Download - Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Transcript
Page 1: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

Lecture 17

Daniel Bump

June 3, 2019

φiφj φk

Page 2: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

Motivation from CFT

The primary fields in a conformal field theory admit anassociative composition law. If φi are a basis of primary fieldsthen the composition is (for certain constants)

φi ? φj = Nki,jφk.

If i→ ∗i is conjugation then Ni,j,k = N∗ki,j will be symmetric underall three parameters. In the physical interpretation, these will bethe 3-point correlation functions at 3 distinct points on apunctured sphere.

φiφj φk

Page 3: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

Affine Lie algebras

Verlinde and Moore and Seiberg showed that the WZW theorieslead to a composition in terms of the representation theory ofaffine Lie algebras. This was turned into a purely mathematicaloperation by Kazhdan and Lusztig. The fields are interpretedas level k integrable representations of an affine Lie algebra.Kazhdan and Lusztig, Finkelberg, Andersen and Paradowskiand Sawin related these fusion rings to the ones constructedfrom representations of quantum groups in Lecture 16.

The modest goal of todays lecture is to describe the fusion ringstructure on integrable highest weight representations of anaffine Lie algebra level k, without proofs. We will not give thedefinition of Kazhdan and Lusztig, but instead describe theS-matrix and define the fusion coefficients by the Verlindeformula.

Page 4: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

Infinite-dimensional Lie algebras

In the 1970’s, people became aware that much of classical Lietheory had an important infinite-dimensional generalizations.For example consider the Weyl denominator formula. If g is acomplex semisimple Lie algebra, say the Lie algebra of the Liegroup G, and if h is the Lie algebra of a maximal torus T of G,we have the Weyl character formula

χ(z) = ∆−1∑w∈W

(−1)`(w)zw(λ+ρ)

where (Weyl denominator formula):

∆ =∑w∈W

(−1)`(w)zw(ρ) = zρ∏α∈∆+

(1 − z−α).

Similar identities were noticed by Macdonald (1972) in which Wis replaced by the affine Weyl group.

Page 5: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

References

Kac, Infinite-Dimensional Lie algebras, particularlyChapter 13.Kac and Peterson, Infinite-dimensional Lie algebras, thetafunctions and modular forms (1984).Affine Root System Basics [Web link]Di Francesco, Mathieu and Sénéchal, Conformal FieldTheory, Chapter 14.Bakalov and Kirillov, Chapter 7.

Page 6: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

The affine Weyl group

The affine Weyl group is the group generated by reflections inthe hyperplanes 〈αv, x〉 = k. The connected components of thecomplement of these hyperplanes are called alcoves. There isa unique alcove that is contained in the positive Weyl chamberthat is adjacent to the origin, namely

Fk = {x|〈α∨i , x〉 > 0, 〈θ∨, x〉 6 k}.

Here αi are the simple roots and θ is the longest root.

The affine Weyl group is a Coxeter group, generated by thereflections si in the hyperplanes bounding the fundamentalalcove. The reflection in 〈θ∨, x〉 = k is denoted s0.

Page 7: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

The fundamental alcove inside the positive Weyl chamber

The SL(3) case

Page 8: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

The Jacobi triple product identity

Jacobi proved∞∑−∞(−1)nqn2

tn =

∞∏m=1

(1 − q2m)(1 − tq2m−1)(1 − t−1q2m−1).

The sum on the left can be considered a sum over the sl2 affineWeyl group. Macdonald noted a similarity to the Weyldenominator formula and gave similar formulas for other affineroot systems.

Meanwhile, infinite-dimensional affine Lie algebras began toappear in mathematical physics as current algebras. Kac (andindependently Moody) developed a general theory thatcontains both the finite-dimensional and affine Lie algebras asspecial cases.

Page 9: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

Generalized Cartan Matrices

Let A = (aij) be a generalized Cartan matrix. The entries areintegers with aii = 2 and aij 6 0 if i 6= j. Things work best if DA issymmetric for some diagonal matrix D. For simplicity we willassume D = I so A is a symmetric matrix. This is thesimply-laced case.

