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The KP Hierarchy and Hurwitz Numbers
Wessel Bindt
July 17, 2013
Projectverslag jaar 3
Supervisor: Loek Spitz, MSc
∧∞2 V ⊇ GL(∞)|0〉
C[x1, x2, . . .] ⊇ {τ -functions} 3 H(β;x1, x2, . . .)
∼
Korteweg-De Vries Instituut voor Wiskunde
Faculteit der Natuurwetenschappen, Wiskunde en Informatica
Universiteit van Amsterdam
Abstract
We give an exposition of the infinite wedge formalism and the theory of the KP hierarchy,and discuss how to construct tau functions of the KP hierarchy by use of the infinitewedge. After this, the Hurwitz numbers are introduced. We relate the Hurwitz numbersto the KP hierarchy by using Goulden and Jackson’s cut-and-join equation to prove thata specific generating series of the Hurwitz numbers is a tau function of the KP hierarchy.Finally, we apply this to Hurwitz-Hodge integrals by trying to generalize Kazarian’s proofthat a certain generating function of Hodge integrals satisfies the equations of the KPhierarchy. This generalization does not lead to interesting results, and we attempt toexplain why.
Title: The KP Hierarchy and Hurwitz NumbersAuthors: Wessel Bindt, [email protected], 6352758Supervisor: Loek Spitz, MScSecond grader: prof. dr. Sergey ShadrinDate: July 17, 2013
Korteweg-De Vries Instituut voor WiskundeUniversiteit van AmsterdamScience Park 904, 1098 XH Amsterdamhttp://www.science.uva.nl/math
ii
Contents
Introduction iv
1 The Infinite Wedge 11.1 The Infinite Wedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Boson-Fermion Correspondence (1.1) . . . . . . . . . . . . . . . . . . . . 61.3 Charge Zero Subspace and the Symmetric Group (1.2) . . . . . . . . . . 13
2 The KP Hierarchy 172.1 Pseudodifferential Operators . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 The KP Hierarchy (2.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3 The Bilinear Identity (2.2) . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Constructing Solutions using∧∞
2 V (1.3, 2.3) . . . . . . . . . . . . . . . . 27
3 Hurwitz Numbers 333.1 Hurwitz Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4 Generating Functions of Hurwitz Numbers 374.1 Hurwitz Numbers and KP Hierarchy (2.4, 3.1) . . . . . . . . . . . . . . . 374.2 A Change of Variables (4.1) . . . . . . . . . . . . . . . . . . . . . . . . . 41
Populaire Samenvatting 45
References 47
iii
Introduction
In the ’80s, the KP equation was extended to an infinite hierarchy of equations calledthe KP hierarchy, exposing the high degree of symmetry of the KP equation. By ex-ploiting this symmetry, one can construct a large class of solutions of the KP hierarchy(and therefore also the KP equation). Though the study of such symmetries appliesto numerous integrable systems, we will restrict our attention to the KP hierarchy. Inchapters 1 and 2, we outline the theory surrounding this hierarchy developed by Satoand others, using the Fermionic Fock space as the natural framework for constructingsolutions.Presently, the KP hierarchy plays a role in diverse areas of mathematics and physics,such as geometry and quantum gravity. In particular, Witten’s conjecture states a cer-tain generating function of intersection numbers of stable classes on the moduli spaceof algebraic curves should satisfy the equations of the KdV hierarchy (which is the KPhierarchy with some additional conditions imposed). First proved by Kontsevich in 1992using matrix integrals, there are now several known proofs of Witten’s conjecture, no-tably one by Kazarian and Lando in 2006. This work later led Kazarian to prove that agenerating series of Hodge integrals on the moduli space of complex stable curves satis-fies the equations of the KP hierarchy.Kazarian’s proof makes use of the ELSV (Ekedahl-Lando-Shapiro-Vainshtein) formula,which expresses Hurwitz numbers (these enumerate branched coverings of P1) in termsof Hodge integrals. The generating function of these Hurwitz numbers satisfies the equa-tions of the KP hierarchy (chapter 3 and the first half of chapter 4 are devoted to this),so by inverting the ELSV formula and using a symmetry of the KP hierarchy, Kazarianproved that the generating function of the Hodge integrals indeed satisfies the equationsof the KP hierarchy.In the second half of chapter 4, using a generalization of the ELSV formula due toJohnson, Pandharipande, and Tseng, we try to generalize Kazarian’s proof to Hodgeintegrals on the moduli space of admissible coverings with monodromy group Zr. How-ever, our calculations seem to indicate that Kazarian’s proof does not carry over directlyto this more general setting.
This thesis is written on a relatively elementary level. Being familiar with the materialin [S1] will more than suffice to understand most of the thesis. For the reader’s conve-nience, we have indicated the prerequisite sections at the end of each section header.
Finally, I wish to express my gratitude towards Loek Spitz, firstly for his invaluableguidance and the amount of time he was willing to spend on shaping this thesis, andsecondly for introducing me to a subject as broad and fascinating as this.
iv
1 The Infinite Wedge
The infinite wedge∧∞
2 V is a certain vector space which has applications in various areasof mathematics. In later chapters we will make use of its ties to integrable systems, andto the representation theory of the symmetric group. Both of these applications arise
from an isomorphism between a space of functions and∧∞
2 V, known as the Boson-Fermion correspondence, which will be defined in section 1.2. In the final section we
discuss the connection of∧∞
2 V with the representation theory of the symmetric group.Our exposition of the infinite wedge largely follows [MJD].
1.1 The Infinite Wedge
In this section we discuss the basics about the infinite wedge. In particular, we definethe standard representation which the infinite wedge carries. We will also define thespace on the other side of the Boson-Fermion correspondence, namely the Bosonic Fockspace and its standard representation.
Definition 1.1. The Heisenberg algebra B is the C-algebra with generators {an, a†n |n ∈ Z>0}, and with relations
[am, an] = 0, [a†m, a†n] = 0, and [an, a
†m] = δmn.
The space C[x1, x2, . . .] of polynomials in infinitely many variables carries a representa-tion of B, defined as follows:
anp = ∂np
a†np = xnp,
where ∂np denotes the derivative of p w.r.t. xn. Clearly the orbit of 1 under this actiongenerates the entire space, so
B1 = C[x1, x2, . . .]. (1.1)
Now suppose that W is a nontrivial B-invariant subspace of C[x1, . . .], and p ∈ W isnonzero. By repeatedly differentiating p (i.e., by applying operators an to p), we find that1 ∈ W , so that W = C[x1, x2, . . .], by equation (1.1). We conclude that the constructedrepresentation is irreducible.Endowed with this representation, C[x1, x2, . . .] is usually called the Bosonic Fock space,and is denoted Fb. The reason for this is that Fb can be used in quantum mechanics to
1
handle situations in which the total number of particles (Bosons, to be more specific) ina given state might vary. For example, the monomial
x1x43x5
describes the state in which there is one particle in the state indexed by 1, four in thatindexed by 3, and one in state 5. Applying a3, we end up with a monomial with oneless particle in state 3. Similarly, a†n adds a particle to state n. For this reason the anand a†n are called annihilation and creation operators, respectively, and 1 is called thevacuum.
Definition 1.2. The Clifford algebra A is the C-algebra generated by the symbols ψn,ψ†n, n ∈ 1
2+ Z, satisfying the relations
[ψn, ψm]+ := ψnψm + ψmψn = 0, [ψ†n, ψ†m]+ = 0, and [ψn, ψ
†m]+ = δnm.
The elements of the formψn1 · · ·ψnsψ†m1
· · ·ψ†mr ,with n1 < · · · < ns and m1 < · · · < mr, form a basis of A.
We now define a space which carries a representation of A, namely the semi-infinite
wedge space∧∞
2 V (infinite wedge for short). This representation is what most of theapplications that we will consider arise from.First some preliminary conventions. Given a set S ⊆ 1
2+ Z, we define S− to be the set
of negative numbers in 12
+ Z not occurring in S, and S+ as the set S ∩ (12
+ Z≥0).
Definition 1.3. First, we define V to be the vector space over C generated by the set{. . . , 3
2, 12,−1
2,−3
2, . . .}. Given a set S = {s1 > s2 > . . .} ⊆ 1
2+ Z with S− is finite, we
definevS = s1 ∧ s2 ∧ . . . ,
and we define the infinite wedge∧∞
2 V to be the vector space based by such vS.
Note that the k are formal symbols, and should not be confused with the numbers 12
+Z.The condition that s1 > s2 > . . . ensures that S+, the set of positive si, is finite.
Let 〈·|·〉 be the unique inner product such that the vS give an orthonormal basis.
Now we can give the standard representation of A on∧∞
2 V. For n ∈ 12
+ Z, define
ψnvS = n ∧ s1 ∧ s2 ∧ . . . ,
and let ψ†n act on∧∞
2 V as the adjoint of ψn w.r.t. 〈·|·〉. It’s easy to see that this extends
to an algebra representation of A on∧∞
2 V.
2
It is useful to have explicit expressions for ψnvS and ψ†nvS:
ψnvS =
{0 if n ∈ S(−1)is1 ∧ . . . ∧ si ∧ n ∧ si+1 ∧ . . . if n 6∈ S and i = max{j | sj < n},
ψ†nvS =
{0 if n 6∈ S(−1)i−1s1 ∧ . . . ∧ si ∧ . . . if n = si.
Given a sequence I = (s1, s2, . . .), we use (s1, s2, . . . , si, . . .) to denote the sequence I,but with si removed.
Similar to C[x1, x2, . . .], the space∧∞
2 V has a physical interpretation in terms of particles.In physics, it is known as Dirac’s electron sea, which is a model devised to deal withthe negative energies of the electron predicted by Dirac’s equation. In this model,the vacuum is redefined to be an infinite sea of negative-energy electrons, instead ofno electrons at all. The absence of a negative-energy electron is then interpreted as thepresence of a positron with positive energy of the same magnitude as that of the removedelectron.The infinite wedge comes into play as follows. We let V describes the possible energylevels of an electron (so 1
2+ Z). Now a state with the energy levels s1 > s2 > s3 > . . .
occupied is described by the element
s1 ∧ s2 ∧ s3 ∧ . . . ∈∧∞
2V .
So the vacuum corresponds to the vector |0〉 := −12∧ −3
2∧ −5
2∧ . . . in
∧∞2 V.
In this physical interpretation, the operator ψn acts (up to a sign change) by creating aparticle of energy n, and ψ†n acts by annihilating one (up to a sign change). The definingrelations of the Clifford algebra imply that ψnψn = 1
2[ψn, ψn]+ = 0, and this shows that
no two particles can occupy the same state. This is Pauli’s exclusion principle, whichreflects the fact that we’re dealing with Fermions rather than Bosons (of which therecan exist an arbitrary number in the same state).
The physical interpretation of∧∞
2 V leads us to define a notion of charge of its elements.For this, we define the charge operator H0 by H0vS = (|S+|−|S−|)vS, where vS is a basis
element of∧∞
2 V. An eigenvector of H0 with eigenvalue q is said to have charge q. Withthis convention, a positive energy electron contributes 1 to the charge, and a positron
(i.e., the absence of a negative-energy electron) contributes −1. We define∧∞2
0 V to be
the space of charge 0 elements of∧∞
2 V.
Another important operator which betrays the physical origins of∧∞
2 V is the energyoperator H. It acts on a basis element vS as follows:
HvS =∑
k∈S+∪S−|k|vS,
3
which is in agreement with the interpretation of V as the possible energy levels of theelectrons.Since we have a basis of
∧∞2 V consisting of simultaneous eigenvectors of H and H0, this
gives us a decomposition∧∞
2 V =⊕
q∈Zd∈ 1
2Z≥0
∧∞2
q, dV .
Here∧∞2q, d V denotes the subspace of charge q and energy d.
An operator L is said to have energy E if it lowers the energy by E, i.e., [H,L] = −EL.Similarly, it has charge q if [H0, L] = −qL. If L has positive energy, then L |0〉 must havenegative energy. But the eigenvalues of H are all nonnegative, so it follows that L |0〉 = 0.This means that for any operator M , the vacuum expectation 〈ML〉 := 〈0|ML |0〉 mustbe 0. In the following section we will compute a lot of vacuum expectations, and thisfact will come in handy.
It is convenient to collect the Fermionic creation and annihilation operators in formalseries,
ψ(k) =∑
n∈ 1
2+Z
ψnkn− 1
2 , and ψ†(k) =∑
n∈ 1
2+Z
ψ†nk−n− 1
2 ,
which are called the Fermionic fields.Given such a formal series F (k) =
∑m∈Z cmk
m with coefficients cm in some ring R (for
instance the ring of operators on∧∞
2 V), we define its residue at infinity as1
Resk = ∞
F (k) = c−1.
More frequently, we will use the notation
∮dk
2πiF (k) = c−1,
which is to be thought of as the integral of F (z) over a positively oriented small circlearound ∞. With this notation, it is obvious that
ψn =
∮dk
2πik−n−
1
2ψ(k), and ψ†n =
∮dk
2πikn−
1
2ψ†(k).
Example 1.4. It is easily verified that 〈ψmψ†n〉 = δmnθ(n < 0), where θ denotes thecharacteristic function which satisfies θ(P ) = 1 when P is true, and θ(P ) = 0 when P
1This definition differs from the complex-analytic definition by a minus sign. Since we will not bemaking use of complex analysis, this shouldn’t be a problem.
