SPECTRAL FLOW IN THE SL(2,R) WZW MODEL Carmen A. Núñez I.A.F.E. & UBA WORKSHOP: New Trends in Quantum Gravity Instituto de Fisica, Sao Paulo Septembre 2005
Dec 31, 2015
SPECTRAL FLOW IN THE SL(2,R) WZW MODEL
Carmen A. Núñez I.A.F.E. & UBA
WORKSHOP: New Trends in Quantum Gravity Instituto de Fisica, Sao Paulo Septembre 2005
MOTIVATIONS
I. SL(2) symmetry is rather general
String Theory on AdS3 SL(2,R) WZW model
Black holes in string theory
Liouville theory of 2D quantum gravity
3D gravity
Certain problems in condensed matter
CFT based on affine SL(2)k, not only for k Z and
unitary integrable representations (j Z or Z+½).
RATIONAL vs NON-RATIONAL CFTs
RCFT finite number of representations of modular group
(e.g. c=1 on circle of rational R2 ; extended algebra).
Non-RCFT are qualitatively different
Verma module is reducible; there are null vectors; free field rep.
II. CFTs with SL(2) symmetry simplest models beyond the well studied RCFT
Continuous families of primary fields No highest or lowest weight representations
No singular vectors fusion rules cannot be determined algebraically
OPE of primary fields involves integrals over continuous sets of operators.
STRING THEORY ON AdS3
This string theory is special in many respects:
Simplest string theory in time dependent backgrounds Concept of time in string theory
String theories in more complicated geometries
In the context of AdS/CFT it is special because Worldsheet theory can be studied beyond sugra
It does not require turning on RR backgrounds
BCFT is 2D infinite dimensional algebra
Important lessons from stringy analyses
Observables in spacetime theory Fundamental string excitations
Worldsheet correlation functions Green’s functions of operators (in flat spacetime interpreted as in spacetime CFT S-matrix elements in target space)
Spacetime CFT has Constraints in worldsheet theory non-local features These restrictions are not understood from the
string theory point of view.
Is string theory on AdSIs string theory on AdS33 consistent (unitary)? consistent (unitary)?
Is the OPA closed over unitary states?Is the OPA closed over unitary states?
)(Oi xx, ),,,(2 zzxxzd i
worldsheetett zdxx FF 2arg ),(
STATUS OF STRING THEORY ON AdS3
Unitary spectrum of physical states (spectral flow symmetry) J. Maldacena, H. Ooguri hep-th/0001053
Modular invariant partition function J. Maldacena, H.Ooguri, J. Son; hep-th/0005183 Product of characters of SL(2,R) representations? D. Israel, C. Kounnas, P.Petropoulos; hep-th/0306053
Correlation functions J. Maldacena, H. Ooguri hep-th/0111180
Analytic continuation of J. Teschner, hep-th/0108121
Generalization of bootstrap to
)2(
),2(3 SU
CSLH
SL(2,R) WZW model WZW model
(actions related by analytic continuation of fields)
States in H of SL(2,R) non-normalizable states in H3+
Not all states in the SL(2,R) WZW model can be obtained by analytic continuation from spectral flowed states
AdS/CFT: Consistency of BCFT implies awkward constraints on worldsheet correlators. Factorization of 4-point functions is not unitary unless external states satisfy certain restrictions with no clear interpretation in worldsheet theory.
