The complete solution of the quartic analogueof the Kontsevich model

Raimar Wulkenhaar

Mathematisches Institut der Westfälischen Wilhelms-Universität Münster

based on joint work withHarald Grosse, Erik Panzer, Alexander Hock, Jörg Schürmann

and work in progress with Johannes Branahl and Alexander HockRaimar Wulkenhaar (Münster) The complete solution of the quartic analogue of the Kontsevich model 0

Introduction Cumulants & Dyson-Schwinger 2-point function Finite N Higher topologies Discussion TR

Introduction

This project started in 1998 as an attempt to understandquantum field theories on noncommutative geometries.

No interacting and mathematically consistent QFT isknown in 4 dimensions.The hope was that the situation could improve onnoncommutative spaces.Many nice results were obtained on renormalisation,β-function, construction in 2D, extension to tensor models.

We know since 2003 that there are parallels with theKontsevich model (which beautifully links mathematics and physics).

Recent progresswhen we understood that topological recursion, a universalstructure behind many developments in algebraic geometry(including the Kontsevich model), is also present in our case.

Raimar Wulkenhaar (Münster) The complete solution of the quartic analogue of the Kontsevich model 1

Introduction Cumulants & Dyson-Schwinger 2-point function Finite N Higher topologies Discussion TR

Solvable matrix models

. . . provide deep relations between many fields of mathematicsand physics:

Hermitian 1-matrix model Z =∫

XNdM e−tr(polynomial(M))

is solvable (XN – space of self-adjoint N × N-matrices)[Brézin-Kazakov, Douglas-Shenker, Gross-Migdal 89/90]

It is deeply equivalent to the [Kontsevich 91]-model. Itproves a conjecture by [Witten 91] about 2D quantum gravity.

The Hermitian 2-matrix model is solvable [many people].

Understanding the common mechanism led to thediscovery of topological recursion by Eynard and others.

We are now sure to have another solvable matrix model: thequartic analogue of the Kontsevich model.

Raimar Wulkenhaar (Münster) The complete solution of the quartic analogue of the Kontsevich model 2

Introduction Cumulants & Dyson-Schwinger 2-point function Finite N Higher topologies Discussion TR

Free Euclidean fields on noncommutative geometriesLet XN be the real vector space of self-adjoint N × N-matrices,and (E1, . . . ,EN) be (increasing) positive real numbers.

Theorem [Bochner 1933, Schur 1911]For any inner product 〈 , 〉 on XN there exists a uniqueprobability measure dµ0 on the dual space X ′N with

exp(−1

2〈M,M〉

)=

∫X ′N

dµ0(Φ) eiΦ(M) ∀M=(Mkl) ∈ X .

Choose 〈M,M ′〉E =1N

N∑k ,l=1

MklM ′lkEk + El

and corresponding dµE ,0

Defines the free Euclidean scalar field on N-dimensionalapproximation of a noncommutative geometry.(E1, . . . ,EN) is truncated spectrum of the Laplacian.

Next: Deform dµE ,0(Φ) and study large-N asymptotics.Raimar Wulkenhaar (Münster) The complete solution of the quartic analogue of the Kontsevich model 3

Introduction Cumulants & Dyson-Schwinger 2-point function Finite N Higher topologies Discussion TR

Two deformations

3 The Kontsevich model dµE ,λ(Φ) =e−

λN3 Tr(Φ

3)dµE ,0(Φ)∫X ′N

e−λN3 Tr(Φ

3)dµE ,0(Φ)

Computes intersection numbers of tautological characteristicclasses on the moduli spaceMg,n of stable complex curves.It is integrable via a relation (suggested by Witten) to theKdV hierarchy. Its moments satisfy topological recursion.

4 A quartic analogue dµE ,λ(Φ) =e−

λN4 Tr(Φ

4)dµE ,0(Φ)∫X ′N

e−λN4 Tr(Φ

4)dµE ,0(Φ)

Although perturbatively far apart, we find very similaralgebraic geometrical structures. Our solutions are exact in λ.