4
is false. Using this, we can compute the vacuum expectation of ψ(p)ψ†(q),
〈ψ(p)ψ†(q)〉 =∑
m,n∈ 1
2+Z
〈ψmψ†n〉pm−1
2 q−n−1
2
=∑
m,n∈ 1
2+Z
δm,nθ(n < 0)pm−1
2 q−n−1
2
=∑
m∈Z≥0
1
p
(q
p
)m=
1
p− q .
Here 1p−q should be understood as its expansion in p and q with |p| > |q|.
This example can be generalized using a result from quantum field theory known asWick’s theorem. This theorem allows one to compute the vacuum expectation of anarbitrary product of creation and annihilation operators, say 〈w1 · · ·wn〉, in terms of thevacuum expectations of all possible pairings wiwj. The idea is to commute operatorswhich send |0〉 to 0 to the right. Since the anticommutators of the ψn and ψ†m arealways either 0 or 1, this reduces the number of factors in the product by at least 1. Forexample,
〈ψ−3/2ψ−1/2ψ†1/2ψ
†−1/2〉 = −〈ψ−3/2ψ
†1/2ψ−1/2ψ
†−1/2〉
= 〈ψ−3/2ψ†1/2ψ
†−1/2ψ−1/2〉 − 〈ψ−3/2ψ
†1/2〉
= −〈ψ−3/2ψ†1/2〉.
By repeatedly reducing the number of factors like this, we finally end up with an ex-pression in terms of the vacuum expectations of pairs of factors.The generalization of the previous example is the following theorem.
Theorem 1.5 (Wick). Let p1, . . . , pn and q1, . . . , qm be formal variables. Then we have
〈ψ(p1) · · ·ψ(pn)ψ†(qm) · · ·ψ†(q1)〉 =
det
(1
pi − qj
)if m = n
0 if m 6= n,
where 1pi−qj is to be understood as the formal series
∑r∈Z≥0
1pi
(qjpi
)r.
The case m = n can be restated using Cauchy’s determinant formula:
〈ψ(p1) · · ·ψ(pn)ψ†(qn) · · ·ψ†(q1)〉 =
∏
1≤i<j≤n
(pi − pj)(qj − qi)∏
1≤i,j≤n
(pi − qj).
5
1.2 Boson-Fermion Correspondence (1.1)
As it turns out, C[x1, x2, . . .][z, z−1] also carries a representation of A (the z and z−1 are
added to account for charge), and∧∞
2 V carries one of B. We start off with describing
the representation of B on∧∞
2 V.Let Hn, for n ∈ Z, be given by
Hn =∑
j∈ 1
2+Z
:ψjψ†n+j:,
where the colons denote the normal product, defined by
:ψiψ†j : =
{ψiψ
†j if j > 0
−ψ†jψi if j < 0.
Roughly speaking, the summands act on vS ∈∧∞
2 V by replacing n+ j with j, resultingin 0 if this is not possible. Evidently, for j � 0 and j � 0 this is not possible, so
∑
j∈ 1
2+Z
:ψjψ†j+n:vS
is a finite sum. We see that the Hn are linear operators on∧∞
2 V. Even more is true, asthe following proposition shows.
Proposition 1.6. The infinite wedge carries a representation of B, given by
an 7→ Hn and a†n 7→ −1
nH−n.
Proof. Using [AB,C] = A[B,C]+ − [A,C]+B, one can easily show that
[Hn, ψm] = ψm−n, and [Hn, ψ†m] = −ψ†m+n. (1.2)
Using this, we can compute [Hm, Hn]. We have
[Hm, Hn] =∑
i>−n
[ψi−mψ
†n+i − ψiψ†m+n+i
]+∑
i<−n
[ψ†m+n+iψi − ψ†n+iψi−m
].
If m 6= −n, then ψiψ†m+n+i = −ψ†m+n+iψi and ψ†n+iψi−m = −ψi−mψ†n+i, so the above
reduces to ∑
i∈ 1
2+Z
ψi−mψ†n+i +
∑
i∈ 1
2+Z
ψ†m+n+iψi,
which, after changing the index on the left sum to j = i−m, is seen to be equal to∑
i∈ 1
2+Z
[ψi, ψ†m+n+i]+ = 0.
6
The sum vanishes because i 6= m+ n+ i, by m 6= −n.When m = −n, a similar calculation shows that [Hm, Hn] = m, so we have
[Hm, Hn] = mδm,−n. (1.3)
It follows that
an 7→ Hn, a†n 7→ −1
nH−n
defines a representation of B on∧∞
2 V.
Example 1.7. We have two separate definitions of H0, the definition from the previoussection, H0vS = (|S+| − |S−|)vS, and the one from this section. We show that theycoincide.To see how H0 acts on a basis element vS of
∧∞2 V, we consider the action of the
terms :ψkψ†k: separately. If k < 0, then :ψkψ
†k: = −ψ†kψk, so :ψkψ
†k:vS = −vS if
k /∈ S, and is 0 otherwise. It follows that∑
k<0 :ψkψ†k:vS = −|S−|vS. Similarly we
find∑
k>0 :ψkψ†k:vS = |S+|vS, so H0vS = (|S+| − |S−|)vS, as in the previous section.
A similar argument shows that the energy operator H is equal to∑
k∈ 1
2+Z k:ψkψ†k:.
Example 1.8. In this example we compute the energies and charges of the Hn. SinceH0 and H commute (the vS are a basis of eigenvectors of both), they have energy andcharge 0. By equation (1.3), the Hn all have charge 0.If we show that ψj and ψ†j+n have energies −j and j + n, then it follows that :ψjψ
†n+j:,
and hence Hn, has energy n. Since :ψiψ†i : = ψiψ
†i − 〈ψiψ†i 〉, and because [A,BC] =
[A,B]+C −B[A,C]+, we have
[ψj, :ψiψ†i :] = [ψj, ψi]+ψ
†i − ψi[ψj, ψ†i ]+ = −δijψi.
It follows immediately that [H,ψj] = jψj. The fact that [H,ψ†n+j] = −(n + j)ψ†n+j canbe shown similarly.
To define a representation of A on C[x1, x2, . . .][z, z−1], it suffices to define how the
Fermionic fields ψ(k) and ψ†(k) act on elements of C[x1, x2, . . .][z, z−1]. When these
are defined, then the action of, for example, ψn on an element f(x, z) can be found by
looking at the coefficient of kn−1
2 in ψ(k)f(x, z).For f ∈ C[z, z−1, x1, x2, . . .], define (eKf)(z, x) = zf(z, x), and (kH0f)(z, x) = f(kz, x).Now ψ(k) and ψ†(k) act on C[z, z−1, x1, x2, . . .] as the operators Ψ(k) and Ψ†(k), whichare defined by
Ψ(k) = exp
(∑
n≥1
xnkn
)exp
(−∑
n≥1
1
nkn∂n
)eKkH0 ,
and
Ψ†(k) = exp
(−∑
n≥1
xnkn
)exp
(∑
n≥1
1
nkn∂n
)e−Kk−H0 ,
7
where e−K and k−H0 denote the inverses of eK and kH0 . Explicitly, the action of ψ(k)on f(z, x) is given by
(ψ(k)f)(z, x) = zeξ(x,k)f(kz, x1 − 1
k, x2 − 1
2k2, . . .
),
where we have introduced the shorthand notation ξ(y, p) =∑∞
n=1 ynpn for any sequence
y = (y1, y2, . . .), and any formal variable p.
The Boson-Fermion correspondence is an isomorphism between C[x1, x2, . . .][z, z−1] and∧∞
2 V which is both A and B-equivariant. It is given by
u 7→∑
q∈Z
zq 〈q| eH(x) |u〉 ,
where |q〉 := q − 12∧ q − 3
2∧ . . ., and H(x) denotes the operator
∑
n≥1
xnHn,
which maps∧∞
2 V into C[x1, x2, . . .]⊗∧∞
2 V (every Hn lowers the energy of a basis elementvS by n, so only finitely many terms in the sum act nontrivially). The exponent eH(x),which is defined to be ∑
n≥0
1
n!(H(x))n ,
also maps∧∞
2 V into C[x1, x2, . . .] ⊗∧∞
2 V. This follows from the fact that every termin (H(x))k has energy ≥ k, which implies that only finitely many powers of H(x) actnontrivially on a basis element vS.
Theorem 1.9 (Boson-Fermion correspondence I). The map
Φ:∧∞
2 V −→ C[x1, x2, . . .][z, z−1], u 7→
∑
q∈Z
zq 〈q| eH(x) |u〉
is an isomorphism of vector spaces, and it is both B and A-equivariant. That is,
Φ(Hnu) =
{∂nΦ(u) if n > 0
−nx−nΦ(u) if n < 0,
andΦ(ψ(k)|u〉) = Ψ(k)Φ(|u〉), and Φ(ψ†(k)|u〉) = Ψ†(k)Φ(|u〉)
for all |u〉 ∈ ∧∞2 V.
8
Proof. Let n be a positive integer. The Hm with m > 0 commute, so ∂neH(x) = eH(x)Hn.
Therefore we have
∂nΦ(u) =∑
q∈Z
zq 〈q| ∂neH(x) |u〉
=∑
q∈Z
zq 〈q| eH(x)Hn |u〉 = Φ(Hnu).
Using the Baker-Campbell-Hausdorff formula and the fact that [H(x), H−n] = nxn, weget
eH(x)H−ne−H(x) = H−n + [H(x), H−n] +
1
2![H(x), [H(x), H−n]] + . . .
= H−n + nxn,
so that
Φ(H−nu) =∑
q∈Z
zq 〈q| eH(x)H−ne−H(x)eH(x) |u〉
=∑
q∈Z
zq 〈q| (H−n + nxn) eH(x) |u〉 .
The fact that H−n has negative energy and charge 0 makes it easy to see that 〈q|H−n =0 (because 〈q| is the lowest energy state with charge q). From this it follows thatΦ(H−nu) = nxnΦ(u).For positive n, Hn |q〉 = 0, so eH(x) |q〉 = |q〉, and Φ(|q〉) = zq. We can now obtain anymonomial of z-degree q in C[x1, x2, . . .][z, z
−1] by successively applying the right H−n to|q〉. This shows that Φ is surjective.The injectivity of Φ can be shown using a combinatorial argument involving the Hilbert-
Poincare series of∧∞
2 V and Fb, or by using the alternative basis of∧∞2
0 V given afterproposition 1.10. The latter argument is much easier, so we postpone the injectivity ofΦ until after proposition 1.10, in remark 1.11.
We only prove the A-equivariance for ψ(k), since the proof for ψ†(k) is similar.Defining ∂ = (∂1,
12∂2,
13∂3, . . .), on the C[z−1, z, x1, x2, . . .]-side we have
Ψ(k)Φ(|u〉) =∑
q∈Z
Ψ(k)zq 〈q| eH(x) |u〉
=∑
q∈Z
zq+1kq 〈q| eξ(x,k)e−ξ(∂,k−1)eH(x) |u〉
=
[∑
q∈Z
zq+1kq 〈q| e−H(ε(k))
]eξ(x,k)eH(x) |u〉 ,
9
where we let H(ε(k)) =∑
n≥1Hn1nkn
.
To rewrite the∧∞
2 V-side, we use the fact that [H(x), ψ(k)] = ξ(x, k)ψ(k) (this fol-lows easily from the commutation relations (1.2)). Now the Baker-Campbell-Hausdorffformula gives
eH(x)ψ(k)e−H(x) = ead(H(x))(ψ(k)) = eξ(x,k)ψ(k), (1.4)
so that
Φ(ψ(k) |u〉) =
[∑
q∈Z
zq 〈q|ψ(k)
]eξ(x,k)eH(x) |u〉 .
Therefore it suffices to show that∑
q∈Z
zq 〈q|ψ(k) =∑
q∈Z
zq+1kq 〈q| e−H(ε(k)),
or, if we look at the components of the zq individually,
〈q|ψ(k) = kq−1 〈q − 1| e−H(ε(k)).
To prove this equality, we can evaluate both sides on a basis element
|v〉 := ψi1 · · ·ψinψ†j1 · · ·ψ†jm|0〉 ,
and show that the outcome is the same. But if we let p1, . . . , pn and q1, . . . , qm be formalvariables, then it is easy to see that |v〉 occurs as a coefficient in the series
S(p1, . . . , pn, qm, . . . , q1)|0〉 = ψ(p1) · · ·ψ(pn)ψ†(qm) · · ·ψ†(q1)|0〉 ,
so it suffices to show that
〈q|ψ(k)S(p1,..., pn, qm,..., q1)|0〉 = kq−1 〈q − 1|e−H(ε(k))S(p1,..., pn, qm,..., q1)|0〉
This is convenient, because now we can express both sides as the vacuum expectation ofa product of Fermionic generating series, which can be computed using Wick’s theorem.
Note that on the left-hand side, we compute the vacuum expectation of |q| + n + 1creation operators2 (|q| from 〈q|, n from S, 1 from ψ(k)), and m annihilation operators.By Wick’s theorem it follows that if m 6= n + |q − 1|, then the left-hand side evaluatesto 0. The same argument shows that the right-hand side is also equal to 0 in this case.In the following calculations we therefore assume m = n+ |q − 1|.
