[Review] DDF states

Posted by Urs Schreiber

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The recent discussion about attempts to quantize the (super-)string by studying its algebra of classical invariants (i.e. of phase space functions that Poisson-commute with the (super-)Virasoro constraints) has shown that perhaps one should not forget about the special role played by the so-called DDF operators in the worldsheet (S)CFT.

These operators, named after DelGuidice, DiVecchia and Fubini, have the special property that they commute with the entire (super-)Virasoro algebra. Applying them to any physical state hence yields another physical state. Indeed, they are even complete in that they generate the spectrum generating algebra, i.e. all physical states of the string can be obtained by acting with DDF operators on a massless (or tachyonic) state.

The DDF operators can furthermore be chosen so as to satisfy the usual (super-)oscillator algebra and hence they neatly encode all the information about the string except for that contained in the worldsheet ground states themselves. It is straightforward to go from the quantum DDF operators to the corresponding classical invariant observables on the string’s phase space, and these DDF invariants should hence be an alternative to the Pohlmeyer charges. (But I have to admit that so far all I know about the theory of Pohlmeyer charges is what is summarized in Thiemann’s paper.)

Before having a closer look at the relation between Pohlmeyer charges and DDF invariants I would like to review the construction of the most general DDF operators in (S)CFT here.

In the standard textbook literature one can find

- in Green, Schwarz & Witten (using non-CFT language) the construction of

- transversal bosonic (section 2.3.2)

- transversal supersymmetric (section 4.3.2)

- longitudinal bosonic (p. 11)

and in Polchinski (using CFT language) the construction of

- transversal bosonic (eq. (8.2.29))

DDF states.

Here I give a summary and derivation (in CFT language) of all

- transversal and longitudinal, bosonic and fermionic

DDF states (for a free supersymmetric worldsheet theory).

This is also summarized in the following notes

Urs Schreiber, DDF-like classical invariants of (super)string.

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Using the usual normalization

(1)

X^\mu(z) X^\nu(0) \sim -\frac{\alpha^\prime}{2}\eta^{\mu\nu} \ln z

(2)

\psi^\mu(z)\psi^\nu(0) \sim \frac{\eta^{\mu\nu}}{z}

for the bosonic and fermionic worldsheet fields, the (super-)Virasoro currents read

(3)

T = -\frac{1}{\alpha^\prime} \partial X \cdot \partial X(z) -\frac{1}{2}\psi \cdot \partial \psi

(4)

T_{\mathrm{F}} = i \sqrt{\frac{\alpha^\prime}{2}} \psi \cdot \partial X \,.

The DDF operators are defined as a set of operators that commute with all modes of $T$ and $T_\mathrm{F}$ (are ‘gauge invariant observables’) and satisfy an algebra that mimics that of creation/annihilation operators.

First of all one needs to single out two linearly independent lightlike Killing vectors $p$ and $k$ on target space, and we choose to normalize them so that $p\cdot k = 2$ . The span of $p$ and $k$ is called the longitudinal space and its orthogonal complement is called the tranverse space.

For

(5)

\mathcal{O}(z) = \sum_{-\infty}^{\infty} \mathcal{O}_m z^{-(m+h)}

a primary field of weight $h$ we shall refer to the OPE

(6)

T(z)\mathcal{O}(0) \sim \frac{h}{z^2}\mathcal{O}(0) + \frac{1}{z} \partial\mathcal{O}(0)

as the tensor law for the sake of conciseness of some of the following formulas.

The modes of $T$ and $T_\mathrm{F}$ are denoted by $L_m$ and $G_{m-\nu}$ as usual.

The crucial idea behind the construction of DDF states is to make use of the fact that the 0-modes of $h=1$ primary fields commute with all Virasoro modes $L_m$ . In canonical language is nothing but the fact that the integral over a unit weight density is reparametrization invariant.

Therefore the task of finding DDF states is reduced to that of finding linearly independent $h=1$ fields that have the desired commutation relations and, in the case of the superstring, are closed with respect to $T_\mathrm{F}$ (see below).

bosonic string:

For the bosonic string the DDF operators $A^\mu_n$ are defined by

(7)

A^\mu_n \propto \oint \frac{dz}{2\pi i} \left( \partial X^\mu + k^\mu \frac{\alpha^\prime}{8} in\, \partial \ln k\cdot \partial X \right) e^{in k\cdot X} \,.

Let’s check that this is really an invariant:

First consider the transverse DDF operators. For $v$ a transverse target space vector the operator $v \cdot A_m$ is manifestly the 0-mode of an $h=1$ primary field (the exponential factor has $h=0$ due to $k\cdot k = 0$ ) and hence invariant.

Furthermore $k\cdot A_n \propto \delta_{n,0} k \cdot \oint \partial X$ also obviously commutes with the $L_m$ .