Raimar Wulkenhaar (Münster) The complete solution of the quartic analogue of the Kontsevich model 4

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Moments and cumulants∫X ′N

dµE ,λ(Φ)n∏

i=1

Φ(eki li ) =:〈 n∏

i=1

eki li〉

=∑

partitionsπ of {1,...,n}

∏blocksβ∈π

〈∏i∈β

eki li〉

c

A culumulant〈∏n

i=1 eki li〉

c is only non-zero if(l1, . . . , ln) = (kσ(1), . . . , kσ(n)) for permutation σ, and onlythe cycle type matters.

Adapted notation for b cycles of lengths n1, . . . ,nb:

Nn1+..+nb〈 b∏

j=1

( nj∏i=1

ek ji kji+1

)〉c

=: N2−bG|k11 ...k1n1 |...|kb1 ...k

bnb|

Expansion Z(M) :=∫

X ′N

dµE,λ(Φ) eiΦ(M)

= 1− 1N2

N∑k,l=1

(NG|kl|

��cycle type (0,1)

MklMlk1! · 21

+ G|k|l|��

cycle type (2,0)

Mkk Mll2! · 12

)+O(M4)

Raimar Wulkenhaar (Münster) The complete solution of the quartic analogue of the Kontsevich model 5

Introduction Cumulants & Dyson-Schwinger 2-point function Finite N Higher topologies Discussion TR

Equations of motionFourier transform Z(M) :=

∫X ′N

dµE ,λ(Φ) eiΦ(M) satisfies(in quartic case)

1i∂Z(M)∂Mab

=iMbaZ(M)

N(Ea + Eb)− λ

i3(Ea + Eb)

N∑k ,l=1

∂3Z(M)∂Mak∂Mkl∂Mlb

.

1N∂Z(M)∂Ea

=( N∑

k=1

∂2

∂Mak∂Mka+

1N

N∑k=1

G|ak | +1

N2G|a|a|

)Z(M)

They give rise to Dyson-Schwinger equations between the G....

Remark: In Kontsevich model (cubic case), first equation reads

1i∂Z(M)∂Mab

=iMbaZ(M)

N(Ea + Eb)− λ

i2(Ea + Eb)

N∑k=1

∂2Z(M)∂Mak∂Mkb

.

For N = 1, λ = i this is the Airy differential equation.Raimar Wulkenhaar (Münster) The complete solution of the quartic analogue of the Kontsevich model 6

Introduction Cumulants & Dyson-Schwinger 2-point function Finite N Higher topologies Discussion TR

Dyson-Schwinger equations [Grosse-W 09, 12]

(Ea + Eb)G|ab| = 1−λ

N

N∑p=1

G|ab|G|ap| +λ

N

N∑p=1

G|pb| −G|ab|Ep − Ea

− λN2(−

G|b|b| −G|a|b|Eb − Ea

+ G|abab| + G|baaa|

+ G|ab|G|a|a| +1N

N∑p=1

G|ab|ap|)− λ

N4G|a|a|ab|

(Ea + Ea)G|a|b| = −λ

N

N∑p=1

G|ap|G|a|b| +λ

N

N∑p=1

G|p|b| −G|a|b|Ep − Ea

+λG|bb| −G|ab|

Eb − Ea− λ

N2(

G|b|aaa| + G|a|abb|

+ 3G|a|b|G|a|a| +1N

N∑p=1

G|a|b|ap|)− λ

N4G|b|a|a|a|

DSEs decouple in formal genus expansion G... =∑∞

g=0 N−2gG(g)...

Raimar Wulkenhaar (Münster) The complete solution of the quartic analogue of the Kontsevich model 7

Introduction Cumulants & Dyson-Schwinger 2-point function Finite N Higher topologies Discussion TR

The planar 2-point function G(0)|ab| (of cycle type (0,1))

. . . extends to holomorphic function G(0) : U × U → C onneighbourhood U of {E1, . . . ,EN} with G(0)(Ea,Eb) = G

(0)|ab|.

