Chapter VII Topological recursion and symplectic invariants We have seen, in almost all previous chapters, that symplectic invariants play an important role. They give the solution to Tutte’s recursion equation for maps, they give the formal expansion of various matrix integrals, including Kontsevich integral, and they also give the asymptotics of large maps. The goal of this chapter is to give their general definition, which is an algebraic geometry notion, and exists beyond the context of combinatorics, and beyond matrix models.

1

Symplectic invariants of spectral curves

Symplectic invariants were introduced in [2], as a common framework for the solution of loop equations of several matrix models: 1-matrix, 2-matrix, matrix with external field (in particular Kontsevich integral), chain of matrices,... , as well as their scaling limits. Then it was discovered that they have many nice properties, in particular symplectic invariance (whence their name), and that they appear in other problems of enumerative geometry. Here we only briefly summarize the construction of [2], and we refer the reader to the original article for more details.

1.1

Spectral curves

Definition 1.1 A spectral curve E = (L, x, y), is the data of a Riemann surface L, and two analytical functions x and y from some open domain of L to C. In some sense, we consider a parametric representation of the spectral curve y(x), where the space of the parameter z is a Riemann surface L. 191

B1

y

A1

A2

B2 y(z)

z x(z)

x

Definition 1.2 If L is a compact Riemann surface of genus g¯, and x and y are meromorphic functions on L, we say that the spectral curve is algebraic. If L = C ∪ {∞} is the Riemann sphere (thus g¯ = 0), we say that the spectral curve is rational. Indeed, for an algebraic spectral curve, it is always possible to find a polynomial relationship between x and y: ∀z ∈ L,

Pol(x(z), y(z)) = 0.

Definition 1.3 A spectral curve (L, x, y) is called regular if: • the differential form dx has a finite number (non vanishing) of zeroes dx(ai ) = 0, and all zeroes of dx are simple zeroes. • The differential dy does not vanish at the zeroes of dx, i.e. dy(ai ) %= 0. This means that near x(ai ), y(x) behaves locally like a square-root, or in other words, that the curve y(x) has a vertical tangent at ai . From now on, we assume that we are considering only regular spectral curves. Symplectic invariants are defined only for regular spectral curves, and they diverge when the curve becomes singular. Examples of singular spectral curves appear in chapter V, where they play a central role in the double scaling limit, i.e. the limit of large maps. ˜ x Definition 1.4 We say that two spectral curves E = (L, x, y) and E˜ = (L, ˜, y˜) are ˜ symplectically equivalent if L = L, and if y˜d˜ x − ydx is an exact form, i.e. it is the differential of a function on L. The group of symplectomorphisms is generated by: • x → x, y → y + R(x), where R(x) = rational function of x. • x→

ax+b , cx+d

y→

(cx+d)2 ad−bc

y.

• x → y, y → −x.

all those transformations conserve the symplectic form dx ∧ dy The main property of the Fg ’s we are going to define, is that they are symplectic invariants, i.e. two regular spectral curves which are symplecticaly equivalent, have the same Fg ’s. 192

1.2

Geometry of the spectral curve

Topology Consider a compact Riemann surface L, of genus g¯. If it is simply connected, the genus is g¯ = 0, and L is the Riemann sphere, i.e. the complex plane compactified with a point at ∞. If it is of genus g¯ ≥ 1, it is not simply connected, and one can find a basis of 2¯ g non-contractible cycles, which can be normalized in order that their intersections (the sign of the intersection corresponds to the orientation of contours, positive if Ai , Bi is direct) are: Ai ∩ Bj = δi,j

Ai ∩ Aj = 0

,

,

Bi ∩ Bj = 0

This choice of basis of non-contractible cycles is called ”symplectic”, and it is not unique.

A1

B1

A2

B2

By definition, L = L \ ∪i Ai ∪i Bi is simply connected. It is called the ”fundamental domain” . Bergman kernel = fundamental form of the 2nd kind When a symplectic basis of cycles is chosen, one defines the Bergman kernel: B(z1 , z2 ) as the unique bilinear differential having one double pole at z1 = z2 (it is called ”2nd kind”) and no other pole, and such that: ! dz1 ⊗ dz2 B(z1 , z2 ) ∼z1 →z2 + reg , ∀i = 1, . . . , g¯, B(z1 , z2 ) = 0 (z1 − z2 )2 z1 ∈Ai