The only subtlety arises for the longitudinal $p \cdot A_n$ . Here, the non-tensor behaviour of

(8)

T(z) p\cdot \partial X e^{in k\cdot X}(w) \sim -\frac{\alpha^\prime}{2} \frac{in}{(z-w)^3} e^{in k\cdot X}(w) + [(h=1){-tensor law}]

is precisely canceled by the curious logarithmic correction term

(9)

\partial \ln k\cdot \partial X(z) = \frac{k\cdot \partial^2 X} {k \cdot \partial X}(z) \,.

Namely because of

(10)

T(z) \partial^2 X^\mu(0) \sim \frac{2 \partial X^\mu(w)}{(z-w)^3} + [(h=2)-{tensor law}]

one has

(11)

\Rightarrow T(z) \, \frac{k\cdot \partial^2 X} {k\cdot \partial X} e^{in k\cdot X}(w) \sim \frac{2 e^{in k\cdot X}}{(z-w)^3}(w) + [(h=1)-{tensor law}] \,,

which hence makes the entire integrand of $p\cdot A_m$ transform as an $h=1$ primary, as desired.

superstring:

The analogous construction for the superstring has to ensure in addition that the DDF operators commute with the supercharges $G_{m-\nu}$ . This is simply achieved by ‘closing’ the integral over a given $h=1/2$ primary field $D(z)$ to obtain the operator

(12)

[G_{-\nu},D_{\nu}] = \left[ \oint \frac{dz}{2\pi i} T_\mathrm{F}\left(z\right) , \oint \frac{dz}{2\pi i} D\left(z\right) \right] \,\,\,\,\, \left\{ \array{ \nu = 0 & {R sector}\\ \nu = 1/2 & {NS sector} } \right. \,.

Here and in the following the brackets denote supercommutators.

The resulting operator is manifestly the zero mode of a weight $h=1$ tensor and hence commutes with all $L_n$ . Furthermore it commutes with $G_{-\nu}$ because of

(13)

\left[G_{-\nu}, \left[G_{-\nu},D_\nu\right]\right] = [L_{-2\nu},D_\nu] = 0 \,,

where we have used the tensor law

(14)

[ L_m, \mathcal{O}_n ] = ((h-1)m-n) \mathcal{O}_{m+n}

in the last step

Since $G_{-\nu}$ and $L_{\pm 1}$ generate the entire algebra the ‘closed’ operator $[G_{-\nu},D_\nu]$ indeed commutes with all $L_n$ , $G_n$ .

It is therefore clear that the superstring DDF operators, which we can define as

(15)

A_n^\mu := \left\{ G_\nu, { \oint \frac{dz}{2\pi i} \psi^\mu e^{i n k\cdot X}\left(z\right) } \right\}

(16)

B_n^\mu := \left[ {G_\nu}, { \oint \frac{dz}{2\pi i} \left( \psi^\mu \,k\cdot \psi - \frac{1}{4} k^\mu \partial \ln k\cdot \partial X \right) \frac{e^{in k \cdot X}} {\sqrt{k\cdot \partial X}} } \right]

commute with the super-Virasoro generators, since the second arguments of the supercommutators are integrals over weight 1/2 tensors. The nature and purpose of the logarithmic correction term in the second line is just as discussed for the bosonic theory above: It cancels the non-tensor-law term in

(17)

T(z) \, p\cdot \psi\, k\cdot\psi \frac{e^{i k\cdot X}} {\sqrt{k\cdot \partial X}}(w) \sim \frac{1}{(z-w)^3} \frac{e^{i n k\cdot X}} {\sqrt{k\cdot \partial X}} + [(h=1/2)-{tensor law}] \,.

Evaluating the supercommutators in the definition of the superstring DDF operators yields the explicit form for $A^\mu_n$ and $B^\mu_n$ :

(18)

A^\mu_n = i \sqrt{\frac{2}{\alpha^\prime}} \oint \frac{dz}{2\pi i} \left( \partial X^\mu + \frac{\alpha^\prime}{2}in \psi^\mu \, k\cdot \psi \right) e^{ink\cdot X} (z)

(19)

B^\mu_n = i \sqrt{\frac{2}{\alpha^\prime}} \oint \frac{dz}{2\pi i} \left( \partial X^\nu \, k\cdot \psi - \psi^\mu k\cdot \partial X + \frac{\alpha^\prime}{4} \psi^\mu \, k\cdot \psi\, k\cdot \partial \psi \frac{1}{k\cdot \partial X} \right) \frac{e^{ink\cdot X}} {\sqrt{k\cdot \partial X}} (z) + i \sqrt{\frac{2}{\alpha^\prime}} \oint \frac{dz}{2\pi i} k^\mu \left( k\cdot \psi f_1({k\cdot X,k\cdot \partial X}) + k\cdot \partial \psi f_2({k\cdot X, k\cdot \partial X}) \right) (z) \,.

It is straightforward to check that the transversal DDF operators have the usual oscillator (super-)commutators when appropriate normalization factors are included.

Posted at January 29, 2004 1:16 PM UTC

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January 29, 2004