Vertex operators and bosonization

Consider first the unit operator. Fields remain holomorphic at the ori-gin, and in particular they are single-valued. From the Laurent expan-sion (10.2.7), the single-valuedness means that the unit operator must be in the NS sector; the conformal transformation that takes the incoming string to the point z = 0 cancels the branch cut from the antiperiodicity.

The holomorphicity of ψ at the origin implies, via the contour argument, that the state corresponding to the unit operator satisfies

ψ^µ_r|1 = 0 , r = 1 2,3

2, . . . , (10.3.1) and therefore

|1 = |0 . (10.3.2)

Since the ψψ OPE is single-valued, all products of ψ and its derivatives must be in the NS sector. The contour argument gives the map

ψ_−r^µ → 1

(r− 1/2)!∂^r−1/2ψ^µ(0) , (10.3.3) so that there is a one-to-one map between such products and NS states.

The analog of the Noether relation (2.9.6) between the superconformal variation of an NS operator and the OPE is

δ_ηA(z, ¯z) = −*^∞

n=0

1 n!



∂ⁿη(z)G_n−1/2+ (∂ⁿη(z))^∗G˜_n−1/2^· A(z, ¯z) . (10.3.4) The R sector vertex operators must be more complicated because the Laurent expansion (10.2.7) has a branch cut. We have encountered this before, for the winding state vertex operators in section 8.2 and the orbifold

10.3 Vertex operators and bosonization 11 twisted state vertex operators in section 8.5. Each of these introduces a branch cut (the first a log and the second a square root) into X^µ. For the winding state vertex operators there was a simple expression as the exponential of a free field. For the twisted state vertex operators there was no simple expression and their amplitudes are determined only with more effort. Happily, through a remarkable property of two-dimensional field theory, the R sector vertex operators can be related directly to the bosonic winding state vertex operators.

Let H(z) be the holomorphic part of a scalar field,

H(z)H(0)∼ − ln z . (10.3.5)

For world-sheet scalars not associated directly with the embedding of the string in spacetime this is the normalization we will always use, corresponding to α^ = 2 for the embedding coordinates. As in the case of the winding state vertex operators we can be cavalier about the location of the branch cut as long as the final expressions are single-valued. We will give a precise oscillator definition below. Consider the basic operators e^±iH(z). These have the OPE

e^iH(z)e^−iH(0) ∼ 1

z , (10.3.6a)

e^iH(z)e^iH(0) = O(z) , (10.3.6b)

e^−iH(z)e^−iH(0) = O(z) . (10.3.6c)

The poles and zeros in the OPE together with smoothness at infinity determine the expectation values of these operators on the sphere, up to an overall normalization which can be set to a convenient value:

i

e^i*iH(zi) 

S2

=

i<j

z_ij^*i*j , ^

i

*_i= 0 . (10.3.7) The *i are±1 here, but this result holds more generally.

Now consider the CFT of two Majorana–Weyl fermions ψ^1,2(z), and form the complex combinations

ψ = 2^−1/2(ψ¹+ iψ²) , ψ = 2^−1/2(ψ¹− iψ²) . (10.3.8) These have the properties

ψ(z)ψ(0) ∼ 1

z , (10.3.9a)

ψ(z)ψ(0) = O(z) , (10.3.9b)

ψ(z)ψ(0) = O(z) . (10.3.9c)

Eqs. (10.3.6) and (10.3.9) are identical in form, and so the expectation values of ψ(z) on the sphere are identical to those of e^iH(z). We will write ψ(z) ∼=e^iH(z) , ψ(z) ∼=e^−iH(z) (10.3.10)

to indicate this. Of course, all of this extends to the antiholomorphic case,

˜

ψ(¯z) ∼=e^{i ˜H(¯z)} , ψ( ) ∼˜ =e^{−i ˜H(¯z)} . (10.3.11)

R and kL can be formed by repeated operator products of e^±iH(z) and e^{±i ˜H(¯z)}, and arbitrary local operators built out of the fermions and their derivatives can be formed by repeated operator products of ψ(z), ψ(z), ˜ψ(¯z), and ˜ψ(¯z), the equivalence of the theories can be extended to all local operators. Finally, in order for these theories to be the same as CFTs, the energy-momentum tensors must be equivalent. The easiest way to show this is via the operator products

e^iH(z)e^−iH(−z) = 1

2z + i∂H(0) + 2zT_B^H(0) + O(z²) , (10.3.12a) ψ(z)ψ(−z) = 1

2z + ψψ(0) + 2zT_B^ψ(0) + O(z²) . (10.3.12b) With the result (10.3.10), this implies equivalence of the H momentum current with the ψ number current, and of the two energy-momentum tensors,

ψψ ∼=i∂H , T_B^ψ ∼=T_B^H . (10.3.13) As a check, e^iH and ψ are both (¹₂, 0) tensors.

