An introduction to the KP hierarchy and Hurwitz numbers with completed cycles

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An introduction to the KP hierarchy and

Hurwitz numbers with completed cycles

Eddo van den Boom

July 14, 2014

Bachelor thesis

Supervisors: prof. dr. Sergey Shadrin, MSc Petr Dunin Barkovskiy

Korteweg-de Vries Instituut voor Wiskunde

Faculteit der Natuurwetenschappen, Wiskunde en Informatica Universiteit van Amsterdam

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Abstract

The goal of this thesis is to give an introduction to the theory of the KP hierarchy and prove that the generating function for Hurwitz numbers with completed cycles is a solution to the KP hierarchy. In the first part, we review the mathematics we need: irreducible representations of the symmetric group, Pl¨ucker relations, the semi-infinite wedge space Λ∞2V and the boson fermion correspondence. Then we explain how the

KP hierarchy can be defined by looking at a representation of the group GL∞ on Λ

∞ 2V .

The last chapter is devoted to Hurwitz numbers. We prove the theorem about Hurwitz numbers with completed cycles and also look at why the operator corresponding to another linear combination of cycles is not a symmetry of the KP hierarchy for a specific case.

Title: An introduction to the KP hierarchy and Hurwitz numbers with completed cycles Authors: Eddo van den Boom, eddovdboom(at)gmail.com, 10200657

Supervisors: prof. dr. Sergey Shadrin, MSc Petr Dunin Barkovskiy Second grader: prof. dr. Jan de Boer

Date: July 14, 2014

Korteweg-de Vries Instituut voor Wiskunde Universiteit van Amsterdam

Science Park 904, 1098 XH Amsterdam http://www.science.uva.nl/math

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1 Introduction

Personal note

Since high school I have been interested in both mathematics and physics. That’s why I chose to follow the double bachelor’s degree in mathematics and physics at the University of Amsterdam (UvA). What I like about mathematics is the precise definitions and the theorems that are proved using only these definitions. Everything that is proved in mathematics is sure and indisputable.

Physics is a very different kind of science: if you show that something is true in a certain situation, it may not be true in a more extreme situation. Take for example Newton’s theory of classical mechanics. This theory is certainly true for everyday life, but Einstein showed that it really is an approximation of his theory of relativity, which also holds for objects moving near the speed of light. What I like about physics is that it describes the real world and not an imaginary world. However, it cannot do this on its own: it needs mathematics. This kind of mathematics is, how I see it, exactly what mathematical physics is about. So in this field I can combine my passions for physics and mathematics.

This thesis is about the KP hierarchy and Hurwitz numbers. Before I started writing this thesis, I knew nothing about the subject, but one of my supervisors, professor Shadrin, introduced me to it when I told him that I would like to write something about a subject in mathematical physics. Unfortunately, I didn’t have time to look at the connection between physics and the subject I studied, but professor Shadrin told me that it has some connections with string theory.

Symmetries of differential equations and the KP

hierarchy

Solving non-linear differential equations is often a very difficult thing to do. One example of a non-linear differential equation is de Korteweg-de Vries (KdV) equation:

ut+ 6uux+ uxxx = 0 (1.1)

where u = u(x, t). The subscripts stand for partial derivatives, for example: ux = ∂u_∂x.

We are going to look at symmetries of certain differential equations. What does this mean? A symmetry is a way of producing new solutions from known solutions. For example, we introduce a new variable s and consider a differential equation ∂u_∂s = ˆK(u)

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such that

if u = u(x, t, s) and u(x, t, 0) is a solution of (1.1) and ∂u

∂s = ˆK(u)

then u(x, t, s) is a solution of (1.1) for any value of s. In this case we say that ∂u_∂s = ˆK(u) is a symmetry of (1.1).

In fact, we will look at symmetries of the KP hierarchy, which is an infinite set of differential equations. We say that a function is a solution to the KP hierarchy if and only if it satisfies all the differential equations that the hierarchy contains. Instead of defining the KP hierarchy as it was first defined historically, we will define it in such a way that the symmetry of the hierarchy is immediately clear from the definitions. More precisely, we will construct a vector space Λ

∞ 2

0 V and an isomorphism of vector spaces

σ : Λ∞2 V → C[x₁, x₂, . . . ] and define an action of the Lie group GL_∞ on Λ ∞

2V . Then we

will see that for τ ∈ C[x1, x2, . . . ] the following two statements are equivalent:

• There exists an element g ∈ GL∞ such that σ−1(τ ) = g · |vaci, where |vaci ∈ Λ

∞ 2

0 V

is the vacuum vector, defined in chapter 4 (σ(|vaci) = 1).

• The function τ is a solution to a certain infinite hierarchy of differential equations. The construction of this hierarchy naturally follows from the first statement and this will be our definition of the KP hierarchy. However, it should be noted that this way of looking at the KP hierarchy was discovered by, among others, M. Sato in the 1980s, see for example [9]. Before that, the KP hierarchy was known, but only from antother viewpoint. Nonetheless, since the viewpoint of Sato will be so important in this thesis, we will only use this viewpoint.

What is so special about the KP hierarchy? It turns out that there’s a close connection to the theory of Hurwitz numbers and moduli spaces. For example, in 1991 Witten conjectured that a generating series for intersection numbers of stable classes on the moduli space of curves is a solution to the KdV-hierarchy, which is closely related to the KP hierarchy. Witten’s cojecture was proved by Kontsevich in 1992. It is also known that a generating series for simple Hurwitz numbers is a solution to the KP hierarchy, see for example [5].

Organization of the thesis

In chapter 2, we classify the irreducible representations of Sn and state some important

results about them, because they are closely related to both the KP hierarchy and Hurwitz numbers. In chapter 3, we give an overview of Pl¨ucker relations, because, as we will see, a function is a solution to the KP hierarchy if and only if its coefficients (in a certain basis) satisfy the Pl¨ucker relations. Then we will introduce the semi-infinte wedge space and state the boson fermion correspondence in chapter 4 and use it to define the KP hierarchy in chapter 5. Finally, in chapter 6 we will define Hurwitz numbers and prove that the operator corresponding to multiplication with a completed cycle is an

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infinitesimal symmetry of the KP hierarchy. This implies that the exponential generating series of Hurwitz numbers with completed cycles is a solution to the KP hierarchy.

Acknowledgements

First of all, I want to thank Petr Dunin Barkovskiy for all the time he has been willing to spend helping me and answering my questions. We talked regularly, almost every week, and that helped me a lot. Secondly, I want to thank professor Sergey Shadrin for introducing me to the subject, which I was not familiar with before I first saw him, and for being always available for questions. Finally, I also want to thank professor Jan de Boer for his willingness to be the second grader of my thesis and to put time in reading and assessing it.

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2 Irreducible representations of S

_n

and their characters

In the study of Hurwitz numbers and the KP hierarchy, the characters of the irreducible representations of the symmetric group often appear in formulas. In this chapter we state and proof a few main results about these representations and their characters. We used [1] as the main reference for this chapter.

2.1 Basic definitions in representation theory and

Maschke’s theorem

Let us first define what a representation of an algebra is.

Definition 2.1. A representation of an algebra A is a vector space V together with a homomorphism of algebra’s ρ : A → End(V ). We will also write av instead of ρ(a)v.

This means that to every element of A we assign a linear map V → V such that the compositions of these maps satisfy the relations of the algebra. A special role in the study of representations is played by irreducible and indecomposable representations, which we will define now.

Definition 2.2. A subrepresentation of a representation ρ : A → End(V ) is a subspace W ⊂ V such that for all operators ρ(a) we have ρ(a)w ∈ W for all w ∈ W .

Definition 2.3. An irreducible representation is a representation without any proper (i.e. unequal to itself or {0}) subrepresentations.

Definition 2.4. An isomorphism of representations of an algebra A is an isomorphism of vector spaces φ : V1 → V2 such that φ(av) = aφ(v) for every a ∈ A, v ∈ V1.