One should keep in mind that the Bergman kernel depends only on L, and not on the functions x and y. Intuitively, the Bergmann kernel can be viewed as the electric field measured in z1 generated by a small unit dipole located at z2 . Or said differently, the integral " z1 " z2 B(z1$ , z2$ ) ln E(z1 , z2 ) = z1! =o1

z2! =o2

(where o1 , o2 are arbitrary base points), is the electric potential measured at z1 , created by a unit charge located at z2 , it satisfies the Poisson equation ∆z1 ln E(z1 , z2 ) = 2iπ δ(z1 − z2 ). Examples: 193

• if L = C ∪ {∞} =the Riemann Sphere (genus g¯ = 0), the Bergman kernel is B(z1 , z2 ) =

dz1 ⊗ dz2 = dz1 dz2 ln (z1 − z2 ) (z1 − z2 )2

• if L = C/(Z + τ Z) =Torus (genus g¯ = 1) of modulus τ , the Bergman kernel is B(z1 , z2 ) = (℘(z1 − z2 , τ ) +

π ) dz1 ⊗ dz2 Imτ

where ℘ is the Weierstrass elliptical function. • if L is a compact Riemann surface of genus g¯ ≥ 1, of Riemann matrix of periods τ = [τi,j ]i,j=1,...,¯g , the Bergman kernel is B(z1 , z2 ) = dz1 dz2 ln (θ(u(z1 ) − u(z2 ) − c, τ )) where u(z) is the Abel map, c is an odd characteristic, and θ is the Riemann theta function of genus g¯ (cf [?, ?] for theta-functions). Branchpoints Branchpoints are the points with a vertical tangent, they are the zeroes of dx. Let us write them ai , i = 1, . . . , #bp. ∀i,

dx(ai ) = 0

Since we consider a regular spectral curve, all branchpoints are simple zeroes of dx, the curve y(x) behaves locally like a square root, near a branchpoint ai , and thus there are exactly two points p and p¯ in the vicinity of ai such that: x(¯ p) = x(p) p¯ is called the conjugated point of p. It is defined locally near each branchpoint ai , and it is not necessarily defined globally.

y a2

z a1

z

x(z) Examples: 194

x

• maps =1-matrix model, 1-cut. In chapter III we have seen that maps’ spectral curves are parametrized by the Zhukovsky variable z. In that case z ∈ L =Riemann sphere, and x(z) and y(z) are rational functions of z. In particular we have x(z) = α + γ(z + 1/z)

dx(z) = x$ (z)dz = γ(1 − z −2 ) dz

,

The zeroes of dx(z) are z = ±1, and we clearly have z¯ = 1/z: a1 = 1, a2 = −1

,

z¯ = 1/z.

• pure gravity (3, 2). In chapter V, we have seen that Liouville pure gravity (3, 2) minimal model, is related to the spectral curve x(z) = z 2 − 2

y(z) = z 3 − 3z

,

z ∈ L = Riemann sphere.

,

We have dx(z) = x$ (z)dz = 2z dz whose only zeroe is z = 0, and we have z¯ = −z: a=0

,

z¯ = −z.

• Ising model (4, 3). The minimal model (4, 3) coupled to Liouville gravity, has central charge c = 1/2 and is also called Ising model. It has the rational spectral curve x(z) = z 3 − 3z

y(z) = z 4 − 4z 2 + 2

,

,

z ∈ L = Riemann sphere,

and thus dx(z) = x$ (z)dz = 3(z 2 − 1) dz

√ whose zeroes are ai = ±1, and near ai = ±1 we have z¯ = − 21 (z − ai 12 − 3z 2 ): ai = ±1

,

√ 1 z¯ = − (z − ai 12 − 3z 2 ) 2

one may verify that a¯i = ai . In this case, z¯ is not defined globally, it is defined only in the vicinity of ai . Recursion Kernel We define the recursion kernel: #z B(z0 , z $ ) 1 z ! =¯ z K(z0 , z) = 2 (y(z) − y(¯ z )) dx(z) K(z0 , z) is a meromorphic 1-form in the variable z0 , it is defined globally ∀z0 ∈ L. 195

On the contrary, in the variable z, K(z0 , z) = K(z0 , z¯) is defined only locally near branchpoints z ∼ ai , and it is the inverse of a form. As we shall see below, K(z0 , z) will always be used only in the vicinity of branchpoints, and only multiplied by a quadratic differential. K(z0 , z) has a simple pole at z = ai , and near z = ai it behaves like: K(z0 , z) ∼

1 B(z0 , z) + reg 2 dy(z) dx(z)

Remember that dx(z) has a simple zero at z = ai and dy(ai ) %= 0. Correlation functions Definition 1.5 We define recursively the following meromorphic forms: (0)

