Kondo flow invariants, twisted K-theory and Ramond-Ramond charges

Article

Reference

Kondo flow invariants, twisted K-theory and Ramond-Ramond charges

MONNIER, Samuel

Abstract

We take a worldsheet point of view on the relation between Ramond-Ramond charges, invariants of boundary renormalization group flows and K-theory. In compact super Wess-Zumino-Witten models, we show how to associate invariants of the generalized Kondo renormalization group flows to a given supersymmetric boundary state. The procedure involved is reminiscent of the way one can probe the Ramond-Ramond charge carried by a D-brane in conformal field theory, and the set of these invariants is isomorphic to the twisted K-theory of the Lie group. We construct various supersymmetric boundary states, and we compute the charges of the corresponding D-branes, disproving two conjectures on this subject. We find a complete agreement between our algebraic charges and the geometry of the D-branes.

MONNIER, Samuel. Kondo flow invariants, twisted K-theory and Ramond-Ramond charges.

Journal of High Energy Physics , 2008, vol. 2008, no. 06, p. 022 - 022

DOI : 10.1088/1126-6708/2008/06/022

Available at:

http://archive-ouverte.unige.ch/unige:8545

Disclaimer: layout of this document may differ from the published version.

arXiv:0803.1565v4 [hep-th] 6 Jul 2008

Ramond-Ramond harges

Samuel Monnier

∗

UniversitédeGenève,SetiondeMathématiques,

2-4RueduLièvre,Casepostale64,1211Genève4,Switzerland

July 6,2008

Abstrat

WetakeaworldsheetpointofviewontherelationbetweenRamond-Ramondharges,

invariantsofboundaryrenormalizationgroupowsandK-theory. InompatsuperWess-

Zumino-Witten models, we show how to assoiate invariants of the generalized Kondo

renormalization group ows to a given supersymmetri boundary state. The proedure

involvedis reminisentof thewayoneanprobetheRamond-Ramond hargearried by

a D-brane in onformal eld theory, and the set of these invariants is isomorphi to the

twistedK-theoryoftheLiegroup. Weonstrutvarioussupersymmetriboundarystates,

andweomputethehargesoftheorrespondingD-branes,disprovingtwoonjetureson

thissubjet. Wendaompleteagreementbetweenouralgebraihargesandthegeometry

oftheD-branes.

∗

samuel.monniermath.unige.h

1 Introdution 3

2 An overview 5

2.1 Theatspaease . . . 5

2.2 Summaryoftheonstrution . . . 6

3 Basi notions 8 3.1 ThesuperWess-Zumino-Wittenmodel . . . 8

3.2 WilsonloopsandKondorenormalizationgroupows . . . 14

4 Boundary states in the sWZWmodel 16 4.1 MaximallysymmetriD-branes . . . 17

4.2 TwistedD-branes . . . 20

4.3 CosetD-branes . . . 21

4.4 TwistedosetD-branes. . . 23

5 SupersymmetriKondo perturbations 24 6 Test states : the ohomologyof thesuperharge 26 6.1 Test states. . . 27

6.2 Theohomologyofthesuperharge. . . 28

6.3 Thegeneriteststates . . . 30

6.4 Gluingonditionsandshiftsonthegroup . . . 31

6.5 Ation ofquantizedWilsonloopoperators . . . 32

7 The harges ofsWZW boundarystates 32 7.1 Thegeneralpresription . . . 33

7.2 Solvingthegluingonditions . . . 33

7.3 Averaging . . . 36

7.4 Normalizationandperiodiity . . . 38

7.5 InvarianeunderthegeneralizedKondoows . . . 39

7.6 MaximallysymmetriD-branes . . . 40

7.7 CosetD-branes . . . 41

7.8 TwistedD-branes . . . 42

7.9 TwistedosetD-branes. . . 43

7.10 Someremarks . . . 43

8 Examples 45 8.1

SU (2)

^{. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46}

8.2

SU (3)

^{. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46}

8.3

SU (4)

^{. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46}

9 Relationto homology 50 9.1 TheKostantonjeture . . . 50

9.2 Baktotheexamples. . . 52

10 Disussionand onlusion 54

IthasbeenknownforadeadethatRamond-RamondhargesofD-branes,invariantsofbound-

aryrenormalizationgroupowsandK-theoryareintimatelylinked. Therststeadyindiation

of the linkbetween Ramond-Ramond harges [1℄ and K-theoryame from a detailed analysis

of the oupling of the gauge and salar elds supported on the D-brane worldvolume to the

Ramond-Ramond elds living inthe bulk [2℄. Along separate line of development, it beame

learthatSen'sonjetures[3 ℄impliedthatthesetofD-braneongurationsmodulotheation

oftheboundaryrenormalizationgroupowsisgivenbyanappropriateversionoftheK-theory

of thetarget spae[4 ,5℄. (See[6,7 ,8℄for aomplete overviewof K-theoryapplied to strings.)

The boundaryrenormalization group ows aretriggered byboundaryperturbations ofthe

onformaleldtheorydesribingtheopenstringmodeslivingontheD-branesonsidered. They

generially map aD-brane ongurationonto another (possiblyempty)D-brane onguration

plus losed string radiation. As losed strings do not arry any harge with respet to the

Ramond-Ramond gauge elds, it is natural to expet that the Ramond-Ramond harges of

the two D-brane ongurations oinide. This property has been heked in [9 ℄ for the at

spae ase, in [10℄ for the ase of orbifolds of toroidal ompatiations, and in [11,12℄ for

Kazama-Suzuki osetmodels. Ramond-Ramondhargesthereforeprovidenatural invariantsof

theboundaryrenormalization groupows.

Theaimofthis paperistoonstrutinvariantsofgeneralized Kondoows[13 ,14,15 ,16 ,17℄

insuperWess-Zumino-Witten(sWZW)modelsonaompatsimplyonnetedLiegroup,using

a proedure that is formally a measurement of the Ramond-Ramond harge of the boundary

state by a Ramond-Ramond test state (see for instane [18 ℄, setion 8). We used formally

in the last sentene beause sWZW models do not ontain massless Ramond-Ramond gauge

elds, so the onept of Ramond-Ramond harge is ill-dened in this setting. The harges

onstruted inthispaperarewell-dened asinvariants oftheboundary renormalizationgroup

ows,however.

