Logarithmic Conformal Field Theories via Logarithmic Deformations

35 Pages·0.49 MB·

by  J. Fjelstad

hep-th/0201091
LOGARITHMIC CONFORMAL FIELD THEORIES
VIA LOGARITHMIC DEFORMATIONS
J. FJELSTAD, J. FUCHS, S. HWANG, A.M. SEMIKHATOV, AND I.YU. TIPUNIN
2
0 Abstra
t. We
onstru
t logarithmi

onformal (cid:28)eld theories starting from an ordinary
on-
V
0 formal (cid:28)eld theory(cid:22)with a
hiral algebra C and the
orresponding spa
e of states (cid:22)via a
2 V EndK[[z,z−1]]
two-step
onstru
tion: i) deforming the
hiral algebra representation on ⊗ ,

n K
where is an auxiliary (cid:28)nite-dimensional ve
torspa
e, and ii) extending C by operators
or-
a EndK K=C2 EndK
J respondingtothe endomorphisms . For , with beingthetwo-dimensional

2 Cli(cid:27)ordalgebra,our
onstru
tion resultsin extending C by an operatorthat
anbe thought of
2 ∂ 1E E (2,p)
−
as , where isa fermioni
s
reening. This
oversthe Virasorominimal models as
sl(2)H
2 well as the WZW theory.
v b
1
9
c=0 N=2
0
Contents 4.2. The theory and the
1
0 super
onformal algebra 19
1. Introdu
tion 1 (2,p)
2 4.3. minimal models 21
0 2. A deformation of the energy-momentum
5. A further deformation example:
h/ tensor 3 sl(2)
logarithmi
23
-t 3. Deformation of a general
hiral algebra 5 bsl(2)
p 5.1. A fermioni
s
reening for 23
3.1. The OPA deformation 6
e 5.2. The deformation and indeb
omposable
h 3.2. Remarks 8
: representations 25
v 3.3. Extension of the state spa
e 9
6. Con
lusions 27
i
X 3.4. Ward identities 12
(2,p) Appendix A. Proof of Theorem 3.1 29
r 4. Logarithmi
Deformations:
a Appendix B. The VOA setting 31
Virasoro Examples 15
c= 2 Referen
es 33
4.1. The − theory: From free fermions
to symple
ti
fermions 15
1. Introdu
tion
It issomewhatmira
ulousthatlogarithmi

onformal(cid:28)eldtheories[1,2,3,4,5,6,7,8℄, although
violating some of the fundamental prin
iples usually assumed in CFT, nevertheless emerge in
many situations where
onformal symmetry plays a role. Appli
ations that have re
ently been
attra
ting attention rea
h from gravitational dressing [9℄, extended symmetries in disordered
systems, polymersandper
olation[10,11,12,13,14℄andWZWmodelsonsupergroups [15,16,17,
18℄ to the D-brane dynami
s in string theory [19,20℄ and the AdS/CFT
orresponden
e [21,22,
23,24,25,26,27℄. On the representation-theoreti
side, logarithmi

onformal (cid:28)eld theories are
of interest be
ause studying nondiagonalizable Virasoro representations [28,29℄ may require an
2 FJELSTAD, FUCHS, HWANG, SEMIKHATOV, AND TIPUNIN
extension(and
ertainly,newappli
ations)oftheideasandmethodsthathave beensu

essfully
used in
onstru
ting mathemati
al foundations of the (cid:16)standard(cid:17)
onformal (cid:28)eld theory.
Although there is no pre
ise de(cid:28)nition of the
lass of logarithmi

onformal (cid:28)eld theories
(cid:21) we
omment on this issue at the end of Se
. 6 (cid:21) the two
hara
teristi
properties are the
nondiagonalizable Virasoro a
tion and the appearan
e of logarithms in
orrelation fun
tions
(these issues are
losely related; see, e.g., [30℄ on how the maximum Jordan
ell size restri
ts
the maximal power of logarithms). A trivial example of logarithms o

