Karlstads universitet 651 88 Karlstad Tfn 054-700 10 00 Fax 054-700 14 60 Faculty of Technology and Science
Department of Physics
Jörgen Gladh
Physics Master Thesis
Date/Term: 2008-08-28 Supervisor: Prof Jürgen Fuchs Examiner: Prof Stephen Hwang Serial Number: 2008:4
tations of the Lie algebras su(2) and sl(2), and the tensor products of the representations.
In the first case I looked at a tensor product involving a representa- tion similar to one that appears in an article by A. van Tonder. This representation and tensor product was investigated mainly to get a good comprehension in the subject and to understand some of the problems that can arise.
In the other case, which is the main problem in this thesis, I looked at a tensor product and representations that appears in an article by M. R.
Gaberdiel. Here we deal with a tensor product of representations of su(2) with a specific value for the level at k = −4/3 and a specific eigenvalue of the Casimir operator at −2/9. This was done in the frame of finite dimensional Lie algebra and affine Lie algebra and not in the case of fusion rules as in the article by M. R. Gaberdiel.
In both cases some of the calculations where done from in situ and the investigation of the representations behaviour due to the step operators, theirs eigenvalue and theirs weight system.
Results and conclusions of the investigations are discussed in the last part of this thesis.
First of all I want to thank my supervisor Professor J¨urgen Fuchs for his advice and for his professional way to guide me in this subject and writing this paper, secondly I want to thank my fellow students for all support through the years.
In addition to the mentioned I want to thank my family, for their patience that they have had with me during this time, especially my beautiful wife Ann-Kristin and my lovely daughters Rebecca and Rene´e.
To my family.
1 Introduction 1
1.1 About this work . . . 1
1.2 Why should we investigate this kind of representations and tensor products . . . 1
2 Some History and decisive breakthroughs 2 2.1 Lie Algebras and Representations . . . 2
3 Methods and Calculation Tools 4 3.1 Lie Algebras . . . 4
3.2 Group Theory . . . 4
3.2.1 Vector space . . . 5
3.2.2 Generators and step operators . . . 6
3.2.3 The Casimir operator . . . 7
3.3 Representations in Physics . . . 8
3.3.1 The adjoint representation . . . 8
3.3.2 Constructing representations . . . 9
3.3.3 Schur’s Lemma . . . 10
3.4 Tensor products . . . 10
3.4.1 The weight system . . . 11
3.4.2 The highest weight . . . 12
4 Conformal field theory, WZW theory and Fusion rules 13 4.1 Conformal field theory . . . 13
4.1.1 2-D Conformal field theory . . . 13
4.2 WZW theories . . . 14
4.3 Fusion rules . . . 15
5 The Tensor product between one finite and one infinite dimen- sional representation. 17 5.1 The general properties . . . 17
5.2 One use of the tensor product vm+⊗ vm−. . . 23
6 The tensor product of two infinite dimensional representations. 24 6.1 Dj± representation . . . 24
6.2 The E representation . . . 26
6.3 The tensor product . . . 27
6.3.1 The sub-representation D0 . . . 27
6.3.2 The sub-representation E . . . 29
7 Summary and Conclusions 32
B The Casimir operator for two infinite tensor representation 37
1 Introduction
1.1 About this work
In this thesis we will look at two kinds of tensor products. First we will investigate the tensor product between a finite and an infinite representation, and there after a tensor product between two infinite representations.
In the first case we have a tensor product between a finite and an infinite representation and we will look at this tensor product mostly from a mathemat- ical point of view. This is because we have to see the behaviour of the tensor product due to the infinite representation. We start with two basis, analyse their behaviour, form a tensor product and from that we work our way up to a full representation in the frame of su(2) Lie algebra. This investigation is mainly to get a good comprehension in the subject and to see what kinds of problem that can arise. Furthermore we will do some comment on article [12], in which a similar representation is involved and have been used.
In the second case we have two infinite representations, we do the same proce- dure as in the first case and make a tensor product of them. From the results in the first case we expand our representation “still in the frame of the Lie algebra su(2)” and look at the results not only from the mathematical point of view, but a more important, in the perspective of physics. This kind of representations is also in frame of current research and we will compare our result to an article [7]
written by M. R. Gaberdiel. Furthermore we will briefly discuss our result and make some conclusions about it.
1.2 Why should we investigate this kind of representa- tions and tensor products
From the history of mankind people have always been interested in symmetries, both for beauty and admiration. It has also been a inspiration to figure out their properties, and how to calculate them. Nowadays calculating symmetries is a science of many areas.
The question in the heading can be answered in many different ways, but if we look at the topic of physics, many areas like theoretical physics, representation and Lie algebra is the main tool.
For example in Conformal Field Theory (CFT) these kinds of tensor products are used in different kinds of calculations, e.g. fusion rules. Fusion rules are one of the issues in [7] and from which the main problem of this thesis is taken.
These kinds of problems also arise in elementary particle physics and in string theory. To overcome these problems we have to deal with representations and Lie algebras.
From the Wess-Zumino-Witten (WZW) theories in CFT we know e.g. how to treat the affine Lie algebra g when the level is k ∈ Z, but we also would like to
know how to treat the algebra when the level is k /∈ Z. This is the motivation for main article [7] to get past the difficulties and to find a way to treat the algebra when k /∈ Z. In article [7] the level is k = −4/3.
2 Some History and decisive breakthroughs
To get a little perspective due to this area, it is a good idea to look at some of the historical findings and ideas of Lie algebras and representation theories.
2.1 Lie Algebras and Representations
In the end of the nineteenth century the Norwegian mathematician Marius Sophus Lie (1842–1899) worked on continuous transformation groups and symmetries. In his honour, Lie algebras are named after him and his pioneering in the study of these mathematical objects.
