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8. K. T. Phelps. A general product construction for error corre cting codes. SIAM J.
Algebraic Discrete Methods , 5(2):224–228, 1984.
9. K. T. Phelps. A product construction for perfect codes over a rbitrary alphabets.
IEEE Trans. Inf. Theory , 30(5):769–771, 1984.
10. V. N. Potapov and D. S. Krotov. On the number of n-ary quasigroups of ﬁnite order.
Submitted. ArXiv:0912.5453
11. V.N.PotapovandD.S.Krotov. Asymptoticsforthenumbero fn-quasigroupsoforder
4.Sib. Math. J. , 47(4):720–731, 2006. DOI: 10.1007/s11202-006-0083-9 tran slated
from Sib. Mat. Zh. 47(4) (2006), 873-887. ArXiv:math/0605104
O. Heden
Department of Mathematics, KTH
S-100 44 Stockholm, Sweden
email:[email protected]
D. Krotov
Sobolev Institute of Mathematics
and
Mechanics and Mathematics Department, Novosibirsk State Univer sity
Novosibirsk, Russia
email:[email protected]
8
arXiv:1001.0002v2 [hep-th] 9 Mar 2010Gravity duals for logarithmic conformal ﬁeld theories
Daniel Grumiller and Niklas Johansson
Institute for Theoretical Physics, Vienna University of Te chnology
Wiedner Hauptstrasse 8-10/136, A-1040 Vienna, Austria
E-mail:[email protected], [email protected]. ac.at
Abstract. Logarithmic conformal ﬁeldtheories with vanishingcentra l charge describe systems
withquencheddisorder, percolation ordiluteself-avoidi ngpolymers. Inthesetheories theenergy
momentum tensor acquires a logarithmic partner. In this tal k we address the construction of
possible gravity duals for these logarithmic conformal ﬁel d theories and present two viable
candidates for such duals, namely theories of massive gravi ty in three dimensions at a chiral
point.
Outline
Thistalk isorganized asfollows. Insection 1werecall sali ent featuresof2-dimensionalconformal
ﬁeld theories. In section 2 we review a speciﬁc class of logar ithmic conformal ﬁeld theories where
the energy momentum tensor acquires a logarithmic partner. In section 3 we present a wish-list
for gravity duals to logarithmic conformal ﬁeld theories. I n section 4 we discuss two examples
of massive gravity theories that comply with all the items on that list. In section 5 we address
possible applications of an Anti-deSitter/logarithmic co nformal ﬁeld theory correspondence in
condensed matter physics.
1. Conformal ﬁeld theory distillate
Conformal ﬁeld theories (CFTs) are quantum ﬁeld theories th at exhibit invariance under angle
preserving transformations: translations, rotations, bo osts, dilatations and special conformal
transformations. In two dimensions the conformal algebra i s inﬁnite dimensional, and thus
two-dimensional CFTs exhibit a particularly rich structur e. They arise in various contexts in
physics, including string theory, statistical mechanics a nd condensed matter physics, see e.g. [1].
The main observables in any ﬁeld theory are correlation func tions between gauge invariant
operators. There exist powerful tools to calculate these co rrelators in a CFT. The operator
content of various CFTs may diﬀer, but all CFTs contain at leas t an energy momentum tensor
Tµν. Conformal invariance requires the energy momentum tensor to be traceless, Tµ
µ= 0,
in addition to its conservation, ∂µTµν= 0. In lightcone gauge for the Minkowski metric,
ds2= 2dzd¯z, these equations take a particularly simple form: Tz¯z= 0,Tzz=Tzz(z) :=OL(z)
andT¯z¯z=T¯z¯z(¯z) :=OR(¯z). Conformal Ward identities determine essentially unique ly the form
of 2- and3-point correlators between theﬂuxcomponents OL/Rof theenergy momentum tensor:∝an}b∇acketle{tOR(¯z)OR(0)∝an}b∇acket∇i}ht=cR
2¯z4(1a)
∝an}b∇acketle{tOL(z)OL(0)∝an}b∇acket∇i}ht=cL
2z4(1b)
∝an}b∇acketle{tOL(z)OR(0)∝an}b∇acket∇i}ht= 0 (1c)
∝an}b∇acketle{tOR(¯z)OR(¯z′)OR(0)∝an}b∇acket∇i}ht=cR
¯z2¯z′2(¯z−¯z′)2(1d)
∝an}b∇acketle{tOL(z)OL(z′)OL(0)∝an}b∇acket∇i}ht=cL
z2z′2(z−z′)2(1e)
∝an}b∇acketle{tOL(z)OR(¯z′)OR(0)∝an}b∇acket∇i}ht= 0 (1f)
∝an}b∇acketle{tOL(z)OL(z′)OR(0)∝an}b∇acket∇i}ht= 0 (1g)
The real numbers cL,cRare the left and right central charges, which determine key p roperties of
the CFT. We have omitted terms that are less divergent in the n ear coincidence limit z,¯z→0 as
well as contact terms, i.e., contributions that are localiz ed (δ-functions and derivatives thereof).
If someone provides us with a traceless energy momentum tens or and gives us a prescription
how to calculate correlators,1but does not reveal whether the underlying ﬁeld theory is a CF T,
thenwecanperformthefollowing check. Wecalculate all 2- a nd3-point correlators of theenergy
momentum tensor with itself, and if at least one of the correl ators does not match precisely with
the corresponding correlator in (1) then we know that the ﬁel d theory in question cannot be a
CFT. On the other hand, if all the correlators match with corr esponding ones in (1) we have
non-trivial evidence that the ﬁeld theory in question might be a CFT. Let us keep this stringent
check in mind for later purposes, but switch gears now and con sider a speciﬁc class of CFTs,
namely logarithmic CFTs (LCFTs).
2. Logarithmic CFTs with an energetic partner
LCFTs were introduced in physics by Gurarie [2]. We focus now on some properties of LCFTs
and postpone a physics discussion until the end of the talk, s ee [3,4] for reviews. There are two
conceptually diﬀerent, but mathematically equivalent, way s to deﬁne LCFTs. In both versions
there exists at least one operator that acquires a logarithm ic partner, which we denote by Olog.
We focus in this talk exclusively on theories where one (or bo th) of the energy momentum
tensor ﬂux components is the operator acquiring such a partn er, for instance OL. We discuss
now brieﬂy both ways of deﬁning LCFTs.
According to the ﬁrst deﬁnition “acquiring a logarithmic pa rtner” means that the
Hamiltonian Hcannot be diagonalized. For example
H/parenleftbigg
Olog
OL/parenrightbigg
=/parenleftbigg
2 1
0 2/parenrightbigg/parenleftbigg
Olog
OL/parenrightbigg
(2)
Theangularmomentum operator Jmay ormay not bediagonalizable. Weconsider onlytheories
whereJis diagonalizable:
J/parenleftbigg
Olog
OL/parenrightbigg

