Worldsheet Supersymmetry and the NSR Action

The bosonic string has two glaring problems if it is meant to describe nature: it has a tachyon, and its perturbative spectrum contains no spacetime fermions. The Neveu-Schwarz-Ramond formulation addresses both issues by adding worldsheet fermions $\psi^\mu$ as superpartners of the embedding coordinates $X^\mu$.

The central idea is modest but powerful:

$$\boxed{ X^\mu(\sigma)\quad\text{and}\quad\psi^\mu(\sigma) \quad\text{form a two-dimensional supersymmetric multiplet.} }$$

Spacetime supersymmetry is not yet manifest in this formulation. It emerges only after the correct projection on the Hilbert space, the GSO projection, which will be discussed later.

The NSR route to superstrings: add worldsheet fermions, impose local worldsheet supersymmetry, quantize NS and R sectors, and finally apply the GSO projection.

Worldsheet spinors

Let $\sigma^a=(\tau,\sigma)$ be Lorentzian worldsheet coordinates. A two-dimensional Majorana spinor can be written as

$$\psi^\mu= \begin{pmatrix} \psi_-^\mu\\ \psi_+^\mu \end{pmatrix},$$

where $\psi_+^\mu$ and $\psi_-^\mu$ are right-moving and left-moving components after imposing the equations of motion. The target-space index $\mu$ tells us that for each spacetime coordinate $X^\mu$ we introduce a worldsheet spinor $\psi^\mu$.

A convenient set of two-dimensional gamma matrices $\rho^a$ satisfies

$$\{\rho^a,\rho^b\}=2\eta^{ab}_{\rm ws}, \qquad \eta^{ab}_{\rm ws}=\operatorname{diag}(-,+).$$

The Dirac adjoint is $\bar\psi=\psi^\dagger\rho^0$. Since the spinor is a worldsheet spinor but a spacetime vector, the NSR matter fields transform as

$$X^\mu: \text{worldsheet scalar, spacetime vector}, \qquad \psi^\mu: \text{worldsheet spinor, spacetime vector}.$$

The worldsheet supermultiplet contains a bosonic embedding coordinate $X^\mu$ and a Majorana fermion $\psi^\mu$. The target-space index is the same on both fields.

The NSR matter action

In flat target space and conformal gauge, the Lorentzian NSR matter action is

$$\boxed{ S_{\rm m} = -\frac{1}{4\pi\alpha'}\int d^2\sigma \left( \partial_aX^\mu\partial^aX_\mu -i\bar\psi^\mu\rho^a\partial_a\psi_\mu \right). }$$

The equations of motion are

$$\partial_a\partial^aX^\mu=0, \qquad \rho^a\partial_a\psi^\mu=0.$$

The fermion equation splits into chiral components:

$$\partial_-\psi_+^\mu=0, \qquad \partial_+\psi_-^\mu=0.$$

Thus, just like the bosonic string coordinate, the fermion decomposes into left- and right-moving pieces:

$$\psi^\mu_+(\tau,\sigma)=\psi^\mu_+(\tau+\sigma), \qquad \psi^\mu_-(\tau,\sigma)=\psi^\mu_-(\tau-\sigma).$$

In Euclidean complex coordinates, the holomorphic fields obey the OPEs

$$X^\mu(z,\bar z)X^\nu(w,\bar w) \sim -\frac{\alpha'}{2}\eta^{\mu\nu}\ln |z-w|^2,$$

and

$$\boxed{ \psi^\mu(z)\psi^\nu(w)\sim\frac{\eta^{\mu\nu}}{z-w}, \qquad \tilde\psi^\mu(\bar z)\tilde\psi^\nu(\bar w) \sim\frac{\eta^{\mu\nu}}{\bar z-\bar w}. }$$

The holomorphic and antiholomorphic fermions are independent on the closed-string worldsheet; for open strings they are related at the boundary.

Rigid worldsheet supersymmetry

The action is invariant under the rigid supersymmetry transformations

$$\boxed{ \delta X^\mu=\bar\epsilon\psi^\mu, \qquad \delta\psi^\mu=-i\rho^a\epsilon\,\partial_aX^\mu, }$$

where $\epsilon$ is a constant anticommuting Majorana spinor. The transformation says that the bosonic coordinate and the worldsheet fermion are superpartners.