Now there exists a finite-dimensional vector space h∗ with h itsdual space, and linearly independent subsets

Π = {α1, · · · ,αs} ⊂ h∗, Π∨ = {α∨1 , · · · ,α∨

s }

(Roots and coroots) such that

〈α∨i ,αj〉 = aij

and dim(h) is as small as possible. These data are called therealization of A.

Page 10: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

The Kac-Moody Lie algebra

Now there exists a Lie algebra g(A) with generators andrelations as follows. We need h, which will be a commutativesubalgebra, and for each 1 6 i 6 s generators Ei, Fi subject to

[H,Ei] = 〈αi,H〉Ei, [H,Fi] = −〈αi,H〉Fi,

[Ei,Fj] = δijα∨i

and Serre relations

ad(Ei)1−αij Ej = ad(Fi)

1−αij Fj = 0.

Such a Lie algebra has a Weyl group, which is a (possiblyinfinite) Coxeter group.

Page 11: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

Extended Dynkin diagrams

A class of Kac-Moody Lie algebras that appears in real life isthe class of (untwisted) affine Lie algebras. If g is afinite-dimensional complex simple Lie algebra of rank r, thenthere is an “extended Dynkin diagram” with r + 1 nodes, fromwhich may be read off a Cartan matrix. For example if g = sl4then the extended Dynkin diagram looks like

α1 α2 α3

α0

Eliminating the α0 node gives the ordinary Dynkin diagram.

Page 12: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

The extended Dynkin-diagram continued

If the root system ∆ of g is known we may describe theextended Dynkin-diagram as follows. Let θ be the highest rootand α0 = −θ.

Note: the meaning of α0 will change later when we embed theroots in a larger space h∗.

To define the extended Dynkin diagram, we note that for everyi, j ∈ {0, 1, · · · , r} either i = j or 〈α0,αj〉 6 0. The extendedDynkin diagram is made by the usual recipe, namely αi and αj

are joined by an edge if they are not orthogonal.

Page 13: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

The Cartan matrix (finite case)

Starting with the Dynkin diagram we may reconstruct theCartan matrix A of g.

aij =

2 if i = j;−1 if αi, αj are adjacent;0 if αi, αj are not adjacent.

α1 α2 α3

The corresponding Kac-Moody algebra is the originalfinite-dimensional g, in this example sl4.

A◦ =

2 −1 0−1 2 −10 −1 2

.

Page 14: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

The Cartan matrix (affine case)

Now the extended Cartan matrix is made the same way fromthe extended Dynkin diagram:

α1 α2 α3

α0

This adds one row and column to the Cartan matrix.

A =

2 −1 0 −1−1 2 −1 00 −1 2 −1−1 0 −1 2

.

Page 15: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

The affine Lie algebra

We may now apply the Kac-Moody construction to both Cartanmatrices.

Applying it to A recovers g.Applying it to A gives the (untwisted) affine Lie algebra g.The dimension of the Cartan algebra h is r + 2.

There is another way of constructing these untwisted affine Liealgebras as the Lie algebra of a loop group. Start with the Liealgebra of vector fields on the circle taking values in g is

g⊗ C[t, t−1]

[X ⊗ tn], [Y ⊗ tm] = [X,Y]⊗ tn+m.

Page 16: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

Central extensions

Suppose g is a Lie algebra and ψ : g× g→ C is a bilinear mapsuch that

ψ(X,Y) = −ψ(Y,X),

ψ([X,Y],Z) +ψ([Y,Z],X) +ψ([Z,X],Y) = 0.

Such a map is called a 2-cocycle. Given such data we mayconstruct a central extension which is g⊕ C · K and the bracketoperation is

[X + aK,Y + bK] = [X,Y] + (a + b +ψ(X,Y))K

and this is a Lie algebra.

Page 17: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

A cocycle for the loop agebra

Start with g⊗ C[t, t−1]. We define a 2-cocycle by

ψ(X ⊗ tn,Y × tm)x = δn,−m n〈X,Y〉

in terms of an invariant inner product on g. We may thenconstruct a central extension g ′ of g. It is possible to enlargethis Lie algebra one dimension further by adjoining a derivationd to obtain the affine Lie algebra g.

