Recurrent Relation for Branching Coeﬃcients of aﬃne Lie Algebras

8/14/2019 Recurrent Relation for Branching Coefficients of affine Lie Algebras

1/22

Recurrent relation for branching coefficients

of affine Lie algebras

Vladimir LyakhovskyTheoretical Department, SPb State University,

198904, Sankt-Petersburg, Russia

e-mail:[email protected]

Anton NazarovTheoretical Department, SPb State University,

198904, Sankt-Petersburg, Russiae-mail:[email protected]

November 9, 2009

Abstract

We present the recurrent relation for the branching coefficients of

affine Lie algebras. Then we describe the algorithm for the decomposi-

tion of integrable highest weight modules of a simple Lie algebra with

respect to its reductive subalgebra which is based upon this recurrent

relation and present some examples. Also we discuss the appearance

of branching coefficients in the physical models.

1 Introduction

The problem of the reduction of Lie algebra representation to the repre-sentations of the subalgebra is studied for several decades and has variousapplications in physics. In the context of finite-dimensional algebras it isimportant for the study of great unification models whilst the problem ofthe branching of the affine Lie algebras emerges in conformal field theory, forexample, in the construction of the modular-invariant partition functions [1].

There exist several approaches to the computation of the branching co-efficients which use the BGG resolution [2] (for Kac-Moody algebras the

1


2/22

algorithm is described in [3],[4]), the Schure function series [5], the BRST

cohomology [6], Kac-Peterson formulas [3, 7] or the combinatorial methodsapplied in [8].

Usually only the embedding of the maximal reductive subalgebras is con-sidered since the case of non-maximal subalgebra can be obtained using thechain of maximal subalgebras. In this paper we present the recurrent for-mula for the branching coefficients which generalises the recurrent relationof the paper [9] to the case of non-maximal reductive subalgebra. Using thisrelation we formulate simple explicit algorithm for the computation of thebranching coefficients which is applicable to the non-maximal subalgebras offinite-dimensional and affine Lie algebras.

We show that our algorithm can be used in the study of the conformal em-beddings where the central charge of the conformal field theory is preserved,but the computation in this case is simplified using physical requirements.

The paper is organised as follows. In the next subsection of the introduc-tion we fix the notation used throughout the paper. Then in section 2 wederive the central recurrent formula for the anomalous branching coefficientsand describe the algorithm for the decomposition integrable highest weightmodules of algebra g to the modules of reductive subalgebra a 2.2. In thenext section 3 we present several examples and discuss some applications inthe physical models 4. We conclude the paper with the review of results and

the discussion of possible future developments 5.

1.1 Notation

Consider the affine Lie algebras g and a with the underlying finite-dimensional

subalgebrasg and

a and an injection a g such that a is a reductive sub-

algebra a g with correlated root spaces: ha

hg

and ha

hg

.

We use the following notations adopted from the paper [9].L (L

a) the integrable module of g with the highest weight ; (resp.

integrable a -module with the highest weight );

r , (ra) the rank of the algebra g (resp. a) ; (a) the root system; + (resp. +

a) the positive root system (of

g and a respectively);mult () (multa ()) the multiplicity of the root in (resp. in (a));

,

a

the finite root system of the subalgebra

g (resp.

a);

N , (Na

) the weight diagram of L (resp. La

) ;W , (Wa) the corresponding Weyl group;C , (Ca) the fundamental Weyl chamber;

2


3/22

C, Ca the closure of the fundamental Weyl chamber; , (a) the Weyl vector; (w) := det (w) ;i ,

(a)j

the i-th (resp. j-th) basic root for g (resp. a); i = 0, . . . , r

, (j = 0, . . . , ra); the imaginary root ofg (and ofa if any);

i ,

(a)j

the basic coroot for g (resp. a) , i = 0, . . . , r ; (j = 0, . . . , ra);

,

(a) the finite (classical) part of the weight P ,

resp. (a) Pa

;

=

; k; n

the decomposition of an affine weight indicating the

finite part

, level k and grade n .P (resp. Pa) the weight lattice;M(resp. Ma) :=

=

ri=1 Z

i

resp.

ri=1 Z

(a)i

for untwisted algebras or A

(2)2r ,r

i=1 Zi

resp.r

i=1 Z(a)i

for A(u2)r and A = A

(2)2r ,

; () :=

wW

(w)ew(+) the singular weight element for the g-module L; ()(a) :=

wWa

(w)ew(+a )a the corresponding singular weight element for the

a-module La

;() ()(a) the set of singular weights P (resp. Pa) for themodule L (resp. L

a) with the coordinates

, k, n, (w ())

|=w()(+),

(resp.