Rigid worldsheet supersymmetry exchanges $X^\mu$ and $\psi^\mu$. Acting twice gives a translation along the worldsheet.

Indeed, the commutator of two such transformations closes on a worldsheet translation:

$$[\delta_{\epsilon_1},\delta_{\epsilon_2}]X^\mu = 2i\bar\epsilon_1\rho^a\epsilon_2\,\partial_aX^\mu,$$

and similarly for $\psi^\mu$, up to the equations of motion. This is the characteristic algebra of supersymmetry: the square of a supercharge is a translation.

Local worldsheet supersymmetry and superconformal gauge

The full NSR string is obtained by coupling the matter fields to two-dimensional worldsheet supergravity. In addition to the metric $h_{ab}$, one introduces a gravitino $\chi_a$, the gauge field for local worldsheet supersymmetry.

The local gauge symmetries are the supersymmetric extension of the Polyakov symmetries:

$$\begin{array}{ccl} \text{diffeomorphisms} &:& h_{ab}\text{ and fields transform geometrically},\\[1mm] \text{Weyl transformations} &:& h_{ab}\mapsto e^{2\omega}h_{ab},\\[1mm] \text{local supersymmetry} &:& \chi_a\text{ gauges supersymmetry},\\[1mm] \text{super-Weyl transformations} &:& \text{remove the gamma-trace of }\chi_a. \end{array}$$

Using these symmetries, one can choose superconformal gauge:

$$h_{ab}=e^{2\omega}\eta_{ab}, \qquad \chi_a=0.$$

Superconformal gauge fixes the worldsheet metric and gravitino. The residual gauge symmetry becomes the superconformal symmetry generated by $T$ and $G$.

The equations obtained by varying $h_{ab}$ and $\chi_a$ survive as constraints:

$$T_{ab}=0, \qquad G_a=0.$$

In the quantum theory these become the super-Virasoro physical-state conditions.

Stress tensor and supercurrent

In holomorphic language, the matter stress tensor is

$$\boxed{ T(z)=T_X(z)+T_\psi(z) =-\frac{1}{\alpha'}:\partial X^\mu\partial X_\mu: -\frac12:\psi^\mu\partial\psi_\mu:. }$$

The holomorphic supercurrent is

$$\boxed{ G(z)=i\sqrt{\frac{2}{\alpha'}}:\psi^\mu\partial X_\mu:. }$$

There are analogous antiholomorphic currents $\tilde T(\bar z)$ and $\tilde G(\bar z)$ for closed strings.

The stress tensor $T$ generates conformal transformations, while the supercurrent $G$ generates worldsheet supersymmetry transformations in the matter CFT.

The dimensions are

$$X^\mu: h=0, \qquad \partial X^\mu: h=1, \qquad \psi^\mu: h=\frac12, \qquad T: h=2, \qquad G: h=\frac32.$$

The basic OPEs imply, for example,

$$T(z)\psi^\mu(w) \sim \frac{\frac12\psi^\mu(w)}{(z-w)^2} + \frac{\partial\psi^\mu(w)}{z-w},$$

so $\psi^\mu$ is a primary field of weight $1/2$. Similarly $\partial X^\mu$ is a primary of weight $1$.

The next page will organize the OPEs of $T$ and $G$ into the $\mathcal N=1$ superconformal algebra.

Critical dimension preview

The matter central charge is the sum of the bosonic and fermionic contributions:

$$c_X=D, \qquad c_\psi=\frac{D}{2}, \qquad c_{\rm matter}=\frac{3D}{2}.$$

Gauge fixing the NSR string also produces two ghost systems:

$$(b,c): c=-26, \qquad (\beta,\gamma): c=11.$$

Thus the total central charge is

$$c_{\rm total}=\frac{3D}{2}-26+11 =\frac{3D}{2}-15.$$

Quantum super-Weyl invariance requires $c_{\rm total}=0$, so

$$\boxed{D=10.}$$

The NSR matter fields have central charge $3D/2$. The reparametrization ghosts and superghosts contribute $-15$, so criticality requires $D=10$.