We will denote by h the Cartan subalgebra of g. It can berealized as h⊕CK ⊕Cd where K is the central element and d isthe derivation.

Page 18: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

Roots and weights

The set of λ ∈ h such that 〈α∨i , λ〉 ∈ Z for all αi are called

weights.

For example the simple roots αi are roots, and one particularroot

δ =

r∑i=0

aiαi

called the nullroot is orthogonal to all the others in an invariantinner product on h∗. The coefficient a0 = 1 and the other ai canbe looked up in Kac, Infinite Dimensional Lie Algebras; forg = slr+1 they are all equal to 1.

Page 19: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

Weight space decompositions

We have Λi ∈ h∗ such that Λi(α∨j ) = δij. Then the weight lattice

is the linear span of ∆ and the Λi. The Λi are the fundamentalweights.

We have a triangular decomposition

g = h⊕ n+ ⊕ n−.

We require modules to have a weight space decomposition

V =⊕λ∈h∗

Vλ.

A module V is called a highest weight module with highestweight λ if it has a vector vλ annihilated by n+ such that

X · vλ = 〈X, λ〉vλ, X ∈ h.

Page 20: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

The affine Weyl group

The Weyl group of a general Kac-Moody Lie algebra g is be thegroup of linear operations on h∗ generated by the simplereflections

si(λ) = λ− 〈α∨i , λ〉αi. (1)

Note that this action fixes a point (the origin) unlike the affineWeyl group defined earlier, which included a translations. Letus relate the two actions. If λ is a weight, we call k = (δ|λ) thelevel of λ. Now we consider the action of the Weyl group W(defined by si as in (1).

Page 21: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

The affine Weyl group (continued)

k = 0

k = 1

k = 2

k = 3

The affine Weyl group fixes the origin. But parallel to the level 0plane, it induces affine linear maps on each level (except level0). The fundamental alcove on the k level set is the intersectionwith the shaded area, consisting of dominant weights.

Page 22: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

Kac-Moody root systems

For every Kac-Moody Lie algebra, there is a root system. Incontrast with finite root systems, infinite Kac-Moody rootsystems include are two kinds of roots. Roots may be classifiedas as real or imaginary.

Macdonald invented affine root systems before it was realizedthat they were root systems of infinite-dimensional Lie algebra.The imaginary roots appeared as factors 1 − q2m in the Jacobitriple product identity, which Macdonald greatly generalized.

∞∑−∞(−1)nqn2

tn =

∞∏m=1

(1 − q2m)(1 − tq2m−1)(1 − t−1q2m−1).

For general Kac-Moody root systems, the imaginary rootsremain somewhat mysterious.

Page 23: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

The Affine Root System

We started with a finite-dimensional semisimple Lie algebra gof rank r having a root system ∆. We produced aninfinite-dimensional (untwisted) Lie algebra g with root system∆.

The real roots are

∆re = {α+ nδ |α ∈ ∆, n ∈ Z}.

The imaginary roots are

∆im = {nδ | n ∈ Z, n 6= 0}.

There is a Weyl vector ρ though it cannot be defined as half thesum of the positive roots.

Page 24: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

Integrable highest weight modules

A weight λ ∈ h∗ is dominant if 〈α∨i , λ〉 is a nonnegative integer.

We may assume that λ =∑

niΛi where ni are nonnegativeintegers. In this case the unique irreducible highest weightrepresentation L(λ) is called integrable because therepresentation can be exponentiated to a representation of theloop group. One implication is that the weight multiplicities areinvariant under the action of the Weyl group.

Let µ be a weight, and we consider the multiplicity m(µ) of µ inL(λ). This is nonzero uness µ lies in a cone

(λ+ ρ|λ+ ρ) − (µ+ ρ,µ+ ρ) > 0.

Page 25: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

The Weyl-Macdonald identities

The results in this slide are valid for an arbitrary Kac-Moody Liealgebra, though we are interested mainly in the affine case.Recall that there is a Weyl vector ρ such that 〈α∨

i , ρ〉 = 1 for allsimple coroots αi. Then∑

w∈W

(−1)`(w)ew(ρ) = eρ∏α∈∆+

(1 − e−α).