, k, n, (wa ())

|=wa()(+a)a ), (this set is similar to P

nice ()

in [4])

m() ,

m

()

the multiplicity of the weight P (resp. Pa) in

the module L , (resp. La

);ch (L) (resp. ch (L

a)) the formal character of L (resp. L

a);

ch (L) =P

wW (w)ew(+)Q+(1e

)mult()= ()

(0) the Weyl-Kac formula.

R :=

+ (1 e)mult() = (0)

resp. Ra :=

+a(1 e)multa() = (0)a

the denominator.

Lga =

P+a

b() La the reduction of the representation;

3


4/22

b() the branching coefficients;

Ca

b() ()(a) =

Pa

k() e

(1)

k the anomalous branching coefficients;It is important to mention that

b() = k() for Ca (2)

2 Recurrent formula for branching coefficients

Here we present the final recurrent relation for the anomalous branchingcoefficients (1).

k() =

1

s (0)

W\W

() dim

La((+))aa

0,a((+)) +

ag

s (+ 0) k()+

(3)

The proof of the formula and definitions of a, W, s(), ag are in thefollowing subsection.

In the next section we describe the algorithm for the computation of

branching coefficients based upon this formula and then present some exam-ple and discuss physical meaning.

2.1 Proof of the recurrent formula

The decomposition of the representation of the algebra to the representationsof the subalgebra can be symbolically written using formal characters andprojection operator:

Lga =

P+ab() L

a

= a(chLg

) =

P+ab() chL

a

(4)

Using the Weyl-Kac formula for the character of the module we obtain theequality

a

W ()e

(+)+(1 e

)mult()

=P+a

b()

Wa

()e(+a)a+a

(1 e)multa()(5)

It is important to mention that the projection of some of the positive rootsof the algebra g can be equal to zero. These roots are orthogonal to the root

4


5/22

space of the subalgebra a embedded into the root space of the algebra g.

Lets denote the subset of these roots by = +g : +a , .We should notice that if the set is non-empty than Weyl reflections

which correspond to the positive roots of generate a subgroup W ofWeyl group W. To prove it we should show that for any two positive roots, and corresponding Weyl reflections , W there existsWeyl reflection = W and the corresponding positive root . Since roots of the subalgebra a are invariant under , theyare also invariant under the action of = . Then the positive rootin+ which corresponds to is orthogonal to the root space of a andhence .

Since W is the subgroup of W and the space spanned over the set is invariant under its action there exists a subalgebra with the root spacespanned over the set which we denote by a.

Now we should discuss when the subset is non-empty and the sub-group W and subalgebra a are non-trivial.

If a is a maximal regular subalgebra of g then rank of a is equal to therank ofg and it is clear that is empty. Then the modules La are trivial,the dimensions are equal to 1 and we get the formula (11) from the paper[9].

Non-maximal regular embedding of a into g can be obtained through

the chain of maximal embeddings a p1 p2 g. Also the maximalregular embeddings are constructed by the exclusion of one or two roots fromthe extended Dynkin diagram of the algebra. Since this process can give usnon-connected Dynkin diagrams we can see which roots are orthogonal tothe root space of non-maximal regular subalgebra a.

Consider for example regular embedding of A1 B2 (su(2) so(5)).The extended Dynkin diagram of B2 We then drop central node and get

Figure 1: Extended Dynkin diagram of B2 and embedding of A1

the embedding A1 A1 B2. Then a = A1 and a = A1.

5


6/22

In the case of special embeddings the set can be empty as for the

special embedding of A1 A2 with the embedding index equal to 4, ornon-empty for example for the embedding su(p) su(p) su(q) TODO

Well, I cant understand how the roots are embedded in the special embed-dings. I need some help here.

