Current algebras

The gauge boson vertex operators in the heterotic string are of the form j(z) ˜ψ^µ(¯z)e^ik·X, where j(z) is either a fermion bilinear λ^Aλ^B or a spin field (11.2.26). Similarly the gauge boson vertex operators for the toroid-ally compactified bosonic string were of the form j(z)¯∂X^µ(¯z)e^ik·X with j being ∂X^m for the Kaluza–Klein gauge bosons or an exponential for the enhanced gauge symmetry (or the same with right and left reversed). All these currents are holomorphic (1, 0) operators. In this section we consider general properties of such currents.

Let us consider in a general CFT the set of (1, 0) currents j^a(z). Con-formal invariance requires their OPE to be of the form

j^a(z)j^b(0)∼ k^ab

z² +ic^ab_c

z j^c(0) (11.5.1)

with k^ab and c^abc constants. Dimensionally, the z⁻² term must be a c-number and the z⁻¹ term must be proportional to a current. The Laurent coefficients

j^a(z) =

∞ m=−∞

j_m^a

z^m+1 (11.5.2)

thus satisfy a closed algebra

[j_m^a, j_n^b] = mk^abδm,−n+ ic^abcj_m+n^c . (11.5.3) In particular,

[j₀^a, j₀^b] = ic^abcj₀^c . (11.5.4) That is, the m = 0 modes form a Lie algebra g, and

c^ab_c = f^abc . (11.5.5)

We focus first on the case of simple g. The j₁^aj^b₀j₋₁^c Jacobi identity requires that

f^bcdk^ad+ f^badk^dc = 0 . (11.5.6) This is the same relation as that defining the Lie algebra inner product d^ab, and since we are assuming g to be simple it must be that

k^ab= ˆkd^ab (11.5.7)

for some constant ˆk. The algebra (11.5.3) is variously known as a current algebra, an affine Lie algebra, or an (affine) Kac–Moody algebra. The currents are (1, 0) tensors, so

[L_m, j_n^a] =−njm+n^a . (11.5.8)

11.5 Current algebras 67 Physically, the j_n^a generate position-dependent g-transformations. This is possible in quantum field theory because there is a local current. The central extension or Schwinger term ˆk must always be positive in a unitary theory. To show this, note that

ˆkd^aa=1| [ j1^a, j₋₁^a ]|1 =  j₋₁^a |1 ² (11.5.9) (no sum on a). For a compact Lie algebra d^aa is positive and so ˆk must be nonnegative. It can vanish only if j₋₁^a |1 = 0, but the vertex operator for j₋₁^a |1 is precisely the current j^a: any matrix element of j^a can be obtained by gluing j₋₁^a |1 into the world-sheet. Thus ˆk = 0 only if the current vanishes identically.

The coefficient ˆk is quantized. To show this, consider any root α. Defining J³= α· H

α² , J^± = E^±α , (11.5.10) one finds from the general form (11.4.12) that these satisfy the S U(2) algebra

[J³, J^±] = J^± , [J⁺, J⁻] = 2J³ . (11.5.11) The reader can verify that the two sets

α· H0

α² , E₀^α , E₀^−α , (11.5.12a) α· H0+ ˆk

α² , E₁^α , E₋₁^−α (11.5.12b) also satisfy the S U(2) algebra. The first is just the usual center-of-mass Lie algebra, while the second is known as pseudospin. From familiar properties of S U(2), 2J3 must be an integer, and so 2ˆk/α² must be an integer. This condition is most stringent if α is taken to be one of the long roots of the algebra (denoted ψ). The level

k = 2ˆk

ψ² (11.5.13)

is then a nonnegative integer, and positive for a nontrivial current.

It is common to normalize the Lie algebra inner product to give the long roots length-squared two, so that ˆk = k is the coefficient of the leading term in the OPE. We will usually do this in examples, as we have done in giving the roots of various Lie algebras in the previous section. Inciden-tally, it follows that with this normalization the generators (11.4.14) are normalized, so the S O(n) inner product is half of the vector representation trace. Similarly the inner product for S U(n) such that the long roots have length-squared two is equal to the trace in the fundamental representation.

In general expressions we will keep the inner product arbitrary, inserting explicit factors of ψ² so that results are independent of the normalization.

±

We will, however, take henceforth a basis for the generators such that d^ab= δ^ab.

