_3   =D_L_1 L_2 L_3 
            	- R_L_1 L_2 L_3 θ(L -L_2) - R_L_1L_3 L_2 θ(L -L_3) + L_1  θ(L-L_2)  θ(L -L_3)   .
        
        The kernels depend on the threshold lengthL that is bounded by
        
            0 < L ≤ 2 sinh^-1 1 ≈ 1.76   ,
        
        for the recursion (<ref>) to hold true. Finally, the functions R_L_1 L_2 L_3 and D_L_1 L_2 L_3 above are the well-known functions that appear in Mirzakhani's recursion for the WP volumes of moduli spaces of Riemann surfaces <cit.> (also see <cit.>)
        
            R_L_1 L_2 L_3 = L_1 - log[cosh(L_2/2) + cosh(L_1 + L_3/2) /cosh(L_2/2)  + cosh(L_1 - L_3/2) ]    ,
            	
               D_L_1 L_2 L_3 = 2log[exp( L_1/2) + exp(L_2 + L_3/2) /exp(-L_1/2) + exp(L_2 + L_3/2) ]   .
        
        
        The central idea behind (<ref>) is to excise certain pants from surfaces that make up ⟨𝒱_g,n (L_i) | (see figure <ref>) to factorize it in terms of lower-order vertices, then perform the moduli integration over the lengths and twists of the excised seams. We need to make sure that each surface in ⟨𝒱_g,n (L_i) | is counted once in the moduli integration after the factorization and this introduces the measure factors R_L_1 L_2 L_3,D_L_1 L_2 L_3 in the integrand (<ref>). The excisions of pants from the surface further requires insertion of 𝔅, see section <ref>.
        
        A careful reader may notice the similarity between (<ref>) and the recursion proposed by Ishibashi <cit.>. Despite the structural similarity, there are a few crucial differences between our constructions. First, Ishibashi's off-shell amplitudes are unconventional from the perspective of covariant CSFT. They do not factorize through using the flat propagators. Instead, the simple closed geodesics of hyperbolic surfaces play the role of the propagator and the degenerations occur when the lengths of simple closed geodesics shrink to zero. This obscures its connection to ordinary CSFT, but it provides the straightforward structure to write a recursion relation for off-shell amplitudes as the moduli integration is performed over the entire moduli space.
        
        On the other hand, ⟨𝒱_g,n (L_i) | in (<ref>) are obtained by performing the moduli integration only over “systolic subsets” and the recursion is formed exclusively among them. A priori, it isn't clear that such a recursion has a right to exist—it requires a nontrivial modification of Mirzakhani's method where the systolic subsets are related to each other rather than the entire moduli spaces. But thanks to the “twisting procedure” of <cit.> and the collar lemma <cit.>, this modification is in fact possible, see section <ref> for details. Combining the twisting procedure and the factorization analysis of amplitudes, we derive a recursion relation (<ref>) without giving up the connection to CSFT.
        
        In order to see the impact of this modification, first recall the hyperbolic CSFT BV master action
        
            S_L[Ψ] = S_0,2 [Ψ] + ∑_g,n1  n! ħ^gκ^2g-2+n ⟨𝒱_g,n (L, ⋯, L_n  times)   | Ψ⟩^⊗ n  ,
        
        when 0 < L ≤ 2 sinh^-1 1 <cit.>. Here S_0,2[Ψ] is the free part of the action and the sum over the non-negative integers g,n above, and henceforth, denotes the sum over 2g-2+n > 0. The recursion (<ref>) then strikingly shows the different order terms in the hyperbolic CSFT action are related to each other and they are entirely determined by the cubic data (<ref>). This manifestly demonstrates the cubic nature of CSFT in the sense of topological recursion. We point out this result is consistent with the famous no-go theorem for having a cubic level-matched covariant  CSFT <cit.>: we don't give such a formulation here, only express a relation between different terms in the action (<ref>).
        
        A similar observation has already been made using the relation between hyperbolic vertices and the classical conformal bootstrap <cit.> by one of the authors. However the equation (<ref>) here actually makes this cubic nature even more apparent.[In fact these features are possibly related to each other, see the discussion on conformal blocks in <cit.>.] We highlight that (<ref>), and its consequences, are background-independent and apply to any bosonic closed string background.
        
        The recursion (<ref>) is stronger than the BV structure of CSFT. There, the focus is on how the Feynman diagrams with propagators are related to the elementary vertices—as stated in terms of the geometric master equation (<ref>). In (<ref>), on the other hand, the elementary vertices themselves relate to each other. Something akin to this feature can be artificially created by deforming a polynomial theory with stubs <cit.>, for which the interactions are obtained through homotopy transfer of the seed theory using a “partial propagator” and the resulting (infinitely many) vertices are related to each other iteratively. This is the primary reason why the stubbed theories are theoretically tractable despite their non-polynomiality. The expectation is that the topological recursion (<ref>) would help to achieve similar feats in CSFT.
        
        In order to demonstrate the power of this extra structure we examine its implication for the classical solutions to CSFT (<ref>). Let us call the stationary point of the classical CSFT action Ψ_∗, keeping its L dependence implicit, and introduce the string field Φ(L_1)
            Φ(L_1) = ∑_n=3^∞κ^n-2 (n-1)! (𝕀⊗⟨𝒱_0,n (L_1, L, ⋯, L_(n-1)  times) | 
            	)  ( | ω^-1⟩ ⊗ | Ψ_∗⟩^⊗ (n-1))   ,
        
        associated with the solution Ψ_∗. Observe that we have
        
            Φ(L_1 = L) = - Q_B  Ψ_∗  ,
        
        by the equation of motion, so Φ(L_1) should be thought as a certain off-shell generalization of the the BRST operator Q_B acting on the solution. It is parameterized by L_1. The recursion (<ref>) implies the following quadratic integral equation for this string field (see (<ref>))
        
            0 = Φ (L_1)
            	- κ 2  
            	A_L_1 L L ( β Φ  (L) , β Φ  (L) )
            	   - κ ∫_-∞^∞ d ℓ  
            	B_L_1 L ℓ(β Φ  (L) , Φ(ℓ))
            	
               - κ 2 ∫_-∞^∞ d ℓ_1  ∫_-∞^∞ d ℓ_2  
            	C_L_1 ℓ_1 ℓ_2(Φ(ℓ_1), Φ(ℓ_2) )   ,
        
        where the 2-products A,B,C are constructed using (<ref>). Here we fixed the gauge by 
        
            β(L)  Ψ_∗ = 0   ,
        
        see (<ref>). The detailed derivation of this equation is given in section <ref>.
        
        The solutions  Φ(L_1)  to (<ref>) (if they exist) are related to the honest CSFT solutions through (<ref>) and they can be used to obtain the cohomology around them. Even though we haven't made any attempt towards realizing this here, we have investigated the counterpart problem in the stubbed cubic scalar field theory as a proof of principle. We construct the nonperturbative vacuum of <cit.> and read the mass of the linear excitations around it entirely using the analog of (<ref>) with some guesswork. In particular we haven't performed a resummation while doing so. This is encouraging for future attempts of solving CSFT with (<ref>): resumming the CSFT action would have been a hopeless endeavor with current technology.
        
        The outline of the paper is as follows. In section <ref> we review hyperbolic geometry and CSFT. In section <ref> we compute the WP volumes of systolic subsets à la Mirzakhani, after reviewing her recursive method for computing WP volumes of the entire moduli spaces. This provides a toy model for the actual case of interest, which is the recursion for hyperbolic string vertices. We derive it in section <ref>. We provide more insight on our results by obtaining an analogous recursion relation for the stubbed cubic scalar theory of <cit.> in section <ref> and investigate its implications. This section is mostly self-contained and should be accessible to readers who are not familiar with CSFT. The solutions of the stubbed theory are shown to obey a quadratic integral equation in section <ref>. We also show the solutions to CSFT also obey a similar equation in section <ref> using similar reasoning. We conclude the paper and discuss future prospects in section <ref>. 
        
        In appendix <ref> we report the local coordinates for the generalized hyperbolic three-vertex of <cit.>. Finally we compute certain systolic volumes using the recursion and the bootstrap methods of <cit.> in appendix <ref> and <ref> respectively. 
        
        
        
        § PRELIMINARIES
        
        
        In this section we review hyperbolic geometry and the construction of CSFT. For a mathematically-oriented introduction to hyperbolic geometry refer to <cit.> and for the basics of CSFT there are many excellent reviews <cit.>. We follow <cit.> for our discussion of hyperbolic vertices.
        
        
        
         §.§ Hyperbolic surfaces and Teichmüller spaces
        
        
        Let S_g,n be an orientable closed two-dimensional marked surface of genus g with n borders that has a negative Euler characteristic[The marking is an orientation-preserving diffeomorphism f : S_g,n→ S_g,n and two markings are equivalent if they are isotopic to each other. The surface together with the isotopy class of the marking [f] is a marked surface.]
            -χ_g,n≡ 2g-2+n > 0   .
        
        By the uniformization theorem, such a surface admits metrics with constant negative Gaussian curvature K = -1 with geodesic borders of length L_i ≥ 0 for i=1, ⋯, n. These are hyperbolic metrics with geodesic borders. We denote the space of all such distinct metrics on S_g,n by 𝒯_g,n(L_i) and call it Teichmüller space. This is same as the set of marked (hyperbolic) Riemann surfaces of genus g with n borders. The border is a hyperbolic cusp when its associated length vanishes.
        
        The ordinary moduli spaces of Riemann surfaces ℳ_g,n(L_i) are obtained by forgetting the marking on the surface S_g,n, i.e., by taking the quotient[We always consider ℳ_g,n (L_i) to be compactified in the sense of Deligne-Mumford. This also requires using the so-called augmented Teichmüller space <cit.>, however we are not going to be explicit about this.]
            ℳ_g,n (L_i) = 𝒯_g,n(L_i)/ MCG(S_g,n)   ,
            	     MCG(S_g,n) = Diff^+(S_g.n) Diff_0^+(S_g,n)  ,
        
        where MCG(S_g,n) is the mapping class group of the surface S_g,n. Here Diff^+(S_g.n) is the set of orientation-preserving diffeomorphisms and 0 subscript denotes those  isotopic to the identity. These diffeomorphisms are taken to be restricted to be the identity on the boundary ∂ S_g.n. In other words MCG(S_g,n) is the group of large diffeomorphisms of S_g,n. The action of MCG(S_g,n) on 𝒯_g,n(L_i) produces the hyperbolic structures on S_g,n that are same up to marking.
        
        The Teichmüller space is somewhat easier to work with relative to the moduli space ℳ_g,n (L_i) since it is simply-connected <cit.>. That is, 𝒯_g,n(L_i) is the universal cover of ℳ_g,n(L_i) and there is a covering map
        
            π : 𝒯_g,n(L_i) →ℳ_g,n(L_i)   .
        
        In order to better appreciate this feature we remind the reader that marked Riemann surfaces admit pants decompositions, which is done by cutting the surface along a set of 3g-3+n disjoint simple closed geodesics <cit.>. The pant decompositions are far from unique. There are infinitely many such decompositions and they are related to each other by large diffeomorphisms.
        
        Given a pants decomposition, however, any marked surface can be constructed uniquely through specifying the tuple
        
            (ℓ_i, τ_i) ∈ℝ_+^3g-3+n×ℝ^3g-3+n  ,
        
        where ℓ_i are the lengths of the seams of the pants and τ_i are the relative twists between them. This provides a particularly nice set of coordinates for Teichmüller space known as Fenchel-Nielsen coordinates. They descend to the moduli spaces by (<ref>), but only locally due to the large diffeomorphisms. It is then apparent that the dimensions of Teichmüller and moduli spaces are given by
        
            _ℂ𝒯_g,n(L_i) = _ℂℳ_g,n(L_i) = 3g - 3 + n ≡ d_g,n  .
        
        Note that we present the complex dimensions as these spaces have complex structures <cit.>.
        
        Teichmüller space further admits a Kähler metric known as the Weil-Petersson (WP) metric. The associated symplectic form is the WP form. Its mathematically precise definition is not important for our purposes, but one of its important facets is that the WP form is MCG-invariant so that there is a globally well-defined symplectic form ω_WP over the moduli space ℳ_g,n(L_i). This is reflected in Wolpert's magic formula <cit.>
            π^∗ ω_WP  = ∑_i=1^3g -3 + n d ℓ_i ∧ d τ_i   ,
        
        that presents the pullback of ω_WP under (<ref>) in Fenchel-Nielsen coordinates (ℓ_i, τ_i). Here the “magic” refers to the fact that the right-hand side holds for the Fenchel-Nielsen coordinates associated with any pants decomposition (i.e., its invariance under the action of the MCG). Using ω_WP, it is possible to construct the WP volume form vol_WP on the moduli space ℳ_g,n(L_i)
            vol_WP = ω_WP^3g-3+n (3g-3+n)!   ,       π^∗ vol_WP = ⋀_i=1^3g -3 + n d ℓ_i ∧ d τ_i    ,
        
        and compute integrals over moduli spaces.
        
        We are not going to delve into the proofs of these statements. However, we highlight one of the common ingredients that goes into their proofs: the collar lemma <cit.>. This lemma essentially states that two sufficiently short closed geodesics on a hyperbolic surface cannot intersect each other. More precisely, defining a collar 𝒞_γ around a simple closed geodesic  γ∈Σ_g,n(L_i) by
        
            𝒞_γ = { 
            	p ∈Σ_g,n(L_i)
            	 |  dist(p, γ) ≤w  2   ,   sinhw  2sinhγ 2  = 1
            	}  ,
        
        the collar lemma states that the collars are isometric to hyperbolic cylinders and are pairwise-disjoint for a pairwise-disjoint set of simple closed geodesics. The lengths of the geodesics are always denoted by the same letter and “dist” stands for the distance measured by the hyperbolic metric. One can analogously define half-collars around the geodesic borders and they will be pairwise-disjoint from other (half-)collars. Here w is the width of the collar and it increases as γ decreases.
        
        The last observation, together with the collar lemma, necessitates the lengths of a simple closed geodesic γ and a (not necessarily simple) closed geodesic δ with γ∩δ≠∅ to obey
        
            sinhγ 2 sinhδ 2 > 1   .
        
        Defining the maximum threshold length
        
            L_∗ = 2 sinh^-1 1   ,
        
        it is apparent that a simple closed geodesic γ with γ≤ L_∗ cannot intersect with a closed (but not necessarily simple) geodesic δ with δ≤ L_∗.
        
        
        
         §.§ String vertices
        
        
        Now we return our attention to the construction of CSFT. The central geometric ingredient is string vertices. They can be succinctly expressed as the formal sum
        
            𝒱 = ∑_g,nħ^g κ^2g-2 + n 𝒱_g,n  .
        
        Here the sum runs over non-negative integers g,n with 2g -2 +n > 0 and ħ and κ (string coupling) are formal variables. The objects 𝒱_g,n are 6g - 6 +2n real dimensional singular chains of the bundle 𝒫_g,n→ℳ_g,n, where ℳ_g,n≡ℳ_g,n(L_i = 0). This bundle encodes a choice of local coordinates without specified global phases around the punctures.
        
        The perturbative consistency of CSFT requires 𝒱 to satisfy the geometric master equation
        
            ∂𝒱 = - 1  2{𝒱 , 𝒱} - ħΔ𝒱  ,
        
        as stated in the introduction. Expanding in ħ, κ, this equation can be seen as a homological recursion relation among 𝒱_g,n. Here  is the boundary operator on the chains, while the anti-bracket {·, ·} and the Laplacian Δ are the following multilinear operations:
        
          *  The chain {𝒱_g_1, n_1, 𝒱_g_2, n_2} is the collection of all surfaces constructed by twist-sewing a puncture in a surface belonging to 𝒱_g_1, n_1 to a puncture in a surface belonging to 𝒱_g_2, n_2 by
        	
            w_1 w_2 = expi θ      where       0 ≤θ < 2 π  .
        
        	Here w_i are the local coordinates around the sewed punctures. The resulting surfaces are of genus g_1+g_1 and have n_1 + n_2 - 2 punctures.
        	
        	
          *  The chain Δ𝒱_g,n is the collection of all surfaces constructed by twist-sewing two punctures of the same surface in 𝒱_g,n. The resulting surfaces are of genus g+1 and have n - 2 punctures. 
        
        These operations, together with an appropriate notion of dot product on the chains over the moduli space of disjoint Riemann surfaces with a choice of local coordinates around their punctures, form a differential-graded Batalin-Vilkovisky (BV) algebra <cit.>.
        
        Any explicit construction of CSFT requires a solution to (<ref>). In this paper we are concerned with the one provided by hyperbolic geometry <cit.>. The primary idea behind its construction is as follows. We first imagine bordered Riemann surfaces endowed with hyperbolic metrics whose borders are geodesics of length L_i and consider
        
            𝒱^L_g,n(L_i) =
            	{ Σ∈ℳ_g,n(L_i)  |  sys(Σ) ≥ L  }⊆ℳ_g,n(L_i)
            	  .
        
        Here sys(Σ) is the systole of the surface Σ—the length of the shortest non-contractible closed geodesic non-homotopic to any border. We denote these subsets as the systolic subsets.
        
        We remark that the set 𝒱^L_g,n(L_i) may be empty. For example, the maximum value of the systole in ℳ_2,0 is 2 cosh^-1 ( 1 + √(2)) ≈ 3.06 and it is realized by the Bolza surface <cit.>. Any choice of L greater than this value leads to an empty set. It is clear that taking the threshold length L sufficiently small always leads to a non-empty 𝒱^L_g,n(L_i) since 𝒱^L_g,n(L_i) covers the entire moduli space ℳ_g,n(L_i) as L → 0. In particular the systolic subsets are always non-empty when L ≤ L_∗.
        