The left-hand side is the easiest to rewrite. By definition we have
〈q| = 〈0|ψ−1/2 · · ·ψq+1/2 = 〈0|m−1∏
j=n+1
∮dpj2πi
pj−n−1j ψ(pj).
2Here we implicitly assume q < 0. The case q ≥ 0 is handled similarly, and therefore omitted.
10
So we can express the left-hand side as a repeated contour integral of a vacuum expec-tation of generating functions, namely
(−1)n(m−n)∮dpn+1
2πi· · ·∮dpm−1
2πip0n+1· · ·pm−n−2m−1 〈ψ(p1) · · ·ψ(pm−1)ψ(k)ψ†(qm) · · ·ψ†(q1)〉.
The factor (−1)n(m−n) appears because we commuted ψ(pn+1) · · ·ψ(pm−1)ψ(k) to theright of ψ(p1) · · ·ψ(pn), and the ψ(pi) anticommute. Wick’s theorem now gives that theleft-hand side is equal to (we rename k as pm for the sake of brevity)
(−1)n(m−n)∮dpn+1
2πi· · ·∮dpm−1
2πip0n+1· · ·pm−n−2m−1
∏
1≤i<j≤m
(pi − pj)(qj − qi)∏
1≤i,j≤m
(pi − qj),
where the factors in the denominator should be expanded as though |pi| > |qj|.We first carry out the integration over pn+1. The part of the integrand which dependson pn+1 is equal to [
n∏
i=1
(pi − pn+1)
][m∏
j=n+2
(pn+1 − pj)]
m∏
j=1
(pn+1 − qj). (1.5)
The numerator is a polynomial in pn+1 of degree m− 1, with leading coefficient (−1)n.
The denominator is an m-fold product of series of the form∑
i≥0
(qjpn+1
)i1
pn+1, so the
term of highest degree in pn+1 is p−mn+1. The integral is defined as the coefficient of p−1n+1,so we find that it equals (−1)n. It follows that the left-hand side is equal to
(−1)n(m−n+1)
∮dpn+2
2πi· · ·∮dpm−1
2πip1n+2· · ·pm−n−2m−1
[ ∏
1≤i<j≤mi 6=n+16=j
(pi − pj)][ ∏
1≤i<j≤m
(qj − qi)]
∏
1≤i,j≤mi 6=n+1
(pi − qj).
Now we integrate over pn+2. The pn+2-dependent part of the integrand is[
n∏
i=1
(pi − pn+2)
][m∏
j=n+3
(pn+2 − pj)]
m∏
j=1
(pn+2 − qj)pn+2,
which again has a polynomial of degree m−1 as the numerator, and a denominator withhighest order term p−mn+2. It follows that the integral is equal to (−1)n.
11
The general pattern is clear. As we evaluate the integrals from left to right, the factorswhich depend on pn+1 are replaced by (−1)n, the factors which depend on pn+2 arereplaced by (−1)n, and so on. So when we integrate over pn+k, the integrand is
[n∏
i=1
(pi − pn+k)][
m∏
j=n+k+1
(pn+k − pj)]
m∏
j=1
(pn+k − qj)pk−1n+k,
of which the integral is (−1)n by the same argument as for pn+1 and pn+2.
We now see that computing the integrals amounts to replacing the factors which dependon pn+1, . . . , pm−1 by (−1)n(m−n−1) (there are m − n − 1 integrals, each gives a factor(−1)n), so the left-hand side is equal to
n∏
i=1
(k − pi)m∏
j=1
(k − qj)
[ ∏
1≤i<j≤n
(pi − pj)][ ∏
1≤i<j≤m
(qj − qi)]
∏
1≤i≤n1≤j≤m
(pi − qj), (1.6)
where we’ve substituted k back for pm.
Now we express the right-hand side as a vacuum expectation of Fermionic generatingseries. To do so, we must get rid of the e−H(ε(k)), so we will commute it to the rightusing the relations
e−H(ε(k))ψ(p) =(1− p
k
)ψ(p)e−H(ε(k)) and e−H(ε(k))ψ†(p) =
1(1− p
k
)ψ†(p)e−H(ε(k)),
where (1− pk)−1 should be read as the series
∑n≥0(pk
)n. These relations can be proved
using equation (1.4) and its ψ†-analogue, and the fact that
exp
(∑
n≥1
1
n
(pk
)n)
=1(
1− pk
) ,
which can be proved by showing that the derivative of (1 − x) exp(∑
n≥11nx)
w.r.t. xvanishes and noting the fact that the zeroth order coefficient in this series is 1.
Since e−H(ε(k)) |0〉 = |0〉 (because H(ε(k)) annihilates |0〉), we see that the right-handside is equal to
n∏
i=1
(k − pi)m∏
j=1
(k − qj)〈q − 1|ψ(p1) · · ·ψ(pn)ψ†(qm) · · ·ψ†(q1)|0〉 .
12
Note that the power of k introduced by writing (1− pk) = 1
k(k − p) cancelled the factor
kq−1 which was already present. Now by writing out the definition of 〈q − 1|, and byexpressing the creation operators as contour integrals of ψ(pn+1), . . . , ψ(pm), we find thatthe right-hand side is given by
(−1)n(m−n)
n∏
i=1
(k − pi)m∏
j=1
(k − qj)
∮dpn+1
2πi· · ·∮dpm2πi
p0n+1· · ·pm−n−1m 〈ψ(p1) · · ·ψ(pm)ψ†(qm) · · ·ψ†(q1)〉.
The integral is now exactly the same as it was on the left-hand side, except for thefactor pm−n−1m we now have, and the fact that we also integrate over pm. As before, aswe calculate the integrals from left to right, the integral over pn+k replaces the factorsinvolving pn+k with (−1)n. The resulting expression is obviously equal to (1.6), whichcompletes the proof.
1.3 Charge Zero Subspace and the Symmetric
Group (1.2)
In this section we show that the action of the Hn on∧∞
2 V gives rise to a connection
between the charge zero subspace of∧∞
2 V and the characters of the symmetric group.Since these characters play an important role in combinatorics, this relates the infinitewedge to combinatorics, which will be useful in chapters 3 and 4 when we discuss Hurwitznumbers. We begin by collecting some basic facts about the charge zero subspace.
We now describe the vS of charge 0 more explicitly. They can be naturally labelled bypartitions as follows:
vλ := λ1 − 12∧ λ2 − 3
2∧ . . . =
[n−1∏
i=0
ψλn−i+i−n+
12ψ†i−n+1
2
]|0〉. (1.7)
Conversely, we give a rough sketch of an algorithm to associate a Young diagram to
vS ∈∧∞2
0 V. Starting at s = +∞, and going through 12
+ Z to −∞, draw line segmentsof a fixed length according to the following prescription:
s ∈ S+ =⇒ down
s ∈ (12
+ Z)>0 \ S+ =⇒ left
s ∈ S− =⇒ left
s ∈ (12
+ Z)<0 \ S− =⇒ down
Since |S+| = |S−|, the sum∑
k∈S+∪S− |k| has an even number of terms, and is thereforean integer. More specifically, one can show that Hvλ = |λ|, using an argument based on
13
the energies of the ψk and equation (1.7).
The action of the Hn on vλ ∈∧∞
2 V has a nice description in terms of the Young diagramof λ. The operator Hn subtracts (adds, if n < 0) boundary strips S of length |n| tothe diagram of λ. The sign given to vλ∪S is (−1)m+1, where m is the number of rowsaffected by the addition of S. For example,
H−2 = − + − + .
Using the interpretation of the Hn as operators on the Young diagrams, it is easy to seethat H†m = H−m.The description of theHn in terms of Young diagrams is also reminiscent of the Murnaghan-Nakayama rule (see [S1] for a proof), a result about the representations of symmetricgroups.
Theorem (Murnaghan-Nakayama rule). Let χλµ denote the character of the Specht mod-ule Rλ evaluated on the conjugacy class of elements of cycle type µ. Then
χλµ =∑
S
(−1)1+sgn(S)χλ\Sµ\µ1,
where the sum is taken over all boundary strips S of λ of length µ1. The sign sgn(S) ofa boundary strip S is defined as the number of rows S spans in the Young diagram of λ.
The following proposition makes the connection between∧∞
2 V and the characters ofthe symmetric group explicit. We make use of the notation Hµ = Hµ1 · · ·Hµn andH−µ = H−µ1 · · ·H−µn for any partition µ.
Proposition 1.10. Let µ be a partition of d. Then
H−µ|0〉 =∑
λ`d
χλµvλ,
and given a fixed partition λ with |λ| = d = |µ|, we have
Hµvλ = χλµ|0〉.
Proof. For the first equation, we use the MN rule, and proceed by induction on thelength of µ.If µ = (d) is of length one, then the LHS is easy to compute using the description ofH−d in terms of Young diagrams. The boundary strips of the empty diagram (i.e., the
14
Young diagram associated to |0〉) are precisely the hook diagrams of length d. So wefind
H−µ1|0〉 =d−1∑
k=0
(−1)k+1v(d−k,1k).
To find the RHS, we need to compute the characters χλ(d) for all partitions λ of length d.If λ is not a hook diagram, then it cannot have a boundary strip of length d, so from MNit follows that χλ(d) = 0. Applying the MN rule to a hook diagram λ = (d− k, 1k) gives
χλ(d) = (−1)k−1χ∅∅ = (−1)k−1. This shows that the first equality holds for µ of length 1.Assume now that the first equation is true for µ of length n− 1. Then we have
H−µ1H−µ2 · · ·H−µn|0〉 =∑
λ`d−µ1
χλµ\µ1H−µ1vλ
=∑
λ`d−µ1
χλµ\µ1
∑
S
(−1)sgn(S)vλ∪S,
where the last sum is taken over all boundary strips S of length µ1, of λ. If λ ` d, thenthe coefficient of vλ in the sum above is given by
∑
S
(−1)sgn(S)χλ\Sµ\µ1 .
And by the MN rule this equals χλµ. Therefore we find
H−µ|0〉 =∑
λ`=d
χλµvλ,
which was to be shown.
The second equation is obtained more easily. First we note that Hmvλ has energy |λ|−mand that Hm leaves the charge invariant, and conclude that Hµvλ must be of energy andcharge 0. That is, it is an element of C |0〉. Taking the inner product with 〈vλ| in thefirst equation, we obtain
〈vλ|H−µ |0〉 = χλµ,
which gives us the coefficient of |0〉 in H†−µvλ. From the fact that H†m = H−m and thefact that Hm and Hk commute when k 6= −m, it now follows that
Hµvλ = χλµ|0〉.
A basic result from representation theory states that for partitions µ, ν ` d,
∑
λ`d
χλµχλν = |Kµ|δµ,ν ,
15
where Kµ ⊆ Sd denotes the conjugacy class of elements of cycle type µ. This shows thatthe character table of Sd has orthogonal rows of nonzero length, and hence is invertible.The matrix which transitions between vλ and H−µ|0〉 is the direct sum of the charactertables of the Sd, and hence is invertible. We can conclude that {H−µ|0〉 | µ a partition}is a basis for
∧∞20 V. Immediately we obtain a basis for
∧∞2 V, namely {T qH−µ|0〉 |
µ a partition, q ∈ Z}. Here T is the translation operator, given by
T(s1 ∧ s2 ∧ s3 ∧ . . .
)= s1 + 1 ∧ s2 + 1 ∧ s3 + 1 ∧ . . . .
Remark 1.11. Using the B-equivariance of the Boson-Fermion correspondence Φ, wecan now show that Φ is injective. The B-equivariance implies that T qH−µ |0〉 is mappedto a non-zero scalar multiple of the monomial zqxµ := xµ1xµ2 · · ·xµn . These monomialsare linearly independent, so the injectivity of Φ now follows from the fact that the
T qH−µ |0〉 form a basis of∧∞
2 V. This completes the proof of theorem 1.9.
The preceding proposition allows us to write down an expression for the Boson-Fermioncorrespondence which is easier to compute than that in theorem 1.9, by using theMurnaghan-Nakayama rule.
Proposition 1.12. Given a partition λ ` d, we have
Φ(vλ) =∑
µ`d
χλµAut(µ)
xµ.
The sum is over all partitions of d, xµ is defined to be xµ1 · · ·xµn, and Aut(µ) is definedas m1(µ)!m2(µ)! · · ·mµ1(µ)!, where mi(µ) is the multiplicity of i in µ.
Proof. Let k ∈ Z≥0. The only nonzero terms in
Φ(vλ) = 〈0| (x1H1 + x2H2 + . . .)k |vλ〉
are the ones of the form 〈0|xm1Hm1 · · ·xmkHmk |vλ〉 such that the mi add up to d. Ifthey would not add up to d, then Hm1 · · ·Hmkvλ would be of nonzero energy and henceorthogonal to |0〉.Applying the previous proposition, we get
〈0|xm1Hm1 · · ·xmkHmk |vλ〉 = χλµxµ,
where µ ` d is the partition obtained by reordering (m1, . . . ,mk). A simple combinatorialargument now gives the result.