In the operator description of the theory, define

ψ(z) ∼=^◦◦e^iH(z)◦◦ . (10.3.14) From the Campbell–Baker–Hausdorff (CBH) formula (6.7.23) we have for equal times|z| = |z^|

◦◦e^iH(z)◦◦◦

◦e^iH(z)◦◦ = exp{−[H(z), H(z^)]}^◦◦e^iH(z)◦◦◦

◦e^iH(z)◦◦

= −^◦◦e^iH(z)◦◦◦◦e^iH(z)◦◦ , (10.3.15) where we have used the fact (8.2.21) that at equal times [H(z), H(z^)] =

±iπ. Thus the bosonized operators do anticommute. This is possible for operators constructed purely out of bosons because they are nonlocal. In particular, note that the CBH formula gives the equal time commutator

H(z)^◦◦e^iH(z)◦◦ = ^◦◦e^iH(z)◦◦



H(z) + i[H(z), H(z^)]^

= ^◦◦e^iH(z)◦◦



H(z)− π sign(σ1− σ1^)



, (10.3.16) so that the fermion field operator produces a kink, a discontinuity, in the bosonic field.

This rather surprising equivalence is known as bosonization. Equiva-lence between field theories with very different actions and fields occurs frequently in two dimensions, especially in CFTs because holomorphicity puts strong constraints on the theory. (The great recent surprise is that it is

¯z

Since arbitrary local operators with integer k

10.3 Vertex operators and bosonization 13 also quite common in higher-dimensional field and string theories.) Many interesting CFTs can be constructed in several different ways. One form or another will often be more useful for specific purposes. Notice that there is no simple correspondence between one-boson and one-fermion states. The current, for example, is linear in the boson field but quadratic in the fermion field. A single boson is the same as one ψ fermion and one ψ fermion at the same point. On a Minkowski world-sheet, where holomorphic becomes left-moving, the fermions both move left at the speed of light and remain coincident, indistinguishable from a free boson.

A single fermion, on the other hand, is created by an operator exponential in the boson field and so is a coherent state, which as we have seen is in the shape of a kink (10.3.16).

The complicated relationship between the bosonic and fermionic spectra shows up also in the partition function. Operator products of e^±iH(z) gen-erate all operators with integer kL. The bosonic momentum and oscillator sums then give

Tr (q^L0) =



kL∈Z

q^kL2/2 _∞

n=1

(1− qⁿ)⁻¹ . (10.3.17)

In the NS sector of the fermionic theory, the oscillator sum gives Tr (q^L0) =

∞ n=1

(1 + q^n−1/2)² . (10.3.18) We know indirectly that these must be equal, since we can use the OPE to construct an analog in the fermionic theory for any local operator of the bosonic theory and vice versa. Expanding the products gives

1 + 2q^1/2+ q + 2q^3/2+ 4q²+ 4q^5/2+ . . . (10.3.19) for each, and in fact the equality of (10.3.17) and (10.3.18) follows from the equality of the product and sum expressions for theta functions, section 7.2.

Note that while bosonization was derived for the sphere, the sewing construction from chapter 9 guarantees that it holds on all Riemann surfaces, provided that we make equivalent projections on the spectra.

In particular, we have seen that summing over integer kL corresponds to summing over all local fermionic operators, the NS sector.

Bosonization extends readily to the R sector. In fact, once we combine two fermions into a complex pair we can consider the more general periodicity condition

ψ(w + 2π) = exp(2πiν) ψ(w) (10.3.20) for any real ν. In ten dimensions only ν = 0,¹₂ arose, but these more gen-eral periodicities are important in less symmetric situations. The Laurent

expansion has the same form (10.2.7) as before, ψ(z) = ^

r∈Z+ν

ψr

z^r+1/2 , ψ(z) = ^

s∈Z−ν

ψs

z^s+1/2 , (10.3.21)

with indices displaced from integers by±ν. The algebra is

{ψr, ψ_s} = δr,−s . (10.3.22) Define a reference state|0 _ν by

ψ_n+ν|0 _ν = ψn+1−ν|0 _ν = 0 , n = 0, 1, . . . . (10.3.23) The first nonzero terms in the Laurent expansions are then r =−1 + ν and s =−ν, so for the corresponding local operator Aν the OPE is

ψ(z)Aν(0) = O(z^−ν+1/2) , ψ(z)Aν(0) = O(z^ν−1/2) . (10.3.24) The conditions (10.3.23) uniquely identify the state|0 _ν, and so the corre-sponding OPEs (10.3.24) determine the bosonic equivalent

exp[i(−ν + 1/2)H] ∼=Aν . (10.3.25) One can check the identification (10.3.25) by verifying that the weight is h = ¹₂(ν−¹₂)². In the bosonic form this comes from the term ¹₂p² in L0. In the fermionic form it follows from the usual commutator method (2.7.8) or the zero-point mnemonic.

The boundary condition (10.3.20) is the same for ν and ν + 1, but the reference state that we have defined is not. It is a ground state only for 0≤ ν ≤ 1. As we vary ν, the state |0 ν changes continuously, and when we get back to the original theory at ν + 1, by the definition (10.3.23) it has become the excited state

|0 _ν+1 = ψ_−ν|0 _ν . (10.3.26) This is known as spectral flow. For the R case ν = 0 there are the two degenerate ground states

|s ∼=e^isH, s =±¹₂ . (10.3.27) For the superstring in ten dimensions we need five bosons, H^a for a = 0, . . . , 4. Then²

2^−1/2(±ψ⁰+ ψ¹) ∼= e^±iH0 (10.3.28a) 2^−1/2(ψ^2a± iψ^2a+1) ∼= e^±iHa , a = 1, . . . , 4 . (10.3.28b)

2The precise operator definition has a subtlety when there are several species of fermion. The H^a for different a are independent and so the exponentials commute rather than anticommute. A cocycle is needed, as in eq. (8.2.22). A general expression will be given in the next section.