Conformal ghosts on the sphere

Conformal Ghosts on the Sphere

Von der Fakultät für Mathematik und Physik

der Gottfried Wilhelm Leibniz Universität Hannover

zur Erlangung des Grades einer

Doktorin der Naturwissenshaften

Dr. rer. nat.

genehmigte Dissertation

von

Dipl. Phys. Kirsten Vogeler

geboren am21. August 1978 in Hanau

2010

Koreferent: Prof. Dr. Maro Zagermann

Tag der Promotion: 16.07.2010

Zusammenfassung

IndieserArbeit behandele ih dieVerbindungvonGeometrie undlogarithmish konformen

Feldtheorien. Dabeibetrahte ih zwei vershiedene geometrishe Situationen: inTeil Idas

topologishe A-Modell mit Einbettungsabbildung

x : R × S ¹ → CP ¹

und inTeil IIkonforme, fermionishe Geister aufdemTorus.

Das A-Modelllässt sih ineineForm bringen,inderdasPfadintegraleine

δ

-Distribution aufdemModulraumderInstantonenist. Integriert mandieAbhängigkeitvon

S ¹

^heraus,er-

hältmaneineMorsetheorieaufderuniversellenÜberlagerung

„ L CP ¹

desLoop-Raumes. Deren Niedrigenergie-Zustandsräume lassen sih in Zellen dieser Mannigfaltigkeit störungstheo-

retishbestimmenunddurhDarstellungsräumedesChiralendeRham-Komplexesbeshrei-

ben. Unter der Annahme, dass die Darstellungstheorien des A-Modelles und des Chiralen

de Rham Komplexes übereinstimmen, betrahte ih im Folgenden den Chiralen de Rham-

Komplex. DieZustandsräumesindlokale,induzierteDarstellungenderSymmetrie,diedurh

das Gradientenfeld der Morsefunktion erzeugt wird. Entsprehend einer Hypothese von E.

Frenkel,A.LosevundN.NekrasovführteineVerallgemeinerungdieserlokalenDarstellungen

als Distributionenauf

„ L CP ¹

zu quantenexakten Zuständender Theorie. Aufdiesen Zustän- denmussderHamiltonoperatordurhzusätzliheTermekorrigiertwerden. Ihdiskutieredie

DarstellungstheoriederquantenexaktenZustände undbestimmedieDeformationstermedes

Hamiltonoperators. Ihzeige, dassdiese eine geometrishe Deutungals Kohomologieopera-

toren in einem Komplex global erweiterter lokaler Darstellungsräume haben. Zuletzt zeige

ih,dassdenzusätzlihenTermenimHamiltonoperatorderMorsetheorieeinelogarithmishe

Erweiterung deshiralende Rham-Komplexesentspriht.

Diekonformen,fermionishenGeisterausTeilIItransformierensihinirreduziblenDarstel-

lungen der Monodromiegruppe

Z ₂

. Ih zeige, dass die durh sie beshriebene konforme Feldtheorie logarithmish erweitert werden muss, sobald man zu den Darstellungen der

Monodromiegruppe Felder assoziiert, die sih frei auf dem Parameterraum

CP ¹ \ {0, 1, ∞ }

bewegen. DasTripletmodell stellt eineminimale logarithmishe Erweiterung dieserTheorie

dar undbildetdieGrundlagemeinesletztenKapitels. Darin drükeih diespektraleKurve

der

SU (2)

-Seiberg-Witten Theorie durh dieCharaktere desTripletmodelles aus, und führe ebenfallsdasPräpotential aufdiesesModellzurük,indem ihes alsFunktiondesModulus

derspektralen Kurve gewinne.

Schlagworte:

Nihtlineares Sigma Modell, Logarithmish Konforme Geister, Seiberg Wit- tenTheorie

Abstract

This thesis is about the relation of geometry and logarithmi onformal eld theories. I

onsidertwodierentgeometri settings: inpartIthe topologialA-modelwithembedding

x : R × S ¹ → CP ¹

,and inpartIIonformal, fermioni ghosts onthetorus.

The A-model an be transformed suh that the path integral yields a

δ

distribution on the moduli spae of instantons. Integrating out the dependeny on

S ¹

^{, one obtains Morse}

theoryon theuniversalover

„ L CP ¹

of loop spae. Itslow-energy statespae an be derived perturbatively in ells of this manifold, and an be modelled by therepresentations of the

hiral de Rham omplex. Assuming that the representation theory of the A-model and

the hiral de Rham omplex are idential, I onsider the hiral de Rham omplex in the

following. Thestatespaes areloal,induedrepresentationsofthesymmetrygeneratedby

thegradient vetor eld ofthe Morsefuntion. Aording to a onjetureof E. Frenkel, A.

Losev and N. Nekrasov, a generalization of these loal representations as distributions on

„ L CP ¹

leads to nonperturbative statesof thetheory. On thesestates, theHamiltonian must beorretedbyadditionalterms. Idisusstherepresentation theoryofthenonperturbative

states and determine the terms whih deform the Hamiltonian. They have a geometri

signiane asohomology operators inaomplex ofglobally extendedloalrepresentation

spaes. Eventually, I prove that a logarithmi extension of the hiral de Rham omplex

orrespondsthe additional termsintheHamiltonian.

The onformal, fermioni ghosts ofpart IItransform inirreduible representationsof the

monodromygroup

Z ₂

. Ishowthat theonformaleld theoryofthese elds hasto beloga- rithmially extended assoon asthe representations ofthe monodromygoup are allowed to

move freely on the parameter spae

CP ¹ \ {0, 1, ∞ }

^{of the torus. The triplet model onsti-}

tutesaminimallogarithmiextensionofthistheoryandisfundamentalformylasthapter.

ThereinIobtainthe spetralurveof

SU (2)

Seiberg-Wittentheoryintermsof haratersof thetripletmodel. Further,Itrae baktheprepotential to thatmodelbyexpressing itasa

funtion ofthe torus modulus ofthespetralurve.

Key words:

^{NonlinearSigmaModel,LogarithmiConformalGhosts,SeibergWittenTheory}

Contents

1 Introduction 1

I Supersymmetric Ghosts with Values on the Sphere 5

2 Morse Theory 7

2.1 ThePath IntegralPoint ofView . . .

7

2.1.1 Making CPT Breaking andLoalization Manifest. . .

9

2.1.2 TheInstanton Moduli Spae . . .

9

2.2 TheCanonial Point ofView. . .

12

2.2.1 Onthe Cohomology. . .

13

2.2.2 Implementing CPT Breaking and Loalization . . .

14

2.2.3 TheInstanton Moduli SpaeRevisited . . .

15

2.2.4 TheOut-States . . .

17

2.3 Summaryof the Constraints on

X

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

19

2.4 MorseTheory on

X = CP ¹

. . .