We can multiply the equation (5) by the term

a

+\

(1 e)multg()

(6)

This term is non-zero.Also we can see that for any formal polynomial or series Q

a(Q)a(1 e) = a

Q (1 e)

(7)

The equation (5) takes the form

a

W ()e(+)

(1 e)mult()

=

a

+\

(1 e)multg()

P+a

b()

Wa

()e(+a)a+a

(1 e)multa()(8)

The right-hand side of this equation can be reorganised as in the paper [ 9].We introduce the anomalous branching coefficients k.

Pa

b() ()(a) =

Pa

k() e (9)

Also we extract the common denominator of the right-hand side of the equa-tion (8)

a

W ()e

(+)

(1 e)mult()

=

a(+\)

(1 e)multg()mult

Pa

k() e (10)

6


7/22

If the set is non-empty then the Weyl reflections corresponding to the

positive roots of generate a subgroup W of Weyl group W.We have denoted the subalgebra with the root space spanned over the set

by a.Then we can reorganise the summation on the left-hand side of the equa-

tion (10) in the following way. We consider the factor-group W\W and theleft conjugate classes. For the class W\W we choose the representative in such a way that ( + ) Ca. Then

a

W ()e

(+)

(1 e)mult()

=

a

W\W

()

W()e(+)

(1 e)mult()

(11)We see that the fraction on the right-hand side of the equation looks like

the character of some representation of the algebra a. In order to write itexplicitly we rewrite ( + ) as

(+) =

(+)a((+))a+a+a((+))

(12)

Since a(( + )) = a(( + )) and ( + ) a(( + )) =

a(( + )), we getW\W

()

W

()e(+)

(1 e)mult()=

W\W

()ea((+))ea

W

()e(a((+))a+a)a

(1 e)mult()=

W\W

()ea((+))eachLa ((+))aa

(13)

The projection a of the character of the highest-weight module La((+))aa

is equal to the dimension of the module multiplied by the identity elementof the algebra of formal exponents. So we have

a

W\W

()ea((+))eachLa((+))aa

=

W\W

() dim

La((+))aa

ea((+)) (14)

7


8/22

These dimensions of the modules could be easily calculated using Weyl for-

mula.This explicit calculation to get the character of the a representation can

be thought as the calculation of the character of the a h-module with thehighest weight ( + ), where h is the Cartan subalgebra ofg.

Thus we have the equalityW\W

()dim

La ((+))aa

ea((+)) =

a(+\)(1 e)multg()multPa k() e (15)We can rewrite the multiplier of the right-hand side as in the paper [9].

a(+\)

1 e

mult()multa()=

Pa

s()e (16)

For the coefficient function s () define ag Pa as its carrier:

ag = { Pa | s () = 0} ; (17)

a(+\)

1 e

mult()multa() = ag

s () e. (18)

So we get the equationW\W

()dim

La ((+))aa

ea((+)) =

=

ag

s () ePa

k() e

= ag

Pa

s () k()

e

(19)

From the equality of the coefficients of the equal formal exponents we getW\W

()dim

La((+))aa

,a((+))+

ag

s() k()+ = 0; Pa

(20)To get the recurrent relation for the anomalous branching coefficients we

should use the following procedure, introduced in the paper [9].

8


9/22

Let 0 be the lowest vector with respect to the natural ordering in

a inthe lowest grade of ag. Decomposing the defining relationa(+\)

1 e

mult()multa

()= s (0) e

0

ag\{0}

s () e,

(21)in (20) we obtain

k() = 1

s (0)

W\W

() dim

La((+))aa

0,a((+)) + ags (+ 0) k

()+

(22)

where the setag = { 0| ag} \ {0} . (23)

So weve obtained recurrent relation for the anomalous branching coeffi-cients. In the next section we describe the algorithm for the computation ofbranching coefficients using the relation (22).

2.2 Algorithm for the recursive computation of the

branching coefficients

We use the recurrent relation (22) to formulate the algorithm for recursivecomputation of the branching coefficients. It is important to mention that thecomputation of the branching coefficients is organised without the explicitconstruction of the module L()g and any of the modules L

()a .

The algorithm is divided into the following steps.

