[PPT] - Non-hiral urrent algeb ras fo r defo rmed sup ergroup WZW PowerPoint Presentation

SLIDE 1 Non- hiral urrent algeb ras fo r defo rmed sup ergroup WZW mo dels Anatoly K

ne hny

Heriot-W att Universit y Septemb er 21, 2011, EMPG semina r, Edinburgh Based

n

a joint w

rk

with Thomas Quella (JHEP03(2011)124)

SLIDE 2 Outline Motivation Results

btained

using p erturbation theo ry Non-p erturbative refo rmulation

f

the mo del Dis ussion Anatoly K

ne hny

Non- hiral urrent algeb ras fo r defo rmed sup ergroup WZW mo dels

SLIDE 3 Motivation and p rio r w

rk

Confo rmally inva riant 2D sigma-mo dels whose ta rget spa e is a sup erspa e emerge in diso rdered systems

f
ndensed

matter theo ry

va

riant quantization

f

sup erstrings

n AdSd × Sd

ba kgrounds The string ba kground with pure Neveu-S hw a rz ux an b e fo rmulated in terms

f

a WZW mo del

n

the sup ergroup

PSU(1, 1|2)

(Berk

vits,

V afa, Witten, 1999). In su h a fo rmulation

ne

an des rib e defo rmations

rresp
nding

to swit hing

n

a mixture

f

N-S and R-R uxes whi h p reserves the full isometry

f

G = PSU(1, 1|2)

whi h is isomo rphi to G × G . In the

p

erato r des ription this is des rib ed b y p erturbing the WZW mo del b y an

p

erato r

:Jaφab ¯ Jb : (z, ¯ z)

where φab is the p rima ry eld

rresp
nding

to the adjoint rep resentation. Anatoly K

ne hny

Non- hiral urrent algeb ras fo r defo rmed sup ergroup WZW mo dels

SLIDE 4 Geometri ally this defo rmation is des rib ed as a p rin iple hiral mo del

n G

with a W ess-Zumino term. It w as rst a rgued in Bershadsky , Zhuk

v,

V aintrob (1999) that su h p rin ipal hiral mo dels a re

nfo

rmal (fo r any value

f

the WZ term) when the sup ergroup has a vanishing Killing fo rm. This happ ens fo r

PSU(1, 1|2)

and fo r its generalizations PSL(n|n) . Their a rguments w ere generalized to general sup ergoups with vanishing Killing fo rm and to their

sets:Kagan,

Y

ung

(2005); Babi henk

(2006)

One an think

f

sup ermatri es from SL(n|n) as

f

blo k 2n × 2n matri es

M =   A | B −− −− −− C | D  

with the

ndition Sdet(M) = det(A)/det(B) = 1

. Anatoly K

ne hny

Non- hiral urrent algeb ras fo r defo rmed sup ergroup WZW mo dels

SLIDE 5 Proje ting

ut

the multiples

f

identit y

ne
btains

the sup ergroup

PSL(n|n)

. The sup ergroup PSU(1, 1|2) is a real fo rm

r PSL(2|2)

. Another interesting defo rmation

f

the WZW theo ries

n

these sup ergroups is an isotropi urrent- urrent defo rmation:

Str(J ¯ J) = κba :Ja ¯ Jb : (z, ¯ z)

This defo rmation

nly

p reserves the diagonal sub roup

f

the isometry sup ergroup G ⊂ G × G . Both defo rmations a re exa tly ma rginal due to the vanishing Killing fo rm. The

nserved

urrents

rresp
nding

to the global symmetries in the defo rmed mo dels do not split into holomo rphi and antiholomo rphi

mp
nents.

The

nservation

equations a re

∂ ¯ Ja(z, ¯ z) + ¯ ∂Ja(z, ¯ z) = 0 .

Anatoly K

ne hny

Non- hiral urrent algeb ras fo r defo rmed sup ergroup WZW mo dels

SLIDE 6 The ab

ve

is usually the ase in massive theo ries with

nserved

urrents. In CFT's this

nly

happ ens when unita rit y is violated. Supp

se φ

is a CFT p rima ry

f

w eights (n, 0) then

||¯ L−1|φ||2 = φ|¯ L1 ¯ L−1|φ = φ|2¯ L0|φ = 0 .

An OPE algeb ra fo r

nserved

urrents in massive theo ries w ere investigated in M.Lues her (1978); D. Berna rd (1991) . The

nstraints

imp

sed

b y urrent

nservation

w ere explo red and an innite to w er

f

non-lo ally

nserved

ha rges w ere

nstra ted.