He worked in a joint project with Felix Klein (1849–1925) of the classifi- cation of all finite-dimensional groups acting on the plane. Lie struggled with this task to a generally non-linear nature of such group actions, and was able to solve this problem by remarking that a transformation group can be locally reconstructed from its corresponding “infinitesimal generators”, that is to say vector fields corresponding to various 1-parameter subgroups. In terms of this geometric correspondence, the group composition operation manifests itself as the bracket of vector fields, and this is very much a linear operation. Thus the task of classifying group actions in the plane became the task of classifying all finite-dimensional Lie algebras of planar vector field; a project that Lie brought to a successful conclusion [1].
This led to combining the transformations in a way that Lie called an in- finitesimal group, but which is not a group with our definition, rather what is today called a Lie algebra [1]. It was during the winter of 1873–74 that Lie began to develop systematically what became his theory of continuous transformation groups, later called Lie groups, leaving behind his original intention of examin- ing partial differential equations. Later Killing was to examine the Lie algebras associated with Lie groups.
Finally a short outline of history and the most important ideas in the subject of representation theory and Lie algebra [9].
• 1873: Lie groups. Sophus Lie (1842–1899)
• 1888: Classification of Lie algebras. Wilhelm Killing (1847–1923)
• 1896: Representations of finite groups, characters. Georg Frobenius (1849–
1917)
• 1897: Integration over compact Lie groups. Adolf Hurwitz (1859–1919)
• 1905: Schur’s lemma. Issai Schur (1875–1941)
• 1913: Highest weight representations of Lie algebras. Eli´e Cartan (1869–
1951)
• 1925: General representation theory of compact Lie groups, Weyl character formula. Hermann Weyl (1885–1955)
• 1925–6: Quantum mechanics. Werner Heisenberg (1901–1976), Erwin Schr¨od- inger (1887–1961), P. A. M. Dirac (1902–1984)
• 1926: Peter-Weyl theorem.
• 1931: Heisenberg algebra and group, Stone-von Neumann theorem. Weyl, Marshall Stone (1903–1989), John von Neumann (1903–1957)
• 1935–38: Clifford algebras and spinors. Richard Brauer (1901–1977), Weyl, Cartan
• 1951: Representations of non-compact semi-simple groups. Harish-Chandra (1923–1983)
• 1954: Borel-Weil theorem. Armand Borel (1923–), Andr´e Weil (1906–1998)
• 1957: Borel-Weil-Bott theorem. Raoul Bott (1923–)
• 1964: Metaplectic Representation. Weil
• 1968: Kac-Moody Lie algebras. Victor Kac (1943–), Robert Moody (1941–)
• 1970: Geometric quantization, method of orbits. Bertram Konstant (1928–
), Alexandre Kirillov (1941–)
• 1974: Highest weight representations of Kac-Moody algebras. Kac
As we can see the list is long over the last century. This findings gives physicist a remarkable tool to explore our world both in micro and in macro.
3 Methods and Calculation Tools
3.1 Lie Algebras
An algebra is a Lie algebra if the bilinear operator possesses two special properties [4]:
Definition 3.1.1 A Lie Algebra is an algebra such that the map [, ] : g × g → g obeys:
[x, x] = 0 ∀ x ∈ g (antisymmetry) and
[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 ∀ x, y, z ∈ g (The Jacobi identity).
The Lie bracket obeys i.e. [ζx + ςy, z] = ζ[x, z] + ς[y, z] and also the analogous relation [x, ζy + ςz]. Plus it satisfies [x, y] = −[y, x] which follows from [x, x] = 0.
Proof:
[x + y, x + y] = [x, x] + [y, y] + [x, y] + [y, x] ⇒ [x, y] = −[y, x] (1) The elements of a basis in g are called for the generators Ta of g and with a expansion coefficient fabc in their Lie brackets [Ta, Tb] = P
cfabcTc is known as the structure constant of the Lie algebra g.
If the Lie algebra g have the property [x, y] = 0 ∀ x, y ∈ g then the Lie algebra is called Abelian.
If a Lie algebra g for which each element can be written as a commutator of two elements of g is called a semisimple Lie algebra. Any Lie algebra which is semisimple is the direct sum of one or more basic constituents, the simple Lie algebras. A example is sl(2) and sl(3) are both simple Lie algebras [5].
The Lie algebras which are of the most important for applications in physics are the Abelian and the simple Lie algebras and their direct sum.
3.2 Group Theory
In the group theory a group is specified in the following way [5]:
• A group is a set G together with a product ‘’, i.e. a map from G × G to G (a ‘binary operation’) which is associative and has a unit element and inverses.
Associativity means that no bracketing is required: x(y z) = xy z = (xy)z for all x, y, z ∈ G. The unit, or identity, element e ∈ G is an element satisfying e x = x = x e x ∈ G; the existence of inverses means that any element x of G there is an inverse element x−1 in G satisfying x x−1 = e = x−1 x.
If the product is commutative i.e. x y = y x ∀x, y ∈ G, then the group is abelian. If there is no unit element or not all elements possess an inverse, but the associativity axiom is fulfilled, then G is called for a semigroup.
3.2.1 Vector space
A vector space V over some field F is a set endowed with a addition and a multi- plication by elements of F , such that distributivity laws and other compatibility properties of the two operations are valid.
The elements of V are called for vectors and the elements of underlying field F are called scalars. The defining properties between V and F is (ζ+η)·v = ζ·v+η·v, ζ · (η · v) = (ζ · η)v, 0 · v = 0, 1 · v = 1 etc, for all v, w ∈ V and for all ζ, η ∈ F .