If Ei has multiplicity ri , with r1+...+rd =N, then G(0) satisfies(ζ+η+

λ

N

d∑k=1

rk G(0)(ζ,Ek ))

G(0)(ζ, η) = 1 +λ

N

d∑k=1

rkG(0)(Ek , η)−G(0)(ζ, η)

Ek − ζ

Alternatively, setting %0(t) = 1N∑d

k=1 rkδ(t − Ek ),(ζ+η+λ

∫dt %0(t)G(0)(ζ, t)

)G(0)(ζ, η) = 1 + λ

∫dt %0(t)

G(0)(t , η)−G(0)(ζ, η)t − ζ

1 With Erik Panzer we solved this equation for %0(t) ≡ 1.2 Alexander Hock noticed a remarkable pattern in our solution.3 His observation gave rise to a general solution method.4 Connections to algebraic geometry (with Jörg Schürmann).

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Solution of the non-linear integral equation

Theorem [Panzer-W 18 for %0 = 1, Grosse-Hock-W 19]

1 Ansatz G(0)(x , y) =eHx [τy (•)] sin τy (x)

Zλπ%0(x)Z=renormalisation

Hx [f ] = 1π −∫ dp f (p)

p−x

2 τy (x) = Im log(y + I(x+i�)

)with I(ζ) = −R(−µ2 − R−1(ζ))

3 R(z) = z − λ(−z)D/2∫ ∞

0

dt %λ(t)(µ2 + t)D/2(t + µ2 + z)

D = 2[ δ2 ] at spectral dimension δ = inf(p :

∫ dt%0(t)(1+t)p/2

Introduction Cumulants & Dyson-Schwinger 2-point function Finite N Higher topologies Discussion TR

D = 4 Moyal space: %0(t) = t [Grosse-Hock-W 19]

%λ(x) ≡ %0(R(x)) = R(x) = x − λx2∫∞

0dt %λ(t)

(µ2+t)2(t+x)

If %λ(t) ∼ %0(t) = t , then R(x) bounded above.Consequently, R−1 would not be globally defined: triviality!Fredholm equation perturbatively solved by iterated integrals:Hyperlogarithms and ζ(2n) which can be summed to

%λ(x) = x · 2F1(αλ,1− αλ

2

∣∣∣− xµ2

)αλ =

{arcsin(λπ)

π for |λ| ≤1π

12 + i

arcosh(λπ)π for λ ≥

1π

Corollary

The interaction alters the spectral dimension to 4− 2arcsin(λπ)πand thus avoids the triviality problem.

Raimar Wulkenhaar (Münster) The complete solution of the quartic analogue of the Kontsevich model 10

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Solution for finite N [Schürmann-W 19]

Recall(η+ ζ+

λ

N

d∑k=1

rk G(0)(ζ,Ek ) +λ

N

d∑k=1

rkEk − ζ

)G(0)(ζ, η)

= 1 +λ

N

d∑k=1

rkG(0)(Ek , η)

Ek − ζ, r1 + · · ·+ rd = N

Assume there is a branched covering R : Ĉ→ Ĉ with1 R has degree d + 1 and maps (for λ > 0) neighbourhood

of R+ bijectively to neighbourhood U 3 ζ, η,Ek of R+2 ζ = R(z), η = R(w), Ek = R(εk ), G(0)(ζ, η) = G(0)(z,w)

3 R(z) +λ

N

d∑k=1

rkG(0)(z, εk ) +λ

N

d∑k=1

rkR(εk )− R(z)

= −R(−z)

Gives (R(w)− R(−z))G(0)(z,w) = 1 + λN

d∑k=1

rkG(0)(εk ,w)

R(εk )− R(z)Take z = −ŵl for other d preimages of η = R(w) = R(ŵ l).Express G(0)(εk ,w) in terms of R(−ŵ l), R(εk ); insert back

Raimar Wulkenhaar (Münster) The complete solution of the quartic analogue of the Kontsevich model 11

Introduction Cumulants & Dyson-Schwinger 2-point function Finite N Higher topologies Discussion TR

Rationality

Using formulae for inverses of Cauchy matrices ( 1ak−bl )kl andtheir row sums [Schechter 59]:

Theorem [Schürmann-W 19, extending Grosse-Hock-W 19]

R(z) = z − λN

d∑k=1

%kz + εk

where Ek = R(εk ), rk = R′(εk )%k

G(0)(z,w)

=

1− λN

d∑k=1

rk(R(z)− R(εk ))(R(εk )− R(−w))

d∏j=1

R(w)− R(−ε̂k j )R(w)− R(εj )

R(w)− R(−z)G(0)(z,w)=G(0)(w , z), and 3 of the ansatz identically satisfied!