16
2 The KP Hierarchy
The KP hierarchy is an infinite system of nonlinear partial differential equations whichplays a role in mathematical physics and geometry. Stating the KP hierarchy takessome effort, and we do so in section 2.2, after we deal with some preliminaries onpseudodifferential operators in section 2.1. In the third section we rephrase the KPhierarchy as an identity known as the bilinear identity. We then use this identity and theBoson-Fermion correspondence to construct solutions of the KP hierarchy in section 2.4.Our exposition mostly follows [D] and [MJD], and more detailed accounts of the theorycan be found in these books.
2.1 Pseudodifferential Operators
We present the basics of pseudodifferential operators, and give propositions which willprove to be useful for defining the KP hierarchy. Throughout this section, x1, x2, . . .are indeterminates, and x denotes (x1, x2, . . .), so that f(x) is a function of the xi. Thederivative ∂/∂xn is denoted by ∂n, and ∂ := ∂1.
Let R be the ring of power series in variables x1, x2, . . . with coefficients in C. A pseu-dodifferential operator is a formal sum of the form
∞∑
n=0
un∂N−n,
where the ui ∈ R, and N ∈ Z. The pseudodifferential operators are supposed to extendthe usual differential operators R[∂] by adding in a formal inverse of ∂, namely ∂−1. Thismotivates the following definition of the product of two pseudodifferential operators:
PQ =
(∑
n≥0
pn∂N−n
)(∑
m≥0
qm∂M−m
)=
∑
s,m,n≥0
(N − ns
)pn∂
s(qm)∂N+M−m−n−s.
Note that this product reduces to the usual Leibniz rule when P and Q only have termsf∂i with i ≥ 0. With this product, the space of all pseudodifferential operators becomesa C-algebra, denoted R[∂, ∂−1). It can be checked that if the leading coefficient ofP ∈ R[∂, ∂−1) is invertible in R, then P has an inverse in R[∂, ∂−1).The degree of a pseudodifferential operator P =
∑i pi∂
i is the highest power of ∂occurring in the sum, and it is denoted by degP . It follows directly from the definition
17
of the product that deg(PQ) = deg(P )+deg(Q), and deg([P,Q]) ≤ deg(P )+deg(Q)−1.By P+ we denote the differential part of a pseudodifferential operator, i.e.,
P+ =∑
i≥0
pi∂i,
and P− := P − P+. The formal residue Res∂
P of a pseudodifferential operator is the
coefficient of ∂−1, that is,
Res∂
∑
i
pi∂i = p−1.
The formal adjoint of P =∑
i≥0 pi∂n−i is defined as P † =
∑i≥0(−∂)n−ipi, where the
product (−∂)n−ipi is understood as the product of (−∂)n−i and pi∂0 as pseudodifferential
operators. Since (−∂)n−ipi has degree n− i, this term only has nonzero contributions tothe coefficient of ∂k with k ≤ n− i in P †. It follows that only finitely many (−∂)n−jpjcontribute to the coefficient of any ∂k, so that P † is well-defined.A rather long, but elementary calculation shows that P †Q† = (QP )†. An immediateconsequence of this is that (P−1)† = (P †)−1. Another fact which can be checked bydirect computation is that (P †)† = P .
Note that the pseudodifferential operators aren’t actual operators; their action on ele-ments of R is only defined when they are strictly differential. However, we define theaction of ∂i on the exponent of the function
ξ(x, k) =∞∑
n=1
xnkn
by ∂ieξ(x,k) = kieξ(x,k). We extend the action to functions of the form Peξ(x,k), with Pa pseudodifferential operator, by Q(Peξ(x,k)) = (QP )eξ(x,k) for all Q ∈ R[∂, ∂−1). Thisdefines the R[∂, ∂−1)-module M of oscillating functions,
M := {Qeξ(x,k) | Q ∈ R[∂, ∂−1)}.
As a vector space, M should be viewed as a subspace of R[[k, k−1]], the space of formalLaurent series in k with power series in x1, x2, . . . as coefficients. The fact that thisaction of R[∂, ∂−1) on M is well-defined follows from the proof of Proposition 2.1.In the next section, we will often encounter the situation where we have w ∈M \ 0 andP ∈ R[∂, ∂−1) such that Pw = 0. In this case we would like to conclude that P = 0,and the following proposition allows us to do just that.
Proposition 2.1. The space M is a one-dimensional free R[∂, ∂−1)-module. SinceR[∂, ∂−1) has no zero divisors, this implies that if Pw = 0, with w ∈ M \ 0, then itfollows that P = 0.
Proof. We will prove that if Peξ(x,k) = 0, then P = 0. Then it follows that if Peξ(x,k) =Qeξ(x,k), then P = Q, so that Peξ(x,k) 7→ P obviously defines a R[∂, ∂−1)-linear bijection
18
between M and R[∂, ∂−1).Define s0, s1, . . . ∈ R by the relation
eξ(x,k) = s0 + s1k1 + s2k
2 + . . . ,
where s0 = 1. So if P =∑
i≤N pi∂i, then we have
Peξ(x,k) =
(∑
i≤N
piki
)eξ(x,k) =
∑
`∈Z
∑
i≥0
p`−isik`,
where we have put p`−i = 0 when ` − i > N . Looking at the coefficients of the k`, for` = N,N − 1, N − 2, . . ., we find that the pn satisfy
0 =
1 s1 s2 · · ·0 1 s1 · · ·0 0 1 · · ·...
......
. . .
pNpN−1pN−2
...
= Sp.
So if we find a left-inverse of S, then it follows that the pn vanish, so that P = 0. It’seasy to determine, by induction, a row vector (A0, A1, . . .) whose inner product with thefirst column of S is 1, and which is orthogonal to the remaining columns of S. Nowobviously
A :=
A0 A1 A2 · · ·0 A0 A1 · · ·0 0 A0 · · ·...
......
. . .
is a left-inverse of S. Therefore P = 0, which was to be shown.
The following lemma is easy to prove, but is quite useful. We record it here for lateruse.
Lemma 2.2. If P and Q are pseudodifferential operators, then
Resk = ∞
[(Peξ(x,k))(Qe−ξ(x,k))
]= Res
∂P ·Q†.
Proof. Let P =∑
i pi∂i, and Q =
∑j qj∂
j. Then, on the left side we have
Resk = ∞
[(Peξ(x,k))(Qe−ξ(x,k))
]= Res
k = ∞
∑
i,j
pikieξ(x,k)(−1)jqjk
je−ξ(x,k)
=∑
i+j=−1
(−1)jpiqj,
all of which follows directly from the definition of the action of P and Q on eξ(x,k) ande−ξ(x,k).Multiplying P and Q†, we get
PQ† =∑
i,j
(−1)jpi∂i+jqj.
19
When i+ j is nonnegative, then ∂i+jqj is a differential operator, so terms with i+ j ≥ 0don’t contribute to the residue. Therefore
Res∂
PQ† = Res∂
∑
i+j<0
(−1)jpi∂i+jqj
= Res∂
∑
i+j<0s≥0
(i+ j
s
)(−1)jpi∂
s(qj)∂i+j−s.
Now if i + j − s = −1 in this sum, then because of s ≥ 0 and i + j < 0, it follows thats = 0, and hence i+ j = −1. Therefore we find
Res∂
PQ† =∑
i+j=−1
(i+ j
0
)(−1)jpiqj.
Noting that(i+j0
)= 1, we arrive at the desired result.
2.2 The KP Hierarchy (2.1)
Now that we know the basics of pseudodifferential operators, we can define the KPhierarchy. Let L ∈ R[∂, ∂−1) be of the form ∂ + u1∂
−1 + u2∂−2 + . . ., and define Bn =
(Ln)+. Now L is a solution of the KP hierarchy if for all n ∈ N, we have
∂n(L) = [Bn, L].
Here ∂n acts on the coefficients of L.Note that the KP hierarchy is really just an infinite system of nonlinear partial differ-ential equations in the coefficients ui of L. For example, for n = 1, the KP hierarchyreads
∂1(L) = [∂, L] = ∂L− L∂ = ∂(L),
so ∂1(ui) = ∂(ui) for all i. After the next proposition, which gives an equivalent setof equations, we will be able to give a non-trivial example of an equation in the KPhierarchy.
Proposition 2.3. A pseudodifferential operator L is a solution to the KP hierarchy ifand only if
∂n(Bm)− ∂m(Bn) = [Bn, Bm]
holds for all n,m ∈ N. These equations are called the Zakharov-Shabat equations.
Proof. First assume that the ZS equations hold, and suppose that
∂n(L)− [Bn, L] = b∂r + . . . ,
20
where b ∈ R is nonzero, the dots denote lower order terms in ∂, and r is some integer.From the fact that [Bn, L] and ∂n(L) satisfy the Leibniz rule as functions of L, i.e.,D(L1L2) = L1D(L2) +D(L1)L2 for D = [Bn, ·] and D = ∂n, one can easily derive that
∂n(Lm)− [Bn, Lm] =
m−1∑
j=0
Lj (∂n(L)− [Bn, L])Lm−j−1.
It follows that
∂n(Lm)− [Bn, Lm] =
m−1∑
j=0
Lj (b∂r + . . .)Lm−j−1
=m−1∑
j=0
(b∂r+m−1 + . . .
)
= mb∂r+m−1 + . . . ,
so ∂n(Lm)− [Bn, Lm] is of degree r +m− 1.
On the other hand, from the ZS equations and the fact that Lm = Bm+(Lm)−, it followsthat
∂n(Lm)− [Bn, Lm] = ∂n(Bm) + ∂n((Lm)−)− [Bn, Bm]− [Bn, (L
m)−]
= ∂m(Bn) + ∂n((Lm)−)− [Bn, (Lm)−],
which is a sum of operators of degrees ≤ n, ≤ −1, and ≤ n− 2, respectively. Thereforewe find that the degree of ∂n(Lm) − [Bn, L
m] is less than or equal to n. But for largeenough m this contradicts the fact that deg(∂n(Lm)−[Bn, L
m]) = r+m−1. We concludethat the ZS equations imply ∂n(L) = [Bn, L] for all n ∈ N.
Conversely, suppose that the equations of the KP hierarchy hold. Then we have
∂n(Bm)− ∂m(Bn)− [Bn, Bm] =(∂n(Lm)− ∂m(Ln)− [Bn, Bm]
)+
. (2.1)
From the fact that [Bn, ·] and ∂n satisfy the Leibniz rule we conclude that
∂n(Lm)− [Bn, Lm] =
m−1∑
j=0
Lj (∂n(L)− [Bn, L])Lm−j−1,
which is zero because we assumed that the equations of the KP hierarchy hold. So wefind(∂n(Lm)− ∂m(Ln)− [Bn, Bm]
)+
=(
[Bn, Lm]− [Bm, L
n]− [Bn, Bm])+
=(
[Lm, Ln]− [Lm, Bn]− [Bm, Ln] + [Bm, Bn]
)+
=(
[Lm −Bm, Ln −Bn]
)+
.
21
This is the differential part of the operator [(Lm)−, (Ln)−]. Since this operator is of
degree < 0, its differential part is 0. Using equation (2.1) we find that
∂n(Bm)− ∂m(Bn) = [Bn, Bm].
Example 2.4. In this example we show that the Kadomtsev-Petviashvili equation ispart of the KP hierarchy, which explains where the name comes from. Some basiccomputations show that B2 = ∂2 + 2u1 and that B3 = ∂3 + 3u1∂ + 3u2 + 3∂1(u1). Itfollows that
[B2, B3] = (3∂21(u1) + 6∂1(u2))∂ + ∂31(u1)− 6u1∂1(u1) + 3∂21(u2).
By applying the Zakharov-Shabat equations with n = 2 and m = 3, we find the system
∂2(u1) = ∂21(u1) + 2∂1(u2)
3∂2(u2) + 3∂2∂1(u1)− 2∂3(u1) = ∂31(u1)− 6u1∂1(u1) + 3∂21(u2)
Now by eliminating u2, renaming u = 2u1, x1 = x, x2 = y, and x3 = t, we obtain
3∂2y(u) = ∂x
(4∂t(u)− ∂3x(u)− 6u∂x(u)
),
which is precisely the KP equation.
We now wish to define a class of solutions of the KP hierarchy which admit so-calledwave functions. To do so, consider the auxiliary system of equations
Lw = kw
Bnw = ∂nw for all n ∈ N
}(2.2)
Here we assume that w ∈M is of the form Weξ(x,k), where W = 1+w1∂−1+w2∂
−2+ . . . .
Proposition 2.5. Suppose L ∈ R[∂, ∂−1) is of the form ∂ + u1∂−1 + u2∂
−2 + . . . , andBn = (Ln)+. If there exists a solution w = Weξ(x,k) ∈ M of system (2.2), then L is asolution of the KP hierarchy. Furthermore, we have L = W∂W−1.
Proof. We first show that the ZS equations hold. By Proposition 2.3 it then follows thatL satisfies the KP hierarchy.We apply ∂m to Bnw, and use the product rule to find
∂m(Bnw) = ∂m(Bn)w +Bn∂m(w).
Since w solves (2.2), we have ∂m(w) = Bm(w), so that
∂m(Bnw) = ∂m(Bn)w +BnBm∂m(w).
22
It follows that
∂n(Bmw)− ∂m(Bnw) = (∂n(Bm)− ∂m(Bn)− [Bn, Bm])w,
the left-hand side of which vanishes, because
∂n(Bmw) = ∂n(∂mw) = ∂m(∂nw) = ∂m(Bnw).