Given any subset M of a vector space V , the set of all linear combinations of elements of M , is called the span of M and is denoted: spanF(M ) ≡ spanF{v|v ∈ M }, and is a linear subspace or sub-vector space of V .
A basis B of a vector space V is a subset which has the properties that is linearly independent, and basis elements spans spanF{v|v ∈ B}. The number of basis elements is independent of the actual choice of basis B, and if this number is a finite number one denotes it by
d ≡ dimFV := |B| (2)
and it is called the dimension of the vector space V . Otherwise V is said to be infinite-dimensional.
V can be compose as a direct sum V1L V2 of two vector spaces V1 and V2 over the same field F is the set of all formal sums v1⊕ v2 of elements of V1 and V2. The scalar multiplication and addition is defined by ζ(v1⊕ v2) = (ζv1) ⊕ (ζv2) and (v1 ⊕ v2) + (w1 ⊕ w2) = (v1+ w1) ⊕ (v2 + w2). For example, if j = 1, 2 the set B = {v(i)[j]}, with i = 1, 2, . . . , dimVi is a basis of Vi, then the set
B := {v[j](i)⊕ 0|i = 1, 2 . . . , dimV1} ∪ {0 ⊕ v(i)[j]|i = 1, 2, . . . , dimV2} (3) is a basis of the direct sum V1L V2.
Now if we consider a Cartesian product or Kronecker product V1 × V2 of the two vector spaces V1 and V2. This is the set of all ordered pairs (v1; v2) of the elements v1 ∈ V1 and v2 ∈ V21. If V1 and V2 are vector spaces over the same field F , then one obtains a new vector space. The set obtained in this way of V1× V2 is called the tensor product V1N V2 of V1 and V2. The elements of they tensor product that is of class of the pair (v1; v2) is denoted by v1 ⊗ v2, and the scalar multiplication then acts as ζ(v1⊗ v2) = (ζv1; v2) = (v1; ζv2).
Then the given basis Bj = {v(i)[j]} of Vj for j = 1, 2 is
B = {v(i)[1]⊗ v(j)[2]|i = 1, 2 . . . , dimV1, j = 1, 2 . . . , dimV2} (4) is a the basis for the tensor product V1N V2. The dimensions of the tensor product V1N V2 is the product of d1· d2 [5].
1Note that the Cartesian product is itself not a vector space.
3.2.2 Generators and step operators
We start this section by giving a definition of a specific basis of the semisimple Lie algebra
Definition 3.2.1 A basis basis of the semisimple Lie algebra g of the form B = {Hi|i = 1, . . . , r} ∪ {Eα|α ∈ Φ} (5) with Hi and Eα obeying
[Hi, Hj] = 0 for i, j = 1, . . . , r (6) and
[Hi, Eα] = αiEα for i, j = 1, . . . , r (7) is called a Cartan-Weyl basis of g.
The generators Eα and E−α are called the step operators or ladder operators associated to the root α.
In quantum mechanics and the standard treatment of angular momentum we have the relation:
[Li, Lj] = iijkLk i, j, k = 1, 2, 3
where ijk is the totally antisymmetric tensor in three dimensions also known as the Levi-Civita symbol.
Now the next step is to consider the linear combinations of the first two generators, and we get:
L± = L1± iL2
L3, L±
= ±2L±. (8)
When we admit a complex linear combinations of the generator L2, the conse- quence are that the operators L± are not hermitian any more which gives
L+†
= L−. (9)
L±is the step operator for the quantum mechanics as E±α is for the Cartan-Weyl basis in Lie algebra.
Moreover the quadratic Casamir operator2 ~L2then gives the following relation h~L2, L3i
= 0
L+, L−
= L3. (10)
2See section 3.2.3 or [4, 5].
Another generator of importants in non-simple affine Lie algebra is the central generator
K := ζ
r
X
i=0
a∨iHi (11)
where ζ is an any constant, a∨i is the dual Coxeter label of g and Hi is the canonical Chevalley generator of g. The normalized eigenvalue of K, k is called for the level of the representation RΛ, and can be defined as
k∨ := k · 2
(¯θ, ¯θ), (12)
where ¯θ is a positive root.
3.2.3 The Casimir operator
Let us first look at theory of angular momentum in quantum mechanics, the square of the operator L is
~L2 ≡ L21 + L22+ L23 = 1
4L23+1
2 L+L−+ L−L+
(13) and the angular momentum vector commutes with all generators Li. Now ~L2 acts as the multiple 14Λ(Λ+2) of the identity on all vectors of any highest weight mod- ule3 VΛ. As a consequence this element is useful when one wants to decompose modules into irreducible modules. As soon two eigenvectors of ~L2 in a module have different eigenvalues, one knows that this module is reducible. A summa- rizing, the finite- dimensional irreducible representations of sl(2) are completely characterized by their spin eigenvalue j(j + 1) = 14Λ(Λ + 2).
Now a generalization of the operator ~L2 to any arbitrary finite-dimensional simple Lie algebra by taking equation (13) to the explicit form of the Killing form4 of sl(2), leads to
C2 :=
d
X
a,b=1
κabTaTb (14)
as a element of U(g), which is known as the quadratic Casimir operator.
The quadratic Casimir operator C2 commutes with all elements of g and Schur’s Lemma5 therefore implies that C2 acts as a constant on each irreducible modules. In particular,
C2v = CΛ· v ∀ v ∈ VΛ (15)
3To clarify the concept of the highest weight module, see section 3.4.1 or more in detail see [4, 5, 6, 8].
4See equation 24 in section 3.3.1.
5See section 3.3.3 or [4, 5].
for irreducible highest weight modules. To compute the eigenvalue CΛ we use the relation [Eα, E−α] = Hα ≡ (αv, H) and definition ρ = 12P
α>0α of the Weyl vector to write C2 as
C2 =X
i,j
GijHiHj +X
α
nαEαE−α = (H, H) +X
α>0
nαE−αEα+ 2(ρ, H), (16)
where Gij is matrix [5].
