The complete solution of the quartic analogue of the Kontsevich … · 2020. 8. 28. · The complete solution of the quartic analogue of the Kontsevich model Raimar Wulkenhaar Mathematisches

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 & DS 2-point function Finite N Higher topologies Discussion TR

IntroductionWe still admire the straightedge and compass constructionsconceived by ancient Greek mathematicians. They wereunsuccessful for others (squaring the circle, trisecting an angle).

Cubic and quartic equations are solvable [Cardano 1539-45].Led to the invention of C. Higher polynomial equations notsolvable by radicals [Ruffini 1799, Abel 1824, Galois 1835].

The two-body problem is solvable [Newton 1687], thethree-body problem is not. This failure pushed thedevelopment of perturbation theory.

Hydrogen atom is solvable [Pauli 1925], other atoms are not.

Many spin models [critical, 3-state Potts, Heisenberg,6-vertex, 8-vertex,. . . ] are solvable. But not all.

Common to all these solvable cases is a certain beauty, and alasting impact in mathematics and physics.

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


Matrix models

Today we have a long list of such nice examples. Some of themcome from matrix models.

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 quantumgravity. We introduce it on the next two slides.

The Hermitian 2-matrix model is solvable [Staudacher 93].Understanding the common mechanism behind the 1- and2-matrix models led to the discovery of topologicalrecursion by Eynard and others.

We are now sure to have another solvable matrix model: thequartic analogue of the Kontsevich model. The talk is about it.



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


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.Moments evaluate to explicitly known rational functions.Integrability is not yet proved, but it seems the onlyreasonable explanation.



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.

In fact only the conjugacy class of σ matters, i.e. its cycletype (`1(σ), . . . , `n(σ)) with

∑i i`i = n.

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

Nn〈 b∏

j=1

( nj∏i=1

ek ji kji+1

)〉c

=: N2−bG|k11 ...k1n1 |...|kb1 ...k

bnb|, k

jnj +1≡k j1

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)



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 gise to Dyson-Schwinger equations between the G....

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


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)...



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).



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)

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


D = 4 Moyal space: %0(t) = t [Grosse-Hock-W 19]%λ(x) ≡ %0(R(x)) = R(x) = x − λx2

∫∞0

dt %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); the latter combine to αλ =

arcsin(λπ)π :

%λ(x) =x

1− αλ

∞∑n=0

α2nλ Hlog(xµ2, [0,−1, . . . ,0,−1︸︷︷︸

2n

])

− x + µ2

1− αλ

∞∑n=0

α2n+1λ Hlog(xµ2, [−1,0,−1, . . . ,0,−1︸︷︷︸

2n+1

])

= x 2F1(αλ,1− αλ

2

∣∣∣− xµ2

), αλ =

{arcsin(λπ)

π for |λ| ≤1π

12 + i

arcosh(λπ)π for λ ≥

1π

The interaction alters the spectral dimension to4− 2arcsin(λπ)π and thus avoids the triviality problem!



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



RationalityUsing 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(z + w)

d∏k ,l=1

(εk + εl)(−ŵk − ẑ l)(εk − ẑ l)(εl − ŵk )

=

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)Relation 3 of the ansatz is identically satisfied!!

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



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



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


c

Ta1,...,an‖b|c| =[ ∫

X ′N

dµE,λ(Φ)Φ(ebb)Φ(ecc)n∏

i=1

( d∑ki =1


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


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



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)

}



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 .

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)

separates the poles. . .



DSE III: Ω(0)2 (v , z) continued

R′(z)Ω(0)2 (v , z) +∂

∂R(v)

( 1v + z

+1

v − z

)=

1G0(z)

[λ

N2

d∑n,k=1

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

+λ2

N2

d∑k,l,m,n=1

Cm,nk,l( 1

(v−ε̂l n)2+ 1

(v−ε̂k m)2)

R′(v)(z+ε̂l n)(z−ε̂k m)

]We do not know what the rhs is, but we know it has at mostpoles at the zeros of G0.Assuming Ω(0)2 (v , z) is regular there, both sides must beconstant in z by Liouville, then zero when z →∞:

PropositionThe cylinder ampitude is

Ω(0)2 (v , z) =

1R′(v)R′(z)

( 1(v + z)2

+1

(v − z)2)

One recognises the Bergmann kernel of topological recursion!Raimar Wulkenhaar (Münster) The complete solution of the quartic analogue of the Kontsevich model 18


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!



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)



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 in 2009.

The breakthrough was to solve the 2D Moyal case with Erikin Les Houches 2018.

After Alex’ insight in that particular solution we solved (withHarald & Alex) last year the general continuous case.

The even deeper discrete case with its link to algebraicgeometry was solved last December with Jörg.

The solution introduced a rational function

R(z) = z − λN

d∑k=1

%kεk + z

with R(εk ) = Ek & %kR′(εk ) = rk .



The solution is completeAll 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 .

The final results for these Ω(g)n are remarkably simple.

ω(g)n (z1, . . . , zn) := (

∏nl=1 R

′(zi))Ω(g)n (z1, . . . , zn) has poles

only at zk = −zl and zk = βi where R′(βi) = 0 (the branchpoints of the curve), but not at zk = ±ẑl

j or zl = ±ε̂k j .Raimar Wulkenhaar (Münster) The complete solution of the quartic analogue of the Kontsevich model 22


Analogies to the Hermitian 2-matrix model

It is more or less obvious that the ω(g)n are the objectsusually studied in topological recursion [Eynard-Orantin 07].

The need to go beyond TR was previously observed in theHermitian 2-matrix model where the boundaries ofRiemann surfaces are bicoloured (or spins ±1: Ising model).

The uniformly coloured sectors H(g)0,m,0 and H(g)0,0,n relate to

topological recursion. The sectors H(g)0,m,n and H(g)1,m,n are

computed in a triangular pattern.

H(g)0,m,0(z1, . . . , zm) and H(g)0,0,n(z1, . . . , zn) have poles only at

branch points of curve x and zk = −zl (not zk = 0).The mixed sectors also have poles at zk = ẑl

j .



Blobbed topological recursion?

At fixed g, our Ω(g)n , T (g) seem to play the same rôle:Ω

(g)n correspond to uniform sector, T (g) to mixed sector.

The part of Ω(1)1 with poles at branch points is identical tothe 2-matrix model when taking the spectral curve

x(z) = R(z), y(z) = −R(−z)On top of it we have an additional pole at z = 0 from G(0)(z|z).

It seems that the 2-matrix model and our quartic model are twoexamples of an extension of topological recursion.

We currently investigate whether it could be the blobbedtopological recursion [Borot-Shadrin 15].Our hierarchy of loop equations generates the blobs asadditional input for the recursion.



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

t i+k =R(k+2)(βi)

R′′(βi), t i−k =

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?



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 branch 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, . . .



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


Main partIntroduction

Cumulants & DS

2-point function

Finite N

Higher topologies

Discussion

TR

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The complete solution of the quartic analogue of the Kontsevich … · 2020. 8. 28. · The complete solution of the quartic analogue of the Kontsevich model Raimar Wulkenhaar Mathematisches