ω1 (z1 ) = y(z1 ) dx(z1 ) (0)

ω2 (z1 , z2 ) = B(z1 , z2 ) and if 2g − 2 + n ≥ 0, and J = {z1 , . . . , zn }: (g) ωn+1 (z0 , J)

=

$ i

g $ % & $ $ (g−1) (h) (g−h) Res K(z0 , z) ωn+2 (z, z¯, J) + ω1+|I| (z, I)ω1+n−|I| (¯ z , J/I)

z→ai

h=0 I⊂J

(VII-1-1) where in the RHS means that we exclude the terms with (h, I) = (0, ∅), and (g, J). This definition is indeed a recursive one, because all the terms in the RHS have a strictly smaller 2g − 2 + n than the LHS. '$

(g)

ωn (z1 , . . . , zn ) is a tensorial product of meromorphic forms in each variables zi . It can be proved by recursion, that it is in fact a symmetric form. Moreover, if 2g−2+n > 0, its only poles are at branchpoints zi → aj , and have no residues. Those properties can be proved recursively, and we refer the reader to [2]. Symplectic invariants (g)

(g)

The previous definition, defines ωn with n ≥ 1. Now, we define Fg = ω0 following:

by the

• Fg for g ≥ 2 Definition 1.6 Symplectic invariants We define for g ≥ 2: Fg =

$ 1 (g) Res Φ(z) ω1 (z) z→a 2 − 2g i i 196

,

dΦ = ydx (VII-1-2)

• Fg for g = 1 Definition 1.7 For g = 1 we define ( * ) 1 F1 = − ln τB ({x(ai )})12 y $(ai ) 24 i where we define: y(z) − y(ai) y $ (ai ) = lim + z→ai x(z) − x(ai ) and τB is the Bergman τ -function, it depends only on the values of x at branch points xi = x(ai ), it is defined by: ∂ ln τB ({xi )}) B(z, z¯) = Res z→ai dx(z) ∂xi • For example, If we have a rational spectral curve with two branchpoints a, b, parametrized by Zhukovsky map x(z) = (a + b)/2 + γ (z + 1/z) where γ = (a − b)/4, we have −dz 2 /z 2 1 1 1 ∂ ln τB (a, b) = Res = = 2 2 z→1 (z − 1/z) γ(1 − 1/z ) dz ∂a 16γ 4(a − b) and similarly

∂ ln τB (a,b) ∂b

=

1 , 4(b−a)

which leads to τB (a, b) ∝ γ 1/4 .

i.e.

1 , 3 $ ln γ y (a) y $(b) 24 . y(z)−y(1) . , and similarly for y $ (b), i.e. Then, notice that y $(a) = lim √ = √1γ dy dz z=1 F1 = −

γ(z+1/z−2)

finally:

. . / 0 . 1 dy .. 2 dy . , F1 = − ln γ 24 dz .z=1 dz .z=−1 which is what we found in chapter III.

• For example, If we have a rational spectral curve with only one branchpoint a, parametrized by x(z) = a + z 2 , we have ∂ ln τB (a) −dz 2 1 = Res =0 2 z→0 ∂a (z + z) 2z dz

and thus and

τB (a) ∝ 1,

/ . 0 1 dy .. F1 = − , ln 24 dz .z=0 which is what we found for (p, q) minimal models and Kontsevich integral in chapter V and chapter VI. 197

• Fg for g = 0 Let αi , i = 1, . . . , npoles be the poles of ydx. - If αi is a pole of degree di of x(z), we define the local coordinate ξi (z) = x(z)−1/di - If αi is a pole not a pole of x(z) (it is thus a pole of y(z)), we define the local coordinate ξi (z) = x(z) − x(αi ) We define the potentials Vi and times ti : ti = Res y(z) dx(z) z→αi

,

/ 0 ξi(z $ ) Vi (z) = Res ln 1 − y(z $ ) dx(z $ ). z ! →αi ξi(z)

They are such that y(z)dx(z) − dVi (z) − ti dξi(z)/ξi (z) is analytical near z = αi . Then, given an arbitrary generic basepoint o ∈ L, we define: " o µi = (y(z)dx(z) − dVi (z) − ti dξi(z)/ξi (z)) + Vi (o) + ti ln ξi (o) αi

Then, we define Definition 1.8 $ $ 1 1% $ F0 = Res Vi (z) y(z)dx(z) + ti µi + 2 i αi 2iπ i i=1 g¯

!