Definition 2.5. If ρ1 : A → End(V1) and ρ2 : A → End(V2) are representations, then

the direct sum representation of V1 and V2 is ρ : A → End(V1 ⊕ V2), ρ(a)(v1 ⊕ v2) =

ρ1(a)v1⊕ ρ2(a)v2.

Definition 2.6. An indecomposable representation is a representation that is not iso-morphic to the direct sum of two representations.

Note that an irreducible representation is automatically indecomposable, but the con-verse is not true in general. However, in some cases, the concon-verse is true, as the next theorem shows.

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Theorem 2.7. (Maschke’s theorem) For a finite group G and a field k whose charac-teristic does not divide |G|, every finite dimensional representation of the group algebra k[G] is isomorphic to a direct sum of irreducible representations.

A proof can be found for example in [1].

2.2 Classification of all irreducible representations

of S

n

We first introduce some notation. A partition λ of n is a tuple of non-increasing positive integers (λ1, . . . , λk) such that Pk_i=1λi = n. In this case, we write λ ` n, |λ| = n and

l(λ) = k. We also write mj(λ) = |{i : λi = j}| and denote the sign of the permutation

σ by (σ).

It is known that for a finite group, the number of irreducible representations is equal to the number of conjugacy classes. Because in Sn the conjugacy classes are exactly the

elements with the same cycle type, we can label the irreducible representations of Sn by

partitions λ ` n. In order to define a representation of Sn, we first need a vector space

and an action of Sn on this vector space. This vector space will be the space of standard

Young tableaux, which are defined below.

Definition 2.8. The Young diagram of a partition λ = (λ1, . . . , λk) is a diagram

con-sisting of squares of the same size such that the i’th row from the top has λi blocks and

the blocks are aligned to the left.

Definition 2.9. A Young tableau of a partition λ ` n is the Young diagram of λ with in each square a number of the set {1, . . . , n} filled in.

Definition 2.10. A standard Young tableau of λ ` n is a Young tableau of λ such that every number of the set {1, . . . , n} is used exactly once.

Example 2.11. We illustrate these definitions with an example. Let λ = (3, 1, 1) ` 5. Then one possible standard Young tableau corresponding to λ is

4 2 1 3 5 _.

It is convenient to choose a specific tableau for every λ, hence the following definition. Definition 2.12. The regular tableau Tλ of λ is the tableau filled in increasing order

from left to right, from top to bottom.

Now we fix a partition λ ` n and let Sn act on standard Young tableaux of λ by

permuting the entries. We also define Aλ and Bλ to be the subgroups of Sn that keep

every number in the same row or in the same column of Tλrespectively. For convenience,

we will also write A and B if λ is fixed. Now we define aλ = X α∈A α, bλ = X β∈B (β)β and cλ = aλbλ

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as elements of CSn. Note that αβ = e if and only if α = e and β = e for α ∈ A, β ∈ B,

so cλ 6= 0. This element is called the Young symmetrizer of λ. Now we can state the

main result of this section.

Theorem 2.13. The irreducible representations of Sn are given by {(CSn)cλ : λ ` n},

with the action of Sn given by left multiplication. We will write (CSn)cλ = Vλ.

The proof of this theorem consists of three parts:

1. Show that Vλ is an irreducible representation of Sn for all λ ` n.

2. Show that Vλ and Vµ are different representations of Sn if λ 6= µ.

3. Show that {Vλ : λ ` n} are all irreducible representations of Sn.

The following lemma is an important part for the first step of the proof.

Lemma 2.14. Let λ ` n and x ∈ CSn. Then x is a scalar multiple of cλ if and only if

(β) · αxβ = x for all α ∈ A, β ∈ B. (2.1) Proof. Suppose that x is a scalar multiple of cλ, so x = k · aλbλ for a certain k ∈ C.

Then for α ∈ A, β ∈ B, we have (β) · αxβ = k · (α · aλ) · (bλ· (β)β) = k · aλbλ = x.

Conversely, suppose that (2.1) holds. We write x = X

g∈Sn

kgg (kg ∈ C)

and note that (2.1) implies that

kαgβ = (β)kg for all α ∈ A, β ∈ B, g ∈ Sn. (2.2)

For g = e, this implies that kαβ = (β)ke. So if we can proof that kg = 0 for all g /∈ AB,

then it follows that x = kecλ.

So if, for a fixed g /∈ AB, we can find a transposition t ∈ A such that g−1_{tg ∈}

B, then we are done, since in that case g = (t)g(g−1tg), so (2.2) implies that kg =

(g−1tg)kg = −kg. We are going to show that if such a t doesn’t exist, then g ∈ AB.

This will complete the proof.

So suppose such a transposition doesn’t exist. Then g cannot send two elements in the same row of T = Tλ to the same column, since then the transposition of these two

elements would be the required t. So all elements in the first row get sent to different columns, so there is an element β1 permuting only columns of gT such that β1gT has the

same unordered first row as T so there exists an element α1 ∈ A such that β1gT and α1T

have the same first row. Note that β1 ∈ gBg−1, because g−1β1g permutes only columns

of T . Also note that we can still not find a transposition that sends two elements in the second row of α1T to the same column of β1gT . So proceding inductively, we can find

α2, . . . , αk and β2, . . . , βk such that βgT and αT have the same k rows, where α =Q αi

and β =Q βi and α ∈ A, β ∈ gBg−1. We apply this procedure until k = l(λ), and then

we have found α, β such that αT and βgT are the same diagrams. So α = βg, so there exists a β0 ∈ B such that α = (gβ0_g−1_{)g = gβ, so g = αβ}−1_{, so g ∈ AB.}

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This lemma implies that in particular cλcλ is proportional to cλ, since this element

satisfies (2.1). For the proof of the first part, we will need to show that it is a non-zero multiple of cλ.

Lemma 2.15. Write c2

λ = k · cλ. Then k = _dim(Vn!

λ).

Proof. We calculate the trace of right multiplication by cλ in two different bases. In the

standard basis of CSn, it is clear that the trace of cλ is n!, since the coefficient of g in

gcλ is 1 and the dimension of CSn is n!. Now choose a basis of Vλ and extend it to a

basis of CSn. Then every element gcλ of the basis of Vλ has eigenvalue k, while all other

basis vectors have eigenvalue 0. So the trace is also equal to k · dim(Vλ) (since the trace

is basis independent). So n! = k · dim(Vλ) and the lemma follows.

We can now proof the first step. Using Maschke’s theorem, it is sufficient to prove that Vλ is indecomposable. So suppose Vλ is decomposable, Vλ = V1⊕ V2. Then, by the

previous lemma, we have that cλVλ = Ccλ is a one-dimensional vector space. Assume

without loss of generality that cλV1 = Ccλ. Then Vλ = CSncλ = CSncλV1 ⊂ V1. So V2 is

trivial and Vλ is indecomposable.

For the second step we need the following definition and lemma.

Definition 2.16. We say that λ > µ if and only if λi > µi for the first value of i where

they are different. This is called the lexicographic ordering partitions. Note that it is a total order, i.e. λ 6= µ ⇒ (λ < µ or µ < λ).

Lemma 2.17. If λ > µ, then aλ· x · bµ = 0 for all x ∈ CSn.

Proof. For the same reason as in lemma 2.14, it is sufficient to show that for g ∈ Sn

arbitrary, there exists a transposition t ∈ Aλ such that g−1tg ∈ Bµ. So if T = Tλ and

T0 = gTµ, we want to find two numbers i, j that are in the same row in T and in the

same column in T0. The proof is more or less a generalization of the pigeonhole principle. If λ1 > µ1, we can immediately apply this principle. Else, λ1 = µ1 and all numbers in

the first row of T are in a different column of T0. So if we define B_µ0 = gBµg−1, then we

can find α1 ∈ Aλ and β1 ∈ Bµ0 such that α1T and β1T0 have the same first row and α1

only permutes entries in the first row of T . Now, if λ2 > µ2, we can find a permutation

(ij) such that i and j are in the second row of α1T and in the same column in β1T0.