urring in
orrelation
fun
tions is provided by the
onformal (cid:28)eld theory of a free boson. However, the logarithm in
this
ase merely
orresponds to in
ludingthe positionas an operator(
onjugate to momentum)
on the state spa
e, and this in
lusion does not lead to any interesting new stru
ture for the
u(1)
representations of the relevant
hiral algebra (the
urrent algebra). On the other hand, in
(cid:16)genuinely(cid:17) logarithmi
models, thelogarithmsare
loselytiedtoredu
ible,butinde
omposable
modules over the Virasoro [28,29℄ or extended
hiral algebras. In su
h modules, the a
tion of
L
0
is not diagonalizable, but rather
onne
ts ve
tors in a Jordan
ell of a
ertain size.
In a number of examples studied in the literature, Jordan
ells o

ur at the top level of the
module, whi
h in the
ase of the va
uum module implies the existen
e of a (cid:16)logarithmi
partner
totheidentity operator.(cid:17) However, this behaviorisnotgeneri
. Moreover, logarithmi
partners
of the energy-momentum tensor need not ne
essarily appear (although their existen
e seems to
have sometimes been taken nearly as the de(cid:28)nition of a logarithmi

onformal (cid:28)eld theory). It
was indeed already noti
ed in [29℄ that Jordan
ells
an (cid:28)rst o

ur at lower levels, giving rise
to the so-
alled staggered modules, in
ontrast to Jordan highest-weight modules [29℄, where
1
Jordan
ells are (cid:16)inherited(cid:17) from the top level of the module. The (cid:16)staggered(cid:17)
ase is in fa
t
most frequent in appli
ations, in parti
ular in logarithmi
ally extended minimal models.
In this paper, we present a
onstru
tion of logarithmi

onformal (cid:28)eld theories via a
ertain
deformation of
onventional
onformal (cid:28)eld theories. We
onsider the
ase where a given
E(z) E

hiral algebra C is a part of a larger algebra A that also in
ludes a (cid:28)eld su
h that
E H
is a s
reening for C. When is in addition a fermion, irredu
ible modules of A de
ompose
d= E
into entire
omplexes of modules over C, with the di(cid:27)erential . Inde
omposable C-
L H
modules are then
onstru
ted by (cid:16)gluing together(cid:17) the modules in two adja
ent terms of the
d R d R d R L
i i+1 i

omplex ··· −→ −→ −→···, in su
h a way that be
omes a submodule of , while
L/R R R
i i+1 i+1
the quotient is isomorphi
to . The pre-image of in the inde
omposable module
∂ 1E
−

an be thought of as des
endants of a (cid:28)eld involving the operator by whi
h the
hiral
algebra is extended. Not surprisingly, extending the
hiral algebra by an operator of the form
∂ 1E
−
leads to the o

urren
e of logarithms in operator produ
ts. We apply this
onstru
tion
1
The
ase studied in [29℄ is that of Jordan Verma modules(cid:22)the (cid:16)logarithmi
version(cid:17) of Verma modules,
whilewhato

ursin manyspe
i(cid:28)
models arenotVerma,but Fo
k(orrather,Feigin(cid:21)Fuks(FF) [31℄)modules.
A

ordingly,thenotionsintrodu
edin[29℄shouldbeappliedmodulothe
orresponding(cid:16)
orre
tion.(cid:17) Compared
to Verma modules, the FF modules are distorted by inverting (cid:16)one half(cid:17) of the embeddings. This relationship

arries over to their logarithmi
versions.
LOGARITHMIC DEFORMATIONS 3
(2,p) sl(2)
of inde
omposable modules to Virasoro minimal models and to in a realization that
b
admits a fermioni
s
reening. In the asso
iated Virasoro representation given by the Sugawara
L 2
0

