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Exact solution of the quartic matrix model

and application to 4D noncommutative QFT

Raimar Wulkenhaar

Mathematisches Institut, Westf

¨alische Wilhelms-Universit¨at M¨unster

joint work with Harald Grosse (Vienna) (based on arXiv:1205.0465v4, arXiv: 1306.2816)

Raimar Wulkenhaar (M¨unster)Exact solution of the quartic matrix model and application to 4D NCQFT

IntroductionSolution of quartic matrix modelMoyalφ44-theory in batrix basisPosition spaceOutlookAppendix

Matrix models

12D quantum gravityis theenumeration of random

triangulations of surfaces.

Its asymptotic behaviour is captured by thematrix

model partition function Z=? dMexp? -N? nt ntr(Mn)? ,M=M??MN(C) ForN → ∞, this series in(tn)is evaluated in terms of theτ-function for the

Korteweg-de Vries (KdV) hierarchy.

22D topological quantum gravityhas correlation functions

which are intersection numbers of complex curves. They can be arranged into a generating functional with series parameters(tn). [Witten, 1990] conjectured that both(tn)-series are the same.

Raimar Wulkenhaar (M¨unster)Exact solution of the quartic matrix model and application to 4D NCQFT

IntroductionSolution of quartic matrix modelMoyalφ44-theory in batrix basisPosition spaceOutlookAppendix

The Kontsevich model

[Kontsevich, 1992] computed the intersection numbers in terms of weighted sums over ribbon graphs. He proved these graphs to be generated from theAiry function matrix model (Kontsevich model)

Z[E] =?

dMexp?-1

2tr(EM2) +i6tr(M3)??

dMexp?-1

2tr(EM2)?

,M=M??MN(C) forE=E?>0 andtn= (2n-1)!!tr(E-(2n-1)). LimitN → ∞ofZ[E]gives the KdV evolution equation, thus proving Witten's conjecture.

Raimar Wulkenhaar (M¨unster)Exact solution of the quartic matrix model and application to 4D NCQFT

IntroductionSolution of quartic matrix modelMoyalφ44-theory in batrix basisPosition spaceOutlookAppendix

A matrix model inspired by noncommutative QFT

The simplestQFT on a 4D noncommutative manifoldcan be written as a matrix model

Z[E,J,λ] =?

dMexp?-tr(EM2) +tr(JM)-λ

4tr(M4)??

dMexp?-tr(EM2)-λ

4tr(M4)?

whereE=E??MN(C)is the 4D Laplacian,λ≥0 and

J?MN(C)generates correlation functions.

In joint work withHarald Grosse[arXiv:1205.0465v4] we achieved the exact solution ofZ[E,J,λ]forN → ∞and after renormalisation ofE,λ

Schwinger functionsdescribe acommutative 4D QFT

[arXiv:1306.2816]. "Particles" interact without momentum transfer. There are non-trivial topological sectors.

Raimar Wulkenhaar (M¨unster)Exact solution of the quartic matrix model and application to 4D NCQFT

IntroductionSolution of quartic matrix modelMoyalφ44-theory in batrix basisPosition spaceOutlookAppendix

Field-theoretical matrix models

classical scalar fieldφ? C0(Rd)? B(H), withm2? R ddxφ2(x) translates totr(φ2)<∞, i.e.nc scalar field is

Hilbert-Schmidt compact operator

on Hilbert spaceH=L2(I,μ) realise as integral kernel operators:M= (Mab)?L2(I×I,μ×μ) product:(MN)ab=?

Idμ(c)MacMcb

trace:tr(M) =?

Idμ(a)Maa

adjoint:(M?)ab=Mba action= non-linear functionalSforφ=φ?in volumeV:

S[φ] =Vtr(Eφ2+P[φ])

E- unbounded positive selfadjoint op. with compact resolvent, P[φ]- polynomial inφwith scalar coefficients partition functionZ[J] =?

Dφexp(-S[φ] +Vtr(φJ))

Raimar Wulkenhaar (M¨unster)Exact solution of the quartic matrix model and application to 4D NCQFT

IntroductionSolution of quartic matrix modelMoyalφ44-theory in batrix basisPosition spaceOutlookAppendix

Topological expansion

Connected Feynman graphs in matrix models areribbon graphs Viewed as simplicial complexes, they encode thetopology (B,g) of agenus-gRiemann surface withBboundary components (or punctures, marked points, holes, faces).

