The ZZ annulus one-point function in non-critical string theory A string ﬁeld theory analysis Dan Stefan Eniceicu1Raghu Mahajan1Pronobesh Maity2Chitraang Murdia34and

2025-05-06 0 0 955.84KB 36 页 10玖币

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The ZZ annulus one-point function in non-critical string theory:

A string ﬁeld theory analysis

Dan Stefan Eniceicu,1Raghu Mahajan,1Pronobesh Maity,2Chitraang Murdia,3,4and

Ashoke Sen2

1Department of Physics, Stanford University, Stanford, CA 94305, USA

2International Centre for Theoretical Sciences, Bengaluru - 560089, India

3Berkeley Center for Theoretical Physics, Department of Physics, University of California, Berkeley,

CA 94720, USA

4Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

eniceicu@stanford.edu, raghumahajan@stanford.edu, pronobesh.maity@icts.res.in,

murdia@berkeley.edu, ashoke.sen@icts.res.in

Abstract

We compute the ZZ annulus one-point function of the cosmological constant operator in non-

critical string theory, regulating divergences from the boundaries of moduli space using string ﬁeld

theory. We identify a subtle issue in a previous analysis of these divergences, which was done in the

context of the c“1string theory, and where it had led to a mismatch with the prediction from the

dual matrix quantum mechanics. After ﬁxing this issue, we ﬁnd a precise match to the expected

answer in both the că1and c“1cases. We also compute the disk two-point function, which is a

quantity of the same order, and show that it too matches with the general prediction.

arXiv:2210.11473v1 [hep-th] 20 Oct 2022

Contents

1 Introduction and summary 2

2 Setup and conventions 5

2.1 Conventions for string amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Minimal string theory and Liouville CFT . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 General predictions for the disk two-point function and the annulus one-point func-

tion 8

4 The disk two-point function 11

5 The annulus one-point function 14

5.1 Brief review of the construction of vertices . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.2 The worldsheet contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5.3 Contribution from ψexchange................................. 20

5.4 Gaugeparameterredeﬁnition ................................. 21

5.5 Final result and concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

A Fixing PSLp2,Rqwith two closed string punctures 27

B Tachyon exchange contribution to the disk two-point function 28

C Normalization of the worldsheet contribution to the annulus one-point function 29

D Normalization of the ψexchange contribution 31

E Normalization of the C-O-O-O amplitude 32

1 Introduction and summary

The study of non-perturbative eﬀects due to ZZ instantons [1] in two-dimensional string theory by

Balthazar, Rodriguez, and Yin [2] has motivated a string ﬁeld theory analysis of IR divergences in

instanton amplitudes. The agreement between the string ﬁeld theory analyses and the predictions from

the dual matrix quantum mechanics is impressive [3,4,5]. The string ﬁeld theory analysis of instanton

amplitudes has been extended to other non-critical string models [6,7,8,9,10] where the computations

agree precisely with the predictions from the dual matrix models. It has also been extended to critical

superstrings [11,12,13,14,15] where the D-instanton eﬀects match precisely with the predictions from

superstring dualities. In [16], worldsheet computations of instanton eﬀects were performed in Calabi-

Yau orientifold compactiﬁcations, which is a new result and was not previously known from a dual

description.

Among all these successes, there is one particular observable that stands out and does not match with

the dual prediction. This is the annulus one-point amplitude in the original c“1string theory com-

putation of [2], which is relevant for computing the ﬁrst subleading-in-gscorrection to the D-instanton

induced n-point amplitude of closed-string operators. This amplitude receives divergent contributions

from integration over the worldsheet moduli near the boundaries of moduli space. This leads to an

additive ambiguous term in the amplitude. Extracting the ﬁnite part of the worldsheet amplitude via

numerical integration over the moduli space, and comparing this with the amplitude from the dual

matrix quantum mechanics leads to a prediction for the ambiguous piece of the worldsheet amplitude

[2,17]. In [4], a string ﬁeld theory analysis was performed to determine the ambiguous piece, but the

result was found to not match with [2,17], leading to a puzzle.

In the present work, we resolve this mismatch. We ﬁrst simplify the model by working with the

că1non-critical string, and we study the integrated correlation functions of the cosmological constant

operator. The ﬁrst simpliﬁcation is that one does not have to worry about the translation zero mode

of the c“1scalar. Second, the one-point annulus amplitude of the cosmological operator has a

simple form that can be obtained by diﬀerentiating the partition function with respect to the world-

sheet cosmological constant µ. Third, because of the Liouville equation of motion, the cosmological

operator is a total derivative and the moduli space integral reduces to just boundary contributions,

obviating the need for numerical integration over the moduli space. In trying to compute the one-point

annulus amplitude in this simpler model, we were able to identify a subtle issue in the computation

in appendix D of [4], which analyzed the disk amplitude with one closed string puncture and three

open string punctures. This was needed for ﬁnding the relation between the string ﬁeld theory gauge

transformation parameter and the rigid Up1qtransformation parameter under which an open string with

one end on the instanton picks up a phase.

Let us brieﬂy explain the subtlety. The disk amplitude with one closed string puncture and three

open string punctures has a two-dimensional moduli space. String ﬁeld theory instructs us to integrate

a two-form on a subset Sof this moduli space that excludes certain regions around the boundaries. The

two-form turns out to be exact. Let’s denote it by dJ, so that the moduli-space integral reduces to the

integral of Jover the boundary BSof S. It so happens that in appendix D of [4], when we go once

around BS, the open string punctures do not return to their original position, but to a conﬁguration

related to the initial one by the one-parameter subgroup of PSLp2,Rqthat keeps the point z“iin

the upper half plane ﬁxed. In such a situation, one must make sure that the contraction of Jwith the

tangent vector along the orbits of this PSLp2,Rqtransformation is zero, something that was not true

for the Jchosen in [4]. It turns that one can add an exact one-form to Jso that it satisﬁes the desired

property. (Alternatively, one can reduce the orbits under discussion to points by ﬁxing the location

of one of the punctures and work directly with the two-dimensional moduli space.) After this ﬁx, the

mismatch goes away and one-ﬁnds agreement with predictions from the dual matrix models both in the

că1and the c“1case.