onstru
tion, the generator then a
quires dimension- Jordan
ells, whi
h
an again be
2
restated as the existen
e of (cid:16)logarithmi
partners(cid:17) to
ertain states.
In its simplest form, our approa
h involves a deformation of the energy-momentum tensor
thateventuallyleadstothe
onstru
tionofinde
omposablerepresentations. Thisdeformationis
des
ribed in Se
. 2. A deformation of general
hiral algebras is
onsidered in Se
. 3. In Se
. 3.1,
we (cid:28)rst re
all some basi
fa
ts about operator produ
t algebras (OPAs). Then we pro
eed
to
onstru
t an OPA derivation and use it to perform the deformation (see Theorem 3.1).
Thinking of this (outer) derivation as an inner one generated by a (cid:28)rst-order pole with some
new operator then leads to an extension of the
hiral algebra. In Se
. 3.3, we show how
the
orresponding spa
e of states is extended su
h that the operator(cid:21)state
orresponden
e is
maintained. The resulting spa
e of states
arries an inde
omposablerepresentation of the
hiral
algebra. The
onstru
tion is reformulated and generalized in terms of vertex operator algebras.
Ward identities for the energy-momentum tensor in the logarithmi
theories
onstru
ted via a
fermioni
s
reening are presented in Se
. 3.4. Spe
i(cid:28)
examples of the general
onstru
tion are
sl(2)
studied inSe
s. 4 (Virasoromodels)and 5 (the WZW theory). Finally,in Se
. 6 we brie(cid:29)y
b
dis
uss how our results (cid:28)t into the quest of gaininga deeper understanding of logarithmi
CFT.
2. A deformation of the energy-momentum tensor
As already mentioned, logarithmi

onformal (cid:28)eld theories are naturally asso
iated with the
presen
e of Jordan
ells in Virasoro modules. In this se
tion, we present a simple re
ipe for
deforming the energy-momentum tensor that leads to the appearan
e of Jordan
ells.
To start, we
onsider the simple problem of (cid:28)nding an energy-momentum tensor of the form
T(z) = T(z)+T (z),
1
(2.1)
T e T
1
where istheenergy-momentumtensorof a(standard)
onformal(cid:28)eld theory and isan (cid:16)im-
provement(cid:17) term that eventually leads to Jordan
ells. Later on, we
onstru
t inde
omposable
T
1
Virasoro representations by allowing to a
t on an auxiliary ve
tor spa
e, but at the present
T
stage, we (cid:28)rst ensure that has the
orre
t operator produ
t to be an energy-momentum
T
e 1
tensor. A simple ansatz is to take to have regular OPE with itself and to be a primary (cid:28)eld
h T
of some weight with respe
t to . In this
ase we have
2T(w)+2hT (w)
1
T(z)T(w) = (T(z)+T (z))(T(w)+T (w)) = +... ,
(2.2) 1 1 (z w)2
−
e e
2
The o

urren
eofJordan
ells asthe
hara
teristi
propertyofinde
omposablerepresentationspertainsto
the Virasoro algebra. The existen
e of Jordan
ells in a Virasoro module asso
iated with an a(cid:30)ne Lie algebra
via the Sugawara
onstru
tion implies that the a(cid:30)ne Lie algebra module is inde
omposable.
4 FJELSTAD, FUCHS, HWANG, SEMIKHATOV, AND TIPUNIN
h=1
whi
h implies . To also have
ompatibility between the grading of Laurent modes, we
T
1
therefore take to be of the form
1
T (z) = E(z),
(2.3) 1 z
E(z) T (z)T (w)
1 1
with a weight-one primary (cid:28)eld. More general solutions, for whi
h does
possess a singular part are also possible. We further analyze the general
ase in Se
. 3. Here
we
ontinue our
onstru
tion with the parti
ular deformation (2.3) for the parti
ular
ase of a
single free boson with a ba
kground
harge. We show that ansatz (2.3) leads to sele
ting the
(1,n) (2,n)
and minimal models and that it in fa
t o(cid:27)ers a
onstru
tion of their logarithmi
extension.
ϕ ϕ(z)ϕ(w)= log(z w)
Let be a
anoni
ally normalized free boson, − . Then the energy-
momentum tensor
T(z) = 1 :∂ϕ(z)∂ϕ(z):+Q∂2ϕ(z)
(2.4) 2
c=1 12Q2 :eγϕ(z):
has the
entral
harge − . The
onformalweight of the vertex operator in this
model is (the
olon indi
ating normal ordering is suppressed in what follows)
h(eγϕ) = 1 γ2 Qγ.
(2.5) 2 −
T (z)
1
We now spe
ialize ansatz (2.3) for as
T (z) = 1 βeγϕ.
(2.6) 1 z
γ β
Here is a number, while we keep the pre
ise meaning of the quantity as yet unspe
i(cid:28)ed. It
may be a
onstant or an operator in a (cid:28)nite-dimensionalve
tor spa
e, in whi
h
ase it a