WZW models areonformally invariant sigmamodels witha Lie group

G

^{astarget spae.}

When

G

^{is ompat, the} orresponding onformal eld theory is rational and an be solved exatly. Theset of braneongurations moduloboundary renormalization group ows ison-

jeturedtobeisomorphitothetwistedK-theoryoftheLiegroup

G

^{,wherethetwistisprovided}

bythelassofthe NS-NS3-form

H

^{(thislassisdeterminedbythelevel}

k

^{oftheWZWmodel).}

Thetwisted K-theoryfor

G

^{ompat,simple andsimplyonneted hasbeen omputed in[19℄,}

and isgiven bythe diretprodutof

2 ^{r − 1}

^{opies of}

Z /M Z

:

K H (G) = ( Z /M Z ) ^{(2 r−1 )} ,

^(1.1)

where

r

^{is therankof the group}

G

^and

M

^{isan integer depending}

G

^{and on thelevel}

k

^{. (The}

expliit expression for

M

^{is not very} enlightening and an be found in[19 ℄.) One remarkable feature of these K-groups is that they have torsion (they are diret produts of nite yli

groups), so that the orresponding D-brane harge is onserved only modulo the integer

M

^.

In partiular,a stak of

M

^{identialD-branes an deay into losed stringradiation. Thefat}

thatWZWmodelsarewell-knownCFTs,thattheirtargetspaes admitanon-trivial(andnon-

torsion)

H

^{eldandthat theirK-theorygroups areylimakesthem} interesting exampleson whih onean test theK-theoryonjeture.

Comparison between the Kondo ow ation on WZW D-brane and the K-theory of the

underlying Lie groups have been performed in several papers [20,21,22 ,23,24 ℄. The general

idea underlying these tests of the K-theory onjeture is the following. One assoiates to the

knownD-branesa

Z /M Z

valuedharge,andonsidertheonstraintsimposedbythemaximally symmetriKondoboundaryowsontheseharges. Theseonstraintsanbesolved,andimpose

thatthevalueof

M

^{oinides withtheresultoftheK-theory}omputation. Given anarbitrary D-brane, one an generate byKondo ows other D-branes inthe same

Z /M Z

fator, and the universality [25 ,26 ℄ of the Kondo renormalization group ows explains why the integer

M

^is

thesameineah fatorof (1.1) . However, itisimpossible to testthenumberof

Z /M Z

fators in(1.1 ) withthesetehniques. Itisalsoimpossible toknowwhethertwoboundary statesarry

thesametypeofharge(ie. lieinthesame

Z /M Z

fator)iftheyarenot linked byaboundary renormalization group ow of the Kondo type. For instane, in the two papers [27,28 ℄, the

authorslaimtodisplayboundarystatesarrying harges fromeahofthefatorsof (1.1 ),but

no steady argument supports these onjetures. We will see thatatually both of them prove

wrong.

This situation makes it desirable to have a well-dened presription to assign to a given

boundary state an element of the K-theory group (1.1 ). In this paper, we will desribe a

proedurethatallows toassoiateto aworldsheetsupersymmetri boundarystate ofthesuper

Wess-Zumino-Witten modela quantityinvariant underthegeneralized Kondo renormalization

groupows. WewillhekthattheseinvariantsformagroupisomorphitothetwistedK-theory

of the Liegroup

G

^.

The paper is organized as follows. In the next setion we review the lassiation of at

spae type IIB D-branes by K-theory, and show how this K-theory harge an be probed by

omputingaouplingoftheboundarystatewithaRamond-Ramondteststate. Wethenreview

brieyandroughlyouronstrution. WealsosummarizethemainresultsabouttheKondoow

invariants that we will build. In setion 3, we review the super Wess-Zumino-Witten model,

aswell astherelation between quantum Wilsonoperatorsand theKondo ows inthebosoni

WZW model.

Insetion4,weonstrutalargelassofworldsheetsupersymmetriboundarystatesforthe

sWZWmodel,usingwell-knownonstrutionsavailableforthebosoniWZWmodel. Insetion

5,weshowhowWilsonoperatorsanbeusedtodesribethesupersymmetriKondoowsofthe

sWZWmodel. Insetion6 ,weompute theohomology of theworldsheet superharge, whih

determines the spae of Ramond-Ramond test states. We also ompute theation of Wilson

operatorson these test states. Setion 7 isdevoted to theonstrution ofthe harges. At the

end of this setion we ompute in full generality the harges of the simplest boundary states

onstrutedinsetion4(namelythemaximallysymmetri,osetandtwistedboundarystates).

Severalspeiexamplesofmoreelaborateboundarystatesin

SU (4)

^{aretreatedinsetion8,as}

anillustrationofourformalism. Insetion9,weusetheso-alledKostantonjeturetoonnet

our proedurewiththe familiarpiture ofthe lassiation ofbraneharges by homology. We

hekthatintheexamplesonsideredinsetion8,thehargesthatwe foundoinidewiththe

geometri intuition. We end withsomeonluding remarksinsetion10.

The aim of this setion is to onvey a general feeling of the ideas that will be relevant to the

understanding ofthe onstrution elaborated inthesetions6 and 7.

2.1 The at spae ase

First, it is instrutive to have a look at the at spae ase. A more detailed aount of the

K-theorylassiation of D-branesinat spae an be foundin[6℄.

So letus onsider typeIIB stringtheoryon

R ¹⁰

,andreall thatthetype IIB D-branesare loated along even dimensional hyperplanes (in spaetime). Aording to [4 ℄, any D-brane in

this bakground an be onstruted from a tahyon eld onguration on a stak of an equal

number (say

n

^{) of D9 and anti-D9 branes. This stak arries a} Chan-Patton bundle with struturegroup

U (n) × U (n)

^{,and thetahyoneld}

T

^{isaomplexLorentzsalar}transforming in the bifundamental representation of

U (n) × U (n)

^{. (It is the eld} orresponding to open strings strethed between a braneand an anti-brane.) Inthis way,a D

p

^{-branesupported on a}

p +1

-dimensional hyperplane of

R ¹⁰

is identied witha solitoni onguration of

T

^vanishing

on thishyperplane, andonstant inthediretionsparallel to it. Suha tahyon prole an be

built out of the Atiyah-Bott-Shapiro onstrution [29 ℄, and these tahyoni ongurations are

lassied bythe K-theorywithompat supportin the diretionsnormal to the D

p

^{-brane [6℄.}

After ompatiation of

R ⁹ − p

by the addition of its sphere at innity, ompatly supported

K-theory is isomorphi to the relative K-theory

K(B ^{9 − p} , S ^{8 − p} ) = Z

(

p

^{odd), where}

B ^m

^and

S ^m

^denotethe

m

-dimensional ballandsphere. ThisK-theorylassiestheharges ofallofthe brane ongurations whih aretrivial along a presribed

p +1

-dimensional hyperplane (inthe sensethatanytwosetionsofsuhongurationsorthogonaltothehyperplanearehomotopi).

We onsiderhere onlybranes belongingto thisfamily.