The level represents the relative magnitude of the z⁻² and z⁻¹ terms in the OPE. For U(1) the structure constant is zero and only the z⁻² term appears. Hence there is no analog of the level. It is convenient to normalize all the U(1) currents to

j^a(z)j^b(0)∼ δ^ab

z² . (11.5.14)

From this OPE and holomorphicity it follows that each U(1) current algebra is isomorphic to a free boson CFT,

j^a = i∂H^a . (11.5.15)

We will often use this equivalence.

The current algebra in the heterotic string consisted of n real fermions λ^A(z). The currents

iλ^Aλ^B (11.5.16)

form an S O(n) algebra. The maximal set of commuting currents is iλ^2K−1λ^2K for K = 1, . . . , [n/2]. These correspond to the generators (11.4.14), which are normalized such that roots (11.4.16) have length-squared two. The level is then the coefficient of the leading term in the OPE; this is 1/z², so the level is k = 1. The case n = 3 is an exception:

there are no long roots, only the short roots ±1, so we must rescale the diagonal current to 2^1/2iλ¹λ² and the level is k = 2.

For any real representation r of any Lie algebra, one can construct from dim(r) real fermions the currents

λ^Aλ^Bt^a_r,AB . (11.5.17) These satisfy a current algebra with level k = Tr/ψ², with Tr defined in eq. (11.4.6). The case in the previous paragraph is the n-dimensional vector representation of S O(n), for which TR = ψ². As another example, nk fermions transforming as k copies of the vector representation give level k.

As a final example consider the S U(2) symmetry at the self-dual point of toroidal compactification. The current is exp[2^1/2iH(z)]. The current i∂H is then normalized so that the weight (from the OPE) is 2^1/2, with length-squared two. The OPE of i∂H with itself starts as 1/z², so the level is again k = 1.

In some cases one may have sectors in which some currents are not periodic, j^a(w + 2π) = R^abj^b(w), where R^ab is any automorphism of the algebra. In these, the modes of the currents are fractional and satisfy a twisted affine Lie algebra.

2 1

11.5 Current algebras 69 The Sugawara construction

In current algebras with conformal symmetry, there is a remarkable con-nection between the energy-momentum tensor and the currents, which leads to a great deal of interesting structure. Define the operator

: jj(z1) : = lim

with the sum on a implicit. We first wish to show that up to normalization the OPE of : jj : with j^a is the same as that of TB with j^a. This takes a bit of effort; the same calculation is organized in a different way in exercise 11.7.

The OPE of the product : jj : is not the same as the product of the OPEs, because the two currents in : jj : are closer to each other than they are to the third current; we must make a less direct argument using holomorphicity. Consider the following product: We have used the current–current OPE to determine the singularities as z3

approaches z1 or z2, with a holomorphic remainder. In this relation take z₂ → z1 and make a Laurent expansion in z21, being careful to expand both the operator products and the explicit z2 dependence. Keep the term of order z⁰₂₁ (there is some cancellation from the antisymmetry of f^cad) to obtain Here h(g) is again the dual Coxeter number. Define

T_B^s(z) = 1

(k + h(g))ψ² : jj(z) : . (11.5.21) The OPE of T_B^s with the current is the same as that of the energy-momentum tensor TB(z),

T_B^s(z)j^c(0) ∼ TB(z)j^c(0) . (11.5.22)

Now repeat the above with j^c(z3) replaced by T_B^s(z3), Again expand in z21 and keep the term of order z₂₁⁰ to obtain

T_B^s(z1)T_B^s(z3)∼ c^g,k

This is of the standard form for an energy-momentum tensor, with central charge c^g,k. The Laurent coefficients

L^s₀ = 1 satisfy a Virasoro algebra with this central charge. The vanishing of the normal ordering constant in L^s₀ can be deduced by noting that holomor-phicity requires L^s₀ and also j_n^a for n≥ 0 to annihilate the state |1 .

We have used the jj OPE to determine the : jj :: jj : OPE. We could not do this directly, because the jj OPE is valid only for two operators close compared to all others, and in this case there are two additional currents in the vicinity. Naive application of the OPE would give the wrong normalization for T^s and c^g,k. The argument above uses the OPE only where it is valid, and then takes advantage of holomorphicity. The operator T_B^s constructed from the product of two currents is known as a Sugawara energy-momentum tensor.

Finding the Sugawara tensor for a U(1) current algebra is easy. With the normalization (11.5.14) it is simply

T_B^s = 1

2 : jj : , (11.5.27)

as one sees by writing the current in terms of a free boson, j = i∂H.

