UTHEP- 773 The Fokker-Planck formalism for closed bosonic strings Nobuyuki Ishibashi

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UTHEP- 773

The Fokker-Planck formalism for closed bosonic strings

Nobuyuki Ishibashi∗

Tomonaga Center for the History of the Universe,

University of Tsukuba

Tsukuba, Ibaraki 305-8571, JAPAN

Abstract

Every Riemann surface with genus gand npunctures admits a hyperbolic metric, if 2g−2+n>0. Such

a surface can be decomposed into pairs of pants whose boundaries are geodesics. We construct a string

ﬁeld theory for closed bosonic strings based on this pants decomposition. In order to do so, we derive

a recursion relation satisﬁed by the oﬀ-shell amplitudes, using the Mirzakhani’s scheme for computing

integrals over the moduli space of bordered Riemann surfaces. The recursion relation can be turned into

a string ﬁeld theory via the Fokker-Planck formalism. The Fokker-Planck Hamiltonian consists of kinetic

terms and three string vertices. Unfortunately, the worldsheet BRST symmetry is not manifest in the

theory thus constructed. We will show that the invariance can be made manifest by introducing auxiliary

ﬁelds.

∗e-mail: ishibashi.nobuyuk.ft@u.tsukuba.ac.jp

arXiv:2210.04134v2 [hep-th] 23 Jan 2023

1 Introduction

For constructing a string ﬁeld theory (SFT), we should specify a rule to cut worldsheets into fundamental

building blocks, i.e. propagators and vertices. A few simple rules were proposed and SFTs for bosonic strings

were constructed following these rules [1–4]. Construction of an SFT for superstrings is more complicated

because of the spurious singularities [5].

The worldsheets of closed strings describing scattering amplitudes are punctured Riemann surfaces. In

mathematics, there exists a convenient way to decompose them into fundamental building blocks. On a

Riemann surface with genus gand nboundaries or punctures, one can introduce a metric with constant

negative curvature, if 2g−2+n>0. Such a metric is called a hyperbolic metric and surfaces with hyperbolic

metrics are called hyperbolic surfaces. With a hyperbolic metric, one can decompose the surface into pairs of

pants with geodesic boundaries. It may be possible to consider the pair of pants as the fundamental building

block of the surface.

The hyperbolic metric was used to construct SFT in [6–8], in which the kinetic term of the action was

taken to be the conventional one ∫Ψc−

0QΨ,(1.1)

so that the propagators correspond to cylinders. The theories include inﬁnitely many vertices besides the

three string vertex and the Feynman graphs have nothing to do with the pants decomposition. In this paper,

we would like to construct an SFT based on the pants decomposition. Namely, we will construct an SFT for

closed bosonic strings regarding the pair of pants as the three string vertex and the cylinders with vanishing

heights as the propagator, as depicted in Figure 1.

In such a theory, a state of string will correspond to a boundary of a pair of pants. Accordingly, the

string ﬁeld should be labeled by an element of the Hilbert space of the ﬁrst quantized strings and the length

Lof the boundary. The external states of the scattering amplitudes are regarded as the limit L→0of such

states. The oﬀ-shell amplitudes may correspond to Riemann surfaces which have geodesic boundaries with

ﬁxed lengths and will be expressed by integrals over the moduli spaces of such surfaces.

Unfortunately, such an approach suﬀers from a problem addressed in [9] (section IV. E). The three

string vertex will be given by the correlation function of the worldsheet theory on hyperbolic pants with

the boundary lengths speciﬁed. Suppose that one calculates the one loop one point function following

the conventional Feynman rules. The amplitude corresponds to the worldsheet in Figure 2 and we should

integrate over the length land the twist angle θ. By doing so, the fundamental domain of the modular group

is covered inﬁnitely many times, as will be seen in section 4.3. The same happens for all the other amplitudes.

Therefore, the conventional Feynman rule with the vertex and the propagator in Figure 1 does not yield the

correct amplitudes.

In order to overcome this problem, we formulate the theory using the Mirzakhani’s scheme [10, 11] for

computing integrals over moduli space of bordered Riemann surfaces. Mirzakhani derived a recursion relation

for the volume of the moduli space. Applying her method to the oﬀ-shell amplitudes of closed bosonic strings,

we derive a recursion relation satisﬁed by these amplitudes.