20

2.4.1 Polynomial Distributions on

CP ¹

. . .

23

2.5 Interpretation oftheExtension . . .

29

2.6 Generalizationto General Target Manifolds . . .

32

2.6.1 ThePerturbative State Spaes . . .

32

2.6.2 TheGrothendiek-Cousin Operators . . .

33

3 From the A-Model to Morse Theory 35

3.1 MassagingtheA-model . . .

36

3.2 TheMorse TheorybehindtheA-model . . .

39

3.2.1 ThePotential . . .

40

3.2.2 Isolatingthe CritialPoints . . .

41

3.3 Perturbative MorseDesription oftheA-Model . . .

42

3.3.1 ThePerturbative State Spaes . . .

43

3.3.2 ThePerturbative State Spaeon

g LX _0,k

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

44

3.3.3 ThePerturbative State Spaeon

g LX _∞ ,k

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

46

3.4 Relationto Conformal SupersymmetriGhosts . . .

46

3.4.1 TheConformal Supersymmetri

bc

^{-System. . . . . . . . . . . . . . . . .}

47

3.4.2 Identifying theState Spaes . . .

51

3.4.3 What iftheGauge Field isAbsent? . . .

52

3.5 Conformal Supersymmetri Ghosts on

CP ¹

. . .

54

3.5.1 TheChiral de RhamComplex . . .

54

3.6 Beyondthe Perturbative Representations . . .

59

3.6.1 Existeneof Grothendiek-CousinOperators. . .

59

3.6.2 Chiral Bosonization . . .

61

3.6.3 TheGCOsand theCohomology Interpretation . . .

67

3.6.4 Conlusion . . .

70

4 The A-Model beyond Topology 71

4.1 TheMethod ofLogarithmi Deformation . . .

71

4.1.1 Extension ofthe Fields . . .

71

4.1.2 Extension ofthe Representation Theory . . .

72

4.1.3 TheFermioni

bc

^{-System . . . . . . . . . . . . . . . . . . . . . . . . . . .}

73

4.2 Introduing the GCOs. . .

74

4.2.1 Extension ofthe Fields . . .

74

4.2.2 Notes ontheSymmetries . . .

76

4.2.3 ExeptionalLogarithmi Partners . . .

76

4.2.4 OntheNeessityto Deformthe Fermions . . .

77

4.2.5 Extension ofthe State Spae . . .

77

4.2.6 Conlusion . . .

80

5 Summary and Conclusion 81 II Conformal Fermionic Ghosts on the Torus 85 6 Motivation 87 7 Fermionic Ghosts on Algebraic Curves 89

7.1 TheAlgebrai Surfaes . . .

89

7.2 TheFermioni

bc

^-Systemon

T ^n,m

. . .

90

7.2.1 AroundtheBranhPoints . . .

90

7.2.2 TheTwistedRepresentations . . .

92

7.2.3 Conlusion . . .

94

8 On Twist Fields and Torus Periods 95

8.1 TheLegendre Family . . .

95

8.1.1 Relationto the LattieTorus . . .

96

8.1.2 A Dierential Equation forthe Periods . . .

97

8.1.3 Solutionsfor thePeriods . . .

98

8.2 LCFT-ationof theLegendre Family . . .

99

8.2.1 A HypergeometriEquation for the Twist Fields . . .

100

8.2.2 TheNeessityof a LogarithmiExtension . . .

101

8.3 TheTripletModel . . .

102

8.3.1 Symmetriesand Representations . . .

102

8.3.2 Realizationof the TripletModel . . .

105

8.3.3 Charaters . . .

106

9 Relation to Seiberg-Witten Theory 109

9.1 SomeWordsonSeiberg-Witten Theory . . .

109

9.1.1 TheSpetralCurve ofSW Theory . . .

110

9.1.2 Modular Transformations . . .

110

9.2 TheSpetralCurve and TripletCharaters . . .

112

9.2.1 TheSpetralCurve inTerms of

τ

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

112

10 Conclusion 117 A Topological Field Theories 121 B From the Sigma to the A-Model 123

B.1 Twisting/Gauging theSigma Model . . .

124

C The Toric CSbc - Unfinished 127

C.1 DeformationbyHolomorphiCompletion . . .

127

C.2 TheCohomology OperatorsinLogarithmi Coordinates . . .

129

Notations

∗

Field-state-orrespondene,93

≺ A ≺ B :

^theset

A

^{isasubsetofodimensiononein thelosureof}

B

^.

˜1(z)

^{Thelogarithmipartnerof}

1(z )

^,100

∆ _T (φ)

^{Conformalweightof}

φ

^{withrespet to}

T

δ

Grothendiek-Cousinoperator,32

[ · , · ] , { · , · }

^{Gradedommutator,}antiommutator

[f ,g ] n (z )

^{Fieldintheoperatorprodutexpansion,72}

F

Disjointunion

| ˜0 〉

^{Logarithmipartnerof}

| 0 〉

^,100

| p 〉 ± , | p, ¯ p 〉 ± , | p, ¯ p 〉

^ChargedrepresentationsoftheCSb,47,49,49

A ^ǫ

RepresentationspaeoftheHeisenbergLie algebra,61

CSb Conformalsupersymmetri

bc

^-system,46

C ₀ CP ¹ \ { ∞ }

C _∞ CP ¹ \ {0}

C ×

Asaset

C × = C \ {0}

^{,asasymmetryf.pg.20}

C [ · ]

Polynomials

C (( · ))

^{Formalpowerseries}

C [[ · ]]

^Powerseries

D , D ^∗

^{Complexunitdisk}with/without thepoint

{0}

D , D ∗

Test funtions,distributions,23

D ₀ , D _∞

Test funtionswithompatsupportin

C ₀ , C _∞ e, ¯ e

^{Extensionofthe}perturbativerepresentations,34,

extensioneld,72

E _λ

Legendrefamily, 95

F ₀ , F _∞ , F × , F ¹

∞

^HolomorphirepresentationsoftheCSb,58,59

f

(Logarithmially)extended eld,30,72 GCO Grothendiek-Cousinoperator

g , g O

^{GCO32,nontrivialpartof}

O = O + g O

³⁰

H

Globallydened states,28

H λ

Non-unitaryHamiltonian,14

H ^(pert)

PerturbativeHamiltonian,18

H _c,n ⁱⁿ , H ⁱⁿ

0,0 , H ⁱⁿ

∞ ,0

PerturbativespetrumoftheTb,44,45,46

H ₀

Morsetheory/CSb,22,51

H _∞

, 22,51

Homogeneity

r

^Aprefatorof

| z | ^r , r ∈ R

in theeldexpansion,45,51

i ifandonlyif

j ^ǫ

^{CurrentsoftheCSb,48}

j _V ^ǫ , j _A ^ǫ

^{VetorialandaxialurrentsoftheCSb,48}

J _µ ^ǫ , J ^ǫ

^{CurrentsoftheHeisenbergLiealgebras,61}

J ⁻

^{Currentofthebosonizedbosons, 64}

J N

^{Currentmeasuringthegradingof}

N (p,p )

