Chapter IV Multicut case In the previous chapter, we have seen that generating functions of maps are special solutions of loop equations, namely 1-cut solutions. However, loop equations have other solutions, which are also formal power series. In this chapter, we explore the other solutions, and their combinatorical meaning. Beside mathematical curiosity, those other multicut solutions play an important role in physics (e.g. particularly in string theory) and mathematics (e.g. asymptotics of orthogonal polynomials).

1

Formal integrals and extrema of V

For example, consider the cubic potential:

0 V (M) =

M2 M3 − 2 3

1

, N

So far, we have defined formal series using the Taylor expansion of e− t Tr V (M ) near the extremum of V (M) at M = 0. However, there is another extremum V " (1) = 0, i.e. another possibility of having no linear term in the Taylor expansion. We may chose another zero of V " (M), and for instance Taylor expand near M = Id. One could also expand around any other zero of V " (M), for instance: n times

N −n times

! "# $ ! "# $ M0 = diag(0, 0, . . . , 0, 1, 1, . . . , 1) 89

and write M = M0 + A. One has: N

e− t

Tr V (M )

N

= e− t

Tr ( 12 M02 − 13 M03 ) − Nt Tr ( 21 −M0 )A2

e

N

e− t

Tr

A3 3

In that case, the quadratic form for A is not definite, it has eigenvalues +1, −1, 0: 1% 1 (1 − (M0 )i,i − (M0 )j,j ) |Ai,j |2 Tr ( − M0 )A2 = 2 2 i,j and we have: 1 − (M0 )i,i − (M0 )j,j ∈ {−1, 0, 1}, and it indeed takes the eigenvalue 0 if there exists i and j such that (M0 )i,i = 0 and (M0 )j,j = 1, i.e. unless n = 0 or n = N. & N 1 2 In other words, the Gaussian integral dA e− t Tr ( 2 −M0 )A is ill-defined unless n = 0 or n = N. So, for arbitrary n ∈ [0, N], we cannot define the formal integral as the exchange of the gaussian integral in A, and the Taylor series. However, there exists another way of defining a formal integral around M0 , it is described below.

1.1

A digression on convergent normal matrix integrals

Consider the convergent integral: ' ( ( N dx1 . . . dxN (xj − xi )2 e− t V (xi ) γN

i0 x

Therefore, there exists a compact Riemann surface L of genus g¯ ≤ d − 2, as well as d − 1 cycles Ai , and two meromorphic functions x and y defined on it, such that:

1 (0) W1 (x(z)) = V " (x(z)) − y(z) 2 1 1 ydx = #i 2iπ Ai The branch-points a1 , . . . , a2¯g+2 are the zeroes of the differential form dx, i.e. the points at which the tangent is vertical, i.e. the points at which y behaves like a squareroot. ∀z ∈ L,

3.4

Geometry of the spectral curve

The curve y(x), which can be written parametrically as: 1 (0) y = V "2 (x) − P1 (x) 4 2

⇔

y(x)

⇔

2

x(z) y(z)

is called the spectral curve, it is an hyperelliptical curve1 . Let us study some of its properties (a more general framework is presented in chapter VII). Bergman kernel On any compact Riemann surface L equipped with a symplectic basis (not unique) of non-contractible cycles Ai ∩ Bj = δi,j

,

Ai ∩ Aj = 0

,

Bi ∩ Bj = 0

∀ i, j = 1, . . . , g¯,

is defined the Bergman kernel: B(z1 , z2 ) as the unique bilinear differential of the 2nd kind, having one double pole at z1 = z2 and no other pole, and such that: 1 dz1 dz2 B(z1 , z2 ) ∼z1 →z2 + reg , B(z1 , z2 ) = 0 (z1 − z2 )2 Ai 1

Any algebraic equation of the form y 2 = Pol(x) is called hyperelliptical. It is called elliptical if deg Pol = 3 or 4, and it is rational if deg Pol ≤ 2.

102

One should keep in mind that the Bergman kernel depends only on L, and not on the functions x and y. It is easy to see that the Bergman kernel is unique, because the difference of two Bergman kernels would be a meromorphic form, with no pole, and with vanishing A-cylce integrals, i.e. it must vanish. Examples: • if L = C∪{∞} =the Riemann Sphere, the Bergman kernel is the rational fraction: dz1 dz2 B(z1 , z2 ) = (z1 − z2 )2 • if L = C/(Z + τ Z) =Torus of modulus τ , the Bergman kernel is an elliptical function: π ) dz1 dz2 B(z1 , z2 ) = (℘(z1 − z2 , τ ) + Imτ where ℘ is the Weierstrass elliptical function. • if L is a compact Riemann surface of genus g¯ ≥ 1, of modulus τi,j , 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 hyperelliptical Riemann theta function of genus g¯ (cf [?, ?] for details).

Branchpoints and conjugated points Branchpoints are the points with a vertical tangent, i.e. they are the zeroes of dx. Let us write them ai , i = 1, . . . , 2¯ g + 2. ∀i,

dx(ai ) = 0

For any z away from branchpoints, there is a unique point z¯ *= z such that: x(¯ z ) = x(z)

,

y(¯ z ) = −y(z)

z¯ is called the conjugated point of z. The branchpoints are the points where z¯ = z.