As was pointed out in [12, 13], Mirzakhani’s recursion relation is related to the loop equation of minimal

string theory. On the other hand, the loop equations for minimal strings can be described by an SFT via

the Fokker-Planck formalism [14, 15]. We will show that the recursion relation of the oﬀ-shell amplitudes

can be described by an SFT using the Fokker-Planck formalism. The Fokker-Planck Hamiltonian consists of

kinetic terms and three string vertices. One can develop perturbation theory which does not suﬀer from the

above mentioned problem. Unfortunately, the worldsheet BRST symmetry is not manifest in the SFT thus

constructed. We will show that we can make the invariance manifest by introducing auxiliary ﬁelds.

The organization of this paper is as follows. In section 2, we deﬁne the oﬀ-shell amplitudes of closed

bosonic string theory based on the moduli space of bordered Riemann surfaces. In section 3, we derive

recursion relations satisﬁed by the oﬀ-shell amplitudes. In section 4, we prove that the oﬀ-shell amplitudes

deﬁned in section 2 can be derived from the Fokker-Planck formalism for string ﬁelds. We show that the

solution of the recursion relations in section 3 satisﬁes the Schwinger-Dyson equation derived from the Fokker-

Planck Hamiltonian. In section 5, we modify the theory by introducing auxiliary ﬁelds and make it manifestly

Figure 1: A pants decomposition.

Figure 2: One loop one point function.

BRST invariant. Section 6 is devoted to discussions and comments. In Appendix A, we present formulas for

the local coordinates on hyperbolic pants. In appendix B, we prove the BRST identity.

2 Oﬀ-shell amplitudes

The oﬀ-shell amplitudes of the theory we will study should correspond to hyperbolic surfaces which have

geodesic boundaries with ﬁxed lengths. In this section, we would like to deﬁne such amplitudes. The

formulation is a modiﬁcation of the conventional ones [4, 16–18].

2.1 The moduli space Mg,n,L

Let Σg,n,Lwith L=(L1,⋯, Ln)be a genus ghyperbolic surface with ngeodesic boundaries (labeled by an

index a=1,⋯, n) whose lengths are L1,⋯, Ln. Cutting the surface Σg,n,Lalong non-peripheral simple closed

geodesics, we can decompose it into pairs of pants Si(i=1,⋯2g−2+n). There are many choices for such

decomposition and here we pick one. The hyperbolic structure of the surface is speciﬁed by the lengths of

the non-peripheral simple closed geodesics and the way how boundaries of Si’s are identiﬁed. Therefore the

hyperbolic structure of Σg,n,Lcan be parametrized by the Fenchel-Nielsen coordinates (ls;τs)(s=1,⋯,3g−3+

n), where lsare the lengths of the nonperipheral boundaries of Siand τsdenotes the twist parameters which

specify how boundaries of diﬀerent pairs of pants are identiﬁed. The Teichmüller space Tg,n,Lcorresponds to

the region 0<ls<∞,−∞<τs<∞. A volume form Ωg,n,Lon Tg,n,Lcalled the Weil-Petersson volume form

is given by

Ωg,n,L=3g−3+n

⋀

s=1[dls∧dτs].

Ωg,n,Ldoes not depend on the choice of the pants decomposition. The moduli space Mg,n,Lis deﬁned as

Mg,n,L≡Tg,n,LΓ,

where Γdenotes the mapping class group. The Fenchel-Nielsen coordinates (ls;τs)can be used as local

coordinates on Mg,n,L. We will deﬁne the oﬀ-shell amplitudes as integrals over Mg,n,L. The space of all

inequivalent hyperbolic structures on a surface is the same as that of the complex structures. Hence the

deﬁnition of the oﬀ-shell amplitudes here can be regarded as the traditional one for the case where the

lengths of the external strings are speciﬁed.

2.2 b-ghost insertions

Let us consider an element Σg,n,Lof Mg,n,L. One can attach a ﬂat semi-inﬁnite cylinder to each boundary [8]

as depicted in Figure 3 and obtain a punctured Riemann surface. The cylinder is conformally equivalent to a

disk with a puncture. Let wa(a=1,⋯n)be a local coordinate on the a-th disk Dasuch that Dacorresponds

to the region wa≤1, the ﬂat metric is given as

ds2=L2

(2π)2dwa2

wa2,

and the a-th puncture corresponds to wa=0. By these conditions, wais ﬁxed up to a phase rotation. wacan

be expressed as a function wa(z)of a local coordinate zon Σg,n,L.wa(z)is holomorphic in a neighborhood

of ∂Da.