Aordingtothe philosophyexposedin[30 ℄,anyobjetthatanbeonstrutedfromtarget

spae onepts should have a ounterpart on the worldsheet, in the onformal eld theory

formalism. SoletusseewhatistheanalogueoftheK-group

K (B ^{9 − p} , S ^{8 − p} ) = Z

fromtheCFT point of view. Closedtype IIB string theoryin

R ¹⁰

is desribed bythe onformal eldtheory (CFT)onsistingoftenfreebosonsandfermionstensoredwithaghostCFT,andsubjettothe

appropriateGSOprojetion. IntheRamond-Ramondsetor,zeromodes

ψ ^µ ₀

^{oftheholomorphi}

andantiholomorphifermionsformtwoantiommutingopies oftheCliordalgebraCl

( R ⁹⁺¹ )

^.

Theysatisfy:

{ ψ ^µ ₀ , ψ ^ν ₀ } = η _µν , { ψ ¯ ₀ ^µ , ψ ¯ ^ν ₀ } = η _µν , { ψ ^µ ₀ , ψ ¯ ^ν ₀ } = 0 ,

where

η _µν

^{isthe standardMinkowski metri. TheGSO projetion keepsonlyoperatorsformed}

by an even number of fermioni zero modes. The massless Ramond-Ramond setor of this

theory (at zero momentum) is given by the even part of a (

Z /2 Z

graded) tensor produt of two Cliord modules. The following representation of this vetor spae is useful. Pik an

orthonormal basis

{ e ^µ }

^{suh that the rst}

p + 1

^{vetors are tangent to the} worldvolume of

p + 1

dimensional hyperplane onsidered above, dene

ψ ^µ _± = √ ¹

2 (ψ ₀ ^µ ± i ψ ¯ ^µ ₀ )

^{,and let}

| 1 i

^{be the}

state ofunitnormsuh that

ψ ₊ ^µ | 1 i = 0

^{for all}

µ

^{. Thenthemassless} Ramond-Ramondsetoris generated bythe states:

| e ^{µ 1} ∧ ... ∧ e ^{µ p+1} i := ψ ^µ ₋ ¹ ... ψ ₋ ^{µ p+1} | 1 i

withodd

p

^{. Wewillallthesestatesteststates,forthefollowingreason. Uptoa}normalization fator,the omponent on the Ramond-Ramond ground statesof theD-branes inour family is

given(inthelightonegauge)by

| e ² ∧ ... ∧ e ^p+1 i + | e ^p+2 ∧ ... ∧ e ⁹ i

^{. The}Ramond-Ramondharge ofagiven D-braneanbeprobedbyomputingtheoverlap

h e ² ∧ ... ∧ e ^p+1 |

^D

p i

^{(seeforinstane}

setion 8 of [18 ℄ for more details). Now it is well-known that boundary states form a lattie

(ie. one an stak only an integer number of D-branes), so the overlap with the test states

are quantized for all of the boundary states. After a suitable normalization, the linear form

h e ² ∧ ... ∧ e ^p+1 |

^{anbeseenasa mapfromthesetof boundary statesinto}

K (B ^{9 − p} , S ^{8 − p} ) = Z

, whih assigns to eah D-brane its K-theory harge. [9 ℄ provided good indiations that these

overlaps between boundary states and test states are invariant under renormalization group

ows. Indeed,the D9and anti-D9 ongurationwith a non-trivial tahyon eld onguration

seems to arrya Ramond-Ramond harge idential to the harge of theequivalent D

p

^-brane.

Thereforeontheworldsheet,themapassoiatingtoaD-braneitsK-theoryhargeisrealizedby

takingtheoverlapoftheorrespondingboundarystatewithateststateintheRamond-Ramond

vauum. Notethatthestudyoftheoupling ofD-branestotest statesisompletelyequivalent

to the study oftheir intersetionform, and yields their harges more straightforwardly.

We will only aim at probing the Ramond-Ramond harges of worldsheet supersymmetri

boundary states, that is, boundary states

| B i

^satisfying

D ₋ | B i = 0

^.

D ₋

^{is a worldsheet}

superharge, and we dene

D + = (D ₋ ) ^†

^{. In} non-trivial onformal eld theories, some of the harges obtained by taking the salar produt of supersymmetri boundary state with

Ramond-Ramond test states may be linearly dependent. Indeed, any test state

| RR i

^{of the}

form

| RR i = D + | RR ^′ i

^{hasvanishing salar produt with}

| B i

^{. There may also exist massless}

Ramond-Ramondstateswhiharenotsupersymmetri. Wewillignore suhstatesandrestrit

ourselvestoteststates

| RR i

^{whihlieinthekerneloftheadjointoftheworldsheetsuperharge:}

D + | RR i = 0

^{. Therefore the set of} Ramond-Ramond test states we propose to onsider is provided by a basisof representatives of theohomology of the superharge

D ₊

^{on the set of}

massless Ramond-Ramond states. Note that ohomology appears here exatly for the same

reasonasinthe BRSTformalism: wearerestritingourselvesto stateslyinginthekernel ofa

nilpotent operator. In theat spaease,

D ₊

^{vanisheson theset of}Ramond-Ramond ground states, whih thereforeoinides withtheohomology. However theworldsheet superharge of

thesuper Wess-Zumino-Witten model doeshave a non-trivial ohomology, and itis neessary

totakethisfatintoaountto obtainagreementwiththepreditionofK-theoryontheharge

group.

2.2 Summary of the onstrution

When trying to apply the test-state proedure reviewed above to the super Wess-Zumino-

Witten model, onefaes at leastthreemajor diulties:

•

^{The physial state spae ofsuperWZW modelsdoesnot ontain any massless Ramond-}

Ramond state [31 ℄. This shows that the very onept of Ramond-Ramond harge is

ill-dened, as there are no massless Ramond-Ramond gauge elds in the target spae

theory. (It is the reason why one should think of theharges to be found below only as

invariants ofthe boundaryrenormalization group ow.)

•

^{When one triesto use the(massive)} Ramond-Ramondground statesto probe D-branes, the harge obtained is not quantized, and moreover it is modied by the ation of the

Kondo ows [32 ℄.

•

^{Finally, to math the K-theory predition (1.1 ), the harge must be identied modulo}

M

^{. It isunlear howone ouldpossiblygeta} periodially identiedhargefrom asalar produtbetween theboundary state anda test state.

There is however away of solving these threeproblems all at one : we have to look for truly

masslessRamond-RamondteststatesoutsidethephysialstatespaeofthesuperWZWmodel.