        The systolic subsets are used to construct hyperbolic string vertices via grafting semi-infinite flat cylinders to each geodesic borders of a surface in 𝒱^L_g,n(L_i)[We use tilde to distinguish systolic subsets before and after grafting. We are not going to make this distinction moving forward unless stated otherwise.]
            𝒱^L_g,n(L_i) = gr'_∞( 𝒱^L_g,n(L_i) 
            	)      where    gr'_∞ : ℳ_g,n(L_i) →𝒫_g,n  .
         
        The grafting map gr'_∞ naturally endows a bordered surface with local coordinates. In fact this map is a homeomorphism <cit.> and 𝒱^L_g,n is a piece of a section over the bundle 𝒫_g,n→ℳ_g,n as a result. 
        
        It can be shown that
        
            𝒱^L = ∑_g,nħ^g κ^2g-2 + n 𝒱^L_g,n (L_i = L)    ,
        
        solves (<ref>) when L ≤ L_∗ = 2 sinh^-1 1 by using  the collar lemma <cit.>. This essentially follows from noticing that 𝒱^L_g,n consists of surfaces where the length of at least one simple closed geodesic is equal to L and these surfaces are in 1-1 correspondence with those constructed using {·, ·} and Δ. The crucial insight behind this proof was using the corollary of the collar lemma stated in (<ref>), which introduces a nontrivial constraint on the threshold length L. Refer to <cit.> for more details.
        
        
        
         §.§ Differential forms over 𝒫_g,n→ℳ_g,n
        
        
        String vertices in bosonic CSFT are the chains over which the moduli integration is performed while the integrand is constructed using a 2d matter CFT of central charge c=26 together with the bc ghost system whose central charge is c = -26. We call their combined Hilbert space ℋ and consider its subspace ℋ whose elements are level-matched
        
            b_0^- |Ψ⟩ = ( b_0 - b_0)   |Ψ⟩  = 0   ,     
            	L_0^-  |Ψ⟩ = ( L_0 - L_0)   |Ψ⟩  = 0   .
        
        Here b_0 and L_0 are the zero modes of the b-ghost and stress-energy tensor respectively. In this subsection we often follow the discussion in <cit.>.
        
        The natural objects that can be integrated over the singular p-chains are differential p-forms. This requires us to introduce a suitable notion of p-forms over the the bundles 𝒫_g,n→ℳ_g,n. There are primarily two ingredients that go into their construction: the surface states and the b-ghost insertions. 
        
        The surface states⟨Σ_g,n| are the elements of the dual Hilbert space (ℋ^∗)^⊗ n that encode the instruction for the CFT correlator over a given Riemann surface Σ_g,n. That is
        
            ⟨Σ_g,n |  Ψ_1  ⊗⋯⊗Ψ_n 
            	 =
            	⟨Ψ_1(w_1 = 0) ⋯Ψ_n(w_n = 0) ⟩_Σ_g,n  ,
        
        for any Ψ_1  , ⋯ , Ψ_n  ∈ℋ. Notice ⟨Σ_g,n | contains the local coordinate data around each puncture. They have the intrinsic ghost number 6g-6 and have even statistics. Because of this, it is sometimes useful to imagine ⟨Σ_g,n (L_i) | encodes the CFT path integral over bordered surfaces and Ψ_i are inserted through grafting to provide boundary conditions for them in the context of hyperbolic vertices. We usually keep the border length dependence of the hyperbolic string amplitudes to distinguish them from the generic ones.
        
        Often times we apply the surface states to an element
        
            Ψ_1 ⊗⋯⊗Ψ_n ∈ℋ^⊗ n  ,
        
        where Ψ_i is understood to be grafted to the i-th border. However, the order of the borders in the surface states may come permuted in our expressions, i.e., ⟨Σ_g,n(L_σ_i) | for σ∈ S_n is a possibility. In this case we have
        
            ⟨Σ_g,n(L_σ_i) |  Ψ_1  ⊗⋯⊗Ψ_n 
            	= ϵ(σ)  ⟨Σ_g,n(L_σ_i) |  Ψ_σ_1⊗⋯⊗Ψ_σ_n  ,
        
        from (<ref>). Here ϵ(σ) is the Koszul sign of the permutation σ after commuting string fields. This amounts to replacing ⊗→∧, however we keep the tensor product notation.
        
        Now for the b-ghost insertions, recall that the defining property of the p-forms is that they produce scalars when they are applied to p vectors and are antisymmetric under exchange of these vectors. The surface states don't have this structure—we need to act on them with appropriate b-ghosts so that we construct the relevant (ℋ^∗)^⊗ n-valued p-forms over 𝒫_g,n→ℳ_g,n. So introduce
        
            B_p (V_1, ⋯ V_p) = b (v_1) ⋯ b (v_p)   ,
        
        where V_1, ⋯, V_p ∈ T 𝒫_g,n are vector fields over the bundle 𝒫_g,n→ℳ_g,n while
        
            b(v_q) = ∑_i = 1^n [
            	∮ dw_i   v_q^(i) (w_i)   b (w_i) + ∮ dw_i  v_q^(i) (w_i)  b (w_i)
            	]   ,     
            	q = 1, ⋯, p   ,
        
        are given in terms of b-ghosts insertions around the punctures. Here 
        
            v_q ∈ T Σ_g,n  ,
        
        are the so-called Schiffer vector fields associated with the vector V_q(Σ_g,n) ∈ T_Σ_g,n𝒫_g,n and the expressions v_q^(i) (w_i) are their presentations in the local coordinates w_i around the punctures. We take 
        
            ∮dz  z = ∮ d zz =1   ,
        
        for a contour oriented counterclockwise around z=0. Notice the object B_p is antisymmetric under exchanging V_i's by the anticommutation of the b-ghosts.
        
        The (generalized) Schiffer vector fields are constructed as follows.[Strictly speaking Schiffer vectors arise only due to the change of local coordinates around the punctures while keeping the rest of the transition functions of the surface fixed. So it is better to call the objects v described here generalized Schiffer vectors. Unless stated otherwise, we simply call them Schiffer vectors as well.] Suppose that we have two coordinates z_i and z_j on the surface with a holomorphic transition map
        
            z_i = f_i j (z_j)   ,
        
        where z_i and z_j can be uniformizing coordinates and/or local coordinates around punctures. Further suppose the vector V(Σ_g,n) ∈ T_Σ_g,n𝒫_g,n is associated with some infinitesimal deformation of the transition map corresponding a change of the moduli of the surface. We keep the coordinate z_i = f_i j (z_j) fixed while taking f_ij→ f^ϵ_ij and z_j → z_j^ϵ for this deformation. Then
        
            z_i= f^ϵ_ij(z^ϵ_j)
            	     
            	z^ϵ_j = (f^ϵ_ij)^-1 (z_i) = (f^ϵ_ij)^-1 (f_i j (z_j) ) 
            	= z_j + ϵ  v^(i) (z_j)   .
        
        Here v^(i) (z_j) defines the Schiffer vector field on the surface associated with V. It is regular on the intersection of coordinate patches z_i, z_j but it can develop singularities away from this region. We further have
        
            f_ij^ϵ (z_j) =  f_ij^ϵ (z^ϵ_j - ϵ  v^(i) (z_j)  )  
            		= f_ij^ϵ (z^ϵ_j ) - ϵ  f_ij (z_j)  z_j  v^(i) (z_j)
            		= z_i -  ϵ  v^(i) (z_i)    , 
               v^(i) (z_i)  =   f_ij (z_j)  z_j  v^(i) (z_j)
            		  ,
        
        after pushing the vector forward to z_i coordinates. Here we abuse the notation and use the arguments of the Schiffer vectors to implicitly remind us the coordinates in which they are expressed. The superscript on v^(i) (z_j) informs which patch is held fixed.
        
        Given the local coordinates t^a on 𝒫_g,n we have the dependence f_ij (z_j) = f_ij(z_j ; t^a). The variation associated with t^a → t^a + δ t^a (which is also associated with the vectors V_a) is then given by 
        
            f_ij (z_j; t^a + δ t^a) = f_ij (z_j; t^a) + δ t^a   f_ij (z_j; t^a)  t^a
            	= z_i + δ t^a   f_ij (z_j; t^a)  t^a  .
        
        Using (<ref>) we find
        
            v^(i)_a (z_i) = -   f_ij (z_j; t^a)  t^a = -   f_ij (f_i j^-1 (z_i); t^a)  t^a
               
            	v^(i)_a (z_j) = - (   f_ij (z_j; t^a)  z_j)^-1 f_ij (z_j; t^a)  t^a  .
        
        From here it is natural to take Schiffer vectors as 1-forms over 𝒫_g,n and express the operator-valued p-form over 𝒫_g,n (<ref>) as
        
            B_p = b(v_ a_1)   d t^a_1∧⋯∧ b(v_a_p)   d t^a_p  ,
        
        in the coordinates t^a. The b-ghost insertions here can be chosen to act either on coordinates z_i or z_j using the first or second expressions in (<ref>)
            b(v_a_q) = [
            	∮ d u   v^(i)_a_q (u)   b (u) + 
            	∮ du v_a_q^(i) (u)  b (u)
            	]    , 
            	     u = z_i, z_j   ,
        
        since this is conformally invariant. We point out the Schiffer vectors v^(i) (u) and v^(i) (u) above may not be complex conjugates of each other.
        
        Given these ingredients, we define
        
            ⟨Ω^(p)_g,n | = (- 2π i)^- d_g,n⟨Σ_g,n |   B_p    .
        
        The prefactor is inserted to generate the correct factorization of the amplitudes. It carries an intrinsic ghost number 6g-6-p and has the statistics determined by (-1)^p. The object ⟨Ω^(p)_g,n | is indeed a (ℋ^∗)^⊗ n-valued p-form over the bundle 𝒫_g,n→ℳ_g,n due to the antisymmetry of exchanging b-ghosts. We call ⟨Ω^(6g-6+2n)_g,n | = ⟨Ω_g,n | the string measure. This is the correct integrand for the moduli integrations in string amplitudes.
        
        
        
         §.§ Closed string field theory action
        
        
        Given p-forms over 𝒫_g,n→ℳ_g,n we are now ready to construct the CSFT action and discuss its perturbative consistency <cit.>. The free part of it is given by
        
            S_0,2 [Ψ] = 1  2 ⟨Ψ | c_0^- Q_B | Ψ⟩  ,
        
        where c_0^- = ( c_0 - c_0 ) /2 is the zero mode of the c-ghosts, Q_B is the BRST operator of matter + ghost CFT,  and ⟨· | ·⟩ is the ordinary BPZ inner product
        
            ⟨Σ_0,2 |  Ψ_1 ⊗Ψ_2 
            	= ⟨Ψ_1|Ψ_2 ⟩ =
            	⟨Ψ_1(z = 0)  Ψ_2(z = 0) ⟩    where    z(z) = 1  z  .
        
        
        Above we have defined ⟨Σ_0,2 | encoding the BPZ product, which carries intrinsic ghost number -6 and is even.  In a given basis ϕ_α of ℋ it can be expressed as
        
            ⟨Σ_0,2 | = ∑_α ⟨ϕ_α | ⊗⟨ϕ_α^c |   , 
            	    
            	| Σ_0,2⟩ = ∑_α | ϕ_α⟩⊗ | ϕ_α^c ⟩  ,
        
        and the conjugate states ϕ_α^c ∈ℋ are given by
        
            ⟨ϕ_α^c | ϕ_β⟩ = 
            	(-1)^ϕ_α⟨ϕ_α | ϕ_β^c ⟩ =
            	δ_αβ  ,
        
        where (-1)^ϕ_α denotes the statistics of the state ϕ_α. Notice the conjugate state has the same statistics as ϕ_α and it has ghost number 6-gh(ϕ_α). The associated partitions of the identity operator 𝕀 acting on ℋ are
        
            𝕀 = ∑_α | ϕ_α⟩⟨ϕ_α^c | 
            	= ∑_α (-1)^ϕ_α | ϕ_α^c ⟩⟨ϕ_α |   .
        
        
        By further introducing the symplectic form <cit.>
            ⟨ω | = ⟨Σ_0,2 |  𝕀⊗ c_0^- ∈ (ℋ^∗)^⊗ 2  ,
        
        the quadratic part of the action (<ref>) can be expressed as
        
            S_0,2 [Ψ] = 1  2 ⟨ω|  Ψ⊗ Q_B Ψ  ,
        
        since Q_B is cyclic under the symplectic form and the string field Ψ is even in the CSFT master action. The object ⟨ω| has odd statistics and carries intrinsic ghost number -5.
        
        Relatedly, we also introduce the Poisson bivector
            |ω^-1⟩   =  𝕀⊗ b_0^- δ(L_0^-)    | Σ_0,2⟩  
            	=
            	 ∑_α (-1)^ϕ_α | ϕ_α⟩ ⊗  b_0^-  δ(L_0^-)   | ϕ_α^c ⟩  ,
        
        which inverts the symplectic form in the following sense <cit.>
            ( ⟨ω | ⊗𝕀)  ( 𝕀⊗  |ω^-1⟩)
            	= b_0^-  δ(L_0^-)   c_0^-   .
        
        Note that b_0^- δ(L_0^-) c_0^-  is the projector onto ℋ so it restricts to the identity 𝕀 there. That is,
        
            b_0^- δ(L_0^-) c_0^-   | Ψ⟩ = | Ψ⟩  , 
            	      ⟨Ψ |   c_0^- δ(L_0^-) b_0^- = ⟨Ψ |    .
        
        for every Ψ∈ℋ. Furthermore the Poisson bivector is annihilated by b_0^- and L_0^- at both entries, hence
        
            |ω^-1⟩∈ℋ^⊗ 2  .
        
        Like ⟨ω|, it has odd statistics and carries intrinsic ghost number 5. The Poisson bivector essentially implements the entirety of the twist-sewing (<ref>), thanks to the combination of these properties <cit.>.
        
        Finally, it is also useful to introduce a (odd, ghost number 5) bivector that implements the sewing with a particular value of twist τ
            | 𝔖 (τ) ⟩   ≡[ 𝕀⊗ b_0^- exp( 2 π i τ L_0^- ℓ) ] | Σ_0,2⟩
               = ∑_α (-1)^ϕ_α | ϕ_α⟩⊗ b_0^- exp(2 π i τ L_0^- ℓ)
            	| ϕ_α^c ⟩∈ℋ^⊗ 2  .
        
        Note that this bivector does not belong to the space ℋ^⊗ 2. Only upon integrating over all twists do we obtain the Poisson bivector
        
            | ω^-1⟩ = ∫_0^ℓ d τℓ  | 𝔖 (τ) ⟩∈ℋ^⊗ 2  .
        
        and we restrict to the level-matched states.
        
        We now include the interactions to the action (<ref>). These are defined by integrating the string measure over the string vertices
        
            ⟨𝒱_g,n | = ∫_𝒱_g,n⟨Ω_g,n | = 
            	(- 2π i )^-d_g,n∫_𝒱_g,n⟨Σ_g,n |   B_6g-6+2n  ,
        
        We denote the resulting bras by the same letter as the integration chain. Combining them for all g,n produces the interacting CSFT action
        
            S[Ψ] = S_0, 2[Ψ] + ∑_g,n1  n! ħ^g κ^2g-2+n ⟨𝒱_g,n | Ψ⟩^⊗ n  .
        
        Observe that each term in the action is finite individually since the moduli integration (<ref>) doesn't get contributions from the degenerating surfaces. The action S[Ψ] can be shown to be BV quantizable as a result of string vertices 𝒱 solving the geometric master equation (<ref>), refer to <cit.> for details.
        
        A somewhat convenient way of writing the action S[Ψ] is
        
            S[Ψ] = 
            	S_0,2[Ψ] 
            	+ ∑_g,n1  n! ħ^g κ^2g-2+n ⟨ω |  Ψ⊗ L_g,n-1(Ψ^(n-1) )   ,
        
        for which we have defined the string productsL_g,n-1 : ℋ^⊗ (n-1)→ℋ
            L_g,n-1 = (  𝕀⊗⟨𝒱_g,n  |  )  ( | ω^-1⟩⊗𝕀^⊗ (n-1))   .
        
        Using them and their cyclicity under the symplectic form, the classical equation of motion and the gauge transformations can be expressed as
        
            Q_B  Ψ + ∑_n=2^∞κ^n-1 n!  L_0,n(Ψ^n) = 0   , 
               δ_ΛΨ = Q_B  Λ 
            	+ ∑_n=2^∞κ^n-1 (n-1)!  L_0,n (Λ, Ψ^n-1)   .
        
        Here Ψ and Λ should be restricted to have ghost numbers 2 and 1 respectively. The latter is also taken to have odd statistics.
        
        We see CSFT is simply a result of  reverse engineering string amplitudes as far as generic string vertices 𝒱 are concerned. It is a natural anticipation that a smart choice of string vertices can manifest extra information about the theory however, as we have argued before. This situation already presents itself for open strings: Witten's vertex is superior relative to other string vertices since it makes the theory's underlying associative algebra manifest and the analytic solution becomes accessible as a result—even though any other choice would have accomplished the BV consistency. 
        