1. Construct the set + of the positive roots of the algebra g.

2. Select the positive roots + which are orthogonal to the rootsubspace of the subalgebra a and form the set .

3. Construct the set () = {( + ) ; W} of the anomalouspoints of the g-module L().

4. Select those weights = ( + ) which lies in the closure of the mainWeyl chamber of the algebra a. Since we have constructed the set we can easily check if the weight lies in the main Weyl chamberof a checking that the scalar product of with the roots of isnon-negative.

9


10/22

5. For = ( + ), Ca calculate the dimensions of the correspond-

ing modules dimLa((+))aa using the Weyl formula with theset .

6. Construct the set (23).

7. Calculate anomalous branching coefficients in the main Weyl chamberof the subalgebra a using recurrent relation (22).

If we are interested in the branching coefficients for the embedding of thefinite-dimensional Lie algebra into the affine Lie algebra we can constructthe set of the anomalous points up to required grade and use steps 4-7 of the

algorithm for each grade. We can also speed up the algorithm by one-timecomputation of the representatives of the conjugate classes W\W.

The next section consists of several examples computed with this algo-rithm.

3 Examples

3.1 Finite dimensional Lie algebras

3.1.1 Regular embedding of A1 into B2

Consider the regular embedding of A1 into B2. Simple roots 1, 2 of B2are drawn as the thick vectors at the figure 2. We denote the correspondingWeyl reflections by 1, 2. Simple root of the embedded A1 is equal to1 + 2.

Lets describe the reduction of fundamental representation ofB2 with thehighest weight (in fundamental weight basis) equal to (1, 0). On the figure2 we have also shown the set of points ( + ), W of fundamentalrepresentation ofB2 with the corresponding determinants of Weyl reflections(). Now we have to factorise the Weyl group W by W = {1}. We getthe following set of anomalous points ( + ) , W\W: We have

also depicted the corresponding a = A1-modules La((+))aa

. Thenwe project these points and dimensions of modules onto the root space ofsubalgebra a = A1 and get the following anomalous points in fundamentalweights basis with corresponding multiplicities:

(1, 2), (0, 3), (4, 3), (5, 2) (24)

For the function s() and the set from the definition (17,23) we have

(0, 1), (1, 2), (2, 1) (25)

10


11/22

+11

+1

1

+1 1

+1

1

Figure 2: Regular embedding of A1 into B2

3 +2

+3

2

0

1

1

2

3

2

3

Figure 3: Anomalous points and the corresponding a = A1-modules

11


12/22

Here the second component denotes the value of s().

Anomalous branching coefficient k(1,0)1 = 2, then for anomalous branching

coefficient k(1,0)0 the formula (22) gives us

k(1,0)0 = 1 k

(1,0)2 + 2 k

(1,01 3 0,0 = 1 (26)

So weve computed the branching coefficients.

3.1.2 Embedding of B2 into B4

Consider the the regular embedding of the subalgebra B2 into the algebra B4.We calculate the branching coefficients for the fundamental representation ofB4. The corresponding Dynkin diagrams are in the figure.

Figure 4: Dynkin diagrams

In the orthogonal basis e1, . . . , e4 simple roots of B4 are

(e1 e2, e2 e3, e3 e4, e4) (27)

Positive roots are

(e1 e2, e2 e3, e3 e4, e4, e1 e3, e2 e4, e3 + e4, e3, e1 e4,

e2 + e4, e2, e1 + e4, e2 + e3, e1, e1 + e3, e1 + e2) (28)

Simple roots of the embedded subalgebra a = B2 are

(e3 e4, e4) (29)

12


13/22

The set is equal to

{e1 e2, e1 + e2, e1, e2} (30)

We see that this is the set of positive roots of the algebra a = B2.To find the branching coefficients we need to compute the anomalous

points ofB4, select point lying in the main Weyl chamber ofa and computethe dimensions of corresponding a-modules.

We consider the B4-module with the highest weight = (0, 1, 0, 2) =2e1 + 2e2 + e3 + e4.

The set of the anomalous points ( + ) , W consists of 384points. We dont show it here.

We need to select those points ( + ) which are projected into the mainchamber of the embedded algebra a. It means that scalar product of thesepoints with all the roots from is non-negative.

To compute dimensions of the corresponding a-modules we need toproject each selected point onto the root space and substract a, thenuse Weyl dimension formula.