Using the fact that M is a free R[∂, ∂−1)-module, we see that the ZS equations hold.
Now for the second assertion, consider the fact that Lw = kw. The right-hand side isobviously equal to W∂eξ(x,k), because ∂eξ(x,k) = keξ(x,k). Therefore we have
(LW −W∂)eξ(x,k) = Lw − kw = 0.
Making use of the freeness of M , we conclude that LW = W∂, and hence L = W∂W−1.
If L and w = (1 +w1k−1 +w2k
−2 + . . .)eξ(x,k) = Weξ(x,k) are as in Proposition 2.5, thenw is called a wave function of the KP hierarchy. If we restrict our attention to solutionsof the KP hierarchy which admit such wave functions, then we can reinterpret the KPhierarchy as an infinite system of differential equations in the wi. By Proposition 2.5,the KP hierarchy for wave functions consists of the equations
(W∂nW−1)+w = ∂nw for all n ∈ N . (2.3)
2.3 The Bilinear Identity (2.2)
In this section we prove an identity for wave functions of the KP hierarchy which com-pletely characterizes such wave functions. This identity can be used to show the existenceof tau functions of the KP hierarchy, and it gives rise to the construction of solutions
using∧∞
2 V.
Given the wave function w(x, k) = Weξ(x,k), we define the adjoint wave function as1
w†(x, k) = (W †)−1e−ξ(x,k). (2.4)
We can now formulate the bilinear identity.
Theorem 2.6 (Bilinear identity). A function w ∈M of the form Weξ(x,k), where W isa pseudodifferential operator of degree 0, is a wave function of the KP hierarchy if andonly if for all x and x′ we have
∮w(x′, k)w†(x, k)dk = 0, (2.5)
where w† is related to w as in equation (2.4).
1Note that w† 6∈ M . In fact, w† resides in the ”adjoint” M† := {f(x, k)e−ξ(x,k) | f ∈ R[[k−1]][k]} ofM . This space is also a one-dimensional free R[∂, ∂−1)-module, in the obvious way.
23
We prove this theorem in several stages. First note that w(x′, k) can be written as theformal expansion
w(x′, k) =∑
I
(x′1 − x1)i1 · · · (x′m − xm)im
i1! · · · im!∂i11 · · · ∂imm (w(x, k)),
where the sum is taken over all sequences I = (i1, . . . , im) of nonnegative integers, of alllengths m ∈ N. It follows that to show that wave functions of the KP hierarchy satisfythe bilinear identity, it suffices to prove the following proposition.
Proposition 2.7. Suppose that w(x, k) = Weξ(x,k) is a wave function of the KP hier-archy. Then for all m-tuples (i1, . . . , im) of nonnegative integers, we have
Resk = ∞
[(∂i11 · · · ∂imm w(x, k))w†(x, k)
]= 0.
Proof. By Lemma 2.2, and the fact that w is a wave function of the KP hierarchy (inparticular, ∂kn(w) = Bk
nw), we have
Resk = ∞
[(∂i11 · · · ∂imm w(x, k))w†(x, k)
]= Res
k = ∞
[(Bi1
1 · · ·Bimm Weξ(x,k))(W †)−1e−ξ(x,k)
]
= Res∂
(Bi11 · · ·Bim
m W )[(W †)−1
]†
= Res∂
Bi11 · · ·Bim
m .
Since the Bn are differential operators, this last residue must be 0, which was to beshown.
Before we can prove the converse, we need a lemma. We make use of the followingnotation:
(x′ − x)J
J !:=
(x′1 − x1)j1 · · · (x′m − xm)jm
j1! · · · jm!, ∂J := ∂j11 · · · ∂jmm ,
for any nonnegative m-tuple J = (j1, . . . , jm).
Lemma 2.8. Suppose that w satisfies the bilinear identity. Then for any differentialoperator Q in the variables x1, x2, . . . , we have, for all x and x′,
∮Q(w(x, k))w†(x′, k) dk = 0.
Also, if w is a power series of the form
w(x, k) =
(∑
`≥1
w`(x)
k`
)eξ(x,k), (2.6)
and ∮w(x, k)w†(x′, k) dk = 0
for all x and x′, then w = 0.
24
Proof. We want to show that∮∂I(w(x′, k))w†(x, k) dk = 0
for all x and x′, and nonnegative m-tuples I = (i1, . . . , im). Formally expanding∂I(w(x′, k)), we find that the left-hand side equals
∑
J
(x′ − x)J
J !
∮∂I+J(w(x, k))w†(x, k) dk.
Therefore it suffices to show that∮∂J(w(x, k))w†(x, k) dk = 0
for all x, and all nonnegative m-tuples J . But by the fact that w satisfies the bilinearidentity, we already know that
0 =
∮w(x′, k)w†(x, k) dk =
∑
J
(x′ − x)J
J !
∮∂J(w(x, k))w†(x, k) dk,
and this immediately implies∮∂J(w(x, k))w†(x, k) dk = 0, which proves the first part
of the lemma.
For the second part, let w†(x, k) = (1 + w†1(x)k−1 + w†2(x)k−2 + . . .)e−ξ(x,k). Then thecoefficient of k−1 in w(x, k)w†(x, k) is w1, so by
∮dk w(x′, k)w†(x, k) = 0 with x′ = x,
we have w1 = 0.Assume that we have shown that wi = 0 for all i < n. Because
∮dk w(x′, k)w†(x, k) = 0
for all x′ and x, an argument similar to the one used to prove the first part of the lemmashows that ∮
∂n−1(w(x, k))w†(x, k) = 0 (2.7)
for all x. But
∂n−1(w)w† =1
kn(∂n−1wn +
∂n−1wn+1
k+ . . .)eξ(x,k)w† + kn−1(
wnkn
+wn+1
kn+1+ . . .)eξ(x,k)w†,
which obviously has wn as the coefficient of k−1. From (2.7) we conclude that wn = 0.The result now follows by induction.
The following proposition finishes the proof of the main theorem of this section, Theo-rem 2.6.
Proposition 2.9. If w(x, k) = Weξ(x,k) satisfies the bilinear identity, i.e.,∮w(x, k)w†(x′, k) dk = 0
for all x and x′, where w†(x, k) := (W †)−1e−ξ(x,k), then w is a wave function of the KPhierarchy.
25
Proof. Define the differential operator Q = ∂n − (Ln)+. Then Qw = ∂n(w) − Ln(w) +(Ln)−(w). By our choice of w and L, we already have Lnw = knw. Therefore
∂n(w)− Ln(w) = ∂n(W )eξ(x,k) +W∂n(eξ(x,k))− knw= ∂n(W )eξ(x,k) + knWeξ(x,k) − knw= ∂n(W )eξ(x,k).
But ∂n(W ) is a pseudodiffererential operator of negative degree, so that (∂n−Ln)(w) is ofthe form (2.6). In addition to this, (Ln)−(w) = ((Ln)−W )eξ(x,k) is also of the form (2.6),because (Ln)−W has negative degree. It follows that Qw is also of the form (2.6).Now since w satisfies the bilinear identity, and because Q is a differential operator, bythe first part of Lemma 2.8 we have
∮(Qw(x, k))w†(x′, k) dk = 0
for all x and x′. But since Qw is of the form (2.6), the second part of Lemma 2.8 showsthat Qw = 0. It follows that ∂n(w) = Bnw, so that w is a wave function of the KPhierarchy.
One of the most remarkable facts about the KP hierarchy is that one can without lossreplace the infinitely many dependent variables w1(x), w2(x), . . . by one function τ(x).We give a partial proof of this fact.
Proposition 2.10. Suppose w(x, k) ∈M is a wave function of the KP hierarchy. Thenthere exists a function τ ∈ R such that
w(x, k) =τ(x1 − 1
k, x2 − 1
2k2, x3 − 1
3k3, . . .
)
τ(x1, x2, x3, . . .)eξ(x,k), (2.8)
and
w†(x, k) =τ(x1 + 1
k, x2 + 1
2k2, x3 + 1
3k3, . . .
)
τ(x1, x2, x3, . . .)e−ξ(x,k). (2.9)
Such a function τ is called a tau function of the KP hierarchy.
Partial proof. Define N(k) =∑
j≥1 k−j−1∂j − ∂
∂k, and consider the 1-form
ω = −∑
n≥1
Resk = ∞
[knN(k) log(w(x, k))] dxn,
where w(x, k) ∈ R[[k−1]] is defined by w(x, k) = w(x, k)eξ(x,k). In [DKMJ] it is shownthat the bilinear identity implies that ω is closed, so there exists an f ∈ R such thatdf = ω. Now τ is given by τ = ef , which defines τ up to multiplication by a constant.
The fact that d log(τ) = ω just means that ∂n(log(τ)) is the coefficient of k−n−1 in−N(k) log(w(x, k)). Since log(τ) does not depend on k, this proves that
N(k) log(τ) = −N(k) log(w(x, k)).
Now we make use of two easily verified facts about N(k):
26
(i) if N(k)P = 0, with P =∑
j≥1 pj(x)k−j, then P = 0,
(ii) N(k)e−ξ(∂,k−1) = 0 (recall that ∂ denotes the sequence (∂1,
12∂2,
13∂3, . . .)).
Applying N(k) to P := log(w(x, k))− log
(τ(x1−
1k,x2−
12k2
,...)
τ(x1,x2,...)
), and using (ii) and the fact
that log(τ(x1 − 1k, x2 − 1
2k2, . . .)) = eξ(∂,k
−1) log(τ(x1, x2, . . .)), we find
N(k)P = N(k) log(w(x, k)) +N(k) log(τ(x)).
But from the definition of τ we have N(k) log(τ(x)) = −N(k) log(w(x, k)), so it followsthat N(k)P = 0.If we show that P is of degree −1 in k, then by (i) it follows that P = 0, whichproves the proposition. Obviously log(w(x, k)) is of degree −1, so we need to show that
log(e−ξ(∂,k
−1)τ(x)τ(x)
)is as well. But this follows directly from rewriting the fraction as
e−ξ(∂,k−1)τ(x)
τ(x)= 1− ξ(∂, k−1)e−ξ(∂,k
−1)τ(x)
τ(x).
For the proof of equation (2.9), the reader is referred to [D].
2.4 Constructing Solutions using∧∞
2 V (1.3, 2.3)
In this section we characterize the set of tau functions of the KP hierarchy as the im-age under the Boson-Fermion correspondence of the orbit of |0〉 under the action of thegroup GL(∞). In order to define GL(∞), we must first define the Lie algebra gl(∞)associated to this group.
To a (12
+ Z)× (12
+ Z)-matrix A = (Aij) we can associate an operator XA on∧∞
2 V by
XA =∑
i,j∈ 1
2+Z
Aij :ψiψ†j :.
However, as it stands, the action of this operator on |u〉 ∈ ∧∞2 V is not necessarily
well-defined, because XA|u〉 may involve infinite sums. To get rid of this problem, werestrict our attention to matrices A which have only finitely many nonzero diagonals.Equivalently, there must exist an N such that Aij = 0 for all i, j with |i− j| > N .We now compute the commutator of two such operators XA and XB. An elementarycomputation using [AB,C] = A[B,C]+− [A,C]+B and the relations imposed on the ψiand ψ†i shows that
[ψiψ†j , ψkψ
†` ] = δjkψiψ
†` − δ`iψkψ†j .
27
Since :ψiψ†j : differs from ψiψ
†j by a complex number (namely 〈0|ψiψ†j |0〉), and because
complex numbers commute with ψiψ†j , we find
[XA, XB] =∑
i,j,k,`
AijBk`[ψiψ†j , ψkψ
†` ]
=∑
i,j,k,`
AijBk`
(δjkψiψ
†` − δ`iψkψ†j
)
=
(∑
i,j,`
AijBj`ψiψ†`
)−(∑
i,j,k
BkiAijψkψ†j
)
=
(∑
i,j,`
AijBj`:ψiψ†` :
)−(∑
i,j,k
BkiAij:ψkψ†j :
)+ C,
where C is some complex number. Because ψiψ†j = :ψiψ
†j :+〈ψiψ†j〉 = :ψiψ
†j :+δijθ(j < 0),
C is given by ∑
i,j
AijBji(θ(i < 0)− θ(j < 0)),
where θ(i < 0) is equal to 1 if i < 0, and otherwise equal to 0. The only nonzero termsin this sum are the ones in which the signs of i and j differ. Because A and B have onlyfinitely many nonzero diagonals, only finitely many of the terms with sgn(i) 6= sgn(j) arenonzero, so C is finite. The two sums to the left of C are easily seen to be the operatorX[A,B]. Therefore [XA, XB] = X[A,B] +C, with C ∈ C. These calculations show that thefollowing definition indeed defines a Lie algebra.
Definition 2.11. We define gl(∞) to be the Lie algebra given by
gl(∞) = {XA | ∃N ∀i, j (|i− j| > N =⇒ aij = 0)} ⊕ C .
To this Lie algebra we associate the group GL(∞), defined by
GL(∞) = {eX1 · · · eXn | X1, . . . , Xn ∈ gl(∞)}.It turns out that the image of GL(∞)|0〉 under the Boson-Fermion correspondence con-sists precisely of the tau functions of the KP hierarchy2. Before we can show this, wecharacterize the vacuum orbit by an identity similar to the bilinear identity for the KPhierarchy.
