FFebruary 2021MIT-CTP/5282
Hyperbolic Three-String Vertex
Atakan Hilmi Fırat Center for Theoretical Physics,Massachusetts Institute of TechnologyCambridge, MA 02139, USA
Abstract
We begin developing tools to compute off-shell string amplitudes with the recently pro-posed hyperbolic string vertices of Costello and Zwiebach. Exploiting the relation betweena boundary value problem for Liouville’s equation and a monodromy problem for a Fuch-sian equation, we construct the local coordinates around the punctures for the generalizedhyperbolic three-string vertex and investigate their various limits. This vertex correspondsto the general pants diagram with three boundary geodesics of unequal lengths. We derivethe conservation laws associated with such vertex and perform sample computations. Wenote the relevance of our construction to the calculations of the higher-order string verticesusing the pants decomposition of hyperbolic Riemann surfaces. Email: fi[email protected] a r X i v : . [ h e p - t h ] F e b ontents Defining off-shell amplitudes in closed string field theory requires selecting a set of string vertices V g,n with g − n > [1, 2]. These are subsets of the moduli spaces (cid:98) P g,n of compact Riemannsurfaces of genus g and n punctures with a choice of local coordinates (defined up to globalphases) around each puncture. String vertices ought to satisfy the geometric master equationin order to define a consistent quantum theory [3–5].There have been few proposals in the past for how to explicitly specify string vertices V g,n .The oldest, and probably the most well-known, is the one that uses the minimal area metrics onRiemann surfaces [2,6]. Using such metrics there is a simple prescription for how to specify stringvertices that solves the geometric master equation [2]. The minimal area metrics for higher genussurfaces, however, are not known explicitly and still lack rigorous proof of existence. Nonetheless,one may expect that these will soon follow in the light of the recent discoveries [7–9].Another proposal for string vertices V g,n that utilizes the fact that the Riemann surfacesconsidered for V g,n admit hyperbolic metrics (of constant negative Gaussian curvature K = − )was recently made by Moosavian and Pius [10,11]. This interesting approach seems particularlypromising considering the rigorously established existence of hyperbolic metrics and the recentdevelopments in evaluating integrals over the moduli spaces of Riemann surfaces using theassociated Teichmüller spaces [12–18]. However, it has been shown that these string vertices1olve the geometric master equation only to the first approximation and they require a correctionat each order of approximation. It is not known that such corrected string vertices always exist.Although they are intriguing in their own rights, we see that two proposals for string ver-tices above suffer from either missing the proof of existence or failing to satisfy the geometricmaster equation exactly, therefore falling short of providing a consistent string field theory.In order to have a consistent string field theory we must guarantee that the string verticesexist on the moduli spaces of Riemann surfaces while exactly satisfying the geometric masterequation. Hyperbolic string vertices by Costello and Zwiebach simultaneously achieved both ofthese conditions recently [19]. To that end, the authors considered Riemann surfaces endowedwith hyperbolic metric with geodesic boundaries of length L , for < L ≤ arcsinh(1), andwith systole greater than or equal to L . Then they specified the string vertices by attachingflat semi-infinite cylinders of circumference L at each boundary component to such surfaces.By the existence of hyperbolic metrics on Riemann surfaces of genus g and n boundaries with g − n > , it was argued that this construction is always possible. Furthermore, it hasbeen shown that the resulting string vertices exactly satisfy the geometric master equation bythe virtue of the collar theorems of hyperbolic geometry [20]. We are going to call the closedbosonic string field theory hyperbolic string vertices define hyperbolic string field theory .Beyond establishing the first rigorous, explicit, and exact construction for the string vertices,using the hyperbolic string vertices also seems promising from the perspective of the aforemen-tioned developments in computing integrals over the moduli spaces of Riemann surfaces byexploiting the underlying hyperbolic geometry, just like in the case of the vertices of Moosavianand Pius. One might imagine (or hope) similar methods can be applied to evaluate the stringamplitudes to arbitrary orders and provide a useful handle for the computations in hyperbolicstring field theory as a result.A natural first step in this direction would be to compute the off-shell three-string amplitudesusing the hyperbolic three-string vertex V , , which is constructed by grafting three flat semi-infinite cylinders to the three-holed sphere (or pair of pants ) equipped with a hyperbolic metric,since V , contains just a single surface. For the sake of generality, we are going to leave thecircumferences of the grafted cylinders arbitrary for this vertex, even though only the case ofequal circumferences is needed for V , [19]. This generalized hyperbolic three-string vertex is ofinterest in the hyperbolic string field theory in the long run on account of the well-known pantsdecomposition of Riemann surfaces [20]. For brevity, we will also denote this generalized vertexas hyperbolic three-string vertex without making a distinction.In order to perform the computations mentioned above using the operator formalism ofconformal field theory (CFT), one needs to obtain the explicit expressions of the local coordinatesaround the punctures for the hyperbolic three-string vertex [1]. In this paper, we find theselocal coordinates, investigate their various limits, and derive the associated conservation lawsby following the procedure in [21].In principle, the local coordinates for the hyperbolic three-string vertex can be obtained bythe following procedure. First, recall that the hyperbolic metric on the three-holed sphere with Systole on a bordered surface is defined as the length of the shortest closed geodesic that is not a boundarycomponent. L i ( i = 1 , , ) is unique up to isometry [20]. So we can simplywrite down this hyperbolic metric as ds = e ϕ ( z, ¯ z ) | dz | , (1.1)on the Riemann sphere minus three disjoint simply connected regions, or holes , unique up toPSL(2, C ) transformations, whose boundaries are geodesics of given lengths L i . From this pointof view, one can obtain the local coordinates by finding how punctured unit disks conformallymap onto these simply connected regions, since a semi-infinite cylinder is conformal to a punc-tured unit disk and it canonically introduces the local coordinates [19]. Note that such conformaltransformations exist by the Riemann mapping theorem.Therefore, we see that the problem of finding the local coordinates for the hyperbolic three-string vertex is a two-step procedure:1. Find an explicit description of the union of three disjoint simply connected regions onthe Riemann sphere whose complement is endowed with a hyperbolic metric (1.1) andboundary components are geodesics of lengths L i ,2. Find the conformal transformations from punctured unit disks to the aforementioned sim-ply connected regions.The first step clearly involves solving a complicated boundary value problem for a partial differ-ential equation, Liouville’s equation , and getting an exact answer is a hard endeavor in general.Luckily, it is known that the solutions for such boundary value problem can be related to amonodromy problem of a particular second-order linear ordinary differential equation with reg-ular singularities, or a
Fuchsian equation , on the complex plane [22,23]. Exploiting this relation,which we review and expand in sections 2 and 3, we find the explicit description of the hyperbolicmetric (1.1) and of the three holes on the Riemann sphere, up to PSL(2, C ) transformations.Furthermore, the second step becomes trivial after we find such explicit description as weargue in section 3. In the end, for the hyperbolic three-string vertex whose grafted flat cylindershave the circumferences L i ≡ πλ i , (1.2)we obtain the following local coordinates w i ( z ) around the punctures at z = 0 , , ∞ respectively: w ( z ) = 1 N exp (cid:18) v ( λ , λ , λ ) λ (cid:19) z (1 − z ) − λ λ (1.3) × (cid:34) F (cid:0) (1 + iλ − iλ + iλ ) , (1 + iλ − iλ − iλ ); 1 + iλ ; z (cid:1) F (cid:0) (1 − iλ + iλ − iλ ) , (1 − iλ + iλ + iλ ); 1 − iλ ; z (cid:1) (cid:35) iλ = 1 N exp (cid:18) v ( λ , λ , λ ) λ (cid:19) (cid:18) z + 1 + λ + λ − λ λ ) z + O ( z ) (cid:19) , We fix the locations of the punctures to z = 0 , , ∞ using PSL(2, C ) transformations without loss of generality.So indices and numbers i, j = 1 , , appearing on the objects denote the punctures z = 0 , , ∞ respectively, unlessotherwise stated. This shall be obvious from the context and we are not going to report it every time. If thereis no label on an object, it should be understood that it has the same value for each puncture. ( z ) = 1 N exp (cid:18) v ( λ , λ , λ ) λ (cid:19) (1 − z ) z − λ λ (1.4) × (cid:34) F (cid:0) (1 + iλ − iλ + iλ ) , (1 + iλ − iλ − iλ ); 1 + iλ ; 1 − z (cid:1) F (cid:0) (1 − iλ + iλ − iλ ) , (1 − iλ + iλ + iλ ); 1 − iλ ; 1 − z (cid:1) (cid:35) iλ = 1 N exp (cid:18) v ( λ , λ , λ ) λ (cid:19) (cid:18) (1 − z ) + 1 + λ + λ − λ λ ) (1 − z ) + O ((1 − z ) ) (cid:19) ,w ( z ) = 1 N exp (cid:18) v ( λ , λ , λ ) λ (cid:19) (cid:18) z (cid:19) (cid:18) − z (cid:19) − λ λ (1.5) × (cid:34) F (cid:0) (1 + iλ − iλ + iλ ) , (1 + iλ − iλ − iλ ); 1 + iλ ; z (cid:1) F (cid:0) (1 − iλ + iλ − iλ ) , (1 − iλ + iλ + iλ ); 1 − iλ ; z (cid:1) (cid:35) iλ = 1 N exp (cid:18) v ( λ , λ , λ ) λ (cid:19) (cid:18) z + 1 + λ + λ − λ λ ) 1 z + O (cid:18) z (cid:19)(cid:19) . Here F ( a, b ; c ; z ) is the ordinary hypergeometric function F ( a, b ; c ; z ) = 1 + abc z + a ( a + 1) b ( b + 1) c ( c + 1) z
2! + · · · , (1.6)and the function v ( λ , λ , λ ) is given in terms of the gamma function Γ( z ) as v ( λ , λ , λ ) ≡ i Log (cid:34)