        As we shall argue, using hyperbolic vertices 𝒱^L (<ref>) for the action (<ref>) manifests a peculiar recursive structure among its elementary interactions. This structure is topological in nature and somewhat resembles the recursive structure of the stubbed theories <cit.>. On the other hand, some of its features are quite distinct due to the presence of closed Riemann surfaces and their modularity properties.
        
        
        
        § THE RECURSION FOR THE VOLUMES OF SYSTOLIC SUBSETS
        
        
        Before we discuss the recursion in hyperbolic CSFT it is beneficial to investigate a simpler analog to communicate some of the central ideas behind its construction. In this section, after reviewing Mirzakhani's method for computing the WP volumes of the moduli spaces of bordered Riemann surfaces ℳ_g,n(L_i) <cit.>, we form a recursion among the WP volumes of the systolic subsets 𝒱_g,n^L(L_i) for L ≤ L_∗ using the twisting procedure developed in <cit.>. The twisting allows one to restrict the considerations to the vertex regions of CSFT, which is one of the most crucial ingredients for deriving the recursion for hyperbolic vertices in the next section.
        
        
        
         §.§ Mirzakhani's recursion for the volumes of ℳ_g,n(L_i)
        
        
        We begin by reviewing Mirzakhani's method for recursively computing the WP volumes of the moduli spaces ℳ_g,n(L_i) in order to set the stage for our discussion <cit.>. The central idea behind her method is to replace the original volume integration with another integration over a suitable covering space—at the expense of introducing measure factors in the integrand that compensate the overcounting when working on the covering space.
        
        Let us demonstrate this procedure by considering a toy example provided in <cit.>. Imagine we have a circle S^1 described by the interval [0,1] with identified endpoints. We would like to evaluate
        
            I = ∫_0^1 f(x)   dx   ,
        
        where f(x) is a periodic integrable function f(x) = f(x + 1) on S^1. This integral can be replaced with another integral over the universal cover of the circle S^1, the real line ℝ, by noticing the partition of unity
        
            1 = ∑_k ∈ℤsin^2 ( π (x-k) ) π^2   (x-k)^2  ,
        
        and inserting it into the integrand of I
            I = ∫_0^1 1 · f(x) dx 
            	   = ∫_0^1  (  ∑_k ∈ℤsin^2 ( π (x-k) ) π^2 (x-k)^2) f(x)   dx 
            	 
               = ∫_0^1 ∑_k ∈ℤsin^2 ( π (x-k) ) π^2 (x-k)^2 f(x-k)   dx 
            	
               =  ∑_k ∈ℤ ∫_0^1  sin^2 ( π (x-k) ) π^2 (x-k)^2 f(x-k)   dx 
            	= ∫_-∞^∞ sin^2 ( π x) π^2 x^2 f(x)   dx   .
        
        Above we have used the periodicity of f in the second line and commuted the sum out of the integral in the third line. Then we replaced the sum and integral with a single integration over ℝ using the translation property of the integrand.  For f(x)=1 we plot the integrand of the last integral of (<ref>) in figure (<ref>). As one can see, not only [0,1] contributes to the integral, but there are contributions from all of its images in ℝ accompanied by a suitable measure factor. Summing all of them adds up to integrating 1 over [0,1]. 
         
        
        We can formalize this result. Consider a space M and its covering space N with the projection π: N → M. Imagine the top form ε on M and its pullback π^∗ε on N. Also imagine a function h on N. This can be pushed forward to M by
        
            π_∗ h (x) = ∑_y ∈π^-1{ x } h(y)   ,
        
        assuming the sum converges. Then we see
        
            ∫_Mπ_∗ h ·ε = ∫_N  h ·π^∗ε  ,
        
        following the reasoning in (<ref>). There, for instance, M=S^1, N = ℝ, ε = dx and h(x) = sin^2 ( π x) f(x) / π^2 x^2. We note the nontrivial step of this method is coming up with the function h in the cover N from a seed function π_∗ h on the original manifold M, i.e., identifying the appropriate partition of unity like in (<ref>).
        
        Remarkably, Mirzakhani found a way to calculate the volumes of ℳ_g,n(L_i) by identifying the appropriate partitions of unity via the hyperbolic geometry of pair of pants <cit.> and using (<ref>) recursively, refer to appendix D of <cit.> for a derivation accessible to a physicist. Before we illustrate this procedure let us introduce the functions which we call the Mirzakhani kernels
            D_L_1 L_2 L_3   = 2log[exp( L_1/2) + exp(L_2 + L_3/2) /exp(-L_1/2) + exp(L_2 + L_3/2) ]   , 
            
            		T_L_1 L_2 L_3   =
            		log[cosh(L_3  2)+cosh(L_1 + L_2  2) 
            		cosh(L_3  2) + cosh(L_1 - L_2  2) ]   , 
            
            		R_L_1 L_2 L_3   = L_1 - log[cosh(L_2/2) + cosh(L_1 + L_3/2) /cosh(L_2/2)  + cosh(L_1 - L_3/2) ]    .
        
        They have the symmetry properties
        
            D_L_1 L_2 L_3 = D_L_1 L_3 L_2  , 
               T_L_1 L_2 L_3 = T_L_2 L_1 L_3  ,  
               D_L_1 L_2 L_3 + T_L_1 L_2 L_3 = L_1 - T_L_1 L_3 L_2
            		= R_L_1 L_2 L_3
        
        and satisfy
        
            D_L_1 L_2 L_3 L_1 = H_ L_2+L_3, L_1   ,
            	     R_L_1 L_2 L_3 L_1 = 1  2  (H_L_3, L_1+L_2+ H_L_3, L_1-L_2 )   ,
        
        where
        
            H_L_1,L_2 = [ 1 + exp(L_1+L_2 2)]^-1
            	+  [ 1 + exp(L_1-L_2  2) ]^-1  .
        
        We can express these kernels compactly as
        
            D_L_1 L_2 L_3 = F_L_1 - L_2 - L_3 - F_-L_1 - L_2 - L_3
               R_L_1 L_2 L_3 = 1  2(
            		F_L_1 + L_2 - L_3 + F_L_1 - L_2 - L_3 - F_-L_1 + L_2 - L_3 - F_-L_1 - L_2 - L_3)   ,
        
        by introducing
        
            F_L = 2 log[ 1 + exp(L  2)]   .
        
        Some of these identities are going to be useful later.
        
        In terms of the kernels D and R, the relevant partition of the length L_1 for a Riemann surface with n borders of length L_i is given by the Mirzakhani–McShane identity <cit.>
            L_1 = ∑_{γ, δ}∈𝒫_1  D_L_1γδ 
            	+∑_i = 2^n ∑_γ∈𝒫_i R_L_1L_i γ  .
        
        Here the sets 𝒫_i are certain collections of simple geodesics. For i=1, it is the set of pairs of simple closed internal geodesics that bound a pair of pants with the first border L_1, and for i = 2, ⋯, n, it is the set of simple closed internal geodesics that bound a pair of pants with the first and i-th borders L_1, L_i. We often imagine these pants are “excised” from the surface as shown in figure <ref> to interpret various different terms in the expressions below.
         
        
        
        The volumes of the moduli spaces ℳ_g,n(L_i) are given by the integrals
        
            V ℳ_g,n(L_i) = ∫_ℳ_g,n(L_i)vol_WP  = 
            	∫_ℳ_g,n(L_i)ω_WP^3g-3+n (3g-3+n)!  ,
        
        for which we take V ℳ_0,3(L_i)= 1 to set the units. This definition is implicitly taken to be divided by 2 for (g,n) = (1,1) due to the presence of a nontrivial ℤ_2 symmetry <cit.>. This simplifies the expressions below.
        
        It is possible to derive the following recursion relation by substituting the partition of identity (<ref>) to the integrand of (<ref>) for 2g -2 + n > 1 and n ≥ 1 to obtain
        
            L_1 · V ℳ_g,n (L_i)    = 
            	∑_i=2^n ∫_0^∞ℓ d ℓ  R_L_1L_iℓ  V ℳ_g,n-1(ℓ, 𝐋∖{L_i }) 
            	
                + 1  2∫_0^∞ℓ_1 d ℓ_1  ∫_0^∞ℓ_2 d ℓ_2  
            	D_L_1 ℓ_1 ℓ_2[
            	V ℳ_g-1, n+1 (ℓ_1, ℓ_2, 𝐋) 
            	+ ∑_stable V ℳ_g_1, n_1 (ℓ_1,  𝐋_1) · V ℳ_g_2, n_2 (ℓ_2,  𝐋_2)
            	]   ,
        
        by 	“unrolling” the integral over the relevant fibers in the covering space, see <cit.> for mathematical details. Here
        
            𝐋 = {L_2, ⋯, L_n }  ,     𝐋_1 ∪𝐋_2 = 𝐋   ,     𝐋_1 ∩𝐋_2 = ∅  ,
        
        and the “stable” in (<ref>) denotes the sum over all g_1,g_2,n_1,n_2,𝐋_1, 𝐋_2 that satisfy
        
            g_1 + g_2 = g  ,    n_1 + n_2 = n + 1   ,    2g_1 - 2 + n_1 > 0   ,    2g_2 - 2 + n_2 > 0    .
        
        The pictorial representations of various terms in the relation (<ref>) along with their associated pants excisions are given in figure <ref>. The first few of the volumes are given in table <ref>. Observe that it is possible to ignore the first term in the second line of (<ref>) (i.e., nonseparating D term) as long as surfaces of genus zero are concerned.
        
        
        
        We note the formula (<ref>) doesn't apply to 1-bordered tori directly and this case requires special treatment. Nevertheless, it can be still found using the partition of unity (<ref>)
            V ℳ_1,1 (L_1) = 1  21  L_1∫_0^∞ℓ  d ℓ D_L_1 ℓℓ·
            	V ℳ_0,3(L_1, ℓ, ℓ) 
            	= π^2  12 + L_1^2  48  ,
        
        and together with Vℳ_0,3(L_i) = 1, the recursion (<ref>) can be used to find the volumes for the rest of the moduli spaces of surfaces with at least one border. We highlight the presence of the extra 1/2 in front of this equation: this comes from the ℤ_2 symmetry of one-bordered tori. 
        
        
        Finally, we point out the formula, known as the dilaton equation,
        
            (2g-2+n)   V ℳ_g,n(L_i) =  1  2 π i V ℳ_g,n+1' (L_i, 2 π i)   ,
        
        can be used for the volumes of the moduli spaces of surfaces without borders <cit.>. Here the prime stands for the derivative with respect to the border length L_n+1.
        
        
        
         §.§ The recursion for the volumes of 𝒱_g,n^L(L_i)
        
        
        In this subsection we generalize Mirzakhani's recursion discussed in the previous subsection to the recursion for the WP volumes of the systolic subsets 𝒱_g,n^L(L_i). Only the cases with L_i = L are relevant for CSFT, but as we shall see, it will be necessary to consider L_i ≠ L to construct the desired recursion. We always take L ≤ L_∗≡ 2 sinh^-1 1 for reasons that are going to be apparent soon, unless stated otherwise. 
        
        The volumes of the systolic subsets are given by
        
            V 𝒱_g,n^L(L_i) = ∫_𝒱_g,n^L(L_i) vol_WP
            	=∫_ℳ_g,n(L_i) 1_𝒱_g,n^L(L_i) ·vol_WP  .
        
        Here 1_𝒱_g,n^L(L_i) is the indicator function for the region 𝒱_g,n^L(L_i), that is
        
            1_𝒱_g,n^L(L_i) (Σ) =
            	
            		0    for  sys(Σ) < L 
            
            		1    for  sys(Σ) ≥ L
            	  .
        
        The volume V 𝒱_g,n^L(L_i) can be understood as integrating the indicator function over the moduli space with respect to the WP volume form vol_WP. The definition (<ref>) is implicitly divided by 2 for (g,n) = (1,1) like before.
        
        It is beneficial to have a “regularized” representation for the indicator function 1_𝒱_g,n^L(L_i)  by considering its slight generalization <cit.>. Let us denote the set S(Σ) to be the set of simple closed “short” geodesics γ of Σ∈ℳ_g,n(L_i) with γ < L. For L ≤ L_∗ the curves in the set S(Σ) cannot intersect by the collar lemma (<ref>), hence
        
            |S(Σ)| ≤ 3g - 3 + n < ∞  ,
        
        and the following holds
        
            Ω_t(Σ) ≡ (1+t)^|S(Σ)|   = ∑_k=0^|S(Σ)||S(Σ)|k  t^k
            	= ∑_μ⊆ S(Σ) t^|μ|
            	= ∑_c ∈ M(Σ) ∏_γ∈π_0(c) t  θ(L - γ ) 
            	  ,
        
        for t ∈ℝ, as each sum and product is finite. Here μ⊆ S(Σ) is a collection of short geodesics on Σ, which we obtained by binomial theorem. It can be empty.
        
        We also define M(Σ) to be the set of primitive multicurves on the surface Σ. A multicurvec is defined as the homotopy class of one-dimensional submanifolds on Σ whose components are not homotopic to borders. Primitive refers to no disjoint component of c ∈ M(Σ) is homotopic to each other. The primitive multicurve can be thought of the union of pairwise-disjoint simple closed geodesics on the surface Σ. From this definition (<ref>) directly follows after employing the step functions. Finally π_0(c) stands for the set of components of the multicurve c.
        
        The equation (<ref>) then shows
        
            1_𝒱_g,n^L(L_i)  (Σ) = lim_t → -1Ω_t(Σ)
            	= Ω_-1(Σ)
            	= ∑_c ∈ M(Σ) ∏_γ∈π_0(c)  (-1)  θ(L - γ )   ,
        
        since the product and sum are finite. The indicator function, and its regularized version, are MCG-invariant because they only depend on the set S(Σ). This is also manifest in (<ref>) since the terms in the sum map to each other under large diffeomorphisms. Based on this analysis the volumes (<ref>) take the form
        
            V 𝒱_g,n^L(L_i)  
            	= ∫_ℳ_g,n(L_i) Ω_-1·vol_WP 
            	= ∫_ℳ_g,n(L_i) ∑_c ∈ M(Σ) ∏_γ∈π_0(c) (-1)  θ(L - γ ) ·vol_WP  .
        
        
        As before, we need to find an appropriate partition of the function Ω_-1(Σ) (<ref>) and apply (<ref>). However we now have a somewhat complicated integrand that involves a sum over the multicurves, rather than a mere constant, and using the Mirzakhani-McShane identity (<ref>) wouldn't be sufficient as it is. We can handle this situation by classifying the multicurves c appropriately and refining (<ref>) with respect to this classification. This would lead to the desired recursion à la Mirzakhani.
        
        Let us investigate this in more detail. We can think of the multicurves c as decomposition of the surface Σ into a disjoint collection of connected surfaces. Having done this, let us isolate the connected surface Σ_1 which contains the border L_1. Then we have the following categorization of multicurves:
        
          *  Those having a component γ that bounds a pair of pants with the border L_1 and L_i. The surface Σ_1 = P_γ i is a pair of pants in this case.
        	
          *  Those having two components γ and δ that together with the border L_1 bound a pair of pants.  The surface Σ_1 = P_γδ is a pair of pants in this case as well.
        	
          *  Those such that Σ_1 is not a pair of pants. 
        
        We remind that each multicurve c may contain any number of components on the remaining surface Σ∖Σ_1. These distinct situations are shown in figure <ref>.
         
        
        Now we evaluate the sum over M(Σ) in (<ref>) for each of these cases. For the first case we find the contribution to the sum over the multicurves is
        
            Ω_-1(Σ)
            	   ⊃ - ∑_i=2^n ∑_γ∈𝒫_iθ(L-γ) ∑_c∈ M(Σ∖ P_γ i)∏_δ∈π_0(c) (-1)  θ(L-δ) 
               =-∑_i=2^n ∑_γ∈𝒫_iθ(L - γ)  Ω_-1(Σ∖ P_γ i)   ,
        
        after singling out the geodesic γ∈𝒫_i that bounds Σ_1=P_γ i in the decomposition (<ref>). Similarly we find the contribution
        
            Ω_-1(Σ)  ⊃∑_{γ,δ}∈𝒫_1θ(L- γ)  θ(L- δ)  Ω_-1(Σ∖ P_γδ)   ,
        
        for the second case, after singling both γ and δ. Note that the surface Σ∖ P_γδ can be disjoint in this case. For those cases we implicitly consider the function Ω_-1(Σ∖ P_γδ) to be the product of Ω_-1 associated with these two disjoint surfaces.
        
        The third case is somewhat more involved. We begin by rewriting their total contribution as
        
            Ω_-1(Σ) ⊃1/L_1∑_c∈ M'(Σ) L_1 ∏_γ∈π_0(c) (-1)  θ(L-γ) ,
        
        where M'(Σ) is the set of primitive multicurves that excludes the components bounding a pair of pants with the border L_1 whose contributions have already been considered in (<ref>) and (<ref>). This form allows us to use the Mirzakhani–McShane identity (<ref>) for 
        
            L_1    =  
            	∑_{γ,δ}∈𝒫_1^(1) D_L_1 γδ +
            	∑_i=2^m_1∑_γ∈𝒫_i^(1) R_L_1 L_i γ + 
            	∑_i=m_1+1^m ∑_γ∈𝒫_i^(1) R_L_1 L_i γ  ,
        
        in the summand (<ref>). This partition is associated with the surface Σ_1 determined by c ∈ M' (Σ) and the superscript on the set 𝒫_i^(1) is there to remind us that we only consider the relevant pair of pants over the surface Σ_1, which is assumed to have m borders. We split the contribution of the R-terms into two: the first m_1 of the borders are assumed to be borders of Σ, while the rest are the components of the relevant multicurve c, see the second and third surfaces in figure <ref>.
         