Thus, planar 2-point function solved by the composition of arational function G(0) with inverse of another rational function R.

Raimar Wulkenhaar (Münster) The complete solution of the quartic analogue of the Kontsevich model 12

Introduction Cumulants & Dyson-Schwinger 2-point function Finite N Higher topologies Discussion TR

Planar 2-point function of cycle type (2,0)

Dyson-Schwinger equation extends meromorphically to

(R(z)−R(−z)

)G(0)(z|w)− λ

N

d∑k=1

rkG(0)(εk |w)R(εk )−R(z)

= λG(0)(z,w)−G(0)(w ,w)

R(z)− R(w)

Proposition [Schürmann-W 19]

G(0)(z|w)

=λ

(R(z)− R(w))2

{G(0)(z,w)

−

(R(z)+R(w)−2R(0))d∏

k=1

(R(z)− R(αk ))(R(w)− R(αk ))(R(z)− R(εk ))(R(w)− R(εk ))

(R(z)−R(−z))(R(w)−R(−w))

}

where {0,±αk} are all roots of R(z)− R(−z) = 0

Raimar Wulkenhaar (Münster) The complete solution of the quartic analogue of the Kontsevich model 13

Introduction Cumulants & Dyson-Schwinger 2-point function Finite N Higher topologies Discussion TR

Higher topologies [Branahl-Hock-W soon]

G(0)(z,w) is topologically a disk (sphere with 1 hole)G(0)(z|w) is topologically a cylinder (sphere with 2 holes)Does the same construction exhaust all higher topologies?

NO! A new structure is needed [Branahl-Hock-W, w.i.p]

Ωa1,...,an =

[ ∫X ′N

dµE,λ(Φ)n∏

i=1

( d∑ki =1

rki Φ(eai ki )Φ(eki ai ))]

c

Ta1,...,an‖bc| =[ ∫

X ′N

dµE,λ(Φ)Φ(ebc)Φ(ecb)n∏

i=1

( d∑ki =1

rki Φ(eai ki )Φ(eki ai ))]

c

Ta1,...,an‖b|c| =[ ∫

X ′N

dµE,λ(Φ)Φ(ebb)Φ(ecc)n∏

i=1

( d∑ki =1

rki Φ(eai ki )Φ(eki ai ))]

c

Complexified Ω(g)n (z1, . . . , zn) relate to topological recursion!Raimar Wulkenhaar (Münster) The complete solution of the quartic analogue of the Kontsevich model 14

Introduction Cumulants & Dyson-Schwinger 2-point function Finite N Higher topologies Discussion TR

Dyson-Schwinger equations I: G from T

(R(w)− R(−z))G(g)(z,w |J )− λN

∑k

rkG(g)(εk ,w |J )R(εk )− R(z)

= δ0,|J |δg,0 − λ{ ∑I]I′=J , h+h′=g

(I,h) 6=(∅,0)

G(h′)(z,w |I′)T (h)(z‖I) + T (g−1)(z‖z,w |J )

+∑

I]I′=J , h+h′=g

G(h)(w |I)G(h′)(z|I′)− G(h

′)(w |I′)R(w)− R(z)

+b∑β=2

nβ∑j=1

G(g)(w , z, zβj , .., zβnβ+j−1

|J \{Jβ})− G(g)(w , zβj , .., zβnβ+j|J \{Jβ})

R(zβj )− R(z)

+G(g−1)(z|w |J )− G(g−1)(w |w |J )

R(w)− R(z)

}Conditions on cardinalities of J , I, I ′Inversion of first line via Cauchy matricesconvention G(g)(z,w |J ) = T (g)(∅‖z,w |J |) on next pages