The general

nstraints
n

OPE w ere mo re re ently generalized to pa rit y-nonp reserving theo ries b y S.Ashok, R. Beni hou and J. T ro

st

(2009). The urrent algeb ra in the G × G

p

reserving defo rmation des rib ed ab

ve

w as studied in S.Ashok, R. Beni hou and J. T ro

st

(2009), (2009); R. Beni hou and J. T ro

st

(2010); R. Beni hou (2011). An innite set

f
nserved

urrents w ere put fo rw a rd fo r this defo rmation. Those

nserved

quantities stem from the quantum Maurer-Ca rtan equation. Our w

rk

w as mu h inspired b y those pap ers. Anatoly K

ne hny

Non- hiral urrent algeb ras fo r defo rmed sup ergroup WZW mo dels

SLIDE 7 Many

f

the results

btained

fo r the G × G

p

reserving defo rmation dep end

n

ertain assumptions involving the adjoint p rima ry

p

erato rφab(z, ¯

z)

whose p rop erties at p resent a re not under analyti

ntrol.

W e study the se ond defo rmation where no su h p roblem a rises. W e fo us

n

the algeb ra generated b y the defo rmed urrents whi h is

f

interest at least fo r three reasons

ne

w

uld

hop e that as in the massive ase studied b y D. Berna rd there is an innite to w er

f
nserved

non-lo al ha rges

nstru ted
ut
f

the urrents. The interpla y b et w een

nfo

rmal symmetry and the integrable stru ture (Y angian) w

uld

b e interesting to explo re

ne
uld

hop e that the non- hiral urrent algeb ra ma y b e useful fo r

rganizing

the sp e trum it w

uld

b e a new OPE algeb ra generalizing vertex

p

erato r algeb ras in unita ry CFT's Anatoly K

ne hny

Non- hiral urrent algeb ras fo r defo rmed sup ergroup WZW mo dels

SLIDE 8 F

r

the urrent- urrent defo rmation w e found that despite the absen e

f

geometri reasons (no Maurer-Ca rtan equation) the quantum equation

f

motion tak es the same fo rm as in D. Berna rd (1991) whi h mak es p

ssible

the

nstru tion
f

Y angian ha rges. Anatoly K

ne hny

Non- hiral urrent algeb ras fo r defo rmed sup ergroup WZW mo dels

SLIDE 9 The mo del The WZW mo del

n

a sup ergroup G has lo al urrents Ja(z) and

¯ Jb(¯ z)

b

eying the OPE

Ja(z)Jb(w) = kκab (z − w)2 + if ab

cJc(w)

z − w + . . . ¯ Ja(¯ z) ¯ Jb( ¯ w) = kκab (¯ z − ¯ w)2 + if ab

c ¯

Jc( ¯ w) ¯ z − ¯ w + . . .

where κab is the non-degenerate inva riant 2-fo rm and f ab

c

a re the stru ture

nstants.

They satisfy

κab = (−1)aκba , f ba

c = −(−1)abf ab c

f ab

df dc e + (−1)c(a+b)f ca df db e + (−1)a(b+c)f bc df da e = 0

the

graded Ja obi identit y . The vanishing

f

the Killing fo rm implies that

f acdf b

dc = 0

Anatoly K

ne hny

Non- hiral urrent algeb ras fo r defo rmed sup ergroup WZW mo dels

SLIDE 10 Exa t t w

-p
int

fun tions

f

urrents In the defo rmed theo ry the

p

erato rs rep resenting the defo rmed urrents a re the ba re WZW urrents inserted into p erturbation theo ry series. The dimension and spin a re

nserved

and there a re no elds with whi h the urrents

uld

fo rm a Joa rdan blo k. F

rmally

the defo rmed t w

-p
int

fun tions a re

Ja(z1, ¯ z1)Jb(z2, ¯ z2)λ = Ja(z1)Jb(z2) exp

− λ

kπ

d2w :Je ¯

Jr: κre

T

w

rema

rks a re in

rder.Firstly

in loga rithmi theo ries t ypi ally

1 = 0

This is not the ase fo r some free eld realizations

f

WZW mo dels;

therwise
ne

should treat su h

rrelato

rs mo re fo rmally as a means to

mpute

OPE's. Anatoly K

ne hny

Non- hiral urrent algeb ras fo r defo rmed sup ergroup WZW mo dels

SLIDE 11 Se ondly , the p erturbation series integrals have divergen es. The integrands (fo r nite sepa ration) a re p rop