We are using eµ as a formal symbol. Although this is a formalidentity, specializing eµ can give an identity betweenconvergent expressions.

For example, in sl2 let ∆ = {±α} be the usual sl2 roots, andα0 = δ− α1.

Page 26: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

The triple product identity

The sl2 positive roots are:

∆re = {αi + nδ | n > 0, i = 0, 1}

∆im = {nδ | n > 1}

Let q, t ∈ C with t 6= 0. To get a convergent series we want|q| < 1. Specialize eµ so that

eα1 = tq, eα0 = t−1q, eα1 = tq, eδ = q2.

Then the Weyl-Macdonald identity∑w∈W

(−1)`(w)ew(ρ) = eρ∏α∈∆+

(1 − e−α)

becomes the Jacobi triple product identity∞∑−∞(−1)nqn2

tn =

∞∏m=1

(1 − q2m)(1 − tq2m−1)(1 − t−1q2m−1).

Page 27: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

The Kac-Weyl character formula

They Weyl character formula is valid for the integrablehighest-weight representation L(λ) of an arbitrary Kac-MoodyLie algebra. The character

χλ =∑µ

m(µ)eµ

equals:D−1

∑w

(−1)`(w)ew(λ+ρ)

where D is the denominator from the last slide:

D =∑w∈W

(−1)`(w)ew(ρ) = eρ∏α∈∆+

(1 − e−α)

Page 28: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

String functions

In the affine case, the character χλ of L(λ) can be written out interms of more basic functions called string functions.

Let µ be a weight. The function m(µ− nδ) is monotone, andzero outside the cone

(λ+ ρ|λ+ ρ) − (µ+ ρ,µ+ ρ) > 0.

Therefore we may find a smallest n such that m(µ− nδ) isnonzero. We call µ− nδ a maximal weight. Define the stringfunction

bλµ =∑

k

m(µ− kδ)e−kδ.

replacing µ by µ− nδ just multiplies the series by ekδ so there isno loss of generality in assuming that µ is maximal.

Page 29: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

String functions (continued)

The characterχλ =

∑µ maximal

bλµ.

Moreover using the fact that δ = w(δ) for all w ∈ W we mayhave w(bλµ) = bλwµ and therefore we may rewrite bλµ by finding aWeyl group element such that wµ is dominant. There are only afinite number of dominant maximal weights. Then

χλ =∑

µ maximal, dominant

∑w∈W/Wµ

ewµbλµ.

(To avoid overcounting Wµ is the stabilizer of the dominantmaximal weight µ.)

Page 30: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

Modular characteristics

Define the modular characteristic (modular anamoly)

sλ =|λ+ ρ|2

2(k + h∨)−

|ρ|2

2k,

We are using the notation ( | ) for an invariant inner product onΛ. Here h∨ is the dual Coxeter number and k = (δ|λ). Also

sλ(µ) = sλ −|µ|2

2k.

Now define

cλµ = e−sλ(µ)δbλµ = e−sλ(µ)δ∑n>0

m(λ− nδ)e−nδ.

These modified string functions are theta functions which aremodular forms, as was proved by Kac and Peterson. Becauseof this, the character χλ is a modular form.

Page 31: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

The fusion product following Kazhdan and Lusztig

We have already mentioned that if φi are the primary fields in aWZW conformal field theories the fusion coefficients Nijk = N∗kijare the three-point correlation functions for primary fields φi onthe Riemann sphere. Note that the group of conformal maps ofthe Riemann sphere P1(C) is 3-transitive, so there is no loss inassuming the points are 0, ∞ and −1.

Therefore the construction as abstracted by Kazhdan andLusztig though purely algebraic, also uses the spherepunctured on three points, an affine variety whose affinealgebra is R = C[t, t−1, (1 + t)−1].