Ai

ydx

!

Bi

& ydx

One may'check that this quantity is independent of the choice of basepoint o (this is because i ti = 0). Notice, that contrarily to all Fg with g ≥ 1, which depend only on the local behavior of x and y near branchpoints, F0 depends on the full spectral curve, and in particular on its local behavior near the poles.

2

Main properties

So, for every spectral curve E = (L, x, y), we have defined some meromorphic forms (g) (g) ωn and some complex numbers Fg = ω0 . They have some remarkable properties (see [2]): (g)

• ωn is symmetric in its n variables (this is proved by recursion). (g)

• If 2 − 2g − n < 0, then ωn is a meromorphic form in each variable, with poles only at the branch-points, of degree at most 6g − 6 + 2n + 2, and with vanishing residue. 198

(g)

• If 2 − 2g − n < 0, ωn is homogeneous of degree 2 − 2g − n: ωn(g) ((L, x, λy); z1, . . . , zn ) = λ2−2g−n ωn(g) ((L, x, y); z1, . . . , zn ), and in particular for n = 0 Fg (L, x, λy) = λ2−2g Fg (L, x, y). In particular, if λ = −1, we see that Fg is invariant under y → −y. • If two spectral curves E = (L, x, y) and E˜ = (L, x ˜, y˜) are symplectically equivalent, i.e. dx ∧ dy = d˜ x ∧ d˜ y, or alternatively ydx − y˜d˜ x = exact, then they have (g) the same Fg ’s (although they do not have the same ωn ’s in general) dx ∧ dy = d˜ x ∧ d˜ y

→

˜ Fg (E) = Fg (E)

and more generally: (g)

(g)

˜ z1 ) = exact form, ω1 (E; z1 ) − ω1 (E; (g)

i.e. the cohomology class of ω1 is symplectically invariant. • Out of the Fg ’s, one can construct ' a formal tau function, which obeys Hirota’s 2−2g equation, it is of the form τ = exp ( ∞ Fg ) Θ, where we refer the reader g=0 N to [] for more details. • Dilaton equation, for 2g − 2 + n > 0: $ (g) Res Φ(zn+1 ) ωn+1 (z1 , . . . , zn , zn+1 ) = (2 − 2g − n) ωn(g) (z1 , . . . , zn ) i

zn+1 →ai

• Their derivatives with respect to any parameter of the spectral curve, are computed below in section 3. • They have many other properties, for instance their modular behaviour satisfies the Holomorphic anomaly equation.

3

Deformations of symplectic invariants

Consider a 1-parameter family of spectral curves Et = (Lt , xt (z), yt (z)), defined for t in a small vicinity of t = 0, and such that at t = 0 it is the spectral curve E: E0 = E

,

(L0 , x0 (z), y0 (z)) = (L, x(z), y(z)).

Our goal is to compute derivatives of the symplectic invariants at t = 0: . ∂ n Fg (Et ) .. . ∂tn .t=0 199

3.1

Spectral curve deformation y

δy (z) z δ x (z) At small time, we have 1

x

xt (z) = x(z) + tδx(z) + O(t2 ) yt (z) = y(z) + tδy(z) + O(t2 )

and the curve Lt itself may not be constant. The spectral curve is the set of points {(x(z), y(z)) , z ∈ L}, and the variable z is only a convenient parameter to describe it. In particular, one may change the parameter z to any z $ = f (z) if f is an analytical bijection L → L. This means that z is not an intrinsic notion, and thus δx(z) and δy(z) are not really intrinsic. Instead, what is intrinsic is the following 1-form: Ω(z) = δx(z) dy(z) − δy(z) dx(z), and we let the reader check that Ω is unchanged under any reparametrization z → f (z). One may notice that the location of branchpoints ai may change with time. Let Xi = x(ai ) be the x−projection of the ith branchpoint ai . Since dx(ai ) = 0 by definition, we have: Ω(ai ) ∂Xi = δXi = . ∂t dy(ai) In particular, the position of branchpoints in the x-plane is unchanged only if Ω(ai ) vanishes. If Ω(ai ) %= 0, we have δXi %= 0, and the conformal structure of the curve L changes. Rauch variational formula [], tells us how the Bergman kernel changes under such a change of conformal structure: $ ∂B(z1 , z2 ) B(z, z1 ) B(z, z2 ) = δB(z1 , z2 ) = δXi Res z→ai ∂t dx(z) i and since δXi =

Ω(ai ) , dy(ai )

and dx(z) is assumed to have a simple zero at ai , we may rewrite

δB(z1 , z2 ) =

$ i

Res

z→ai

Ω(z) B(z, z1 ) B(z, z2 ) . dx(z) dy(z)

200

(VII-3-1)

Since we now know the variation of the spectral curve, and the variation of the Bergman kernel, we may deduce the variation of the recursion kernel K(z0 , z), and by (g) recursion, the variation of every ωn . This can be understood in a very geometrical way as follows.