Γ ( − iλ ) Γ ( iλ ) γ (cid:0) (1 + iλ + iλ + iλ ) (cid:1) γ (cid:0) (1 + iλ − iλ + iλ ) (cid:1) γ (cid:0) (1 − iλ − iλ + iλ ) (cid:1) γ (cid:0) (1 − iλ + iλ + iλ ) (cid:1) (cid:35) ,γ ( x ) ≡ Γ(1 − x )Γ( x ) . (1.7)The factors N i above will be called scale factors . They are fixed by integer ˜ l i ∈ Z via N i = exp (cid:20) πλ i (cid:18) ˜ l i + 12 (cid:19)(cid:21) . (1.8)Tilde on the integers ˜ l i comes from our construction. As we will see, among the sets of integers l i ,only specific ones, l i = ˜ l i , would give the correct scale factor. The integers ˜ l i can be determinedfor a given set of λ i ’s in principle. Even though we couldn’t find a closed-form expression forthe integers ˜ l i ’s for arbitrary λ i ’s, one can still easily find them by investigating numerical plotsof the local coordinates. In certain symmetric situations it is possible to find the integers ˜ l i ’swithout resorting the plots. For example, when ≤ λ i = λ ≤ for the grafted cylinders, wefind ˜ l i = ˜ l = − . This result is anticipated to hold for all values of λ i = λ .Since we have the explicit expressions of the local coordinates for the hyperbolic three-vertex (1.3-1.5), we can check their consistency with the other local coordinates in the literatureby investigating their limiting behaviors [10, 24, 25]. For instance, as argued in [19], this vertexmust produce the three-string vertex obtained from the minimal area metric as all λ i → ∞ at the same rate, which we are going to denote as the minimal area limit . In the light of thisfact, we consider the minimal area limit of the coordinates (1.3-1.5) and show that the local4oordinates for the minimal area three-string vertex and those for the hyperbolic three-stringvertices with λ → ∞ match perturbatively to the order O ( z ) in section 4. We then discussthe possibility of extending our argument to all orders in z .Moreover, the hyperbolic three-string vertex must reduce to the three-string vertex consid-ered by Moosavian and Pius [10] as λ i → after a suitable modification, since the geodesicboundaries become cusps in this regime and this is exactly what is considered there. We ar-gue that this limiting behavior indeed holds in section 4. Lastly, we consider the situation λ = λ + λ with λ i → ∞ for which the geometry resembles the light-cone vertex [25]. Weshow that the hyperbolic three-string vertex reduces to the light-cone vertex in this limit, inaccord with our expectations.Having an explicit expression for the local coordinates (1.3-1.5) also means that it is possibleto derive the conservation laws for the hyperbolic three-string vertex in the spirit of [21], which wedo in section 5. Again, we can investigate various limits of these conservation laws. Especiallywe observe that all of our expressions in section 5 reduces to their respective counterpartsin [21] in the minimal area limit. This is consistent, since the open string Witten vertex and itsclosed string analog must generate the same conservation laws. It is known that conservationlaws provide an systematic and easily implementable procedure for the computations in thecubic open string field theory, especially for the level truncation [26], and we hope that theseexpressions will accomplish the same in the hyperbolic string field theory in the future.As a sample computation using the local coordinates (1.3-1.5), we calculate the t term inthe closed string tachyon potential V with t is the zero-momentum tachyonic field in the caseof λ i = λ . Remember this is the case that appears in the string action. We find ( α (cid:48) = 2 ) κ V = − t + 13 t r + · · · = − t + 13 exp (cid:20) v ( λ, λ, λ ) + 3 πλ (cid:21) t + · · · . (1.9)Here κ is the closed string coupling constant and r is the mapping radius of the local coordinates,whose inserted expression is derived in section 3. Note that this calculation is exactly likein [27–29], the only difference being the mapping radii we used for the expression above.In order to get a sense of its value, let us set λ = L ∗ / (2 π ) = arcsinh (1) /π ≈ . , which isthe largest value of λ for which the hyperbolic vertices solves the geometric master equation [19].Substituting this value and evaluating, we obtain the closed string tachyon potential V in thehyperbolic string field theory is given by κ V = − t + (cid:0) . × (cid:1) t + · · · . (1.10)The coefficient for the t term is quite large compared to the corresponding one in the minimalarea three-string vertex, which is approximately equal to . [27–29]. However, this coefficientin fact has the expected order of magnitude. We can see this by considering the coefficientobtained from the minimal area three-string vertex with stubs of length π , which roughly “lookslike” a hyperbolic three-string vertex geometrically. The coefficient for the case with stubsis easily obtained by observing that adding stubs scales mapping radii by e − π , and in turnmultiplies the no-stub coefficient by e π by the first equality in (1.9). This gives approximately e π · . ≈ . × , which is close to the value given in (1.10).5he outline of the paper is as follows. In section 2 we introduce the boundary value problemfor the hyperbolic metric with geodesic boundaries of fixed lengths on the Riemann sphereminus three holes and its relation to Fuchsian equations. In section 3 we consider the relevantmonodromy problem in order to find the explicit description of the holes on the Riemann sphere.The results of these two sections are well-established in the literature [22, 23], but we provide aself-contained discussion where we emphasize and investigate the resulting hyperbolic geometryin more detail. Additionally, we construct the local coordinates around the punctures for thehyperbolic three-string vertex in section 3 and later in section 4 we investigate their variouslimits. Lastly, we obtain the conservation laws associated with the hyperbolic three-string vertexin section 5. We conclude the paper and discuss the possible future directions in section 6. In this section, we describe the problem of finding an explicit description of the hyperbolicmetric on the three-holed sphere with geodesic boundaries of lengths L i on the Riemann sphere,which will help us obtain the shapes and locations of the geodesic boundaries and the localcoordinates later on. As we mentioned briefly, this is equivalent to solving Liouville’s equationwith specified boundary conditions on the Riemann sphere minus three holes. This problem ishard by itself, so instead we introduce a stress-energy tensor (in the sense of Liouville theory)and consider its associated Fuchsian equation, which we define below. The properties of thisequation is investigated. Most importantly, we show that its multi-valued solutions can berelated to hyperbolic metrics. The results of this section are well-known in the literature in thecontext of Liouville theory and the uniformization problem [22, 23, 30–40], but we are going toprovide a self-contained review that focuses on the issues relevant to us.As noted above, our first goal is to solve Liouville’s equation ∂ ¯ ∂ϕ ( z, ¯ z ) = e ϕ ( z, ¯ z ) , (2.1)on the three-holed sphere X whose boundaries are chosen to be geodesics of lengths L i ofthe metric (1.1). It can be easily seen that satisfying Liouville equation is equivalent to themetric (1.1) having constant negative curvature K = − . We will call e ϕ ( z, ¯ z ) the conformalfactor and take ϕ ( z, ¯ z ) ∈ R always to define a real metric. Like we mentioned before, we canthink the surface X endowed with the metric (1.1) as the Riemann sphere (cid:98) C = C ∪ {∞} withthree disjoint simply connected regions taken out and this understanding will be implicit. So ( z, ¯ z ) will denote the complex coordinates on X ⊂ (cid:98) C .Solving the boundary value problem described above directly is non-trivial and we won’tattempt to do that. Instead, we are going to relate this problem to solving a more manageablelinear ordinary differential equation. In order to do that, let the factor ϕ denote a solution ofLiouville’s equation (2.1) and define the (holomorphic) stress-energy tensor associated with ϕ as follows [37]: T ϕ ( z ) ≡ − ( ∂ϕ ) + ∂ ϕ = − e ϕ ∂ e − ϕ . (2.2)Observe that we only wrote the dependence on z , and not on ¯ z , of the stress-energy tensor since6t can be shown that T ϕ is holomorphic, ¯ ∂T ϕ = 0 , using Liouville’s equation (2.1). Furthermore,the converse of this statement holds as well: If T ϕ = T ϕ ( z ) is holomorphic, then the factor ϕ defined by (2 . solves the Liouville’s equation. Lastly, we note that T ϕ ( z ) is the (classical) stress-energy tensor in the context of Liouville theory and it transforms under conformal transformation z → ˜ z ( z ) as follows [37]: T ϕ ( z ) = (cid:18) ∂ ˜ z∂z (cid:19) (cid:101) T ϕ (˜ z ) + { ˜ z, z } . (2.3)Here tilde on the stress-energy tensor indicates that it is written in the ˜ z coordinates. We cansimilarly define the anti-holomorphic stress-energy tensor T ϕ by replacing ∂ → ¯ ∂ in (2.2).Now consider the following second-order linear ordinary differential equation constructedwith the stress-energy tensor T ϕ ( z ) above [22, 23]: ∂ ψ ( z ) + T ϕ ( z ) ψ ( z ) = 0 . (2.4)We will call this the holomorphic Fuchsian equation associated with T ϕ ( z ) . The reason for thename “Fuchsian” will be justified in section 3 when we show that the relevant T ϕ ( z ) containsat most double poles, so that the equation (2.4) has only regular singularities (i.e. Fuchsian ).Similarly, we can define the anti-holomorphic Fuchsian equation associated with T ϕ (¯ z ) . Con-sidering (2.3), in order to make the equation (2.4) conformal invariant, we are going to take theobject ψ ( z ) transforms as a conformal primary of dimension ( − , . That is we demand ˜ ψ (˜ z ) = (cid:18) ∂ ˜ z∂z (cid:19) ψ ( z ) . (2.5)under conformal transformation z → ˜ z ( z ) .Now suppose we have solved the Fuchsian equation and found two linearly independent,not necessarily single-valued, complex-valued solutions ψ + ( z ) and ψ − ( z ) . We are going toalways assume these solutions are normalized appropriately, in the sense that their Wronskian W ( ψ − , ψ + ) is equal to one: W ( ψ − , ψ + ) ≡ ( ∂ψ + ) ψ − − ψ + ( ∂ψ + ) = 1 . (2.6)Now define the ratio A ( z ) of these solutions and observe that we have the relations A ( z ) ≡ ψ + ( z ) ψ − ( z ) ⇐⇒ ψ + ( z ) = A ( z ) (cid:112) ∂A ( z ) , ψ − ( z ) = 1 (cid:112) ∂A ( z ) . (2.7)From this, we immediately see the stress-energy tensor can be written as follows: T ϕ ( z ) = − ∂ ψ − ψ − = − ∂A ) ∂ ( ∂A ) − = ( ∂A ) ∂ (cid:16) ( ∂A ) − ∂ A (cid:17) = ∂ A ( z ) ∂A ( z ) − (cid:18) ∂ A ( z ) ∂A ( z ) (cid:19) ≡ { A ( z ) , z } = ⇒ T ϕ ( z ) = { A ( z ) , z } . (2.8)7ere {· , ·} is the Schwarzian derivative. In general, it is highly non-trivial to find the function A ( z ) for a given T ϕ ( z ) satisfying (2.8) above. However, if we know the solutions to the Fuchsianequation (2.4), we see that A ( z ) is determined by (2.7) up to Möbius transformations. That isone utility of the Fuchsian equation. Moreover, given A ( z ) satisfying T ϕ ( z ) = { A ( z ) , z } , we canfind the normalized solutions for the Fuchsian equation from (2.7) as well. Note that A ( z ) is ascalar under conformal transformations as can be seen from (2.5) and (2.7).Also we can see that putting the stress-energy tensor T ϕ ( z ) in the form (2.8) and knowingsuch A ( z ) is advantageous on the account of the transformation property of the