This is the superstring analogue of the bosonic result $D=26$. The removal of the tachyon and the emergence of spacetime supersymmetry require additional input: the NS/R sectors and the GSO projection.

Exercises

Exercise 1. Equations of motion of the NSR action

Starting from

$$S_{\rm m} = -\frac{1}{4\pi\alpha'}\int d^2\sigma \left( \partial_aX^\mu\partial^aX_\mu -i\bar\psi^\mu\rho^a\partial_a\psi_\mu \right),$$

show that the equations of motion are $\partial^2X^\mu=0$ and $\rho^a\partial_a\psi^\mu=0$.

Solution

Varying $X^\mu$ gives

$$\delta S_X =-\frac{1}{2\pi\alpha'}\int d^2\sigma\,\partial_a\delta X^\mu\partial^aX_\mu.$$

After integrating by parts and dropping the boundary term, this becomes

$$\delta S_X =\frac{1}{2\pi\alpha'}\int d^2\sigma\,\delta X^\mu\partial_a\partial^aX_\mu,$$

so $\partial_a\partial^aX^\mu=0$.

For the fermion, varying $\bar\psi^\mu$ gives

$$\delta S_\psi =\frac{i}{4\pi\alpha'}\int d^2\sigma\,\delta\bar\psi^\mu\rho^a\partial_a\psi_\mu.$$

Thus

$$\rho^a\partial_a\psi^\mu=0.$$

Exercise 2. Closure of rigid worldsheet supersymmetry

Using

$$\delta X^\mu=\bar\epsilon\psi^\mu, \qquad \delta\psi^\mu=-i\rho^a\epsilon\partial_aX^\mu,$$

show that two supersymmetry transformations close on a translation of $X^\mu$.

Solution

Acting twice on $X^\mu$ gives

$$\delta_1\delta_2X^\mu =\bar\epsilon_2\delta_1\psi^\mu =-i\bar\epsilon_2\rho^a\epsilon_1\partial_aX^\mu.$$

Antisymmetrizing the two transformations and using the anticommuting nature of the spinor parameters gives

$$[\delta_1,\delta_2]X^\mu =2i\bar\epsilon_1\rho^a\epsilon_2\partial_aX^\mu.$$

This is a translation with parameter

$$v^a=2i\bar\epsilon_1\rho^a\epsilon_2.$$

Exercise 3. The fermion conformal weight

Use the OPE

$$\psi^\mu(z)\psi^\nu(w)\sim\frac{\eta^{\mu\nu}}{z-w}$$

and

$$T_\psi(z)=-\frac12:\psi^\mu\partial\psi_\mu:$$

to show that $\psi^\mu$ has conformal weight $1/2$.

Solution

In the OPE $T_\psi(z)\psi^\mu(w)$, there are two contractions. The result is

$$T_\psi(z)\psi^\mu(w) \sim \frac{\frac12\psi^\mu(w)}{(z-w)^2} + \frac{\partial\psi^\mu(w)}{z-w}.$$

This is the primary-field OPE with conformal weight

$$h=\frac12.$$

Exercise 4. Matter central charge

Show that $D$ free bosons and $D$ free Majorana fermions have matter central charge $c_{\rm matter}=3D/2$.

Solution

Each free boson contributes $c=1$, so $D$ bosons contribute

$$c_X=D.$$

Each real chiral Majorana fermion contributes $c=1/2$, so $D$ such fermions contribute

$$c_\psi=\frac{D}{2}.$$

Therefore

$$c_{\rm matter}=c_X+c_\psi =D+\frac{D}{2} =\frac{3D}{2}.$$

Exercise 5. Critical dimension of the NSR string

Using $c_{\rm matter}=3D/2$, $c_{bc}=-26$, and $c_{\beta\gamma}=11$, derive the critical dimension of the NSR string.

Solution

The total central charge is

$$c_{\rm total}=\frac{3D}{2}-26+11=\frac{3D}{2}-15.$$

Quantum super-Weyl invariance requires

$$c_{\rm total}=0.$$

Therefore

$$\frac{3D}{2}=15,$$

and hence

$$D=10.$$