3.2

Form-cycle duality

Let us assume that Ω(z) is a meromorphic form. It may have poles αi of some degrees di . It is customary to classify meromorphic forms into 4 kinds: • Exact forms. Ω(z) is2an exact form if and only if, for any closed contour γ on C avoiding the poles, we have γ Ω = 0. In that case, there exists a meromorphic function f (z) such that Ω(z) = df (z). Notice that, since B(z, z $ ) has a double pole at z = z $ , we have: $ $ $ df (z) = Res f (z ) B(z, z ) = − Res f (z $ ) B(z, z $ ). ! ! z →z

i

z →αi

• 1st kind differential. Ω(z) is said to be 1st kind, if it has no pole. On a Riemann surface L of genus g¯, the vector space of 1st kind forms, is of dimension g¯. A basis is given by: ! 1 vi (z) = B(z, z $ ) , i = 1, . . . , g¯ . 2iπ z ! ∈Bi This choice of basis is such that ! vj (z) = δi,j

,

i, j = 1, . . . , g¯ .

z∈Ai

• 3rd kind differentials. Ω(z) is said to be 3rd kind, if it has only simple poles. We may 2 add any 1st kind form without changing that property, so we will also assume that Ai Ω = 0 for every i = 1, . . . , g¯. Notice that since the sum of all residues of a differential form must vanish, a 3rd kind differential must have at least two poles. A basis of 3rd kind differentials is: " z1 dSz1 ,z2 (z) = B(z, z $ ). z ! =z2

dSz1 ,z2 (z) clearly has a simple pole at z = z1 , with residue +1, and a simple pole at z = z2 with residue −1: Res dSz1 ,z2 (z) = 1 = − Res dSz1 ,z2 (z).

z→z1

z→z2

• 2nd kind differentials. It is more or less everything remaining, i.e. Ω(z) is said to be 2nd kind, if it has poles of higher degrees, with vanishing residues, and vanishing integrals around Ai cycles: ! Res Ω = 0 , Ω = 0. αi

Ai

201

One can prove that if Ω(z) is 2nd kind, there always exist an analytical function f (z), locally defined near the poles αi (not necessary defined globally on L), such that, ∀ z ∈ L: $ Ω(z) = Res B(z, z $ ) f (z $ ). ! i

z →αi

In the end, we see, that for any meromorphic form Ω(z), there exists an integration contour Ω∗ ⊂ L, and a function fΩ (z), such that " Ω(z) = B(z, z $ ) fΩ (z $ ). z ! ∈Ω∗

This leads to the notion of form-cycle duality: Definition 3.1 (Form-cycle duality) . If Ω(z) is a meromorphic form, there exists a path (or a linear combination of paths) Ω∗ ⊂ L, and an analytical function fΩ (z) defined on a vicinity of Ω∗ (not necessarily defined elsewhere in L), such that: " Ω(z) = B(z, z $ ) fΩ (z $ ). z ! ∈Ω∗

The path Ω∗ , together with fΩ , is called the dual contour to the form Ω. The pairing between a path and its dual, is realized by the Bergman kernel. Remark 3.1 One may get rid of the function fΩ , #by changing the variable z → ζ(z), whose

jacobian cancels the fΩ (z), and then write Ω(z) =

3.3

z ! ∈Ω∗

B(z, z $ ) in the new variable ζ.

(g)

Variation of ωn

Using eq.(VII-3-1) for the variation of the Bergman kernel, we find that: " B(z, z1 ) B(z, z2 ) B(z, z $ ) fΩ (z $ ) δB(z1 , z2 ) = Res z→ai dx(z) dy(z) ! ∗ z ∈Ω "i $ $ B(z, z ) B(z, z1 ) B(z, z2 ) = fΩ (z $ ) Res z→ai dx(z) dy(z) ! ∗ i "z ∈Ω (0) = fΩ (z $ ) ω3 (z $ , z1 , z2 ) $

z ! ∈Ω∗

We thus get: (0)

Theorem 3.1 The variation of ω2 , is the integral of ω3 (0) on the dual cycle to Ω: (0)

∂ω2 (z1 , z2 ) (0) = δω2 (z1 , z2 ) = ∂t

"

202

(0)

z ! ∈Ω∗

fΩ (z $ ) ω3 (z $ , z1 , z2 ).