8. K. T. Phelps. A general product construction for error corre cting codes. SIAM J.

Algebraic Discrete Methods , 5(2):224–228, 1984.

9. K. T. Phelps. A product construction for perfect codes over a rbitrary alphabets.

IEEE Trans. Inf. Theory , 30(5):769–771, 1984.

10. V. N. Potapov and D. S. Krotov. On the number of n-ary quasigroups of ﬁnite order.

Submitted. ArXiv:0912.5453

11. V.N.PotapovandD.S.Krotov. Asymptoticsforthenumbero fn-quasigroupsoforder

4.Sib. Math. J. , 47(4):720–731, 2006. DOI: 10.1007/s11202-006-0083-9 tran slated

from Sib. Mat. Zh. 47(4) (2006), 873-887. ArXiv:math/0605104

O. Heden

Department of Mathematics, KTH

S-100 44 Stockholm, Sweden

email:[email protected]

D. Krotov

Sobolev Institute of Mathematics

and

Mechanics and Mathematics Department, Novosibirsk State Univer sity

Novosibirsk, Russia

email:[email protected]

8

arXiv:1001.0002v2 [hep-th] 9 Mar 2010Gravity duals for logarithmic conformal ﬁeld theories

Daniel Grumiller and Niklas Johansson

Institute for Theoretical Physics, Vienna University of Te chnology

Wiedner Hauptstrasse 8-10/136, A-1040 Vienna, Austria

E-mail:[email protected], [email protected]. ac.at

Abstract. Logarithmic conformal ﬁeldtheories with vanishingcentra l charge describe systems

withquencheddisorder, percolation ordiluteself-avoidi ngpolymers. Inthesetheories theenergy

momentum tensor acquires a logarithmic partner. In this tal k we address the construction of

possible gravity duals for these logarithmic conformal ﬁel d theories and present two viable

candidates for such duals, namely theories of massive gravi ty in three dimensions at a chiral

point.

Outline

Thistalk isorganized asfollows. Insection 1werecall sali ent featuresof2-dimensionalconformal

ﬁeld theories. In section 2 we review a speciﬁc class of logar ithmic conformal ﬁeld theories where

the energy momentum tensor acquires a logarithmic partner. In section 3 we present a wish-list

for gravity duals to logarithmic conformal ﬁeld theories. I n section 4 we discuss two examples

of massive gravity theories that comply with all the items on that list. In section 5 we address

possible applications of an Anti-deSitter/logarithmic co nformal ﬁeld theory correspondence in

condensed matter physics.