3.5

Cylinder generating function

Theorem 3.3 The cylinder generating function is the Bergman kernel: * + 1 (0) (0) ω2 (z1 , z2 ) = W2 (x(z1 ), x(z2 )) + dx(z1 )dx(z2 ) = B(z1 , z2 ) (x(z1 ) − x(z2 ))2

where B(z1 , z2 ) is the Bergman kernel on L.

proof: The proof is very similar to that of theorem III-2-6 in chapter ??. We first show that this expression is a meromorphic function, and it can have a pole only at z1 = z2 and no other pole, and then that its Ai cycle integral vanish. The only differential form having that property is the Bergman kernel. ! 103

3.6

Higher correlation functions

It can be seen recursively from the loop equations, that Lemma 3.2 If 2g − 2 + k > 0, (g)

(g)

ωk (z1 , . . . , zk ) = Wk (x(z1 ), . . . , x(zk ))dx(z1 ) . . . dx(zk ) is an hyperelliptical meromorphic form on L with poles only at the branch-points. proof: The proof is quite easy, indeed the loop equations says that y(z) Wn(g) (z, z2 , . . . , zn ) = R.H.S where the Right hand side RHS is a meromorphic form on L by recursion hypothesis (recursion on 2g + n − 2), having poles only at the branchpoints. Then dividing by y(z) gives poles at branchpoints and possibly simple poles at the double zeroes (i.e. the (g) zeroes of M(x(z))). However, from lemma 3.1, we see that the residues of Wn at those (g) simple poles must vanish, and therefore, the only poles of Wn are at branchpoints. ! Let us define the ”3rd kind differential form”: ' z1 dSz1 ,z2 (z0 ) = B(z " , z0 ) z2

which is a meromorphic differential form in the variable z0 , with a simple pole of residue +1 at z0 = z1 , and a simple pole of residue −1 at z0 = z2 , and it is normalized on A-cycles: 1 dSz1 ,z2 = 0. Ai

On the other hand, regarded as a function of z1 , dSz1 ,z2 (z0 ) is a scalar, and it is only defined on a fundamental domain (i.e. L/(∪i Ai ∪i Bi ) which is simply connected). Let o ∈ L be an arbitrarily fixed origin on the spectral curve. Since dSz,o(z0 ) has a simple pole at z = z0 , we can write Cauchy theorem 1 1 (g) (g) (g) ωk+1 (z0 , J) = − Res dSz,o(z0 ) ωk+1 (z, J) = − dSz,o(z0 ) ωk+1(z, J) z→z0 2iπ Cz0 where Cz0 is a small circle surrounding z0 . We can move the integration contour in the fundamental domain. The only poles of the integrand are at z = z0 or z = ai , and the Riemann bilinear identity (see (g) for instance [?, ?]) says that if the A-cycle integrals of ωk+1 and B vanish, then the boundaries of the fundamental domain don’t contribute, therefore: % (g) (g) ωk+1 (z0 , J) = Res dSz,o(z0 ) ωk+1(z, J) i

z→ai

104

Now, we use the loop equation eq.III-3-1, i.e. dSz,o (z0 ) , (g−1) = Res ω ˆ n+2 (z, z, J) z→ai 2y(z)dx(z) i g % " % (h) (g−h) + ω ˜ 1+|I| (z, I) ω ˜ 1+k−|I|(z, J/I) %

(g) ωk+1 (z0 , J)

h=0 I⊂J

where (g)

(g)

ω ˜ k = ωk −

1 dx(z1 )dx(z2 ) δk,2 δg,0 2 (x(z1 ) − x(z2 ))2

and (g)

(g)

ω ˆ k = ωk − δk,2δg,0

dx(z1 )dx(z2 ) . (x(z1 ) − x(z2 ))2

One should notice that we have the parity property: (g)

(g)

z , J) ω ˜ k+1 (z, J) = −˜ ωk+1 (¯ where z¯ *= z is solution of x(z) = x(¯ z ). This allows to symmetrize the integrand, and get: (g) ωk+1 (z0 , J)

dSz,¯z (z0 ) , (g−1) 1% Res ωn+2 (z, z¯, J) = − 2 i z→ai 2y(z)dx(z) g " % % (h) (g−h) + ω ˜ 1+|I| (z, I) ω ˜ 1+k−|I|(¯ z , J/I) h=0 I⊂J

(g)

Because of parity, it is possible to change ω ˜ → ω, and all the generating functions ωk with 2 − 2g − k < 0 are given by:

Theorem 3.4 The generating functions of nodal maps of genus g with k boundaries, obey the recursion: (g)

1% dSz,¯z (z0 ) , (g−1) Res ωn+2 (z, z¯, J) 2 i z→ai 2y(z)dx(z) g " % % (h) (g−h) + ω1+|I| (z, I) ω1+k−|I|(¯ z , J/I)

ωk+1 (z0 , J) = −

(IV-3-2)

h=0 I⊂J

This theorem shows that the generating functions of nodal maps are a special case of the symplectic invariants of [2], presented in chapter VII.

4

Maps without boundaries

The generating functions of maps with no boundaries are given by: 105

Theorem 4.1 ∀g≥2

,

Fg =

1 % (g) Res Φ(z) ω1 (z) 2 − 2g i z→ai

where dΦ = ydx. Expressions for F0 and F1 can be found in chapter VII. proof: Since the multicut case is not so relevant for the combinatorics of maps, we let the reader find the proof in the literature []. ! Again, the generating functions of nodal maps are a special case of the symplectic invariants of [2], presented in chapter VII.

106