rtional

to

Ja(z1)Jb(z2)Je1(w1) . . . Jen(wn)0 ¯ Je1( ¯ w1) . . . Jen( ¯ wn)0

Ea h

rrelato

r is given b y the sum

f

singula r terms in the OPE

ntra tions

and thus lo

ks

lik e a rational fun tion times an inva riant tenso r

nstru ted

from κab and f abc . Fixing the metho d

f

subtra tion is equivalent to dening the

mp
site
p

erato r :Je ¯

Jr: κre

, i.e. w e dene its

rrelation

fun tions as distributions. Conta t term ambiguities in these

rrelato

rs a re related b y

upling
nstant

redenitions. There a re no infra red divergen es. Anatoly K

ne hny

Non- hiral urrent algeb ras fo r defo rmed sup ergroup WZW mo dels

SLIDE 12 Noting that f abc is the

nly

inva riant 3-tenso r fo r the sup ergroups

f

interest, and that

ne

annot fo rm any inva riant s ala r quantities from f abc due to the vanishing

f Cad

,

ne
nly

needs to sum the ab elian pa rt

f

the p erturbation series ab

ve.

A pa rti ula rly ni e p res ription fo r p erturbative integrals in the ab elian theo ry w ere w

rk

ed

ut

b y G. Mo

re

(1993).F

r

to ri the

rresp
nding
uplings
in ide

with ertain anoni al

rdinates
n

the mo duli spa e, su h as Klein

rdinates

fo r T 2 . W e adopt Mo

re's
rdinate λ.

Summing up the p erturbation series w e

btain

Ja(z1, ¯ z1)Jb(z2, ¯ z2)λ = kκab (1 − λ2)z2

12

, ¯ Ja(z1, ¯ z1) ¯ Jb(z2, ¯ z2)λ = kκab (1 − λ2)¯ z2

12

, Ja(z1, ¯ z1) ¯ Jb(z2, ¯ z2)λ = 0 .

Anatoly K

ne hny

Non- hiral urrent algeb ras fo r defo rmed sup ergroup WZW mo dels

SLIDE 13 Exa t three-p

int

fun tions

f

urrents It w as a rgued b y Bershadsky , Zhuk

v

and V aintrob that to

mpute

a defo rmed 3-p

int

fun tion it su es to retain terms in p erturbation theo ry

ntaining
nly
ne

fa to r

f f abc

. That a rgumant w

rks

if w e assume that there a re

nly

three tra eless inva riant four-tenso rs:

fabefcde , facefbde , κabκcd + (−1)bcκacκbd + κadκbc

Any 3-tenso r resulting from a

ntra tion
f

mo re than t w

stru ture
nstants

an b e rep resented diagrammati ally as Anatoly K

ne hny

Non- hiral urrent algeb ras fo r defo rmed sup ergroup WZW mo dels

SLIDE 14 The blob

ntaining

four external lines must

rresp
nd

to a tra eless tenso r as there a re no

rre tions

to the inva riant metri

κab

Ea h

f

the ab

ve

three tra eless tenso rs vanishes when

ntra ted

with the stru ture

nstants
n

any t w

indi es.

Extra ting fa to rs

f f abc

via the singula rities in the OPE's and summing the remaining ab elian p erturbation series using Mo

re's

p res ription w e

btain

Ja(z1, ¯ z1)Jb(z2, ¯ z2)Jc(z3, ¯ z3)λ = 1 − λ3 (1 − λ2)3 −ikf abc z12z23z31

Ja(z1, ¯

z1)Jb(z2, ¯ z2) ¯ Jc(z3, ¯ z3)λ = λ(1 − λ) (1 − λ2)3 −ikf abc¯ z12 z2

12¯

z23¯ z31

Anatoly

K

ne hny

Non- hiral urrent algeb ras fo r defo rmed sup ergroup WZW mo dels

SLIDE 15 Using the ab

ve

metho ds w e an

nly
btain 1/k

expansion fo r four and higher n -p

int

fun tions. Thus w e

mputed

Ja(z1, ¯ z1)Jb(z2, ¯ z2)Jc(z3, ¯ z3)Jd(z4, ¯ z4)λ = k2R1 + kR2 +kλ2(1 − λ)2 (1 − λ2)5