The tensor T_B^s may or may not be equal to the total TB of the CFT.

Define

T_B^ = T_B− TB^s . (11.5.28)

11.5 Current algebras 71 Since the T_B and T_B^s OPEs with j^a have the same singular terms, the product

T_B^(z1)j^a(z2)∼ 0 (11.5.29) is nonsingular. Since T_B^s itself is constructed from the currents, this implies T_B^sT_B^ ∼ 0. Then

T_B^(z)T_B^(0) = T_B(z)T_B(0)− T_B^s(z)T_B^s(0)− T_B^(z)T_B^s(0)− T_B^s(z)T_B^(0)

∼ c^ 2z⁴ + 2

z²T_B^(0) +1

z∂T_B^(0) , (11.5.30)

the standard T T OPE with central charge

c^ = c− c^g,k . (11.5.31)

The internal theory thus separates into two decoupled CFTs. One has an energy-momentum tensor T_B^s constructed entirely from the current, and the other an energy-momentum tensor T_B^ that commutes with the current. We will use the term current algebra to refer to the first factor alone, since the two CFTs are completely independent. For a unitary CFT c^ must be nonnegative and so

c^g,k≤ c , (11.5.32)

and T_B^ is trivial precisely if

c^g,k= c , (11.5.33)

in which case T_B = T_B^s.

We now consider examples. The dual Coxeter number can be writ-ten as a sum over the roots. For any simply-laced algebra, h(g) + 1 = dim(g)/rank(g), and so

c^g,k = k dim(g) rank(g)

dim(g) + (k− 1)rank(g) . (11.5.34) For any simply-laced algebra at k = 1, the central charge is therefore

c^g,1 = rank(g) . (11.5.35)

For the E8×E8and S O(32) heterotic strings, this is the same as the central charge of the free fermion representation, and for the free boson repre-sentation of the next section: these are Sugawara theories. The operator : jj : looks as though it should be quartic in the fermions, but by using the OPE and the antisymmetry of the fermions one finds that T_B^s reduces to the usual −¹₂λ^A∂λ^A.

Another example is S U(2) = S O(3), for which c^g,k= 3k

2 + k = 1, 3 2, 9

5, 2, 15

7 , . . .→ 3 . (11.5.36)

We have seen the first CFT in this series (the self-dual point of toroidal compactification) and the second (free fermions). Most levels do not have a free-field representation. For any current algebra the central charge lies in the range

rank(g)≤ c^g,k≤ dim(g) . (11.5.37) The first equality holds only for a simply-laced algebra at level one, and the second only for an Abelian algebra or in the limit k→ ∞.

Primary fields

By acting repeatedly with the lowering operators j_n^a with n > 0, one reaches a highest weight or primary state of the current algebra, a state annihilated by all the j_n^a for n > 0. It is therefore also annihilated by the L^s_n for n > 0, eq. (11.5.26), so is a highest weight state of the Virasoro algebra. The center-of-mass generators j₀^atake primary states into primary states, so the latter form a representation of the algebra g,

j₀^a|r, i = |r, j t^ar,ji , (11.5.38) with r (not summed) labeling the representation. It then follows that

L^s₀|r, i = 1

(k + h(g))ψ²|r, k t^ar,kjt^a_r,ji

= Q_r

(k + h(g))ψ²|r, i , (11.5.39) with Qr the Casimir (11.4.7). The weights of the primary fields are thus determined in terms of the algebra, level, and representation,

h_r= Q_r

(k + h(g))ψ² = Q_r 2ˆk + Qg

, (11.5.40)

where Qg is the Casimir for the adjoint representation. For S U(2) at level k, the weight of the spin-j primary is

hj = j(j + 1)

k + 2 . (11.5.41)

It is also true that at any given level, only a finite number of represen-tations are possible for the primary states. For any root α of g and any weight λ of r, the S U(2) algebra (11.5.12b) implies that

r, λ| [ E1^α, E₋₁^−α]|r, λ = 2 r, λ|(α · H0+ ˆk)|r, λ /α²

= 2(α· λ + ˆk)/α² . (11.5.42) The left-hand side isE₋₁^−α|r, λ ² ≥ 0, and so ˆk ≥ −α · λ. Combining this with the same for−α gives

ˆk ≥ |α · λ| (11.5.43)

In document Polchinski J. String Theory. Vol. 2. Superstring Theory and Beyond (Page 88-95)