In this way, from Σg,n,L, we obtain a punctured Riemann surface Σg,n with local coordinates around

punctures, which are speciﬁed up to phase rotations. To Σg,n thus obtained, one can associate a surface

state, picking a local coordinate waas above for each Da. Let us denote this surface state by Σg,n,L. By

deﬁnition, we have Σg,n,LΨ1⋯Ψn=n

∏

a=1

w−1

a○OΨa(0)Σg,n ,(2.1)

where OΨadenotes the operator corresponding to the state Ψaand ⋅Σg,n denotes the correlation function

on Σg,n. Under a phase rotation wa→eiαawa,Σg,n,Ltransforms as

Σg,n,L→Σg,n,L∏

eiαa(L(a)

0−¯

L(a)

0).

The correlation function Σg,n,LΨ1⋯Ψnis invariant under the phase rotation, if

(L0−¯

L0)Ψa=0.

In order to deﬁne the amplitudes, we need to construct a top form on the moduli space Mg,n,Lfrom

the b-ghost. A deformation of the hyperbolic structure of a surface induces that of the complex structure.

Therefore we can construct the b-ghost insertion corresponding to a tangent vector of Mg,n,L, following the

procedure given in [4,16,18,19]. Let zibe a local coordinate on the pair of pants Si, such that the hyperbolic

metric on Siis in the form

ds2=eϕdzi2.

Each boundary of Siis either shared by another pair of pants Sj(j≠i)or is equal to one of the boundaries

of Σg,n,L. In the former case, the local coordinate zjon Sjand ziare related by

zi=Fij (zj),(2.2)

Figure 3: Attaching ﬂat semi-inﬁnite cylinders to Σg,n,L.

in a neighborhood of Si∩Sj=Cij . If the boundary of Sicoincides with ∂Da,ziand waare related by

zi=fia(wa),(2.3)

in a neighborhood of ∂Da. The transition functions Fij , fia describe the moduli of Σg,n,L.

Suppose that under an inﬁnitesimal change of moduli, zi, wa, Fij , fia change as

zi→zi+εvi,

wa→wa,

Fij →Fij +δFij ,

fia →fia +δfia .

Eqs. (2.2), (2.3) imply

zi+εvi=(Fij +δFij )(zj+εvj).

zi+εvi=(fia +δfia)(wa),

and we obtain

ε(vi−dzi

dzj

vj)=δFij (zj),

εvi=δfia(wa),

in a neighborhood of Cij , ∂Darespectively. One can take vito be holomorphic in neighborhoods of boundaries

of Siand smooth inside. For such vi, we deﬁne

b(v)≡∑

i∮∂Si

dzi

2πi vi(zi)b(zi)−∮∂Si

d¯zi

2πi ¯vi(¯zi)¯

b(¯zi).(2.4)

Here the integration contours are taken so that they run along ∂Sikeeping Sion the left for zi.

For our purpose, we need to make the formulas (2.2), (2.3) and (2.4) more explicit. Siitself is a hyperbolic

surface with three boundaries and by attaching ﬂat semi-inﬁnite cylinders to the boundaries as above, we

get a three punctured sphere with local coordinates Wk(k=1,2,3). Therefore Siis conformally equivalent

to C−⋃3

k=1Dkwhere Dkare the disks corresponding to the cylinders. We choose the local coordinate zion

Sito be the complex coordinate zon Csuch that the three punctures are at z=0,1,∞. The explicit forms

of Wk(zi)are given in [20, 21], which are presented in appendix A. There is a freedom in choosing which of

∂Dkcorresponds to each boundary of Si, but (A.3) implies Wk(zi)’s are related by SL(2,C)transformation

of ziand a phase rotation and the choice does not change the result. If the boundary ∂Daof Σg,n,Lcoincides

with Wk(zi)=1, we can take wato be equal to Wk(zi). Then the explicit form of (2.3) becomes

zi=W−1

k(wa).(2.5)

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UTHEP-773TheFokker-PlanckformalismforclosedbosonicstringsNobuyukiIshibashi*TomonagaCenterfortheHistoryoftheUniverse,UniversityofTsukubaTsukuba,Ibaraki305-8571,JAPANAbstractEveryRiemannsurfacewithgenusgandnpuncturesadmitsahyperbolicmetric,if2g2nA0.Suchasurfacecanbedecomposedintopairsofpantswhosebou...

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UTHEP- 773 The Fokker-Planck formalism for closed bosonic strings Nobuyuki Ishibashi

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