        
        After this substitution we commute the sums over 𝒫_i^(1) in (<ref>) with the sum over M'(Σ) in  (<ref>). Note that this is allowed since M'(Σ) is a finite set. To do this, we point out that there are three independent cases for a simple closed geodesic γ to bound a pant with the border L_1 in Σ_1:
        
          *  The geodesic γ bounds a pant P_γδ with the border L_1 and some other internal simple closed geodesic δ of Σ_1. These are not components of the multicurve c ∈ M'(Σ) by design.
        	
        	
          *  The geodesic γ bounds a pant P_γ i with the border L_1 and L_i of Σ_1, which is also a border of Σ. This border is not a component of a multicurve c ∈ M'(Σ) by design.
        	
        	
          *  The geodesic γ bounds a pant P_γδ with the border L_1 and δ of Σ_1, which is not a border of Σ. This border is a component of c ∈ M'(Σ) by design.
        
        These can be associated with the three sums in (<ref>), also refer to figure <ref>. Note that the geodesic γ above is not a component of a multicurve by the construction. These cases yield the total contributions
        
            Ω_-1(Σ)    ⊃1  L_1∑_{γ,δ}∈𝒫_1 D_L_1 γδ Ω_-1(Σ∖ P_γδ)   , 
            Ω_-1(Σ)     ⊃1  L_1∑_i=2^n ∑_γ∈𝒫_i R_L_1 L_i γ Ω_-1(Σ∖ P_γ i)   , 
            Ω_-1(Σ)    ⊃
            		- 1  L_1∑_{γ, δ}∈𝒫_1[R_L_1 γδ θ(L-γ) +
            		R_L_1 δγ θ(L-δ)] Ω_-1(Σ∖ P_γδ)   ,
        
        respectively after summing over the multicurves c ∈ M'(Σ) following the reasoning around (<ref>). Notice  the symmetrization over γ and δ in the last equation after the sum is performed. This is due to {γ, δ}∈𝒫_1  being an unordered set, while both of the situations where only one of them belongs to c ∈ M'(Σ) contributes.
        
        Combining the exhaustive contributions (<ref>) (<ref>) and (<ref>) we find the desired partition
        
            L_1 ·Ω_-1(Σ) = 
            	∑_i=2^n ∑_γ∈𝒫_iR_L_1 L_i γ Ω_-1(Σ∖ P_γ i)
            	+∑_{γ, δ}∈𝒫_1D_L_1γδ Ω_-1(Σ∖ P_γδ)   ,
        
        for which we defined the twisted Mirzakhani kernels
            R_L_1 L_2 L_3 = R_L_1 L_2 L_3 - L_1  θ(L - L_3)   , 
               D_L_1 L_2 L_3  = D_L_1 L_2 L_3  - R_L_1 L_2 L_3  θ(L - L_2 )  
            	- R_L_1 L_3 L_2  θ(L - L_3 )  + L_1  θ(L - L_2 )  θ(L - L_3 )  
            	   .
        
        We highlight that the symmetries in (<ref>) are also satisfied and taking L→ 0 reduces them to the Mirzakhani kernels. Similarly the partition (<ref>) reduces to (<ref>) in this limit.
        
        Upon inserting the partition (<ref>) to (<ref>) and exactly repeating Mirzakhani's analysis in <cit.> we conclude for 2g -2 + n > 1 and n ≥ 1 the systolic volumes satisfy a recursion relation among themselves
        
            L_1 · V 𝒱^L_g,n (L_i)    = 
            	∑_i=2^n ∫_0^∞ℓ d ℓ R_L_1L_iℓ  V 𝒱^L_g,n-1(ℓ, 𝐋∖{L_i }) 
            	
                + 1  2∫_0^∞ℓ_1 d ℓ_1  ∫_0^∞ℓ_2 d ℓ_2  D_L_1 ℓ_1 ℓ_2[
            	V 𝒱^L_g-1, n+1 (ℓ_1, ℓ_2, 𝐋) 
            	+ ∑_stable V 𝒱^L_g_1, n_1 (ℓ_1,  𝐋_1) · V 𝒱^L_g_2, n_2 (ℓ_2,  𝐋_2)
            	]   ,
        
        whenever L ≤ L_∗. Like before, we take V 𝒱^L_0,3 (L_i) = V ℳ_0,3 (L_i) = 1 to set the units. We are going to discuss the case (g,n)=(1,1) shortly.
        
        The first few volumes V 𝒱^L_g,n (L_i) are listed in table <ref>. The results of this systolic recursion can be shown to be independent of the choice of the border L_1 following the arguments for the counterpart statement for (<ref>) <cit.>. The dilaton equation (<ref>) applies to the systolic volumes as well and can be used to find the systolic volumes for the surfaces without borders. We remark in passing that this particular twisting procedure has been used in <cit.> to establish that hyperbolic vertices (<ref>) satisfy the geometric master equation (<ref>), providing an alternative to the argument of <cit.>.
        
        
        Take note that using step functions θ(L-L_i) wasn't essential for the twisting procedure itself. For instance, we could have formed a recursion similar to (<ref>) for the integrals of the functions <cit.>
            ζ (Σ) = ∑_c ∈ M(Σ) ∏_γ∈π_0(c)  f(γ)   ,
         
        provided f is a sufficiently well-behaved real integral function. In the twisted Mirzakhani kernels (<ref>) this simply amounts to replacing -θ(L-γ) with f(γ). In particular we can use a suitable family of functions f(γ) that limits to -θ(L-γ), which may be particularly helpful to evaluate (<ref>) efficiently.
        
        In order to get an intuition for the systolic volumes it is instructive to evaluate (<ref>) for the first nontrivial case, (g,n)=(0,4),
        
            V 𝒱^L_0,4 (L_i)    = 
            	1  L_1∑_i=2^4 ∫_0^∞ℓ d ℓ R_L_1L_iℓ  
            	= V ℳ_0,4 (L_i) - ∑_i=2^4 ∫_0^L  ℓ d ℓ
               
            	= V ℳ_0,4 (L_i) - 3  2 L^2 
            	= 2 π^2 + 1  2∑_i=1^4 L_i^2 - 3  2 L^2   .
        
        Note that for all L_i ≥ 0
            V 𝒱^L_0,4 (L_i)  ≥ 2 π^2 - 3  2  L^2 > 0     when     0 ≤ L ≤  L_∗ = 2 sinh^-1 1   .
        
        which shows this systolic volume is always positive in the allowed regime, as it should be. 
        
        Let us illustrate another way to perform the same computation. Recall that we exclude surfaces whose systoles are smaller than L from the systolic subset 𝒱^L_0,4 (L_i). This means we need to subtract the WP volume associated with the region 
        
            0 <  ℓ≤ L ≤ L_∗  ,      0 ≤τ < ℓ  ,
        
        from the the WP volume V ℳ_0,4 (L_i) as part of evaluating 𝒱^L_0,4 (L_i).  Here (ℓ, τ) are the Fenchel-Nielsen coordinates determined by a particular way to split a four-bordered sphere into two pairs of pants with an internal simple closed geodesic ℓ.
        
        The volume of this excluded region is L^2/2 since the MCG acts freely there, i.e., it always maps surfaces outside of (<ref>)—there isn't any multiple counting of surfaces. Recall that the effect of MCG is to exchange geodesics on the surface with possible twists. The restriction on the twist τ has already eliminated the possibility of the Dehn twists. Further, two short geodesics never intersect on the surface by the collar lemma (<ref>), so short geodesics always get exchanged with the longer ones, i.e., the MCG takes us outside of the region (<ref>). For the additional multiplication by 3, notice the decomposition (<ref>) is just one way to split a four-bordered sphere: there are two other distinct channels to perform a similar decomposition. So one should subtract 3L^2/2 from the total volume to obtain (<ref>). We emphasize again that these regions can't intersect by the collar lemma and there was no oversubtraction in our results.
        
         
        Now we turn our attention to (g,n)=(1,1) required to run (<ref>). This case requires special attention like (<ref>). The partition of Ω_-1 for a one-bordered torus Σ is
        
            L_1 ·Ω_-1(Σ) = ∑_γ∈𝒫_1[ D_L γγ - L_1  θ(L-γ) ]
            	 Ω_-1(Σ∖ P_γγ)   .
        
        This can be argued similarly to (<ref>) after observing there is only one geodesic γ that has been cut, which is either part of a primitive multicurve (the second term) or isn't (the first term), refer to figure <ref>. Using this partition we then find
        
            V 𝒱^L_1,1 (L_1)    = 1  21  L_1 ∫_0^∞ℓ d ℓ [ D_Lℓℓ - L_1 θ(L- ℓ) ]
            	· V 𝒱^L_0,3(L, ℓ, ℓ) 
            	= V ℳ_1,1(L_1) - 1  4 L^2 
            	  .
        
        This result can be alternatively argued from the geometric perspective similar to V 𝒱^L_0,4 (L_i). We also have
        
            V 𝒱^L_1,1 (L_1) ≥π^2  12 -1  4 L^2 > 0       when    
            	0 ≤ L ≤ L_∗ = 2 sinh^-1 1   ,
        
        for all L_1 ≥ 0 and this volume is positive in the allowed range.
        
        For the higher-order cases the systolic volumes are somewhat more complicated. We subtract the volumes of the regions with short geodesics from the the total volume of the moduli space, however the MCG action is not necessarily free in these subtracted regions and the evaluations of their volumes requires suitable weighting of the integrand by Mirzakhani kernels as a result. The computations for some sample systolic volumes are given in appendix <ref>.
        
        We can further crosscheck the results for V 𝒱_1,1(L_1) and  V 𝒱_0,4(L_i) by numerically integrating the WP metric derived using the Polyakov conjecture <cit.>. The dependence of V 𝒱^L_1,1(L_1) on L for L_1 = 0 and L_1 = π/2, along with their fits to the quadratic polynomial
        
            f(L) = c_1 + c_2   L   ,
        
        are shown in figure <ref> for instance. It is apparent that we obtain consistent results. For the details of this computation refer to appendix <ref>.
        
        
        As a final remark in this section, we highlight the systolic recursion (<ref>) simplifies upon restricting to genus 0 surfaces. Like in Mirzakhani's recursion, we don't need to consider the first term in the second line of (<ref>). This means for n > 3
            L_1 · V 𝒱^L_0,n (L_i)    = 
            	∑_i=2^n ∫_0^∞ℓ d ℓ R_L_1L_iℓ  V 𝒱^L_0,n-1(ℓ, 𝐋∖{L_i }) 
            	
                + 1  2∫_0^∞ℓ_1 d ℓ_1  ∫_0^∞ℓ_2 d ℓ_2  D_L_1 ℓ_1 ℓ_2∑_stable V 𝒱^L_0, n_1 (ℓ_1,  𝐋_1) · V 𝒱^L_0, n_2 (ℓ_2,  𝐋_2)
            	  .
        
        This again corresponds to considering only the first and second types of pants excisions in figure <ref>. 
        
        Remember it is possible to take 0 < L ≤∞ for string vertices in the classical CSFT <cit.>. It is then natural to ask whether the condition on the threshold length, L ≤ L_∗, can also be eliminated for (<ref>). However, this doesn't follow from our construction. Clearly we can't take 0 < L ≤∞ for the general border lengths: taking L too large while keeping L_i small would eventually make systolic volumes negative, see table <ref>. So the bound for them should persist in general. Although we can argue it can be relaxed to
        
            0 < L ≤ 2 L_∗ = 4 sinh^-11 ≈ 3.53   ,
        
        since two intersecting closed geodesics always traverse each other at least twice. We point out (<ref>) remains positive with this weaker bound, but (<ref>) doesn't apply anymore. It is still in the realm of possibilities (<ref>) to hold when the border lengths L_i scale with L while taking some of them the same however. We comment on this possibility more in the discussion section <ref>. 
        
        
        
        § THE RECURSION FOR HYPERBOLIC STRING VERTICES
        
        
        We now turn our attention to deriving the recursion relation satisfied by the elementary vertices of hyperbolic CSFT. We remind that they are given by integrating the string measure over the systolic subsets[Strictly speaking, only the amplitudes ⟨𝒱_g,n(L_i) | with L_i = L for 0<L≤ L_∗ form string vertices. However we also denote the cases with generic L_i and L as string vertices and/or elementary interactions for brevity.]
            ⟨𝒱_g,n(L_i) | =
            	∫_𝒱^L_g,n (L_i) ⟨Ω_g,n (L_i)  | 
            	=  ∫_ℳ_g,n (L_i)1_𝒱_g,n^L(L_i)  ·⟨Ω_g,n (L_i)  |
        
        and the resulting recursion would be among ⟨𝒱_g,n (L_i)  | ∈ (ℋ^∗)^⊗ n.  It is useful to introduce
        
            ⟨ω_g,n (L_i) | = 1_𝒱_g,n^L(L_i)  ·⟨Ω_g,n (L_i)  |   ,
        
        in this section. The indicator function 1_𝒱_g,n^L(L_i) is already given in (<ref>).
        
        The integrations (<ref>) above take place in the moduli space of Riemann surfaces with geodesic borders of length L_i, ℳ_g,n (L_i).  In the context of CSFT, on the other hand, it is often considered that such integrations take place in the sections of moduli space of punctured Riemann surfaces endowed with suitable local coordinates around the punctures 𝒫_g,n.  In hyperbolic CSFT, however, they are equivalent since both the string measure (and the local coordinate data it contains) and the regions of integrations are defined by the bordered Riemann surfaces through grafting. More precisely, gr_∞' ℳ_g,n (L_i) is a section of 𝒫_g,n by gr_∞' being a homeomorphism <cit.>. We are performing the integration over this section of 𝒫_g,n.
        
        
        
         §.§ b-ghost insertions for the Fenchel-Nielsen deformations
        
        
        In our arguments we are going to need to excise pairs of pants from the surface states. For this we ought to understand the Schiffer vectors corresponding to the Fenchel-Nielsen deformations. So consider a Riemann surface Σ_g,n(L_i) together with its pants decomposition. The lengths ℓ_i and the twists τ_i of the seams of the pants define local coordinates over the moduli space (<ref>).  Let us denote the Schiffer vectors corresponding to the coordinate vectors / τ_i and / ℓ_i by u_i and v_i respectively. In order to derive their expression, and subsequently the associated b-ghost insertions, we need to find the transition functions between two coordinate patches for pants after twisting and increasing the length of the seams. Similar analysis has been considered with different levels of detail in <cit.>, however we are going to be more explicit.
        
        Let z_L, z_R be the coordinate patches on the left and right pants L,R for the semi-infinite flat cylinder grafted i-th seam whose length is ℓ_i. They are related to each other by (observe figure <ref>)
        
            w_L(z_L; L^(L)_j)     w_R(z_R; L^(R)_j) = e^2 π iτ_i  / ℓ_i
                z_L = w_L^-1(  e^2 π iτ_i  / ℓ_i w_R(z_R ; L^(R)_j) ; L^(L)_j
            	) = f_LR (z_R ; L^(L)_j, L^(R)_j)
            	  ,
        
        where w_L, w_R are the local coordinates for the generalized hyperbolic three-string vertex around the puncture at z=0, whose expression is given in appendix <ref>. The formula above is just a consequence of performing the sewing fixture (<ref>) for the i-th seam. We importantly highlight that the definition of coordinates depends on the lengths of the borders of the left (L^(L)_j) and right (L^(R)_j) pairs of pants with j=1,2,3. However it doesn't depend on the relative twist τ_i between them. We take L^(L)_1 = L^(R)_1 = ℓ_i.[If the surface is a one-bordered torus, we take z = z_L = z_R and w_R= w_R(z) to be the local coordinate around z=1 instead. We also specify the border lengths by L^(L)_j = L^(R)_j= (ℓ_i, ℓ_i, L_3).] 
        
        The aim is finding the Schiffer vectors v_i and u_i and their associated b-ghost insertions based on (<ref>) now. Begin by investigating the twist deformation τ_i →τ_i + δτ_i while keeping all other coordinates the same. Using (<ref>) we simply find
        
            u_i(w_R) = - 2 π i ℓ_i  w_R   .
        
        The corresponding b-ghost insertion according to (<ref>) is then given by
        
            b(u_i) = - 2 π i ℓ_i[
            	∮ dw_R   w_R   b(w_R) - ∮  dw_R  w_R  b(w_R)
            	] = - 2 π i ℓ_i  b_0^-   ,
        
        applied to the states inserted in the local coordinates w_R around the puncture z_R=0. It has the same expression in w_L by L ↔ R symmetry. This is a well-known result from the ordinary factorization analysis <cit.>.
        
        Now we return our attention to the variation ℓ_i →ℓ_i + δℓ_i. This is slightly more involved compared to the twist deformations since the definition of the local coordinates themselves depends on the pair of pants. Nonetheless we can express the associated Schiffer vector v_i as a sum of the following vectors using (<ref>)
            v_i =
            	v^(L)_i(w_L) + v^(R)_i(w_R)+ 2 π i τ_i ℓ_i^2  w_R   ,
        
        where
        
            v^(L)_i (w_L) = -  w_L (z_L(w_L )  ; L^(L)_j )  ℓ_i  .
        
        This holds similarly for L ↔ R and their anti-holomorphic counterparts.
        