^,65

J(φ)

^{Chargeoftheeld}

φ

^{, 50}

Λ ^a,b

Basisofexteriorforms,23

LX

^Loopspaeof

X

^,39

g LX

^{Universaloverof}

LX

^,40

g LX n

^Sheetof

LX

ⁱⁿ

g LX

^,41

g LX c,n

^{Desendingmanifoldwithritialpoint}

x c

^,44

M ^ǫ (p) , ¯ M ^ǫ (p) , M ^ǫ (p, ¯ p )

^ChargedrepresentationsoftheCSb,48,49

M _E

ParameterspaeoftheLegendreFamily96

M (α,β)

^{Instantonmodulispae,10}

µ

^TheA-model gauge-eldstrenght41,twisteldonthetorus100

ν ^ǫ _{p, ¯ p}

^{Highest weightstateoftheHeisenbergLiealgebra,61}

N (p ) , ¯ N ( ¯ p) , N (p, ¯ p)

^Extendedrepresentationsofthebosonizedbosons,64

N (p ), ¯ N ( ¯ p) , N (p, ¯ p)

Perturbativerepresentations , 66

N L (p, ¯ p )

^{Logarithmiextensionof}

N (p − 1,p − 1)

^,67

O ^(naive)

Naiveoperator,30

OPA OperatorProdutAlgebra

OPE OperatorProdutExpansion

PER Physially EligibleRepresentation,102

P ₀

Polynomialof theeld modesintheCSb,54

p, ˜ p

^{InpartII,projetion}

T ^n,m → CP ¹ \ {e i }

^{, 89}

q

^{Bakgroundharge,anomaly,49}

Q ,Q 0

^{BRSThargeinMorsetheory,12,intheTb,37,53}

Q (z , ¯ z) , Q (z )

Superhargeeld, 49,itsholomorphipart,48

SQM Superquantummehanis

Tb Topologial

bc

^{-system,ungauged38,gauged42}

V ^ǫ (r, z)

^{FieldsoftheHeisenbergLiealgebra,62}

X n

^Subspae

X n ≃ X

ⁱⁿ

g LX n

^,44

X c,n X n ∩ g LX c,n

^,44

Introduction 1

Thisthesiswasinitiatedbymyinterestintherelationbetweengeometryandphysis. Itwas

sine I got to know the publiation of V.G. Knizhnik [Kni87 ℄ thatI wanted to investigate

the geometri signiane of theaspetswhih render aonformal eldtheorylogarithmi.

Knizhnikonsidersholomorphidierentialformsonalgebraisurfaeswhiharebranhed

overingsof

CP ¹

andhaveaglobal

Z _n

monodromygroup. Thedierential formsanbeiden- tiedwithonformalfermioni ghosts,and themonodromygrouphasanindued ationon

theseelds,whihthusfallinto

n

^irreduiblerepresentations. Inthespiritofonformaleld theory(CFT),theserepresentationsarerealizedbyloatingtheonformalelds isomorphi

to the respetive highest weight vetors at the branh points. In mathematial terms, this

amounts to restriting the dierential forms to a neighborhood of a branh point and to

onsidering representation theorythereon.

Ifthealgebraisurfaehasbranhpoints

e _i , i ∈ {1, . . . , 2N }

^,

N ≥ 2

^{,onemayturnthesurfae}

intoafamilyoftopologiallyequivalentsurfaesbyallowing

2N − 3

^{branhpointstovaryover}

CP ¹ \ S _{2N − 3}

i = 1 {e i }

^{. This helps to extrat further geometri} information, suh as degeneraies when branh points arefusing, or periods, whih satisfy dierential equations withrespet

to theoating parameters.

Although my investigations started withthe work of Knizhnik, I will disuss this setting

inthe seond partof my thesis. There, I will onsider theCFT realization of both, degen-

eraies and periods for thealgebrai surfae being atorus. The dierential equationfor its

periods is realized as the nullstate ondition for the odd representation of the monodromy

group

Z ₂

. Therefore, the four-point funtion of the so-alled twist eld orresponding to this representation is proportional to the periods of the torus. In partiular, it ontains

logarithmsand thefusionof twobranh points, whih issimulated bytheoperatorprodut

expansion(OPE)oftwosuhelds,yieldsa doubletrepresentationofthesymmetriesofthe

onformal fermioni ghost system. The Hamiltonian is not diagonalizable on this doublet,

whih signies that the CFT has to be extended to a logarithmi onformal eld theory

(LCFT). Theminimalisti way to dothis will leadto thetriplet model, asexplainedbyM.

Flohrin[Flo98 ℄.

Thissettinghasbeen the startingpointfor mypubliationwithM. Flohr[VF07℄. Asthe

torusisthespetralurveofpuregauge,

SU (2)

Seiberg-Wittentheory,wewantedtoexpress theprepotential intermsof haraters of the tripletmodel. Although we onlyobtained the

prepotential in termsof the torus modulus, whih equals theratio of twist eld four-point

funtions, we have been able to determine the spetral urve by means of suh haraters.

Thiswill be the subjet ofhapter 9inpartII.

of E. Frenkel, A. Losev and N. Nekrasov [FLN06 , FLN08 ℄, who investigated Morse theory

and the topologial A-model beyond their topologial setors. What is implied by those

onsiderations?

(Cohomologial)topologialeldtheoriesdealwithglobalgeometriobjetsonmanifolds,

in partiular with dieomorphism invariants that are inthe ohomology of some nilpotent

operator

Q

^{, alled} Behi-Rouet-Stora-Tyutin (BRST) harge due to its properties. It has an ation on the elds and state spaes of the theory and the elements in its ohomology

lasses omprise whatis alledthe topologial setorofa eldtheory.

Underertainirumstanesaeldtheoryhasinaddition toitstopologial setorfurther

dynamial states and observables. While the ohomology of

Q

^{is invariant under dieo-}

morphisms, this is not the ase for the dynamial setor. Hene, the dynamial degrees

of freedom should in priniple desribe part of the loal geometry of the target or domain

manifold.

In[FLN06 ℄,Frenkel, LosevandNekrasovonsiderthesituationdesribedaboveforMorse

theorywitharstorderLagrangian onaKählermanifold

X

^{withsaled metri}

λg, λ ∈ R ^{> 0}

. The perturbative spetrumof this theory inludes topologial as well as dynamial states.

