1 Introduction

UTHEP-683

Worldsheet theory of light-cone gauge noncritical strings on higher genus Riemann surfaces

Nobuyuki Ishibashi***e-mail: ishibash@het.ph.tsukuba.ac.jp and Koichi Murakamie-mail: koichi@kushiro-ct.ac.jp

Graduate School of Pure and Applied Sciences, University of Tsukuba,

Tsukuba, Ibaraki 305-8571, Japan

National Institute of Technology, Kushiro College,

Otanoshike-Nishi 2-32-1, Kushiro, Hokkaido 084-0916, Japan

It is possible to formulate light-cone gauge string field theory in noncritical dimensions. Such a theory corresponds to conformal gauge worldsheet theory with nonstandard longitudinal part. We study the longitudinal part of the worldsheet theory on higher genus Riemann surfaces. The results in this paper shall be used to study the dimensional regularization of light-cone gauge string field theory.

1 Introduction

Since light-cone gauge string field theory is a completely gauge fixed theory, there is no problem in formulating it in noncritical dimensions. It should be possible to find the worldsheet theory in the conformal gauge describing such a string theory, in which the spacetime Lorentz invariance shall be broken. In [Baba:2009ns, Baba:2009fi], we have constructed the longitudinal part of the worldsheet theory which we call the CFT. The CFT turns out to be a conformal field theory with the right central charge so that the whole worldsheet theory is BRST invariant. The light-cone gauge superstring field theory in noncritical dimensions can be used [Baba:2009kr, Baba:2009zm, Ishibashi:2010nq, Ishibashi:2011fy] to regularize the so-called contact term divergences [Greensite:1986gv, Greensite:1987hm, Greensite:1987sm, Green:1987qu, Wendt:1987zh], in the case of tree level amplitudes. The supersymmetric CFT plays crucial roles in studying such a regularization.

In this paper, we would like to study the CFT on higher genus Riemann surfaces. In a previous paper [Ishibashi:2013nma], we have dealt with the bosonic CFT on higher genus Riemann surfaces, but we have not investigated its properties in detail. In this paper, we will define and calculate the correlation functions of bosonic and supersymmetric CFT on higher genus Riemann surfaces and explore various properties of the theory. The results in this paper will be used in a forthcoming publication, in which we discuss the dimensional regularization of the multiloop amplitudes of light-cone gauge superstring field theory.

The organization of this paper is as follows. In section 2, the bosonic CFT is studied. We calculate the correlation functions based on the results in [Ishibashi:2013nma]. In section LABEL:sec:Supersymmetric--CFT, we deal with the supersymmetric CFT. In [Baba:2009fi], we have given a way to calculate the correlation functions of the supersymmetric CFT on a surface of genus 0, but it is a bit unwieldy. In this paper, we develop an alternative method to calculate them, apply it to higher genus case and explore various properties of the supersymmetric CFT. Section LABEL:sec:Discussions is devoted to discussions. In the appendices, we give details of definitions and calculations which are not included in the text.

2 Bosonic X± Cft

It is straightforward to calculate the amplitudes of light-cone gauge bosonic string field theory perturbatively by using the old-fashioned perturbation theory and Wick rotation. Each term in the expansion corresponds to a light-cone gauge Feynman diagram for strings. A typical diagram is depicted in Figure 1. A Wick rotated -loop -string diagram is conformally equivalent to an punctured genus Riemann surface . The amplitudes are given by an integral of correlation functions of vertex operators on over the moduli parameters.

As has been shown in [Baba:2009ns, Ishibashi:2013nma], the amplitudes in dimensions can be cast into the conformal gauge expression using the worldsheet theory with the field contents

 X+,X−,Xi,b,c,¯b,¯c, (2.1)

in which the reparametrization ghosts and the longitudinal variables are added to the original light-cone variables . The worldsheet action for the longitudinal variables is given by

 S[^gz¯z,X±]=−14π∫dz∧d¯zi(∂X+¯∂X−+∂X−¯∂X+)+d−2624Γ[^gz¯z,X+]. (2.2)

Here the metric on the worldsheet is taken to be and is the Liouville action

 Γ[^gz¯z,X+]=−12π∫dz∧d¯zi(∂ϕ¯∂ϕ+^gz¯z^Rϕ), (2.3)

where the Liouville field is given by

 ϕ≡ln(−4∂X+¯∂X+)−ln(2^gz¯z), (2.4)

and is the scalar curvature derived from the metric . The theory with the action (2.2) turns out to be a conformal field theory which we call the CFT.