(g)

We shall generalize this theorem to any ωn . First, let us rewrite: " (0) δB(z1 , z2 ) = fΩ (z $ ) ω3 (z $ , z1 , z2 ) "z ! ∈Ω∗ $ = fΩ (z $ ) Res K(z1 , z) (B(z, z $ ) B(¯ z , z2 ) + B(z, z2 ) B(¯ z , z $ )). z ! ∈Ω∗

Then, since

i

z→ai

#z

B(z0 , z $ ) K(z0 , z) = , 2(y(z) − y(¯ z )) dx(z) z ! =¯ z

and since we know the derivatives of B and y and x, we find (we leave it to the reader, or otherwise look in []), that for any quadratic differential f (z, t) defined in the vicinity of branchpoints ai such that f (¯ z , t) = f (z, t), that ∂ $ Res K(z0 , z) f (z, t) ∂t i z→ai

=

$

∂f (z, t) Res K(z0 , z) z→ai ∂t i$ + Res Res K(z0 , z $ ) K(z $ , z) f (z, t) Ω(z $ ). ! i,j

z→ai z →aj

Then, by an easy recursion on 2g − 2 + n using the topological recursion, we find: (g)

Theorem 3.2 (Form-cycle duality variation) The variation of ωn , is the integral of ωn+1 (g) on the dual cycle to Ω: (g)

∂ωn (z1 , . . . , zn ) = δωn(g) (z1 , . . . , zn ) = ∂t

"

z ! ∈Ω∗

(g)

fΩ (z $ ) ωn+1 (z $ , z1 , . . . , zn ).

In particular, for n = 0: ∂Fg = δFg = ∂t

"

(g)

z ! ∈Ω∗

fΩ (z $ ) ω1 (z $ ).

Remark 3.2 This property is often referred to as ”special geometry” in the context of string theory. If we specialize it to the case n = 0, g = 0, we get: " ∂F0 = δF0 = fΩ (z $ ) y(z $ ) dx(z $ ). ∂t ! ∗ z ∈Ω

Remark 3.3 Prepotential Let us specialize this theorem further. Let us denote ! 1 $i = ydx. 2iπ Ai If we want to vary $i , while keeping all other $j unchanged, and all poles of ydx (and all the negative part of their Laurent series expansion near poles) unchanged, we find that Ω(z) must have no pole, and that ! 1 Ω = δi,j 2iπ Aj

203

thus, Ω(z) = 2iπvi (z) is a first kind differential, and is dual to the cycle Bi Ω(z) = 2iπ vi (z) =

!

B(z, z $ ).

z ! ∈Bi

We thus have: (g)

∂ωn (z1 , . . . , zn ) = ∂$i

!

z ! ∈Bi

(g) ωn+1 (z $ , z1 , . . . , zn )

∂Fg = ∂$i

,

!

Bi

(g)

ω1 .

And in particular: ∂F0 = ∂$i

!

ydx

1 where $i = 2iπ

,

Bi

!

ydx. Ai

This property is the characterization of the prepotential in Seiberg-Witten theory. This is why, we claim that F0 is the prepotential.

4

Diagrammatic representation (g)

The recursive definitions of ωk and Fg can be represented graphically. (g) We represent the multilinear form ωk (p1 , . . . , pk ) as a blob-like “surface” with g (g) holes and k legs (or punctures) labeled with the variables p1 , . . . , pk , and Fg = ω0 with 0 legs and g holes. p1 p2 (g) ωk+1(p, p1 , . . . , pk ) := p

Fg :=

, pk

(g)

(g)

(0)

We represent the Bergman kernel B(p, q) (which is also ω2 , i.e. a blob with 2 legs and no hole) as a straight non-oriented line between p and q B(p, q) := p

q .

We represent K(p, q) as a straight arrowed line with the arrow from p towards q, and with a tri-valent vertex whose left leg is q and right leg is q q

K(p, q) :=

p

q

204

.