This implies that i and j are also in the same row in T and in the same column in T0, so in this case (ij) is the required permutation. If λ2 = µ2, we can apply this procedure

inductively, until the first part where λ and µ differ.

Now we can proof the second step. Let λ and µ be two different partitions and suppose without loss of generality that λ > µ. We already know that cλVλ 6= 0. By the previous

lemma, we have cλVµ= cλCSncµ = aλ(bλCSnaµ)bµ= 0. So an isomorphism of

represen-tations Φ : Vλ ∼= Vµcannot exist, because in that case, the action of the element cλshould

commute with this isomorphism, which is not possible, because Φ(c2_λ) = cλΦ(cλ) = 0,

while c2 λ 6= 0.

The third part automatically follows from the first and second part, because we have found different irreducible representations for every conjugacy class of Sn, so we found

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2.3 Theorems about characters of V

_λ

In this section, we state two important results about the characters of the irreducible representations of the symmetric group. We will not prove the theorems in this chapter, but we will give examples to illustrate both theorems.

Theorem 2.18. (Frobenius formula) Let χλ

µ be the character of the representation Vλ

evaluated at the conjugacy class Cµand let k = l(λ) and d = |λ| = |µ|. Write mi = mi(µ)

for i = 1, . . . , d. Then χλ

µ is equal to the coefficient of x λ1+k−1 1 x λ2+k−2 2 . . . x λk k in ∆(x) · (x1+ · · · + xk)m1(x21+ · · · + x 2 k) m2_{. . . (x}d 1+ · · · + x d k) md_,

where ∆ is the Vandermonde determinant: ∆(x) =Q

i<j(xi− xj).

Example 2.19. Let us calculate χλ

µ for µ = [31] and λ = [22]. In this case, k = 2, d = 4

and (m1, m2, m3, m4) = (1, 0, 1, 0). So we have calculate the coefficient of x31x22 in

(x1− x2)(x1+ x2)(x31+ x32)

and it’s easy to see that this coefficient is −1. So in this case χλ

µ = −1.

The following theorem gives yet another way to compute χλ_µ. We first need the fol-lowing definitions:

Definition 2.20. A rim hook tableau of shape λ and content µ = (µ1, . . . , µk) is a Young

tableau T of λ such that the following conditions are satisfied: • The number of i’s in T is equal to µi.

• The rows and columns of T are weakly decreasing (from left to right, from top to bottom).

• Every component Tj of T (that is, the diagram consisting of boxes occupied by j)

is connected.

• No component contains a 2 × 2 square.

Definition 2.21. Let T be a rim hook tableau consisting of components T1, . . . , Tk. The

height h(Tj) of a component Tj is the number of rows in Tj decreased by one. The height

h(T ) of T is defined to be h(T ) = k X i=1 h(Ti).

Theorem 2.22. (Murnaghan-Nakayama rule) One has χλ_µ=X

T

(−1)h(T ),

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Corollary 2.23. For µ = (1, . . . , 1) ` n and λ ` n arbitrary, χλ_µ is equal to the number of standard Young tableaux of shape λ by the above theorem. On the other hand, it is also equal to dim(Vλ), since µ can be represented as the identity matrix, which has trace

dim(Vλ). So the number of standard Young tableaux of shape λ is equal to dim(Vλ).

The previous theorem is actually a corollary of the next theorem, for which we first need a definition.

Definition 2.24. The border of a Young diagram is the set of boxes at positions (i, j) such that there is no box at position (i + 1, j + 1). A border strip of a Young diagram is a connected piece of the border such that the remaining diagram is still a Young diagram. Theorem 2.25. One has

χλ_µ=X

S

(−1)h(S)χλ−S_µ−µ

1,

where the sum runs over all border strips S of length µ1 of the Young diagram of λ.

Example 2.26. Let λ = (4, 3, 3) and µ = (6, 3, 1). Then there is only one rim hook tableau T of shape λ and content µ:

3 2 1 1 2 2 1 1 1 1

The heights of the tableaux T1, T2, T3 are respectively 2, 1 and 0, so the total height of T

is 2 + 1 + 0 = 3. So in this case the character χλ_µ is equal to (−1)3 = −1. Alternatively, we can apply theorem 2.25 and we find that χλ

µ = (−1)2· (−1)1· (−1)0· χ (0)

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3 Finite dimensional Pl¨

ucker

relations

We will see that the solutions of the KP hierarchy can be seen as elements of an infinite dimensional Grassmanian. In this chapter we will consider how in the finite dimensional case the Grassmanian Gr(m, V ) can be viewed as a subset of the exterior algebra ΛmV . Our main references for this chapter are [8] and [2].

3.1 Pl¨

ucker embedding

Let V be a vector space of dimension N . The Grassmanian Gr(m, V ) is defined to be the space of m-dimensional subspaces of the vector space V . Since a vector space is completely determined (up to isomorphism) by its dimension, we will also write this as Gr(m, N ).

We consider the case m = 1 seperately, because it will be useful later on. In this case, Gr(1, V ) is called the projective space over V and is also denoted by P(V ) or PN −1(C)

(we assume that V is a vector space over C). This is the space of all lines through the origin and an element is written as [v1 : · · · : vN]. This notation means that an element

of the projective space can be considered as an equivalence class of vectors in V under the following equivalence relation:

(v1, . . . , vN) ∼ (w1, . . . , wN) ⇐⇒ ∃λ ∈ C : vi = λwi ∀i.

Now we define a map α : Gr(m, V ) → P(ΛmV ) by α({v1, . . . , vm}) = [v1 ∧ · · · ∧ vm].

Here {v1, . . . , vm} is a basis for the associated subspace, so we have to check that this

map is independent of the choice of basis. Indeed, it is one of the properties of the wedge product that it gets multiplied by the determinant of the change-of-basis matrix when changing bases. We make this precise in the following lemma.

Lemma 3.1. Let V be a vector space over C of dimension N and let m ≤ N . Suppose that {w1, . . . , wm} and {w01, . . . , w0m} are two sets of linearly independent vectors that

span the same subspace of V . Then we can write w0_i =Pm

j=1hijwj for certain constants

hij ∈ C with det h 6= 0. In this setting, we have w01∧ · · · ∧ w 0

m = det h · w1∧ · · · ∧ wm.

Proof. We write out the wedge product w0₁∧ · · · ∧ w0

m in terms of the vectors wi:

w0₁∧ · · · ∧ w0_m = m X j1=1 h1,j1wj1 ∧ · · · ∧ m X jm=1 hm,jmwjm

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= X j1,...,jm h1,j1. . . hm,jm· wj1 ∧ · · · ∧ wjm = X σ∈Sn h1,σ(1). . . hm,σ(m)· wσ(1)∧ · · · ∧ wσ(m) = X σ∈Sn h1,σ(1). . . hm,σ(m)· (σ) · w1∧ · · · ∧ wm = det h · w1∧ · · · ∧ wm.

Here we used the fact that wj1 ∧ · · · ∧ wjm 6= 0 if and only if {j1, . . . , jm} = {1, . . . , m}

and for the last equation, we used the Leibniz formula for determinants.

Now we know that the map α is well defined, we want to show that we do not lose information when applying it.

Lemma 3.2. The map α : Gr(m, V ) → P(ΛmV ) is injective.

Proof. A nice way to show injectiveness of a map is to construct explicitly a ’left-inverse’ and that is what we will do here, following [2]. We define the map φ : P(Λm_{V ) →}

Gr(m, V ) by φ([x]) = {v ∈ V : v ∧ x = 0}. It is easy to check that this map is well defined and that φ(α(U )) = U for all U ∈ Gr(m, V ).

This lemma rectifies the name Pl¨ucker embedding that is given to this map. What about surjectiveness? It is useful to introduce a new definition here.

Definition 3.3. A vector ω ∈ ΛmV is decomposable if there exist vectors v1, . . . , vm ∈ V

such that ω = v1∧ · · · ∧ vm.