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Dyson-Schwinger equations II: T from Ω(R(w)− R(−z))T (g)(u1, ...,um‖z,w |J |)−

λ

N

∑k

rkT (g)(u1, ...,um‖εk ,w |J |)

R(εk )− R(z)

= δ0,mδ0,|J |δg,0 − λ

{ ∑K]K′={u1,..,um}

h+h′=g, (K,h)6=(∅,0)

T (h′)(K′‖z,w |J |)Ω(h)|K+1|(K, z)

+∑

I]I′=J , I6=∅K]K′={u1;..;um}, h+h′=g

T (h′)(K′‖z,w |I ′|)T (h)(K, z‖I|)

+m∑

i=1

∂

∂R(ui)T (g)(u1, .., ǔi , ..,um‖ui ,w |J |)

R(ui)− R(z)+ T (g−1)(u1, ...,um, z‖z,w |J |)

+T (g−1)(u1, ...,um‖z|w |J |)− T (g−1)(u1, ...,um‖w |w |J |)

R(w)− R(z)

+b∑

β=2

nβ∑j=1

T (g)(K‖w , z, zβj , .., zβnβ+j−1|J \{J

β}|)− T (g)(K‖w , zβj , .., zβnβ+j|J \{Jβ}|)

R(zβj )− R(z)

∣∣∣K={u1,..,um}

+∑I]I′=J

K]K′={u1,..,um}, h+h′=g

T (h)(K‖w |I|)T(h′)(K′‖z|I ′|)− T (h′)(K′‖w |I ′|)

R(w)− R(z)

}

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Dyson-Schwinger equation III: Ω(0)2 (v , z)

Ω(0)2 (v , z)R

′(z)G0(z)−λ

N2

d∑n,k=1

rk rnT (0)(v‖εk , εn|)(R(εk )− R(z))(R(εn)− R(−z))

= − ∂∂R(v)

(G(0)(v , z) + G(0)(v ,−z)

)where G0(z) = Resw→−z G(0)(z,w).

Seems to need T (0)(v‖εk , εn|) which itself needs Ω(0)2 .

But poles separate by partial fraction decomposition

G(0)(z, v) = G0(z)v + z

+λ2

N2

d∑k ,l,m,n=1

Cm,nk ,l(z + ε̂l n)(z − ε̂k m)(v − ε̂l n)

Proposition

Ω(0)2 (v , z) =

1R′(v)R′(z)

( 1(v − z)2

+1

(v + z)2)

One recognises the Bergman kernel of topological recursion!Raimar Wulkenhaar (Münster) The complete solution of the quartic analogue of the Kontsevich model 17

Introduction Cumulants & Dyson-Schwinger 2-point function Finite N Higher topologies Discussion TR

Dyson-Schwinger equation IV: Ω from lower T ,Ω

R′(z)G0(z)Ω(g)m+1(u1, ...,um, z) +

λ

N2∑n,k

rnrkT (g)(u1, ...,um‖εk , εn|)

(R(εk )− R(z))(R(εn)− R(−z))

=δg,0δm,1

(R(z)− R(u1))2−

∑K]K′={u1;..;um}

(K,h)6=(∅,0)6=(K′,h′)h+h′=g

Ω(h′)|K′+1|(K

′, z)λ

N

∑n

rnT (h)(K‖z, εn|)R(εn)− R(−z)

−m∑

i=1

∂

∂R(ui)

λN∑

n rnT (g)(u1,..,ǔi ,..,um‖ui ,εn|)

R(εn)−R(−z)

R(ui)− R(z)− λ

N

∑n

rnT (g−1)(u1, ...,um, z‖z, εn|)

R(εn)− R(−z)

− λN

∑n

rnT (g−1)(u1, ...,um‖z|εn|)− T (g−1)(u,; ...,um‖εn|εn|)

(R(εn)− R(z))(R(εn)− R(−z))

−m∑

i=1

∂

∂R(ui)T (g)(u1, .., ǔi , ..,um‖ui , z|) + T (g−1)(u1, ...,um‖z|z|)

Can be solved without knowing the green term!Miraculously, all poles on rhs (other than ±ε̂nj ) have prefactor G0!