Thekthboundary component carries acycle

J

Nkp1...pNk:=?Nkj=1Jpjpj+1

ofNkexternal sources,Nk+1≡1. ExpandlogZ[J] =?1SV2-BG|p1...pN1|...|q1...qNB|JN1p1...pN1···JNBq1...qNB according to the cycle structure.

Raimar Wulkenhaar (M¨unster)Exact solution of the quartic matrix model and application to 4D NCQFT

IntroductionSolution of quartic matrix modelMoyalφ44-theory in batrix basisPosition spaceOutlookAppendix

Ward identity

Unitary transformationφ?→UφU?leads toWard identity 0=? Dφ

Eφφ-φφE-Jφ+φJ?

exp(-S[φ] +Vtr(φJ)) that describes howE,Jbreak the invarianceof the action. ...chooseE(but notJ) diagonal, useφab=∂

V∂Jba:

Proposition [Disertori-Gurau-Magnen-Rivasseau, 2006] The partition functionZ[J]of the matrix model defined by the external matrixEsatisfies the|I| × |I|Ward identities 0=? n?I? (Ea-Ep)V∂ ForEof compact resolvent we can always assume that m?→Em>0 is injective!

Raimar Wulkenhaar (M¨unster)Exact solution of the quartic matrix model and application to 4D NCQFT

IntroductionSolution of quartic matrix modelMoyalφ44-theory in batrix basisPosition spaceOutlookAppendix

We turn the Ward identity forEinjective into formula for? n?I∂2Z[J] ∂Jan∂Jnp. TheJ-cycle structure in logZcreates singular contributions≂δap regular contributionspresent for alla,p

Theorem (Ward identity for injectiveE)

n?I∂ 2Z[J] ∂Jan∂Jnp=δap?V? (K)J

P1···JPK

SK? n?IG |an|P1|...|PK|+G|a|a|P1|...|PK| r≥1? q

1....qr?IG

|q1aq1...qr|P1|...|PK|Jrq

1...qr?

V2? (K),(K?)J

P1···JPKJQ1···JQK?

SKSK?G|a|P1|...|PK|G|a|Q1|...|QK?|?

Z[J] +V

Ep-Ea?

n?I? J

Raimar Wulkenhaar (M¨unster)Exact solution of the quartic matrix model and application to 4D NCQFT

IntroductionSolution of quartic matrix modelMoyalφ44-theory in batrix basisPosition spaceOutlookAppendix

How to use the Ward identity

WriteS=V2?

a,b(Ea+Eb)φabφba+VSint[φ]. Functional integration yields, up to irrelevant constant,

Z[J] =e-VSint[∂V∂J]eV2?J,J?E,?J,J?E:=?

m,n?IJ mnJnmEm+En

Example:G|ab|(fora?=b)

G |ab|=1

VZ[0]∂

2Z[J]∂Jba∂Jab???

J=0 1

VZ[0]?

∂Jbae-VSint?∂V∂J?∂ ∂JabeV2?J,J?E? J=0 1

Ea+Eb+1(Ea+Eb)Z[0]??φab∂(-VSint)

∂φab

V∂J??

Z[J]???J=0

∂(-VSint) ∂φabcontains, for anyP[φ], the derivative? n∂2∂Jan∂Jnp

Raimar Wulkenhaar (M¨unster)Exact solution of the quartic matrix model and application to 4D NCQFT

IntroductionSolution of quartic matrix modelMoyalφ44-theory in batrix basisPosition spaceOutlookAppendix

Schwinger-Dyson equations (forSint[φ] =λ4tr(φ4)) The previous formula lets the usually infinite tower of

Schwinger-Dyson equations collapse:

after genus expansionG...=?∞g=0V-2gG(g)...:

1. Aclosed non-linear equation forG(0)

ab(planar+regular): G(0) |ab|=1Ea+Eb-λV(Ea+Eb)? p?I? G(0) |ab|G(0) |ap|-G(0) |pb|-G(0) |ab|Ep-Ea?

2. Forevery otherG(g)

a

1...aNan equation which only depends on

G(g) G(h) this dependence is linear in the top degree(N,g)

SomeG...need renormalisationofE,M, andλ!

Raimar Wulkenhaar (M¨unster)Exact solution of the quartic matrix model and application to 4D NCQFT

IntroductionSolution of quartic matrix modelMoyalφ44-theory in batrix basisPosition spaceOutlookAppendix

Exact solution forφ=φ?