ounts
forthepresen
e of thelatter auxiliaryspa
e withwhi
h the representation spa
es of the original
β
theory are to be tensored (
an then be regarded as a kind of zero mode operator, similar to
the
enter of mass
oordinate of a free boson or to the gamma matri
es in the Ramond se
tor
γ T
1
of fermions). We now determine the most general value of for this ansatz to work. For to
3
have a regular OPE with itself, i.e.,
1
T (z)T (w) = β2(z w)γ2eγ(ϕ(z)+ϕ(w)) = 0,
1 1
(2.7) zw −
γ2=p 0,1,2,... β2=0
we must require that ∈ { } or that .
γ=√p eγϕ
We begin with the (cid:28)rst possibilityand take . The requirement that is a weight-one
operator then yields
√p 1
Q = ,
(2.8) 2 − √p
c=13 3p 12 T β
and hen
e − − p . Obviously, for 1 to be bosoni
, must be bosoni
(fermioni
) if
eγϕ
and only if is bosoni
(fermioni
). We must therefore distinguish two
ases:
3 A(z)B(w)
As usual, in the left-hand side of operator produ
ts stands for the singular part of the radially
A(z) B(w)
ordered operator produ
t of and .
LOGARITHMIC DEFORMATIONS 5
β p=2q q=1,2,3,... (1,q)
1. is bosoni
and with . Then (2.8) gives the minimal series
1 1
Q = (√q )
(2.9) √2 − √q
with the
entral
harge
6
c = 13 6q .
(2.10) − − q
β
In this
ase,
an in parti
ular be just a
onstant.
β p=2r+1 r=0,1,2,... (2,p)
2. is fermioni
and with . This gives the minimal series with
the
entral
harge
12
c = 10 6r .
(2.11) − − 2r+1
c= 2
Note that the value − is
ontained in both series. In the (cid:28)rst series it
orresponds to
γ2=4 γ2=1
using the vertex operator with ((cid:16)long s
reening(cid:17)), while in the se
ond
ase one has
((cid:16)short s
reening(cid:17)).
β2=0 β
The se
ond option that follows from (2.7) is . The situation where is a nilpotent
boson does not lead to any restri
tions on the
entral
harge, and we do not
onsider it here.
β
A more natural
ase is that of a fermioni
, whi
h brings us ba
k to the above
ase 2.
T(z) (1,q)
To summarize, ansatz (2.6) for leads to two possible series of solutions: the
(2,p) p
minimal series, and the series with odd.
β
The idea on how this deformation gives rise to a logarithmi
extension is to allow to be
K T
an operator a
ting on a suitable auxiliary spa
e su
h that furnishes an inde
omposable
e
representation of the Virasoroalgebra. The
onstru
tion of the state spa
e then depends on the
β β EndK dimK>2
representation
hosen for . In parti
ular, for ∈ with , generi
ally we must
deal with Jordan
ells that
an have size larger than two. (Correspondingly, higher powers of
logarithmso

ur inthe operatorprodu
ts.) In
ontrast, inthe fermioni

ase we onlyen
ounter
Jordan
ells of size two.
3. Deformation of a general
hiral algebra
Inthepreviousse
tion,we
onstru
tedaparti
ulardeformationofaspe
i(cid:28)

