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O. Heden
Department of Mathematics, KTH
S-100 44 Stockholm, Sweden
D. Krotov
Sobolev Institute of Mathematics
and
Mechanics and Mathematics Department, Novosibirsk State Univer sity
Novosibirsk, Russia
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
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