We show the result of this procedure on the figure 5.Then we should construct the fan and use the recurrent relation for the

computation of anomalous branching coefficients.Using the definition (23) we get the following set of the points with the

corresponding values s( + 0) We use the recurrent relation (22) and getfollowing branching coefficients:

a

chL

(0,1,0,2)B4

= 6 chL

(0,0)B2

+60 chL(0,2)B2

+30 chL(1,0)B2

+19 chL(2,0)B2

+40 chL(1,2)B2

+10 chL(2,2)B2

(31)

The dimension of the highest-weight B4-module L(0,1,0,2)B4

is equal to 2772. Itis easy to see, that right-hand side of the equation (31) gives the same result.

3.2 Affine Lie algebras

3.2.1 Embedding of the affine algebra into affine algebra

Consider the affine extension of the example 3.1.1. Since this embedding isregular, the level of the representations of the subalgebra is equal to the levelof the representation of the algebra.

The set of the orthogonal positive roots with the zero projection onthe root space of the subalgebra A1 is the same as in the finite-dimensionalcase.

Consider the level one representation of the algebra g = B2 with the high-est weight w1 = (1, 0, 1, 0), where the first to components are the coordinates

13


14/22

10

10

81 10

154 10

81

105 81

10

154

220 154

10

81

105

105

81

10

165 154

220

220

154

1081

105

105

81

165

165

154

220

220

154

81105

105

165

165

154220

220

105

165

165

220

165

6

4

2

0

2

4

6 4 2 2 4

Figure 5: Anomalous points with the dimensions of corresponding a-modules.

of the classical part in the orthogonal basis e1, e2, the third is the grade ofthe weight and the fourth is the level.

The set of the anomalous points of this representation up to sixth gradeis depicted in the figure 7 and in each grade it looks like in the figure 2.

As the next step of our algorithm 2.2 we project the anomalous points tothe weight space of the subalgebra A1 and calculate the dimensions of the

corresponding a-modules La((+))aa

. The result of this computationup to the twelfth grade is presented at the figure

Then we should construct the fan and use the recurrent relation for thecomputation of anomalous branching coefficients.

Using the definition (23) we get the following set of the points with thecorresponding values s(+ 0): 9. Here we restricted the computation to thetwelfth grade.

14


15/22

6

416

24

61

4

24 4 16

4

4

24

16

16

24 6

416

14

4

36

0

1

2

3

4

0 1 2 3 4

Figure 6: Fan for B2 B4

Also we should mention that the lowest vector of the fan 0 is equal tozero, since we have excluded all the roots of from the defining relation(23).

Using the recurrent relation for the anomalous branching coefficients weget the following result

Selecting the points inside the main Weyl chamber of the subalgebra A1

15


16/22

8

6

4

2

0

2

4

6

56 5 4 3 2 1

Figure 7: The anomalous points of the (1, 0, 1, 0) representation of the algebraB2

we get the following results for the branching coefficients up to twelfth grade

Lw1B2A1

= 2Lw1A1

(0) 1Lw0A1

(0) 4Lw0A1

(1)

2Lw1A1

(1) 8Lw0A1

(2) 8Lw1A1

(2) 15Lw0A1

(3)

12L

w1

A1(3) 26L

w1

A1(4) 29L

w0

A1(4) 51L

w0

A1(5)42Lw1

A1(5) 78Lw1

A1(6) 85Lw0

A1(6) 120Lw1

A1(7)

139Lw0A1

(7) 202Lw1A1

(8) 222Lw0A1

(8) 306Lw1A1

(9)

346Lw0A1

(9) 530Lw0A1

(10) 482Lw1A1

(10) 714Lw1A1

(11)

797Lw0A1

(11) 1080Lw1A1

(12) 1180Lw0A1

(12) (32)

This result can be expressed using the power series expansion of the branching

16


17/22

3

13

2

5

10

6

2

1113

3

3

2

3

5

13

6

11

5

10

2

6

10 2

2

5

3

10 2

6

3

5

14

6

14

11

3

2

11

13

6

6

6

3

5

8

6

4

2

0

2

4

6

12 10 8 6 4 2 0

Figure 8: Projected anomalous points and the dimensions of a-modules.

functions [3].

b(w1)0 = 1 + 4 q1 + 8 q2 + 15 q3 + 29 q4 + 51 q5 + 85 q6 + 139 q7+

222 q8 + 346 q9 + 530 q10 + 797 q11 + 1180 q12 + . . .(33)

b(w1)1 = 2 + 2 q1 + 8 q2 + 12 q3 + 26 q4 + 42 q5 + 78 q6 + 120 q7+

202 q8 + 306 q9 + 482 q10 + 714 q11 + 1080 q12 + . . .(34)

Here the lower index of the branching function denotes the number of thecorresponding A1 fundamental weight w0 = 0 = (0, 1, 0), w1 = /2 =(1, 1, 0).