Lemma 3.4. The image of the map α is exactly the subset of decomposable vectors of P(ΛmV ).

Proof. This follows from the definitions.

In general, this map is not surjective, because not every vector of Λm_{V is}

decom-posable. For example, if m = 2 and V = C4_{, then the element ω = e}

1 ∧ e2 + e3∧ e4

is not decomposable, because ω ∧ ω 6= 0, while this equality obviously holds for all decomposable vectors ω.

3.2 Pl¨

ucker relations

In this section, it will be convenient to choose a specific basis B = {v1, . . . , vN} of V .

We will also use the corresponding basis C = {vα1 ∧ · · · ∧ vαm : α1 < · · · < αm} of Λ

m_{V .}

Now fix a subspace W ⊂ V spanned by the vectors {w1, . . . , wm} and write the vectors

wi in the basis B as wi =P vijvj. We now calculate α(W ) in the basis C of ΛmV :

α(W ) = w1∧ · · · ∧ wm =

X

α1<···<αm

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where vα1,...,αn = det(vi,αj). This calculation is done in the same way as in lemma 3.1

and we will not do it here.

Given a vector ω ∈ Λm_{V , it might be not immediately clear if it is decomposable or}

not. If ω ∧ ω 6= 0, then ω is not decomposable, but if ω ∧ ω = 0, then ω doesn’t have to be decomposable, so this criterium is not sufficient. The next theorem classifies all decomposable vectors.

Theorem 3.5. Let W ⊂ V be a subspace and let vα1,...,αm denote the coeficient of

vα1∧ · · · ∧ vαm in α(W ), as in (3.1). Then for any two subsets A = {α1, . . . , αm−1} and

B = {β1, . . . , βm+1} of {1, . . . , N } such that A ∩ B = ∅, the following relation holds: m+1

X

i=1

(−1)i−1vα1,...,αm−1,βivβ1,..., ˆβi,...,βm+1 = 0,

where ˆβi means that we omit βi. These relations are called the Pl¨ucker relations.

The converse is also true: if the Pl¨ucker relations are satisfied, then the corresponding element of Λm_{V is a decomposable vector. Proofs of these statements can be found for}

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4 Boson fermion correspondence

In this chapter we will introduce the semi-infinite wedge space Λ∞2 V and discuss an

isomorphism σ between the spaces F = Λ∞2V and B = C[z, z−1; x₁, x₂, . . . ] called the

bosonfermion correspondence (F stands for fermion and B for boson, of course). The τ -functions that are solutions to the KP hierarchy are elements of the space B. We will see that under σ, these τ -functions correspond exactly to the decomposable vectors of the space F . Moreover, there is a group G acting transitively on the set of all decomposable vectors, so that any τ -function of the KP hierarchy can be written as τ = σ(g |vaci) for a certain g ∈ G, since |vaci is a decomposable vector.

4.1 Semi-infinite wedge space

Definition 4.1. Let V be the vector space L

i∈Z+1₂ Cvi. Let B be the set of formal

symbols of the form vi1∧ vi2∧ vi3∧ . . . , with ij ∈ Z +

1

2 with the following two properties:

(i) ij+1 < ij ∀j ∈ N,

(ii) ∃N ∈ N : ij+1 = ij− 1 ∀j ≥ N.

Now we define the semi-infinite wedge space Λ∞2 V to be the vector space over C with

basis B.

Remark 4.2. The vectors in Λ∞2 V can be represented by so-called Maya diagrams, see

for example [8]. An example of a Maya diagram can be found at the cover of this thesis. Definition 4.3. For a set S = {i1 > i2 > i3 > . . . } ⊂ Z +1₂, we define the following two

sets: S+ _{= S\(Z}

≤0−1₂) and S−= (Z≤0−₂1)\S. Note that if v = vi1∧ vi2∧ vi3∧ . . . and

S = {i1 > i2 > i3, . . . } (we write v = vS and S = Sv), both sets S− and S+ are finite.

The vector v for which S_v+= S_v−= ∅ will play a special role and is called the vacuum vector. We will denote it by |vaci.

Definition 4.4. For k ∈ Z +12, we define the operators ψk and ψ ∗ k as follows: ψkvS = ( 0 if k ∈ S (−1)jvi1∧ · · · ∧ vk∧ vij+1 ∧ . . . if k /∈ S and j = max{l : il < k}, ψ_k∗vS = ( 0 if k /∈ S (−1)j−1_v i1 ∧ · · · ∧ vij−1 ∧ vij+1. . . if k = ij.

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A more convenient way of writing these operators is as follows: ψkvS = vk∧ vS and ψk∗vS =

∂ ∂vk

vS,

where we keep in mind that by switching two adjacent symbols in order to arrange the vector so that it is in the form of definition 4.1, we get a minus sign, that is:

vi1 ∧ · · · ∧ vij ∧ vij+1∧ · · · = −vi1 ∧ · · · ∧ vij+1∧ vij ∧ . . . .

Lemma 4.5. The operators ψk and ψk∗ obey the following commutation relations:

[ψi, ψj]+ = 0, [ψi∗, ψ ∗

j]+ = 0 and [ψi, ψj∗]+ = δij,

where [A, B]+ def

= AB + BA for two operators A, B and δ is the Kronecker delta.

Proof. Using the convenient notation, we see that [ψi, ψj]+v = vi∧ vj∧ v + vj∧ vi∧ v =

vi ∧ vj ∧ v − vi ∧ vj ∧ v = 0. For the second relation, suppose that i, j ∈ S and write

v = vi∧ vj∧ w using the convention mentioned above. Now note that ∂i∂jvi∧ vj∧ w +

∂j∂ivi ∧ vj ∧ w = −∂i∂jvj ∧ vi ∧ w + ∂j∂ivi ∧ vj ∧ w = −w + w = 0. If on the other

hand either i or j is not in S, then it also acts as zero (trivially). Finally, for the third relation, note that if i = j, then [ψi, ψ∗j]+v = ψiψ∗iv + ψ

∗

iψiv. If i ∈ S, then the first

term gives v and the second term gives zero, while if i /∈ S, the first term gives zero and the second term gives v. So this expression is always equal to v if i = j. If i 6= j, we can easily check that [ψi, ψj∗]+= 0 in the same way as above.

This lemma shows that we can make V into a representation of the Clifford algebra (because this is the alegbra generated by {ψk, ψk∗}k∈N with exactly these defining

rela-tions). This representation also has a physical meaning. Acting with ψi can be thought

of as adding a fermion in state i and acting with ψ∗_i as removing a fermion in state i. A simple consequence of these relations is that ψ2

i = ψj2 = 0, which is known to all

physicists as Pauli’s exclusion principle: no two fermions can be in the same state. Definition 4.6. The charge operator C is defined as follows:

C = X

k∈Z+1₂

Ekk,

where Eij =: ψiψ∗j : is the normally ordered product of ψi and ψ∗j:

: ψiψ∗j : =

(

ψiψj∗ if j > 0

−ψ∗

jψi if j < 0.

Definition 4.7. The energy operator H is defined as follows: H = X

k∈Z+1₂

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The energy operator will be useful later on. Right now we will focus our attention on the charge operator. We can easily check that Cv = (|S+

v| − |S − v |)v, by writing C as follows: C = X k∈Z+1 2 k>0 Ekk+ X k∈Z+1 2 k<0 Ekk= X k∈Z+1 2 k>0 ψkψ∗k− X k∈Z+1 2 k<0 ψ_k∗ψk.

Now we derive a formula that is useful in the next lemma. Suppose that v = vi1 ∧

· · · ∧ vin ∧ vin+1 ∧ · · · ∧ viN ∧ . . . is a vector, where we wrote the vector in such a way

that in > 0, in+1 < 0 and N is a number satisfying (ii) of definition 4.1. Then by

counting the elements of S+

v and S −

v using definition 4.3, we get that |Sv+| = n and

|S− v | = −12 − iN − (N − (n + 1)) = 1 2 − iN − N + n, so that |S_v+| − |S_v−| = N + iN − 1 2.