stress-energytensor (2.3). The relation (2.8), combined with the transformation property of the Schwarzianderivative and the stress-energy tensor, allows us to find the explicit expression of the stress-energy tensor T ϕ in other coordinates. We will see the benefit of this observations in the nextsection.Another utility of the Fuchsian equation (2.4) can be understood as follows. We can easilysee that ψ = e − ϕ ( z, ¯ z )2 solves (2.4) using the second equality in (2.2). This solution of the Fuchsianequation is real and single-valued because the metric (1.1) itself is real and single-valued. Itis important to observe that such factor solves the Fuchsian equation, because this allows usto relate the linearly independent, normalized solutions ψ ± ( z ) of the Fuchsian equation to thehyperbolic metric (1.1): knowing ψ ± ( z ) would suffice to construct the metric.Before we do that more precisely, we should first describe the multi-valuedness of the solu-tions ψ ± ( z ) . For our purposes, it is going to be sufficient to assume that the multi-valuednessof the solutions ψ ± ( z ) are described by SL(2, R ) transformations, in the sense that when we goaround any point z = u ∈ (cid:98) C by ( z − u ) → e πi ( z − u ) the solutions are taken to be transformingas follows: (cid:34) ψ + ψ − (cid:35) → (cid:34) a bc d (cid:35) (cid:34) ψ + ψ − (cid:35) where a, b, c, d ∈ R , ad − bc = 1 , (2.9)unless otherwise stated. That is, we assume the values that the functions ψ ± ( z ) attain at agiven point are related by SL(2, R ) transformations like above. From this, it is easy to see thatthe solution e − ϕ ( z, ¯ z )2 of (2.4) is given by the following linear combination of ψ + ( z ) and ψ − ( z ) : e − ϕ ( z, ¯ z )2 = C i ψ − ( z ) ψ + ( z ) − ψ + ( z ) ψ − ( z )) , (2.10)since this is the unique real linear combination of the solutions ψ ± ( z ) that is invariant underSL(2, R ) transformations (i.e. single-valued). As usual, the bar over the solutions denotes thecomplex conjugation. Here C is a real constant, which turns out to be C = ± , as we will showit shortly. With this, the following metric has constant negative curvature K = − : ds = e ϕ ( z, ¯ z ) | dz | = − | dz | ( ψ − ( z ) ψ + ( z ) − ψ + ( z ) ψ − ( z )) . (2.11)Note that a version of these expressions appears in the context of Liouville theory [37]. There,the solutions ψ ± ( z ) are interpreted as spin- / representations of SL(2, R ) and their physicalmeaning is discussed.The main takeaway from the discussion in the previous paragraphs is that the hyperbolic8etric on a three-holed sphere X can be related to the solutions of the Fuchsian equation usinga suitable T ϕ ( z ) . From the expression in (2.2), it might seem that finding T ϕ ( z ) as a functionof z is as hard as finding the explicit form of the metric (1.1). However, as we will see insection 3, T ϕ ( z ) can be found without knowing the metric. Then we can deduce the form ofthe hyperbolic metric by solving the associated Fuchsian equation through the relation (2.11),which will eventually lead us to the local coordinates. Before we conclude this section, we need to show C = ± as we claimed. It is clear thatnot every value of a priori unfixed C ∈ R can define a hyperbolic metric with K = − , so weneed to choose the right value(s). This is essentially the reflection of the fact that the Fuchsianequation is linear: Every scaling of e − ϕ ( z, ¯ z )2 is also a solution of (2.4), even though the scaledones don’t define a hyperbolic metric with K = − because the Liouville’s equation (2.1) isnon-linear.We can fix such C once and for all as follows. First note that the conformal factor e ϕ ( z, ¯ z ) = λ | ∂f ( z ) | | f ( z ) | sin ( λ log | f ( z ) | ) = | ∂ ( λ log( f ( z ))) | sin ( λ log | f ( z ) | ) . (2.12)always defines a (possibly singular) hyperbolic metric with K = − , or equivalently, ϕ abovesolves the Liouville’s equation (2.1) for an arbitrary holomorphic function f ( z ) and an arbitrary λ ∈ R ≥ , as one can check by explicit calculation. Now take the function f ( z ) to be equal to f ( z ) = A ( z ) iλ = (cid:18) ψ + ( z ) ψ − ( z ) (cid:19) iλ . (2.13)We will denote the right-hand side as the scaled ratio . After substituting this expressioninto (2.12) we exactly get the metric (2.11). This shows C = ± . For us, the equivalencebetween (2.11) and (2.12), with the choice (2.13), is going to be extremely useful and we willuse both forms interchangeably in our arguments.In summary, we have seen that we can relate the hyperbolic metric on a three-holed sphere X to the solutions of the Fuchsian equation (2.4) through (2.11). Not only this will provide us asolution for the Liouville’s equation (2.1), but, more importantly, it will be also used to make theboundaries of X geodesics of the metric (1.1). After all, that’s the whole reason we are takingthis detour into Fuchsian equations. We have already seen that the conformal factor (2.12)always defines a (possibly singular) hyperbolic metric for any given f ( z ) , but the boundariesof X are going to be geodesics only when we relate it to a particular set of solutions for theFuchsian equation through the relation (2.13), as we shall see. In the next section, we are goingto focus on the three-punctured sphere (cid:101) X = C \ { , , ∞} , rather than a three-holed sphere X ,since it is simpler to deal with initially. Then we will cut open appropriate holes around thepunctures in (cid:101) X to return back to X ⊂ (cid:98) C and graft flat semi-infinite cylinders to these holes toconstruct the local coordinates for the hyperbolic three-string vertex. These relations hold for other hyperbolic Riemann surfaces with geodesic boundaries as well. But we willrestrict our discussion for three-holed sphere, since it is the simplest case to perform these computations explicitly. A Monodromy Problem of Fuchsian Equation
In this section we find the hyperbolic metric on a three-holed sphere X by investigating acertain monodromy problem of the Fuchsian equation (2.4) on the three punctured sphere (cid:101) X and construct the local coordinates for the hyperbolic three-string vertex. First, we describe therelevant monodromy problem and solve the Fuchsian equation on (cid:101) X accordingly. Then we findthe explicit form of the (singular) hyperbolic metric on (cid:101) X by the relations given in section 2.The resulting geometry looks like three semi-infinite series of hyperbolic cylinders, attachedwhere they flare up, connected to each other while keeping the curvature constant and negative.Next, we cut these hyperbolic cylinders out from the geometry appropriately, which leaveus with a three-holed sphere X . This procedure doesn’t change the hyperbolic metric, so atthe end we obtain an explicit description of the hyperbolic metric with geodesic boundarieson a three-holed sphere. Moreover, we describe the holes on the Riemann sphere explicitlyby investigating the simple closed geodesics of this hyperbolic metric. After that, grafting flatsemi-infinite cylinders needed for the construction of the local coordinates amounts to simpleconformal transformations of the punctured unit disks to these holes.Most of the results from this section (except for subsection 3.4) are from [22, 23], for whichwe provide a detailed summary. However, we elaborate the geometric picture coming from thehyperbolic metric in more detail and prove some important results necessary for the explicitconstruction of the local coordinates. Consider the three-punctured sphere (cid:101) X = C \ { , , ∞} and suppose that the solutions of theFuchsian equation (2.4) have hyperbolic SL(2, R ) monodromy around each puncture. That is,as we go around a puncture by ( z − z j ) → e πi ( z − z j ) , we demand that the solutions for theFuchsian equation ψ ± ( z ) change as, (cid:34) ψ + ψ − (cid:35) → M j (cid:34) ψ + ψ − (cid:35) where M j ∈ SL(2, R ) , | Tr M j | > . (3.1)Note that the condition on the trace makes the matrix M j a hyperbolic element of SL(2, R ) andthat’s why we say we have a hyperbolic monodromies around the puncture z = z j . Realizingthis structure for the solutions to the Fuchsian equation and finding them is our monodromyproblem . This problem is first considered in [22] in the context of Liouville theory. We will calla puncture hyperbolic singularity if the solutions of the Fuchsian equation have a hyperbolicSL(2, R ) monodromy around it.In order to solve the monodromy problem, we need to first determine appropriate T ϕ ( z ) as a function of z (if exists) so that the solutions of the Fuchsian equation can realize thesemonodromies around the punctures. Then declaring that particular T ϕ ( z ) to be equal to (2.2)coming from Liouville theory and using the reasoning in section 2 we can extract the possiblysingular hyperbolic conformal factor on (cid:101) X with the solutions that realize these monodromies.As explained above the equation (2.10), this metric is going to be single-valued by SL(2, R ) X .Before we do that, let us investigate an individual hyperbolic singularity. We begin bypicking a puncture, say z = 0 , and choosing a normalized basis of solution ψ ± ( z ) for which themonodromy around z = 0 is diagonal as follows: (cid:34) ψ +1 ψ − (cid:35) → (cid:34) − e − πλ − e πλ (cid:35) (cid:34) ψ +1 ψ − (cid:35) ⇐⇒ ψ ± ( z ) = e ± iv √ iλ z ± iλ (1 + O ( z )) . (3.2)The solutions always can be put into this form around z = 0 since hyperbolic elements ofSL(2, R ) can be diagonalized by conjugation, which amounts to performing a SL(2, R ) change ofbasis of the solutions. Here λ ∈ R will be called the geodesic radius associated with the z = 0 puncture and the reason for its name will be apparent shortly. Without loss of generality wewill take λ > . Note that the Wronskian of these solutions is equal to thanks to the factor / √ iλ in front. Furthermore, we also included the factors e ± iv , with v ∈ C , to account forthe multiplicative constant that is not fixed by the Wronskian condition (2.6). As we shall see,the constant v will be fixed below by demanding SL(2, R ) monodromies around each puncture.Using (3.2), we can write the scaled ratio associated with the puncture z = 0 , as in (2.13), ρ ( z ) ≡ (cid:18) ψ +1 ( z ) ψ − ( z ) (cid:19) iλ = e v λ ( z + O ( z )) . (3.3)Note that this series expansion converges only on the open unit disk D = { z ∈ C | | z | < } ,around the puncture z = 0 , since outside D the scaled ratio ρ ( z ) is multi-valued by thesolutions ψ ± ( z ) having a non-diagonal monodromy around the punctures at z = 1 , ∞ . Wecan analytically continue the scaled ratio defined above outside the disk D , but inevitablythis will require us to choose a branch for which ρ ( z ) is continuous across ∂D except at thepunctures/branch cuts. We will choose the branch cut (cid:101) L of ρ ( z ) to extend from 1 to ∞ alongthe real axis and take this to be the principal branch of ρ ( z ) . Thus, we conclude that the scaledratio ρ ( z ) can be defined analytically on the set S = C \ (cid:101) L , (3.4)with the expansion (3.3). When we mention the scaled ratio, we will consider the principalbranch implicitly henceforth, unless otherwise stated. Lastly, note that the scaled ratio is ananalytic scalar under conformal transformations, just like the ratio A ( z ) in (2.7).Now by performing the conformal transformation z → ρ = ρ ( z ) on S and using theequation (2.8) along with the properties of the Schwarzian derivative we see T ϕ ( z ) = { A ( z ) , z } = (cid:26) ψ +1 ( z ) ψ − ( z ) , z (cid:27) = { ρ ( z ) iλ , z } = ( ∂ρ ) { ρ iλ , ρ } + { ρ , z } . (3.5)Comparing the final form with (2.3) we read that the stress-energy tensor (cid:102) T ϕ ( ρ ) in the ρ -plane11akes the following form: (cid:102) T ϕ ( ρ ) = { ρ iλ , ρ } = ∆ ρ where ∆ ≡