−f ab

rf rcd

1 z2

34z2 12

ln

z13z24

z23z14

2

+(−1)a(b+c)f ad

rf bcr

1 z2

14z2 23

ln

z24z13

z12z34

2

−(−1)abf ac

rf brd

1 z2

13z2 24

ln

z23z14

z12z34

2

+ O(k0)

where R1 and R2 a re rational fun tions

f z12, z23, z34

kno wn expli itly and to all

rders

in λ. Anatoly K

ne hny

Non- hiral urrent algeb ras fo r defo rmed sup ergroup WZW mo dels

SLIDE 16 OPE algeb ra

f

urrents F

r

the defo rmed OPE's

f

urrents w e an tak e a basis

f

defo rmed

p

erato rs lab elled b y

p

erato rs in the va um se to r

f

the WZW theo ry . Note that :Ja ¯

Jb : (z, ¯ z)

stands fo r an

p

erato r in the defo rmed theo ry whose

rrelato

rs a re

btained

b y inserting the ba re

mp
site :Ja ¯

Jb :

into the p erturbation seris and taking integrals using Mo

re's

p res ription

r

its extension. Su h an

p

erato r in general will not

in ide

with the no rmal

rdered

p ro du t

f

the defo rmed urrents:

:Ja ¯ Jb : (z, ¯ z) = lim

:z1→z2: Ja(z1, ¯

z1) ¯ Jb(z2, ¯ z2) .

T

mpute

the leading

rder

OPE

e ients

w e used the metho d

f

R. Guida and N. Magnoli (1996). Anatoly K

ne hny

Non- hiral urrent algeb ras fo r defo rmed sup ergroup WZW mo dels

SLIDE 17 F

r

the JJ OPE w e an write the folo wing ansatz

Ja(z1, ¯ z1) Jb(z2, ¯ z2) = kκab(λ) (z1 − z2)2 + if ab

c(λ)Jc(z2, ¯

z2) z1 − z2 +gab

cd(λ) :JcJd : (z2, ¯

z2) + hab

c (λ)∂Jc(z2, ¯

z2) + ¯ z1 − ¯ z2 z1 − z2 tab

cd(λ) :Jc ¯

Jd : (z2, ¯ z2) + ¯ z1 − ¯ z2 (z1 − z2)2 uab

c (λ) ¯

Jc(z2, ¯ z2) +(¯ z1 − ¯ z2)2 (z1 − z2)2 vab

c (λ)¯

∂ ¯ Jc(z2, ¯ z2) + (¯ z1 − ¯ z2)2 (z1 − z2)2 wab

cd(λ) : ¯

Jc ¯ Jd : (z2, ¯ z2) +

higher dimension elds The z

dep

enden e

ntains

rational pa rts di tated b y the spin and s aling dimensions, whi h ho w ever an b e de o rated b y loga rithms. This do es not happ en fo r some

f

the leading singula rities whi h a re xed b y the exa t three-p

int

fun tions. Thus w e have

κab(λ) = κab 1 − λ2 , f ab

c(λ) =

1 − λ3 (1 − λ2)2 f ab

c ,

uab

c (λ) = if ab c

λ(1 − λ) (1 − λ2)2

Anatoly K

ne hny

Non- hiral urrent algeb ras fo r defo rmed sup ergroup WZW mo dels

SLIDE 18 The rest

f

the terms in the ab

ve

ansatz w e evaluated to the rst

rder

in λ. W e found that

p

erato rs(−1)bdf a

dgf gb c :Jc ¯

Jd :

and

f ab

c ¯

∂ ¯ Jc

also app ea r

n

the RHS. Simila rly fo r the J ¯

J

OPE w e nd

Ja(z1, ¯ z1) ¯ Jb(z2, ¯ z2) = Bab

c (λ)

z1 − z2 ¯ Jc(z2, ¯ z2) + Cab

c (λ)

¯ z1 − ¯ z2 Jc(z2, ¯ z2) −z1 − z2 ¯ z1 − ¯ z2 iλf ab

c ∂Jc(z2, ¯

z2) −(−1)fs λ k κeff ae

cf bf d ln |z1 − z2|2

ǫ2 :Jc ¯ Jd : (z2, ¯ z2) + O(λ2) .

where

Cab

c (λ) = Bab c (λ) = if ab c

λ(1 − λ) (1 − λ2)2

a re kno wn to all

rders

in λ. Anatoly K

ne hny

Non- hiral urrent algeb ras fo r defo rmed sup ergroup WZW mo dels

SLIDE 19 Equal time

mmutato

rs The equal time

mmutato

r algeb ra

f

the urrents an b e

btained

from the most singula r terms in the OPEs via the Bjo rk en-Johnson-Lo w limit:

Ja

µ(σ1, 0), Jb ν(σ2, 0)

= lim

ǫ→0

Ja

µ(σ1, iǫ)Jb ν(σ2, 0)−Jb ν(σ2, iǫ)Ja µ(σ1, 0)

W

e further

mpa tify

the spa ial dire tion

n

a ir le:

σ ∼ σ + 2π

and intro du e the F

urier

mo des

Ja(σ, τ) = i

n∈Z

e−inσJn(τ) , ¯ Ja(σ, τ) = −i

n∈Z

e−inσ ¯ Jn(τ) .