17


18/22

3

2

2

2

1

2

2

3

2

1

2

1

2

2

2

2

2

2

2

1

2

2

2

2

2

2

2

1

2

2

2

3

2

2

2

2

2

2

2

1

4

2

2

2

2

2

2

3

2

2 22

1 1

12

2

2

1

4

2 2

1

1

2

1

1

2

2

2

2

2

2

2

4

1

1

2

2

23

1

1

2

1

2

4

1

1

2

2

2

2 2

1 1

2

3

2

2

2

2

1

2

2

2

1

3

2

1

0

1

2

3

4

2 4 6 8 10 12

Figure 9: Fan for A1 B2

0

00

0

12

0

78

120

0

0

202

15

12

0

0

0

29

0

4

714

0

0

8

1080

15

8

2

0

714

0

0

0

1080

0

2

0

2

26

1

0

0

0

0 0

0

0

0

0

0

0

0

0

0

0

8

0

8

0

0

42

15

139

0

0

78

4

0

85

0

222

0

0

0

0

8

0

4

2

0

0

0

0

0

0

0

0

0

0

0

0

26

4

0

0

26

482

0

85

0

0

222

8

0

4

0

0

0

0

1180

0

120

0

0

530

15

0

0

0

0

26

0

8

0

0

29 1

8

8

0

306

0

0

0

0

120

85

0

0

42

0

0

0

0

0

0

306

0

0

0

120

797

2

0

482

482

0

4

202

120

8

0

8

0

51

0

0 0

0

0

0

0

0

0

0

530

0

0

0

0

714

0

0

2

0

202

0

1

530

0

2

0

346

0

0

0

0

0

4

0

306

0

139

0

0

85

0

346

714

51

2

0

0

1180

0

1

0

12

0

12

85

8

2

0

0

0

00

0

0

0

51

0

42

0

306

346

222

0

0

0

797

0

0

0

0

0

0

0

12

2

139

0

0

0

0

202

0

0

1

0

0

139

1

29

0

2

0

29

8

0

0

2

29 8

15

139

0

0

482

12

0

0

0

0

0

0

12

0

222

0

2

0

0

51

0

0

0

15

0

0

26

0

0

0

0

0

0

42

222

0

139

0

0

0

0

0

26

222

120

0

2

78

0

0

12

0

0

0

0

0

0

0

0

0

51

0

42

0

0

8

8

4

0

0

2

26

0

0

346

26

0

0

1

2

0

0

0

51

85

0

029

0

78

0

42

78

78

0

0

530

0

0

0

0

1

0

8

6

4

2

0

2

4

6

12 10 8 6 4 2 0

Figure 10: Anomalous branching coefficients for A1 B2

18


19/22

4 Physical applications

Here we want to discuss possible applications of the described techniques inphysical models.

Branching coefficients for the embedding of affine Lie subalgebra intoaffine Lie algebra can be used to construct modular invariant partition func-tions of Wess-Zumino-Novikov-Witten models ([1], [10], [11], [12]).

But for this construction to work the embedding is required to be confor-mal, which means that the central charge of the subalgebra is equal to thecentral charge of the algebra.

c(a) = c(g) (35)

The class of the conformal embeddings is rather small, the complete clas-sification is given in the paper [12]. The requirement (35) allows to reducethe task of the computation of the branching coefficients of affine Lie algebrasto the computation of the branching coefficients of the finite-dimensional Liealgebras.

Here we describe this procedure and discuss how the requirement (35)can be used to simplify the algorithm 2.2.

Conformal embeddings should preserve conformal invariance, so Sug-awara central charge should be the same for enveloping and embedded theory.