Lemma 4.8. Cv = 0 ⇐⇒ v is a linear combination of vectors of the form vλ = vλ1−1₂∧

v_λ

2−3₂ ∧ . . . , where λ = (λ1 ≥ λ2 ≥ . . . ) is a partition, i.e. ∃N ∈ N : λi = 0 ∀i ≥ N .

Proof. Suppose that Cv = 0 for a basis vector v. Then (|S_v+|−|S−

v |)v = 0, so |Sv+| = |Sv−|.

Let N be a number that satisfies (ii) of definition 4.1, so that we can write v = vi1 ∧ · · · ∧ viN −1∧ viN ∧ viN−1∧ . . . .

Then by by using the formula above we find that iN = −N + 1₂. Now if we write

v = v_λ

1−1₂ ∧ vλ2−3₂ ∧ · · · ∧ vλj−j+1₂ ∧ . . . (by choosing λj = ij + j −

1

2) then we see

that (λ1, λ2, . . . ) is indeed non-increasing (because (i1, i2, . . . ) is strictly decreasing) and

λi = 0 for all i ≥ N .

For the reverse statement, let v be a vector of the given form and let N be such that λi = 0 for all i ≥ N . Then iN = −N + 1₂, so that |Sv+| = |S

−

v |, again using the formula

above.

We will denote the subspace of elements of charge 0 by Λ

∞ 2

0 V . By the above lemma,

we have a bijection between the basis vectors that span Λ

∞ 2

0 V and all partitions.

4.2 Boson fermion correspondence

Definition 4.9. We define the operators {Hn}n∈Z on the Fermionic Fock space as

fol-lows: Hn=P_j∈Z+1

2 Ej,j+n.

Remark 4.10. Note that the operator H0 is equal to the charge operator we introduced

previously.

These operators are well defined, because for a basis vector v = vi1 ∧ vi2 ∧ vi3 ∧ . . .

and N a number that satisfies (ii) of definition 4.1, we have Ej,j+nv = 0 for j > N − n

and for j < i1, so the sum that appears to be infinite contains only finitely many terms

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The calculation of the commutator of these operators with the standard fermionic operators ψm and ψm∗ is straightforward and the result is:

[Hn, ψm] = ψm−n and [Hn, ψm∗] = −ψ ∗

m+n. (4.1)

We now define H(x) =P∞

n=1xnHn. Note that this operator is also well defined, because

for the same basis vector v as above we have Hnv = 0 for n > N − i1. By eH(x) we mean

the sum P∞

n=0 1 n!H(x)

n_{. This is also a well defined operator, because for n > 0, H} n

lowers the energy by n and preserves the charge of a vector (if it does not act trivially), but since there is a minimum energy for each charge, the sum eH(x) is finite. So we can define the following function:

σ : F → B : |ui 7→X

l∈Z

zlhl|eH(x)_{|ui ,}

extended linearly to F .

Example 4.11. Consider the vector vλ where λ = (3, 1). To calculate σ(vλ), we list all

the possibilities of obtaining the vacuum vector after applying operators Hn(n > 0) to

vλ: H4vλ = − |vaci , H1H3vλ = 0, H2H2vλ = − |vaci , H1H1H2vλ = |vaci , H₁4vλ = 3 |vaci .

Noting that the operators {Hn : n > 0} commute, it follows that σ(vλ) = −x4 − x2 2 2! + x21x2 3! + 3x41 4! .

Theorem 4.12. (Boson fermion correspondence) The map σ defined above is an iso-morphism of vector spaces and moreover, we have

σ(Hn|ui) =

( _∂

∂xnσ(|ui) if n > 0

−nx−nσ(|ui) if n < 0.

(4.2) So instead of acting with xn or _∂x∂_n on a function and then looking at the image under

σ−1, we can first look at the image under σ−1 of this function and then acting with −1

nH−n or Hn respectively.

Proof. It is not difficult to prove the relations in (4.2), see for example [8]. The map is linear by construction, so we only need to show that it is bijective. For surjectiveness, note that the vector |mi gets mapped to zl, so by (4.2), the vector Hi1

−1. . . H ik

−k gets

mapped to a constant times xi1

1 . . . x ik

k · z

l_{. For injectiveness, note that a vector v} λ gets

mapped to a monomial of weight |λ| and the number of linearly independent polynomials of weight |λ| is exactly the number of partitions of |λ|. Together with surjectiveness, this proves injectiveness.

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5 KP hierarchy

Finally in this chapter we describe how to obtain the differential equations of the KP hierarchy by looking at the group action of GL∞ on Λ

∞

2 V . We mainly follow [3].

5.1 Representations of Lie algebra gl

_∞

and Lie

group GL

_∞

on Λ

∞2

V

The vector space gl_∞ _{consists of all Z × Z matrices A = (a}ij)i,j∈Z that have only finitely

many non-zero entries. So it is a vector space with basis {Eij : i, j ∈ Z}, where Eij

denotes the matrix with a one at position (i, j) and zero’s elsewhere. We now give gl∞

the structure of a Lie algebra by defining the Lie bracket to be simply the commutator of matrices. The space gl_∞ naturally acts on V by setting

Eijvk = δjkvi.

The Lie algebra gl_∞ can be interpreted as the Lie algebra of the group GL∞, which is

the group of matrices A = (aij)i,j∈Z such that A is invertible and

|{(i, j) ∈ Z2 _{: a}

ij − δij = 0}| < ∞.

For gl_∞ and GL∞, we define representations r and R respectively on the space Λ

∞ 2 V as follows: r(a)(vi1 ∧ vi2 ∧ . . . ) = X j vi1 ∧ vi2 ∧ . . . ∧ avj ∧ vj+1∧ . . . and R(A)(vi1 ∧ vi2 ∧ . . . ) = Avi1 ∧ Avi2 ∧ . . . .

This are representations, because [r(Eij), r(Ekl)] = [Eij, Ekl] and R(A)R(B) = R(AB)

for A, B ∈ GL∞.

Remark 5.1. The group action of GL∞ will only give us solutions to the KP hierarchy

in the space B. However, generating series of Hurwitz numbers are formal (infinite) power series and thus not elements of B. To include the formal power series solutions, one can look at the group action of a larger group, see for example [8]. However, since the arguments are very similar for both groups, in this thesis we will only consider the group GL∞.

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5.2 Bilinear identity

We are interested in the orbit of the vacuum vector under the action of the group GL∞.

For convenience, we will write this orbit as Ω. We first need a definition and a lemma. Definition 5.2. For v ∈ V , we define the wedging operator ˆv on Λ∞2 V as

ˆ

v(vi1 ∧ vi2 ∧ . . . ) = v ∧ vi1 ∧ vi2 ∧ . . . .

For f ∈ V∗, we define the contracting operator ˇf on Λ∞2 V as

ˇ

f (vi1 ∧ vi2 ∧ . . . ) = f (vi1)vi2 ∧ vi3 ∧ · · · − f (vi2)vi1 ∧ vi3 ∧ · · · + . . . .

Lemma 5.3. Let A ∈ GL∞, v ∈ V and f ∈ V∗. Then the operators ˆv and ˇf have the

following transformation properties under R(A): R(A)ˆvR(A)−1= cAv and

R(A) ˇf R(A)−1= ˇg, where g = (At)−1f.

Proof. Let us denote the matrix elements of A by aij. To prove the first equation, we

have to show that

R(A)ˆv = cAv · R(A).

So we have to show that applying both sides to a basis vector |ui = vi1 ∧ vi2 ∧ . . . gives

the same result. The left hand side applied to |ui immediately gives Av ∧ R(A) |ui = P

jajivj ∧ R(A) |ui and this is indeed the same as the right hand side applied to |ui.

