From theorems 2.6 and 2.7, we have cell decompositions of the space M˜(F) = ˜T(F)/M C(F).
Since M(F) and ˜M(F) are homotopy equivalent, the Penner-Strebel cell de- composition can be considered as the cell decomposition ofM(F).
Surprisingly the problem of enumerating orientable fatgraphs turns out to be solvable by considering the integrals of certain functions over GUE, the ensemble of Hermitian matrices ([1, 25]). To illustrate this remarkable connection, let us start with a toy example given in [64]. A fundamental result for this section and beyond is Wick’s theorem, which states the expectation value of a product of Gaussian random variables can be computed as the sum over all pairings of product of expectation values of pairs;
Theorem 2.8. (Wick; see Bessis et al. [21]) Let x= (x1, . . . , xn)∈Rn and let A be a symmetric, positive definiten×nmatrix. Consider the integral
hxi1xi2· · ·xiki=
Rxi1xi2· · ·xikexp
−12
P
µ,νxµAµνxν
dx R exp
−12
P
µ,νxµAµνxν
dx .
Then we have
hxi1xi2· · ·xi2k+1i= 0, hxi1xi2i=
A−1
i1i2
, hxi1xi2· · ·xi2ki= X
parings
Y
(s,t)
hxisxiti,
where the sum is over all pairings of the indicesi1, . . . , i2k.
For the toy example, letx, y∈Rn and (x, y) be the Euclidean inner product in Rn. For a functionf(x) onRn, set
hfi= R
Rnexp −12(x, x)
f(x) dx R
Rnexp −12(x, x) dx . Then we see that
hxixji=δij. So Wick’s theorem implies, for example,
hxixjxkxli=hxixjihxkxli+hxixkihxjxli+hxixlihxjxki=δijδkl+δikδjl+δilδjk. The different pairings of indicesi, j, k, lcan be represented as the different ways of pairing the four indexed half-edges of a four-valent vertex (figure 11).
i
k
j l
i
k
j l
i
k
j l
Figure 11: Three different pairings of indicesi, j, k, l.
Now considerEN2, the ensemble ofN×N Hermitian matrices and letf be a conjugation-invariant function on it. We will consider the expectation off(M),
hf(M)i= R
E2Nexp(−12TrM2)f(M) dM R
EN2 exp(−12TrM2) dM .
Sincef is conjugation-invariant, and the measure dM is invariant under changes of bases, this is just a Gaussian integral, and we may apply Wick’s theorem.
Since
TrM2=X
i,j
MijMji, (4)
we deduce that the so-calledWick contraction is given by hMijMkli=δilδkj.
This can be seen by writing the matrix M in vector form, and expressing the quadratic form TrM2 as aN2×N2matrix. This matrix Ahas the form
AN(i−1)+j,N(j−1)+i= 1, 1≤i, j≤N
and 0 everywhere else. In particular, A−1=A and the index that corresponds to MijMkl is (N(i−1) +j, N(k−1) +l). Thus by Wick’s theorem, we obtain (4).
Again using Wick’s theorem, we compute DTrM4E
=
*X
i,j,k,l
MijMjkMklMli
+
= X
i,j,k,l
hMijMjkMklMlii
= X
i,j,k,l
hMijMjkihMklMlii+hMijMklihMjkMlii+hMijMliihMjkMkli
= X
i,j,k,l
δikδjjδkiδll+δilδkjδjiδlk+δiiδjlδjlδkk
(5)
=
2N3+N
. (6)
For the graphical representation of the different pairings of entries in the matrix M, we use the “fattened” version of the four-valent vertex we used above, with the half-edges now shown as double lines labelled by double indices. Each half- edge has one line oriented away from, and one oriented towards the vertex. We also require that the orientation of the half-edges are consistent, so that the inward-pointing edge iis connected to the outward pointing edge of the same label (figure 12). Now we can interpret the first summand in equation (5) as connectingi-out withk-in,j-out withj-in,k-out withi-in, andl-out withl-in.