Letusexplainthisidea. ReallthatthestatespaeofthesuperWess-Zumino-Wittenmodel

on a simplyonneted Liegroup inthe Ramond-Ramond setoris a diretsum of irreduible

modules oftheform :

V _λ = H _λ ^g ⊗ H ¯ _λ ^g ∗ ⊗ F _R ⊗ F _R ,

where

H _λ ^g

^and

H ¯ _λ ^{g ∗}

^{areintegrablemodulesfor aKa-Moodyalgebra}

ˆ g _k

^{assoiatedto thegroup}

G

^,and

F _R

^{areFokmodules for}

d =

^dim

G

^{freefermions intheRamondsetor. The ondition}

thattheRamond-Ramond groundstatesbe masslessreads (seesetion 6.2 ):

(λ, λ + 2ρ) + h ^∨ d 12 = 0 ,

where

h ^∨

^{is the dualCoxeternumber of}

G

^and

ρ

^{its Weyl vetor (halfthesum of thepositive}

roots). Asany integrable module for

ˆ g _k

^satises

(λ, ρ) ≥ 0

^{, thephysial setorof the sWZW}

model does not ontain any massless Ramond-Ramond state. However, the ansatz

λ = − ρ

satises the equation and is the unique solution (by Freudenthal-de Vries strange formula).

Whilethenon-integrablemodule

V _{− ρ}

^{doesnotappearinthephysialstate spaeofthetheory,}

it doesarry a representation of thespetrumgenerating algebra of the sWZWmodel, soone

an inpartiular study the ohomology of the superharge. It turns out thatthis ohomology

hasdimension

2 ^r

^,where

r

^istherankof

G

^{,andissupportedonahighestgradesubspaeof}

V _{− ρ}

^,

for some appropriategrading. This spaewill be our spaeof Ramond-Ramondtest states.

Topursuethe proedure skethed aboveinthe aseofatspae, onewouldliketo measure

thehargeofagivenboundarystatebytakingitssalarprodutwithagivenRamond-Ramond

teststate. Thisisnotreadilypossiblebeausetheboundarystatedoesnothaveanyomponent

along

V _{− ρ}

^{. One should thereforeomplete theboundary state in thevirtual setor}

V _{− ρ}

^{. This}

ompletion an be performed in a onsistent way as follows. Reall that boundary states are

linear ombinations of Ishibashi states, whih are themselves a set of linearly independent

solutions to the gluing onditions imposed on the boundary state. These gluing onditions

speify how the bulk elds of the theory are reeted at the boundary, and they an also

be solved in

V ₋ ρ

^{1. While there may be several linearly} independent solutions to the gluing onditionsin

V _{− ρ}

^{,onlyoneofthemintersetsthe}representativesoftheohomology,anddoesso alongaone-dimensionalsubspae,sothegluingonditionsseletanelementoftheohomology

up tonormalization.

1

Tobepreise,wewillhavetosolvetheminabundlewhihbersareomposedofhighestweightmodules

V _−ρ

with twisted ationof the hiralalgebra of the model. This is neessaryto preserve the global

G × G

symmetriesofthemodel,seesetion6.3 .

Wilson operators enodethereetion oeientsof theboundary state (theprefators ofthe

Ishibashi states). But they are also normal-ordered series in the Ka-Moody urrent, whih

haveawell-dened ationonanyhighestweightmodule. Thisimportant propertyanbeused

to perform an appropriate ontinuation of the omponents of the boundary states from their

valueinthephysial statespae to

V _{− ρ}

^.

The harge ofa boundary state arethenmeasured bytakingthesalarprodutof arepre-

sentative of the ohomology withthe ompleted boundary state.

Wewill show that:

•

^{Theresulting harges arequantized (that is, integer up to} normalization). (Setion 7.4)

•

^{The ation of Wilson operators naturally imposes that these harges are periodi, with}

theright periodiity

M

^{,sothey an atually be taken to lie in}

Z /M Z

. (Setion7.4 )

•

^{Whenever a} generalized Kondo ow sends a brane onguration onto another one, the hargesofthetwo ongurationsareequal. Thesehargesarethereforetheinvariantswe

arelooking for. (Setion7.5 )

•

^{Half of the linearly} independent test states annot ouple to any boundary state, so the number of independent harges is

2 ^{r − 1}

^{. This yields a harge group of the form :}

( Z /M Z ) ^{(2 r− 1 )}

^{,whih oinides withthetwisted K-theorygroup (1.1 ) of}

G

^{. (Setion9)}

•

^{There isa}distinguished basisoftherepresentative of theohomologyof thesuperharge that an be naturally identied with the generators of the homology of the Lie group.

Thisprovidesthelinkbetweenour algebraipitureoftheharges andthemorefamiliar

geometri piture, in term of homology lasses. We will hek in several examples that

thealgebrai harge oinides withthegeometri one. (Setion9)

Wewill nowmake thesestatementspreise.

3 Basi notions

3.1 The super Wess-Zumino-Witten model

We start by reviewingthe superWess-Zumino-Witten(sWZW) model [33,34,35 ,36,37,31 ℄.

The hiral algebra

WedesribeherethehiralalgebraofthesWZWmodel. Tosimplifythenotationsinthispaper,

we will notdistinguish typographiallythevarious innitedimensionalLie algebrasfromtheir

respetive vertexalgebras.

Let

G

^{beaompat,simpleandsimplyonnetedLiegroupofdimension}

d

^{. Let}

g =

^Lie

(G)

be its Lie algebra, and

ˆ g _{˜ k}

^the orresponding Ka-Moodyalgebra at level

˜ k = k + h ^∨

^,

k > 0

^,

where

h ^∨

^{is the dual Coxeter number of}

g

^{. Let us hoose an} orthonormal basis

{ e ^a } ^d a=1

^of

g

withrespetto the Killing form, let

{ J ^a (z) }

^{be the omponents oftheKa-Moodyurrent on}

this basis, and

{ J _n ^a }

⁽

n ∈ Z

) their Laurent modes. All the sums on the Lie algebra indies

a, b, c, ...

^{willbeimpliit.}

Let

ˆ f

^betheLiesuperalgebrageneratedby

d

^freefermions

{ ψ ^a (z) }

^withantiperiodi(Neveu- Shwarz) or periodi (Ramond) boundary onditions, and

ψ ^a _r

^{their Laurent modes. Here}

r ∈ Z + ¹ ₂

^{in the} Neveu-Shwarz setor, and

r ∈ Z

in the Ramond setor. We will adopt this onvention throughout therest of this paper. We will sometimes see the

d

^{free fermions}

{ ψ ^a }

as a single

g

^{-valued fermioni eld}

ψ

^{. The hiral algebra}

ˆ c

^{of thelevel}

˜ k

^superWess-Zumino- Witten(sWZW)modelisgivenbythe semidiretprodut

ˆ c = ˆ g ˜ k ⋉ˆ f

^{,wheretheationof}

g ˆ ˜ k

^on

ˆ f

isgiven by(3.1 ) below. Expliitly,thegenerators satisfythefollowing ommutation relations :

[J _n ^a , J _m ^b ] = f _abc J _m+n ^c + ˜ knδ _ab δ _n+m,0 , { ψ ^a _r , ψ ^b _s } = δ _ab δ _r+s,0 ,

[J _n ^a , ψ _r ^b ] = f _abc ψ ^c _n+r ,

^(3.1)

where

f _abc

^{arethe struture onstants of}

g

^.