If

X

^is supplemented with an additional struture, these states have their support on the desending manifolds of the gradient vetor eld of the Morse funtion. Moreover, the

submanifoldsyield adisjoint overof

X

^{,and sodothe} perturbative state spaes.

The loal geometry of

X

^{an be aessed employing the dynamial states. For}

λ → ∞

^,

the Hamiltonian beomes the Lie derivative in diretion of the gradient vetor eld of the

Morse funtion. The perturbative state spaes whih survive that limit turn into loally

dened indued representations of the symmetry generated by the gradient eld. This is,

metaphorially, what an observer loated on a desending manifold would expet to see.

However, Frenkel,Losevand Nekrasovlaimthatthere arenonperturbative eets through

whih the observer obtains additional insights into the loal representations of the Hamil-

tonian on

X

^{. They propose that the} nonperturbative state spaes are obtained by ex- tending the perturbative state spaes as distributions to

X

^{and their analysis shows that}

thethus globalizedrepresentations aretheloalohomology groupsinaomplex alledthe

global Grothendiek-Cousin omplex, [Kem78℄. This omplex has a ohomology operator,

theGrothendiek-Cousinoperator(GCO),whihompoundstheloalrepresentationspaes

and appears as an additional term in the Hamiltonian. The observer is thus onfronted

withaHamiltonianwhihannot be diagonalizedonall dynamialstatesasituation well

knownin thetheory oflogarithmi CFTs.

My initial interest in thework of Frenkel, Losevand Nekrasov [FLN06℄ arose from their

proposal that the topologial A-model in the large volume limit is an LCFT beyond its

topologial setor. In [FLN08 ℄, they reduethe A-modelwithembedding

x : R ¹ × S ¹ → CP ¹

to the Morse theory of [FLN06 ℄ by integrating out the dependene on

S ¹

^{. In partiular,}

one an derive the perturbative state spaes and it appears that they an be modelled by

representationspaesoftheonformalsupersymmetrighosts(CSb)withtargetspae

CP ¹

. Itis nowsuggestive to assumethatat leastthe representation theoryof theA-modelinthe

largevolume limitequalsthatoftheCSbandthetheoriesan,aordingly,besubstituted.

Furthermore, Frenkel, Losev and Nekrasov propose thedeformation of the Hamiltonian,

but do not analyze the extension of therepresentation spaes indetail. Moreover, in order

toprovetheironjeturethattheA-modelisanLCFTinthelargevolume limitandbeyond

its topologial setor, it is not suient to onsider the underlying Morse theory. A loga-

rithmi deformation oftheCSb hasto befound,whih yieldstheorretextensions ofthe

perturbative representation spaes and addsthe deformation terms to the Hamiltonian. It

is only then, thatthe Grothendiek-Cousin operators an be interpretedas thezero modes

ofthelogarithmi improvement termswhihdeform theenergy momentumtensor. Partsof

those onsiderations have been addressedinmyseondpubliation withM. Flohr[VF09℄.

Asmentionedabove,thisthesishastwoparts,thersttreatsthelogarithmiextensionof

theCSbunderlyingtheA-model,theseondisaboutfermionighostsonthetorusandtheir

relationtoSeiberg-Wittentheory. BeforeIstartwithanoutline,Iwillbrieyommentonthe

appendix,whihservestosupplementthemainpart. InappendixAIsummarizeandspeify

the basi ingredients of a topologial eld theory [BBRT91 , Wit82 , Wit88a , Wit88b℄. In

appendixB.1IbrieyexplainhowthetopologialA-modelisobtainedbytwistingan

N = 2

supersymmetrisigmamodelandnotedownthesupersymmetryofthistheory[Mar05℄. The

last appendix Cisthe foundation of another publiation,wherein Istudy thepossibilityto

generalize the approah ofFrenkel,Losevand Nekrasov[FLN08 ℄,bywhih theydeform the

Hamiltonian oftheA-model, toa deformation ofthe assoiated CSb.

Part I

^{Inthefollowinghapter2,IwillstartwithadisussionofMorsetheory. Therein,the}

geometri origin of the deformation operatorsis disussed and theonditions on thetarget

spaemanifoldarexed. ThishapterfollowsthepubliationofFrenkel,LosevandNekrasov

[FLN06 ℄, but some subtle pointsare treated inmore detail. Inpartiular this onerns the

extension ofthe perturbative representation spaes. Iwillpropose analternative ansatz for

the extension, whih relies on a priniple bywhih I an enlarge the representation spaes.

Thisansatz isappliable intheontext ofthe A-model.

Inhapter3,IwillintroduetheA-modelwithtargetspae

CP ¹

andtakethelargevolume limit. ReduingthethusobtainedtheorytoMorsetheory,Iwillderivetheperturbativestate

spaes andexplainwhytheyan be modelled bytheCSb. BeausetheA-modelisdened

on

CP ¹

,itisneessarytomakeharttransitions. FortheCSb,thesetransitionsaredened through the hiral de Rham omplex, whih I will also introdue. My method to derive

thedeformation of the Hamiltonian diers again fromthat ofFrenkel, Losev andNekrasov

thattheholomorphiandanti-holomorphihalves oftheCSbareonsideredtogether,not

onlybeause ofanomalies ourringbutalsobeausethe GCOsareomposedofbothparts.

Indeed, I will explain that this omposition onstrains the representation spaes and the

symmetries ofthe theory.

Having determinedthe perturbativerepresentationspaes, theirextensions, andtheGro-

thendiek-Cousin operatorsthat mediate between them, Iwill thenmove bak fromMorse

theory to the onformal eld theory. In hapter 4, I will use the method of Fjelstad et

al. [FFH

+

02 ℄ to deform the CSb logarithmially. I will do that in suh a way that the

representation spaesareextendedonsistently andthattheGCOsareaddedto theHamil-

tonian. Thishasan eet on the operator produt algebra of theelds, but neitheron the

supersymmetrynor the onformalsymmetryofthe CSb.

I willonlude this partof the thesiswitha briefsummaryand disussioninhapter 5.

Part II

^{Inparttwo Iwillonentrateonthefermionionformalghosts onbranhed over-}

ingsof

CP ¹

[Kni87 ℄. Afterabriefmotivationinhapter6,Iwillspeifythealgebraisurfaes under onsideration and introdue theonformal ghosts inhapter 7. Sine they will have

nontrivial operator produtexpansionsin a neighborhood of a branh point it is neessary

to extend therepresentation spaes by therepresentationsof themonodromygroup.

In thethe subsequent hapter8, Iwill derive bygeometri argumentsthat thefermioni

ghosts on thetorus neessarilyomprise a logarithmi onformaleld theory. The minimal

versionis thetripletmodel[Flo98 ℄,whih Iwill introdue inhapter8.3.