3.3 Representations in Physics
In Lie algebras as defined in section 3.1 the contact in to physical systems these objects have to act or operate in a more concrete manner; on some space, a space of physical states. To be more precise, by acting on some space V we mean that to any element x of g there is a associated map:
R(x) : V → V (17)
such that for any elements e.g. x, y of g the commutator of R(x) and R(y) exists and reproduces Lie bracket of x and y in g [5]:
R(x) ◦ R(y) − R(y) ◦ R(x) = R ([x, y]) . (18) Here the ’◦’ denotes the composition of maps and this composition endows the set of maps from V to V .
If one is given a set V and mappings R(x) which obey the relation in equa- tion(18), one says that V is a representation space of g and R calls a representa- tion of g.
However in a more particular interesting case is when V is a vector space and the R(x) are a linear mapping. A situation that arises natural in quantum mechanics where the space of physical states is a Hilbert space and therefore carries a linear structure.
If the vector space V on which the representation acts is called for a repre- sentation space or a module. The dimension of a module is the same as it is for the vector space.
By examining representations we can learn more about the structure of the Lie algebra itself [5].
3.3.1 The adjoint representation
From what we know the Lie algebra g is itself a vector space and it is possible to represent g on itself. Then we have obtained new representation known as the adjoint representation. The adjoint representation exists for any Lie algebra independent of its dimension or structure and is defined by
Rad : g→ gl(g), x 7→ adx, (19)
with the map adx is defined as
y 7→ adx := [x, y]. (20)
The dimension of the adjoint representation is equal to the dimension of the algebra. In the representation Rad the Lie brackets are the commutators [adx, ady] ≡ adxady− adyadx, and one has
([adx, ady]) (z) = [x, [y, z]] − [y, [x, z]], (21) so that the antisymmetry and the Jacobi identity for the Lie brackets of g imply that
([adx, ady]) (z) = [[x, y], z] = ad[x,y](z). (22) In terms of generators the adjoint representation is given by the structure constants
adTa(Tb) = [Ta, Tb] =X
c
fabcTc. (23)
The so-called Killing form is defined by the trace of the adjoint representation, i.e.
κ(x, y) := tr(adx◦ ady), (24) where ‘◦’ denotes the composition of maps and ‘tr’ is the trace of linear maps.
Further important properties of the Killing form are its invariance
κ([x, y], z) = κ(x, [y, z]) (25) which follows with help of equation (22) which gives ad[x,y] = adx◦ ady− ady◦ adx [5].
3.3.2 Constructing representations
Considering one representation of g, then one can construct more representation using the machinery from linear algebra. Assume that we are given a matrix representation R on some vector space V . Then we can define a new set of matrices by
R+(x) := −(R(x))t ∀ x ∈ g, (26)
where ‘t’ stands for the transposition of the matrices. The representation R+ is called the representation conjugate to R. The vector space on which this representation acts has the same dimensions as V . Because this is a transposed matrices they are seen as a dual vector space V∗.
Given two modules V and W , one can represent g on the vector space direct sum V L W , namely by
RV RW : (RV RW)(x)(v ⊕ w) := RV(x)v + RW(x)w, (27)
and on the tensor product V N W by
RV RW : (RV RW)(x)(v ⊗w) := (RV(x)v)⊗w +v ⊗(RW(x)w). (28) These representations are called the direct sum and the tensor product of the representations RV and RW, more about this in section 3.4.
3.3.3 Schur’s Lemma
To get a overview how all possible representations up to isomorphism are re- lated one would like to identify the fundamental building blocks on which other representations can be decomposed, but which cannot be further decomposed themselves. This leads to irreducible representations or module.
Now an irreducible module of a Lie algebra is by definition a module which does not contain a non-trivial submodule, otherwise the module is reducible.
Irreducible modules are distinguished by a number of properties which are a consequence of Schur’s Lemma.
Schur’s Lemma says:
Schur’s Lemma 3.3.1 Any intertwiner between two irreducible modules V and W is an isomorphism or zero.
It is important that this assertion is valid for any Lie algebra over an arbitrary field F ; in particular it holds for both real and complex Lie algebras.
The situation of main practical interest of Schur’s Lemma are the cases where F = R or F = C, and where in addition the irreducible modules are finite- dimensional.
3.4 Tensor products
Two modules V and W of a Lie algebra g can be represented as a tensor product of a vector space V ⊗ W as:
RV ⊗ RW :
RV ⊗ RW(x)
(v ⊗ w) :=
RV(x)v
⊗ w + v ⊗
RW(x)w
. (29) This representation is called the tensor product of the g-representations RV and RW; analogously one speaks of the product module V ⊗ W of the g-modules V and W [5].
If vi ∈ W and wi ∈ W is countable bases of V and W , respectively then the basis of V ⊗ W is of the form vi⊗ wj is different from the vector space W ⊗ V . So the basis for the tensor product is
B = {vi⊗ wj|vi ∈ V, wj ∈ W }. (30)
In the special case when V = W , the vector space can be split into two invariant subalgebras
Vs:= {v ⊗ v0+ v0⊗ v|v, v0 ∈ V }, Va := {v ⊗ v0− v0⊗ v|v, v0 ∈ V } (31) of symmetric respectively antisymmetric elements.
It follows from the linearity of the tensor product that these representations are subspaces and again a g-modules and can be decomposed as a direct sum as R ⊗ R = Rs⊗ Ra, V ⊗ V = Vs⊗ Va. (32) In physics these two sub-representations of R ⊗ R are often referred to as the symmetric and antisymmetric coupling of R with itself [5].