onformal(cid:28)eld(cid:22)
T
the energy-momentum tensor (cid:22)in a spe
ial
lass of models. We now extend the
onstru
tion
beyond the one-boson
ase, not relyingon the presen
e of a free-(cid:28)eld realization, and deforming
all (cid:28)elds in the
hiral algebra. This is done in su
h a way that the OPEs are preserved, but
β
is interwoven in the deformed operators su
h that new, inde
omposable representations are
K β EndK
obtained by tensoring with an auxiliary ve
tor spa
e and taking ∈ . The two basi
ingredients of the
onstru
tion (cid:22) the deformation and the extension of the spa
e of states (cid:22)
are des
ribed in Se
s. 3.1 and 3.3, respe
tively.
6 FJELSTAD, FUCHS, HWANG, SEMIKHATOV, AND TIPUNIN
3.1. The OPA deformation. Instead of working with operator produ
t expansions of the
form
[A,B] (w)
A(z)B(w) = n ,
(3.1) (z w)n
Xn 1 −
≥
we use operator produ
t algebras (OPAs) in the form introdu
ed in [32℄. This
onstitutes a
on-
venient formulation of the properties of operator produ
ts that fa
ilitates expli
it
al
ulations,
and
an be regarded as an adaptation of (a part of) the axioms of vertex operator algebras
(VOAs) [33,34,35℄.
The essential ingredients of the OPA setting in [32℄ are as follows. The operators (or (cid:28)elds)
A V Z V
form a (super)ve
tor spa
e , whi
h is graded over . The OPA stru
ture on is given by
a
olle
tion of bilinear operations
[ , ] : V V V, n Z,
(3.2) n × → ∈
4
that are
ompatible with the grading by
onformal weight and satisfy the (cid:16)asso
iativity(cid:17)
on-
dition [32, Eq.(2.3.21)℄
q 1
A,[B,C] = ( 1)AB B,[A,C] + − [[A,B] ,C] .
(3.3) h piq − h qip Xℓ 1 (cid:18)ℓ−1(cid:19) ℓ p+q−ℓ
≥
( 1)AB 1 A B +1
(The sign fa
tor − is equal to − if and are both fermioni
, and to otherwise.)
1 V
There is also a distinguished element ∈ su
h that
[1,A] = δ A
(3.4) n n,0
A V ∂ : V V
forall ∈ , andan even linearmapping → su
h that the(cid:16)
ommutativity(cid:17) property [32,
Eq.(2.3.16)℄
( 1)i
[B,A] = ( 1)AB − ∂i n[A,B]
(3.5) n − (i n)! − i
Xi n −
≥
A B
is satis(cid:28)ed. For any two operators and , their OPE is then re
onstru
ted from the produ
ts
[ , ]
n as in (3.1). In other words, operator produ
t (3.1) should be regarded as a linearmapping
V V V C[[z,z 1]]
C −
(3.6) ⊗ → ⊗
that furnishes a generating fun
tion for the in(cid:28)nite family of produ
ts in Eq. (3.2).
E V
To perform the deformation, we (cid:28)x an operator ∈ and de(cid:28)ne the operation
( 1)n+1 [E,A] (z)
∆ : A(z) logz [E,A] (z)+ − n+1
(3.7) E 7→ 1 n zn
Xn 1
≥
A V
on ∈ .
E V ∆ V
E
Theorem 3.1. For any ∈ , the operation is a superderivation of the OPA .
4 A h B h [A,B] h +h n
A B n A B
That is, when has weight and weight , then the (cid:28)eld has weight − . The
T V
existen
e of the energy-momentum tensor ∈ is understood here; we omit the obvious details.
LOGARITHMIC DEFORMATIONS 7
The proof is obtained by dire
t
al
ulation; it is given in Appendix A.
E β
It follows from Theorem 3.1 that if is fermioni
and a

ordingly is also fermioni
, so
β2=0
that , deformation (3.7) gives rise to an OPA isomorphism
A A+β∆ (A).
E
(3.8) 7→
E β βN=0 N
If is bosoni
and is bosoni
and nilpotent, for some integer , then
A exp(β∆ )A,
E
(3.9) 7→
∆
E
where the right-hand side is a polynomial in , is again an isomorphism. (In a number of
E
spe
i(cid:28)
examples, e.g. for models based on an a(cid:30)ne Lie algebra, and with taken to be a
β β