Realityimplies invariance under orientation reversal empty forG|ab| cancellations in(Ea+Eb1)Gab1b2...bN-1-(Ea+EbN-1)GabN-1...b2b1

Theorem (universal algebraic recursion formula)

G|b0b1...bN-1|

= (-λ)N-2 2? l=1G (Eb0-Eb2l)(Eb1-EbN-1) V N-1? k=1G (Eb0-Ebk)(Eb1-EbN-1) Last line increases the genus and is absent inG(0) |b0b1...bN-1|

Raimar Wulkenhaar (M¨unster)Exact solution of the quartic matrix model and application to 4D NCQFT

IntroductionSolution of quartic matrix modelMoyalφ44-theory in batrix basisPosition spaceOutlookAppendix

Further observations

Non-planar contributions with genusg≥1 are suppressed byV-2g. In limitV→ ∞, full function and its restriction to planar sector satisfy the same equations.

Thenon-linearequation

G (0) |ab|=1

Ea+Eb-λV(Ea+Eb)?

p?I? G(0) |ab|G(0) |ap|-G(0) |pb|-G(0) |ab|Ep-Ea? is not algebraic and to be solved case by casefor givenE. Divergent index sumscan possibly be renormalised by Pattern extends toB≥2 boundary components: Equation for (N1+...+...NB)-point functions

G|p11...p1N1|...|pB1...pBNB|is

1universally algebraic if oneNi≥3

The coefficients are known by induction.

Raimar Wulkenhaar (M¨unster)Exact solution of the quartic matrix model and application to 4D NCQFT

IntroductionSolution of quartic matrix modelMoyalφ44-theory in batrix basisPosition spaceOutlookAppendix

Renormalisation theorem

The renormalisation leaves algebraic equations invariant:

Theorem

Given a real scalar matrix model withS=Vtr(Eφ2+λ4φ4)and m?→Eminjective, which determines the set

G|p11...p1N1|...|pB1...pBNB|

of (N1+...+...NB)-point functions. E a?→Z(Ea+μ2

2-μ2bare2)andλ?→Z2λ.

Then all functions with one

Ni≥3

1are finitewithout further need of a renormalisation ofλ, i.e.

all renormalisable quartic matrix models have vanishing

β-function

2aregiven by algebraic recursion formulaein terms of

Raimar Wulkenhaar (M¨unster)Exact solution of the quartic matrix model and application to 4D NCQFT

IntroductionSolution of quartic matrix modelMoyalφ44-theory in batrix basisPosition spaceOutlookAppendix

Graphical realisation(B=1,g=0)

Gb0b1b2b3= (-λ)Gb0b1Gb2b3-Gb0b3Gb2b1(Eb0-Eb2)(Eb1-Eb3)=-λ?• G b0...b5=λ2? b i bj=Gbibjleads tonon-crossing chord diagrams; these are counted by the

Catalan numberCN

2=N!(N2+1)!N2!

b ibj=1Ebi-Ebjleads torooted treesconnecting theevenorodd vertices, intersecting the chords only at vertices Open Problem (Combinatorics):Which trees arise for a given chord diagram?

Raimar Wulkenhaar (M¨unster)Exact solution of the quartic matrix model and application to 4D NCQFT

IntroductionSolution of quartic matrix modelMoyalφ44-theory in batrix basisPosition spaceOutlookAppendix

φ44on Moyal space with harmonic propagation

Moyal product(f?g)(x) =?

R d×Rddx dk(2π)df(x+12Θk)g(x+y)ei?k,y?

S[φ]=64π2?

d (x) renormalisable as formal power seriesinλ[Grosse-W., 2004] (renormalisation of

μ2bare,λ,Z?R+andΩ?[0,1])

means: well-defined perturbativequantum field theory

Langmann-Szabo duality (2002): theories atΩandΩ?=1Ωare the same; self-dual caseΩ =1ismatrix model

β-function vanishes to all ordersinλforΩ =1 [Disertori-Gurau-Magnen-Rivasseau, 2006] means: almost scale-invariant

Is the self-dual (critical) model integrable?

Raimar Wulkenhaar (M¨unster)Exact solution of the quartic matrix model and application to 4D NCQFT

IntroductionSolution of quartic matrix modelMoyalφ44-theory in batrix basisPosition spaceOutlookAppendix

Matrix basis and thermodynamic limit

Moyal algebra has matrix basis [Gracia-Bond´ıa+V´arilly, 1988]:

φ(x)=?

m ,n?N2φ m nfmn(x),fmn(x) =fm1n1(x0,x1)fm2n2(x3,x4) fmn(y0,y1)=2(-1)m?m! n!? 2

θy?

n-mLn-mm?2|y|2)θ? e-|y|2

θ,y=y0+iy1

satisfies(fkl?fmn)(x) =δmlfkn(x),? R

4dx fmn(x) = (2πθ)2δmn

previous action becomes forΩ =1

S[φ] =V

m ,n?N2N

Emφmnφnm+Z2λ

4? m,n,k,l?N2Nφ m nφnkφklφlm

V=?θ4?