The states for the theory that corresponds to the algebra g

Ja1n1Ja2n2

. . . | n1 n2 > 0 (36)

For sub-algebra a g

Ja1n1J

a2n2 . . . |a() (37)

Here Jajnj are the generators of a and a is the projection of g to a. g-

invariance of vacuum entails its a-invariance, but it is not the case for energy-momentum tensor. So energy-momentum tensor of bigger theory shouldconsist only of generators of a. Then Tg = Ta c(g) = c(a). This leads to

equationk dim g

k + g=

xek dim a

xek + a(38)

Here xe is the embedding index and g, a are dual Coxeter numbers of corre-sponding algebras.

It can be shown that solutions of equation (38) exist only for level 1 k = 1[1].

If we have modular-invariant partition function for the fields described bythe representation of the algebra g this modular invariance is preserved by

19


20/22

the projection on the subalgebra a, but we need also the preservation of the

conformal invariance. So we should select only those highest-weight modulesof the subalgebra a for which the relation (38) holds.

Since in the decomposition (4) the highest weight of the subalgebra

module belongs to some grade n of projected algebra module a L()g , from

the relation (38) one can obtain the following requirement on the conformaldimensions of the corresponding fields

a + n = (39)

It leads to the relation on the classical parts of the corresponding weights:

(,

+ 2)

2(1 + g) + n =

(,

+ 2a)

2(xe + a) (40)

There exits the finite reducibility theorem for the conformal embeddingswhich states that only finite number of the branching coefficients is non-zeroin the case of the conformal embedding a a.

Then after we have found all such weights and the correspondingbranching coefficients b

() we can substitute the sums

P+a

b() over the

modified characters of the corresponding a-modules in place of the char-acters of the g-modules in the diagonal modular-invariant partition function

Z() = P+g ()() (41)Thus we obtain the non-diagonal modular-invariant partition function forthe theory with the current algebra a.

Za() =

,P+a

()M() (42)

4.1 Example

Consider the level ten embedding of the affine A1 into the level one affineC

2. The relation (40) and the finite reducibility theorem greatly simplify the

calculation with the algorithm 2.2, since we know that there exists only finitenumber of the non-zero branching coefficients and the maximal grade of thiscoefficients is restricted.TODO

Do the calculation, show the restriction on the grade and the final result

Z =[10,0] + [4,6]2 + [7,3] + [3,7]2 + [6,4] + [0,10]2 (43)

20


21/22

5 Conclusion

We have proved the recurrent relation on the branching coefficients and pro-posed practical algorithm of the reduction of the representations. Also wehave discussed the application of this algorithm to the physical problem ofconstruction of modular-invariant partition functions in the conformal fieldtheory. This method of conformal embeddings is well-known but may be ac-tual in the study of WZW-models emerging in the context of the AdS/CFTcorrespondence [13, 14, 15].TODO

What about acknowledgements?

References

[1] P. Di Francesco, P. Mathieu, and D. Senechal, Conformal field theory.Springer, 1997.

[2] I. Bernstein, M. Gelfand, and S. Gelfand, Differential operators onthe base affine space and a study of -modules, Lie groups and theirrepresentations, in Summer school of Bolyai Janos Math.Soc. Halsted

Press, NY, 1975.[3] V. Kac, Infinite dimensional Lie algebras. Cambridge University Press,

1990.

[4] M. Wakimoto, Infinite-dimensional Lie algebras. AmericanMathematical Society, 2001.

[5] B. Fauser, P. Jarvis, R. King, and B. Wybourne, New branching rulesinduced by plethysm, J. Phys A: Math. Gen 39 (2006) 26112655.

[6] S. Hwang and H. Rhedin, General branching functions of affine Lie

algebras, Arxiv preprint hep-th/9408087 (1994) .

[7] T. Quella, Branching rules of semi-simple Lie algebras using affineextensions, Journal of Physics A-Mathematical and General 35(2002) no. 16, 37433754.

[8] B. Feigin, E. Feigin, M. Jimbo, T. Miwa, and E. Mukhin, Principalsl3 subspaces and quantum Toda Hamiltonians, arxiv 707 .

21


22/22

Documents

Recurrent Relation for Branching Coeﬃcients of aﬃne Lie Algebras