Now we present the main result of this work. Let V“e2bφ denote the bulk cosmological constant

vertex operator in minimal string theory. The integrated correlation functions of Vcan be obtained by

taking µ-derivatives of the partition function. Let g´1

sbe the tension of the ZZ brane. Further, let gsf

denote the disk two-point of V, divided by the square of the disk one-point function of V, and let gsg

denote the annulus one-point of V, divided by the disk one-point function of V. By taking µ-derivatives

of the partition function and setting µ“1

π, one ﬁnds

f“2b

Q´1, g “1

2.(1.1)

Our goal will be to reproduce both of these results via explicit integrals over the relevant moduli spaces,

using string ﬁeld theory to regulate divergences from the boundaries.

The rest of the paper is devoted to setting up the problem in the că1case and explaining the

above remarks in more detail. In section 2, we brieﬂy review the că1non-critical string theory,

and also certain conventions for the Liouville and ghost CFTs and for string amplitudes that will be

important for us. In section 3, we provide the general analysis that leads to a concrete prediction for

the disk two-point function and the annulus one-point function of the cosmological constant operator,

leading to the predictions (1.1). The key point is that the correlation functions of the cosmological

constant operator can be related to µ-derivatives of the partition function. In section 4, we compute the

disk two-point function of the cosmological constant operator by integrating over the one-dimensional

moduli space, which matches with the prediction (1.1). The disk two-point function contributes at the

same order as the annulus one-point function, and was already found to match the matrix quantum

mechanics result in the c“1case [2,17,4]. Finally, in section 5, we compute the annulus one-point

function exploiting the total derivative nature of the cosmological constant operator to integrate over

the two-dimensional moduli space. The subtlety in appendix D of [4] is explained in section 5.4. The

analysis of section 4and section 5requires computing string ﬁeld theory Feynman diagrams to get ﬁnite

results. Since the analysis of [4] was quite lengthy overall, we will not repeat all the computational

details of the various contributions and instead emphasize the conceptual points that are diﬀerent in

our analysis. The appendices contain details of various overall normalizations of string amplitudes that

are important for our work.

2 Setup and conventions

2.1 Conventions for string amplitudes

We will follow the conventions of [12]. One important convention is that the integrated closed string

vertex operators are integrated with the measure dxdy

π. For the upper half plane geometry, the open

string punctures are integrated along the real axis with measure dx.

We take the three-point function of the c-ghost in the upper half plane to be

xcpz1qcpz2qcpz3qyUHP “ ´pz1´z2qpz2´z3qpz1´z3q.(2.1)

This normalization, with the string ﬁeld theory path integral being weighted as expr1

2xΨ|QB|Ψys, gives

rise to the path integral weight expr´1

2hbφ2

bsfor a Siegel-gauge bosonic ﬁeld φbwith L0“hb.1For

example, in the case of the the tachyon φ1c1|0y, the action evaluates to 1

2φ2

1x0|c´1c0L0c1|0y “ 1

2φ2

which is the correct result since hb“ ´1for the tachyon. For later use, we also need the out-of-Siegel

gauge ﬁeld ψthat appears in the string ﬁeld as

Ψ“φ1c1|0y ` iψc0|0y ` . . . (2.2)

The path integral weight of ψis expp´ψ2q, so that the propagator of ψequals 1

We deﬁne gsso that nonperturbative contributions to string amplitudes carry an overall factor of

expp´g´1

sq. In other words, the action of the instanton is ´g´1

In the conventions of [12], the one point function of a closed string vertex operator ψcon the upper

half plane is given by

Adiskpψcq “ 1

4gs@pBc´ BcqψcDUHP .(2.3)

The upper half plane amplitude for nclosed string punctures and mopen string punctures, with one

closed puncture and one open string puncture ﬁxed is

Adiskpψn

cψm

oq “ iπ

gsżxψn

cψm

oyUHP .(2.4)

The factor of iwas explained in appendix A of [14]. In this paper, we will instead be interested in the

case where the PSLp2,Rqsymmetry is ﬁxed by ﬁxing the position of one closed string puncture at z“i

and ﬁxing the x-coordinate of another closed string puncture to zero. In appendix Awe show that the

amplitude in this gauge ﬁxing takes the form

Adiskpψn

cψm

oq “ i

2gsżxψn

cψm

oyUHP ,(2.5)

where one closed string puncture is ﬁxed at z“iand takes the form ccV piq, and a second closed string

puncture is integrated with measure dyfrom y“0to y“1and takes the form pc`cqVpiyq. The

important point about (2.5) is the precise overall numerical factor.

1In our convention where the path integral is weighted by exponential of the action, an n-point interaction term in the

action gives a contribution to the n-point amplitude without any extra minus sign or factor of i.

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TheZZannulusone-pointfunctioninnon-criticalstringtheory:AstringeldtheoryanalysisDanStefanEniceicu,1RaghuMahajan,1PronobeshMaity,2ChitraangMurdia,3;4andAshokeSen21DepartmentofPhysics,StanfordUniversity,Stanford,CA94305,USA2InternationalCentreforTheoreticalSciences,Bengaluru-560089,India3BerkeleyCent...

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The ZZ annulus one-point function in non-critical string theory A string ﬁeld theory analysis Dan Stefan Eniceicu1Raghu Mahajan1Pronobesh Maity2Chitraang Murdia34and

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