2is forΩ =1 thevolumeof the noncommutative

manifold which is sent to∞in the thermodynamic limit. We do this in ascaling limitN⎷V= Λ2μ2=const

Raimar Wulkenhaar (M¨unster)Exact solution of the quartic matrix model and application to 4D NCQFT

IntroductionSolution of quartic matrix modelMoyalφ44-theory in batrix basisPosition spaceOutlookAppendix

Integral equations

Matrix indices become continuous|p|⎷V?→μ2pwithp?[0,Λ2]

Normalised planar 2-point functionGab=μ2G(0)

|ab|,a,b?[0,Λ2] Difference of eqns forGabandGa0cancels worst divergence normalisation conditionsG00=1anddGab db??a=b=0=-(1+Y) Integral equation for H¨older-continuousGabandΛ→ ∞ ?b a+1+λπaHa?

G•0?

aGa0

Dab-λπHa?D•b?=-Ga0

where

Dab:=ab(Gab-Ga0),Y=-λ?∞

0dppDp0

Hilbert transformHa[f(•)] :=1πlim?→0?

?a-? 0 a+?? f(q)dqq-a

Raimar Wulkenhaar (M¨unster)Exact solution of the quartic matrix model and application to 4D NCQFT

IntroductionSolution of quartic matrix modelMoyalφ44-theory in batrix basisPosition spaceOutlookAppendix

The Carleman equation

Theorem [Carleman 1922, Tricomi 1957]

The singular linear integral equation

h(x)y(x)-λπHx[y] =f(x),x?[-1,1] is for h(x)continuous + H¨older near±1 andf?Lpsolved by y(x)=sin(?(x))λπ? f(x)cos(?(x)) +eHx[?]Hx? e-H•[?]f(•)sin(?(•))?

CeHx[?]

1-x ?(x) =arctan [0,π]?

λπh(x)?

,sin(?(x)) =|λπ|?(h(x))2+ (λπ)2 where

Cis an arbitrary constant.

Assumption:

C=0

Raimar Wulkenhaar (M¨unster)Exact solution of the quartic matrix model and application to 4D NCQFT

IntroductionSolution of quartic matrix modelMoyalφ44-theory in batrix basisPosition spaceOutlookAppendix

Solution

angle?b(a) :=arctan[0,π]?

λπab+1+λπaHa[G•0]Ga0?

Ga0is solved for?0(a):Ga0=sin(?0(a))|λ|πaeHa[?0(•)]-H0[?0(•)]

Addition theorems and Tricomi's identity

e-Ha[?b]cos(?b(a)) +Ha? e -H•[?b]sin(?b(•)? =1give:

Theorem

2

Consequence:Gab≥0!

Y=λ?

2

Raimar Wulkenhaar (M¨unster)Exact solution of the quartic matrix model and application to 4D NCQFT

IntroductionSolution of quartic matrix modelMoyalφ44-theory in batrix basisPosition spaceOutlookAppendix

The self-consistency equation

Given boundary valueGa0,

ab 2Λ 2 0 Gab

Carleman computesGab,

in particular G0b symmetry forcesGb0=G0b

Master equation

The theory is completely determined by the solution of thefixed point equationG=TG

Gb0=11+bexp?

b 0 dt?

0dp(λπp)2+?t+1+λπpHp[G•0]

Gρ0

?2?

Raimar Wulkenhaar (M¨unster)Exact solution of the quartic matrix model and application to 4D NCQFT

IntroductionSolution of quartic matrix modelMoyalφ44-theory in batrix basisPosition spaceOutlookAppendix

Existence proof

The operatorTsatisfies assumptions ofSchauder fixed point theorem . Define

Kλ:=?

1+b+Cλ?f(b)?

withCλfrom2λP2λ(1+Cλ)eCλPλ=1atPλ=exp(-1λπ2)⎷1+4λ. Then:

1Kλconvex

2TKλ? Kλ

?TKλis relatively compact inKλby variant of Arzel´a-Ascoli

4T:Kλ→ Kλis continuous

This provides exact solution ofφ4-QFT on 4D Moyal space atθ→∞quotesdbs_dbs14.pdfusesText_20

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