In order for the action to be well-defined, should be a well-defined metric on the worldsheet at least at generic points. Hence we should always consider the theory in the presence of the vertex operator insertions

 N∏r=1e−ip+rX−(Zr,¯Zr), (2.5)

with and . The amplitudes with such insertions correspond to light-cone diagrams with external lines at which have string length . With the insertion of these vertex operators, possesses a classical background

 X+cl(z,¯z)=−i2(ρ(z)+¯ρ(¯z)), (2.6)

where is given by

 ρ(z)=N∑r=1αr[lnE(z,Zr)−2πi∫zP0ω1ImΩIm∫ZrP0ω]. (2.7)

Here is the prime form, is the canonical basis of the holomorphic abelian differentials and is the period matrix of the surface.111For the mathematical background relevant for string perturbation theory, we refer the reader to [D'Hoker:1988ta]. The base point is arbitrary. For notational convenience, we introduce

 g(z,w)≡lnE(z,w)−2πi∫zP0ω1ImΩIm∫wP0ω, (2.8)

so that (2.7) can be expressed as

 ρ(z)=N∑r=1αrg(z,Zr). (2.9)

Notice that is a function of and not , but that of both of and .

coincides with the coordinate on the light-cone diagram defined as follows. A light-cone diagram consists of cylinders which correspond to propagators of closed strings. On each cylinder, one can introduce a complex coordinate whose real part coincides with the Wick rotated light-cone time and imaginary part parametrizes the closed string at each time. The ’s on the cylinders are smoothly connected except at the interaction points and we get a complex coordinate on . is not a good coordinate around the punctures and the interaction points on the light-cone diagram. The interaction points are characterized by the equation

 ∂ρ(zI)=0. (2.10)

Since

 ds2=−4∂X+cl¯∂X+cldzd¯z=|∂ρ|2dzd¯z (2.11)

provides a well-defined metric on the worldsheet except for the points , , we can make well-defined.

2.1 Correlation functions on higher genus Riemann surfaces

As has been demonstrated in [Baba:2009ns], all the properties of the worldsheet theory of the longitudinal variables can be deduced from the correlation function of the form

 ⟨N∏r=1e−ip+rX−(Zr.¯Zr)M∏s=1e−ip−sX+(ws.¯ws)⟩X±^gz¯z (2.12) ≡(ZX[^gz¯z])−2∫[dX+dX−]^gz¯ze−S±[^gz¯z]N∏r=1e−ip+rX−(Zr,¯Zr)M∏s=1e−ip−sX+(ws,¯ws).

The correlation functions are normalized by being divided by the factor

 (ZX[^gz¯z])2≡(8π2det′(−^gz¯z∂z∂¯z)∫dz∧d¯z√^g)−1, (2.13)

which coincides with the partition function of the worldsheet theory when As is explained in appendix LABEL:sec:Definition-of-the, taking the integration contours of appropriately, we can evaluate it and obtain

 ⟨N∏r=1e−ip+rX−(Zr,¯Zr)M∏s=1e−ip−sX+(ws,¯ws)⟩X±^gz¯z (2.14)

Therefore we need to calculate to get the correlation function.

Since the metric (2.11) is singular at as mentioned above, one gets a divergent result if one naively substitutes into in (2.3). One way to deal with the divergences may be to regularize them as was done in [Mandelstam:1985ww]. An alternative way is to integrate the variation formula

 δ(−Γ)=∑IδTI∮CIdz2πi1∂ρTLiouville(z)+c.c.. (2.15)

Here labels the internal lines of the light-cone diagram and denotes the contour going around it as depicted in Figure 2. is defined as

 TI=TI+iαIθI, (2.16)

where denotes the length of the -th internal line and , denote the string-length and the twist angle for the propagator. ’s and ’s should satisfy some linear constraints so that the variation corresponds to that of the shape of a light-cone diagram. denotes the energy-momentum tensor corresponding to the Liouville action (2.3) given as

 TLiouville(z)=(∂ϕ(z))2−2(∂−∂ln^gz¯z)∂ϕ(z), (2.17)

where is now given as

 ϕ=ln|∂ρ|2−ln(2^gz¯z). (2.18)

It is possible to calculate the right hand side of (2.15) and integrate it with respect to the variation to get . By doing so, we can fix the form of as a function of the parameter ’s. Imposing the factorization conditions in the limit where some of the ’s become infinity, it is possible to fix completely. By this method, we can calculate without encountering divergent constants.

Such a computation was performed in [Ishibashi:2013nma] and we can evaluate by using the results. The energy-momentum tensor (2.17) with in (2.18) can be rewritten as

 TLiouville = (∂ln|∂ρ|2)2−2∂2ln|∂ρ|2−(∂ln^gz¯z)2+2∂2ln^gz¯z (2.19) = −2{ρ,z}−((∂ln^gz¯z)2−2∂2ln^gz¯z),

where

 {ρ,z}=∂3ρ∂ρ−32(∂2ρ∂ρ)2 (2.20)

is the Schwarzian derivative. In [Ishibashi:2013nma], we have calculated which satisfies

 (2.21)

where denotes the expectation value of the energy-momentum tensor of a free boson . On the other hand, the partition function satisfies

 δlnZX[^gz¯z]=∑IδTI∮CIdz2πi1∂ρ(⟨TX(z)⟩+124((∂ln^gz¯z)2−2∂2ln^gz¯z))+c.c.. (2.22)

Comparing (2.21), (2.22) and (2.15), we get

 e−Γ∝ZLC(ZX)−24, (2.23)

up to a possibly divergent multiplicative factor.