        We can understand the reasoning behind v_i (<ref>) as follows. When we vary the length of the seam ℓ_i, the local coordinate w_L (w_R), as a function of z_L (z_R) changes, see (<ref>). Keeping the coordinates w_L, w_R and their identification (<ref>) fixed, the vector v^(L)_i (v^(R)_i) is then simply one part of the Schiffer vector v_i. Clearly, the parts associated with L and R add up since we consider infinitesimal changes. Finally we have another part associated with changing ℓ_i in the identification (<ref>). However, this produces a b_0^- insertion like in (<ref>), which would vanish upon multiplying the twist's insertion b(u_i) (<ref>). These together produce the Schiffer vector v_i.
        
        The nonvanishing part of the b-ghost insertion for ℓ_i →ℓ_i + δℓ_i is then given by
        
            b(v_i) = b(v^(L)_i) + b(v^(R)_i)   ,
        
        where
        
            b(v^(L)_i) 
            	   = - ∮ d w_L   w_L (z_L (w_L ) ; L^(L)_j)  ℓ_i  b(w_L)  
            	 - ∮ d w_L  w_L (z_L (w_L ) ; L^(L)_j)  ℓ_i b (w_L) 
            	 
               = 1  2 π[
            	 1 ρ_1∂ρ_1 ∂λ_1( b_0 + b_0 )
            	 + λ_1 ( λ_2^2 - λ_3^2)  (1+ λ_1^2)^2ρ_1 
            	 ( b_1 + b_1 )
            	 + ⋯]   ,
        
        for which we take
        
            2 πλ_j  = L_j^(L)  ,     
            	2 πλ_1 = ℓ_i   ,
        
        to simplify the presentation. Here ρ_1 = ρ_1 (λ_j) is the mapping radius associated with the local coordinate around the puncture z_L =0, see (<ref>). This b-ghost insertion (<ref>) acts on the left pair of pants but  similar considerations apply after exchanging L ↔ R for the right one. 
        
        The same b-ghost insertion applies to changing the border length of any pair of pants. This has already been presented in (<ref>), which we report here again for L_i > 0
            ⟨Σ_0,3( L_1, L_2 , L_3 )| (   𝔅⊗𝕀⊗𝕀)
            	
               = 
            	⟨Σ_0,3( L_1, L_2 , L_3 )|
            	[ 1  2 π(
            	1 ρ_1∂ρ_1 ∂λ_1 ( b_0 + b_0 )
            	+λ_1    (λ_2^2  - λ_3^2)  (1+ λ_1^2)^2 ρ_1 
            	( b_1+ b_1 )
            	+ ⋯) ⊗𝕀⊗𝕀]
            	  .
        
        In these general situations we denote b(v_i) by 𝔅. There may be multiple applications of 𝔅 to the surface states if we consider the deformation of lengths of multiple borders simultaneously. In such cases 1,2,3 labels in 𝔅 should be permuted accordingly and 𝔅 should act on the border whose length is deformed. 
        
        We emphasize strongly that the 𝔅 insertion depends on all of the border lengths of a given pair of pants—not just to the border whose length has shifted. This is quite unorthodox, especially when contrasted to the ordinary factorization analysis for off-shell amplitudes <cit.>. For the latter the associated b-ghosts insertion b_0^- b_0^+ doesn't depend on the details of the surface they are acting on whatsoever. They take a universal form. The b-ghost insertion due to the Fenchel-Nielsen deformation 𝔅 b_0^-, on the other hand, naturally depends on the particular pants decomposition.
        
        One may worry that such dependence may obstruct factorizing generic off-shell amplitudes by requiring “global” knowledge of the surface. However, in the view of (<ref>) and (<ref>), we can decompose the b-ghost insertions into two disjoint parts where each part exclusively depends on either the left or right pant—but not both of them simultaneously. So after cutting the surface along the i-th seam any dependence on the opposite pants disappears. While they still depend on the pants decomposition itself, the parts of the b-ghost insertions can be separated in a way that the knowledge of the opposite part of the surface is irrelevant at least. This will be sufficient for our purposes.
        
        
        
         §.§ The recursion for hyperbolic vertices
        
        
        In the previous subsection we have derived the b-ghost insertions associated with the Fenchel-Nielsen deformations (<ref>). Using such insertions we can factorize the string measure ⟨Ω_g,n (L_i) | according to the different types of pants excisions shown in figure <ref>. They take the form (twists are taken respect to the L pants):
        
          * R-term for the i-th border:
        	
            ⟨Ω_g,n(L_i)| =  d ℓ∧d τℓ ∑_ϵ = ±    [ 
            		⟨𝒱_0,3(L_1, L_i, ϵℓ) | 
            		⊗⟨Ω_g, n - 1(- ϵℓ, 𝐋∖{ L_i }) |
            		 ] | 𝔖 (τ) ⟩  .
        
          * Nonseparating D-term:
        	
            ⟨Ω_g,n(L_i)|    = d ℓ_1 ∧d ℓ_2∧ d τ_1 ℓ_1 ∧d τ_2 ℓ_2
               ∑_ϵ_1,ϵ_2 = ± [ 
            	⟨𝒱_0,3(L_1, ϵ_1 ℓ_1, ϵ_2  ℓ_2) | 
            	⊗⟨Ω_g-1, n + 1(- ϵ_1 ℓ_1, - ϵ_2 ℓ_2, 𝐋 ) |
            	 ]   | 𝔖 (τ_1)  ⟩_1   | 𝔖 (τ_2)  ⟩_2    .
        
          * Separating D-term:
        		
            ⟨Ω_g,n(L_i)|    = d ℓ_1 ∧  d ℓ_2 ∧d τ_1 ℓ_1∧d τ_2 ℓ_2
               ∑_ϵ_1,ϵ_2 = ±[ 
            		⟨𝒱_0,3(L_1, ϵ_1 ℓ_1, ϵ_2  ℓ_2) | 
            		⊗∑_stable⟨Ω_g_1,n_1(-ϵ_1 ℓ_1, 𝐋_1) | ⊗⟨Ω_g_2,n_2(-ϵ_2 ℓ_2,  𝐋_2) |
            		 ] | 𝔖 (τ_1)  ⟩_1  | 𝔖 (τ_2) ⟩_2    .
        
        
        These results can be established following a similar analysis to <cit.> while being mindful about various signs, for example (<ref>). We point out this decomposition only holds locally on ℳ_g,n (L_i). Above we have adopted the conventions stated in (<ref>) and (<ref>)
            ⟨𝒱_0,3 ( L_1, L_2, L_3) |  ≡⟨Σ_0,3( |L_1|, |L_2| , |L_3| )|  
            	 (  𝔅^⊗ θ(-L_1)⊗𝔅^⊗ θ(-L_2)⊗𝔅^⊗ θ(-L_3))   , 
               ⟨Ω_g,n ( L_1, ⋯, L_n) | ≡⟨Ω_g,n (|L_1|, ⋯, |L_n|)  |  (𝔅^⊗ θ(-L_1)⊗⋯⊗𝔅^⊗ θ(-L_n))   ,
        
        of having negative lengths. Since the 𝔅 insertion either acts on the left or right of the seam (but not both at the same time), having a negative length interpretation is quite natural in this context and is associated with the orientation of the pants with respect to each other.
        
        Observe that we insert the non-level-matched bivector | 𝔖 (τ) ⟩ (<ref>) to the entries of the bras in (<ref>), which are supposed to act only on the level-matched states (<ref>). This means that factorizations above come with global phase ambiguities. As we shall see shortly, however, this ambiguity is going to disappear for the final expression.[The same ambiguity also presents itself in the ordinary factorization analysis of the off-shell amplitudes with the string propagator before summing over the twists and disappear only afterward. We observe an analogous behavior. For the recent attempts of relaxing level-matching condition (<ref>) in CSFT, see <cit.>.]
        
        Now, we can promote (<ref>) to the partition of the hyperbolic string measure (<ref>) as
        
            |L_1| ·⟨ω_g,n (L_i) |    = 
            	∑_i=2^n  ∑_γ∈𝒫_i∑_ϵ_γ = ± d γ∧d τ_γγ [ ⟨ℜ( L_1, L_i, ϵ_γγ) |  ⊗⟨ω_g, n - 1(- ϵ_γγ, 𝐋∖{ L_i }) |
            	] | 𝔖(τ_γ) ⟩_γ
               +
            	∑_{γ, δ}∈𝒫_1∑_ϵ_γ, ϵ_δ = ±
            	d γ∧ d δ∧d τ_γγ∧d τ_δδ [
            	⟨𝔇( L_1, ϵ_γγ,  ϵ_δδ) |
            	⊗(
            	 ⟨ω_g-1, n + 1(- ϵ_γγ, - ϵ_δδ, 𝐋 ) |
            	
               +∑_stable⟨ω_g_1,n_1(-ϵ_γγ, 𝐋_1) | ⊗⟨ω_g_2,n_2(-ϵ_δδ,  𝐋_2) |
            	)
            	] | 𝔖(τ_γ) ⟩_γ  | 𝔖(τ_δ) ⟩_δ  ,
        
        in light of (<ref>). Above we have collected the cubic parts into the string kernels (<ref>)
            ⟨ℜ(L_1,L_2,L_3) |    = R_|L_1| |L_2| |L_3| ⟨𝒱_0,3(L_1,L_2,L_3)|   ,
            ⟨𝔇(L_1,L_2,L_3) |    = D_|L_1| |L_2| |L_3| ⟨𝒱_0,3(L_1,L_2,L_3)|   .
        
        There is an implicit dependence on the threshold length L in these objects.
        
        Upon integrating (<ref>) over the moduli space ℳ_g,n(L_i) and repeating the unrolling trick (<ref>) for evaluating such integrals, we obtain the desired recursion relation for 2g-2+n > 1 and n ≥ 1 (refer to figure <ref>)
        
            |L_1| ·⟨𝒱_g,n (L_i) | = ∑_i=2^n  ∫_-∞^∞  d ℓ [ ⟨ℜ (L_1, L_i, ℓ) | ⊗ ⟨𝒱_g, n-1 (-ℓ, 𝐋∖{L_i }) |
            	] |ω^-1⟩
                + 1  2∫_-∞^∞  d ℓ_1   ∫_-∞^∞  d ℓ_2  [⟨𝔇 (L_1, ℓ_1, ℓ_2) | ⊗ (⟨𝒱_g-1,n+1 (-ℓ_1, -ℓ_2, 𝐋 )| 
            	
                +
            	∑_stable⟨𝒱_g_1,n_1 (-ℓ_1, 𝐋_1 ) | ⊗ ⟨𝒱_g_2,n_2 (-ℓ_2, 𝐋_2 ) |
            	)
            	] |  ω^-1⟩_1  |  ω^-1⟩_2   ,
        
        given that the string amplitudes should be independent of the particular marking on the surface. Above we have evaluated the twist integrals, taken to be running from 0 to ℓ's, using (<ref>). After such integration the global phase ambiguity mentioned below (<ref>) disappears since only the level-matched states appear in the decomposition. This is the main result of this paper. Similar logic can be used to establish the following identity for the one-bordered torus (see (<ref>))
        
            |L_1| ·⟨𝒱_1,1 (L_1) | = 1  2∫_-∞^∞   d ℓ (
            	 D_|L_1| |ℓ| |ℓ| - | L_1| ·θ(L - |ℓ|) 
            	 )   ⟨𝒱_0,3(L_1, ℓ,-ℓ) |
            	 (𝕀⊗ | ω^-1⟩)   .
        
        We again included 1/2 due to the nontrivial ℤ_2 symmetry.
        
        A few remarks are in order here. We first highlight that (<ref>) relates hyperbolic vertices with each other iteratively in the negative Euler characteristic -χ_g,n = 2g - 2 +n. This makes it a topological recursion. Furthermore, it contains more information on the nature of vertices compared to the ordinary loop L_∞ relations satisfied by CSFT interactions <cit.>. The latter follow from the constraint satisfied by the boundaries of string vertices ∂𝒱 as a consequence of the geometric master equation—they don't provide any information on the behavior of the interior of 𝒱 whatsoever. On the other hand (<ref>) also relates the interiors.
        
        As mentioned earlier, a similar recursive formulation has been carried out by Ishibashi <cit.>. Despite its inspiring features, such as its connection with the Fokker-Planck formalism, Ishibashi's off-shell amplitudes don't factorize according to the prescription provided by covariant CSFT for which the Feynman diagrams contain flat cylinders. This obstructs their field theory interpretation and degenerating behavior of the amplitudes. Our recursion overcomes these particular issues by relating ⟨𝒱_g,n (L_i) |, for which the string measure is integrated over the  systolic subsets 𝒱^L_g,n(L_i).
        
        Precisely speaking, the difference mentioned above follows from the distinct choices of kernels for (<ref>)—contrast with the equation (3.20) in <cit.>. The impact of this modification particularly presents itself when the length of the seams becomes smaller than L ≤ L_∗. For example
        
            R(L_1, L_2, ℓ) = L_1 - 
            	sinh( L_1  2)/sinh(L_1  2) + cosh(L_2  2)  ℓ + 𝒪 (ℓ^2)   ,
        
        while in the twisted version the leading term L_1 disappears and the expansion holds for 0 ≤ℓ≤ L (<ref>). Similar expressions can be worked out for D(L_1, L_2, L_3).
        
        Thanks to the twist (<ref>), we subtract the Feynman region contributions while maintaining the recursive structure, resulting in a recursion for hyperbolic CSFT. However these subtracted contributions are not ordinary Feynman diagrams that contain flat propagators: rather they are hyperbolic amplitudes as well. As an example, consider (g,n)=(0,4) in (<ref>). From the expansion (<ref>) the subtracted term is
        
            ∫_-L^L   d ℓ [  ⟨𝒱_0,3 (L_1, L_2, ℓ) | 
            	⊗⟨𝒱_0,3 (-ℓ, L_3, L_4) | ] | ω^-1⟩  ,
        
        for one of the channels. This is the hyperbolic contribution from this channel's associated Feynman region. Upon subtracting the other channel's contributions one is left with purely the vertex region as desired.[Even though the final result is finite, this is not necessarily the case for the individual subtracted terms, due to the |ℓ| → 0 regime. They may need analytic continuation.] The form of the subtraction for the higher-order vertices would be more complicated in a similar way to systolic volumes.
        
        The recursion relation (<ref>) works for any bordered hyperbolic surface, but it can also be used for obtaining the vacuum vertices ⟨𝒱_g,0 | ∈ℋ^⊗ 0 for g ≥ 2 using the dilaton theorem <cit.>. The dilaton theorem in the context of hyperbolic CSFT states
        
            ⟨𝒱_g,n+1 (L_i, L_n+1 = 2 π i) | 
            	  ( 𝕀⊗⋯⊗𝕀_n  times ⊗  | D ⟩)
            	= (2g-2+n)  ⟨𝒱_g,n (L_i) |   ,
        
        where
        
            |D ⟩ = ( c_1   c_-1 - c_1  c_-1) | 0 ⟩  ,
        
        is the ghost-dilation. Taking n=0 in (<ref>) and using (<ref>) for ⟨ A_g,1 (2 π i) | D ⟩ leads to the desired recursion. 
        
        Observe, however, D is not a (0,0)-primary and (<ref>) implicitly contains a specific prescription for its insertions. In (<ref>) the ghost-dilaton is grafted to the border of formal length L_k+1 = 2 π i.[This corresponds to the border degenerating to a cone point with the opening angle 2 π and becoming a regular point on a surface with one less border <cit.>.] For this, the choice of local coordinates is made according to section (2.2) of <cit.> using the regular hyperbolic metric on the surface. Since the other borders are geodesics, there are no contributions from the integrals of c_0^-, see <cit.>.
        
        Clearly (<ref>) can be restricted to apply  for only genus 0 surfaces like in (<ref>). We report this case for the completeness (n>3)
        
            |L_1| ·⟨𝒱_0,n (L_i) | = ∑_i=2^n  ∫_-∞^∞  d ℓ [ ⟨ℜ (L_1, L_i, ℓ) | ⊗ ⟨𝒱_0,n-1 (-ℓ, 𝐋∖{L_i }) |
            	] |ω^-1⟩
                + 1  2∫_-∞^∞  d ℓ_1   ∫_-∞^∞  d ℓ_2  [⟨𝔇 (L_1, ℓ_1, ℓ_2) | ⊗ ∑_stable⟨𝒱_0,n_1 (-ℓ_1, 𝐋_1 ) | ⊗ ⟨𝒱_0,n_2 (-ℓ_2, 𝐋_2 ) |
            	] |  ω^-1⟩_1  |  ω^-1⟩_2   .
        
        
        We finally highlight that (<ref>) holds for every quantum hyperbolic CSFT given the former works for L ≤ L_∗ which is the regime for which the quantum hyperbolic vertices 𝒱^L obey the geometric master equation (<ref>) <cit.>. The condition on L can be relaxed slightly for (<ref>), but it can't be entirely disposed of as in (<ref>). But recall the classical CSFT is consistent for any choice of L and it is in principle possible to relate different choices of L through field redefinitions. A question is what is the fate of (<ref>) for generic classical CSFT.
        
        In a wider context, it is possible to modify string vertices 𝒱 while maintaining that they solve the relevant geometric master equation, which is equivalent to performing field redefinitions in CSFT <cit.>. Under generic field redefinitions the form of the recursion (<ref>) or (<ref>) gets highly obstructed. Nevertheless, its observable consequences should remain the same. This relates back to our emphasis on using the “correct” string vertices: we would like to manifest these special structures as much as possible in order to ease the extraction of physics.
        
