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Physics Letters B 770 (2017) 352–356

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Stability of the nonperturbative bosonic string vacuum

J. Ambjørn

a,b

, Y. Makeenko

a,c,∗

The Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, DK-2100 Copenhagen, Denmark

IMAPP, Radboud University, Heyendaalseweg 135, 6525 AJ, Nijmegen, The Netherlands

Institute of Theoretical and Experimental Physics, B. Cheremushkinskaya 25, 117218 Moscow, Russia

a r t i c l e i n f o a b s t r a c t

Article history:

Received

27 March 2017

Received

in revised form 8 May 2017

Accepted

8 May 2017

Available

online 10 May 2017

Editor: N.

Lambert

Quantization of the bosonic string around the classical, perturbative vacuum is not consistent for

spacetime dimensions 2 < d < 26. Recently we have showed that at large d there is another so-

called

mean-ﬁeld vacuum. Here we extend this mean-ﬁeld calculation to ﬁnite d and show that the

corresponding mean-ﬁeld vacuum is stable under quadratic ﬂuctuations for 2 < d < 26. We point out the

analogy with the two-dimensional O (N)-symmetric sigma-model, where the 1/N-vacuum is very close

to the real vacuum state even for ﬁnite N, in contrast to the perturbative vacuum.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction

The action of the Nambu–Goto string is the area of the string

world sheet. It is highly nonlinear in the embedding-space coordi-

nates.

Making use of diffeomorphism invariance and ﬁxing a gauge

makes the action quadratic but nonlinearities are now hidden in

the dependence of the cutoff on the metric induced at the world

sheet.

one uses the Polyakov formulation [1] of string theory, the

embedding-space coordinates and the intrinsic world sheet metric

are independent. The action is quadratic in the embedding-space

coordinates and in the path integral one can in principle perform

the integration over these coordinates. The dependence of this part

of the path integral on the world sheet metric is determined by

the conformal anomaly. In the conformal gauge this leads to the

celebrated

Liouville action whose solution [2] about the classical

(perturbative) vacuum is consistent only for d ≤ 2. For 2 < d < 26

the

solution is not real which may indicate an instability of the

vacuum.

the work [3] we constructed another vacuum state of the

Nambu–Goto string by introducing an independent intrinsic metric

and the corresponding Lagrange multiplier λ

and then inte-

grating

over the d target-space coordinates X

. The correspond-

ing

effective action is a functional of ρ

and λ

which do not

ﬂuctuate in the mean-ﬁeld approximation that becomes exact at

large d. A vacuum state can be found by minimizing the effective

Corresponding author.

E-mail

addresses: ambjorn@nbi.dk (J. Ambjørn), makeenko@nbi.dk

(Y. Makeenko).

action and it is a genuine quantum state because we have taken

into account the quantum ﬂuctuations of the target-space coordi-

nates X

This

approach is quite similar to the well-known introduction

of a Lagrange multiplier ﬁeld in the two-dimensional O (N) sigma-

model.

In this model one integrates over ﬁelds



n obeying the re-

striction



= 1. One gets rid of the constraint by introducing the

Lagrange multiplier ﬁeld u. After integration over



n one obtains

an effective action as a functional of u. The minimum of this ef-

fective

action determines the exact vacuum state for inﬁnite N.

For ﬁnite N the quantum ﬂuctuations of u have to be included,

but they are small even at N = 3. The reason is roughly speaking

that there is only one u, while the effective action is proportional

to N, i.e. parametrically large, which is what is needed for a sad-

dle

point. That is not the case for the N ﬁelds



n which ﬂuctuate

strongly. The perturbative vacuum



= (1, 0, ..., 0) possesses an

O (N − 1) symmetry and is far away from the genuine nonper-

turbative

O (N)-symmetric vacuum, while the mean-ﬁeld vacuum

obtained via the Lagrange multiplier approach possesses the right

symmetry and is close to the exact vacuum even at ﬁnite N.

Fluctuations of u about the mean-ﬁeld value are systematically

tractable within the 1/N-expansion.

the present Paper we construct a nonperturbative mean-ﬁeld

vacuum state for the Nambu–Goto string at ﬁnite d, show that

it is energetically preferable to the perturbative classical vacuum

and discuss two possible scaling limits. We calculate the effective

action which governs ﬂuctuations of ρ

and λ

about their mean-

ﬁeld

values, repeating pretty much the original computation of the

conformal anomaly in Ref. [1] for the Polyakov string. Fixing the

conformal gauge we evaluate the determinants coming from path-

integration

over X

and ghosts, in order to compute the effective

http://dx.doi.org/10.1016/j.physletb.2017.05.021

0370-2693/

SCOAP

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