Subalgebras

Thehiralalgebra

ˆ c

^{denedaboveontainsseveralimportant}subalgebras(inthesenseofvertex algebras).

First,

ˆ f

^{ontainsasubalgebraisomorphito}

ˆ g _h ^∨

^{,the Ka-Moodyalgebra basedon}

g

^atlevel

h ^∨

^{. The generators forthis subalgebra are:}

(J _ψ ) ^a _n = − 1

2 f _abc X

r

ψ _r ^b ψ ^c _{n − r} .

^(3.2)

Onean thendene the bosoni urrent

J

:

J = J − J _ψ .

^(3.3)

ItgeneratesaKa-Moodysubalgebra

ˆ g _k

^of

ˆ c

^{whihhastheruialpropertyofommutingwith}

ˆ f

^:

[ J ^a _n , ψ ^b _r ] = 0 .

Thisshows that

ˆ c

^{isinfatisomorphi to}

ˆ g _k ⊕ ˆ f

^.

ˆ c

^{also ontains aopyof the}

N = 1

superonformal algebra, withgeneratorsgiven by:

L _n = 1 2˜ k

X

m

: J ^a _m J ^a _n

− m : + 1 2

X

r

r : ψ ^a _{n − r} ψ ^a _r :

+ d 16 δ _n,0

,

^(3.4)

G _r = − 1 p ˜ k

X

m

J ^a _m ψ ^a _{r − m} − 1

6 f _abc X

s,t

ψ ^a _s ψ ^b _t ψ ^c _{r − s − t}

!

,

^(3.5)

where theindies

r

^,

s

^and

t

^{are summed over}

Z

or

Z + ¹ ₂

^{inthe Ramond and} Neveu-Shwarz setor,and

m

^and

n

^{arealwayssummedover}

Z

. Thetermbetween parenthesisinthedenition

of

L 0

^{ispresent only inthe Ramondsetor. Theysatisfy:}

[L _m , L _n ] = (m − n)L _m+n + c

12 (m ³ − m)δ _m+n,0 , { G r , G s } = 2L r+s + c

12 (4r ² − 1)δ r+s,0 , [L _m , G _r ] = m − 2r

2 G _m+r ,

(3.6)

where

c = _2(k+h ^{3k+h ∨} ∨ ) d

^{isthe entral harge. Theiration onthegenerators of}

ˆ c

^{is givenby:}

[L _m , J _n ^a ] = − nJ _m+n ^a , [L _m , ψ ^a _n ] = − 2n + m

2 ψ ^a _m+n , [G _m , J _n ^a ] = p

˜ knψ _m+n ^a , { G _m , ψ ^a _n } = − 1 p k ˜

J _m+n ^a .

Finally,we notethat thebosoni part of

ˆ f

^{isisomorphi to theKa-Moody algebra}

so(d) ˆ ₁

at levelone,withgenerators :

J _so ⁱ (z) = 1

2 t ⁱ _ab ψ ^a (z)ψ ^b (z) , i = 1, ..., 1

2 (d ² − d) ,

^(3.7)

where

t ⁱ _ab

^{arethe matrixelementsofthegenerators}

{ t ⁱ }

^of

so(d)

^inthedeningrepresentation.

Thebosoni partof thehiral algebra is thereforegivenby

ˆ g _k ⊕ so(d) ˆ ₁

^.

The full spetrum generating algebra is the diret sum

ˆ c ⊕ ˆ c

^{of a holomorphi and anti-}

holomorphi opy of

ˆ c

^{. Here we understand the diret sum in a}

Z /2 Z

-graded sense, so that holomorphiandantiholomorphielementshavingbothoddfermionnumberantiommute. The

elds intheholomorphi setorwill be denotedasabove, andtheones inthe antiholomorphi

setorwill arryabar (

J ¯

^,

ψ ¯

^,...).

The state spae

We start by desribinghighest weight modules for

ˆ g _k

^and

ˆ f

^.

The highest weight modules for

ˆ g _k

^{that will be relevant to the onstrution of the state}

spae ofthe sWZWmodel arethe integrablemodules. They areindexedbythe (nite)set of

integrable dominant weightsat level

k

^,

P _k ⁺ ⊂ h ^∗

^where

h

^{is theCartansubalgebraof}

g

^{. Given}

anintegrableweight

λ ∈ P _k ⁺

^{,wewilldenotethe}orrespondingintegrablehighestweightmodule by

H _λ ^g

^{, with a supersript indiating the} orresponding Ka-Moody algebra in the situations whenanambiguitymightour. Integrablehighestweightmodulesarryahermitian invariant

bilinear form. The elements of the ompat form of the Ka-Moody algebra are anti-self-

adjoint withrespetto this form, and the adjoints of the Ka-Moody generators are given by

(J _n ^a ) ^† = − J ₋ ^a _n

^.

There are only two irreduible highest weight modules for

ˆ f

^{, one in the} Neveu-Shwarz setor, andone inthe Ramondsetor, and we denotesthem respetively by

F _{N S}

^and

F _R

^.