12 + λ . (3.6)Here the real number ∆ will be called the weight . As a result of this, the Fuchsian equation inthe ρ -plane takes a very simple form and we can easily obtain its solutions: ∂ ˜ ψ ( ρ ) ∂ρ + ∆ ρ ˜ ψ ( ρ ) = 0 = ⇒ (cid:101) ψ ± ( ρ ) = ρ ± iλ √ iλ . (3.7)Here, (cid:101) ψ ± ( z ) are normalized solutions that are chosen to have diagonal monodromy around thepuncture z = 0 , or equivalently ρ = 0 . Here we set the phase factor not fixed by Wronskianequal to one for convenience. Note that the scaled ratio of these two solutions is simply (cid:18) ψ +1 ( z ) ψ − ( z ) (cid:19) iλ = ρ . (3.8)As a result the hyperbolic metric that the Fuchsian equation produces in the ρ -and z -plane aresimply, by the relation (2.12) with the choice f ( ρ ) = ρ , ds = λ | ρ | sin ( λ log | ρ | ) | dρ | = λ | ∂ρ ( z ) | | ρ ( z ) | sin ( λ log | ρ ( z ) | ) | dz | . (3.9)There are two important things we should notice here. First, the metric takes the form of a seriesof hyperbolic cylinders that are attached to each other where they flare up in the ρ -plane, andby the expansion (3.3), when we are sufficiently close to z = 0 in the z -plane. We will explainthis fact, along with the closed geodesics/singularities of this metric in more detail after weobtain the explicit form for ρ ( z ) .Secondly, the z -plane metric is smooth (except for the singularities) not only over S butacross the branch cut (cid:101) L as well. The reason is simply that we demanded SL(2, R ) monodromyaround each puncture and we know that the metric above is invariant under the monodromiesof that kind by the equivalent form in (2.11). So we can use the metric above in the entiretyof the z -plane minus punctures as long as we guarantee the SL(2, R ) monodromies around allpunctures simultaneously.Since we are also demanding hyperbolic SL(2, R ) monodromies for the remaining punctures,the two facts above hold for them without too much modification. We just have to change ρ with appropriate ρ j . Moreover, these produce the same hyperbolic metric when we pullbackthem to the z -plane from any ρ j -plane. This can be easily seen by noticing the fact that theappropriate SL(2, R ) change of basis of solutions ψ ± ( z ) can diagonalize the monodromy aroundanother puncture, by the fact that hyperbolic elements in SL(2, R ) are conjugate to a diagonalmatrix. Such transformations of the solutions don’t affect the metric as we argued before.In conclusion, we see the motivation behind using the Fuchsian equation with correct mon- Considering this factor just adds a phase shift for the sine that appears in (3.9), which would be unimportantfor our considerations in this subsection. yperbolic cylinderspair of pants Figure 1:
Sketch of the hyperbolic metric described by the Fuchsian equation with three hyperbolicsingularities. The smooth hyperbolic pair of pants with geodesic boundaries is going toconnect the hyperbolic cylinders, as we shall see more explicitly. odromy structure in more detail from these comments. Even though any choice of holomorphicfunction works in (2.12) to define a hyperbolic metric, using the scaled ratio coming from theFuchsian equation with the monodromy data above will guarantee to generate the hyperbolicmetric (3.9) on the z -plane where three series of attached hyperbolic cylinders connected to eachother with hyperbolic pair of pants (i.e. three-holed sphere endowed with a hyperbolic metric),as shown in Figure 1. Moreover, it is easy to see from Figure 1 that one can obtain a descrip-tion of the hyperbolic pair of pants by taking out the hyperbolic cylinders and considering theremaining connected region only. This justifies why we considered this particular monodromyproblem of Fuchsian equation on (cid:101) X : It is a natural starting point to generate the hyperbolicmetric with geodesic boundaries on X . Before we describe the hyperbolic pair of pants, we are going to get an explicit expression for thehyperbolic metric on the three-punctured sphere (cid:101) X resulting from three hyperbolic singularities.First, we solve the monodromy problem. That is, we find T ϕ ( z ) for which the solutions of theFuchsian equation can realize the monodromy structure described in (3.1). Then we solve theresulting Fuchsian equation with these prescribed monodromies and proceed to construct themetric by finding the scaled ratio.In order to find T ϕ ( z ) as a function of z , observe that when we are close to the puncture z = 0 , i.e. ρ = 0 , the stress-energy tensor in (3.5) takes the form T ϕ ( z ) = ( ∂ρ ) ∆ ρ + { ρ , z } = ( e v λ + . . . ) ∆ ( e v λ z + . . . ) + { e v λ z + . . . , z } = ∆ z + O ( 1 z ) , (3.10)using (3.3) and (3.6). From this we see that T ϕ ( z ) must have at most double poles of residues ∆ , ∆ and ∆ at z = 0 , , ∞ , respectively in order to have a hyperbolic singularity. One caneasily show that the unique T ϕ ( z ) that has such structure is T ϕ ( z ) = ∆ z + ∆ ( z − + ∆ − ∆ − ∆ z ( z − , (3.11)with ∆ i = (1 + λ i ) / . Clearly we have at most double poles at z = 0 , with appropriate13esidues. Using the inversion map z → ˜ z = 1 /z , along with { ˜ z, z } = 0 , we can easily see thatwe have the correct structure at infinity, i.e. ˜ z = 0 , as well (cid:102) T ϕ (˜ z ) = 1˜ z (cid:2) ∆ ˜ z + ∆ ˜ z + (∆ − ∆ − ∆ )˜ z + O (˜ z ) (cid:3) = ∆ ˜ z + O ( 1˜ z ) . (3.12)The stress-energy tensor T ϕ ( z ) in (3.11) solves the monodromy problem. In order to seethat, first observe the Fuchsian equation in this case takes the form ∂ ψ ( z ) + 12 (cid:20) ∆ z + ∆ ( z − + ∆ − ∆ − ∆ z ( z − (cid:21) ψ ( z ) = 0 . (3.13)This is the hypergeometric equation, written in the so-called Q -form. The solutions of thisequation and their properties are well tabulated (see Schwarz’s function in [41], also [22, 30]).They are, with proper normalization and assignment of diagonal monodromy around z = 0 , ψ ± ( z ) = e ± iv ( λ ,λ ,λ √ iλ z ± iλ (1 − z ) ∓ iλ (3.14) × F (cid:18) ± iλ ∓ iλ ± iλ , ± iλ ∓ iλ ∓ iλ ± iλ ; z (cid:19) . Here F ( a, b ; c ; z ) is the ordinary hypergeometric function (1.6). Using the transformationproperties of these solutions (2.5) and appropriately exchanging punctures, we can also findthe normalized solutions having a diagonal monodromy around z = 1 and z = ∞ . They are,respectively, ψ ± ( z ) = i e ± iv ( λ ,λ ,λ √ iλ (1 − z ) ± iλ z ∓ iλ (3.15) × F (cid:18) ± iλ ∓ iλ ± iλ , ± iλ ∓ iλ ∓ iλ ± iλ ; 1 − z (cid:19) ,ψ ± ( z ) = ( iz ) e ± iv ( λ ,λ ,λ √ iλ (cid:18) z (cid:19) ± iλ (cid:18) − z (cid:19) ∓ iλ (3.16) × F (cid:18) ± iλ ∓ iλ ± iλ , ± iλ ∓ iλ ∓ iλ ± iλ ; 1 z (cid:19) . We should emphasize again that the constant v ( λ , λ , λ ) = v above is not fixed by the Wron-skian and we will determine it below by demanding hyperbolic SL(2, R ) monodromies aroundall the punctures. We will call this compatibility of monodromies. Notice that compatibilityis not guaranteed a priori. This is because when we demand a SL(2, R ) monodromy around apuncture, the monodromies around remaining punctures are elements of SL(2, C ), rather thanSL(2, R ), in general. So, actually, in order to solve the monodromy problem completely, wemust show that the compatibility is achievable for the Fuchsian equation (3.13).In order to ensure compatibility, first observe that we have some SL(2, C ) monodromy around It can still have unit determinant without loss of generality if one assumes appropriately normalized solutionsin the sense of (2.6). = 1 if we use the basis ψ ± ( z ) . That is, as (1 − z ) → e πi (1 − z ) , we have (cid:34) ψ +1 ψ − (cid:35) → M (cid:34) ψ +1 ψ − (cid:35) where M ∈ SL (2 , C ) . (3.17)Here, and throughout, we are going to denote the monodromy of the solutions ψ ± i ( z ) around thepuncture z = z j as M ji . In order to have hyperbolic SL(2, R ) monodromy around z = 1 whilesimultaneously having hyperbolic SL(2, R ) monodromy around z = 0 , we have to make sure that M ∈ SL(2, R ) and | Tr M | > by adjusting v appropriately. To that end, first notice that wehave a diagonal hyperbolic SL(2, R ) monodromy around z = 1 if we use the basis ψ ± ( z ) : (cid:34) ψ +2 ψ − (cid:35) → M (cid:34) ψ +2 ψ − (cid:35) = (cid:34) − e − πλ − e πλ (cid:35) (cid:34) ψ +2 ψ − (cid:35) . (3.18)Secondly, notice that two basis ψ ± ( z ) and ψ ± ( z ) are related via the connection formulas for thehypergeometric function (see section 2.9 in [41], also [22, 30]) (cid:34) ψ +1 ψ − (cid:35) = S (cid:34) ψ +2 ψ − (cid:35) = (cid:112) λ λ (cid:34) e i v − v g − e i v v g + − e − i v v g + − e − i v − v g − (cid:35) (cid:34) ψ +2 ψ − (cid:35) , (3.19)here v = v ( λ , λ , λ ) and the functions g ± are given by g ± = Γ ( iλ ) Γ ( ± iλ )Γ (cid:16) iλ ± iλ + iλ (cid:17) Γ (cid:16) iλ ± iλ − iλ (cid:17) . (3.20)Using them, we observe the monodromies in two basis are related by the following conjugation: M = SM S − = λ λ e − πλ | g − | − e πλ | g + | − (cid:0) e πλ − e − πλ (cid:1) e iv g + g − (cid:0) e πλ − e − πλ (cid:1) e − iv g + g − e πλ | g − | − e − πλ | g + | . (3.21)Here we used the fact | g + | − | g − | = 1 λ λ , (3.22)which can be derived from the expression (3.20).Now it is a simple calculation using (3.21) and (3.22) to check that det M = 1 . Therefore inorder to have M ∈ SL(2 , R ) it is enough to make sure the entries of M are real. That meanswe have e iv ( λ ,λ ,λ ) = g + g − g + g − = Γ ( − iλ ) Γ ( iλ ) γ (cid:16) iλ + iλ + iλ (cid:17) γ (cid:16) iλ − iλ + iλ (cid:17) γ (cid:16) − iλ − iλ + iλ (cid:17) γ (cid:16) − iλ + iλ + iλ (cid:17) , (3.23)with the function γ ( x ) defined as γ ( x ) ≡ Γ( x )Γ(1 − x ) . (3.24)The equality (3.23) fixes the exponent v ( λ , λ , λ ) = v , but in a rather complicated way, and15hows that it is real. Moreover, we can also easily observe Tr M = − πλ ) using (3.21)and (3.22), which unsurprisingly shows the monodromy is still hyperbolic. Thus, we concludethat we can have hyperbolic SL(2 , R ) monodromy around z = 1 while having a hyperbolicSL(2 , R ) monodromy around