In a more general way the tensor product can be rewrite into a direct sum RΛ⊗ RΛ0 =M
Λ00
LΛΛ0Λ00RΛ00 (33)
where RΛ00is a indecompsible representations, and LΛΛ0Λ00is called the Littlewood- Richardson coefficients. If none of the Littlewood-Richardson coefficients of a Kronecker (tensor) product is larger then unity, then the product is said to be simply reducible. The multiplicity of RΛ+Λ0 in RΛ⊗ RΛ0 is always equal to one, so instead of equation (33) one use a shorthand notation
Λ ⊗ Λ0 =M
i
LiΛi. (34)
The description of tensor products does not generalize from finite-dimensional simple to affine Lie algeras as nicely as many other aspects of the representation theory. The reason is that any non-trivial module of an affine algebra is infinite- dimensional, which in general leads to infinite multiplicities in tensor decompo- sitions. To handle such infinite multiplicities, one must keep track of the grade at which an irreducible module arises in the tensor product; at any fixed grade the multiplicity is still finite [5].
3.4.1 The weight system
In any representation R and the bracket relation [Hi, Eα] = αiEα implies that R(Hi)(R(Eα)vλ) = (αi+ λi)R(Eα)vλ or in other words, that
R(Eα)vα∝ vλ+α. (35)
This means that any weight of an module VΛ is of the form λ = Λ − β where β is the sum of the positive roots.
Now any positive root is a non-negative integral linear combination of simple roots, the same holds for β, i.e. that β = Pr
i=1niα(i) ni ∈ Z≥0. Whenever two
weights λ, µ satisfy λ − µ = β with β of this special form, one writes µ ≤ λ; this provides a partial ordering on the weight system of VΛ. The height of the root lattice element β, i.e. the numberPr
i=1ni, is called the depth of the weight λ.
So for any irreducible hight weight module VΛ there is a unique weight of the depth zero, the highest weight Λ [5].
3.4.2 The highest weight
If we consider the generators E± and the H of the representation theory A1 ∼= sl2, obeying
[H, E±] = ±2E±, [E+, E−] = H, (36) and assume that H acts diagonally on any module, H · v = λv for some complex number λ. Thus for each module R of A1 we get a decomposition of R into weight spaces R(λ) of the form
R =M
λ
R(λ), R(λ) = {v ∈ R|R(H) · v = λv, λ ∈ C}. (37)
The numbers λ appearing in equation (37) are called the weights of the module R.
Now if R is irreducible there is exactly one maximal weight, and the different weights of R form an arithmetic with difference 2
λ = −Λ, −Λ + 2, . . . , Λ − 2, Λ (38) with each weight occurring with multiplicity one. In this case Λ is called a highest weight and R the highest weight module RΛ [4].
4 Conformal field theory, WZW theory and Fu- sion rules
In this section we will briefly discuss conformal field theory (CFT), WZW theory and fusion rules. It is from this fields our main problem has its origin. In the first part we will look at conformal field theory in a more general way, and in the second part we will discuss one of the most essential parts in CFT, the Wess- Zumino-Witten (WZW) theories. In the last part we will look at fusion rules, it’s also from this part that has the most relevance to this thesis, and as we mention before one of the main topics in article [7].
4.1 Conformal field theory
A conformal field theory is a quantum field which behaves covariant under the conformal group [4]. Conformal field theories can also be regarded as a relativis- tic quantum field theory with additional space-time symmetries and conformal symmetries. The conformal transformations are defined as the transformations which preserve angles [3]. Those general coordinate transformations of space- time which preserves the angles between any two vectors change the space-time metric only by a local factor.
gµν(x) = σ(x)gµν. (39)
In quantum theories one will consider the action of the conformal transfor- mation on the operators as projective, with the commutators
[Ln, Lm] = (n − m)Ln+m+ 1
12n(n2− 1)δn+mC
[Ln, C] = 0. (40)
This algebra is called for the Virasoro algebra.
The conformal field theory is often reserved for two-dimensional theories but the dimensions can be kept arbitrary to higher dimensions. The problem with the higher dimensional conformal field theories is to actually find a solution.
Conformal transformations contain in particular scaling, so conformal field theories are scale invariant. Conversely, for sufficiently short-ranged interactions, scale invariance implies conformal invariance.
4.1.1 2-D Conformal field theory
In two-dimensional conformal field theory many important areas in physics are developed as in critical phenomena in condensed matter physics, statistical me- chanics in general, and string theory [3]. In two-dimensional conformal field
theory the special features are related to a large number of conformal transfor- mations and the relation to complex analysis.
The Virasoro modules to any given conformal field theory must all have the same eigenvalue of the central generator K = k1. A further restriction on the modules is obtained by considering the particular mode L0+ ¯L0 of the energymo- mentum tensor; this mode generates dilatation and hence, upon identifying radial ordering as time ordering, plays the role of the energy operator of the theory so that its spectrum must be bounded from below.
This implies that the Virasoro modules corresponding to the fields must in fact be highest weight modules of both the holomorphic and the antiholomorphic Virasoro algebras, and this fields corresponding to the highest weight vectors of the modules are called the primary fields [4].
4.2 WZW theories
Wess-Zumino-Witten (WZW) theories, theories whose furnish irreducible highest weight modules of an affine Lie algebra and which possess an energy-momentum tensor of the Sugawara form [4].
The use of affine Lie algebras in WZW theories require an extra term in the Lie brakets for the generators6 Tma. The relation is then written as
[Tma, Tnb] = fabcTm+nc + m¯κabδm+n,0K. (41) where fabc is the structure constant and ¯κab is the Killing form of the under- lying finite-dimensional Lie algebra. In conformal field theory one introduces a generating function for the generators Tma, i.e. a single object depending on a for- mal parameter z such that the generators Tma are obtainable as Fourier-Laurent coefficients of this object. Thus we define
Ja(z) :=X
n∈Z
Tnaz−n−1. (42)
Any quantity depending on the parameter z will be called a field or field operator.