Raimar Wulkenhaar (Münster) The complete solution of the quartic analogue of the Kontsevich model 18

Introduction Cumulants & Dyson-Schwinger 2-point function Finite N Higher topologies Discussion TR

DSE V: cycle type split (0,1) to (2,0)

(R(z)− R(−z))T (g)(u1, ...,um‖z|J |)−λ

N

∑k

rkT (g)(u1, ...,um‖εk |J |)

R(εk )− R(z)

= −λ{ ∑

I]I′=J , I6=∅K]K′={u1,..,um}, h+h′=g

T (h′)(K′‖z|I ′|)T (h)(K, z‖I|) +∑

K]K′={u1,..,um}h+h′=g, (K,h)6=(∅,0)

T (h′)(K′|z|J |)Ω(h)|K+1|(K, z)

+m∑

i=1

∂

∂R(ui)T (g)(u1, .., ǔi , ..,um‖ui |J |)

R(ui)− R(z)+ T (g−1)(u1, ...,um, z‖z|J |)

+b∑

β=2

nβ∑j=1

T (g)(K‖z, zβj , .., zβnβ+j−1|J \{J

β}|)− T (g)(K‖zβj , .., zβnβ+j|J \{Jβ}|)

R(zβj )− R(z)

∣∣∣K={u1,..,um}

}

G(g)(z|J ) = T (g)(∅‖z|J ) (m = 0)first line inverted via Cauchy matrices at z = αknecessary input for genus increase Ω(g)(z) 7→ Ω(g+1)(z)

Raimar Wulkenhaar (Münster) The complete solution of the quartic analogue of the Kontsevich model 19

Introduction Cumulants & Dyson-Schwinger 2-point function Finite N Higher topologies Discussion TR

The solution is complete

We study the quartic analogue of the Kontsevich model dµE ,λ,parametrised by coupling constant λ and spectral valuesE1, . . . ,Ed of multiplicities r1, . . . , rd with r1 + · · ·+ rd = N.

Main obstacle was the non-linear equation for the planar2-point function obtained with Harald Grosse in 2009.

The breakthrough was to solve the 2D Moyal case withErik Panzer in Les Houches 2018.

We found soon the solution in full generality. The discretecase links to algebraic geometry.

The solution introduced a rational function

R(z) = z − λN

d∑k=1

%kεk + z

with R(εk ) = Ek & %kR′(εk ) = rk .

Raimar Wulkenhaar (Münster) The complete solution of the quartic analogue of the Kontsevich model 20

Introduction Cumulants & Dyson-Schwinger 2-point function Finite N Higher topologies Discussion TR

The solution is complete

All other correlation functions (or cumulants) satisfy (aftermeromorphic extension and genus expansion) affine equations.

To solve them we need partial sums T (g)(u1, ...,um‖z,w |....|),T (g)(u1, ...,um‖z|w |....|) and Ω

(g)n (u1, ...,um).

We know how to invert the linear operators in all these loopequations. This is very easy for T (g) (unless one wants tosimplify the expression and to make the symmetry manifest).

Extracting Ω(g)n in practice becomes difficult for large2g + n. So far we have Ω(0)2 ,Ω

(0)3 ,Ω

(0)4 ,Ω

(1)1 and a third of Ω

(1)2 .

Raimar Wulkenhaar (Münster) The complete solution of the quartic analogue of the Kontsevich model 21

Introduction Cumulants & Dyson-Schwinger 2-point function Finite N Higher topologies Discussion TR

Blobbed topological recursion?Final results for these Ω(g)n are remarkably simple. Define

ω0,1(z) := −R(−z)R′(z)dz

ωg,n(z1, . . . , zn) := (−λ)2−2g−nΩ(g)n (z1, . . . , zn)

n∏l=1

R′(zi)dzi

For 2g + n − 2 < 0, these ωg,n seem to have poles only atzk = −zl and zk = βi where R′(βi) = 0 (the ramification pointsof the curve), but not at zk = ±ẑl

j or zl = ±ε̂k j .The parts of ωg,n with poles at ramification points seem tofollow the universal formula of topological recursion.The other poles could signal an extension of TR.