This results in the first surface in figure 13. Similarly the second and third summands correspond to the second and third surfaces in figure 13. Note in equation (6), the powers ofN correspond to the number of boundary cycles in the associated surfaces, and the integral coefficients ofNjis the number of ways of obtaining the same graph by gluing the available half-edges, i.e. #Glu. We can in general obtain the number of fatgraphs with onem-valent vertex in this manner.
i j i
j k
k l
l
Figure 12: “Fattened” four-valent vertex with the oriented dou- ble half-edges.
i j i j k
k l
l
i j i j k
k l
l
i j i j k
k l
l
Figure 13: Three different surfaces obtained by connecting the half-edges of the fattened four-valent vertex in figure 12.
Lemma 2.9. Let M ∈EN2, the ensemble of N ×N Hermitian matrices, and j∈N. Then we have
DTrMjE
=X
τ
#Glu(τ)Nn(τ),
wheren(τ)is the number of boundary cycles of fatgraphτ, and the sum is over all topologically distinct fatgraphs with one j-valent vertex and no unconnected half-edges.
Wick’s theorem further implies that products of traces correspond to mul- tiple vertices, valencies of which are determined by the powers of M. In other
words, *m
Y
k=1
TrMjk +
=X
τ
#Glu(τ)Nn(τ), (7)
where the sum is now over all fatgraphs withmvertices of valenciesj1, j2, . . . , jm, which may not be connected.
Let us consider another integral by setting V(M) = N
2M2 and
hfi= 1 Z0
Z
e−NTrM
2
2 f(M) dM, whereZ0is the normalisation constant, given in this case by
Z0= Z
EN2
dM e−NTrM22 = 2N π
N N22
.
Then we have
MijMkl
= 1 Nδilδjk.
Suppose we have kj j-valent vertices, j = 1, . . . , m. Using lemma 2.1, we can express (7) in another way;
*m Y
j=1
1 kj!(tN
j TrMj)kj +
=X
τ
1
#Aut(τ)Nχ(τ)tPjkj, (8) where χ(τ) is the Euler characteristic, and the sum is now over all fatgraphs with m vertices, mj of which are j-valent. For the power of N, we have a negative contribution from 1/N in the Wick contraction counting the number of edges. By lemma 2.9, we have a positive contribution from the number of boundary components, and fromtN in equation (8), which counts the number of vertices. In other words, the power ofN is given by
#vertices−#edges + #boundaries =χ(τ).
Let
V(M) =M2 2 −
X∞ j=3
tj
jMj (9)
and define a formal matrix integral by setting 1
Z0
Z
formal
dM e−NTrV(M)=X
τ
1
#Aut(τ)Nχ(τ)tkjj, (10) where the sum is over all fatgraphs, with any number of vertices, and any combinations of valencies. It is important to note that this is a formal integral, with no implication that the integral is convergent. However, the definition is a natural one, if we think of it as expanding the exponential of the non-quadratic part of V(M) and then exchanging the integral and the summation;
1 Z0
Z
formal
dM e−NTrV(M)= 1 Z0
Z
formal
dM e−NTrM
2
2 eNP∞j=3tjjTrMj
“=”
*∞ X
k=0
1 k!
X∞
j=3
N tj
j TrMj
k+
“=”
X∞ k=0
1 k!
X
j1,...,jk
* k Y
l=1
N tjl
jl
TrMjl +
,
where the second sum is over all values of jl, l = 1, . . . , k. This sum con- tains permutations ofjl’s that do not change the number ofjl’s with the same value, which correspond to the relabelling of vertices of the same valencies. The number of such permutations is given by Qk!kj!, which cancels the k! in the de- nominator and replaces it with Q 1
kj! in equation (8). Thus we see that the definition of the formal integral (10) is indeed a natural one.
Up to now we have only considered GUE,EN2 in our formal integral. Similar constructions are possible for the two other ensembles,E1N andEN4. In general,
as discussed in [38], forβ ∈ {1,2,4}, with the potential V(M) =M2β
4 −
X∞ j=3
βtj
2j Mj, we get the Wick contraction
MijMkl
=1δilδkj−(1+2)δikδlj, where
1= 1
N, 2=− 2 N β.
Graphically, we can represent the second term in the propagator δikδjl by a twisted band. This allows us to extend the formal integral to count the number of non-orientable graphs. We see thatEN2 is in fact a special case where1+2
vanishes.