In the last hapter 9, I will explain how the spetral torus of puregauge Seiberg-Witten

theory an be obtained from ertain haraters of the triplet model and note down an

expressionoftheprepotentialwhihisgivenompletelyintermsofquantitiesofthisLCFT.

The thesis will be onluded with a summaryand a disussion of open questions in the

last hapter 10.

I

Supersymmetric Ghosts with Values

on the Sphere

Morse Theory 2

Thishapterhasthreeparts. MystartingpointwillbeMorsetheoryonageneralRiemannian

surfae

X

^{withsaled metri}

λg

^{and sympletiform}

ω

^.

Firstly,IwillpreparethetopologialsetorofthistheorybybreakingCPT invarianeand

by making loalization on the instantons expliit. This amounts to onseutively putting

onstraintson

X

^{. Theonstraintswillbesuhthattheinstantonsetorsarewelldenedand}

thatthegradienteldorrespondingtotheMorsefuntiondeomposes

X

^intosubmanifolds, to eah of whih one an perturbatively assoiate a state spae. Among those, there are

exited stateswhihare not saledout inthelarge volume limit

λ → ∞

^.

Frenkeletal.proposed[FLN06 ℄thatthestatespaesinthelimit

λ → ∞

^,whengeneralized as distributions on

X

^{, omprise the} nonperturbative low energy spetrum. In setions 2.4 and2.5IwilldisusssomeonsequenesofthisassumptionforMorsetheoryon

CP ¹

,mainly following their publiation but also with an additional disussion of theohomology of the

superharge,aswellasadierentmethodforextendingthestatespaesasdistributions. The

most important observation will be that observables whih inlude exterior derivatives are

nolongerdiagonalizable onallstates. Inpartiular,this onernstheHamiltonianandthus

draws asimilarityto logarithmi onformaleld theories. Rather, those operators intermix

the state spaes whihformerlyhave beenloatedindierent harts.

Finally, I will disuss the physial and geometrial meaning of this sort of non-loality,

whih isdue tothe non-topologial states.

Thishapterwillbeonludedwithageneralizationofthetoymodeltoalassofmanifolds

X

^{and will be the basis for an} understanding and analysis of the Morse theory underlying

thetopologial A-model. Myexplanationsrely mostlyon[FLN06, BBRT91, Wit82℄.

2.1 The Path Integral Point of View

In terms of the strutures just introdued, the Morse theory I will onsider onsists of a

Riemannian surfae

X

^{, a smooth embedding}

x : Σ ⊆ R → X

^{, its Grassmann valued super-}

partner

ψ

^{and anotherGrassmann valued quantity}

π

^{,whih istheonjugate momentumof}

ψ

^{. The Eulidean metri}

g

^on

X

^{is saled bysome parameter}

λ ∈ R ^{> 0}

and, without lossof generality, Ixa onnetion

D

^{to be the}Levi-Civitaonnetion, dened withpositivesign on

∂

∂x ^µ : D ν ∂

∂x ^µ = _∂x ^∂ λ Γ ^λ

νµ

^.

Let

f : X → R

be Morse, i.e.single valued and withisolated ritial points

x _c : d f (x _c ) = 0

^,

anddenote furtherby

D t ψ ^µ = ^dψ _dt ^µ + Γ ^µ

λσ dx ^λ

dt ψ ^σ

^{the pullbakof}

D

^to

Σ

andby

∇ ^µ f : = g ^µν ∂ ν f

thegradient of

f

^{. Inloal}oordinates, the ation Iaminterested inis

S _λ = Z

Σ

³ 1 2 λg µν

dx ^µ dt

dx ^ν dt + 1

2 λg ^µν ∂ µ f ∂ ν f + iπ _µ ∇ t ψ ^µ − iπ _µ ¡

D α ∇ ^µ f ¢

ψ ^α + 1

2λ R _αβ ^µν π _µ π _ν ψ ^α ψ ^β ´ dt .

(2.1.1)

In thefollowing setions I will extrat its topologial setor, seleting eitherthe instantons

or anti-instantons and by speifyingseveralonditions on

X

^.

Sine

df (x c ) = 0

^,theHessian

H(x)[γ] : = D γ (d f )(x)

^,

γ ∈ T x X

^{doesnot depend on thehoie}

of the onnetion at a ritial point

x c

^{. In loal oordinates it reads}

H µν (x c ) = ∂ µ ∂ ν f (x c )

^.

There existsa basis

e _µ

^{of tangent vetors at}

T _{x c} X

^{in whih it is diagonal with} eigenvalues

κ c µ

^:

H (x c ) e µ = κ c µ e µ

^{. The onditionthattheritial points areisolatedis equivalent to}

theonditionthat

H(x c )

^{hasnozeroeigenvalues. SinetheHessiandoesnotdepend onthe}

onnetion, itisreasonable to dene an index foreveryritial point

ind(x c ) = #{µ : κ c µ < 0} , (2.1.2)

whih isatopologial invariant.

In order to see what the lassial solutions are, I will for a moment onentrate on the

bosoni part. One an apply the so-alled Bogomlny trik to nd theabsoluteminima of

theation:

S bos = Z

Σ

à λ 2

µ dx ^µ dt ∓ ∇ ^µ f

¶ 2

± λ df dt

!

dt . (2.1.3)

Sineit waspositivesemi-denite before, Iobtain alower bound

S bos ≥

¯ ¯

¯ ¯ Z

Σ

d f

¯ ¯

¯ ¯ , (2.1.4)

whih issatisedbythe gradient trajetories

dx ^µ

dt ± ∇ ^µ f = 0 . (2.1.5)

Thesearethelassial bosoni solutions to

δS = 0

^{. There arethree kinds,depending onthe}

boundaryonditions. The vauumongurations aresolutions of

dx ^µ

dt = 0 ∧ ∇ ^µ f (x) = 0 , (2.1.6)

whih issatisedbyonstant loops,i.e. theritialpoints

x c

^{. Ifthere existsmorethan one}

ritialpoint, say

{x ₊ , x ₋ }

^{,therearealsoinstanton (}

−∇ f

^)andanti-instanton ongurations (

+∇ f

^{) :}

dx ^µ

dt ± ∇ ^µ f (x) = 0 , x( ±∞ ) = x _± (2.1.7)

where w.l.o.g. I xed some initial and nal time. From (2.1.4) one an onlude that the

instantonssatisfy

f (x ₊ ) > f (x ₋ )

^{and the} anti-instantons

f (x ₊ ) < f (x ₋ )

^.

2.1.1 Making CPT Breaking and Localization Manifest

The anti-instantons an be exluded from the lassial minima by subtrating

λ R

d f

^from

the ation (2.1.1). This term does not depend on the metri and is hene topologial. It,

however, breaksCPT invariane asonewould expetfor atheorywithout anti-instantons.