A natural generalization of the analysis of an affine algebra g is to study its universal enveloping algebra U(g). It has turned out that the subspace of quadratic elements generated by Tma and Tnb is of a particular interest. In terms of current fields, one should thus consider the ordered product :Ja(z)Jb(z): of two currents. This can be split into an antisymmetric part and a symmetric part.
Now the antisymmetric part is actually fixed uniquely from
: [Ja(z), Jb(z)] : = fabc∂Jc(z), (43)
6Confirm the relations in section 3.1 and section 3.2.2.
where fabc is the structure constant. A new field denoted here by T can be introduced
: [Ja(z), Jb(z)] : = 1
2fabcJc(z) + κ¯ab
ξdT (z). (44)
where d is the dimension of g, and ξ is a constant incorporated to tune the normalization of the field T (z) in a convenient way [4].
It can be shown that the zero mode of the field T (z) can be identified with mi- nus the derivation of g. But even the whole field T has a concise meaning in phys- ical applications, namely as the energy-momentum tensor of a two-dimensional conformal field theory. One can also think of equation (44) as a generalization of the quadratic Casimir operator. This particular construction of T (z) in terms of currents Ja(z) is called the Sugawara constuction.
The name WZW theory has the following origin. Most of these theories can be realized in a lagrangian formulation as two-dimensional nonlinear sigma modules for which the target space is a compact Lie group manifold and which are supplemented by a Wess-Zumino term which, as realized by Witten, guarantees conformal invariance of the theory at the quantum level. The action of such WZW sigma model reads
SW ZW(G, k) = 2πk Sσ+ SW Z
(45) where
Sσ = 1 32π2
Z
d2ztr(∂µγ∂µγ−1) (46) and
SW Z = 1 48π2
Z
d3yµντtr( ˙γ−1∂µ˙γ ˙γ−1∂ν˙γ ˙γ−1∂τ˙γ). (47) Here γ takes values in the Lie group G that corresponds to the finite dimensional part of g and k is the level [4].
4.3 Fusion rules
In conformal field theory a substantial amount of information is provided by the so-called fusion rules[4].
φi∗ φj =X
k
Nijkφk (48)
where {φi} is the set of primary fields of the conformal field theory, and Nijk is a non-negative integers. This situation is analogous to the one the encountered when calculating the Kronecker product for simple Lie algebras.
The fusion rule coefficient Nijk is a analogue of the Littlewood-Richardson coefficient Lijk which contains the information on the decomposition of the Kro- necker product into (irreducible) submodules according to equation (34), in sec- tion 3.4. In the WZW theories the fusion rule coefficient and Littlewood-Richardson
coefficient has the following relation,
NΛΛ0Λ00 ≤ LΛΛ0Λ00. (49) There is no general procedure to determine the fusion rules of a given con- formal field theory. However, there are classes of theories for which one does know how to compute the fusion rules, and one of the most important classes is the WZW theories. For a WZW theory the fusion rule coefficients can be com- pletely determined with the help of the depth . For example the analysis of the case when g = A1 the Clebsch-Gordan coefficients are known explicitly and the criterion reads
Λ¯1+ ¯Λ2+ ¯λ1+ ¯λ2 ≤ 2(k − ¯Λ). (50) The depth rule then leads immediately to the following fusion rules for the A(1)1 WZW theory:
φΛ∗ φΛ0 =
min{ ¯Λ0,k−2 ¯Λ}
X
j=0
φΛj (51)
here Λj stands for the affine weight with a horizontal part
Λ¯j = ¯Λ − ¯Λ0+ 2j (52)
and without loss of generality we have assumed that ¯Λ ≥ ¯Λ0 to simplify the notation. Thus the fusion rule coeficients are
NφΛφ
Λ0
φΛ00 =
1 for ( ¯Λ + ¯Λ0+ ¯Λ00) ∈ 2Z
and | ¯Λ − ¯Λ0| ≤ ¯Λ00≤ k − |k − ¯Λ − ¯Λ0| 0 otherwise.
(53)
Finally, the determination of the fusion rules is a major step in the process of solving a conformal field theory [4].
5 The Tensor product between one finite and one infinite dimensional representation.
Here in this section we will look at the tensor product of one unitary spin rep- resentation and a infinity dimensional negative spin representation and check it out from the su(2) Lie algebra. We will analyze how the operators C2, J3 and the step operators J± will behave on this tensor product. We will also do some comment on article [12] where a similar tensor product has been used.
5.1 The general properties
Let us start in the general perspective and denote a basis for the unitary spin vector representation as {v±1/21/2 } and for the infinity dimension negative spin vec- tor representation as {vm−1/2}. If we start with the two-dimensional unitary spin representation and look at it’s properties, we got the following:
D1/2− J+v1/2−1/2 = v1/21/2 ;D1/2− J+v1/21/2= 0
D1/2− J−v1/2−1/2 = 0 ;D1/2− J−v1/21/2 = v−1/21/2 (54) Because {v±1/21/2 } has both a highest and a lowest representation Dj±, we will use the same sign as the negative representation. The infinity dimensional represen- tation have only a lowest weight it has the following properties:
m = −j, −j + 1, −j + 2, · · · Dj− J+v(j)m = i (j + m + 1) |j − m1| 12
vm+1(j) Dj− J−v(j)m+1 = i (j + m + 1) |j − m1| 12
vm(j)
for j < 0 (55)
Now in the case for m = n ± 1/2 and n ∈ Z≥0, we can get a tensor product of this basis vectors and a so-called ground state from the lowest weights, and this ground state will be denoted
|0i := v1/2−1/2⊗ v−1/21/2 , (56) and for all the other n states
|ni := an
iv−1/21/2 ⊗ v−1/2n+1/2+ v1/21/2⊗ vn−1/2−1/2
(57) with the liner combination
|˜ni := ˜an
−iv−1/21/2 ⊗ v−1/2n+1/2+ v1/21/2⊗ vn−1/2−1/2
, (58)
here an and ˜an is any arbitrary coefficient.