The second equation can be proved in a similar way, using that for f ∈ V∗ the identity f (Av) = At_{f (v) holds.}

Corollary 5.4. The operators ψi and ψi∗ have the following transformation properties

under R(A): R(A)ψiR(A)−1= X j ajiψj and R(A)ψ_i∗R(A)−1=X j aijψ∗j,

where aij are the entries of A and aij are the entries of A−1.

Now the following theorem tells us exactly when a vector is part of Ω.

Theorem 5.5. (Bilinear identity) A vector |ui ∈ F(0) _{is an element of Ω if and only if}

the following identity holds in F(1)_{⊗ F}(−1)_:

X

i∈Z+1 2

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|ui = R(A) |vaci for a certain A ∈ GL∞. Applying R(A) to (5.1) with |ui = |vaci, we

obtain

X

i

R(A)ψiR(A)−1|ui ⊗ R(A)ψ∗iR(A)

−1_{|ui = 0,}

since |vaci = R(A)−1|ui. Now we can use the corollary 5.4 to conclude that X i X j ajiψj|ui ⊗ X k aikψk∗ = 0,

which is exactly what we need, since P

iajiaik = (AA −1₎

jk = δjk. For the proof of the

converse, we refer the reader to the proof of proposition 7.2 of [3].

In chapter 6, we will use a lemma from [8] that rewrites the left hand side of (5.1) as a condition on the coefficients of vλ in |ui.

Lemma 5.6. A vector |ui =P

λc(λ)vλ satisfies the bilinear identity (5.1) if and only if ∞

X

j=0

(−1)jc(γ γj)c(δ ⊕ γj) = 0

for all Maya diagrams γ and δ of charge respectively 1 and −1. Here γ γj is defined

as ψ_γ∗

j · γ with always a positive sign and δ ⊕ γj as ψγj · δ with its usual sign.

Proof. This follows from the fact that

{α ⊗ β : α, β are Maya diagrams with charge resp. 1 and -1}

forms a basis for F(1)_⊗F(−1)_{, so that every coefficient of α⊗β in (5.1) should vanish.}

The identity (5.1) is clearly equivalent to the identity

[k−1](ψ(k) |ui ⊗ ψ∗(k) |ui) = 0, (5.2) where ψ(k) and ψ∗(k) are the generating series of the fermions

ψ(k) = X i∈Z+12 ψiki− 1 2 and ψ∗(k) = X i∈Z+12 ψ∗_ik−i−12

and [k−1] means the coefficient of k−1 in the expression that follows. Now using the boson fermion correspondence, it is possible to ’translate’ this to a set of differential equations in the Bosonic space. This set of equations is exactly the KP hierarchy.

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5.3 Operators ψ(k) and ψ

∗

(k) in Bosonic space

From the equations (4.1) we immediately get the commution relations of ψ(k) and ψ∗(k) with Hn:

[Hn, ψ(k)] = knψ(k) and [Hn, ψ∗(k)] = −knψ∗(k). (5.3)

Now defining the vertex operators Γ(k) and Γ∗(k) as the corresponding operators on B by

Γ(k) = σψ(k)σ−1 Γ∗(k) = σψ∗(k)σ−1,

we conclude from equations (5.3) and theorem 4.12 that the following relations hold between Γ(k) and the operators xn and _∂x∂_n

∂ ∂xn , Γ(k) = knΓ(k) and [xn, Γ(k)] = k−n n Γ(k) (5.4) and similar equations for Γ∗(k). These commutation relations are sufficient to determine the operators Γ(k) and Γ∗(k):

Theorem 5.7. Restricted to the space B(m)_{, the operators Γ(k) and Γ}∗_{(k) have the}

following form: Γ(k; x) = kmz exp X i≥1 kixi ! exp −X i≥1 k−i i ∂ ∂xi ! Γ∗(k; x) = k−mz−1exp −X i≥1 kixi ! exp X i≥1 k−i i ∂ ∂xi ! .

We write Γ(k; x) to be able to distinguish polynomials in different variables: this will be useful later on.

Proof. We restrict our attention to the operator Γ(k). The proof for Γ∗(k) is similar. Because the coefficient of kl_{in Γ(k) corresponds to the action of σψ}

l+1₂σ

−1_{, this coefficient}

is a linear operator from B(m) _{to B}(m+1)_{. So we can write this coefficient as a sum of}

operators, each of which is of the form z · xi1. . . xik

∂ ∂x_j1 . . .

∂ ∂x_jl.

Now we introduce the operator Tk by (Tkf )(x1, x2, . . . ) = f (x1 + k−1, x2 + k

−2

2 , x3 + k−3

3 , . . . ). Note that Tk = exp

P i≥1 k−i i ∂ ∂xi

(this is a fancy way of writing Taylor’s theorem). Using the second equation in (5.4), the commutation relation

[xj, Tk] = −

k−j j Tk

and the simple relation [A, BC] = [A, B]C + B[A, C], it is now a straightforward calcu-lation to check that

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This holds for all j ∈ N, so the operator Γ(k)Tk does not have a differential part. So we

can write Γ(k) = c(k) · z · f (k, k−1; x1, x2, . . . )T_k−1. Similarly, it is easy to show that

" ∂ ∂xj , exp −X i≥1 kixi ! Γ(k) # = 0.

It follows that f (k, k−1; x1, x2, . . . ) = exp

P i≥1k i_x i, so that Γ(k) = c(k) · z · exp X i≥1 kixi ! exp −X i≥1 k−i i ∂ ∂xi ! .

To determine c(k), we look at the action of Γ(k) on the polynomial zm. Taking the coefficient of km_{, this corresponds to acting with ψ}

m+1₂ on |mi, which results in |m + 1i,

corresponding to zm+1_{. So the coefficient of k}m _{should be multiplication by z. The}

only term where multiplication by z appears is the term c(k) · z · 1 · 1 (we pick 1 in the expansion of both exponentials). It follows that c(k) = km and this completes the proof for Γ(k).

5.4 KP hierarchy

The boson fermion correspondence naturally extends to an isomorphism F(0)_{⊗ F}(0) ∼₌

C[x01, x02, . . . ; x001, x002, . . . ]. Combining this with theorem 5.7, we find that (5.2) is

equiva-lent to

[k−1]Γ(k; x0)Γ∗(k; x00)τ (x0)τ (x00) = 0,

because τ (x0) and Γ∗(k; x00) commute. Subsituting the equations for Γ(k) and Γ∗(k) from theorem 5.7 for m = 0 (because τ is an element of B(0)_{), we find that this is equivalent}

to [k−1] exp X i≥1 ki(x0_i− x00_i) ! exp X i≥1 k−i i ∂ ∂x00_i − ∂ ∂x0_i ! τ (x0)τ (x00) = 0. Now introducing the variables x and y by

x0 = x − y and x00 = x + y,

we see that the orbit Ω in the Bososnic space B can be characterized as follows. Theorem 5.8. A function τ ∈ C[x1, x2, . . . ] is an element of Ω if and only if

[k−1] exp −X i≥1 kiyi ! exp X i≥1 k−i i ∂ ∂yi ! τ (x − y)τ (x + y) = 0

This can be further rewritten as a set of Hirota bilinear equations, which gives an infinite set of differential equations for the function τ . The following theorem can be found for example in [4] or in [3].

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Theorem 5.9. A function τ ∈ C[x1, x2, . . . ] is an element of Ω if and only if the

coefficients of all monomials in yr in the expression ∞ X j=0 Sj(−2y)Sj+1( ˜D) exp X r≥1 yrDr ! τ · τ (5.5)

are equal to zero.

Here Sj are Schur functions, defined as

exp ∞ X k=1 xkzk ! = ∞ X j=0 Sj(x)zj

and for a polynomial P , the expression P (D)f · g is defined as P ∂ ∂u1 , ∂ ∂u2 , . . . f (x1− u1, x2− u2, . . . )g(x1+ u1, x2+ u2, . . . ) u=0. (5.6)

We call the set of differential equations that is obtained in this way the KP hierarchy. Example 5.10. To obtain one of the equations of the KP hierarchy, we calculate the coefficient of y3 in (5.5). Since only the Schur polynomial S3 contains an y3-term, we

only have to consider (5.5) for j = 0 and j = 3. The first term gives 1 · D1· y3D3, since

S1( ˜D) = D1. The second term gives

−2y3· D4 1 24 + D2 1D2 4 + D2 2 8 + D1D3 3 + D4 4 · 1.