Proposition 2.12 (Bilinear identity). An element |u〉 ∈ ∧∞2 V is in the vacuum orbit
GL(∞)|0〉 if and only if |u〉 has charge 0 and∑
i∈ 1
2+Z
ψi|u〉 ⊗ ψ†i |u〉 = 0.
2The action of GL(∞) on∧∞
2 V is not well-defined, as for example eH−1 |0〉 must be an infinite sum of
basis elements of∧∞
2 V. It follows that we must work with the completion∧∞
2 V instead of∧∞
2 V.Though all proofs given carry over quite easily to the completion, we will ignore this subtle issue
28
Proof. Using the fact that :ψiψ†j : differs from ψiψ
†j by a complex number, and the relation
[AB,C] = A[B,C]+ − [A,C]+B, one can show that
[XA, ψi] =∑
j
Ajiψj and [XA, ψ†i ] = −
∑
j
Aijψ†j . (2.10)
We will use these relations to show that for any |u〉 ∈ ∧∞2 V and g ∈ GL(∞) there holds
∑
i
gψi|u〉 ⊗ gψ†i |u〉 =∑
i
ψig|u〉 ⊗ ψ†i g|u〉 . (2.11)
From this it follows immediately that the bilinear identity is invariant under the actionof GL(∞), so that the bilinear identity holds for all |u〉 ∈ GL(∞)|0〉 (of course thebilinear identity holds for |0〉, because for any i, either ψi|0〉 = 0 or ψ†i |0〉 = 0).If we show that (2.11) holds for g of the form eXA , then it follows for general ele-ments eXA1 · · · eXAn by successively applying (2.11) with g = eXAi . By Baker-Campbell-Hausdorff we have∑
i
gψi|u〉 ⊗ gψ†i |u〉 =∑
i
gψig−1g|u〉 ⊗ gψ†i g−1g|u〉
=∑
i
ead(XA)(ψi)g|u〉 ⊗ ead(XA)(ψ†i )g|u〉
=∑
i
∞∑
k=0
1
k!
k∑
j=0
(k
j
)ad(XA)j(ψi)g|u〉 ⊗ ad(XA)k−j(ψ†i )g|u〉 .
So if we show that
Sk :=∑
i
k∑
j=0
(k
j
)ad(XA)j(ψi)g|u〉 ⊗ ad(XA)k−j(ψ†i )g|u〉 = 0
for all k ≥ 1, then we’re done.By using (2.10), we find
Sk =k∑
j=0
(k
j
)(−1)k−j
∑
i,n1,...,nk
Anjnj−1· · ·An1iψnjg|u〉 ⊗ Ainj+1
· · ·Ank−1nkψ†nkg|u〉 .
Renaming n1, . . . , nj as nj−1, . . . , n0, respectively (so ni becomes nj−i), and renaming ias nj, we find
Sk =k∑
j=0
(k
j
)(−1)k−j
∑
n0,n1,...,nk
An0n1 · · ·Anj−1njψn0g|u〉 ⊗ Anjnj+1· · ·Ank−1nkψ
†nkg|u〉
=
(k∑
j=0
(k
j
)(−1)k−j
)( ∑
n0,n1,...,nk
An0n1 · · ·Ank−1nkψn0g|u〉 ⊗ ψ†nkg|u〉)
,
29
which is equal to 0 because of the binomial theorem and the fact that k ≥ 1. Thus wehave proved (2.11), and hence one direction of the proposition.
For the converse, let |u〉 ∈ ∧∞20 V satisfy
∑i ψi|u〉 ⊗ ψ†i |u〉 = 0. Since |u〉 is of charge
0, it is a linear combination of basis elements of the form ψm1 · · ·ψmrψ†n1· · ·ψ†nr |0〉 with
n1 < . . . < nr < 0 < mr < . . . < m1. Pick one of these terms with r minimal. We wishto find an element of GL(∞) which transforms this term to a scalar multiple of |0〉.Consider the operators φij = ψi + ψj and φ†ij = ψ†i + ψ†j , with i 6= j. Then using
[φij, φ†ij]+ = 2, it immediately follows that (φ†ijφij)
n = 2n−1φ†ijφij, which implies that
1− φ†ijφij = eiπφ†ijφij/2.
Now it follows that 1 − φ†ijφij ∈ GL(∞), because iπφ†ijφij/2 ∈ gl(∞). A direct compu-tation now shows that
(1− φ†m1n1φm1n1)ψm1 · · ·ψmrψ†n1
· · ·ψ†nr |0〉 = (−1)r+1ψm2 · · ·ψmrψ†n2· · ·ψ†nr |0〉 .
So by repeatedly applying group elements of this form, we can reduce the ψi and ψ†ioccurring in the minimal term in |u〉 until we end up with an element of the form
|v〉 = |0〉+∑
i,j
cijψiψ†j |0〉+ . . . ∈ GL(∞)|u〉 .
Here the ellipsis denotes a linear combination of terms ψm1 · · ·ψmrψ†n1· · ·ψ†nr |0〉 with
r ≥ 2. We also made implicit use of the fact that GL(∞) contains the scalar operatorsbecause C ⊆ gl(∞), so we could fix the sign of |0〉 in |v〉.In the first part of the proof we showed that the relation
∑i ψi|u〉⊗ψ†i |u〉 = 0 is GL(∞)-
invariant, so we have∑
i ψi|v〉 ⊗ ψ†i |v〉 = 0. Clearly there holds ψ†i |v〉 6= 0 for i < 0, sofrom the bilinear identity it follows that ψi|v〉 = 0 for negative i. Similarly we concludethat ψ†i |v〉 = 0 for positive i. But then we have 〈vS|v〉 = 0 for nonempty S (recall our
notation for basis elements of∧∞
2 V), so it follows that |v〉 = |0〉, which shows that |u〉is in the vacuum orbit.
Let g ∈ GL(∞). The image of |u〉 := g|0〉 under the Boson-Fermion correspondence Φis
τ(x) = 〈0|eH(~x)|u〉 .To τ we associate functions w(x, k) ∈M and w†(x, k) ∈M †, defined by
w(x, k) :=τ(x1 − 1
k, x2 − 1
2k2, . . .
)
τ(x1, x2, . . .)eξ(x,k),
and
w†(x, k) :=τ(x1 + 1
k, x2 + 1
2k2, . . .
)
τ(x1, x2, . . .)e−ξ(x,k) =
30
Using the fact that Φ is A-equivariant, it can be shown that these functions are givenby
w(x, k) =〈1|eH(x)ψ(k)|u〉〈0|eH(x)|u〉 , (2.12)
and
w†(x, k) =〈−1|eH(x)ψ†(k)|u〉〈0|eH(x)|u〉 . (2.13)
Theorem 2.13. An element |u〉 of∧∞
2 V is in the vacuum orbit GL(∞)|0〉 if and only ifits image under the Boson-Fermion correspondence is a tau function of the KP hierarchy
Proof. Let |u〉 be an element of the vacuum orbit, and let τ be its image under theBoson-Fermion correspondence, and let w and w† be the wave functions associated toτ . Then a quick computation using equations (2.12) and (2.13) shows that
∮dk
2πiw(x, k)w†(x′, k) =
∑
i
〈1|eH(x)ψi|u〉〈−1|eH(x′)ψ†i |u〉〈0|eH(x)|u〉〈0|eH(x′)|u〉 . (2.14)
By Proposition 2.12, the right-hand side is 0, so by the bilinear identity in Theorem 2.6,τ is a tau function of the KP hierarchy.
Now suppose that τ is a tau function of the KP hierarchy, and let |u〉 be its inverseimage under the Boson-Fermion correspondence. From equation (2.14) we obtain that
∑
i
〈1|eH(x)ψi|u〉〈−1|eH(x′)ψ†i |u〉 = 0
for all x and x′. Looking at the coefficient of xµx′ν on the left-hand side, where µ and ν
are partitions, we find (using the fact that [Hm, Hn] = 0 for positive m and n)∑
i
〈0|ψ†1/2Hµψi|u〉 〈0|ψ−1/2Hνψ†i |u〉 = 0.
Since H−µ is the adjoint of Hµ, the expression 〈0|ψ†1/2Hµψi|u〉 is the inner product of
H−µψ1/2|0〉 with ψi|u〉. Similarly, 〈0|ψ−1/2Hνψ†i |u〉 is the inner product ofH−νψ
†−1/2|0〉 with
ψ†i |u〉. We know that elements of the form H−µ|0〉 are a basis of∧∞2
0 V (see the discussionpreceding remark 1.11), so it’s easy to see that {H−µψ1/2|0〉 | µ a partition} generates the
space of charge 1 elements, and that {H−νψ†−1/2|0〉 | ν a partition} generates the charge
−1 subspace of∧∞
2 V. Therefore, if |v1〉 and |v2〉 have charge 1 and −1, respectively,then we have ∑
i
〈v1|ψi|u〉 〈v2|ψ†i |u〉 = 0.
Since the left-hand side is 0 for |v1〉 and |v2〉 of different charge (because ψi|u〉 has charge1, and ψ†i |u〉 has charge −1), it follows that |u〉 satisfies
∑
i
ψi|u〉 ⊗ ψ†i |u〉 = 0.
31
Because τ has no z-dependence, |u〉 has charge 0, so by Proposition 2.12, |u〉 ∈ GL(∞)|0〉.
This theorem allows us find tau functions of the KP hierarchy which appear literally outof thin air. We simply take an element of GL(∞), apply it to the vacuum |0〉, and theimage under the Boson-Fermion correspondence is a tau function of the KP hierarchy.We illustrate this procedure in the following examples.
Example 2.14. For any integer n > 0 and c ∈ C the operator cnH−n is in gl(∞), so
ecH−n/n ∈ GL(∞). Since Φ(H−n|0〉) = nxn, we know that
Φ(ecH−n/n|0〉
)= ecxn
is a tau function of the KP hierarchy. It easily follows that for any f which is linear inthe xi and which depends on finitely many xi, the function ef is a tau function.
Example 2.15. Consider the operator
M0 =1
2
∑
i,j≥1
(ijxixj∂i+j + (i+ j)xi+j∂i∂j) ,
called the cut-and-join operator. In section 4.1 we will show that M0 ∈ gl(∞), sofor every complex number β, we have exp(βM0) ∈ GL(∞). Now suppose that P =∑
m≥0 Pmβm/m! is power series with Pm ∈ R for all m ≥ 0, which satisfies the so-called
cut-and-join equation∂P
∂β= M0P .
This equation immediately implies that Pm+1 = M0Pm, so that we have
P =∑
m≥0
Pmβm
m!=∑
m≥0
βm
m!Mm
0 P0 = exp(βM0)P0.
Since exp(βM0) is a symmetry of the KP hierarchy, it follows that if P0 is a tau functionof the KP hierarchy, then P is as well3.For instance, by the previous example the series P (β;x1, x2, . . .) = exp(βM0)e
xr is a taufunction, for all r ≥ 1.
3What we really get is a 1-parameter family of tau functions (β being the parameter), but we willrefer to such families simply as tau functions
32
3 Hurwitz Numbers
The Hurwitz numbers are combinatorial constants which were already studied by Hur-witz in the nineteenth century. Advances made in mathematical physics in the ’90sresparked the interest in these numbers. In this chapter we give the definition of thesenumbers, and establish a combinatorial result which will allow us to prove the cut-and-join equation in chapter 4.
3.1 Hurwitz Numbers
Definition 3.1. Let µ, ν ` n be partitions, and let m be some nonnegative integer. Wedefine the double Hurwitz number Hm;µ,ν as
Hm;µ,ν =1
n!|{(σ, η1, . . . , ηm) | σ ∈ Kν and ηi ∈ K(2,1,...,1) and ση1 · · · ηm ∈ Kµ}|,
where Kλ denotes the conjugacy class in S|λ| of elements of cycle type λ. By abuse ofnotation we will often use K2 instead of K(2,1,...,1).Given r ≥ 1, there is a subclass of the double Hurwitz numbers called the orbifoldHurwitz numbers, denoted Hr
m;µ, defined by
Hrm;µ =
{Hm;µ,(r,...,r) if r divides |µ|0 otherwise.
One further subclass of interest are the single Hurwitz numbers, denoted Hm;µ. Theseare obtained by putting r = 1 in the definition of the orbifold Hurwitz numbers, so theyare defined by Hm;µ = H1
m;µ.
Remark 3.2. Note that we allow µ ` 0 in this definition, i.e., µ is the empty partition.In this case the double Hurwitz numbers are defined by
Hm;µ,ν =
{1 if m = 0
0 otherwise.
Consider the group algebra C[Sn] of Sn. We use Cµ to denote the sum∑
σ∈Kµ σ. Obvi-
ously the Cµ are in the center ZSn of C[Sn]. Furthermore, they form a basis of ZSn, sofor any Z ∈ ZSn, we can define complex numbers [Cµ](Z) by
Z =∑
µ`n
[Cµ](Z)Cµ.
33
Now the double Hurwitz numbers can be conveniently expressed in terms of products inZSn, namely
Hm;µ,ν =1
|µ|! |Kµ|[Cµ](CνCm2 ). (3.1)
To better understand the Hurwitz numbers, we analyze multiplication by a transposition,following [GJ]. To do so, recall the cut-and-join operator from example 2.15,
M0 =1
2
∑
i,j≥1
(ijxixj∂i+j + (i+ j)xi+j∂i∂j) .