All of these modules inherit a grading gr

L 0

^{from the adjoint ation of}

L 0

^{and their om-}

ponents with negative grade are trivial. Note that the grade zero subspae

(F _{N S} ) ₀

^of

F _{N S}

^is

onedimensional, whereas

(F R ) 0

^{isanirreduibleCliordmoduleforthe}

d

-dimensional Cliord algebra Cl

(d)

^{generated bythezeromodes}

ψ ₀

^{of thefermions intheRamondsetor.}

WewantnowtoonstrutthestatespaeofthesWZWmodel. Thisstatespaeanbexed

by the requirement that thetorus partition funtion be modular invariant when themapping

lass group

SL(2, Z )

^{of the torus ats. Thismodular invariane ondition fores us to impose}

a GSOprojetion thatbreaksthe spetrumgenerating algebra

ˆ c ⊕ ˆ c

^{to its bosoni part.}

We are working with a ompat simple Lie group of arbitrary dimension, so generially

only the type 0 GSO projetion is available. The type 0 GSO projetor is given by

P _GSO =

1

2 (1 + ( − 1) ^{( F + ¯ F )} )

^{, where}

F

^and

F ¯

^{denote the fermion numbers in the holomorphi and anti-}

holomorphi setors,ie. the

Z /2 Z

gradingsoming fromthesuperalgebrastruturesoftheleft andright fatorof

ˆ c ⊕ ˆ c

^{. Theoperatorsoftotal gradezeroaretheonly oneswhih survivethe}

projetion, sothe remaining hiral algebra isgiven by

ˆ g _k ⊕ so(d) ˆ ₁ ⊕ ˆ g _k ⊕ so(d) ˆ ₁

^{. Notethatthe}

superonformal generators

G

^and

G ¯

^{areprojeted out.}

It turns out that the problem of onstruting a partition funtion for

P _GSO (ˆ c ⊕ ˆ c)

^whih

hastherequiredmodular invariane properties boilsdown toonstruting a modularinvariant

partition funtion for the bosoni WZW model based on the hiral algebra

ˆ g _k ⊕ so(d) ˆ ₁

^{. We}

will onstrutthese partitionfuntions, and thenommenton whytheunderlying moduleis a

modulefor

P _GSO (ˆ c ⊕ ˆ c)

^.

The state spae of the

ˆ g _k ⊕ so(d) ˆ ₁

^{WZW model fatorizes into}

ˆ g _k

^{-modules and}

so(d) ˆ ₁

^-

modules 2

:

H = H g ⊗ H so ^X ,

^(3.8)

where

X =

^{0A,0Bor 0indexesdierent possiblehoies,seebelow. Wewillhoosetheharge}

onjugation modular invariant for the

ˆ g _k

^theory:

H g = M

λ ∈ P _k ⁺

H _λ ⊗ H _λ ^∗ ,

where

λ ^∗

^{is the weight onjugate to}

λ

^{. In the}

k → ∞

^{limit, the WZW model with harge}

onjugation modularinvariant desribesa stringevolvinginthesimplyonneted Liegroup

G

onstruted from

g

^{. The extension of our results to models with dierent modular invariants}

maybenon trivial.

so(d) ˆ ₁

^{hasfour integrable}representationswhen

d

^{iseven,namelytheones}orrespondingto thetrivial(

t

^{),thedening(or}fundamental)(

f

^{),andthetwospinorial(}

s

^and

s ^′

⁾representations of

so(d)

^{. Forodd}

d

^{thetrivialanddening}representationsarestillpresent,butthereisasingle spinorial representation, thatwe denoteby

s

^{. Ineven}

d

^,

so(d)

^{admitsan outer}automorphism, sowe an onstruta modular partition funtion using either this outer automorphism or the

trivial one. The two modular invariants obtained orrespond respetively the type 0A and

0B GSO projetions of string theories. In odd dimension, there is a unique type 0 modular

invariant. Thestate spaefor the

so(d) ˆ ₁

^{parttherefore reads:}

H ⁰ so = (H t ⊗ H t ) ⊕ (H _f ⊗ H _f ) ⊕ (H s ⊗ H s ) , d

^odd,

H ^0A so = (H t ⊗ H t ) ⊕ (H _f ⊗ H _f ) ⊕ (H _s ^′ ⊗ H s ) ⊕ (H s ⊗ H _s ^′ ) , d

^even,

H so ^0B = (H t ⊗ H t ) ⊕ (H _f ⊗ H _f ) ⊕ (H s ⊗ H s ) ⊕ (H _s ^′ ⊗ H _s ^′ ) , d

^even.

(3.9)

2

Thisisnotneessarilytheasefornonsimplyonnetedgroups,see[38 ,39 ℄.

Bystandard CFTarguments(forinstane[40℄,hapter17),thestatespaes

H g

^,

H ⁰ so

^,

H ^0A so

^,

H ^0B so

^{all yield modular invariant partitions funtions. Therefore}

H

^{as dened in (3.8 ) yields a}

modular invariant partition funtionineah ase.

To see that these state spaes are really modules for our spetrum generating algebra

P _GSO (ˆ c ⊕ ˆ c)

^{,notehowthehighestweight modulesfor}

ˆ f

^deomposeinto

so(d) ˆ ₁

^{modules :}

F _{N S} → H _t ^so ⊕ H _f ^so ,

F _R → H _s ^so ⊕ H _s ^{so ′} (d

^even),

F _R → H _s ^so (d

^odd

).

In the deomposition of

F _{N S}

^,

H _t ^so

^{appears at grade zero, while}

H _f ^so

^{appears at grade}

¹ ₂

^{. The}

two spinorial

so(d) ˆ ₁

^{-modules are already present at grade zero inthe} deomposition of

F _R

ⁱⁿ

even dimension. Onewayofhekingtheserelations istoompute thedimensionsofthegrade

0and

1

2

^{subspaesofthe relevant modules,and thenreallthatanyprodutofanevennumber}

offermioni generatorsan beexpressedintermofthe urrentsof

so(d) ˆ ₁

^{. Undertheoperator-}

statemappingofthe vertexalgebra

ˆ f

^,

H _t ^so

^{orrespondsto operatorswithevenfermionnumber,}

while

H _f ^so

^{orresponds to operators with odd fermion number. For even}

d

^{, one hasthe same}

pitureinthe Ramondsetor, where thetwo

so(d) ˆ ₁

^{spinorial modulesorrespondto even and}

odd fermionnumber operators (whih iswhih depends on how we hoose thegrading on the

gradezero subspae). For odd

d

^{, however,}

F _R

^{doesnot arry any}

Z /2 Z

grading. Wehave the same pitureinthe antiholomorphi setor.

Nowbytheremarkaboveand(3.9),weseethatthepostulatedstatespaes(3.8 )aremodules

for

P GSO (ˆ c ⊕ ˆ c)

^{. Operators with}

F = ¯ F = 0

^{preserve the}

so(d) ˆ 1

^{modules, while those with}

F = ¯ F = 1

^{permute the summands in(3.9 ).}

Supersymmetri states

It will be ruial for us to have a notion of supersymmetri state. The problem is that the

superonformalgenerators

G

^and

G ¯

^{dened in(3.5 )donotatonthestatespaeofthesWZW}

model. Theyhave oddfermion number andareprojetedout bytheGSO projetion.