z = 0 .Note that that guaranteeing a hyperbolic SL(2 , R ) monodromies around z = 0 , simultane-ously with the correct choice of v ( λ , λ , λ ) would be sufficient for guaranteeing a hyperbolicSL(2 , R ) monodromy around z = ∞ as well, which is the only remaining point where we have anontrivial monodromy around. This is because we can imagine a contour that surrounds both z = 0 and z = 1 whose associated monodromy would be a product of two hyperbolic SL(2 , R )matrices, which is another SL(2 , R ) matrix. Furthermore, this monodromy would be clearly hy-perbolic by construction. As a result, the solutions would have the desired monodromy structurearound z = ∞ as well when we think this contour to surround z = ∞ instead. So we concludethat the solutions of the Fuchsian equation (3.13), with the right choice of v ( λ , λ , λ ) , canrealize hyperbolic SL(2 , R ) monodromies around each puncture and they are compatible. Wesolved the monodromy problem.Finally, we can list the scaled ratios ρ i = ( ψ + i ( z ) /ψ − i ( z )) /iλ i associated with each puncture.They are: ρ ( z ) = e v ( λ ,λ ,λ λ z (1 − z ) − λ λ F (cid:16) iλ − iλ + iλ , iλ − iλ − iλ ; 1 + iλ ; z (cid:17) F (cid:16) − iλ + iλ − iλ , − iλ + iλ + iλ ; 1 − iλ ; z (cid:17) iλ , (3.25) ρ ( z ) = e v ( λ ,λ ,λ λ (1 − z ) z − λ λ , F (cid:16) iλ − iλ + iλ , iλ − iλ − iλ ; 1 + iλ ; 1 − z (cid:17) F (cid:16) − iλ + iλ − iλ , − iλ + iλ + iλ ; 1 − iλ ; 1 − z (cid:17) iλ ,ρ ( z ) = e v ( λ ,λ ,λ λ (cid:18) z (cid:19) (cid:18) − z (cid:19) − λ λ F (cid:16) iλ − iλ + iλ , iλ − iλ − iλ ; 1 + iλ ; z (cid:17) F (cid:16) − iλ + iλ − iλ , − iλ + iλ + iλ ; 1 − iλ ; z (cid:17) iλ . From the above it is clear that ρ ( z ) and ρ ( z ) can be obtained from ρ ( z ) by exchanging punc-tures, as well as their associated λ j ’s, (1) ↔ (2) and (1) ↔ (3) respectively while keeping theremaining puncture fixed. Moreover, one can also show that the scaled ratio associated with thefixed puncture remains invariant (up to a sign) under this exchange, either by reasoning throughour construction above or by checking it directly using the identities for hypergeometric func-tions [41]. In any case, we see that the set of three scaled ratios given above would be invariant(up to a sign) under the permutation group S acting on the positions and the parameters ofthe punctures. This fact will eventually lead us to a similar symmetry for the local coordinatesof the hyperbolic three-string vertex.As we already argued in the previous subsection, these scaled ratios will define the followingsingle-valued, singular, hyperbolic metric on the whole three-punctured sphere (3.9): ds = λ j | ∂ρ j ( z ) | | ρ j ( z ) | sin ( λ j log | ρ j ( z ) | ) | dz | = λ j | ρ j | sin ( λ j log | ρ j | ) | dρ j | , (3.26)16or which we have three semi-infinite series of attached hyperbolic cylinders connected to eachother. Again, each j = 1 , , defines the same metric. Before we construct the local coordinates, we should understand the geometry of (3.26) betterand show that it looks exactly like in Figure 1 as we have claimed. In order to do that, focus onthe set S , which was the complex plane with a cut from to ∞ (see (3.4)). This will be mappedto the set ρ ( S ) in the ρ -plane. The rough sketch of this region, based on numerics, but not on scale, is given in Figure 2. We will consider and explain this geometry on the ρ -plane fornow, but geometries on the other ρ j -planes are analogous.As we mentioned previously, the metric on the ρ -plane (3.26) takes the form of the hyper-bolic metric of series of attached hyperbolic cylinders. Indeed, we see that the line singularities(where the metric blow up on a curve) and the simple closed geodesics surrounding the origin ρ = 0 of the hyperbolic metric (3.26) are located atLine singularities: | ρ | = e πl λ , Simple Closed Geodesics: | ρ | = e πλ ( l + ) , (3.27)where l ∈ Z . We can see these by noting that the sine in the denominator of the metric (3.26)is equal to zero in the case of line singularity by sin( πl ) = 0 and one in the case of simpleclosed geodesics by sin ( πl + π/
2) = 1 , which makes the metric (3.26) blow up and minimizerespectively.Notice that the line singularities and simple closed geodesics form alternating, exponentiallyseparated circles around the origin on the ρ -plane, as shown in Figure 2 with green and purplerespectively; except for the geodesic colored with magenta which will turn out to be special.Additionally, it is clear that every simple geodesic surrounding the origin has the length πλ by the metric (3.26), which justifies the name geodesic radii for λ j . Obviously we can pull-back these curves to the z -plane with a cut from 0 to ∞ , which will result in closed, simplegeodesics/line singularities around the puncture z = 0 by ρ (0) = 0 . These are shown in Figure 3correspondingly.Observe that the lines just above/below the branch cut of ρ ( z ) , denoted as (cid:101) L ± and shownin Figure 3, are mapped to the red/blue curves ρ ( (cid:101) L ± ) in ρ ( S ) . These curves are shown inFigure 2. They are symmetric with respect to the real axis on the ρ -plane by the choice of theprincipal branch for the scaled ratio. The set S is mapped between ρ ( (cid:101) L +1 ) and ρ ( (cid:101) L − ) , whichis the shaded region in Figure 2. Moreover, if we identify the two curves ρ ( (cid:101) L ± ) , the whole z -plane minus the punctures maps to the region between them. But, in any case, we indicatedwhere the punctures z = 1 and z = ∞ are heuristically getting mapped to in Figure 2: z = 1 ismapped to the right-side infinity and z = ∞ is mapped to the left-side infinity.Now observing Figure 2, we see that some simple closed geodesics/line singularities don’tintersect ρ ( (cid:101) L ± ) . As a result, we see ∃ ˜ l ∈ Z such that the geodesic at | ρ | = e πλj ( ˜ l j + ) doesnot intersect ρ ( (cid:101) L ± ) and surrounds all the closed simple geodesics/line singularities that do It can be shown that this map is invertible, see [41]. So this mapping would be bijective. igure 2: The rough sketch of the geometry on the ρ -plane. The meaning of the curves are explainedin the text. The coloring conventions for the curves will be the same for all figures in thissubsection. Dashed curves indicate the line singularities. Figure 3:
The corresponding geometry on the z -plane after we pullback the metric (3.26) from the ρ -plane above. Note that the gray region would be endowed with the hyperbolic metricwith geodesic boundaries Γ i . ρ ( (cid:101) L ± ) (i.e. those with l ≤ ˜ l ). The closed geodesic with l = ˜ l is shownwith magenta instead of purple in Figure 2 in order to differentiate it from the others. At thisstage nothing prevents us to having a line singularity that surrounds this geodesic and doesn’tintersect ρ ( (cid:101) L ± ) , but this turn out not to be the case as we will prove it shortly. We just assumethis is the case for now.We can pullback the geodesic with l = ˜ l described above to the z -plane, which we denoteit by Γ . Defining the closed geodesics homotopic to the puncture z = 0 as separating geodesicsof z = 0 , we see the simple closed geodesic Γ would be the separating geodesic farthest awayfrom z = 0 by construction. So we will call Γ as the most-distant separating geodesic of z = 0 .This geodesic is shown in Figure 3 with magenta as well.From this, we see that the simply-connected region H on the z -plane surrounded by Γ con-tains every geodesic/line singularity with l ≤ ˜ l . Furthermore, as l → −∞ , the geodesics/linesingularities get closer to the puncture. So we conclude that the geometry on H looks like a se-ries of semi-infinite hyperbolic cylinders attached at where they flare up, like shown in Figure 1.The places where they flare up are the line singularities of the metric.We can repeat the same procedure for the other punctures and obtain their most-distantseparating geodesics Γ j , associated simply-connected regions H j , and integers ˜ l j . Note that Γ i ∩ Γ j = ∅ for i (cid:54) = j , by Γ i ’s being simple geodesics of the same metric. Hence, the resultinggeometry on the z -plane would indeed look like in Figure 3. Again, the most-distant separatinggeodesics Γ j are shown with different colors. In this figure, we also see there are alternatingclosed curves around each puncture representing the simple closed geodesics/line singularitiessurrounding them. These can be related to the geodesics/line singularities that intersect ρ ( (cid:101) L ± ) on the ρ -plane (hence their colors), but this wouldn’t be necessary for our purposes.Now let us inspect how the most-distant separating geodesics of the punctures z = 1 and z = ∞ , denoted as Γ and Γ respectively, look like on ρ ( S ) . In order to do that, let uscall the line singularity with l = ˜ l + 1 to be the first line singularity of z = 0 . Clearly, thefirst line singularity encloses ρ (Γ ) and is enclosed by every other line singularity that encloses ρ (Γ ) on the ρ -plane, hence the name “first.” Moreover, it is clear that the first line singularityintersects with the curves ρ ( (cid:101) L ± ) by definition. We define the first geodesic of a puncture insimilar fashion.Now, we will find the shortest geodesic that is enclosed by the first line singularity of z = 0 and stretches between the curves ρ ( (cid:101) L ± ) for both right/left of the origin ρ = 0 on ρ ( S ) , whichwe will call ρ (Ω ) and ρ (Ω ) respectively. Clearly, ρ (Ω ) and ρ (Ω ) can be made shorterby eliminating any self intersections, so we will consider the simple geodesics without loss ofgenerality. Moreover, ρ (Ω ) and ρ (Ω ) can be made shorter by making them intersect ρ ( (cid:101) L ± ) perpendicularly, which we will also take to be the case.There might be multiple curves satisfying the definition for ρ (Ω ) and ρ (Ω ) above. How-ever, this cannot be the case since their pullbacks on the z -plane would correspond to closedsimple geodesics without a line singularity between them around the punctures z = 0 and z = ∞ , and we know that this can’t happen as we saw above. So ρ (Ω ) and ρ (Ω ) are uniquefor the left and right side. This is shown in Figure 2. Additionally, this argument shows that Ω and Ω are the the most-distant separating geodesics for the punctures z = 1 and z = ∞ Γ = Ω and Γ = Ω , since there are no geodesics that surround them andseparate from the other punctures.Keeping this in mind, we can now demonstrate that the there is no line singularity thatsurrounds the geodesic ρ (Γ ) and doesn’t intersect ρ ( (cid:101) L ± ) on the ρ -plane, which we onlyassumed previously. For the sake of contradiction, suppose there is one and call it ρ (Λ ) , whichis shown in Figure 4. Then it is clear by above that the geodesic ρ (Γ ) around the puncture z = 1 would be a piece of the first geodesic of z = 0 . Now going to the ρ -plane after we pullbackthis geometry to the z -plane, we see that Λ maps to a piece of a line singularity ρ (Λ ) on the ρ -plane stretching between ρ ( (cid:101) L ± ) , while Γ maps to a circle around the origin and doesn’tintersect ρ ( (cid:101) L ± ) . Similar to the arguments above, we can always find a simple geodesic ρ (Ω ) between these two, but this leads to a contradiction with the fact that Γ being the most-distantseparating geodesic of z = 0 since the separating geodesic Ω would be enclosing Γ . Clearlythis argument can be repeated for other punctures, so what we have assumed regarding having aline singularity that surrounds the most-distant separating geodesic and doesn’t intersect ρ ( (cid:101) L ± ) was justified. Figure 4:
The illustration of the geometry described in the argument above on the ρ -plane (left) and ρ -plane (right). Here the line singularity Λ is shown with dark green, while the geodesic Ω is shown with brown. In order to complete our construction, we now need to find the integers ˜ l j . For that, firstnotice the following inequality is satisfied: | ρ j ( z ) | = exp (cid:20) πλ j (cid:18) ˜ l j + 12 (cid:19)(cid:21) ≤ min z ∈ (cid:101) L j | ρ j ( z ) | , (3.28)with (cid:101) L j denoting the branch cut of the function ρ j ( z ) . This inequality is evident since wedemanded above that the geodesic ρ j (Γ j ) , located at | ρ j | = e πλj ( ˜ l j + ) , is not intersecting thecurves ρ j ( (cid:101) L ± ) on the ρ j -plane. Note that | ρ j ( z ) | would be single-valued on the branch cut (cid:101) L j because of the choice of the principal branch. From (3.28) and noting that ˜ l j is the greatestinteger that satisfies it by definition, we can write a prescription for ˜ l j as follows: ˜ l j = (cid:36) λ j π log min z ∈ (cid:101) L j | ρ j ( z ) | − (cid:37) . (3.29)20ere (cid:98)·(cid:99) : R → Z denotes the floor function. We couldn’t be able to find an explicit expressionfor this in terms of λ j ’s. However, determining the exact values of the integers ˜ l j numericallyfor given λ j is trivial by the expression above and using the scaled ratios (3.25).Although it is hard to find an expression for ˜ l j in terms of arbitrary λ j ’s, we can stillmake some progress for the case where two of the λ j ’s are equal by exploiting the permutationsymmetry. In order to do that, suppose we want to find ˜ l in the case of λ = λ = λ . Now recallthat three scaled ratios (3.25) are invariant under the permutations of the punctures and theirassociated geodesic radii up to a sign. Specifically, in the case where we exchange the puncturesat z = 1 , ∞ while keeping z = 0 fixed, which is implemented by the conformal transformation z → zz − , we get the following relation for ρ ( z ) on S ρ ( z ) = − ρ (cid:18) zz − (cid:19) . (3.30)Note that it was essential to take λ = λ to establish this relation.Clearly, z = 2 is the fixed point of the transformation z → zz − . One consequence of this isthat | ρ ( z ) | when restricted to the branch cut (cid:101) L is symmetric around z = 2 when we apply thetransformation z → zz − . Then using this fact and analyticity of ρ ( z ) , it can be shown thatthe point z = 2 would be where | ρ ( z ) | attains its global minimum on the branch cut. (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:6)(cid:1) - (cid:1)(cid:7)(cid:4)(cid:1) - (cid:1)(cid:7)(cid:8)(cid:5) - (cid:1)(cid:7)(cid:8)(cid:4) - (cid:1)(cid:7)(cid:8)(cid:3) - (cid:1)(cid:7)(cid:8)(cid:2) - (cid:1)(cid:7)(cid:8)(cid:1) λ ℒ λ (cid:1) λ (cid:1) λ ) Figure 5:
The plot of L ( λ, λ, λ ) as a function of λ . Note that the floor of this function gives − in therange shown. An analytic proof for ˜ l = − for any λ > would require better understandingof L ( λ, λ, λ ) , especially for the large values of λ . So we see that permutation symmetry of the situation λ = λ = λ allow us to find theglobal minimum of | ρ ( z ) | on its branch cut (cid:101) L , which is at z = 2 . Now define the followingfunction and notice L ( λ , λ, λ ) ≡ λ π log | ρ (2) | −
12 = ⇒ ˜ l = (cid:98)L ( λ , λ, λ ) (cid:99) , (3.31)using (3.29). This expression is certainly more manageable then what has been given in (3.29).Obviously, we can get similar expressions for the other punctures when the remaining punctures21as equal λ j ’s. As an example for what we have discussed so far, we plotted L ( λ, λ, λ ) inFigure 5. This suggests ˜ l = − , and by symmetry ˜ l = ˜ l = − , for < λ < .In summary, we see the geometry of the metric (3.26) is indeed given by Figure 1. Remem-ber the metric (3.26) was on the three-punctured sphere (cid:101) X , but, clearly, we can now obtain thehyperbolic metric with geodesic boundaries on a three-holed sphere by restricting to the region X = (cid:98) C \ ( H ∪ H ∪ H ) , which is shaded gray in Figure 3. On the ρ j -plane this correspondsto the region between the most-distant geodesics with the curves ρ j ( L ± j ) are identified, which isalso shaded gray in Figure 2. Note that X is still endowed with K = − metric (3.26), but nowfree from singularities, and it is clear by the construction that its boundaries Γ j are geodesics.In other words, we performed a “surgery” where we amputated the hyperbolic cylinders aroundthe hyperbolic singularities and left with the geodesic boundaries instead while keeping every-thing the same. Four examples of such region on the z -plane are shown in Figure 6. In thenext subsection, we are going to graft flat semi-infinite cylinders into the places of amputatedhyperbolic cylinders in order to construct the local coordinates explicitly. In this subsection, we describe how to construct the local coordinates around the punctures forthe hyperbolic three-string vertex by attaching flat semi-infinite cylinders of radius λ j at eachgeodesic boundary component of X . First, note that when we perform the surgery describedabove to obtain the geodesic boundaries, we essentially take out the disk ρ j ( H j ) = (cid:26) ρ j ∈ C | | ρ j | < exp (cid:20) πλ j (cid:18) ˜ l j + 12 (cid:19)(cid:21)(cid:27) , (3.32)from the ρ j -plane for each j = 1 , , . Now imagine we have a punctured unit disk < | w j | ≤ with the metric ds = λ j | dw j | | w j | , (3.33)which describes a flat semi-infinite cylinder ( K = 0 ) of radius λ j . We can map this puncturedunit disk into the hole ρ j ( H j ) on the ρ j -plane with a simple scaling: ρ j = exp (cid:20) πλ j (cid:18) ˜ l j + 12 (cid:19)(cid:21) w j = N j w j , N j ≡ exp (cid:20) πλ j (cid:18) ˜ l j + 12 (cid:19)(cid:21) . (3.34)We will call N j the scale factor . Above we haven’t considered the overall rotations of thepunctured unit disk, w j → e iθ w j , while we are mapping to ρ j ( H j ) , since such global phasefactors are not relevant in closed string field theory.Clearly, the flat metric (3.33) does not change under this scaling. Furthermore, the flatmetric (3.33) and the hyperbolic metric (3.26) for the pair of pants as well as their first derivativesmatch at the circular seams ρ j (Γ j ) of radius λ j . As a result, we fill the regions ρ j ( H j ) with flatsemi-infinite cylinders and discontinuity first appears in the curvature as we desire. Note thatthe metric we obtain after grafting these semi-infinite flat cylinders is a Thurston metric on thethree-punctured sphere [19].Now we can pullback these filled ρ j ( H j ) to the otherwise empty holes H j on the z -plane22 ≈ λ ≈ λ ≈ X Γ Γ Γ - (cid:1)(cid:2) - (cid:3)(cid:2) - (cid:4)(cid:2) (cid:2) (cid:4)(cid:2) (cid:3)(cid:2) (cid:1)(cid:2) - (cid:1)(cid:2) - (cid:3)(cid:2) - (cid:4)(cid:2)(cid:2)(cid:4)(cid:2)(cid:3)(cid:2)(cid:1)(cid:2) (cid:1)(cid:2) ( (cid:3) ) (cid:1) (cid:2) ( (cid:3) ) X Γ Γ Γ λ = λ = λ = - (cid:1) - (cid:2) - (cid:3) (cid:4) (cid:3) (cid:2) (cid:1) - (cid:1) - (cid:2) - (cid:3)(cid:4)(cid:3)(cid:2)(cid:1) (cid:1)(cid:2) ( (cid:3) ) (cid:1) (cid:2) ( (cid:3) ) X Γ Γ Γ λ = λ = λ = - (cid:1) - (cid:2) (cid:3) (cid:2) (cid:1) - (cid:1) - (cid:2)(cid:3)(cid:2)(cid:1) (cid:1)(cid:2) ( (cid:3) ) (cid:1) (cid:2) ( (cid:3) ) X Γ Γ Γ λ = λ = λ = - (cid:1) - (cid:2) (cid:3) (cid:2) (cid:1) - (cid:1) - (cid:2)(cid:3)(cid:2)(cid:1) (cid:1)(cid:2) ( (cid:3) ) (cid:1) (cid:2) ( (cid:3) ) Figure 6:
Four examples for the pants diagram region X on the z -plane in the case of equal geodesicradii. Punctures are located at z = 0 , , ∞ , and indicated by black dots. We only show themost-distant separating geodesics Γ j because we performed a surgery and take out everythingsurrounded by them. The region remaining X is endowed with the hyperbolic metric (3.26)and Γ j are its geodesics by construction. Note that the geodesics Γ and Γ in the criticalcase λ = arcsinh(1) /π ≈ . are so small that they haven’t rendered in the top-left figure. with the maps ρ j ( z ) to construct the local coordinates around the punctures z = 0 , , ∞ de-scribing three semi-infinite flat cylinders grafted on to the hyperbolic pair of pants on (cid:98) C . Thus,from (3.34), we see that the local coordinates w j around the punctures w j = 0 are given by w j = exp (cid:20) − πλ j (cid:18) ˜ l j + 12 (cid:19)(cid:21) ρ j ( z ) = N − j ρ j ( z ) , (3.35)with | w j | ≤ . This yields the local coordinates (1.3-1.5) using (3.25). Equivalently, we can write z = ρ − j ( N j w j ) on the coordinate patches H j with the punctures are located at z = z j . Notethat | w j | = 1 maps to ∂H j = Γ j by construction. Obviously, we can get the anti-holomorphiclocal coordinates w j (¯ z ) in similar fashion. Moreover, we see that they satisfy w j (¯ z ) = w j ( z ) w j ( z ) in z can be chosen to be real.As can be seen from (1.3-1.5), and alluded before, the local coordinates are invariant underpermutations of the punctures and their associated λ j . Adding the scale factor N j doesn’t spoilthis symmetry, since its value is getting permuted as well. Moreover, when we take all λ j = λ equal (recall this is the version that appears in the string action), this vertex becomes cyclic inthe technical sense [24]. These results are certainly consistent with what is expected form thegeometry of the hyperbolic pair of pants with three grafted flat cylinders.As a final note, the mapping radius r j = (cid:12)(cid:12)(cid:12) dzdw j (cid:12)(cid:12)(cid:12) w j =0 for this local coordinates can be easilyread from (1.3-1.5), and they are w j = e − π (˜ lj + 12 ) λj e vjλj ( z − z j ) + · · · = ⇒ r j = exp (cid:34) π (˜ l j + ) λ j − v j λ j (cid:35) = N j exp (cid:20) − v j λ j (cid:21) . (3.36)Remember both v j and N j depends on the circumferences of the grafted cylinders as can beseen from (3.23) and (3.34). In this section, we investigate various limits of the local coordinates (3.35) to check that theyare consistent with the literature [10,24,25]. We show that it is possible to produce the minimalarea three-string vertex, Kleinian vertex, and the light-cone vertex as different limits of thehyperbolic three-string vertex.