For our purposes it’s easier to work with formal variables pj, defined by pj = jxj. Interms of these variables, M0 becomes
1
2
∑
i,j≥1
((i+ j)pipj∂i+j + ijpi+j∂i∂j)
where we use ∂n to denote ∂/∂pn. To a permutation σ of cycle type µ, we associate aproduct of the pi as follows:
p(σ) := pµ = pµ1 · · · pµn .
We extend this linearly to define p(X) for all X ∈ C[Sn]. Using M0, we can nowsuccinctly express what happens when you multiply an element of C[Sn] by C2.
Proposition 3.3. If X ∈ C[Sn], then we have
M0p(X) = p(XC2).
Proof. Obviously it suffices to prove that for every σ ∈ Sn we have M0p(σ) = p(σC2).The proposition then follows by linearity.
If τ is a transposition, then it can be directly verified that one of two possible situationsoccur. Either multiplication from the right by τ removes one and adds two cycles to σ,or multiplication from the right results in the removal of two and the addition of onecycle to σ. In the former case, τ is said to be a cut for σ (because it cuts one cycle of σup into two cycles), and in the latter case τ is said to be a join for σ.We denote the set of transpositions which join two cycles of length i and j by Ji,j, andthe set of transpositions which cut a cycle up into two cycles of length i and j by Ci,j.We can also define these sets in terms of p(σ) as follows:
Ji,j =
{τ ∈ K2 | p(στ) =
p(σ)pi+jpipj
},
Ci,j =
{τ ∈ K2 | p(στ) =
p(σ)pipjpi+j
}.
34
So because K2 =⊔i≥1j≥i
(Ji,j t Ci,j), we have
p(σC2) =∑
i≥1j≥i
(|Ji,j|
p(σ)pi+jpipj
+ |Ci,j|p(σ)pipjpi+j
).
Basic combinatorics shows that
|Ji,j| ={ijmi(σ)mj(σ) if i 6= j,12i2mi(σ)(mi(σ)− 1) if i = j,
where mi(σ) denotes the number of cycles of length i in σ. It’s also easy to see that
|Ci,j| ={
(i+ j)mi+j(σ) if i 6= j,12(i+ j)mi+j(σ) if i = j.
Therefore we have
p(σC2) =1
2
∑
i,j≥1
(ijmi(σ)(mj(σ)− δi,j)
p(σ)pi+jpipj
+ (i+ j)mi+j(σ)p(σ)pipjpi+j
),
where δi,j denotes the Kronecker delta. The fact that M0p(σ) = p(σC2) now easilyfollows from the definition of M0.
For completeness’ sake, we mention a different but equivalent definition of the Hurwitznumbers. In what follows, we assume basic familiarity with the theory of Riemann sur-faces. Standard introductions to this subject are [F] and [M].
Let P1 denote the Riemann sphere, and let q1, . . . , qm ∈ P1 \{0,∞} be fixed distinctpoints. The double Hurwitz number Hm;µ,ν is defined as the weighted number of iso-morphism classes of branched coverings f : X → P1 satisfying
• f has simple branching over the points q1, . . . , qm;
• f has branching profiles µ and ν over 0 and ∞, respectively;
• f is unbranched over P1 \{0,∞, q1, . . . , qm}.Each covering is weighted by the size of its automorphism group. The equivalence ofthis definition to 3.1 was already known to Hurwitz, and a proof can be found in [O].
Given a possibly disconnected Riemann surface X, we define its genus g by
g = 1− dim(H0(X,OX)) + dim(H1(X,OX)),
where OX is the sheaf of holomorphic functions on X. When X is connected (i.e., whendim(H0(X,OX)) = 1), this definition coincides with the usual definition of genus.
35
If f : X → P1 is a branched covering as described above, then m, the number of pointsaside from 0 and ∞ with simple branching, is related to µ, ν, and the genus of X bythe Riemann-Hurwitz formula:
m = 2g − 2 + `(µ) + `(ν), (3.2)
where `(µ) and `(ν) denote the respective lengths of µ and ν. This formula permits usto write the Hurwitz numbers as Hg,µ,ν instead of Hm;µ,ν .
This topological setting also suggests a natural other type of double Hurwitz numbers,namely the connected double Hurwitz numbers. We will denote these as hm;µ,ν , andthey differ from the regular Hurwitz numbers by the additional requirement that weonly count coverings X → P1 with X connected.
We can give an interpretation of connectedness in terms of factorizations of permutations,as follows. Consider a factorization ρ = ση1 · · · ηm ∈ Sn, where σ and ρ have respectivecycle types ν and µ, and the ηi are transpositions. To this factorization we associatethe graph Γ with vertices 1, . . . , n, and its edges given by the action of the ηi and σ on{1, . . . , n}. So if ηi = (ab) or if σ maps a to b, then {a, b} is an edge in Γ. The connecteddouble Hurwitz numbers are now the same as the double Hurwitz numbers, except werequire that Γ is connected.It’s obvious that the connectedness of Γ is equivalent to the fact that the subgroup〈σ, η1, . . . , ηm〉 ⊆ Sn is transitive. This shows that the following definition makes sense.
Definition 3.4. Let µ, ν ` n be partitions, and let m be some nonnegative integer. Wedefine the connected double Hurwitz number hm;µ,ν as
hm;µ,ν =1
n!
∣∣∣∣{
(σ, η1, . . . , ηm)
∣∣∣∣σ ∈ Kν , ηi ∈ K(2,1,...,1), ση1 · · · ηm ∈ Kµ,〈σ, η1, . . . , ηm〉 ⊆ Sn is transitive
}∣∣∣∣ .
The connected orbifold and simple Hurwitz numbers, hrm;µ and hm;µ, are defined similarly.
36
4 Generating Functions of HurwitzNumbers
In this chapter we unify the preceding chapters by showing that a generating series ofthe orbifold Hurwitz numbers is a tau function of the KP hierarchy. The key to thisresult is the cut-and-join equation of example 2.15. The combinatorial part of this resultis contained entirely in Theorem 3.3, so the proof is reduced to some basic manipulationsof formal power series. The first section is devoted to this proof. In the final section,we apply the result to abelian Hurwitz-Hodge integrals, by unsuccessfully trying togeneralize computations made by Kazarian in [K].
4.1 Hurwitz Numbers and KP Hierarchy (2.4, 3.1)
We collect the orbifold Hurwitz numbers Hrm;µ in generating series, one for each value
of r:
Hr(β; p1, p2, . . .) =∑
m,n≥0µ`n
Hrm;µpµ
βm
m!.
Using Proposition 3.3 we can quite easily prove that Hr satisfies the cut-and-join equa-tion from example 2.15.
Theorem 4.1 (cut-and-join equation). The generating series Hr satisfies the cut-and-join equation, i.e.,
M0Hr =∂Hr
∂β.
Proof. To prove this, we make use of the expression of the Hurwitz numbers in termsof the group algebra of Sn in equation (3.1), and Proposition 3.3. On the left-hand sidewe have
∂βHr =∑
m,n≥0
∑
µ`n
Hrm+1(µ)pµ
βm
m!
=∑
m,n≥0
1
n!
∑
µ`n
|Kµ|[Cµ](C(r,...,r)Cm+12 )pµ
βm
m!.
Using the fact that C(r,...,r)Cm2 =
∑ν`n[Cν ](C(r,...,r)C
m2 )Cν (this follows from the fact that
the Cλ form a basis of ZSn), and p(Cµ) = |Kµ|pµ, we can rewrite this as
∑
m,n≥0
1
n!
∑
µ,ν`n
[Cµ](CνC2)[Cν ](C(r,...,r)Cm2 )p(Cµ)
βm
m!.
37
By Proposition 3.3 and equation (3.1), the right-hand side is equal to
M0Hr = M0
∑
m,n≥0
1
n!
∑
µ`n
[Cµ](C(r,...,r)Cm2 )p(Cµ)
βm
m!
=∑
m,n≥0
1
n!
∑
µ`n
[Cµ](C(r,...,r)Cm2 )p(CµC2)
βm
m!.
Now the fact that CµC2 =∑
ν`n[Cν ](CµC2)Cν implies
M0Hr =∑
m,n≥0
1
n!
∑
µ,ν`n
[Cµ](C(r,...,r)Cm2 )[Cν ](CµC2)p(Cν)
βm
m!.
Interchanging µ and ν shows that Hr satisfies the cut-and-join equation, which was tobe shown.
If we write Hr =∑
m≥0Hr,mβm
m!, then example 2.15 and theorem 4.1 give us an explicit
expression for Hr in terms of Hr,0:
Hr = exp(βM0)Hr,0.
From equation (3.1) and the fact that |K(rk)| = (kr)!rkk!
, we immediately obtain that
Hr0;µ =
1
rkk!if µ = (rk)
0 otherwise.
So we have
Hr,0 =∑
k≥0
1
rkk!pkr = epr/r,
which by example 2.15 implies that if M0 ∈ gl(∞), then Hr is a tau function of the KPhierarchy.
The rest of this section is devoted to proving that M0 is indeed an infinitesimal symmetry
of the KP hierarchy. We first show that it acts diagonally on the vλ in∧∞2
0 V.Recall that Φ denotes the Boson-Fermion correspondence. Furthermore, we use Rλ todenote the Specht module associated to λ. Given an element X ∈ C[S|λ|], we denote thetrace of X viewed as an operator on Rλ as trλ(X).
Proposition 4.2. The action of M0 on vλ is given by
Φ−1M0Φvλ = f2(λ)vλ
where f2(λ) is defined as
f2(λ) =1
dimRλ
trλ(C2).
38
Proof. By proposition 1.12, it suffices to show that
M0sλ = f2(λ),
where sλ is defined assλ =
∑
µ`n
χλµp(Cµ).
This sλ is just the image of n!vλ, which can be seen by using pi = ixi and the fact that
|Kµ| =n!
Aut(µ)∏
i µi.
Computing M0sλ can be done using proposition 3.3, which gives
M0sλ =∑
µ
χλµp(CµC2)
=∑
µ,ν
χλµ[Cν ](CµC2)p(Cν)
=∑
µ,ν
|Kν |[Cν ](CµC2)χλµpν
=∑
µ,ν
n!H1;ν,µχλµpν ,
where the last equality follows from equation (3.1).
To compute the other side, f2(λ)sλ, we need to consider the action of C2 on Rλ. SinceC2 is an element of ZSn, we know that it acts on Rλ as some scalar c ∈ C. In fact, wemust have
c =1
dimRλ
trλ(C2),
which is precisely f2(λ). Therefore we have
f2(λ)sλ = f2(λ)∑
µ
trλ(Cµ)pµ
=∑
µ
trλ(f2(λ)Cµ)pµ
=∑
µ
trλ(C2Cµ)pµ
=∑
µ,ν
[Cν ](C2Cµ) trλ(Cν)pµ
=∑
µ,ν
|Kν |[Cν ](C2Cµ)χλνpµ
=∑
µ,ν
n!H1;ν,µχλνpµ.
39
Now by interchanging µ and ν, and using the obvious fact that H1;µ,ν = H1;ν,µ, weconclude that M0sλ is indeed equal to f2(λ)sλ.
To prove that M0 ∈ gl(∞), we use the following theorem of Frobenius (see for instanceDon Zagier’s appendix in [LZ] for a proof):
Theorem (Frobenius). If λ is a partition, S(λ) = {(λi− i+ 12) | i ≥ 1}, and S−(λ) and
S+(λ) denote the sets (12
+ Z)<0 \ S(λ) and (12
+ Z)>0 ∩ S(λ), then we have
f2(λ) =1
2
∑
k∈S+(λ)
k2 −∑
k∈S−(λ)
k2
.
We have two different ways of indexing basis elements of∧∞2
0 V. One uses partitionsλ, and the other uses subsets S of 1
2+ Z. Now if vλ = vS, then it’s easy to see that
S = S(λ), so by Frobenius’ theorem we have
Φ−1M0ΦvS =1
2
(∑
k∈S+
k2 −∑
k∈S−k2
)vS.
Now since :ψiψ†i :vS is equal to vS if i ∈ S+, and −vS if i ∈ S−, and 0 otherwise, it follows
that
Φ−1M0Φ =1
2
∑
i∈ 1
2+Z
i2:ψiψ†i :,
which is obviously in gl(∞). This completes the proof of the following theorem:
Theorem 4.3. The generating series Hr of orbifold Hurwitz numbers is a tau functionof the KP hierarchy.
If we define the generating series of orbifold connected Hurwitz numbers by
hr(β; p1, p2, . . .) =∑
m,n≥0µ`n
hrm;µpµβm
m!,
then a basic combinatorial relationship between connected and disconnected objectstells us that Hr = exp(hr). This allows us to restate Theorem 4.3 in terms of connectedHurwitz numbers.
Theorem 4.4. The generating series hr of connected orbifold Hurwitz numbers is thelogarithm of a tau function of the KP hierarchy.
40
4.2 A Change of Variables (4.1)
In [K], Kazarian presented a proof of a Witten-Kontsevich type theorem making use ofthe ELSV formula. First he inverts the ELSV formula to express a generating series h1of the simple connected Hurwitz numbers in terms of intersection numbers
〈λjτk1 · · · τkn〉 :=
∫
Mg,n
λjψk11 · · ·ψknn .