1

Inorderto maketheloalizationpropertymanifest,Imassagetheation

S − λ R

df

^intoa

rstorderform,byintroduingaLagrangianmultiplier

p µ

^{. Viewedaspartofthe}integration kernel

exp{ − S}

^{inthe path integral, I maynow onsider,} equivalently to (2.1.1):

S λ = Z

Σ

³

− ip µ

µ dx ^µ

dt − g ^µν ∂ _ν f

¶ + 1

2λ g ^µν p µ p ν

+ iπ _µ ¡

D _t ψ ^µ − (D _α ∇ ^µ f )ψ ^α ¢ + 1

2λ R _αβ ^µν π _µ π _ν ψ ^α ψ ^β ´ dt .

(2.1.8)

Inthelimit

λ → ∞

^{,theintegralkernelturnsintoa}

δ

distributiononinstantonmodulispae, whih makes loalization expliit. Indeed, for nite

λ

^{, the instantons still ontribute with}

a weight fator

e ^{− 2λ} | ^{f (x + ) − f (x − )} |

to orrelation funtions, but for

λ → ∞

^their ontribution disappears. On the ontrary, the instantons ontribute witha onstant weight fator

1

^for

any value of

λ

^.

Let

v ^µ (x) : = ∇ ^µ f (x)

^{be the vetor eld assoiated with}

f

^and

p _µ ^′ : = p µ + Γ ^λ _µν ψ ^ν π _λ

^{. The}

ation inthe large volume limitan nowbewritten as:

S _∞ = − i Z

Σ

µ p ^′ _µ

µ dx ^µ dt − v ^µ

¶

− π µ

µ dψ ^µ

dt − ψ ^α ∂ α v ^µ

¶¶

dt . (2.1.9)

It isinvariant underthefollowing susytransformations

[Q , x ^µ ] = ψ ^µ

^,

[Q, ψ ^µ ] = 0 [Q ^∗ , x ^µ ] = 0

^,

[Q ^∗ , ψ ^µ ] = v ^µ

[Q ,π µ ] = p ^′ _µ

^,

[Q, p ^′ _µ ] = 0 [Q ^∗ , π µ ] = 0

^,

[Q ^∗ , p ^′ _µ ] = 0 (2.1.10)

andmoreover,theLagrangianis

Q

^-exat,

L = − i[Q , π _µ ³

dx ^µ dt − v ^µ ´

]

^andthusistheHamiltonian.

This isroughly the modelI amgoing to onsider. However, Iwill need some more infor-

mations onthe instanton moduli spae, espeiallyinorder to ndonstraintson thetarget

manifold. Therewill beserveral obstaleswhih have toberesolvedand Iwill listthemup,

wheneverI enounter one. Inthe following andfor onveniene,I willleaveawaytheprime

for

p ^′ _µ

^.

2.1.2 The Instanton Moduli Space

Theinstanton equation

dx ^µ

dt = v ^µ (x)

^{givesrise to a} sympletormorphismof

X

^,i.e.

L _v ω = 0

^:

φ v : X × Σ → X x 7→ φ v (x, t ) = x(t ) , (2.1.11)

1 Though for the model under consideration CPT is really CT, I will follow the terminology of Frenkel, Losev and

Nekrasov [FLN06]. For a more detailed discussion of CPT breaking, c.f. section 2.2.4.

where

x(t )

^{is an instanton solution and}

φ _v ( · , t )

^{determinesa one parameter group in}

t

^{. By}

means of this ow equation of

v

^{one an try to nd a partition of}

X

^into submanifolds whih isgenerated bythe xedpointsof

v

^{. Thesewill bethedesending}

X c

^{and asending}

manifolds

X ^c

^:

X _c ^(c) : = (

x ∈ X : lim

t → ⁽⁺⁾ ₋ ∞

φ _v (x, t ) = x _c )

. (2.1.12)

If

x _c

^isanondegenerateritialpointand

φ _v

^adieormorphism,theyareindeedsubmanifolds [AR67,pg.87f℄and inheritthe tangent spaesdened bytheowlines.

ForthefollowingreasonIdemandthatadeompositionof

X

^{intodesendingandasending}

manifoldsexists. Insetion2.2.4Iwillexplainthatthe statespaeswillbeloalizedaround

thexed pointsof

v

^{. A} deompositionof

X

^{intermsof,say,desendingmanifolds isuseful}

beauseone an thenanonially assoiate to eah suh submanifold astate spae

F _α

and these over

X

^{. Therefore:}

❏

^{Thetarget manifold}

X

^hasa(Bialyniki-Birula)deomposition

X = F

α ∈ A X α = F

α ∈ A X ^α

^withrespetto

v

^.

Theinstanton modulispaesaredenedbymeans ofdesendingand asendingmanifolds

M (α,β) : = X α ∪ X ^β , (2.1.13)

and under further onditions it is possible to alulate thedimension of this moduli spae.

Let

x c

^{bearitialpoint,Ianhooseloaloordinatessuhthatitisloatedat theorigin.}

In its neighborhood I an approximate a solution of the instanton equation by a line ele-

ment

y = x c + x

^{and bymakinga Taylor expansionaround theritialpoint. Thisyields to}

lowest order

d t x ^µ − H _ν ^µ (0)x ^ν = 0

^{,whith Hessian}

H

^{evaluated at}

x c = 0

^{. Thus, loally around}

the xed point, the diretions along whih

H

^{has positive} eigenvalues span the tangent spae of the desendingmanifold while the others span thetangent spae of the asending

manifold. Therefore, at least in a neighborhood of a xed point

x c

^,

T X c ≃ R ^{dimX − ind(x c )}

or

≃ C ^{dim C X − 1} 2 ind(x c )

while for the asendingmanifold

T X ^c ≃ R ^{ind(x c )}

or

≃ C ¹ 2 ind(x c )

. The general-

izationof this onditionisasfollows:

❏

^Let

( f , X , λg)

^allowfor Morse-Smaletransversality,i.e.

∀ x ∈ M (α, β), ∀ α, β : dim T x X α + dim T x X ^β − dim X = dim ¡

T x X α ∪ T x X ^β ¢

.

Onean nowalulate

dim ^R M (α,β) = ind(β) − ind(α) . (2.1.14)

The Morse-Smale ondition yiels a nie desription of the tangent spaes of

X

^{in terms of}

instanton ow lines. Espeially the dimensions of the instanton moduli spaes are natural

numbersinludingzero,restritedbythedimension ofthetarget manifold,andthereareno

dimensional degeneraies. Sine it is expressedbythe Morse indees, the dimension of the

instanton moduli spae is a topologial invariant. Morse-Smale transversality does further

index.