Anatoly K

ne hny

Non- hiral urrent algeb ras fo r defo rmed sup ergroup WZW mo dels

SLIDE 20 The answ er lo

ks

most simple in terms

f

the mo des

la

n(τ) = Ja n(τ) − λ ¯

Ja

n(τ) ,

ra

n(τ) = ¯

Ja

n(τ) − λJa n(τ)

W e get t w

mmuting
pies
f

the ane urrent algeb ra:

la

n(τ), lb m(τ)

= kκabnδn,−m + if ab

clc n+m(τ) ,

ra

n(τ), rb m(τ)

= −kκabnδn,−m + if ab

crc n+m(τ) ,

ra

n(τ), lb m(τ)

= 0 .

Anatoly K

ne hny

Non- hiral urrent algeb ras fo r defo rmed sup ergroup WZW mo dels

SLIDE 21 Equations

f

motion Based

n

dimension, spin

nservation

and global symmetry the defo rmed equations

f

motion must b e

f

the fo rm

¯ ∂a(z, ¯ z) = −∂ ¯ Ja(z, ¯ z) = iG(λ)f a

bc :Jc ¯

Jb : (z, ¯ z)

where w e also used the fa t that f abc is the

nly

inva riant 3-tenso r. Using the same diagrammati reasoning as Bershadsky et al. w e an sho w that

f a

bc :Jc ¯

Jb : (z, ¯ z) = lim

w→z f a bcJc(w, ¯

w) ¯ Jb(z, ¯ z)

where no singula rities

ur

in taking the limit. Using this rep resentation and the exa t OPE

e ients
f

the urrents w e an nd G(λ) exa tly b y mat hing the leading singula rities in the OPE with the urrents

n

b

th

sides

f

the equation

f

motion. Anatoly K

ne hny

Non- hiral urrent algeb ras fo r defo rmed sup ergroup WZW mo dels

SLIDE 22 W e

btain

¯ ∂a(z, ¯ z) = −∂ ¯ Ja(z, ¯ z) = −i λ (1 + λ)kf a

bc :Jc ¯

Jb : (z, ¯ z)

The fo rm

f

this equation

f

motion is the same as the

ne
nsidered

b y D. Berna rd up to res aling the urrents. W e thus exp e t

ur

mo del to p

ssess

the same Y angian symmetries as in D. Berna rd (1991). Anatoly K

ne hny

Non- hiral urrent algeb ras fo r defo rmed sup ergroup WZW mo dels

SLIDE 23 Viraso ro algeb ra The Hamiltonian densities giving the equation

f

motion a re

T(σ, τ) = 1 − λ2 2k

κdc :JcJd : (σ, τ) ,

¯ T(σ, τ) = 1 − λ2 2k

κdc : ¯

Jc ¯ Jd : (σ, τ)

so that

¯ ∂Ja(z, ¯ z) = i 2π dσ ¯ T(σ, τ), Ja(z, ¯ z)

,

∂ ¯ Ja(z, ¯ z) = i 2π dσ T(σ, τ), ¯ Ja(z, ¯ z)

.

W e found an a rgument demonstrating that b

th

generato rs remain holomo rphi (anti-holomo rphi resp e tively) to all

rders

in λ. Anatoly K

ne hny

Non- hiral urrent algeb ras fo r defo rmed sup ergroup WZW mo dels

SLIDE 24 Op en questions/F uture dire tions the p erturbation theo ry metho ds an b e applied to

btain

the diagonal pa rts

f

the defo rmed

nfo

rmal dimensions, w e a re urrently trying to he k those fo rmulas agains the latti e data fo r the sup ersphere sigma mo dels. (W

rk

in p rogress with Candu, Quella, S homerus) integrabilit y and its interpla y with

nfo

rmal symnmetry needs to b e tho roughly investigated string theo ry interp retation

f

the G

p

reserving defo rmation w

uld

b e interesting to w

rk
ut.

Squashed AdS3 with uxes? it w

uld

b e desirable to get b etter analyti

ntrol
ver

the

φab

p rima ry to push further the results available fo r the

G × G

p

reserving defo rmation Anatoly K

ne hny

Non- hiral urrent algeb ras fo r defo rmed sup ergroup WZW mo dels