In order to produce the minimal area three-string vertex from the hyperbolic three-sting vertex,we set the lengths of the boundary components of X the same, λ = λ = λ = λ , and take λ → ∞ . Since the lengths of the boundaries of the pair of pants get larger at the same ratewhile the area of the pair of pants remains constant by the Gauss-Bonnet Theorem in this limit,the pair of pants shrinks and it becomes like a ribbon graph of vanishing width. As a result,after grafting the flat cylinders and rescaling their circumferences, we obtain the three-vertexobtained from the minimal area metric [19]. Therefore, we see that this is indeed the correctlimit to generate the minimal area three-string vertex and we will call it minimal area limit .Note that this limiting behavior is also evident from the examples given in Figure 6. We seemto get the usual representation of the minimal area three-string vertex as λ gets larger [1].In order to consider the minimal area limit explicitly, first notice that we have lim λ →∞ exp (cid:20) v ( λ, λ, λ ) λ (cid:21) = 3 √ . (4.1)This can be obtained from the expression (3.23) for the function v ( λ, λ, λ ) and evaluating itslimit in Mathematica. 24ext, we need to find the limiting value of N = N = N = N in the minimal area limit.Already from Figure 5 and (3.34) it can be visually argued that N → as λ → ∞ , but herewe are going to provide an additional heuristic argument why this expectation is correct in caseFigure 5 is misleading in large λ . To that end, we should first understand the minimal area limitof the hyperbolic metric (3.26). This metric certainly diverges in the minimal area limit, butsince we are going to rescale our cylinders at the end, this overall divergence is not a problem.Ignoring this divergence, indicated by ∼ , the hyperbolic metric, now formally on the ribbongraph of vanishing width, takes the following form in the minimal area limit: ds ∼ | ∂ρ i ( z ) | | ρ i ( z ) | sin ( ∞ log | ρ i ( z ) | ) | dz | . (4.2)Above ∞ in the denominator indicates infinite oscillations of the metric as λ → ∞ except forwhen | ρ i ( z ) | = 1 . But note that if we have such infinite oscillations, the metric would certainlybe ill-defined. The only time it is well-defined is when we have | ρ i ( z ) | = 1 , which produces justa divergence and that is acceptable as we mentioned. Thus, we conclude that the shape of theribbon graph of vanishing width is described by | ρ i ( z ) | = 1 in the minimal area limit, becausethis is the only time we have a meaningful limit of the geometry. Now note that this ribbongraph at | ρ i ( z ) | = 1 can be thought as the union of Γ i , which is described by | ρ i ( z ) | = N , in theminimal area limit by shrinking hyperbolic pair of pants. This gives lim λ →∞ N = lim λ →∞ exp (cid:20) πλ (cid:18) ˜ l + 12 (cid:19)(cid:21) = 1 . (4.3)So our expectation above was indeed correct.Using the two limits we argued above, we see that the local coordinate around z = 0 (1.3)has the following expansion in the minimal area limit: w = 3 √ z + 3 √ z + 27 √ z + 57 √ z + 231 √ z + 459 √ z + 7275 √ z + 14493 √ z + 58077 √ z + 116565 √ z + O ( z ) . (4.4)We obtained this expression by expanding (1.3) in z first, then taking the minimal area limit.One can easily observe that the local coordinates around z = 0 of the minimal area three-stringvertex, as given in equation (2.19) of [24] with a = 1 , z = i (cid:16) − i √ zz − (cid:17) / − (cid:16) i √ zz − (cid:17) / (cid:16) − i √ zz − (cid:17) / + (cid:16) i √ zz − (cid:17) / , (4.5)also has the same expansion (4.4) after an unimportant phase rotation z → − z . So, unsur-prisingly, these local coordinates match in the minimal area limit.Comparison was perturbative in z above, however, we think this limiting behavior holds forall orders in z . That is w = − z exactly in the minimal area limit. The best way to show We also checked the similar results hold for the other punctures. We omit reporting them to avoid repetition. λ is large to generate the expression (4.5), similar to the cases given in [42]. In any case,this perturbative analysis would be sufficient for our purposes. In conclusion, we see that thehyperbolic three-string vertex reduces to the minimal are three-string vertex in the limit λ → ∞ . Now we consider the opposite limit for which λ j = λ → . Clearly, the grafted flat cylindersdisappears in this limit and instead we are left with a purely hyperbolic metric on the three-punctured sphere. So the local coordinates for the hyperbolic three-string vertex is expected tobe related to the Kleinian vertex of [24] in this limit, whose local coordinates z i are given by z = e iπτ ( z ) , z = e − iπ/τ ( z ) , z = e − iπ/ ( τ ( z ) ± , (4.6)around the punctures z = 0 , , ∞ respectively, since it involves the same hyperbolic geometryin its construction which emphasized more recently in [10, 11]. Here the function τ ( z ) is theinverse of the modular λ -function, whose explicit expression is known to be [41] τ ( z ) = i F ( , , , − z ) F ( , , , z ) = − iπ log (cid:16) z (cid:17) + O ( z ) . (4.7)We will denote the limit λ j = λ → as the Kleinian limit .In order to argue for this limit, first notice that the function τ ( z ) satisfies the followingequality [36] { τ, z } = 12 z + 12( z − − z ( z − , (4.8)But recall from (3.5) and (3.11) we also have lim λ → { ρ iλj , z } = lim λ → T ϕ ( z ) = 12 z + 12( z − − z ( z − , (4.9)So from these two we immediately conclude { τ, z } = lim λ → { ρ iλj , z } = { lim λ → log( ρ j ) , z } = ⇒ lim λ → log( ρ j ( z )) = aτ ( z ) + bcτ ( z ) + d . (4.10)Above we moved the limit inside the Schwarzian derivative and used the fact that two equalSchwarzian derivatives must be related to each other by a PGL(2, C ) transformation. So here a, b, c, d ∈ C and ad − bc (cid:54) = 0 . Note that we can easily determine these constants by expandingboth sides of (4.10) to leading order in z .Take z = 0 for instance. We already know the expansion of log( ρ ( z )) around z = 0 from (3.3). In the Kleinian limit this would then yield lim λ → log( ρ ( z )) = log (cid:16) z (cid:17) + O ( z ) , (4.11) Since this is the case, this naive limit of the local coordinates (3.35) seems actually ill-defined. We are goingto comment on this point below. v ( λ, λ, λ ) . Comparing thisto (4.7), we see the constants above get fixed and we obtain the following in the Kleinian limit: a = iπ, b = c = 0 , d = 1 = ⇒ lim λ → ρ ( z ) = e iπτ ( z ) = z . (4.12)Note that we can repeat the same procedure for the other punctures and would similarly obtain lim λ → ρ j = z j up to an unimportant phase factor. We explicitly checked this is indeed the case.Now observe the scale factor N = e − π λ that relates the actual local coordinates w j to ρ j by N w j = ρ j approaches to zero as λ → , which is essentially a consequence of shrinking graftedcylinders. So in order to get a well-defined limit, it is necessary to place a cut-off on the scalefactor N which we can do it as follows. We know λ → would make the length of the boundarygeodesics L smaller. So, as we take this limit, we choose some value of L = (cid:15) (cid:28) that we putin N and keep using it for any L < (cid:15) . In other words, we take N ≈ e − π (cid:15) for sufficiently small L = 2 πλ ≥ instead of what is given before. Note that this procedure essentially mirrors whatis done in [10, 11].With this cut-off in place, we now have N w j = ρ j = z j in the Kleinian limit. Like in [10,11],we will multiply the original local coordinates z j for the Kleinian vertex given in (4.6) by N − and define a new set of local coordinates z (cid:48) j ≡ N − z j in order to use the standard plumbingparameters. With this, we get lim λ → w j = z (cid:48) j and see the scaled local coordinates for theKleinian vertex matches what we find from the Kleinian limit of the hyperbolic three-stringvertex as anticipated. Lastly, consider the situation λ = rλ, λ = λ, λ = (1 − r ) λ, (4.13)for < r < and take λ → ∞ . Having λ = λ + λ while all of them being large, this limitclearly produces the local coordinates for the light-cone vertex [25], by using similar geometricreasoning given in subsection 4.1. Thus, we are going to call this limit the light-cone limit .In order to understand this limit better, first note that the restriction λ = λ + λ alwaysmakes one of the first two arguments of the hypergeometric function appearing in local coordi-nates (1.3-1.5) independent of λ and finite as λ → ∞ . This is crucial because then a genericterm in the expansion of these hypergeometric function around the puncture z = z j takes thefollowing form: term ∼ λ n + . . . λ n + . . . ( z − z j ) n . (4.14)Here denotes some numbers while dots denote the lower order terms in λ . The importantpoint here is that since one of the arguments of the hypergeometric function is independentof λ , the same power of λ appears in the numerator and the denominator of the coefficient of ( z − z j ) n . Therefore, these coefficients remain finite as we take λ → ∞ .On the other hand, observe that the ratio of hypergeometric functions is raised to the power27 /iλ in (1.3-1.5) and this exponent approaches to in the light-cone limit. But as we havejust argued, the expansion of the hypergeometric functions remains finite in this limit. So weconclude that the part depending on the hypergeometric functions must completely drop out.The resulting limit gives, after taking the limits of prefactors like in subsection 4.1, w ( z ) = r − ( r − r − r z (1 − z ) − λ λ , (4.15) w ( z ) = r r ( r − − r ( z − z − λ λ , (4.16) w ( z ) = r rr − ( r − − ( z − − λ λ z λ λ . (4.17)From this, it is clear that if we relate the lengths of strings πλ j = L j to the light-cone momenta p + j in the usual fashion after an infinite rescaling, i.e. p + j ∼ L j , and include the appropriatesigns for the incoming/outgoing momenta, we arrive the light-cone vertex given in [25] up to anunimportant phase ambiguity. Therefore, the hyperbolic three-string vertex indeed reduces tothe light-cone vertex in the light-cone limit. In this section we derive the conservation laws associated with the hyperbolic three-string vertexin the spirit of [21]. Let us denote the hyperbolic three-string vertex with the geodesic bound-aries of length L = 2 πλ as (cid:104) V , ( λ ) | . This should be thought as an element of three-string dualFock space, so it takes 3 states in Fock space and maps to a complex number. For simplicityof reporting, we set all the boundary lengths equal and report the holomorphic Virasoro con-servation laws, but arguments here can be extended trivially to the cases with unequal lengths;ghosts and current conservation laws; and/or anti-holomorphic analogues.First, let us put the punctures at z = √ , , −√ in order to be consistent with [21] andreport the expansions for z in terms of the local coordinates w j : f ( w ) = √ √ N e − vλ w + 3 √ N e − vλ w + √ (cid:0) λ + 139 (cid:1) λ + 4) N e − vλ w + O ( w ) , (5.1) f ( w ) = 12 √ N e − vλ w − √ (cid:0) λ + 1 (cid:1) N e − vλ