Graphs Definition 4.1 For any k ≥ 0 and g ≥ 0 such that k + 2g ≥ 3, we define: (g) Let Gk+1 (p, p1 , . . . , pk ) be the set of connected trivalent graphs defined as follows: 1. there are 2g + k − 1 tri-valent vertices called vertices. 2. there is one 1-valent vertex labelled by p, called the root. 3. there are k 1-valent vertices labelled with p1 , . . . , pk called the leaves. 4. There are 3g + 2k − 1 edges. 5. Edges can be arrowed or non-arrowed. There are k + g non-arrowed edges and 2g + k − 1 arrowed edges. 6. The edge starting at p has an arrow leaving from the root p. 7. The k edges ending at the leaves p1 , . . . , pk are non-arrowed. 8. The arrowed edges form a ”spanning1 planar2 binary skeleton3 tree” with root p. The arrows are oriented from root towards leaves. In particular, this induces a partial ordering of all vertices. 9. There are k non-arrowed edges going from a vertex to a leaf, and g non arrowed edges joining two inner vertices. Two inner vertices can be connected by a non arrowed edge only if one is the parent of the other along the tree. 10. If an arrowed edge and a non-arrowed inner edge come out of a vertex, then the arrowed edge is the left child. This rule only applies when the non-arrowed edge links this vertex to one of its descendants (not one of its parents).

(2)

Example of G1 (p) (2) As an example, let us build step by step all the graphs of G1 (p), i.e. g = 2 and k = 0. We first draw all planar binary skeleton trees with one root p and 2g + k − 1 = 3 arrowed edges: p

,

1

p

.

It goes through all vertices. planar tree means that the left child and right child are not equivalent. The right child is marked by a black disk on the outgoing edge. 3 a binary skeleton tree is a binary tree from which we have removed the leaves, i.e. a tree with vertices of valence 1, 2 or 3. 2

205

Then, we draw g + k = 2 non-arrowed edges in all possible ways such that every vertex is trivalent, also satisfying rule 9) of definition.4.1. There is only one possibility for the first graph and two for the second one: p

p

,

p

,

.

It just remains to specify the left and right children for each vertex. The only possibilities in accordance with rule 10) of def.4.1 are4 :

p

,

p

,

p

,

p

p

.

In order to simplify the drawing, we can draw a black dot to specify the right child. This way one gets only planar graphs.

p

,

p

,

p

p

,

p

Remark that without the prescriptions 9) and 10), one would get 13 different graphs whereas we only have 5. Weight of a graph (g)

Consider a graph G ∈ Gk+1 (p, p1 , . . . , pk ). Then, to each vertex i = 1, . . . , 2g + k − 1 of G, we associate a label qi , and we associate qi to the beginning of the left child edge, and q i to the right child edge. Thus, each edge (arrowed or not), links two labels which are points on the spectral curve L. • To an arrowed edge going from q $ towards q, we associate a factor K(q $ , q). • To a non arrowed edge going between q $ and q we associate a factor B(q $ , q). 4

Note that the graphs are not necessarily planar.

206

• Following the arrows backwards (i.e. from leaves to root), for each vertex q, we take the sum over all branchpoints ai of residues at q → ai . After taking all the residues, we get the weight of the graph: w(G) which is a multilinear form in p, p1 , . . . , pk . Similarly, we define weights of linear combinations of graphs by: w(αG1 + βG2 ) = αw(G1 ) + βw(G2) and for a disconnected graph, i.e. a product of two graphs: w(G1 G2 ) = w(G1)w(G2 ). Theorem 4.1 We have: (g)

ωk+1(p, p1 , . . . , pk ) = (g)



$

G∈Gk+1 (p,p1 ,...,pk )

 w(G) = w 

(g)



$

G∈Gk+1 (p,p1 ,...,pk )

 G

proof: This is precisely what the recursion equations VII-1-1 of def.1.5 are doing. Indeed, one can represent them diagrammatically by

+

=

.

! Such graphical notations are very convenient, and are a good support for intuition and even help proving some relationships. It was immediately noticed after [?] that those diagrams look very much like Feynman graphs, and there was a hope that they could be the Feynman’s graphs for the Kodaira–Spencer quantum field theory. But they ARE NOT Feynman graphs, because Feynman graphs can’t have non-local restrictions like the fact that non oriented lines can join only a vertex and one of its descendent. Those graphs are merely a notation for the recursive definition VII-1-1. Lemma 4.1 Symmetry factor: The weight of two graphs differing by the exchange of the right and left children of a vertex are the same. Indeed, the distinction between right and left child is just a way of encoding symmetry factors. proof: This property follows directly from the fact that K(z0 , z) = K(z0 , z¯). ! 207

4.1

Examples.