        
        
         §.§ Recursion as a differential constraint
        
        
        An alternative and more succinct way to encode (<ref>) and (<ref>) is through a second-order differential constraint on an appropriate generating function. So imagine the string field
        
            Ψ(ℓ) ∈ℋ⊗ℝ  ,     Ψ (ℓ)  = Ψ (-ℓ)    ,
        
        that has a dependence on a real parameter and introduce the generating function Z[Ψ (ℓ)] and its associated free energy W[Ψ (ℓ)] as a functional of Ψ (ℓ)[This partition function is formal since the integral may diverge around ℓ = 0.]
            Z[Ψ (ℓ)]    ≡exp[ W[Ψ (ℓ)]]
            	
               ≡exp[
            	 ∑_g,n 1  n! ħ^g-1 κ^2g-2+n∫_-∞^∞ d ℓ_1 ⋯∫_-∞^∞ d ℓ_n  ⟨𝒱_g,n (ℓ_i) | (   Ψ (ℓ_1)  ⊗⋯⊗Ψ  (ℓ_n) )
            	 ]   .
        
        We always consider Ψ (ℓ) to have even statistics (but arbitrary ghost number) in order to interpret Z[Ψ (ℓ)] as a “partition function”. This is in the same vein of taking the string field even in the BV master action (<ref>). As we shall see this will be sufficient for our purposes.
        
        We further introduce the 2-products A,B,C: ℋ^⊗ 2→ℋ
            A_L_1 L_2 L_3 =
            	[ 𝕀⊗⟨𝒱_0,3 (L_1, L_2, L_3) | ] 
            	[ | ω^-1⟩⊗𝕀⊗𝕀]   , 
               B_L_1 L_2 L_3 =
            	[ 𝕀⊗⟨ℜ (L_1, L_2, L_3) | ] 
            	[ | ω^-1⟩⊗𝕀⊗𝕀]   , 
               C_L_1 L_2 L_3 =
            	[ 𝕀⊗⟨𝔇 (L_1, L_2, L_3) | ] 
            	[ | ω^-1⟩⊗𝕀⊗𝕀]   ,
        
        and the 0-product D:  ℋ^⊗ 0→ℋ
            D_L_1 =
            	[ 𝕀⊗⟨𝒱_1,1 (L_1) | ] | ω^-1⟩  .
        
        We point out these products depend on the length of the borders so they are not graded-symmetric. However A,B are symmetric if we also exchange L_2 ↔ L_3 thanks to the symmetry (<ref>). When L_1 = L_2 = L_3 = L, the products A and D become part of the ordinary L_∞ products of CSFT, see (<ref>).
        
        The final object we introduce is “differentiation” with respect to the level-matched string fields
        
            δΨ'(ℓ') δΨ (ℓ)  ≡δ | Ψ'(ℓ') ⟩δ( ⟨Ψ (ℓ) | c_0^- )      = δδ( ⟨Ψ (ℓ) | c_0^- )   ( 𝕀⊗⟨Ψ'(ℓ') | ) | Σ_0,2⟩
               = δδ( ⟨Ψ (ℓ) | c_0^- )   ( 𝕀⊗⟨Ψ'(ℓ') |   c_0^- δ(L_0^-)  b_0^-  ) | Σ_0,2⟩
               =( 𝕀⊗δ [ Ψ'(ℓ') - Ψ (ℓ)] )   | ω^-1⟩∈ℋ^⊗ 2  ,
        
        where we have used in order, (<ref>); (<ref>); (<ref>) and the fact that c_0^- doesn't support a cohomology. Notice the right-hand side contains the delta functional in the space ℋ⊗ℝ. This derivative obeys the Leibniz rule. The effect of acting the derivative on (<ref>) is
        
            δ Z δΨ (ℓ)    = δ W  δΨ(ℓ) · Z
        
        where
        
            δ W  δΨ(ℓ ) = 
            	∑_g,n ħ^g-1 κ^2g-2+n (n-1)!    ∫_-∞^∞ d ℓ_2 ⋯∫_-∞^∞ d ℓ_n ( 𝕀⊗⟨𝒱_g,n (ℓ, ℓ_2, ⋯ℓ_n)  ) 
            	
               (|  ω^-1⟩⊗Ψ  (ℓ_2) ⊗⋯⊗Ψ  (ℓ_n) 
            	)   ,
        
        after applying the Leibniz rule and symmetry of the vertices. Note that taking this derivative produces a state in ℋ.
        
        One can similarly find the second derivative
        
            δ^2   Z δΨ' (ℓ')  δΨ (ℓ)
            	= [
            	 δ^2   W  δΨ'(ℓ')  δΨ(ℓ)  + δ W  δΨ'(ℓ')  δ W  δΨ(ℓ) ] · Z   ,
        
        where
        
            δ^2 W  δΨ'(ℓ')  δΨ(ℓ)   = 
            	∑_g,n    ħ^g-1 κ^2g-2+n (n-2)! ∫_-∞^∞ d ℓ_3 ⋯∫_-∞^∞ d ℓ_n 
               ( ⟨𝒱_g,n (ℓ', ℓ,  ℓ_3, ⋯ℓ_n)   | ) 
            	  | ω^-1⟩_ℓ'  | ω^-1⟩_ℓ ( Ψ  (ℓ_3)  ⊗⋯⊗Ψ  (ℓ_n) 
            	)   .
        
        This object belongs to ℋ^⊗ 2. Observe how the Poisson bivector acts here for lack of a better notation.
        
        Upon defining the string field | Ψ (ℓ) ⟩-valued operator
        
            𝒟_Ψ(L_1)≡[ ħ δδΨ (L_1)   - κ 2 ∫_-∞^∞ d ℓ_1  ∫_-∞^∞ d ℓ_2  
            	A_L_1 ℓ_1 ℓ_2 (Ψ(ℓ_1), Ψ(ℓ_2))
            	
               - ħκ∫_-∞^∞ d ℓ_1  ∫_-∞^∞ d ℓ_2  
            	B_L_1 ℓ_1 ℓ_2(Ψ(ℓ_1),  δδΨ(ℓ_2))
            	
               - ħ^2 κ 2∫_-∞^∞ d ℓ_1  ∫_-∞^∞ d ℓ_2  
            	C_L_1 ℓ_1 ℓ_2(δ^2 δΨ(ℓ_1)  δΨ(ℓ_2))
            	- ħκ D_L_1]    ,
        
        the differential constraint reads
        
            ∀ Ψ(L_1) ∈ℋ⊗ℝ      𝒟_Ψ(L_1)· Z[Ψ(ℓ)] =0   .
        
        It is not difficult to establish this is equivalent to the recursion (<ref>) using the derivatives above. Observe that (<ref>) admits a trivial solution Z = 1, on top of those provided by the hyperbolic amplitudes. This case corresponds to having no interactions in CSFT that trivially realizes (<ref>).
        
        It is important to point out the uncanny resemblance between (<ref>)- (<ref>) and the differential operators that form quantum Airy structures and the unique function that is annihilated by them <cit.>. This is not coincidental: quantum Airy structures can be used to encode topological recursion as differential equations and we have established the latter already. However the relation between quantum airy structures and CSFT can be more than what meets the eye initially. For example, turning the logic on its head, one may imagine CSFT as something that is constructed by solving (<ref>) perturbatively. We are going to comment on these points at the end of the paper.
        	
        We can also state the recursion restricted to genus 0 surfaces (<ref>) as a first-order differential constraint. This can be simply obtained by taking the classical ħ→ 0 limit of (<ref>)
            0 = δ W δΨ (L_1)   - κ 2∫_-∞^∞ d ℓ_1  ∫_-∞^∞ d ℓ_2  
            	A_L_1 ℓ_1 ℓ_2(Ψ(ℓ_1), Ψ(ℓ_2) )
            	
               - κ∫_-∞^∞ d ℓ_1  ∫_-∞^∞ d ℓ_2  
            	B_L_1 ℓ_1 ℓ_2(Ψ(ℓ_1), δ W δΨ(ℓ_2))
            	
               - κ 2∫_-∞^∞ d ℓ_1  ∫_-∞^∞ d ℓ_2  
            	C_L_1 ℓ_1 ℓ_2(δ W δΨ(ℓ_1) , δ W δΨ(ℓ_2) )    ,
        
        for all Ψ (L_1)  ∈ℋ⊗ℝ. It is further possible to restrict the string fields to be ghost number 2 in this equation. This particular form is going to be useful in section <ref>.
        
        
        
        § COMPARISON WITH A STUBBED THEORY
        
        
        Before we proceed to investigating the implications of the recursion relation described in the previous section, let us show that similar structures can be exhibited in theories with stubs <cit.> and investigate their consequences to get an intuition for the recursion (<ref>). We work with the classical stubbed cubic scalar field theory of <cit.> for simplicity and adopt its conventions unless stated otherwise. This section is mostly self-contained.
        
        
        
         §.§ The stubbed cubic scalar field theory
        
        
        We consider the real scalar field theory in D dimensions
        
            I[ϕ] = ∫ d^D x (- 1  2 ∂_μϕ ∂^μϕ - V(ϕ))   ,
        
        with the cubic potential
        
            V(ϕ) = - μ^2  2!ϕ^2 + κ 3!ϕ^3   .
        
        Here μ^2 > 0 is the mass parameter and κ is the coupling constant. This potential has an unstable perturbative vacuum at ϕ = 0 and the stable nonperturbative tachyon vacuum ϕ = ϕ_∗ residing at
        
            ϕ_∗ = 2 μ^2 κ  ,     
            	V(ϕ_∗) = - 2  3μ^6 κ ^2  .
        
        The squared mass of the linear fluctuations around this vacuum are V”(ϕ_∗) = μ^2 > 0.
        
        We would like to deform this theory by stubs <cit.>. Focusing on the zero-momentum sector of the theory for simplicity, i.e., spacetime-independent configurations, we find the potential of the theory deforms to
        
            V(ϕ; Λ)     = - 1  2!μ^2 ϕ^2 
            	+ 1  3!κ(e^Λμ^2  2ϕ)^3
            	- 1  4! 3 κ ^2 (e^Λμ^2 - 1 μ^2 ) (e^Λμ^2  2ϕ)^4
            	+ ⋯
               = - 1  2!μ^2 ϕ^2  
            	+ ∑_n=3^∞1  n! b_n-2κ ^n-2(1 - e^Λμ^2μ^2 )^n-3(e^Λμ^2  2ϕ)^n  ,
        
        after including stubs of length Λ / 2. The positive integers b_n are the number of unordered rooted full binary trees with n+1labeled leaves, whose formula is given by a double factorial
        
            b_n = (2 n - 1)!! = (2n)!  2^n   n != 1 × 3 × 5 ×⋯× (2 n - 1)   ,
        
        for n ≥ 1 and it is 1 when n=0. The first few terms are
        
            b_n = 1, 1,3,15, 105, ⋯  .
        
        These numbers come from the combinatorics of the Feynman diagrams. Stubs instruct us to treat the Feynman diagrams whose propagator's proper time is shorter than Λ as an elementary interaction. That means we need to include each topologically distinct Feynman diagrams with these “partial propagators” to the potential. Our conventions in (<ref>) and (<ref>) suggests us to consider the labeled Feynman diagrams.[In other words we perform the homotopy transfer with the partial propagator in the context of the homotopy Lie algebras instead of their associative counterpart like in <cit.>.]
        
        We point out the form of this stubbed potential is slightly different from <cit.>. Nevertheless they can be related to each other after setting
        
            κ ^here = 2 g^there  ,       ,
        
        compare (<ref>) with equation (2.2) of <cit.>, since
        
            1  n!  b_n-2  2^n-2 
            	=   1  n! (2n - 4)!  2^n-2 (n-2) !  2^n-2 
            	= 1  n (2n-4)!  (n-1)! (n-2)! = 1  n  C_n-2  ,
        
        where C_n are the Catalan numbers. Plugging this into (<ref>) one indeed obtains the stubbed potential of <cit.>, see equation (2.24). Taking this slightly different convention would help us to draw parallels with the topological recursion presented in the previous section more directly.
        
        Resumming the expansion (<ref>) gives
        
            V(ϕ; Λ) = - μ^2  2 ϕ^2   e^Λμ^2  f_3(x_3) - 1  e^Λμ^2  - 1  ,
        
        where
        
            f_3(x) = 6 x - 1 + (1-4x)^3/2 6x^2   ,
            	    
            	x_3 = - κ 2e^Λμ^2 -1 μ^2 e^Λμ^2  2ϕ  ,
        
        and the nonperturbative  vacuum is shifted exponentially far away
        
            ϕ_∗(Λ) = 2 μ^2 κexp( Λμ^2  2)   ,
        
        in the stubbed theory while the depth of the potential remains the same. We also point out the expansion (<ref>) in ϕ has the radius of convergence
        
            r(Λ) = 
            	μ^2  2 κ e^-  3Λμ^2   2 1 - e^- Λμ^2  ,
        
        due to the fractional power in (<ref>).
        
        
        
         §.§ Topological recursion for the stubs
        
        
        Now we would like to obtain the nonperturbative solution (<ref>) of the stubbed theory using an alternative perspective. From the form of the series in (<ref>) and the identity 
        
            b_n+1 = (2n+1)   b_n   ,
        
        it shouldn't be surprising that there is a recursion among the elementary vertices of the stubbed theory. Introducing
        
            W_n(s_i)  =
            	b_n-2(1 - e^Λμ^2μ^2 )^n-3∏_i=1^n exp( s_i  μ^2  2)   ,
            	    
            	s_i ≥ 0   ,
        
        one can form the topological recursion
        
            W_n(s_i)     =∑_i=2^n ∫_0^∞  dt   r_s_1 s_i t  W_n-1( t, 𝐬∖{ s_i } )  
            	
               +
            	1  2 ∫_0^∞  dt_1  ∫_0^∞  dt_2   ∑_stable  d_s_1 t_1 t_2 
            	W_n_1 (t_1,  𝐬_1)   W_n_2 (t_2,  𝐬_2)   ,
        
        for n > 3, where the stub kernels are given by
        
            r_s_1 s_2 s_3   = (-1 + θ(s_3 - Λ) )   a_s_1 s_2 s_3 
            
            		d_s_1 s_2 s_3   = (1- θ(s_2 - Λ)- θ(s_3 - Λ) + θ(s_2 - Λ)  θ(s_3 - Λ) )   a_s_1 s_2 s_3  ,
        
        and
        
            a_s_1 s_2 s_3 = W_3(s_1, s_2, s_3) =  ∏_i=1^3 exp( s_i  μ^2  2)   .
         
        Notice the functions W_n are totally symmetric in their arguments, whereas the stub kernels r and d have the expected counterparts of the symmetry properties (<ref>). Comparing with the twisted kernels (<ref>) we can roughly identify Λ∼ 1/L.
        
        The expression (<ref>) can be derived by the recursive structure of the labeled binary trees in figure <ref>. From the labeled binary trees we read the identity
        
            b_n = 1  2 ∑_i=1^n n+1i  b_i-1  b_n-i
            	= ∑_i=2^n+2 b_n-1 + 1  2∑_stable b_n_1 b_n_2  ,
        
        which can be also derived using (<ref>). Together with (<ref>), it is possible to include the rest of the terms in (<ref>) through integrals like in (<ref>). We run the bounds of integration from 0 to ∞, which requires introducing step functions as in (<ref>). So we see the stubbed scalar theory obeys a (classical) topological recursion (<ref>) with the twisted kernels (<ref>).[Here we considered  the “classical” recursion for simplicity, however similar considerations can also be applied to the “quantum” recursion after one considers cubic graphs instead of trees.] 
        
        It is useful to look at two limiting cases of (<ref>). First, we see that Λ = 0 trivializes the recursion, W_n(s_i) = 0 for n > 3 and W_3(s_i) = 1, which corresponds to having the polynomial and manifestly local formulation. Such a limit doesn't exist in the stringy analog (<ref>) since there is an upper-bound bound on (<ref>)L ∼ 1/ Λ. This makes sense: the ordinary CSFT can't admit a polynomial and local formulation <cit.>. On the opposite end, Λ→∞, we see (<ref>) becomes well-defined upon analytic continuation. The stringy analog of this case is Ishibashi's recursion <cit.>.
        
        We can similarly cast the recursion (<ref>) in the form of a second-order differential constraint. Introducing the following functionals of ϕ(s)
            z[ϕ(s)] ≡exp( w[ϕ(s)])    ≡exp( ∑_n=3^∞κ^n-2 n!∫_0^∞ d s_1 ⋯∫_0^∞  ds_n  
            	W_n(s_1, ⋯, s_n)  ϕ(s_1)  ⋯ ϕ(s_n) 
            	)   ,
        
        the relation (<ref>) can be expressed compactly as
        
            0 = δ w δϕ(t_1)   - κ 2 ∫_0^∞ dt_2 ∫_0^∞ dt_3   a_t_1 t_2 t_3 ϕ(t_2)  ϕ(t_3) 
            	-  κ ∫_0^∞ dt_2 ∫_0^∞ dt_3   r_t_1 t_2 t_3 ϕ(t_2)  δ w δϕ(t_3)
                -  κ 2 ∫_0^∞ dt_2 ∫_0^∞ dt_3    d_t_1 t_2 t_3( δ w δϕ(t_2))
            	( δ w δϕ(t_3))   .
        
        This is the analog of (<ref>). Note that we didn't need to place any constraint like (<ref>) on the value of Λ in contrast to its stringy counterpart—we can have any stub length. Note
        
            V(ϕ; Λ ) = - 1  2! μ^2 ϕ^2 + w [ϕ(s) = ϕ δ(s - Λ)]    ,
         
        see (<ref>) and (<ref>).
        