urrent in a nilpotent subalgebra, the right-hand side is a polynomial in without requiring
to be nilpotent.)
E 1 [E,T] =δ E n>0
If is a weight- primary (cid:28)eld (and hen
e, n n,2 for ), the deformation of the
energy-momentum tensor is given by
T = exp(β∆ )T = T + 1 βE +β2(...)+... .
(3.10) E z
β e
The term linear in reprodu
es the parti
ular deformation in Eq. (2.3) that was
onsidered in
β
the one-boson
ase in Se
. 2. In Se
. 3.3, we show that taking to be an endomorphism of a
:= exp(β∆ )
E
(cid:28)nite-dimensionalspa
e results in an inde
omposable module over the algebra C C
T
e
that in parti
ular
ontains the Virasoro algebra
orresponding to .
e
In general, the proposed deformation (3.7) maps ea
h (cid:28)eld in the algebra into a linear
ombi-
z logz
nation of (cid:28)elds weighted with powers of and . The produ
t (3.6) of the operator algebra
V V
is thereby lifted to an operator algebra for whi
h the operator produ
ts furnish a mapping
5
e
(
f. [30℄)
V V V C[[z,z 1]][logz].
C −
(3.11) ⊗ → ⊗
logz e e e [E,A] =0
The terms involving do not appear in the transformed (cid:28)elds if and only if 1 for
A V E
all ∈ , or in other words, if and only if is a s
reening operator for the
hiral algebra (in
expβ∆ H V V
E
that
ase, although does not preserve the ve
tor spa
e , itmaps intoan isomorphi
V E
OPA ). Whenever is in addition a fermioni
s
reening, it
an serve as a di(cid:27)erential in a
H
e

omplex; the logarithmi
deformationsof modules that appear in Se
. 3.3 then involve elements
of the adja
ent members of the
omplex.
In appli
ations, s
reenings are often dealt with as follows. One starts by taking some (cid:16)large(cid:17)
algebra, then
hooses in it (cid:28)eld(s) whose integrals are de
lared to be s
reening(s), and (cid:28)nally
sele
ts a (cid:16)small(cid:17) algebra as the
entralizer (
ommutant) of the s
reenings in the large algebra.
E
Verifyingthat agivenoperator givesas
reening uponintegrationinvolves onlythe(cid:28)rst-order
[E, ] n [E, ] n 2
poles · 1, while all the order- poles · n with ≥
an be arbitrary and are irrelevant
5 ∆
E
Sin
e the operation a(cid:27)e
ts the Laurent expansion of (cid:28)elds, it drasti
ally
hangesthe operator produ
t
expansions (3.1). To put it di(cid:27)erently, we deal with a deformation of the operator produ
t expansion (3.1),
[ , ]
whi
h is howevernot arbitrary, but su
h that the produ
ts n in Eq. (3.2) are preserved.
8 FJELSTAD, FUCHS, HWANG, SEMIKHATOV, AND TIPUNIN
E
to the properties of a
onformal (cid:28)eld theory in the
ommutant of . However, these poles
H
are (cid:16)reanimated(cid:17) by the logarithmi
deformation pres
ription, where their a
tual role is to be
responsible for (cid:16)logarithmi
deformation dire
tions(cid:17) of the given
onformal (cid:28)eld theory.
3.2. Remarks.
A
a. We
an think of the mapping (3.7) as indu
ed by the operator produ
t of with some new
α ∂α(z)= E(z)
(cid:28)eld, whi
h we
all . Taking the derivative, we then see that − . We therefore
∂ 1E(z) α(z)
−
use the symbol e as a suggestive notation for − . We
aen then write
( 1)n+1 [E,A] (w)
∂ 1E(z)A(w) = [E,A] log(z w)+ e − n+1 .
(3.12) − 1 − n (w z)n
Xn 1 −
≥
∂α(z)= E(z) α(z)
However, the relation − does not determine the zero mode of ; as we see in
the next subse
tion, su
he zero modes are expressed in terms of endomorphismes of an auxiliary
spa
e.
E ∂E
We note that if is repla
ed with , the
orresponding derivation
an be written as
∆ A = [[ ,E] ,A] ,
∂E O1 0 1
(z)= 1
with O1 z. This operation is obviously a derivation, be
ause the (cid:28)rst-order pole with
∆ A
E
any operator is a derivation of the OPA. In
ontrast, expressing in a similar way as
∆ A= [ ,∂ 1E] ,A V E /∂V
E O1 − 0 1 involves an operator not from whenever ∈ .
(cid:2) (cid:3)
E 1 β
b. If in (3.7) is not of weight , the parameter must be dimensionful, and the
onstru
tion
exp(β∆ ) β z 1
E −
of as a series in involves growing orders of derivatives and growing powers of .
For instan
e, taking the Virasoro algebra, whi
h in the OPA language is generated by a single
T
element su
h that
c
[T,T] = ∂T , [T,T] = 2T , [T,T] = ,
(3.13) 1 2 4 2
we have
2 c
∆ T = logz∂T + T + ,
T
(3.14) z 6z3
logz 4 2logz 1 1 logz
(∆ )2T = (logz)2∂2T +5 ∂T + − T +c 3 − 2 ,
T
(3.15) z z2 z4
(logz)2 7(logz)2 19logz
(∆ )3T = (logz)3∂3T +9 ∂2T − ∂T
T
(3.16) z − z2
4(logz)2 14logz +8 2(logz)2 17 logz + 2
+ − T +c − 6 3 .
z3 z5
LOGARITHMIC DEFORMATIONS 9
exp(β∆ )
E