Combining this with the remark that we can drop the odd terms in the polynomial P in (5.6), because these terms give trivial equations, we obtain the well-known KP equation, which is the simplest equation in the KP hierarchy:

(D4₁+ 3D₂2− 4D1D3)τ · τ = 0.

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6 Relation between Hurwitz

numbers and KP hierarchy

One can define general Hurwitz numbers roughly as follows: given a permutation µ and cycle types C1, . . . , Ck, in how many ways can we write µ as a product of permutations

σ1◦ · · · ◦ σk such that σi ∈ Ci for all i ∈ {1, . . . , k}? However, these general Hurwitz

number are very complicated and don’t seem to have any nice properties. That’s why a lot of other kind of Hurwitz numbers exist that are special cases of general Hurwitz numbers, e.g. simple Hurwitz numbers, double Hurwitz numbers and Hurwitz numbers with completed cycles. All these special Hurwitz numbers do have nice properties: their generating series are solutions to integrable hierarchies. Our references for this chapter are [10, 8, 5].

6.1 Two definitions of Hurwitz numbers

Hurwitz numbers can be viewed from both a combinatorial and a topological viewpoint. We will introduce both viewpoints in this section, beginning with the combinatorial one. Definition 6.1. Let n ∈ N and µ ` n. Then the simple Hurwitz number h◦m;µ is defined

as

h◦_m;µ = 1

n!|{(η1, . . . , ηm) : η1 ◦ · · · ◦ ηm ∈ Cµ and ηi ∈ C2 for all i}|.

The connected simple Hurwitz number is defined in the same way, except that we only count the tuples (η1, . . . , ηm) such that the group hη1, . . . , ηmi they generate acts

transi-tively on the set {1, . . . , n}.

For the other definition of Hurwitz numbers, we will need the definition of a ramified covering. Following [6], we will first explain what an unramified covering is.

Definition 6.2. Let X and Y be path connected topological spaces and let f : X → Y be a continuous map. Then the triple (X, Y, f ) is called an unramified covering if for every point y ∈ Y there exists an open neighbourhood U of Y such that f−1(U ) is a disjoint union of sets in X, each of which is mapped homeomorphically to U by f . Remark 6.3. Unramified coverings (X, Y, f ) appear in the study of the fundamental group of a space Y and in this setting, they are called coverings. This is what we will do too. So when we write ’covering’ without an adjective, what we really mean is ’unramified covering’.

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Because of the definition of a covering, the set An ⊂ X of points that have n preimages

is open. Hence it has to be the whole space X, because X is connected (because it is path connected). Hence the following definition makes sense:

Definition 6.4. The degree of a covering (X, Y, f ) is defined as the number of preimages of a point in Y .

Example 6.5. If X = Y = D2_{\{0} ⊂ C, n ∈ N and f : X → Y : z 7→ z}n_{, then (X, Y, f )}

is an unramified covering of degree n.

We can make the covering in the previous example into an ramified covering by adding the point 0 to both X and Y . In this case the point 0 ∈ X is called a critical point of order n and the point 0 ∈ Y a critical value or ramification point or branch point of the ramified covering. If a branch point y ∈ Y has k corresponding critical points with orders respectively µ1, µ2, . . . , µk, then we say that y is a branch point of type

µ = (µ1, µ2, . . . , µk).

Definition 6.6. We define an equivalence relation on the set of ramified coverings: two ramified coverings (X, Y, f ) and (X, Y, g) are isomorphic if there exist homeomorphisms φ : X → X and ψ : Y → Y such that ψf = gφ.

Definition 6.7. A deck transformation of a ramified coverings (X, Y, f ) is a homeomor-phism φ : X → X such that f = f ◦ φ. These transformations form a group, called the deck transformation group of the covering (X, Y, f )

Definition 6.8. A simple ramification point is a ramification point of type (2, 1, 1 . . . , 1). Now we can give the other definition of Hurwitz numbers. Again, we will only give the definition for simple Hurwitz numbers

Definition 6.9. The simple Hurwitz number h◦_m;µ is the number of equivalence classes of ramified coverings from a surface of genus g to a sphere with one point of ramification given by µ and m more simple branch points, where each equivalence class is weighted with the inverse of the order of the corresponding deck transformation group. The genus g is given by the Riemann-Hurwitz formula.

The equivalence of the two definitions of Hurwitz numbers can be obtained by looking at the action of the monodromy group of a covering, see for example [6] or [5].

6.2 Hurwitz numbers with completed cycles

For η ∈ Sn, we define a function fη on all partitions λ as follows: for |λ| ≥ n, we set

fη(λ) = |λ| |η| |Cη| χλ_η∗ dim(λ)

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and for |λ| < n we set fη(λ) = 0. Here η∗ is the cycle type (η, 1, 1, . . . , 1) in S|λ|.

Furthermore, we define the shifted symmetric powersum pk as follows:

pk(λ) = ∞ X i=1 " λi− i + 1 2 k − −i +1 2 k# .

Furthermore, for µ = [µ1, . . . , µl] a partition, we define

pµ(λ) = l

Y

i=1

pi(λ).

It is known (see [10]) that the functions fµ are shifted symmetric and that the map

φ :L

n≥0ZCSn→ Λ ∗

: Cµ7→ fµ (extended linearly) is an isomorphism. This guarantees

that the following definition makes sense.

Definition 6.10. The completed µ-conjugacy class Cµ is defined as Cµ= φ−1(pµ). An

important case appears when µ = (r) and in this case, we write C(r) = (r) and we call

(r) the completed r-cycle.

Now we have defined completed cycles, we can look at the operators Qr that

cor-respond to multiplication by the completed cycle (r). What makes this operators so special is that they are infinitesimal transformations of the KP hierarchy. We prove this in the following theorem.

Theorem 6.11. If τ satisfies the KP hierarchy, then τ + Qiτ also satisfies the KP

hierarchy for all i up to first order in .

Proof. From the definitions it easily follows that that σ−1Qiσ = 1_i!

P

m∈Z0miEmm def

= Fi,

because the eigenvalue of Fi corresponding to vλ is exactly pi(λ). Suppose that σ(|ui)

is a solution and write |ui =P c0(α) |αi. Then by lemma 5.6 we have ∞

X

j=1

(−1)jc0(γ γj)c0(δ ⊕ γj) = 0 (6.1)

for all Maya diagrams γ, δ with charge respectively 1 and −1. Now let γ and δ be two such Maya diagrams. Then we have to show that

∞

X

j=1

(−1)j[c0(γ γj)ci(δ ⊕ γj) + ci(γ γj)c0(δ ⊕ γj)] = 0,

where the coefficients ci are defined by Fi|ui = P ci(α) |αi. So if we write Fi|αi =

λi(α) |αi, then we have to show that ∞

X

j=1

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So it is sufficient to show that λi(γ γj) + λi(δ ⊕ γj) is a constant (independent of j),

because in that case, we can take it out of the sum and then (6.2) follows from (6.1). Without loss of generality, we may assume that γj > 0. Then λi(γ γj) + λi(δ ⊕ γj) =

(λi(γ) −_i!1γji) + (λi(δ) + 1_i!γji) = λi(γ) + λi(δ). So it is independent of j.

This theorem implies that Hurwitz numbers with completed cycles, which are defined in [10], have nice properties. Specifically, it implies that

[β] X j ψjeβQi|ui ⊗ ψj∗e βQi_|ui ! = 0 (6.3) for all solutions |ui of the KP hierarchy. Furthermore, it is proved in [10] that the exponential generating series Hi for Hurwitz numbers with completed cycles satisfies

the generalized cut-and-join equations ∂Hi

∂β = QiHi.