Thereisanother,physiallyinspiredwaytoalulatethedimensionoftheinstantonmod-

uli spae [H

+

03,se. 10.5.2℄. Consider an instanton solution

x : d t x ^µ − v ^µ (x) = 0, x ^µ ( −∞ ) = x ^µ _α , x ^µ ( ∞ ) = x _β ^µ

^{. Again, I will move in the solution spae of this dierential operator to}

another solution

y = x + ηz

^,where

η > 0

^{is an} innitesimally small number. The urve

y

^is

an instanton solution if the displaement

z

^satises

D ₋ z : = ( d t − H (x(t)) ) z = 0, z( ±∞ ) = 0

to the order

η

^{. For every}

t

^{Imay hoose abasisof} eigenvetorsof

H(x(t ))

^witheigenvalues

κ µ (t )

^{whih spans the tangent spae}

T x(t ) X

^{. The operator}

D ₋

^{is diagonal inthis basisand}

hashomogeneous solutions

z _µ (t ) = e _µ exp(

Z _t

0

κ _µ (τ)dτ) , (2.1.15)

where

e µ

diagonalizes

D ₋

^at

t = 0

^{. These solutions have the orretboundary onditions if}

κ µ ( −∞ ) > 0

^and

κ µ ( ∞ ) < 0

^.

There are two possible senarios. The rst is that the dimension of the solution spae

equals the dimension of the eigenspae of the Hessian. This is the ase if none of the

eigenvalues

κ µ (t )

^{hanges its sign from a negative to a positive value when passing from}

t = −∞

^to

t = ∞

^{. Ifthisissatised,}

dim ^R M (α, β) = ind(β) − ind(α) = #{µ : κ µ ( −∞ ) > 0, κ µ ( ∞ ) <

0} = dim ker D ₋

^{. Intheseondsenariothereexist} eigenvalueswhih hange their signsfrom negative topositvevalue. Theybelongtohomogeneoussolutionsofthedierential operator

D ₊ : = d t + H(x(t ))

^{. Inthatgeneral ase, thedierene}

ind(β) − ind(α)

^{an bewritten as}

dim R M (α,β) = dim ker D ₋ − dim ker D ₊ . (2.1.16)

The operators

D _∓

^{appear in the equations of motion for the fermions}

ψ ^µ

^and

π _µ

^{, re-}

spetively. Under the assumption that the dimension of theinstanton moduli spae equals

dim ker D ₋

^{, it further equals the number of linear} independent solutions of

D ₋ ψ _0,l = 0, l = 1 . . .d

^,

d = dim M (α,β)

^,whereas

π _µ

^{hasnozeromodes. Thisleadstotheseletionrulethat}

observableshave toontaina produt

Q d

l = 1 ψ 0,l

^{,iftheorrelationfuntion isnot tobezero.}

Thereason isthat thepath integral isa

δ

distributionon thehomogeneous solutionsof

D ₋

and theinstanton ongurations

x 0

〈 O 〉 = Z

M (α,β)

Y

l = 1...d

ψ _0,l O | ^M (α,β) . (2.1.17)

AnintegraloverGrassmannvariablesiszeroiftheintegrandisnotavolumeform,andinthe

nextsetionIwillmakelearthat,indeed,thezeromodesof

ψ

^{have ageometri meaningas}

dierentials on

X

^{. Fromthe disussion above Ionlude that they arephysially signifying}

the presene of instantons, and the numberof fermioni insertions ounts thedimension of

their moduli spae.

2

2 In the fermionic bc-system, that I will discuss in the next chapter, it will also be necessary to insert "zero-modes" in

2.2 The Canonical Point of View

The Morse ation (2.1.9) hasan immediate interpretation interms of geometri quantities

of thetarget manifold

X

^{. Thebestplae to understand this istheanonial} formulation of thetheory. Reshueling thetermsin(2.1.9), I anread othelassialHamiltonian inthe

large volume limit 3

H _∞ = v ^µ (ip µ ) + ψ ^α ∂ α v ^µ (iπ µ ) . (2.2.1)

Reonsidering(2.1.10),an immediate hoie howto quantize onsistsinrelatingtheeld-

oordinateswithgeometri quantities inthefollowing way:

bosons: fermions:

x ^µ x ^µ ψ ^µ dx ^µ

ip µ ∂ µ iπ µ ι µ

(2.2.2)

TheHamiltonian above and thesuperharges

Q

^and

Q ^∗

^{an nowbe rewritten as}

Q = d , Q ^∗ = ι v , H _∞ = L _v = {Q,Q ^∗ } , (2.2.3)

and they have a anonial ation on dierential forms on

X

^{. The geometri data satisfy}

the usual quantization rules

[p µ , x ^ν ] = − iδ ^ν _µ , [π µ , ψ ^ν ] = − iδ ^ν _µ

^{for the} superbraket, and in partiular

Q = iψ ^µ p µ . (2.2.4)

InthefollowingIwillreproduethedeformationsdesribedforthepathintegralansatzfor

theanonial formalismof Morse theory. The ideabehindthis isto seewhat thespetrum

oftheHamiltonianinthelargevolumelimitlookslikeandtoinvestigateifthereremainwell

dened exited states in this limit. I will again start with the ation (2.1.1) before taking

the large volume limit and thetarget manifold

(X ,λg )

^{, endowed withan inner produt on}

dierential forms

η,χ ∈ Ω ^• (X )

〈 η, χ 〉 : = Z

X

( ⋆ η) ¯ ∧ χ . (2.2.5)

Thebar denotesomplexonjugation, ifneessary,and

⋆

theHodgeoperator.

4

TheHamil-

tonian orresponding to the ation (2.1.1) with Morse funtion

f

^{is obtained from the}

correlation functions. These do, however, not represent instantons because they are mappings between isomor- phic representation spaces, cf. section 3.4.1 and section 8.3. On the contrary, instantons relate different vacuum configurations (they are highest weight vectors of different representations).

3 This classical Hamiltonian is not bounded from below. However, in section 2.4, I will derive it from the canonically quantized Hamiltonian with λ 6= 0 by deforming the spectrum in a specific way, cf. [FLN06]. Thereby one obtains states which are not in the closure of Ω •

d (X ) with respect to the L ² norm, but on which one can define an orthog- onal pairing and whose eigenvalues with respect to the canonically quantized H _∞ are positive semidefinit (when considered perturbatively, c.f. section 2.5). Analogous will be satisfied for the A-model.

4 On volume elements ⋆ dx ^{µ 1} ∧ ··· dx ^{µ k} = p | g |

(dim R X − k)! ǫ ^µ _ν ^{1 ··· µ k}

k+1 ··· ν _{dim X} dx ^{ν 1} ∧ ··· dx ^{ν k} and ǫ µ ₁ ··· µ _dim

R X = + 1 for even per-

mutations.