Page 32: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

The fusion product following Kazhdan and Lusztig (continued)

In Kazhdan and Lusztig (though modifying their notation), theaffine algebra is a central extension

0→ C · 1→ g→ g⊗ C((t))→ 0,

where C((t)) is the field of formal power series∑∞

k=−N aktk. If V,W are modules of level k, restrict their action to the preimage(g⊗ R)∧ in g of g⊗ R. (Recall R = C[t, t−1, (1 + t)−1].)This is a Lie algebra. Now there are two homomorphismsR→ C((t)) in which t 7→ t − 1 and t 7→ t−1 − 1. If ξ ∈ (g⊗ R)∧ letξ ′ and ξ ′′ be the images of ξ under these two homomorphisms.

Page 33: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

The fusion product following Kazhdan and Lusztig (continued)

Consider the space of bilinear forms λ : V ⊗W → C in which ξacts by

ξλ(x, y) = λ(ξ ′x, y) − λ(x, ξ ′′y).

Now considering the subspace in which the subalgebra of ggenerated by the Ei (including E0) acts nilpotently, this gives amodule V ◦W. The composition law actually is(V,W)→ (V ◦W)∗.

We will not follow up on this description. Instead, we will give adifferent description of the fusion product, based on theS-matrix, modularity and the Verlinde formula.

Page 34: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

The group SL(2,Z)

Let q = e2πiτ where H = {τ = x + iy ∈ C|y > 0}. The groupSL(2,R) and its discrete subgroup SL(2,Z) act on H via(

a bc d

): τ 7→ aτ+ b

cτ+ d.

The subgroup Γ = SL(2,Z) acts discontinuously.

A modular form is a function f that satisfies

f (z) = (cz + d)−kf(

az + bcz + d

)for(

a bc d

)in SL(2,Z) or a subgroup of finite index.

Page 35: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

Two important elements

Note that −I ∈ SL(2,Z) acts trivially on H, so the action is notfaithful. Let

S =

(−1

1

), T =

(1 1

1

)These generate SL(2,Z). Since S2 = −I, S has order 4 as anelement of the group but order 2 in its action on H. NoteS : τ 7→ − 1

τ .

T : τ→ τ+ 1 is the translation by 1. If a holomorphic function fis invariant under SL(2,Z) it is invariant under T and so it has aFourier expansion:

f (τ) =∑

anqn, q = e2πiτ.

Page 36: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

Digression on SL(2,Z) (continued)

Here is the well-known fundamental domain for SL(2,Z):

e2πi/3 i

We have marked the fixed points i and e2πi/3 of S and ST.

S =

(−1

1

), T =

(1 1

1

)

S : z→ −1z, T : z→ z + 1.

Page 37: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

Jacobi forms

One also encounters modular forms involving both the modularparameter τ and one or more auxiliary complex parametersz = (z1, · · · , zr). This leads to the notion of Jacobi modularforms in which the relevant group is a semidirect product ofSL(2,Z) and a Heisenberg group of degree 2r + 1. The action is(

a bc d

): (τ, z) 7→

(−

1τ,

zcτ+ d

).

Examples are found in the theory of theta functions. Weconsider this action on C⊗ h∗.

Page 38: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

The S-matrix

Now the S-matrix is essentially the scattering matrix for thecharacters of the level k. The integrable highest weightrepresentations of level k are parametrized by dominantweights of level k, which lie in the fundamental alcove of level k.

We use the following coordinates on h and its complexification.Choose an orthonormal basis v1, · · · , vr of h and then we willdenote

(τ, z, u) =∑

zivi − τΛ0 + uδ

where Λ0 is the affine fundamental weight. We will write thecharacter χλ in these coordinates. Then

χλ

(−

1τ,

zτ, u −

|z|2

)=

∑µ

Sλ,µχµ(τ, z, u).

Page 39: Lecture 17 - Stanford Universitysporadic.stanford.edu/quantum/lecture17.pdf · Lecture 17 Daniel Bump ... The modest goal of todays lecture is to describe the fusion ring structure

Affine Lie algebras The Fusion Ring

Verlinde formula

The identity element in the fusion ring is the representation withhighest weight kΛ0. So we will denote SkΛ0,λ as just S0,λ.Knowledge of the S-matrix is sufficient to reconstruct the fusioncoefficients Nνλ,µ by the Verlinde formula

Nνλ,µ =∑σ

Sλ,σSµ,σSν,σS0,σ