. As an example that
learly displays the isomorphism property of (although is not
E
relevant to logarithmi
theories), we
onsider the situation where is a bosoni

urrent, that
1
is, a weight- (cid:28)eld with the OPEs
κ
E(z)E(w) = ,
(3.17) (z w)2
−
λ E(w) ∂E(w)
T(z)E(w) = + +
(3.18) (z w)3 (z w)2 z w
− − −
λ E
(where the
onstant is a possible (cid:16)anomaly(cid:17)), and moreover, is diagonal on all primaries of
the theory, i.e.,
q Ψ (w)
i i
E(z)Ψ (w) = .
i
(3.19) z w
−
exp(β∆ )
E
The operation is then readily evaluated to a
t as
κ
exp(β∆ )E(z) = E(z)+β ,
(3.20) E z
βE(z) βλ+β2κ
exp(β∆ )T(z) = T(z)+ + ,
(3.21) E z 2z2
exp(β∆ )Ψ (z) = zβqi Ψ (z).
E i i
(3.22)
logz zβqi β
(In the last formula, an in(cid:28)nite series of powers of is summed up to .) With being
β End(C)
just a number (or, in the setting of the next subse
tion, ∈ ), we re
ognize this as the
E
spe
tral (cid:29)ow transform asso
iated with the
urrent .
3.3. Extension of the state spa
e. The observation leading to Eq. (3.12) indi
ates that
E /∂V
for ∈ , the deformation
an be des
ribed in terms of a new (cid:28)eld introdu
ed into the OPA
∆
E
su
h that the outer derivation be
omes an inner one. We therefore
onsider an extended
OPA
ontaining this new (cid:28)eld, assuming a de(cid:28)nite pres
ription for the zero mode. In order
to preserve the (cid:28)eld(cid:21)state
orresponden
e, we must then also extend the spa
e of states (the
va
uum module).
The
ase
onsidered in (3.10) already
aptures
ru
ial features of the general situation. We
V V Ω
V
take the va
uum module of the OPA , with the highest-weight ve
tor , and extend it by
K
taking the tensor produ
t with some auxiliary (cid:28)nite-dimensional ve
tor spa
e . As a module
V V K dimK V End(K)
over , ⊗ isjust the dire
t sum of
opiesof , withoperators in generi
ally
ω K Ω
V
a
ting between di(cid:27)erent
opies. For a
hosen ∈ , we
an then identify with the ve
tor
Ω = Ω ω β End(K)
V
⊗ . In the deformation formulas, we now take to be an element of su
h
that
βΩ=0.
(3.23)
End(K) E
We give the operators in the same parity (bosoni
or fermioni
) as (so that in
βE α End(K)
parti
ular is always bosoni
). Now, let ∈ satisfy
[α,β] = 1
(3.24)
10 FJELSTAD, FUCHS, HWANG, SEMIKHATOV, AND TIPUNIN
[ , ] E 1
with denoting the super
ommutator. From (3.10) (where is a weight- (cid:28)eld), the modes
L T αΩ
n
of the deformed energy-momentum tensor a
t on as
1
e L αΩ = (T +e βE) αΩ.
(3.25) n · z n ·
e
In parti
ular,
L αΩ = E Ω.
1 1
(3.26) − · − −
e
Via the (cid:28)eld(cid:21)state
orresponden
e, the state in the right-hand side is asso
iated to the (cid:28)eld