Q = d _λ = e ^{− λ f} de ^{λ f} = d + λ d f ∧ , Q ^† = d ^† _λ = e ^λf d ^† e ^{− λf} = 1

λ d ^† + ι _{∇ f} ,

(2.2.6)

as

H = ∆ _λ = 1

2 {Q,Q ^† } = 1 2

¡ λ ^{− 1} ∆ + λ k d f k ² + K _f ¢

, (2.2.7)

where,

k df k ² = ι _{∇ f} d f

^,

K f = L _{∇ f} + L ^†

∇ f

^,

L ^†

∇ f = {d ^† , df }

^and

∆ = {d, d ^† }

^. Conjugation

†

^is

dened with respet to the inner produt. Let me emphasize, that up to now CPT is not

broken and the two superharges areindeed onjugate. However, inthelarge volume limit

CPT will be violated and this makes the dierene between the dagger and the star, for

instanefor the superharge in(2.2.3).

2.2.1 On the Cohomology

As I explained in the introdution and in appendix A, the topologial states are in the

ohomologyof thesuperharge

Q

^{. Underertainonditions on}

X

^{,thatIwill onentrateon}

inthis setion, theohomology of

Q

^{isisomorphito thekernel of the}Hamiltonian.

The superharges above are obtained by a similarity transformation of

d

^and

d ^†

^{, and I}

anhenearryovertheresultsonthedeRhamdierentialto themoregeneral situationin

Morse theory, inpartiular that

H _d ^•

λ (X ) ≃ H _d ^• (X )

^{. If}

X

^{isa real manifold whih is moreover}

ompat, orientedand without boundary,there existsaunique Hodge deomposition

Ω ^k

d λ (X ) = d λ Ω ^{k − 1}

d λ (X ) ⊕ d ^† _λ Ω ^{k + 1}

d λ (X ) ⊕ Ω ^k

∆ _λ (X ) , (2.2.8)

where

Ω ^k

∆ _λ (X)

^{denotes the harmoni forms on}

X

^{withrespet to}

H = ∆ _λ

[Nak03 ℄. If suh a deomposition existsandmoreoveraninnerprodutlike(2.2.5)oneanshowthat

H _d ^•

λ (X ) ≃ Ω •

∆ _λ (X )

^{.5 Thus, in order to identify the ohomology of the superharge with the ground}

states of the Hamiltonian itwould be sensible to invoke thatwhenever

X

^{is real, itshould}

also be ompat,orientedand withoutboundary.

If

X

^{is a ompat Kähler manifold there exist unique, orthogonal Hodge} deompositions for theDolbeault derivatives

∂ _λ

^and

∂ ¯ _λ

^{. Notiethatinthis ase}

d _λ = ∂ _λ + ∂ ¯ _λ

^{and similarfor}

the onjugate. Sine

∆ _d

λ = 2 ∆ _∂

λ = 2 ∆ _¯

∂ λ

^{[Nak03 ℄, one nds that}

H _∂ ^p,q

λ (X ) ≃ Ω ^p,q

∆ _d

λ (X)

^{and the}

same istrue fortheonjugate dierential forms. Therefore:

❏

^Let

X

^{beaompatKählermanifoldor,ifreal,ompat,orientedandwithoutbound-}

ary.

5 Let ω ∈ Ω •

∆ λ (X), then 〈 ω, ∆ _λ ω 〉 = 0 = k d _λ ω k ² + k d ^†

λ ω k ² and this proves that a harmonic form is closed under d _λ and

d ^† _λ . The Hodge decomposition is orthogonal and therefore the harmonic forms are not exact with respect to d _λ .

and thekernel ofthe Hamiltionian will survive CPT breakingif

λ < ∞

^{. For}

λ → ∞

^{this will}

still be trueat leastfor

X = CP ¹

and Iwill prove this insetion2.4.1.

2.2.2 Implementing CPT Breaking and Localization

The transformations I have done on the path integral insetion 2.1.1 an be translated to

the anonial point of view by onsidering orrelation funtions of topologial observables

and states

〈 ω, e ^{(t n − t + )H} O _n e ^{(t n−1 − t n )H} . . .e ^{(t 1 − t 2 )H} O ₁ e ^{(t − − t 1 )H} · χ 〉 = Z

X × X

[ ⋆ ω(x ¯ ₊ )] ∧ χ(x ₋ ) Z

Σ → X : x(t ₋ ) = x ₋ , x(t ₊ ) = x ₊

O _n (t n ) ∧ ··· ∧ O ₁ (t 1 )e ^{− S} . (2.2.9)

Sine the topologial setor is supposed to be invariant under subtrating the exat term

R _{x +}

x ₋ df = f (x ₊ ) − f (x) + f (x) − f (x ₋ )

^{fromthe ation,thismusthave aneetontheoperators}

and states. Expetation values of topologial observables, alulated with an Hamiltonian

in whih CPT is manifestly broken by the additional term, must equal the undeformed

expetation values. Therefore, the states and observables in the CPT-broken phase are

subjet to thefollowing transformations ofthe physial ounterparts 6

χ 7→ e ^{λ f} χ

⋆ ω ¯ 7→ e ^{− λ f} ⋆ ω ¯ O 7→ e ^{λ f} O e ^{− λf}

and inpartiular

Q 7→ d

Q ^† 7→ Q _λ ^∗ = 2ι v + λ ^{− 1} d ^† H 7→ H λ = L _v + _2λ ¹ ∆

(2.2.10)

Let me emphasize that all operators transform in the same way and the mappings above

arenot similaritytransformations. Therefore,thenewHamiltonianisnotself-onjugateany

more andI ratherput a

∗

^{than a}

†

^.

One maynowallowthe transformed Hamiltoniannot only to at ontopologial but also

ondynamialstates. Still,fornitevaluesof

λ

^,thenewHamiltonianhasthesamespetrum as

H

^{beause the in-states have justgained a phase. In partiular, the isomorphy between}

the superharge ohomology and the ground states is still valid, though the theory is not

unitary any more and the in- and out-states are no longer onneted by an inner produt

(I will disuss the out states in setion 2.2.4). The Morse theory with broken CPT and

the one determined by(2.2.7) have thesame ohomologieswithrespetto thesuperharge,

sine

H _d ^•

λ ≃ H _d ^•

^{. Moreover, for nite}

λ

^,

H _d ^• ≃ Ω •

∆ _λ ≃ Ω • H λ

, suh that

dim Ω •

∆ _λ = dim H λ

^{. These}

dimensionsare atopologial invariantsand thus shouldnot be aetedbytaking

λ → ∞

^.

6 The exponent e ^λf : = e ^{λ(f (x) − f (x − ))} for the “ket” and e ^{− λf} = e ^{− λ(f (x) − f (x + ))} for the “bra”.