However, we will be able to dene supersymmetri states if we onstrut a

Z /2 Z

-graded module

H ^′

^for

ˆ c ⊕ ˆ c

^{suhthatthestatespaeof thesWZWmodeloinides withtheeven part}

of

H ^′

^:

H = ( H ^′ ) ₀

^{. Denote by}

i _H

^{this embedding. Then the} supersymmetry generators map

( H ^′ ) 0

^to

( H ^′ ) 1

^{,and we an dene astate}

| X i ∈ H

^{to be}supersymmetri if:

(G r − iǫ G ¯ ₋ r )i _H ( | X i ) = 0

ⁱⁿ

H ^′ . ǫ = ± 1

^{depends onthe superharge hosento be preserved.}

The module

H ^′

^isstraightforward to onstrutwhen

d

^{iseven. One antake inthisase :}

H ^′ = H g ⊗ (F _{N S} ⊗ F _{N S} ⊕ F _R ⊗ F _R ) ,

^(3.10)

whereitisunderstoodthatinthefermionimodulesoftheform

F ⊗ F

^{,theholomorphimodes}

ψ ^a _n

^{aton therst fatorthrough theation of}

ˆ f

^{,and the}anti-holomorphimodes atson the rst fator by

( − 1) ^F

^{and on theseond through the usual ation of}

ˆ f

^{. The} non-trivial ation ofthe anti-holomorphimodesontherstfator isneessarytomakethem antiommutewith

the holomorphi modes, rather than ommute. Depending on the hoie of

Z /2 Z

-grading on thetwo opies of

F _R

^{we getanextension of thestate spaeof the0Aor 0BsWZW model.}

H ^′

^{is less easy to onstrut when}

d

^{is odd. By what was said above, we see readily that}

a onstrution analogous to the one inthe even ase isimpossible, as

F R

^{does not admit any}

Z /2 Z

-grading in the odd ase. However, as long as we do not impose the reality ondition

z ^∗ = ¯ z

^{on the worldsheet} oordinates, the holomorphi and antiholomorphi modes of the fermionsformaLiealgebra isomorphitothe Liealgebraofthemodesof

2d

^{hiral fermions. A}

Ramondmodule

F _R ^2d

^for

2d

^{hiralfermionsadmitsa}

Z /2 Z

-grading,namelythefermionnumber.

Therefore,the following isa modulefor

ˆ c ⊕ ˆ c

^:

H ^′ = H g ⊗ (F N S ⊗ F N S ⊕ F _R ^2d ) .

Theationof

ˆ c ⊕ ˆ c

^{doesnotsplitintoaholomorphiand}antiholomorphimoduleintheRamond setor, aswasthe ase intheeven

d

^{ase. But aftertheGSO projetion,theevenpart of}

F _R ^2d

beomesaspinorialmodule

H _s ^2d

^for

so(2d) ˆ ₁

^{,whihisisomorphito}

H _s ⊗ H _s

^asa

so(d) ˆ ₁ ⊕ so(d) ˆ ₁

^-

module (as an be seen by a omputation of the dimensions of the grade zero subspaes, for

instane). Sotheholomorphi/antiholomorphi splittingisreoveredaftertheGSOprojetion.

Atually, the same onstrution an be applied in the even ase, and is equivalent to the

one we usedbeause for

d

^even,

F _R ^2d ≃ F _R ⊗ F _R

^as

ˆ f ⊕ ˆ f

^-modules.

Wehave nowawell-denednotionofasupersymmetristatein

H

^{. Intheremaining ofthis}

paper, we will omit to write expliitly themap

i _H

^{to avoidluttering the notation too muh.}

Butit shouldbeunderstoodeah time an odd operator ats ona state of

H

^.

Wenowdesribeausefulparametrization ofthegradezerosubspae of

H ^′

^{generatedbythe}

zero modes of the holomorphi and antiholomorphi fermions in theRamond-Ramond setor.

Thesezeromodessatisfythefollowing relations :

{ ψ ^a ₀ , ψ ^b ₀ } = δ _ab , { ψ ¯ ₀ ^a , ψ ¯ ₀ ^b } = δ _ab , { ψ ₀ ^a , ψ ¯ ^b ₀ } = 0 .

Fromthedisussionabove,they generate aCliordmodule

F ₀ ^2d

^{for theCliord algebraCl}

(2d)

indimension

2d

^{. Wean make thefollowing hange of basis:}

ψ ₀₊ ^a = 1

√ 2 (ψ ₀ ^a + i ψ ¯ ^a ₀ ) , ψ ₀ ^a ₋ = 1

√ 2 (ψ ₀ ^a − i ψ ¯ ^a ₀ ) .

^(3.11)

Thenew generators satisfy:

{ ψ ₀₊ ^a , ψ ₀₊ ^b } = 0 , { ψ ^a ₀₊ , ψ ^b _{0 −} } = δ _ab , { ψ ^a _{0 −} , ψ ^b _{0 −} } = 0 .

Dening

| 1 i

^{tobethestatewithunitnormsuhthat}

ψ ^a ₀₊ | 1 i = 0

^forall

a = 1, ..., d

^,theCliord

module

F ₀ ^2d

^{is freely generated from}

| 1 i

^{by the set}

{ ψ ₀ ^a ₋ }

^{. We an therefore} parametrize the vetorsin

F ₀ ^2d

^{by elementsof theexterioralgebra}

V

g

^:

e ^{a 1} ∧ ... ∧ e ^{a p} 7→ | e ^{a 1} ∧ ... ∧ e ^{a p} i := ψ ₀ ^{a 1} ₋ ...ψ ₀ ^a ₋ ^p | 1 i ,

^(3.12)

where we denotedthe produtinthe exterioralgebra by

∧

^.

Finally,letusnotethat,bydenition,thestate

| e ^{a 1} ∧ ... ∧ e ^{a p} i

^{satisesthefollowingrelations:}

(ψ ₀ ^a + iǫ ψ ¯ ₀ ^a ) | e ^{a 1} ∧ ... ∧ e ^{a p} i = 0 ,

^(3.13)

with

ǫ = − 1

^si

a ∈ { a ₁ , ..., a _p }

^and

ǫ = 1

^else.

WedeneheretheKondoperturbationinthebosoniaseandreallhowthexedpointsofthe

indued boundary renormalization group ow an be identied by mean of quantized Wilson

operators.

The Kondo perturbation

Consider apurely bosoni WZW modelwithholomorphi urrent

J ∈ ˆ g _k

^{,dened on asurfae}

(possibly with boundaries)

Σ

^{, with an embedded time-like yle}

C

^{. Let}

A : g → C ⁿ

a

n

^-

dimensionalrepresentationof

g

^,and

A ^a = A(e ^a )

^{. Letus tensorthestatespae}

H g

^oftheWZW

modelwith

C ⁿ

. OneanperturbtheWZWationwiththefollowingterm,atingon

H g ⊗ C ⁿ

:

∆S = l Z

C

dσJ ^a (σ)A ^a ,

^(3.14)

where

l

^{is aoupling, and}

σ

^aparametrization of

C

^{. Onean seethis} perturbation asapoint- like harged defet with worldline

C

^{and spin}

A

^{, whih interats minimally with the urrent}

J

^.