        We remark that (<ref>) was a consequence of the geometric interpretation of stubs in terms of suitable binary trees. If we had used a different way of integrating out UV modes, which is choosing a different field parametrization, this geometric presentation would have been highly obstructed. This is the avatar of what we have discussed for the stringy case at the end of subsection <ref>. But again, the implications of the recursion are supposed to stay the same. 
        
        
        
         §.§ The implications
        
        
        Let us now investigate the consequences of (<ref>). First notice the functional derivative of the free energy w[ϕ(s)] evaluates to
        
            δ w δϕ(t_1) = 
            	∑_n=3^∞ κ^n-2 (n-1)! ∫_0^∞ d s_1 ⋯∫_0^∞  ds_n   W_n(t_1, 𝐬)  ϕ(s_2)  ⋯ ϕ(s_n)   ,
        
        after using the symmetry property of W_n. In particular consider taking ϕ(s_i) = ϕ_∗(Λ) δ(s_i - Λ), where ϕ_∗(Λ) is a solution to the stubbed theory of stub length Λ/2. We have
        
            φ(t_1; Λ) 
            	   ≡δ w δϕ(t_1; Λ)|_ϕ→ϕ_∗
            	 = ∑_n=3^∞κ^n-2 (n-1)!  W_n(t_1, Λ)  ϕ_∗(Λ)^n-1  ,
        
        where we introduced φ(t_1) for convenience. Here Λ indicates that all remaining entries are equal to Λ here. We often suppress the Λ dependence to simplify the presentation of the expressions.
        
        Importantly, when t_1= Λ (<ref>) evaluates to (see (<ref>))
        
            φ(t_1 = Λ) 
            	   =∑_n=3^∞κ^n-2 (n-1)!  W_n(s_i = Λ)  ϕ_∗^n-1
               = ∂ V ∂ϕ|_ϕ = ϕ_∗  + μ^2 ϕ_∗
            	= μ^2 ϕ_∗  ,
        
        since ϕ_∗ is a solution and it extremizes the potential (<ref>) by construction. Therefore one can think of φ(t_1) as a function that captures the solution when t_1 = Λ, but generalizes it otherwise. The definition of this function is schematically shown in figure <ref>. 
         
        
        Taking ϕ(s_i) = ϕ_∗ δ(s_i - Λ) =( φ(Λ) / μ^2 ) δ(s_i - Λ) in (<ref>) then implies 
        
            0 = φ(t_1)
            	- κ 2 μ^4  a_t_1 ΛΛ φ(Λ)^2
            	   - κμ^2 ∫_0^∞ dt_3    r_t_1 Λ t_3 φ(Λ)  φ(t_3)
            	
                -κ 2  ∫_0^∞ dt_2 ∫_0^∞ dt_3    d_t_1 t_2 t_3 φ(t_2)  φ(t_3)   ,
        
        and we obtain a quadratic integral equation for φ(t_1)
            0 = φ(t_1)
            	   - κ 2 μ^4 exp( t_1 μ^2  2 + Λμ^2 )  φ(Λ)^2
            	
               + κμ^2 exp( t_1 μ^2  2 + Λμ^2  2)
            	 φ(Λ)  [ ∫_0^Λ dt'  exp(t' μ^2  2)  φ(t') ]
            	
               -κ 2 exp(t_1 μ^2  2)  [ ∫_0^Λ dt'  exp( t' μ^2  2)
            	φ(t') ]^2
            	  ,
        
        after substituting the stub kernels (<ref>). The schematic representation of this quadratic integral equation is shown in figure <ref>.
        
        Observe that setting t_1 = Λ = 0 above produces V'(ϕ) = 0, see (<ref>). This suggests (<ref>) can be understood as a resummation of the equation of motion V'(ϕ; Λ) = 0 of the stubbed theory. We can iteratively insert φ(t_1) into itself and expand in κ to find 
        
            0 
            	   = φ(t_1) 
            	- κ 2 exp( t_1 μ^2  2 + Λμ^2)  ϕ_∗^2
            	- κ^2  2 exp( t_1 μ^2  2 + 3 Λμ^2  2)
            	 ( 1- e^Λμ^2μ^2)
            	 ϕ_∗^3 
            	+ ⋯  .
        
        Upon taking t_1 = Λ and using (<ref>) one indeed obtains V'(ϕ; Λ) = 0 in the expanded form, see (<ref>). The argument here can be generalized to higher orders in κ.
         
        
        In fact, upon close inspection we see
        
            φ(t_1) = 2 μ^4 κexp( t_1 μ^2  2)   ,
        
        satisfies (<ref>) and we again obtain the solution (<ref>) for the stubbed theory
        
            ϕ_∗ = 1 μ^2 φ(t_1 = Λ ) 
            	= 2 μ^2 κexp( Λμ^2  2)   .
        
        This demonstrates the differential constraints encoding the topological recursion (<ref>) in a theory can be used to find its solutions. Of course, we have found φ(t_1) by guesswork, while it can be quite challenging to do so for more complicated situations. We highlight resumming the potential V(ϕ; Λ) (<ref>) as in <cit.> wasn't necessary to obtain this solution: we were able to access beyond the radius of converges ϕ > r (Λ) (<ref>) directly.
        
        This approach, however, does not give a novel method for finding the energy difference between the solutions since this calculation still has to involve the resummed potential. On the other hand, it is possible to find the mass of the linear fluctuations around the vacuum (<ref>) based on the recursion alone. In order to do that we vary φ→φ + δφ in (<ref>) for which we find
        
            - 1 μ^2 M^2 (t_1)  δφ(t_1)  =
            	δφ(t_1) 
            	   - 2  exp( t_1 μ^2  2 + Λμ^2  2)
            	δφ(Λ)
            	
               + 2  μ^2 exp(t_1 μ^2  2)  ∫_0^Λ dt'  exp(t' μ^2  2)  δφ(t')   ,
        
        using (<ref>). We suggestively named this variation M^2 (t_1) δφ(t_1) /μ^2 given that (<ref>) was essentially the resummed V'(ϕ; Λ) = 0. Its variation can be understood as the resummed version of -V”(ϕ; Λ) and it should encode the spectrum of the linear fluctuations around ϕ = ϕ_∗. The overall sign follows from the choice of signs in (<ref>) and the division by μ^2 is to get the correct units for M^2. We have δφ(Λ) = μ^2 δϕ for the genuine fluctuations.
        
        Indeed, by taking 
        
            δφ(t_1) = 
            	exp( t_1 μ^2  2) ϵ  ,
        
        with infinitesimal ϵ without dependence on t_1, we see
        
            - 1 μ^2 M^2  δφ(Λ ) = - δφ(Λ) 
            	    
            	M^2  δϕ = μ^2  δϕ  ,
        
        so the squared mass of the fluctuations around the nonperturbative vacuum is M^2 = μ^2 > 0—as it should be. We guess the particular form of (<ref>) for δφ(t_1) as a function of t_1 by getting inspired from (<ref>), but the linearity in δφ(t_1) of (<ref>) justifies it a posteriori in any case. We again emphasize that we didn't need to resum the potential to derive (<ref>). In fact, obtaining this result from the resummed form (<ref>) would have been quite intricate due to the complicated higher derivative structure of the stubbed theory.
        
        Given that the integral equation (<ref>) is a consequence of the differential constraint and topological recursion, it is natural to wonder whether there is an analog of the integral equation in hyperbolic CSFT as a result of (<ref>). Indeed, we are going to see there is such an equation in the upcoming section.
        
        
        
        § QUADRATIC INTEGRAL EQUATION FOR HYPERBOLIC CSFT
        
        
        Now we apply the reasoning from the last section to derive a quadratic integral equation for hyperbolic CSFT following from (<ref>). We adapt the conventions from before and we are going to be brief in our exposition since the manipulations are very similar to the previous section. 
        
        We begin with the derivative of W[Ψ(ℓ)]. Upon substituting
        
            Ψ(ℓ_i)  =  Ψ_∗ δ(ℓ_i - L)   ,
        
        into (<ref>) we get
        
            Φ (L_1)
            		≡δ W δΨ(L_1)|_Ψ→Ψ_∗   = ∑_n=3^∞κ^n-2 (n-1)! ( 𝕀⊗⟨𝒱_0,n (L_1, 𝐋) | )  ( | ω^-1⟩⊗ | Ψ_∗⟩^⊗ (n-1)) 
            		
               = ∑_n=2^∞κ^n-1 n !  L_0,n( Ψ_∗^n; L_1 )    .
        
        Here Ψ_∗ = Ψ_∗(L) is a critical point of the classical CSFT action and we defined the string field Φ (L_1) = Φ (L_1; L) accordingly. These string fields are even and assumed to have ghost number 2.
        
        We also introduce a new set of (graded-symmetric) string products L_g,n-1(L_1): ℋ^⊗ (n-1)→ℋ
            L_g, n-1 (L_1) =
            	( 𝕀⊗⟨𝒱_g,n (L_1, 𝐋) | )  ( | ω^-1⟩⊗𝕀^⊗ (n-1))    ,
        
        that generalize the ordinary string products L_g,n (<ref>) such that the border length of the output is L_1 instead of L. They are equivalent when L_1 = L. Note that we have
        
            Φ (L_1 = L)   = - Q_B  Ψ_∗  ,
        
        by the equation of motion. We therefore see Φ (L_1 = L) is related to Q_B acting on the solutions. This definition can also be schematically understood like in figure <ref>.
        
        We remind the reader there is a gauge redundancy for the solutions Ψ_∗ (<ref>), which indicates there should be a redundancy in the choice of Φ (L_1) as well. We demand this is given by
        
            δ_Λ Φ (L_1)  =
            	∑_n=2^∞κ^n-1 (n-1)!  L_0,n( δ_ΛΨ_∗, Ψ_∗^n-1; L_1 )   ,
        
        so that the definition (<ref>) is invariant under (<ref>). Here δ_ΛΨ_∗ is given by (<ref>). Upon taking L_1 = L and using the L_∞ relations <cit.> this gauge transformation simplifies further and can be seen to be consistent with (<ref>).
        
        We need to fix this gauge symmetry in order to invert (<ref>) and write down an integral equation. We do this by assuming a grassmann odd operator β (L_1) = β(L_1; L), possibly to be constructed with b-ghost modes and depending on L_1, satisfying[We take β = β(L_1) so that it indicates β is to be used for fixing the gauge redundancy (<ref>) for all L_1.]
            { Q_B, β(L_1 = L) } = -P    ,
        
        where P is the projector to the complement of the cohomology of Q_B. We impose the gauge
        
            β(L_1 = L)  Ψ_∗  = 0   ,
        
        on the CSFT solutions. As a result the equation (<ref>) gets inverted to
        
            β Φ (L_1 = L)  =  Ψ_∗  .
        
        Note that we have β Φ (L_1) ≠ 0 and { Q_B, β(L_1) }≠ - 1 for L_1 ≠ L in general. We have taken P → 1 above and ignored the subtleties associated with the on-shell modes.
        
        Taking (<ref>) in the differential constraint (<ref>) then implies 
        
            0 = Φ (L_1) 
            	- κ 2  
            	A_L_1 L L ( β Φ (L) , β Φ (L) )
            	   - κ ∫_-∞^∞ d ℓ  
            	B_L_1 L ℓ( β Φ (L), Φ(ℓ))
            	
               - κ 2 ∫_-∞^∞ d ℓ_1  ∫_-∞^∞ d ℓ_2  
            	C_L_1 ℓ_1 ℓ_2(Φ(ℓ_1), Φ(ℓ_2) )   ,
        
        where the 2-products A,B,C are already defined in (<ref>). The similarity to (<ref>) is apparent, also observe the structure in figure <ref>. This is the quadratic integral equation and it makes the cubic nature of CSFT manifest.
        
        Like earlier, iteratively substituting Φ (L_1) into this equation, taking L_1 = L, and expanding in κ we obtain the equation of motion for CSFT (<ref>) using (<ref>).  A similar phenomenon occurs for the stubs (<ref>) and it is associated with resummation as discussed. So we again interpret (<ref>) as the resummation of the CSFT equation of motion. Having such a form is reassuring: trying to resum the action (<ref>) as in (<ref>) would have been quite a daunting task, bordering on the impossible. 
        
        Being able to recast the CSFT equation of motion to a quadratic integral equation (<ref>) is encouraging. It is explicitly and exclusively given  in terms of the generalized hyperbolic three-vertex of <cit.> and the twisted Mirzakhani kernels (<ref>). It may be viable to attempt to solve this integral equation in the future after identifying a convenient gauge-fixing operator β(L_1). Furthermore, it should be possible to read the spectrum of the linear excitations (i.e., cohomology) around the solution by varying (<ref>) like in (<ref>) and repeating a similar analysis.[On the other hand we don't have a new way to compute the on-shell value of the action as before. This quantity vanishes up to boundary terms in CSFT <cit.>.] However, we expect the form of solutions and their variations would be more complicated than the stub counterparts (<ref>) and (<ref>). We comment on this approach more in the next section.
        
        
        
        § DISCUSSION
        
        
        In this paper:
        
          *  We identified a background-independent topological recursion relation satisfied by the elementary vertices of hyperbolic closed string field theory, refer to (<ref>) or (<ref>). This explicitly demonstrates closed string field theory has an underlying cubic structure and makes its features manifest. The moduli integrations are simplified considerably. The recursion relation satisfied among the volumes of the systolic subsets (<ref>) provides the essential ingredients for its closed string field theory counterpart.
        		
        	
          *  We showed a suitable generalization of the classical solutions to closed string field theory obey a quadratic integral equation (<ref>) as a consequence of the aforementioned topological recursion. We discussed how this equation can be used to construct analytic solutions in principle. 
        	
         There are numerous interesting directions left to be investigated in future work. We list the ones that we find the most appealing and exciting:
        
        
          *  Developing a systematic approach for solving the quadratic integral equation (<ref>) is the natural next step. Even though this equation appears to contain sufficient information to construct solutions based on our investigations in the stubbed scalar theory, it is far from clear how to obtain them beyond guessing judiciously. Identifying a convenient gauge choice (<ref>) and the smallest subspace for which a solution to (<ref>) exists are possibly among the first prerequisites for making any further progress. Understanding the underlying algebraic structure to our reformulation may also help.
        	
        	
          *  Even if a solution is constructed, it is somewhat unclear how to probe its physics. For example, it has been established that the on-shell action vanishes up to boundary terms in closed string field theory <cit.>, thanks to the dilaton theorem <cit.>. We don't have a novel way to test this. However, investigating the spectrum around the background may still be doable with a correct approach. One may try to establish (or rule out) the closed string version of Sen's third conjecture <cit.> for the vanishing cohomology around the tachyon vacuum for instance.
        	
        	
          *  The form of the differential constraint encoding the topological recursion (<ref>) is suggestive. We have already pointed out its possible interpretation as a quantum Airy structure <cit.>. It would be interesting to investigate this further to see whether a deeper principle and/or theory is lurking behind here and how it relates to the nonperturbative structure of closed string field theory. There may be connections to the ideas of <cit.> for example. 
        	
        	
          *  It may also be useful to illuminate how our approach compares to the themes in the relation between matrix models <cit.> and Jackiw-Teitelboim gravity <cit.>. Here, the initial point of investigation would be understanding and clarifying the matrix model interpretation of the systolic volumes and their recursion (<ref>).
        	
        	
          *  We have seen the string fields are naturally endowed with an additional positive real parameter: the length of the geodesic border L_i (<ref>). A similar feature also appears in the lightcone SFT <cit.>, where the lightcone momentum k_- relates to the border length. It may be useful to understand how similar their roles are in the interactions.
        	
        	
          *  Physics following from the recursion shouldn't change under field redefinition even though a particular form of the relations will be no longer manifest. Nevertheless, it is still rather counter-intuitive that our derivation falls short of establishing a recursion for all classical hyperbolic vertices—especially for the polyhedral vertices based on Strebel quadratic differentials <cit.>. Such a recursion may still exist considering the work <cit.> and it may be possible to find a generalization to our argument to cover these scenarios as well. Relatedly, it is desirable to establish a recursion relation for other hyperbolic theories <cit.> and/or their supersymmetric versions <cit.>.
        	
          We hope these results help construct closed string field theory solutions some day.
        
        
        
        § ACKNOWLEDGMENTS
        
        
        We are grateful to Scott Collier, Harold Erbin, Ted Erler, Daniel Harlow, David Kolchmeyer, Raji Mamade, and Edward Mazenc for many enlightening discussions; Barton Zwiebach for his comments on the early draft and discussions; and Nobuyuki Ishibashi for the correspondence. NVM in particular thanks Daniel Harlow for the freedom and encouragement to explore. 
        
        The work of AHF is supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics of U.S. Department of Energy under grant Contract Number DE-SC0012567, DE-SC0009999, and the funds from the University of California. NVM is supported by the Hertz Foundation and the MIT Dean of Science Fellowship.
        
        
        
        § HYPERBOLIC THREE-STRING VERTEX
        
        
        In this appendix we summarize the local coordinates for the generalized hyperbolic string vertex ⟨Σ_0,3 (L_1, L_2, L_3)| for L_1,L_2,L_3 ≥ 0. The reader can refer to <cit.> for the details of the derivation using the connection between hyperbolic geometry on the three-bordered sphere and the hypergeometric equation. We also comment on various branch choices in this appendix.
        