E(z) α(z) αΩ
. We therefore
on
lude that the operator
orresponding to the state satis(cid:28)es
∂α(z) =e E(z),
(3.27) −
α(z)e= ∂−1E(z)
whi
h we also write suggestively as − .
e
This illustrates how the spa
e of states (and hen
e, the (cid:28)elds)
an be extended. We now
β
analyze the
onstru
tion systemati
ally, still assuming to be an endomorphism of an aux-
K h =1
E
iliary (cid:28)nite-dimensional ve
tor spa
e (with ). The ne
essary manipulations are best
des
ribed in the language of vertex operator algebras [33,34,35℄, whi
h we summarize in Ap-
pendix B.
K
We begin with making the auxiliary ve
tor spa
e into a (cid:16)toy(cid:17) vertex operator algebra.
[ , ]
Having done so allows us to use the produ
ts n of the
orresponding (cid:16)(cid:28)elds,(cid:17) see Eqs. (B.4)
End(K)[[z,z 1]]
−
and (B.5). These produ
ts on , satisfying all the properties of the OPA opera-
∆
E
tions ex
ept the (cid:16)
ommutativity(cid:17)
ondition, then allow us to
onstru
t a generalization of
in (3.7).
ω K K
We (cid:28)rst
hoose an arbitrary element ∈ , whi
h we
all the va
uum ve
tor of . This
EndK=(EndK) +(EndK) (EndK)
+ +
gives rise to a ve
tor spa
e de
omposition −, where is
(EndK) ω=0 S (EndK)
+
de(cid:28)ned by the requirement that . We next (cid:28)x a set of elements from −
K k=κω κ S
su
h that every ve
tor in
anbe written asa linear
ombinationof ve
tors with ∈ .
(EndK) S
+
Elementsof aresaidtobeannihilationoperatorsandelementsof
reationoperators.
End(K)
From the (super)
ommutator in and the normal ordering asso
iated with the
hosen
End(K)[[z,z 1]]
−
va
uum ve
tor, we
an then de(cid:28)ne an OPA stru
ture on by Eqs. (B.4)(cid:21)(B.6).
Y(k,z)=κ k=κω κ S K
Moreover, setting for with ∈ de(cid:28)nes a (toy) vertex operator algebra
K K=(K,Y,ω)
with the spa
e of states given by . The data satisfy the list of properties of
z Y
VOAs ex
ept those related to the Virasoro algebra. Here, is a dummy variable be
ause
K End(K) K (EndK)
maps into and the
orresponding OPA a
tually
oin
ides with − (with
[A,B] (EndK)
the only nonzero bra
ket being just 0 given by the multipli
ation in −).
V V
′ ′
Returning to the (cid:16)genuine(cid:17) VOAs, we (cid:28)x a VOA with the spa
e of states and the
V 1 E(z) V
′ ′

orresponding OPA . Sele
ting a weight- (cid:28)eld ∈ (in general, several su
h (cid:28)elds),
V V
′
we de(cid:28)ne a VOA ⊂ as the
ommutant (
entralizer) of its zero mode (in general, of the
[E,A] =0 A V
subalgebra of zero modes); we thus have 1 for every (cid:28)eld ∈ . For a VOA module
W W W V V
′ ′ ′
, we similarly sele
t a submodule ⊂ . Clearly, is a module over .