Let us present some examples of correlation functions and free energy for low order. Correlation functions. To leading order, one has the first correlation functions given by: (0)

ω2 (p, q) = B(p, q).

p

p

1

(0)

ω3 (p, p1 , p2 ) =

1

p

+ p

p

p

2

=

2

Res K(p, q) [B(q, p1 )B(q, p2 ) + B(q, p1 )B(q, p2 )] q→a

= −2 Res K(p, q) [B(q, p1 )B(q, p2 )] q→a

= =

B(q, p) B(q, p1) B(q, p2) Res q→a dx(q) dy(q) $ B(ai , p) B(ai , p1 ) B(ai , p2 ) 2 dy(ai) dzi(ai )2

i

where zi (z) =

+

x(z) − z(ai ) is a local coordinate near ai .

p (0)

ω4 (p, p1 , p2 , p3 ) =

3

p

+

p

p

2

1

p +

perm. (1,2,3)

3

p

+

perm. (1,2,3)

p

p

2

1

= Res Res K(p, q) K(q, r) [B(q, p1 )B(r, p2 )B(r, p3 ) q→a r→a

+B(q, p1 )B(r, p2 )B(r, p3 ) + B(q, p2 )B(r, p1 )B(r, p3 ) +B(q, p2 )B(r, p1 )B(r, p3 ) + B(q, p3 )B(r, p2 )B(r, p1 ) +B(q, p3 )B(r, p2 )B(r, p1 )] + Res Res K(p, q) K(q, r) [B(q, p1 )B(r, p2 )B(r, p3 ) q→a r→a

+B(q, p1 )B(r, p2 )B(r, p3 ) + B(q, p2 )B(r, p1 )B(r, p3 ) +B(q, p2 )B(r, p1 )B(r, p3 ) + B(q, p3 )B(r, p2 )B(r, p1 ) 208

+B(q, p3 )B(r, p2 )B(r, p1 )] First orders for the one point correlation function read:

(1)

ω1 (p) =

p

= Res K(p, q) B(q, q) q→a

(2)

p

ω1 (p) =

+ p

+ p

+ p

+ p

= Res Res Res K(p, q)K(q, r)K(q, s) B(r, r)B(s, s) q→a r→a s→a

+ Res Res Res K(p, q)K(q, r)K(r, s) B(r, q)B(s, s) q→a r→a s→a

+ Res Res Res K(p, q)K(q, r)K(r, s) [B(q, r)B(s, s) q→a r→a s→a

+B(s, q)B(s, r) + B(s, q)B(s, r)] = 2 p

+2

p

+ p

where the last expression is obtained using lemma 4.1. Free energy. The second order free energy reads − 2F2 =

Res Res Res Res Φ(p) K(p, q) K(q, r)K(q, s) B(r, r)B(s, s) p→a q→a r→a s→a

+ Res Res Res Res Φ(p)K(p, q)K(q, r)K(r, s) B(r, q)B(s, s) p→a q→a r→a s→a

209

+ Res Res Res Res Φ(p)K(p, q)K(q, r)K(r, s) [B(q, r)B(s, s) p→a q→a r→a s→a

+B(s, q)B(s, r) + B(s, q)B(s, r)]

4.2

Remark: Teichmuller pants gluings

Every Riemann surface of genus g with k punctures can be decomposed into 2g + k pants whose boundaries are 3g+k closed geodesics (in the metric with constant negative curvature). The number of ways (in the combinatorial sense) of gluing 2g + k pants (g) by their boundaries is clearly the same as the number of diagrams of Gk , and each diagram corresponds to one pant decomposition. Indeed, consider the first boundary labelled by z1 , and attach a pair of pant to this boundary. Draw an arrowed propagator from the boundary to the first pant. Then, choose one of the other boundaries of the pair of pants (there are thus 2 choices), it must be glued to another pair of pants (possibly not distinct from the first one). If this pair of pants was never visited, draw an arrowed propagator, and if it was already (g) visited, draw a non-arrowed propagator. In the end, you get a diagram of Gk . This (g) procedure is bijective, and to a diagram of Gk , one may associate a gluing of pants. Example with k = 1 and g = 2:

(2)

ω1 =

5

+2

+2

Exercises

Exercise 1: Compute F0 for the Kontsevich’s spectral curve: 1 x(z) = z 2 ˜ 'N EK : 1 y(z) = z + 2N i=1

1 Λi (z−Λi )

Hint: there are N + 1 poles: α0 = ∞, and αi = Λi for i = 1, . . . , N. The potentials 3 are V0 (z) = 23 x(z) 2 , and Vi (z) = 0 for i = 1, . . . , N.

210