        We begin with a three-punctured sphere whose punctures are placed at z=0,1,∞ and there are coordinate patches
        
            D_i = { z ∈ℂ  |    0 < |w_i(z; L_1, L_2, L_3)| ≤ 1 }   ,
        
        around them such that the surface
        
            Σ_0,3(L_1, L_2, L_3) = ℂ∖⋃_i=1^3 D_i   ,
        
        endows a regular hyperbolic metric with geodesic borders of length L_i = 2 πλ_i. The local coordinate maps w_1(z; L_1, L_2, L_3) in (<ref>) are given by
        
            w_1(z; L_1, L_2, L_3)    = 
            	 z   (1-z)^-λ_2 / λ_1ρ(L_1, L_2, L_3 ) [
            	
            		_2F_1(  1+ i λ_1 - i λ_2 + i λ_3  2, 
            		 1+ i λ_1 - i λ_2 - i λ_3  2;
            		1 + i λ_1; z )  
            		_2F_1(  1- i λ_1 + i λ_2 - i λ_3  2, 
            		 1- i λ_1 + i λ_2 + i λ_3  2;
            		1 - i λ_1; z ) 
            	]^1 / i λ_1
               = 1 ρ_1 [ z + 
            	1 + λ_1^2 + λ_2^2 - λ_3^2  2   (1+λ_1^2)   z^2 + ⋯]   ,
        
        where the expression for the mapping radius ρ_1 = ρ (L_1, L_2, L_3 ) is
        
            ρ (L_1, L_2, L_3 ) = e^-π / 2 λ_1[
            	Γ(1-i λ_1)^2 Γ(1+i λ_1)^2 γ(1 + iλ_ 1 + i λ_2 + i λ_3  2)
            		γ( 1 + iλ_ 1 - i λ_2 + i λ_3  2)
            		γ( 1 - iλ_ 1 - i λ_2 + i λ_3  2)
            		γ( 1 - iλ_ 1 + i λ_2 + i λ_3)  2 )  ]^ i / 2 λ_1  .
        
        Here _2F_1(a,b;c;z) is the hypergeometric function and Γ(x) is the gamma function. The latter also goes into the definition of
        
            γ(x)  = Γ(x) Γ(1-x)  .
        
        Take note the mapping radius ρ_1 diverges as ∼ e^- 1 / L_1 as L_1 → 0. In the inverted form, the expansion (<ref>) is given by
        
            z(w_1; L_1, L_2, L_3) = 
            	ρ_1  w
            	- 1 + λ_1^2 + λ_2^2 - λ_3^2  2   (1+λ_1^2) 
            	(ρ_1 w )^2 
            	+ ⋯  .
        
        Unfortunately the closed-form expressions for this inverse function is not known except for L_i → 0. The expressions for the local coordinates w_2 and w_3 around z=1 and z=∞ are given by
        
            w_2(z; L_1, L_2, L_3) = w_1 (1-z   ; L_2, L_1, L_3)   , 
            	
               w_3(z; L_1, L_2, L_3) = w_1 ( 1  z  ; L_3, L_2, L_1 )   ,
        
        up to possible global phases. 
        
        It is important to be mindful about the various branch choices for the imaginary exponents in the mapping radius (<ref>). This has been partially investigated in <cit.>, however we briefly comment on them again here. First notice there is an overall ambiguity in the mapping radii
        
            ρ(L_1,L_2, L_3) ≃exp(  n  πλ_1 )  ρ(L_1,L_2, L_3)       
            	n ∈ℤ  .
        
        as a result of the imaginary exponent. If one would like to work in a particular branch for the local coordinates then one has to find the correct integer n = n (L_1,L_2,L_3). It has been argued that the correct choice is n=0 when L_1=L_2 = L_3 together with the choice of principal branch for the branches <cit.>. However there may be branch crossings for generic border lengths. Since the mapping radius should be continuous in L_i the integer n should be adjusted accordingly when a branch cut is crossed. In the expressions (<ref>) we assume such adjustments are implicitly present.
        
        
        
        § SAMPLE COMPUTATIONS FOR THE SYSTOLIC VOLUMES
        
        
        In this appendix we perform sample computations for the systolic volumes using the recursion (<ref>). We have already computed (g,n)=(0,4),(1,1) in the main text, see (<ref>) and (<ref>). Using them we can compute the systolic volumes for the surfaces with χ_g,n = -3 and n ≥ 1:
        
          * Five-bordered sphere g=0, n=5. Writing the recursion (<ref>) explicitly we have
        	
            L_1 · V 𝒱_0,5 (L_i)    = 
            		∑_i=2^5 ∫_0^∞ℓ d ℓ R_L_1L_iℓ  V 𝒱^L_0,4(ℓ, 𝐋∖{L_i }) 
            		+ 3∫_0^∞ℓ_1 d ℓ_1  ∫_0^∞ℓ_2 d ℓ_2  D_L_1 ℓ_1 ℓ_2  ,
        
        	after taking V 𝒱^L_0, 3 (L_i) =1.  This can be expressed as
        	
            L_1 · V 𝒱^L_0,5 (L_i)     = L_1 · V ℳ_0,5 (L_i) 
            		- 3  2  L^2  ∑_i=2^5  ∫_0^∞ℓ d ℓ  R_L_1L_iℓ 
            		- L_1  ∑_i=2^5  ∫_0^L  ℓ d ℓ 𝒱^L_0,4(ℓ, 𝐋∖{L_i }) 
                - 6  ∫_0^L ℓ_1 d ℓ_1  ∫_0^∞ℓ_2 d ℓ_2   R_L_1 ℓ_1 ℓ_2
            		+  3 L_1  ∫_0^L ℓ_1 d ℓ_1  ∫_0^L ℓ_2 d ℓ_2   ,
        
        	using the twisted Mirzakhani kernels (<ref>). Focus on the integral
        	
            I_L_1 L_i≡∫_0^∞ℓ d ℓ  R_L_1 L_i ℓ 
            		= ∫_0^∞ℓ d ℓ[
            		L_1 -
            		log(cosh(L_i  2) + cosh(L_1 + ℓ 2) cosh(L_i  2) + cosh(L_1 - ℓ 2))
            		]   .
        
        	Clearly this integral converges. It can be evaluated by taking the derivative of both sides with respect to L_1 and using the identity (<ref>)
            I_L_1 L_i L_1   = 1  2∫_0^∞ℓ d ℓ∑_ϵ_1, ϵ_i = ±[ 1 + exp(1  2(ℓ + ϵ_1 L_1+ ϵ_i L_i )) ]^-1
            		 = 1  2( L_1^2 + L_i^2) + 2 π^2  3
                I_L_1 L_i = 1 6 L_1^3  + 1  2L_1 L_i^2 + 2 π^2  3 L_1
            		  .
        
        	The integration constant fixed by noticing I_L_1=0,L_i =0, see (<ref>). Then we have
        	
            ∫_0^L ℓ_1 d ℓ_1    I_L_1 ℓ_1 = 
            		1  12 L_1^3 L^2 + 1  8L_1 L^4  + π^2  3  L_1 L^2   ,
        
        	and
        	
            ∫_0^L ℓ d ℓ 𝒱^L_0,4(ℓ, 𝐋∖{L_i })
            		   = 	∫_0^L ℓ d ℓ[
            		2 π^2 +1  2ℓ^2 +  1  2∑_j=2 
              j ≠ i^5 L_j^2 - 3  2 L^2
            		]  
               = π^2 L^2 +  1  4 L^2 ∑_j=2 
              j ≠ i^5 L_j^2 - 5  8 L^4   .
        
        	Combining these integrals we obtain
        	
            L_1 · V 𝒱^L_0,5 (L_i)     = L_1 · V ℳ_0,5 (L_i) 
            		- 3 2 L^2 ∑_i=2^5 [ 1 6 L_1^3  + 1  2 L_i^2 L_1 + 2 π^2  3 L_1]
            		
                - L_1 ∑_i=2^5 [
            		π^2 L^2 +  1  4 L^2 ∑_j=2 
              j ≠ i^5 L_j^2 - 5  8 L^4
            		] 
            		- 6 [ 1  12 L_1^3 L^2 + 1  8L_1 L^4  + π^2  3  L_1 L^2 ] 
            		+ 3   4 L_1 L^4
            		
               = L_1 · V ℳ_0,5 (L_i) - L_1^3 L^2 - 3  4 L_1 L^2 ∑_i=2^5 L_i^2 - 4 π^2 L_1 L^2
            		- 4 π^2 L_1 L^2 
            		
                - 3  4 L_1 L^2 ∑_i= 2^5 L_i^2 + 5  2 L_1 L^4
            		- 1  2 L_1^3 L^2 - 3  4 L_1 L^4 - 2π^2 L_1 L^2 + 3  4 L_1 L^4
            		
               = L_1 · V ℳ_0,5 (L_i)  - 3  2L_1^3 L^2 - 3  2 L_1 L^2 ∑_i=2^5 L_i^2 - 10π^2 L_1 L^2 + 5  2 L_1 L^4   ,
        
        	and the final result is
        	
            V 𝒱^L_0,5 (L_i)   =
            		V ℳ_0,5 (L_i)  - L^2  [ 3  2∑_i=1^5 L_i^2  + 10 π^2 ]
            		+ 5  2 L^4   .
        
        	This expression is symmetric under permutations of L_i, as it should be. Observe that the term subtracted from V V ℳ_0,5 (L_i) to get V 𝒱^L_0,5 (L_i) is quite nontrivial unlike (<ref>) and (<ref>).
        	
        	
          * Two-bordered torus g=1, n=2. We begin by writing (<ref>) specific to this case
        	
            L_1 · V 𝒱^L_1,2 (L_i)    = 
            		∫_0^∞ℓ d ℓ R_L_1L_2ℓ  V 𝒱^L_1,1(L_1 ) 
            		+ 1  2∫_0^∞ℓ_1 d ℓ_1  ∫_0^∞ℓ_2 d ℓ_2  D_L_1 ℓ_1 ℓ_2  .
        
        	after using V 𝒱^L(ℓ_1, ℓ_2,L_i) = 0. The twisted Mirzakhani kernels (<ref>) produce
        	
            L_1  · V 𝒱^L_1,2 (L_i)    = 
            		L_1 · V ℳ_1,2 (L_i)
            		- 1  4 L^2 ∫_0^∞ℓ dℓ R_L_1 L_2 ℓ
            		- L_1 ∫_0^L ℓ d ℓ  V 𝒱^L_1,1(ℓ) 
            		
               -∫_0^L ℓ_1 d ℓ_1  ∫_0^∞ℓ_2 d ℓ_2  
            		R_L_1 ℓ_1 ℓ_2
            		+  1  2  L_1 ∫_0^L ℓ_1 d ℓ_1  ∫_0^L ℓ_2 d ℓ_2
            		  .
        
        	This is almost the same as the case considered above, the only difference being that we have to use V 𝒱^L_1,1(ℓ) = π^2/12 + ℓ^2/48 - L^2 /4 (<ref>) now. We perform the integral
        	
            ∫_0^L ℓ d ℓ  V 𝒱^L_1,1(ℓ) =
            		∫_0^L ℓ d ℓ[ π^2  12 +ℓ^2  48 -L^2  4]
            		= π^2  24 L^2 - 23  192 L^4  ,
        
        	and obtain
        	
            L_1 · V 𝒱^L_1,2 (L_i)    =
            		L_1 · V ℳ_1,2 (L_i) 
            		- 1  4 L^2 [ 1 6 L_1^3  + 1  2 L_1 L_2^2 + 2 π^2  3 L_1]
            		- L_1 [π^2  24 L^2  - 23  192 L^4 ]
            		
               
            		- [1  12 L_1^3 L^2 + 1  8L_1 L^4  + π^2  3  L_1 L^2] 
            		+  1 8 L_1 L^4
            		
               = L_1 · V ℳ_1,2 (L_i) - 1  24 L_1^3 L^2 - 1  8 L_1 L_2^2 L^2 - π^2  6 L_1 L^2
            		- π^2  24 L_1 L^2  + 23  192L_1 L^4
            		
               
            		- 1  12 L_1^3 L^2 - 1  8 L_1 L^4 - π^2  3 L_1 L^2 + 1  8 L_1 L^4
            		
               = L_1 · V ℳ_1,2 (L_i) - 1  8 L_1 L^2 ∑_i=1^2 L_i^2  + 23  192 L_1 L^4
            		- 13 π^2  24 L_1 L^2   .
        
        	Then the final result is
        	
            V 𝒱^L_1,2 (L_i) = V ℳ_1,2 (L_i) - 
            		L^2 [ 1  8∑_i=1^2 L_i^2 +  13 π^2  24]
            		+ 23  192 L^4    .
        
        	This is also invariant under exchanging L_1 ↔ L_2. Defining
        	
            x ≡ L_1^2 + L_2^2    ,
        
        	we can alternatively express (<ref>) as 
        	
            V 𝒱^L_1,2 (x)  = 1  192 (x + 4 π^2) (x + 12 π^2) - 1  8 L^2 x 
            		- 13 π^2  24 L^2
            		+  23  192 L^4    .
        
        	which can be plotted as a function of the threshold length L and x, see figure <ref>. Take note that the quantity is positive for all 0 ≤ L ≤ 2 sinh^-1 1. 
        	
        
        § NUMERICAL EVALUATION OF V 𝒱^L_1,1(L_1)
        
        
        Following <cit.>, it is possible to work out the WP geometry of ℳ_0,4(L_i) and ℳ_1,1(L_1) explicitly using classical conformal blocks <cit.>. Despite the fact that this procedure doesn't have a mathematically rigorous basis in general, there is overwhelming evidence that it holds true by the aforementioned works. We can use this approach to numerically compute the volumes 𝒱^L_1,1(L_1) to provide numerical evidence for the systolic volumes (<ref>). The reader can refer to <cit.> for a deeper exposition. 
        
        We begin by reminding the WP metric on the moduli space ℳ_1,1(L_1), parameterized by the moduli τ∈ℍ / PSL(2,ℤ), is given by
        
            g^(1,1)_ττ(L_1) = - 4 π_τ_τ S_HJ^(1,1) (τ, τ; L_1)   .
        
        Here the Kähler potential S_HJ^(1,1) (τ, τ; L_1) is a suitably-regularized on-shell Liouville action that can be found as an expansion in q = e^2 π i τ, see equation (3.25) of <cit.>.  The Kähler form associated with the metric g^(1,1)(L_1) is related to the WP form on ℳ_1,1(L_1). We can use this to evaluate V 𝒱^L_1,1(L_1) by performing the integral
        
            𝒱^L_1,1(L_1) = i  4∫_𝒱^L_1,1(L_1) g_ττ(L_1)   d τ∧ d τ  .
        
        Note the factor of 2 difference between the conventions for the volumes here and in <cit.>. This is a result of our conventions explained below (<ref>).
        
        The integral for L=0 has been numerically evaluated in <cit.> and a good agreement with the analytic results has been observed. Here we repeat an analogous computation after placing a cutoff on the moduli space to find the volumes of the systolic subsets. This first requires us to find the demarcation curve ∂𝒱^2 πλ_1,1(L_1), as a function of the moduli τ, that separates the systolic subset from the region whose volume we subtract. This curve is
        
            ρ(2 πλ, 2 πλ_1, -2 πλ)^λ = | 
            	exp f_λ'^λ∂λ'  (q) 
            	|_λ' = λ    where    
            	q = e^2 π i τ  ,
        
        and the left-hand side of (<ref>) is given by the classical torus conformal blocks <cit.>
            f_λ'^λ (q) = λ'^2  4log q +  (1+λ^2)^2  8   (1+ λ'^2) q + 𝒪(q^2)   ,
        
        and the right-hand side is given by the mapping radius (<ref>).
        
        Now it is possible to approximate the integral (<ref>) numerically, for which we use Monte-Carlo (MC) integration and consider the representative cases L_1 = 0, π/2. We approximate the curve (<ref>) by keeping only the log q term, as the shape of the curve is observed to remain almost the same including the higher order terms in q. To this order ∂𝒱^2 πλ_1,1(L_1) is simply described by the line
        
            Im τ = 1 π logρ(2πλ, 2πλ_1, -2πλ)  .
        
        In our evaluation, we sampled points between this line and Im τ = 40 with the restriction -1/2 ≤Reτ≤ 1/2 and evaluated the volume of this region. We then subtracted it from the total volume of the moduli space V ℳ_1,1(L_1) to find V 𝒱^L_1,1(L_1). We repeated the MC integration five times for each value of L.  The results are already shown in figure <ref> and it is consistent with the analytical result (<ref>). We can also repeat a similar numerical evaluation to argue V 𝒱^L_0,4(L_i) is given by (<ref>) after constructing the WP metric on the four-bordered sphere <cit.>.
        
        We emphasize the restriction L ≤ L_∗ is necessary to cover the region ℳ_1,1 (L_1) ∖𝒱^L_1,1 (L_1) once and only once. This can be seen from setting L=L_∗. The symmetric punctured-torus (L_1 = 0, τ =i) saturates the systolic condition in this case <cit.> and any increase in the threshold length L would overcount the region ℳ_1,1 (L_1) ∖𝒱^L_1,1 (L_1) as a result <cit.>. This implies the formula (<ref>) no longer holds and the form of the subtraction term should be modified from L^2/4 appropriately. Hence the condition L ≤ L_∗ is necessary. Having L_1 > 0 doesn't lead to a stricter condition. Notice this argument doesn't inform whether L ≤ L_∗ is sufficient or not. This follows from the collar lemma instead.
        
        
        
        
        
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