CORRELATION FUNCTIONS
)2(),2(
SUCSL
)2(),2(
SUCSL
WZW MODEL for SL(2, R)WZW MODEL for SL(2, R)kk
)])()([(Tr12
)])([(Tr8
111ijk3
12
ggggggxdk
ggdk
S
kjiV
WZW
Infinitely many symmetries generated by currents Ja(z), Ja(z),
a=,3
ggk
zJggk
zJ aa )(2
)(,)(2
)( 11
)()2(;),2()( zgzgRSLzg
k : level of the representation
Symmetry Algebra: Virasoro Kac-Moody
0,3
30,
33
2],[
],[;2
],[
mnmnmn
mnmnmnmn
knJJJ
JJJnk
JJ
0,3 )(
12)(],[ nmnmnm mm
cLnmLL
anm
anm nJJL ],[
Sugawara relation:::
)2(1 a
ma
mnn JJk
L
23
k
kc
And similarly for n
an LJ ,
Lie algebra of SL(2,R) can be represented by differential operators
jxx
xDjx
xDx
D 2,, 23
x: isospin coordinate
PRIMARY FIELDSPRIMARY FIELDS ),;,( zzxxj
keep track of SL(2) weights),( xx ),( mm
AdS/CFT interpretation location of operator in dual BCFT
3,,),;,(),;,()(
azzxxwz
DzzxxzJ j
a
ja
),;,(||
),( ,2
2
,;, zzxxxxx
xdzz jj
mjmjmjmj
Form representations of the Lie algebra generated by J0a(z)
0,0| ,; nJ mmjan
jm Unitary representations of SL(2,R)Dj
+: m = j, j+1,…Dj
-: m = –j, – j – 1,…
Cj: , m= , +1,…
21
21 k
j
,21
ij
SPECTRAL FLOWSPECTRAL FLOW
wnnn
nnnn
JJJ
wk
JJJ
~2
~0,
333
23
4
~w
kwJLLL nnnn Sugawara
The transformation
with w Z, preserves the SL(2,R) commutation relations
obey Virasoro algebra with same cnL~
wjD̂ w
jC ,ˆ
The spectral flow automorphism generates new representations
and
Hilbert space of SL(2,R) WZW modelHilbert space of SL(2,R) WZW model
wj
wji
wj
k
wj CCddjDDdj
,
1
0
,2
1
2
1
2/1
wˆˆˆˆH
wjD̂ is an irreducible infinite dimensional representation of the SL(2,R)
algebra generated from highest weight state |j;w> defined by
wjwk
jwjJ
nwjJwjJwjJ nwnwn
;|2
;|
)0(0;|,0;|,0;|
30
31
w Z is the spectral flow parameter or winding number
21
21 k
j
0,ˆ jC
is generated from |j,;w> , 0< <1) and
wjwk
wjJ
nwjJwjJ nwn
;,|2
;,|
)0(0;,|,0;,|
30
3
And the Casimir is
wjjjwjJJJJwk
JwjLk wwww ;|)1(;|21
2;|)2(
2300
0ˆjD and are conventional discrete and continuous represent.
wjC ,ˆ
,21
ij
wjD̂ w
jC ,ˆ and are obtained by spetral flow
CFTs based on affine SL(2)k are well known in the case of
Unitary integrable representations of SU(2) k and integer and half integer spins A.B.Zamolodchikov & V.A.Fateev (1986)
Highest weight representations: k C\{0} and
Admissible representations:
Rational level k+2 = p/q, p,q coprime integers
V.G.Kac & D.A.Kazhdan F.G.Malikov, B.L.Feigin & D.B.Fuchs
H.Awata & Y.Yamada
All these are RCFT Null vector method applies.
0,0)(;0,0)(,,,12 , srIIsrIZsrstrj sr
CORRELATION FUNCTIONSCORRELATION FUNCTIONS
The correlation functions in WZW theory obey linear differentialequations which follow from the Sugawara construction of T(z).Knizhnik-Zamolodchikov equation:
0),()...,()2(
),()...,()(
11011
11
1
1
NNjjaa
NNjj
xzxzJJLk
xzxzzT
N
N
In SU(2) there are null vectors which impose extra constraints andallow to determine the fusion rules. But the space of vectors of theunitary representations of SL(2,R)
21
21 k
j ,21
ijwjD̂ w
jC ,ˆ with and with
contains no null vectors. However the spectral flow plays their rol.