He then transforms h1 to a generating series for these intersection numbers, using achange of variables which preserves tau functions of the KP hierarchy. In this way, heobtains a generating series for the intersection numbers which is the logarithm of a taufunction of the KP hierarchy.In a similar setting, Shadrin showed in [S2] that a specific generating series for certainconjectural intersection numbers is the logarithm of a tau function of the KP hierarchy.He does this by inverting the GJV formula, which expresses connected one-part doubleHurwitz numbers1 in terms of these intersection numbers, and then making a change ofvariables which is an automorphism of the KP hierarchy.
Johnson, Pandharipande, and Tseng proved the following ELSV-type formula for con-nected orbifold Hurwitz numbers:
Theorem ( [JPT] ). If µ is a partition, and r is some positive integer, then
hrm;µ
m!=
1
Aut(µ)r1−g+
|µ|r
`(µ)∏
i=1
µibµic
bµic!
∫
Mg,[−µ](B Zr)
∑k≥0(−r)kλUk∏i(1− µiψi)
where [−µ] = [−µ1 mod r, . . . ,−µ`(µ) mod r], the Riemann-Hurwitz formula relates mto g, we use µi to denote µi
r, and Mg,[−µ](BZr) is a moduli space of stable maps into the
classifying space BZr of the cyclic group Zr with r elements. The ψi are pullbacks ofthe usual ψ-classes on Mg,n under the forgetful map, and λUk is the k-th Chern class ofthe generalized Hodge bundle associated to the representation U : 1 mod r 7→ exp
(2πir
)
of Zr on C∗.
For a more complete explanation of this terminology, the reader is referred to [JPT]or [DLN].
Armed with this formula and Theorem 4.4, one might hope to generalize Kazarian’scomputations to the case r ≥ 1 (simple Hurwitz numbers are just orbifold Hurwitznumbers with r = 1). There is one obvious way of generalizing his proof. However, forr > 1 this path leads to a result which seems uninteresting.
1connected one-part double Hurwitz numbers are just connected double Hurwitz numbers with theadditional requirement that the ramification profile over 0 is (d), where d ∈ N
41
Rather than summing over partitions µ in hr, it’s convenient to sum over arbitrary tuplesb = (b1, . . . , bn). By rearranging the terms of b, we obtain a partition µ(b), and we definehrm;b1,...,bn
= hrm;µ(b). This way, hr becomes
hr =∑
n≥1
1
n!
∑
m,b1,...,bn
Aut(µ(b))hrm;b1,...,bnpµ(b)
βm
m!,
where the numerical factors are added to account for the number of permutations of µcontributing to this sum, namely n!
Aut(µ).
We consider the generating function of connected orbifold Hurwitz numbers hrm;µ with`(µ) = n fixed. Using the JPT formula, and the fact that
m = 2g − 2 + n+|µ|r
= 23(3g − 3 + n) +
n∑
i=1
(µi + 1
3
)
= 23
dimMg,[−µ](BZr) +n∑
i=1
(µi + 1
3
),
and similarly
1− g + |µ|r
= −13
dimMg,[−µ](BZr) +n∑
i=1
(µi + 1
3
),
we get
Pn :=∑
m,b1,...,bn
Aut(µ(b))hrm;b1,...,bnpµ(b)
βm
m!
=∑
g,b1,...,bn
n∏
i=1
bi
bbicpbiα
bi+13
bbic!
∫
Mg,[−µ(b)](B Zr)
∑k≥0(−1)kα
2k3 λUk
∏i(1− biα
23ψi)
,
where α = rβ. Since the integrals only depend on µ mod r, we can write bi = qir + ci,and sum over n-tuples c ∈ Znr = {0, . . . , r− 1}n of rest classes mod r, and the quotientsqi. The integral doesn’t depend on the qi, so we have
Pn =∑
c∈Znrg
∫
Mg,−c(B Zr)
n∏
i=1
∑
q≥1
(q + ci)
qprq+ciαq+ci+
13
q!
∑k≥0(−1)kα
2k3 λUk
∏i(1− (q + ci)α
23ψi)
=∑
c∈Znr
⟨(∑
k≥0
(−1)kα2k3 λUk
)n∏
i=1
∑
d≥0
Td,ciψd
i
⟩c
,
where Td,ci is defined by
Td,ci =∑
q≥1
(q + ci)q+d
b!pqr+ciα
q+ci+13+23d,
42
and for each monomial ω in ψi and λUk we denote by 〈ω〉c its integral overMg,−c(BZr),for g ≥ 0 such that the dimension is equal to the degree of the monomial.
We make a short digression on changes of variables. Let x(z) be a series with acompositional inverse z(x). Now from this series we obtain a change of variablespb 7→ pb(p
′1, p′2, . . .) given by
pb =∑
k≥b
cbkp′k, (4.1)
where cbk is defined by xb =∑
k≥b cbkz
k. More algorithmically, we use the correspondence
pb ↔ xb, write xb in terms of z, and then use the correspondence zk ↔ p′k.
Theorem ( [K] ). There is a quadratic function Q(p1, p2, . . .) such that if T is the trans-formation in (4.1), then P 7→ T (P +Q) sends tau functions to tau functions.
In what follows, we ignore the α-dependence, as it makes the computations easier tooversee.At this point of the proof, Kazarian applies the transformation given by
z(x) =∑
b≥1
bb
b!xb,
which has compositional inverse x(z) = z1+z
e− z1+z . The key in this choice of series is the
fact that z(x) is precisely T0,0 under the correspondence xb ↔ pb, and that Td,0 can beobtained from T0,0 by applying (x∂x)
d.Putting r = 1, it’s easy to see that
z(x) =∑
b≥0
(rb+ 1)b
b!xrb+1, (4.2)
so for general r this is a natural choice for z, and it reduces to Kazarian’s choice whenr = 1.Let us try to invert z(x). In [DLN], it is shown that the series
w(x) =∑
b≥0
(rb+ 1)b−1
b!xrb+1
satisfies x = w(x)e−w(x)r. From this relation and the fact that z(x) = x∂x(w(x)), we
obtain z(x) = w(x)1−rw(x)r . So a series x(z) is the compositional inverse of z(x) if and only
if
z =w(x(z))
1− rw(x(z))r. (4.3)
Now we can find p by solving this equation for w(x(z)) and then applying the fact thatx(z) = w(x(z))e−w(x(z))
r. It is not clear how to solve (4.3) for general r, so we discuss
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some specific solutions, for r = 1 and 2.
For r = 1 we get w(x(z)) = z1+z
, which leads to x(z) = z1+z
e− z1+z , as in Kazarian. Now
we can prove thatx∂x = z(1 + z)2∂z
by exploiting the fact that ∂x(z(x)) = [∂z(x(z))]−1. This expression is crucial, becausesince Td,0(p1, p2, . . .) is obtained from T0,0(p1, p2, . . .)↔ z(x) by repeatedly applying x∂x,this expression implies that Td,0(p1, p2, . . .) transforms under (4.1) to a finite linear com-bination of the p′b.
For r = 2, equation (4.3) has two solutions, only one of which admits an expansion innon-negative powers of z. This solution is
w(x(z)) =1−√
1 + 8z2
4z= z − 3z3 + 29
2z5 + . . . ,
so in this case x(z) is given by
x(z) =1−√
1 + 8z2
4zexp
−
(1−√
1 + 8z2
4z
)2 ,
and from this it follows that
x∂x =4z3√
1 + 8z2
−1 +√
1 + 8z2∂z =
(z + 6z3 − 4z5 + . . .
)∂z.
This reveals one of the main problems with the change of variables given by (4.2).Since Td,0 is obtained from T0,0 by applying (x∂x)
d, we see that for d > 0, the seriesTd,0(p1, p2, . . .) transforms to an infinite linear combination of the p′i.
This is not the only problem for r > 1. In the case r = 1, the inversion of the ELSVformula leads to a generating series for the Hodge integrals of the form
∑
j,k0,k1,...
(−1)j〈λjτ k00 τk11 · · · 〉β
j3T k00,0
k0!
T k11,0
k1!· · · ,
but since for r > 1 we have more than one possibility for the second index on Td,c, it isnot immediately clear how to obtain a similarly nice generating series for the Hurwitz-Hodge integrals. We conclude that Kazarian’s proof does not seem to generalize directlyto the case r ≥ 1.
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Populaire Samenvatting
De KP (Kadomtsev-Petviashvili) vergelijking is een differentiaalvergelijking die golvenf(x, y, t) in ondiep water beschrijft. Hier zijn x en y de positie, t is de tijd, en f(x, y, t) dehoogte van de golf. Er bestaat een differentiaalvergelijking voor een functie g(x, y, t1, t2),zodanig dat voor elke vaste waarde t2, de functie f(x, y, t) := g(x, y, t, t2) een oplossingvan de KP vergelijking is. Zo heb je dus niet een enkele oplossing van KP, maar eenhele familie van oplossingen, geındexeerd door de parameter t2. Het blijkt dat we opdeze manier de KP vergelijking kunnen uitbreiden voor oneindig veel parameters t = t1en t2, t3, . . . . Het zo verkregen oneindige stelsel differentiaalvergelijkingen heet de KPhierarchie.Het is a priori niet duidelijk wat het nut van deze uitbreiding is. De toevoeging vande nieuwe parameters stelt ons in staat symmetrieen van de KP hierarchie te vinden.Dit zijn bewerkingen waarmee je uit een oplossing van KP andere oplossingen kan con-strueren. Dus met een triviale oplossing, zoals de stilstaande golf f(x, y, t1, t2, . . .) = 1,kunnen we niet-triviale oplossingen vinden door met die symmetrieen op de triviale op-lossing te werken. De KP hierarchie is dus handig om oplossingen van de KP vergelijkingte vinden, maar inmiddels vindt de hierarchie ook toepassingen in andere gebieden, zoalsde kwantumzwaartekracht.
Een permutatie op n tekens is een manier om n tekens te rangschikken. Bijvoorbeeld,4362157 is een permutatie van de eerste 7 positieve gehele getallen. Merk op dat je dezepermutatie als een soort functie kan zien. De functie f die bij de voorgaande permutatiehoort, voldoet bijvoorbeeld aan f(4) = 1 (want 4 staat op de eerste plek), f(3) = 2,et cetera. Er is een speciaal soort permutatie, een cykel geheten, waar een bepaaldenotatie bij hoort. We noteren dit soort permutaties als bijvoorbeeld (1, 3, 6, 5). Dezepermutatie zet 1 op de derde plek, 3 op de zesde plek, 6 op de vijfde plek, 5 op de eersteplek, en alle andere getallen blijven op hun eigen plek. Net als functies kunnen we dezecykels samenstellen, bijvoorbeeld
(1, 3)(2, 5, 3)[5] = (1, 3)[3] = 1.
(De rechte haakjes noteren dat we de functie toepassen op 5, omdat normale haakjes indit geval ambiguıteit veroorzaken.) Deze samenstelling lijkt een beetje op het verme-nigvuldigen van getallen; de samenstelling van twee permutaties is weer een permutatie.Zoals we getallen kunnen ontbinden in priemgetallen, kunnen we ons afvragen hoe wepermutaties kunnen ontbinden. Specifiek richten we ons op ontbindingen in cykels vanlengte 2, zoals (5, 3). Dit is niet uniek, bijvoorbeeld geldt (1, 2, 3) = (1, 3)(1, 2) =(2, 1)(2, 3) = (3, 2)(3, 1), zoals je eenvoudig zelf na kan gaan. Nu is het natuurlijk om
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je af te vragen op hoeveel manieren dit kan, en dit is precies hoe de Hurwitz getallengedefinieerd zijn. Het Hurwitz getal dat bij een permutatie σ hoort, genoteerd als H(σ),telt het aantal manieren waarop we σ kunnen ontbinden in cykels van lengte 2. Bijvoor-beeld, H((1, 2, 3)) = 3, zoals we zojuist zagen.
In de combinatoriek (de wiskunde van het tellen) is het niet ongebruikelijk om getallenals de Hurwitz getallen bij te houden in een functie F die een genererende functie heet. Inhet geval van de Hurwitz getallen hangt deze functie af van de variabelen x, y, t1, t2, . . . .Als je de combinatoriek van de Hurwitz getallen nader bestudeert, dan kan je met enigemoeite een vrij onverwacht resultaat bewijzen:
Stelling. De genererende functie van Hurwitz getallen is een oplossing van de KPhierarchie.
Het doel van deze scriptie is om die stelling te bewijzen.
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References
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[DLN] N. Do, O. Leigh, P. Norbury. Orbifold Hurwitz numbers and Eynard-Orantin invariants. arXiv:1212.6850.
[F] O. Forster. Lectures on Riemann Surfaces. Springer, 1981.
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[K] M. Kazarian. KP hierarchy for Hodge integrals. Adv. Math., 221 (2009),no. 1, 1–21.
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[S1] B. E. Sagan. The Symmetric Group: Representations, Combinatorial Algo-rithms, and Symmetric Functions. Springer, 2001.
[S2] S. Shadrin. On the structure of Goulden-Jackson-Vakil formula. Math. Res.Lett., 16 (2009), no. 4, 703–710.
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