Taking to be the Arakelov metric , was calculated in [AlvarezGaume:1987vm, Verlinde:1986kw, Dugan:1987qe, Sonoda:1987ra, Wentworth:1991, Wentworth:2008] and its explicit form is

 ZX[gAz¯z]24=ecge2δ(Σ)\leavevmode\nobreak , (2.24)

where is the Faltings’ invariant [Faltings:1984] defined by

 e−14δ(Σ) = (detImΩ)32|θ[ζ](0|Ω)|2∏gi=1(2gA^zi¯^zi)∣∣detωj(^zi)∣∣2 (2.25) ×exp[−∑i

and is a numerical constant which depends on . Here and are arbitrary points on , and

 ζ≡g∑i=1∫^ziP0ω−∫^wP0ω−Δ\leavevmode\nobreak . (2.26)

denotes the vector of Riemann constants for . The definitions of the Arakelov metric and the Arakelov Green’s function are given in appendix LABEL:sec:Arakelov-metric-and. Also taking to be the Arakelov metric, we obtain [Ishibashi:2013nma]

 ZLC=1(32π2)4he2δ(Σ)e−W∏re−2Re¯Nrr00∏I∣∣∂2ρ(zI)∣∣−3, (2.27)

where

 −W ≡ −2∑I

denotes one of the Neumann coefficients and is given by

 ¯Nrr00 ≡ limz→Zr[ρ(zI(r))−ρ(z)αr+ln(z−Zr)] (2.29) = ρ(zI(r))αr−∑s≠rαsαrlnE(Zr,Zs)+2πiαr∫ZrP0ω1ImΩN∑s=1αsIm∫ZsP0ω\leavevmode\nobreak ,

and denotes the coordinate of the interaction point at which the -th external line interacts. Therefore we get

 e−Γ ∝ ZLCe−2δ(Σ) (2.30) = 1(32π2)4ge−W∏re−2Re¯Nrr00∏I∣∣∂2ρ(zI)∣∣−3,

and fix the right hand side of (2.14) to be

 ⟨N∏r=1e−ip+rX−(Zr,¯Zr)M∏s=1e−ip−sX+(ws,¯ws)⟩X±^gz¯z (2.31) = (2π)2δ(∑sp−s)δ(∑rp+r)∏se−p−sρ+¯ρ2(ws,¯ws) ×(1(32π2)4ge−W∏re−2Re¯Nrr00∏I∣∣∂2ρ(zI)∣∣−3)d−2624\leavevmode\nobreak .

Once we know the correlation function of the form (2.31), it is possible to calculate other correlation functions by differentiating it with respect to . For example,

 ⟨∂X+(w)N∏r=1e−ip+rX−(Zr,¯Zr)M∏s=1e−ip−sX+(ws,¯ws)⟩X±^gz¯z (2.32) =i∂w0∂p−0⟨N∏r=1e−ip+rX−(Zr,¯Zr)M+1∏s=0e−ip−sX+(ws,¯ws)⟩X±^gz¯z∣∣ ∣∣p−0=0,w0=w,

where we take . From (2.31), we get

 ⟨∂X+(w)N∏r=1e−ip+rX−(Zr,¯Zr)M∏s=1e−ip−sX+(ws,¯ws)⟩X±^gz¯z (2.33) =−i2∂ρ(w)⟨N∏r=1e−ip+rX−(Zr,¯Zr)M∏s=1e−ip−sX+(ws,¯ws)⟩X±^gz¯z.

It is easy to see that for any functional that can be expressed in terms of the derivatives of and the Fourier modes satisfying

 F[X++c]=F[X+](c=const.), (2.34)

the following equation holds,

 ⟨F[X+]N∏r=1e−ip+rX−(Zr,¯Zr)M∏s=1e−ip−sX+(ws,¯ws)⟩X±^gz¯z (2.35)

This implies that the expectation value of is equal to .

The correlation functions involving can be evaluated in the same way. For example,

 ⟨∂X−(z)N∏r=1e−ip+rX−(Zr,¯Zr)M∏s=1e−ip−sX+(ws,¯ws)⟩X±^gz¯z (2.36) =i∂Z0∂p+0⟨N+1∏r=0e−ip+rX−(Zr,¯Zr)M∏s=1e−ip−sX+(ws,¯ws)⟩X±^gz¯z∣∣ ∣∣p+0=0,Z0=z,

with and we get from (2.31)

 ⟨∂X−(z)N∏r=1e−ip+rX−(Zr,¯Zr)M∏s=1e−ip−sX+(ws,¯ws)⟩X±^gz¯z (2.37) = [M∑s=1(−ip−s)∂zg(z,ws)+i∂Z0∂p+0(−d−2624Γ[^gz¯z,−i2(ρ′+¯ρ′)])∣∣∣p+0=0,Z0=z]

Here we have introduced

 ρ′(z) ≡ N+1∑r=0αrg(z,Zr) (2.38) = ρ(z)+α0(g(z,Z0)−g(z,ZN+1)).

(2.37) can be rewritten as

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