02
;2
|1
km
kjJ
),(2
xzkj
THETHE SPECTRAL FLOW OPERATORSPECTRAL FLOW OPERATOR
),(),(lim),(2
1120
,1, zxzyxyyydzx kj
mjmjmmjwJJ
2,
2k
mJk
mJ
This is an auxiliary field (not physical) which allows to construct operators in sectors w = 1 and w = –1 from operators in w = 0 as follows
),(2
xzk
It satisfies the primary state conditions with
02
;2
|1
km
kjJ
0),(),()(2)(1
12
2
1
N
nnnjk
jnn
nn
N
n n
zxzxxxjx
xxzz n
NULL VECTOR METHODNULL VECTOR METHOD
0)()...()( 1
2
1 1
Njjk zzzJN
),;,(2
),;,()(2
zzxxwz
jxxzzxxzJ j
xj
One can apply the null vector method to correlators containing ),(2
xzk
What information can be obtained from this null vector?
)2
(),(),(),( 212111
221
kjjzxzxzx jjk
This coincides with analytic continuation of Teschner’s result. However it does not determine the fusion rules need 4-point functions
N=2 SL(2,C) conformal invariance of the worldsheet and target space determines the x and z dependence
12
2
12
2
12
2
121221
32
)()()(
)()()(),,2
(),(),(),(
2121
221
22
21213322
2
kkk
zzzzzz
xxxxxxjjk
Czxzxzxjj
kjj
kjj
k
jjk
3-POINT FUNCTIONS3-POINT FUNCTIONS
4-POINT FUNCTIONS4-POINT FUNCTIONS
SL(2,C) conformal invariance of the worldsheet and target space non-trivial dependence on cross ratios
))(())((
;))(())((
2431
4321
2431
4321
xxxxxxxx
xzzzzzzzz
z
Teschner applied generalization of bootstrap for
Maldacena & Ooguri analyzed analytic continuation. Null vector method?
),(
),(),(),(),(
3124214323421
3214142324321
432
31412
4343
31412
4243443322111
xzzzzz
xxxxzxzxzxzx jjjjjjjjjjjjjjjjj
F
),(12
1),( xz
zQ
zP
kxz
zFF
)2(),2(
SUCSL
A closed form for F(z,x) is not known for generic values of ji
KZ reduces to:
Null vector method for 4-point functionsNull vector method for 4-point functions
If one operator is there is one extra equation),(2
xzk
0),(),()(2)(1
3
12
24
1
n
nnjkj
nnn
nn n
zxzxxxjx
xxzz n
and KZ equation simplifies because
02
;2
|1
km
kjJ
The spectral flow operator is not physical.It changes the winding number of another operator by one unit.
This gives a 3-point function violating winding number conservation by one unit.
2|
2| 3
11
kjJ
kjL
N-point functions may violate winding number conservation
up to N-2 units Determined by SL(2,R) algebra
Result agrees with free field approximation (Coulomb gas
formalism). G. Giribet and C.N., JHEP06(2000)010; JHEP06(2001)033
Supersymmetric extension D. Hofman and C.N., JHEP07(2004)019
Need 5-point functions to get information for 4-point function
Comments
Coulomb gas is more practical method than bootstrap of BPZ It works in minimal models and SU(2) CFT due to singular vectors. Extension to SL(2,R) requires analytic continuation in the number of screening operators. It worked for 3-point functions, but this is an experimental fact. There is no theoretical proof.
OPEN PROBLEMSOPEN PROBLEMS
Computation of 4-point functions in w 0 sectors and factorization properties. Closure of OPA on unitary states
Interpretation of unitarity constraints on worldsheet correlators
They do not correspond to well defined objects in BCFT
if nkjn
ii
1
nkjn
ii
1
2
321
kjj
2
343
kjj
Factorization of 4-point functions is not unitary unless
and
j1
j2
j3
j4
J
Non-physical J not well defined objects in BCFT
Each leg imposes additional constraints
Modular properties?
Factorization properties?
Higher genus Riemann surfaces
)()(),(,
RLRL
LRNZ
Verlinde theorem?
THE END