Together with the initial conditions, this completely determines the function Hi:

Hi = eβQiep1.

We know that eβQi ∈ GL

∞ (see above) and ep1 ∈ GL∞ (trivially). So this implies that

Hi is a solution to the KP hierarchy.

But what is really so special about completed cycles? Is it not possible that the operator corresponding to multiplication by another linear combination of cycles is also an infinitesimal transformation of the KP hierarchy? We will illustrate that at least the completed 3-cycle is indeed the only linear combination of cycles that satisfies this property (negelecting the cycles (2) and (1), because these cycles are both completed cycles and thus can always be added without destroying symmetry).

Remark 6.12. We write W (µ) for the operator corresponding to multiplication by µ. Example 6.13. Let

µ = α[1] + β[2] + γ[11] + δ[3] + [21] + ζ[111].

In this example, we proof that if the operator W (µ) is an infinitesimal transformation of the KP hierarchy, then γ = δ and = ζ = 0. We use the table for values of φR(∆)

from [7]. Suppose τ is a solution to the KP hierarchy and write τ = X

λ

c(λ)sλ

as a linear combination of Schur polynomials. Then τ + Qrτ is a solution up to first

order in if and only if

∞

X

j=1

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where vµ def

= αv[1]+ · · · + ζv[111]. Writing out this condition for the Pl¨ucker relation

c(2, 2)c(0) − c(2, 1)c(1) + c(2)c(11) = 0, one obtains γ + ζ = δ, because the equation

a1· c(2, 2)c(0) − a2· c(2, 1)c(1) + a3· c(2)c(11) = 0

implies that a1 = a2 = a3. Likewise, for the Pl¨ucker relation c(3, 2)c(0) − c(3, 1)c(1) +

c(3)c(11) = 0, one obtains the extra condition that = −ζ and for the Pl¨ucker relation c(2, 2, 1)c(0) − c(2, 1, 1)c(1) + c(2)c(1, 1, 1) = 0, one obtains the extra condition that = ζ. So it follows that γ = δ and = ζ = 0.

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7 Conclusion

After two introductory chapters about the irreducible representations of the symmetric group and finite dimensional Pl¨ucker relations, we introduced the semi-infinite wedge space Λ∞2 V and explained the boson fermion correspondence, which is an isomorphism

σ : Λ∞2 V → C[z, z−1; x₁, x₂, . . . ].

Then we gave a representation of the Lie group GL∞ consisting of Z × Z matrices

with a finite number of non-zero entries and we looked at the orbit Ω = GL∞· |vaci.

Subsequently, we proved that the image σ(GL∞|vaci) ⊂ C[x1, x2, . . . ] is exactly the set

of functions that satisfy an infinite hierarchy of differential equations, which we call the KP hierarchy. The space of solutions is exactly the set of decomposable vectors and thus can be thought of geometrically as an infinite dimensional Grassmanian.

We ended with a chapter about Hurwitz numbers and completed cyles. We showed that if τ is a solution to the KP hierarchy, then τ + Qiτ is also a solution up to first

order in , where Qi is the operator that corresponds to multiplication with a completed

cycle. In other words Qi is an infinitesimal symmetry of the KP hierarchy. This theorem

implies that Hurwitz numbers with completed cycles have nice properties. Now because completed cycles are certain linear combinations of conjugacy classes, a natural question is if this is the only linear combination that satisfies this property. This would say something about the integrability of the generating function of different kind of Hurwitz numbers. We did not prove this, but we gave an example how to calculate why it is true for the case of the completed 3-cycle.

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Populaire samenvatting

In deze bachelorscriptie wordt het verband tussen de KP-hi¨erarchie en Hurwitzgetallen bestudeerd. Dit zijn twee wiskundige objecten die voorkomen in de mathematische fys-ica. Dit is het vakgebied dat de wiskunde bestudeert die voortkomt uit natuurkundige vraagstukken en nodig is voor het oplossen hiervan. Hieronder leggen we in simpele bewoordingen uit wat de KP-hi¨erarchie en Hurwitzgetallen zijn. Hoewel er op het eerste gezicht geen verband lijkt te zijn tussen deze twee onderwerpen, blijkt er toch een op-vallende connectie zijn, die we aan het einde van deze samenvatting uit zullen leggen.

De KP-hi¨erarchie is een bepaald oneindig stelsel differentiaalvergelijkingen binnen de wiskunde. Dat betekent dat het oneindig veel differentiaalvergelijkingen zijn en dat een functie een oplossing is voor de hi¨erarchie alleen als het een oplossing is voor al deze vergelijkingen. Wat is een differentiaalvergelijking? Dit is een vergelijking waarin een functie en zijn afgeleides voorkomen. Er zijn talloze voorbeelden in de natuurkunde, waarvan een belangrijk voorbeeld de derde wet van Newton is. Stel dat we de kracht op een deeltje weten als functie van de positie van het deeltje, dan volgt dat

F (x) = m¨x,

waar x = x(t) de positie van het deeltje is als positie van de tijd t en ¨x staat voor de tweede afgeleide van x naar t (dit is de versnelling). Dit geeft een differentiaalver-gelijking voor x en het oplossen van zo een differentiaalverdifferentiaalver-gelijking is het vinden van alle mogelijke functies x(t) die aan deze vergelijking voldoen. In specifieke gevallen zijn differentiaalvergelijkingen exact op te lossen, maar in het algemeen is dit heel lastig.

Een manier om een differentiaalvergelijking proberen op te lossen is het vinden van symmetrie¨en voor de vergelijking. Een symmetrie genereert uit een bepaalde oplossing een nieuwe oplossing. Je kan je de oplossingen van een differentiaalvergelijking bijvoor-beeld voorstellen als punten op een bol. Als je begint met een bepaald punt op de bol en vervolgens de bol roteert, dan krijg je een nieuw punt op de bol. Als je alle rotaties van de bol kunt beschrijven en je weet minstens een oplossing, dan volgen alle andere oplossingen hieruit. In het geval van de KP-hi¨erarchie blijkt dit object niet een bol te zijn, maar een Grassmanniaan.

Om uit te leggen wat Hurwitzgetallen zijn, moeten we eerst uitleggen wat een permu-tatie is.

Definitie. Een permutatie is een manier om de getallen 1, . . . , n door elkaar te husselen. Een voorbeeld voor n = 6 is 123456 → 412653. Een transpositie is een permutatie waarbij slechts twee getallen met elkaar verwisseld worden.

Simpele Hurwitzgetallen geven voor een gegeven permutatie µ en getal m aan op hoeveel manieren je µ kan realiseren als het m keer achter elkaar uitvoeren van transposities.

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Voorbeeld. We bekijken de permutatie µ : 1234 → 2314 en berekenen het Hurwitzgetal voor m = 2. Omdat we maar twee transposities mogen doen, moet het getal 4 wel op zijn plaats blijven. Als we eerst 1 en 2 verwisselen dan moeten we daarna 1 en 3 verwisselen, als we eerst 1 en 3 verwisselen dan moeten we daarna 2 en 3 verwisselen en als we eerst 2 en 3 verwisselen dan moeten we daarna 1 en 2 verwisselen. Dus het bijbehorende Hurwitzgetal is 3.

Simpele Hurwitzgetallen kunnen bij elkaar worden gevoegd in een reeks. Het blijkt dat deze reeks een oplossing is van de KP-hiërarchie. Naast simpele Hurwitzgetallen bestaan er ook andere soorten Hurwitzgetallen. In het algemeen hoeven de bijbehorende reeksen niet een oplossing te zijn van de KP-hiërarchie. Aan het eind van deze scriptie worden ’Hurwitz numbers with completed cycles’ ge¨ıntroduceerd en wordt bewezen dat de reeks die hoort bij dit type Hurwitzgetallen wel een oplossing is van de KP-hiërarchie.

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An introduction to the KP hierarchy and Hurwitz numbers with completed cycles