Fromthestringtheorypointofview,the Kondoperturbationhasadierentinterpretation.

Consider an open string ylinder amplitude between two D-branes. We have

Σ = S ¹ × [0, 1]

^,

withworldsheet timerunning along

S ¹

^{. Letus hoose}

C = S ¹ × { 0 }

^{,so thatthe}perturbation is supportedon one of theboundaries of theylinder.

H g ⊗ C ⁿ

an now be interpreted asthe statespaeforopenstringsstrethedbetween astakof

n

^{identialD-branesat}

S ¹ × { 0 }

^anda

givenD-brane at

S ¹ × { 1 }

^{. This}perturbationamounts toturning on aonstanteld

A

^onthe

stakof D-branes[14 ℄. For generi

l

^,this perturbation breaksthesuperonformal symmetryof the model, and one an study the boundary renormalization group ow that it triggers. The

IR xed point of this ow is desribed bythe Aek-Ludwig presription [13℄, and is again a

WZW model, with a new boundary ondition on theboundary initiallyperturbed. When the

perturbedstakofD-braneisomposedof

n

^{maximallysymmetribranes oflabel}

λ ∈ P _k ⁺

^,the

nal D-brane onguration is given by a set of

N _λµ ^ν

^{maximally symmetri branes of label}

ν

^,

where

µ

^{is the highestweight of the}

g

-representation

A

^and

N _λµ ^ν

^{arethe fusion oeients of}

ˆ g _k

^{. Arigorous justiationofthe Aek-Ludwig priniple an befound in[17℄,setion5.}

It ishowever instrutiveto lookat Kondo perturbationsfrom theworldsheet dualtheory.

Quantum Wilson loops

By open-losed string duality, one an onsider the same setting, but now with worldsheet

time running along the non-periodi diretion of the ylinder. This amplitude has now the

interpretationofalosedstringexhangebetweenthebranessittingateahendoftheylinder.

Theyle

C

^{isspaelike, andthe} perturbation an be seenasa defetsupported on

C

^.

Classially,this defetisa Wilson loophaving thefollowing expression:

w(µ, l) =

^Tr

C ⁿ

^P

exp

il Z

C

dσj ^a (σ)A ^a

,

^(3.15)

where

P

^{denotes the} path-orderedexponential,

µ

^{is thehighest weight ofthe}representation

A

of

g

^onC

ⁿ

^and

j ^a (σ)

^{aretheomponentsofthe lassialurrent}

j

^{. Theselassial observables}

are topologial : they depend only on the homotopy lass of

C

^{. When}

l = _k ¹

^,

w(µ, l)

^even

preservesthefull symmetryof theWZW model. Indeed,ithasvanishing Poissonbraketwith

thelassial urrent

j

^.

To understand the Kondo perturbation from the losed string point of view, one needs a

quantized versionof thelassial Wilson loop. This quantization wasperformed in[17 ℄ inthe

ase

l = ¹ _k

^{. Thequantized Wilson loop}

W _µ

orresponding to thelassialWilson loop

w(µ, ¹ _k )

is a normal-ordered series in the quantum Ka-Moody urrent

J

^{. The speial symmetries of}

w(µ, _k ¹ )

^{are preserved by this} quantization proedure, whih means that

W _µ

^{ommutes with}

every element of

ˆ g _k

^{. Heneit ats bysalar} multipliation onanyirreduible

ˆ g _k

^{-module. The}

power of the quantization proedure of [17℄ is that the spetrumof

W _µ

^{is obtained expliitly.}

Let

η

^{beanyweightatlevel}

k > − h ^∨

^,and

M _η

^{isthe Vermamoduleofhighestweight}

η

^{. Then:}

W _µ = χ _µ

− 2πi

k + h ^∨ (η + ρ)

1 on

M _η ,

^(3.16)

where

χ _µ

^{is the}

g

^{-harater of the} representation with highest weight

µ

^{. On the integrable}

highestweight module

H _λ

^{and for integrable}

µ

^{,the eigenvalue an be written as:}

W _µ = S _µλ

S _0λ

^{1 on}

H _λ ,

where

S _µλ

^isthemodular

S

^-matrixof

ˆ g _k

^{. However,thefatthattheationof}

W µ

^iswell-dened

on anyhighestweight moduleat level

k

^{will be ruialto our argument.}

We have now a well-dened expression for the quantized Wilson operator at the speial

oupling value

l = ¹ _k

^{. Thisvalueorrespondsto the(lassial) IRxed point of therenormal-}

izationgroupowequationstartingfromtheUVxedpoint

l = 0

^{. Moreover,as}

W _µ

^ommutes

with

ˆ g _k

^{,it also ommutes with the assoiated Virasoro algebra. Therefore the theorydened}

on the ylinder inwhih

W µ

^{is inserted in all the amplitudes is still a onformal eld theory,}

whih still has a

ˆ g _k ⊕ ˆ g _k

^{symmetry. As was shown in [17 ℄, it is atually the Aek-Ludwig}

xed point of the Kondo ow. Put dierently, when the Wilson loop

W _µ

^{ats on a boundary}

state

| B i

^{,ityieldstheinfraredxedpointoftheRGowtriggeredbythe}orrespondingKondo perturbationon

d _µ | B i

^{. Beausethespetrumof}

W _µ

^{isompletelyexpliit, thisprovidesavery}

simple and eient way of investigating Kondo ows. We will repeatthis argument in detail

belowinsetion5 , inthe ase ofsupersymmetri Kondo perturbations.

Thedisussionaboveisnotrestritedtomaximallysymmetriboundarystates,beausethe

onstrution oftheWilsonoperator wasompletely independentfromtheboundaryonditions

imposedat the ends of theylinder. We an see theKondo owas ating on defetoperators

as

d _µ

¹

7→ W _µ

^{. Thisowon defetoperatorsturnsinto a boundary ow whenwe letthese two}

operators aton a given boundary state. Wilson operatorstherefore provide a generalization

of the Aek-Ludwig presription : they desribes universal Kondo ows starting from any

D-brane. (See [26 ℄for a deeperdisussionof theuniversalproperties ofthese ows.)

Let us add here an important remark. The Wilson operators desribed above form a ring

isomorphitotherepresentation ringof

g

^,beausetheir eigenvalues(3.16 )aregivenbyhara- ters of

g

^{. However, the physial state spae}

H ^g

^{of the sWZW modeldeomposes into a diret}