32 ( λ + 4) w + O ( w ) , (5.2) f ( w ) = −√ √ N e − vλ w − √ N e − vλ w + √ (cid:0) λ + 139 (cid:1) λ + 4) N e − vλ w + O ( w ) . (5.3)These are based on inverting (1.3-1.5) respectively after performing the global conformal trans-formation z → − z − √ z + √ , (5.4)that makes the monodromies around z = √ , , −√ non-trivial. Here we will refer functionsfrom the w j -plane to the z -plane as f j , f j ( w j ) = z . Like before, here N = e − π λ and v = v = v = v . The global phase of the local coordinates w j are not important as usual, so we used28his freedom to put f j ’s into the rather symmetric form shown above. We are going to workperturbatively in w i below.Notice that when we consider the minimal area limit, these expressions reduce to the onegiven in (2.11) of [21]. This limiting behavior is expected, since the vertex given there, openstring Witten vertex, when considered in the entirety of the complex plane becomes the closedstring minimal area three-vertex, and we know from the previous sections that’s what thehyperbolic three-string vertex approaches in the minimal area limit. So it shouldn’t be toosurprising that the identities we will write below reduces to their counterparts given in [21] inthe minimal area limit.As an example of conservation laws, we derive the Virasoro conservation laws by which wemean the identities of the type, for k > , (cid:104) V , ( λ ) | L (2) − k = (cid:104) V , ( L ) | A k ( λ ) · c + (cid:88) n ≥ a kn ( λ ) L (1) n + (cid:88) n ≥ c kn ( λ ) L (2) n + (cid:88) n ≥ d kn ( λ ) L (3) n . (5.5)Here A k , a kn , c kn , d kn are some functions of λ that we are going to explicitly derive, L n are Virasorogenerators, and the superscript denotes the slot that they apply in (cid:104) V , ( λ ) | . By cyclicity of thehyperbolic three-vertex similar identities holds as we permute (1) → (2) , (2) → (3) , (3) → (1) .So it would be sufficient to report the form above. The idea here is to exchange the negatively-moded Virasoro charges with the positively-moded ones plus the central term.Now let v ( z ) be a vector field holomorphic everywhere except for the punctures. That is, itchanges as v ( z ) → ˜ v (˜ z ) = ( ∂ ˜ z ) v ( z ) under z → ˜ z . Note that v ( z ) should be regular at z = ∞ byits definition, so we must ensure z − v ( z ) is finite as z → ∞ by the inversion map z → ˜ z = 1 /z .It is important to note that the object v ( z ) T ( z ) dz is almost a 1-form, where T ( z ) is stress-energy tensor. Under z → ˜ z it transforms as, v ( z ) T ( z ) dz = ˜ T (˜ z )˜ v (˜ z ) d ˜ z − c { z, ˜ z } ˜ v (˜ z ) d ˜ z. (5.6)As we see above, we have an extra contribution from the central term. Nonetheless, we canintegrate this object on the complex plane on contours and use the usual properties of thecomplex integration, as long as we keep track of this additional term under the change ofcoordinates.In order to derive the Virasoro conservation laws, the following equality is crucial: (cid:104) V , ( λ ) | (cid:73) C dzv ( z ) T ( z ) = 0 . (5.7)Here, C is a contour that surrounds the three punctures, oriented counterclockwise. This is ashorthand notation for the vanishing of the correlator of the integral (cid:72) C v ( z ) T ( z ) dz with anythree vertex operator placed at the punctures. Note that this correlator vanishes because wecan push the contour to shrink around z = ∞ by the inversion map. In this case, the central Not to be confused with v appearing in the local coordinates. In this section T ( z ) will denote the stress-energy tensor of an arbitrary CFT with central charge c , not tobe confused with the stress-energy tensor T ϕ ( z ) we previously considered. C to separate it into positively oriented, disjoint contours C i around each punctures and write down the expression above in terms of the local coordinatesas follows: (cid:104) V , ( λ ) | (cid:88) i =1 (cid:73) C i dw i v ( i ) ( w i ) (cid:104) T ( i ) ( w i ) − c { f i ( w i ) , w i } (cid:105) = 0 , (5.8)using the transformation property of v ( z ) T ( z ) dz given in (5.6). Here v ( i ) ( w i ) denotes the com-ponents of the vector field v ( z ) ∂∂z in the local coordinates w i and similarly for the stress-energytensor.We will clearly need to find { f i ( w i ) , w i } because of (5.8). This is easy to do: { f i ( w i ) , w i } = − (cid:0) λ + 1 (cid:1) λ + 4) N e − vλ + 135 (cid:0) λ + 1 (cid:1) (cid:0) λ + 19 λ − (cid:1)
64 ( λ + 4) ( λ + 16) N e − vλ w i + · · · . (5.9)This is the same for each puncture because of the cyclicity, which we explicitly checked. Un-surprisingly, in minimal area limit we arrive the expression given in equation (3.8) of [21]. Bythis expansion it is easy to see this term only appears if we have odd-powered poles around apuncture by (5.8).Now remember L ( i ) − k = (cid:73) C i dw i w − k +1 i T ( i ) ( w i ) , (5.10)so we need a vector field that behaves like v (2) ∼ w − k +12 for k > around the puncture (2) whilebehaves like v (1) ∼ w and v (2) ∼ w around the other punctures in order to put the Virasorogenerators in the form given in (5.5). Additionally, we have to ensure the regularity at infinity.For k = 1 case, all of these can be achieved with the following globally defined holomorphicvector field: v ( z ) = − N e − vλ √ (cid:0) z − (cid:1) . (5.11)Normalization is chosen to get the convention in (5.5) and in the minimal area limit this reducesto one given in (3.10) of [21]. This has the following expansion in the local coordinates w i v (1)1 ( w ) = − N e − vλ w + 12 N e − vλ w − (cid:0) λ + 1 (cid:1) λ + 4) N e − vλ w + O ( w ) , (5.12) v (2)1 ( w ) = 1 + (cid:0) λ − (cid:1)
16 ( λ + 4) N e − vλ w + 5 (cid:0) − λ − λ + 3 λ + 1 (cid:1)
128 ( λ + 4) ( λ + 16) N e − vλ w + O ( w ) , (5.13) v (3)1 ( w ) = N e − vλ w + 12 N e − vλ w + 5 (cid:0) λ + 1 (cid:1) λ + 4) N e − vλ w + O ( w ) . (5.14)Unsurprisingly, these reduce to the equation (3.11) of [21] in the minimal area limit. Aftersubstituting this into (5.8), each integration amounts to doing the replacement w ni → L ( i ) n − by30he residue theorem. Therefore we get (cid:104) V , ( λ ) | (cid:32) − N e − vλ L + 12 N e − vλ L − (cid:0) λ + 1 (cid:1) N e − vλ λ + 4) L + 5 (cid:0) λ + 1 (cid:1) N e − vλ
32 ( λ + 4) L + . . . (cid:33) (1) + (cid:104) V , ( λ ) | (cid:32) L − + (cid:0) λ − (cid:1) N e − vλ
16 ( λ + 4) L + 5 (cid:0) − λ − λ + 3 λ + 1 (cid:1) N e − vλ
128 ( λ + 4) ( λ + 16) L + . . . (cid:33) (2) + (cid:104) V , ( λ ) | (cid:32) N e − vλ L + 12 N e − vλ L + 5 (cid:0) λ + 1 (cid:1) N e − vλ λ + 4) L + 5 (cid:0) λ + 1 (cid:1) N e − vλ
32 ( λ + 4) L + . . . (cid:33) (3) . (5.15)Again, this reduces to (3.12) of [21] in the minimal area limit. Note that this doesn’t have anycentral charge contribution since the vector v ( z ) does not have a pole around the punctures.We can continue to generate identities of the form (5.5) by using the following vector fields: v ( z ) = − N e − vλ z − z , (5.16) v ( z ) = − √ N e − vλ z − z − λ )16(4 + λ ) N e − vλ v ( z ) . (5.17)They produce the following identities respectively, (cid:104) V , ( λ ) | (cid:32) − N e − vλ L + 54 N e − vλ L − (cid:0) λ + 23 (cid:1)
16 ( λ + 4) N e − vλ L + . . . (cid:33) (1) + (cid:104) V , ( λ ) | (cid:32) L − + 5 (cid:0) λ + 1 (cid:1)
32 ( λ + 4) N e − vλ c + (cid:0) λ + 1 (cid:1) λ + 4) N e − vλ L + . . . (cid:33) (2) (5.18) + (cid:104) V , ( λ ) | (cid:32) − N e − vλ L − N e − vλ L − (cid:0) λ + 23 (cid:1)
16 ( λ + 4) N e − vλ L + . . . (cid:33) (3) , (cid:104) V , ( λ ) | (cid:32) (cid:0) λ − (cid:1)
16 ( λ + 4) N e − vλ L + 15 (cid:0) λ + 9 (cid:1)
32 ( λ + 4) N e − vλ L + . . . (cid:33) (1) + (cid:104) V , ( λ ) | (cid:32) L − − (cid:0) λ + 9 (cid:1) (cid:0) λ − (cid:1) (cid:0) λ + 1 (cid:1)
128 ( λ + 4) ( λ + 16) N e − vλ L + . . . (cid:33) (2) (5.19) + (cid:104) V , ( λ ) | (cid:32) − (cid:0) λ − (cid:1)
16 ( λ + 4) N e − vλ L + 15 (cid:0) λ + 9 (cid:1)
32 ( λ + 4) N e − vλ L + . . . (cid:33) (3) . We explicitly checked these reduces to their counterparts in [21] in the minimal area limit. Notethat we can continue generating similar identities for L − k recursively by using vector fields v k ( z ) ∼ ( z − z − k +1 and appropriately subtracting previous ones. Doing this allows us toput the identities in the form (5.5) for which only a single negatively-moded Virasoro generatorappears in the left-hand side. 31 Remarks and Open Questions
In this paper, we constructed the local coordinates for the hyperbolic three-string vertex firstdescribed in [19] and investigated its various limits explicitly. We calculated the t term in theclosed string tachyon potential and developed the conservation laws associated with such vertexin the spirit of [21]. We conclude by providing some final remarks and highlighting possiblefuture directions relevant to us:1. Since we now know the local coordinates for the hyperbolic three-string vertex, it is possibleto construct the Feynman diagrams by identifying them as w j w (cid:48) j = exp (cid:20) − πsL j + iθ (cid:21) with s ∈ R ≥ , θ ∈ [0 , π ) (6.1)using the local coordinates w j and w (cid:48) j associated to boundaries of equal length on not-necessarily-distinct pair of pants. Making this identification corresponds to having a fi-nite flat cylinder of circumference L j and length s with a twist θ stretching betweennot-necessarily-distinct pair of pants and it has the natural interpretation of the stringpropagator.As usual, we must consider every possible value of ( s, θ ) A when we are computing thestring amplitudes. Here we added an index A to indicate there are generally more thanone propagator in the Feynman diagrams. It would be interesting the study the Feynmanregions these diagrams cover in the moduli spaces of Riemann surfaces of genus g and n punctures M g,n to see if they provide a piece of a section over the bundle (cid:98) P g,n → M g,n ornot. The simplest Feynman regions to study would be for four-string scattering or stringtadpole interaction. Note that with the metric we constructed on the hyperbolic pair ofpants, it is possible to describe a Thurston metric of [19] explicitly on Riemann surfaces.2. The local coordinates (3.35) we constructed in this paper also can be used for the open-closed hyperbolic string vertices without moduli [43]. There are two additional verticeswithout moduli on top of the sphere with three closed string punctures in this situation.They are disk with three open string punctures and disk with one open string punctureand one closed string puncture. Note that if we cut open the hyperbolic three-closedstring vertex along a geodesics connecting all punctures for the former and one punctureconnecting back to itself for the latter, we generate these additional cases exactly. Fromthis construction it is clear that these would carry hyperbolic metric with appropriatelygrafted flat strip/cylinder parts and would be the same as what is constructed in [43]. Sowe can still use the local coordinates (3.35) for these additional cases.3. The primary method we used in this paper, that is relating Liouville’s equation on aspecified domain to a monodromy problem, can be generalized to construct the localcoordinates of the classical ( g = 0 ) hyperbolic n -string vertices in principle. For this,instead of (3.11) we should take the stress-energy tensor of Liouville’s equation to be T ϕ ( z ) = n (cid:88) i =1 (cid:20) ∆ i ( z − z i ) + c i ( z − z i ) (cid:21) , (6.2)32ith punctures positioned at z = z i and use this stress-energy tensor to generate the localcoordinates. Here c i ∈ C are so-called accessory parameters [23].There are two important problems with this approach. First, after we fixed the positions ofthree punctures by PSL(2, C ) symmetry, assigned prescribed weights at all punctures, anddemanded regularity at infinity, we would still have n − unfixed c i parameters functionsof n − unfixed positions z i , the usual moduli for the n -punctured sphere. It is argued thatsuch accessory parameters can be fixed in terms of moduli using the action of Liouvilletheory so that the metric associated to T ϕ ( z ) is smooth and hyperbolic [23], which goesunder the name Polyakov Conjecture . Computing these parameters exactly is not known,so this is the first problem. However, some numerical results are available in the case ofvanishing L i , see [34].Secondly, even if we find the correct c i , the remaining problem is that guaranteeing thecorrect monodromy structure for the resulting Fuchsian equation with n regular singular-ities is impractical. This is because of the lack of analogous formulas given in (3.19) forthe solutions to the Fuchsian equation associated with (6.2). It seems to us this is not thedirection one should pursue if their goal is to do practical computations.4. It would seem more promising to evaluate all higher elementary string interactions byexploiting the pants decomposition of the (marked) Riemann surfaces and their associatedTeichmüller spaces [20]. The idea would be to decompose the contribution from a givenRiemann surface as sums of products of cubic interaction of appropriate string fieldsdictated by a given pair of pants decomposition of such Riemann surface and to use anappropriate region in Teichmüller space to perform the moduli integration, similar to whatis suggested in [10, 11]. Both of these steps need further study. Related to this idea, itmay be possible to form a recursion relations in the similar vein of [12, 13, 15, 18].From the possibility of using pants decomposition we see the relevance of the hyperbolicthree-string vertex with unequal L i we considered so far. After the pants decomposition,we would only need to use the hyperbolic three-string vertex of arbitrary L i to computeCFT correlators and the rest of the computation would presumably just involve combiningthem together in correct fashion. We leave investigating this to a future work. Acknowledgments
I would like to thank Barton Zwiebach for suggesting me this problem and his guidance inthe writing process. This material is based upon work supported by the U.S. Department ofEnergy, Office of Science, Office of High Energy Physics of U.S. Department of Energy undergrant Contract Number DE-SC0012567.
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