To start analysing this tensor product we let the step operator operate on the different vectors. From the unitary representation the operators will produce as we sew in equation (59), and for the infinity dimensional representation the step operators will produce7
D1/2− J+v−n−1/2 = inv−n+1/2 ;D1/2− J+vn+1/2− = i(n + 1)vn+3/2− D1/2− J−v−n−1/2 = i(n − 1)vn−3/2− ;D1/2− J−vn+1/2− = invn−1/2− (59) With this in mind we let the step operator now operate on the |ni state, and we get
D1/2− J+ |ni = i(n + 1) an
an+1 |n + 1i , (60)
and for the negative step operator J− we get D1/2− J− |n + 1i = inan+1
an |ni . (61)
If we do the same procedure for the linear combination |˜ni we get D1/2− J+ |˜ni = i˜an
−i(n + 1)v+−1/2⊗ v−n+3/2+ (n − 1)v1/2+ ⊗ vn+1/2−
(62) and
D1/2− J−| ]n + 1i = ˜an+1
(n + 2)v−1/2+ ⊗ vn+1/2− + inv1/2+ ⊗ vn−1/2−
(63) To get a better expression and to see the correlation between |ni and |˜ni on J± operating on |˜ni we set the equation A |ni + B |˜ni = 0 and calculate the coefficients and we get
A |ni + B |˜ni = Aan
iv+−1/2⊗ v−n+1/2+ v1/2+ ⊗ vn−1/2− +B˜an
−iv−1/2+ ⊗ vn+1/2− + v1/2+ ⊗ vn−1/2−
= i(Aan− B˜an)v−1/2+ ⊗ v−n+1/2
+(Aan+ B˜an)v+1/2⊗ vn−1/2− (64) this gives for J+
Aan −B˜an −i˜an(n + 1) Aan B˜an i˜an(n − 1)
=⇒
A 0 −iaa˜n
n
0 B in
. (65)
7For simplicity we have dropped the −1/2 over the unitvector and replaced it only with a
‘-’, and the same for 1/2 with a ‘+’, in the rest of our calculations.
If we use the result in equation (65) and put in the equation (62) plus extending the equation we get
D1/2− J+ |˜ni =
−i ˜an
an+1 |n + 1i + in ˜an
˜
an+1 | ]n + 1E
. (66)
Using the same procedure for J− we get
Aan+1 −B˜an+1 −i˜an+1(n + 2) Aan+1 B˜an+1 i˜an+1n
=⇒
A 0 −i˜aan+1
n+1
0 B i(n + 1)
(67)
and take this result from equation (67) as in the same way as for J+ put it in to equation (63) and extending, we get
D1/2− J−| ]n + 1i =
−i˜an+1
an |ni + i(n + 1)˜an+1
˜ an |˜ni
(68) This tensor product will produce a weight system from the lowest weight up to infinity. We can see this system in figure (1)8.
s s s s
s s s
p p p
p p p p
p p
p p
p
1 2 4 6
3 5 7
|0i |1i |2i |3i
|˜1
|˜2
|˜3
- - -
@
@
@
@
@
@
@@ I
@
@
@
@
@
@
@@ I
@
@
@
@
@
@
@@ I
- -
Figure 1: In this figure can we see a schematic sketch off the weight system for the tensor product v−1/2+ ⊗ v1/2− . The black arrows marks the phase for J+ on |ni, the blue arrows for J+ on |˜ni and the green arrows are J− on |ni and the red arrows for the J− on |˜ni.
Now let use further analyse this tensor product to see if it holds for C2 |0i = 0 and for C2 |ni = 0, but before we do that let use first look at behavior of J3 and
8Note, that the colored arrows may look similar in a greyscale printing.
see what it gives.
D1/2− J3 |0i = D1/2− J3
v−1/2+ ⊗ v1/2−
= D1/2− J3v+−1/2⊗ v−1/2+ v+−1/2⊗D1/2− J3v−1/2
= −1
2v−1/2+ ⊗ v1/2− + v−1/2+ ⊗1 2v−1/2
=
−1 2 +1
2
v−1/2+ ⊗ v1/2−
= 0. (69)
And for the | ni state we get D1/2− J3| ni = D1/2− J3
v+−1/2⊗ v−n+1/2+ v1/2+ ⊗ vn−1/2−
= D1/2− J3v+−1/2⊗ vn+1/2− + v−1/2+ ⊗D1/2− J3vn+1/2− +D1/2− J3v+1/2⊗ v−n−1/2+ v1/2+ ⊗D1/2− J3vn−1/2−
=
−1
2 + n +1 2
v+−1/2⊗ v−n+1/2+ 1
2+ n − 1 2
v1/2+ ⊗ v−n−1/2
= n
v−1/2+ ⊗ vn+1/2− + v+1/2⊗ vn−1/2−
= n| ni , (70)
and for the | ˜ni we get
D1/2− J3| ˜ni = D1/2− J3
−v−1/2+ ⊗ vn+1/2− + v+1/2⊗ vn−1/2−
= −D1/2− J3v−1/2+ ⊗ v−n+1/2− v−1/2+ ⊗D1/2− J3vn+1/2− +D1/2− J3v1/2+ ⊗ vn−1/2− + v1/2+ ⊗D1/2− J3vn−1/2−
= 1
2 − n − 1 2
v−1/2+ ⊗ vn+1/2− + 1
2 + n − 1 2
v+1/2⊗ vn−1/2−
= n
−v+−1/2⊗ v−n+1/2+ v1/2+ ⊗ v−n−1/2
= n| ˜ni . (71) This results is in good agreement of what we can expect to get. Now let use carry on with the analysis and see if it also holds for D1/2− C2| 0i = 0 and for D1/2− C2 |ni = 0, then we get
D1/2− C2 |0i = D1/2− J32+ J+J−− J3 |0i
= D1/2− J32 |0i +D1/2− J+J− |0i −D1/2− J3 |0i ...