1. Conformal ﬁeld theory distillate

Conformal ﬁeld theories (CFTs) are quantum ﬁeld theories th at exhibit invariance under angle

preserving transformations: translations, rotations, bo osts, dilatations and special conformal

transformations. In two dimensions the conformal algebra i s inﬁnite dimensional, and thus

two-dimensional CFTs exhibit a particularly rich structur e. They arise in various contexts in

physics, including string theory, statistical mechanics a nd condensed matter physics, see e.g. [1].

The main observables in any ﬁeld theory are correlation func tions between gauge invariant

operators. There exist powerful tools to calculate these co rrelators in a CFT. The operator

content of various CFTs may diﬀer, but all CFTs contain at leas t an energy momentum tensor

Tµν. Conformal invariance requires the energy momentum tensor to be traceless, Tµ

µ= 0,

in addition to its conservation, ∂µTµν= 0. In lightcone gauge for the Minkowski metric,

ds2= 2dzd¯z, these equations take a particularly simple form: Tz¯z= 0,Tzz=Tzz(z) :=OL(z)

andT¯z¯z=T¯z¯z(¯z) :=OR(¯z). Conformal Ward identities determine essentially unique ly the form

of 2- and3-point correlators between theﬂuxcomponents OL/Rof theenergy momentum tensor:∝an}b∇acketle{tOR(¯z)OR(0)∝an}b∇acket∇i}ht=cR

2¯z4(1a)

∝an}b∇acketle{tOL(z)OL(0)∝an}b∇acket∇i}ht=cL

2z4(1b)

∝an}b∇acketle{tOL(z)OR(0)∝an}b∇acket∇i}ht= 0 (1c)

∝an}b∇acketle{tOR(¯z)OR(¯z′)OR(0)∝an}b∇acket∇i}ht=cR

¯z2¯z′2(¯z−¯z′)2(1d)

∝an}b∇acketle{tOL(z)OL(z′)OL(0)∝an}b∇acket∇i}ht=cL

z2z′2(z−z′)2(1e)

∝an}b∇acketle{tOL(z)OR(¯z′)OR(0)∝an}b∇acket∇i}ht= 0 (1f)

∝an}b∇acketle{tOL(z)OL(z′)OR(0)∝an}b∇acket∇i}ht= 0 (1g)

The real numbers cL,cRare the left and right central charges, which determine key p roperties of

the CFT. We have omitted terms that are less divergent in the n ear coincidence limit z,¯z→0 as

well as contact terms, i.e., contributions that are localiz ed (δ-functions and derivatives thereof).

If someone provides us with a traceless energy momentum tens or and gives us a prescription

how to calculate correlators,1but does not reveal whether the underlying ﬁeld theory is a CF T,

thenwecanperformthefollowing check. Wecalculate all 2- a nd3-point correlators of theenergy

momentum tensor with itself, and if at least one of the correl ators does not match precisely with

the corresponding correlator in (1) then we know that the ﬁel d theory in question cannot be a

CFT. On the other hand, if all the correlators match with corr esponding ones in (1) we have

non-trivial evidence that the ﬁeld theory in question might be a CFT. Let us keep this stringent

check in mind for later purposes, but switch gears now and con sider a speciﬁc class of CFTs,

namely logarithmic CFTs (LCFTs).

2. Logarithmic CFTs with an energetic partner

LCFTs were introduced in physics by Gurarie [2]. We focus now on some properties of LCFTs

and postpone a physics discussion until the end of the talk, s ee [3,4] for reviews. There are two

conceptually diﬀerent, but mathematically equivalent, way s to deﬁne LCFTs. In both versions

there exists at least one operator that acquires a logarithm ic partner, which we denote by Olog.

We focus in this talk exclusively on theories where one (or bo th) of the energy momentum

tensor ﬂux components is the operator acquiring such a partn er, for instance OL. We discuss

now brieﬂy both ways of deﬁning LCFTs.

According to the ﬁrst deﬁnition “acquiring a logarithmic pa rtner” means that the

Hamiltonian Hcannot be diagonalized. For example

H/parenleftbigg

Olog

OL/parenrightbigg

=/parenleftbigg

2 1

0 2/parenrightbigg/parenleftbigg

Olog

OL/parenrightbigg

(2)

Theangularmomentum operator Jmay ormay not bediagonalizable. Weconsider onlytheories

whereJis diagonalizable:

J/parenleftbigg

Olog

OL/parenrightbigg