We currently investigate whether it could be the blobbed topo-logical recursion [Borot-Shadrin 15]. Our hierarchy of loop equa-tions generates the blobs as additional input for the recursion.

Raimar Wulkenhaar (Münster) The complete solution of the quartic analogue of the Kontsevich model 22

Introduction Cumulants & Dyson-Schwinger 2-point function Finite N Higher topologies Discussion TR

Integrability?

Understanding better our recursion should give access to thepartition function itself. It is a function of λ and the spectrum of E .

At fixed genus it is very likely a polynomial with rationalcoefficients in two sets of variables

xk ,i =R(k+2)(βi )

R′′(βi ), yk ,i =

R(k+1)(−βi )R′(−βi )

These contain the λ-dependence (exactly).Because our recursion inserts blobs at every next step (ina controlled way!), it will differ from known examples.

Main questionIs it nevertheless a τ -function for a Hirota equation,i.e. is it integrable?If so, are the rational coefficients intersection numbers ofsome characteristic classes on a moduli space?

Raimar Wulkenhaar (Münster) The complete solution of the quartic analogue of the Kontsevich model 23

Introduction Cumulants & Dyson-Schwinger 2-point function Finite N Higher topologies Discussion TR

Backup: Topological recursion [Eynard, Orantin 07]Starting from a spectral curve consisting of

a branched covering x : Σ→ Σ0 of Riemann surfaces,meromorphic differentials ω0,1 on Σ and ω0,2 on Σ× Σ,

recursively construct family ωg,n of meromorphic n-differentialson Σn, with poles only at ramification points of x , by

ωg,n(z1, ..., zn) =∑

a

Resz→a

K (z1, z, σa(z))dz(ωg−1,n+1(z, σa(z), z2, ..., zn)

+ ′∑

g1+g2=g

I1]I2={z2,...,zn}

ωg1,1+#I1 (z, I1)ωg2,1+#I2 (σa(z), I2))

[sum over ramification points a of x ; local involution x(z) = x(σa(z))

near a; recursion kernel K (z1, z2, z3) =12

∫ z2z′=z3

ω0,2(z1,z′)

ω0,1(z2)−ω0,1(z3)

]Examplesone- and two-matrix models, Kontsevich model, Weil-Peterssonvolumes, Hurwitz numbers, Gromov-Witten numbers, . . .

Raimar Wulkenhaar (Münster) The complete solution of the quartic analogue of the Kontsevich model 24

Introduction Cumulants & Dyson-Schwinger 2-point function Finite N Higher topologies Discussion TR

Topological recursion of the Kontsevich model

branched cover x : Ĉ 3 z 7→ z2 ∈ Ĉ, where z = (4ζ2 + c)1/2

ω0,1(z) = 2zy(−z)dz with −y(−z) = z + 1N∑N

k=11

2εk (z+εk )

(related to planar 1-point function), εk = (4E2k + c)1/2

ω0,2(z, z ′) = dz dz′

(z−z′)2 (related to planar 1+1-point function)

Meromorphic differentials relate to higher correlation functionsωg,n(z1, . . . , zn) = G(g)(z1| . . . |zn)

∏ni=1 d(x(zi)) , where

G(g)(z1| . . . |zn) = (2−t3)2−2g−n∑

l1,...,ln

〈ψl11 · · ·ψ

lnn e

∑k t̂kκk

〉g,n

n∏i=1

(2li +1)!!z2li +3i

ψi , κk are tautological characteristic classes onMg,n and〈. . . 〉g,n their intersection numbers

e−∑

k t̂k u−k

= 1− 12∑

l(2l+1)!!t2l+1u−l , tl = 1N

∑Nk=1 ε

−2l−1k

Raimar Wulkenhaar (Münster) The complete solution of the quartic analogue of the Kontsevich model 25

Main partIntroduction

Cumulants & Dyson-Schwinger

2-point function

Finite N

Higher topologies

Discussion

TR