= 1 4 +1
2
+ 1
4− 1 2
− 1 2+ 1
2− 1 2
v−1/2+ ⊗ v1/2−
= 0 (72)
and
D1/2− C2 |ni = anD1/2−
J32+ J+J−− J3
iv−1/2+ ⊗ v−n+1/2+ v1/2+ ⊗ vn−1/2− ...
= an
"
i 1 4+
n + 1
2
2
+ 2
−1 2
n + 1
2
− n2+1 2 −
n + 1
2
+ n
v+−1/2⊗ v−n+1/2
+ 1
4 +
n − 1
2
2
+ 2 1 2
n − 1
2
+ 1
− (n − 1)2− 1 2−
n − 1
2
− n
v+1/2⊗ v−n−1/2
= 0 (73) For details see the Appendix A.
From this we can draw the conclusion that the tensor product are in agreement with the relations of su(2), and that the weight system is fairly more complicated.
From these results we can now construct matrix representations for J+, J− and J3. The matrices should be of the form9,
J+ =
| 0i | 1i | ˜1
| 2i | ˜2
| 3i | ˜3
| 4i | ˜4 · · ·
| 0i 0 0 0 0 0 0 0 0 0 · · ·
| 1i i 0 0 0 0 0 0 0 0
| ˜1
0 0 0 0 0 0 0 0 0
| 2i 0 2i −i 0 0 0 0 0 0
| ˜2
0 0 i 0 0 0 0 0 0
| 3i 0 0 0 3i −i 0 0 0 0
| ˜3
0 0 0 0 2i 0 0 0 0
| 4i 0 0 0 0 0 4i −i 0 0
| ˜4
0 0 0 0 0 0 3i 0 0
... ... . ..
, (74)
9In the first presentation of the matrices they are not normalized, but they are in the final equation.
J− =
0 0 −i 0 0 0 0 0 0 · · ·
0 0 0 i −i 0 0 0 0
0 0 0 0 2i 0 0 0 0
0 0 0 0 0 2i −i 0 0
0 0 0 0 0 0 3i 0 0
0 0 0 0 0 0 0 3i −i
0 0 0 0 0 0 0 0 4i
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
... . ..
(75)
and
J3 = (0) ⊕
∞
M
n=1
n 0 0 n
(76) From the matrices can we than construct C2 from the relation C2 = Jx2+Jy2+Jz2 = J+J−+ Jz2− Jz which gives10
C2 =
0 0 0 0 0 · · ·
i 0 0 0 0
0 0 0 0 0
0 √
2i −i
2√
2 0 0
0 0 √
2i 0 0
... . ..
·
0 0 −i 0 0 · · ·
0 0 0 √
2i 2−i√2
0 0 0 0 √
2i
0 0 0 0 0
0 0 0 0 0
... . ..
+
"
(0) ⊕
∞
M
n=1
n 0 0 n
#2
−
"
(0) ⊕
∞
M
n=1
n 0 0 n
#
= (0) ⊕
∞
M
n=1
0 1 0 0
(77)
and from the result can we see that the Casimir operator C2 is nilpotent, because
C22
= 0.
10Note that this matrices in equation (77) is now normalized, and the coefficient an and ˜an
is set to√
n and 1/(2√
n) respectively.
5.2 One use of the tensor product v
m+⊗ v
m−.
A. van Tonder have in article [12], used a similar kind of tensor product to construct a representation theory for a negative spin in su(2) Lie algebra.
The purpose to construct this representation theory is from the quantum field theory [11]. In the investigation of electromagnetic field in the quantum field theory one uses a non-Abelian gauge field, also called the Yang-Mills field.
A powerful and general way to treat first class constraints is the BRST11 formalism. In the BRST formalism one introduces extra degrees of freedom with the opposite of the Grassman parity to the constraints. These extra degrees of freedom are needed in the construction of the theory, invariant under the BRST symmetry. It is this symmetry that will project out these unphysical degrees of freedom [10].
When you use a technique called the Faddeev-Popov method, you reduce these degrees of freedom, with the so-called ghosts or ghost fields.
It is in this frame the negative spin representation is created to suit A. van Tonder purposes. In his paper he use a different notation, a ket notation for all vectors denoted
v±1/2+ =
±1 2
s
(78) v−n±1/2 =
n ± 1 2
g
. (79)
For the groundstate or the vacuum state this then given by this tensor product
| 0i ≡
−1 2
s
⊗
1 2
g
, (80)
and for all the other n states is given by
| ni ≡√ n i
−1 2
s
⊗
n + 1 2
g
+ 1 2
s
⊗
n − 1 2
g
!
, (81)
here an=√
n and the linear combination | ˜ni were ˜an= 1/(2√
n) gives
| ˜ni ≡ 1 2√
n −i
−1 2
s
⊗
n + 1 2
g
+ 1 2
s
⊗
n − 1 2
g
!
. (82)
11For further information on this subject see for example [2, 5, 11]
