Worldsheet Supersymmetry and the RNS Action

The bosonic string already contains the two signatures of string theory: an infinite tower of higher-spin states and, in the closed-string sector, a massless spin-two excitation. It also contains two serious defects. First, the ground state is tachyonic. Second, the spectrum has no spacetime fermions. A theory meant to include gravity and quantum matter cannot stop there.

The cure is supersymmetry, but there are two rather different things one might mean by that phrase:

• Spacetime supersymmetry acts on the target-space coordinates and turns spacetime bosons into spacetime fermions.
• Worldsheet supersymmetry acts on the two-dimensional fields living on the string worldsheet.

The Ramond—Neveu—Schwarz, or RNS, formulation begins with the second idea. It pairs each embedding coordinate $X^\mu(\sigma,\tau)$ with a two-dimensional Majorana fermion $\psi^\mu(\sigma,\tau)$. The index $\mu$ is still a target-space vector index; the spinor index is a worldsheet spinor index. This distinction is essential. RNS fermions are not yet spacetime spinors. Spacetime fermions arise after quantizing the Ramond sector, and spacetime supersymmetry appears only after the GSO projection.

This page builds the classical and conformal-field-theory foundation for that story. The next pages will quantize the RNS system, introduce NS/R boundary conditions, and extract the superstring spectrum.

From the bosonic string to a supersymmetric string

The bosonic open string has mass formula

$$\alpha' M^2=N-1,$$

while the closed bosonic string has

$$\alpha'M^2=4(N-1), \qquad N=\tilde N.$$

Thus both theories contain tachyons. The closed string also contains a massless scalar, antisymmetric two-form, and graviton at level $(N,\tilde N)=(1,1)$. In a hadronic interpretation, open strings would model mesons and closed strings would model glueballs; in a gravitational interpretation, the closed-string spin-two state is the great prize. The tachyon and the absence of fermions tell us that the bosonic string is not the final theory.

A tempting direct route is the Green—Schwarz idea: enlarge spacetime itself to superspace by adding target-space spinor coordinates $\theta^A$ to $X^\mu$. The natural supersymmetric one-form is schematically

$$\Pi_a^{\mu} =\partial_aX^\mu-i\bar\theta\Gamma^\mu\partial_a\theta,$$

because under a rigid spacetime supersymmetry transformation

$$\delta\theta=\epsilon, \qquad \delta X^\mu=i\bar\epsilon\Gamma^\mu\theta,$$

one has $\delta\Pi_a^\mu=0$. This makes spacetime supersymmetry manifest. The price is that the action has a local fermionic gauge symmetry, kappa symmetry, and covariant quantization becomes subtle.

The RNS route takes the opposite bargain. It makes the two-dimensional theory almost free and therefore easy to quantize, but spacetime supersymmetry is hidden at first. Instead of a target-space spinor $\theta$, introduce a worldsheet spinor field $\psi^\mu$ for each coordinate $X^\mu$.

The RNS formulation places $D$ matter supermultiplets on the worldsheet. The fermions $\psi^\mu_\pm$ are spinors on the worldsheet but vectors in spacetime.

The basic RNS matter multiplet is therefore

$$\bigl(X^\mu,\psi^\mu_+,\psi^\mu_-\bigr), \qquad \mu=0,1,\ldots,D-1.$$

Here $\psi_+^\mu$ and $\psi_-^\mu$ are the two chiral components of a two-dimensional Majorana spinor. They will become the left-moving and right-moving worldsheet fermions after Wick rotation.

Worldsheet spinors and gamma matrices

Let the Lorentzian worldsheet coordinates be

$$\sigma^0=\tau, \qquad \sigma^1=\sigma,$$

with metric

$$\eta_{ab}=\operatorname{diag}(-,+).$$

A convenient real representation of the two-dimensional gamma matrices is

$$\rho^0= \begin{pmatrix} 0&-1\\ 1&0 \end{pmatrix}, \qquad \rho^1= \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix},$$

so that

$$\{\rho^a,\rho^b\}=2\eta^{ab}.$$

The chirality matrix is

$$\rho^3=\rho^0\rho^1= \begin{pmatrix} -1&0\\ 0&1 \end{pmatrix}.$$

A Majorana spinor may be written as

$$\psi^\mu= \begin{pmatrix} \psi_-^\mu\\ \psi_+^\mu \end{pmatrix}, \qquad \bar\psi^\mu=(\psi^\mu)^T\rho^0.$$

The notation is deliberately suggestive: in light-cone coordinates

$$\sigma^\pm=\tau\pm\sigma, \qquad \partial_\pm={1\over2}(\partial_\tau\pm\partial_\sigma),$$

the two chiral components obey first-order equations. One chirality depends only on $\sigma^+$ and the other only on $\sigma^-$. Different books attach the words “left-moving” and “right-moving” to these components in slightly different ways, but the equations themselves are unambiguous.

The free RNS matter action

In conformal gauge the flat-worldsheet RNS action is

$$S_{\rm RNS} =-{1\over4\pi\alpha'}\int d^2\sigma\, \left( \partial_aX^\mu\partial^aX_\mu -i\bar\psi^\mu\rho^a\partial_a\psi_\mu \right).$$

The bosonic term is the Polyakov action in conformal gauge. The new term is first order in derivatives, as appropriate for fermions. With the spinor decomposition above, the action can be written equivalently as

$$S_{\rm RNS} ={1\over2\pi\alpha'}\int d^2\sigma\, \left( \partial_+X\cdot\partial_-X +{i\over2}\psi_+\cdot\partial_-\psi_+ +{i\over2}\psi_-\cdot\partial_+\psi_- \right),$$

up to the harmless overall sign convention associated with Lorentzian continuation. The equations of motion are

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

Thus

$$X^\mu(\sigma,\tau)=X_L^\mu(\sigma^+)+X_R^\mu(\sigma^-),$$

while

$$\psi_+^\mu=\psi_+^\mu(\sigma^+), \qquad \psi_-^\mu=\psi_-^\mu(\sigma^-).$$

The fermions are as free as the bosons, but their first-order nature is the key novelty. In quantization, their modes obey anticommutation relations rather than commutation relations. This one change is responsible for the Ramond-sector spacetime spinors and for the eventual cancellation of the bosonic tachyon.

Global worldsheet supersymmetry

The flat action is invariant under the rigid worldsheet supersymmetry transformations

$$\delta X^\mu=i\epsilon^+\psi_+^\mu+i\epsilon^-\psi_-^\mu,$$ $$\delta\psi_+^\mu=-2\epsilon^+\partial_+X^\mu, \qquad \delta\psi_-^\mu=-2\epsilon^-\partial_-X^\mu,$$

where $\epsilon^+$ and $\epsilon^-$ are constant anticommuting parameters. In spinor notation this is often written compactly as

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

The physical meaning is simple: $X^\mu$ and $\psi^\mu$ are superpartners on the worldsheet. Acting once with supersymmetry turns a bosonic embedding fluctuation into a worldsheet fermion. Acting again gives a worldsheet translation. Schematically,

$$\{Q_+,Q_+\}\sim P_+, \qquad \{Q_-,Q_-\}\sim P_-, \qquad \{Q_+,Q_-\}=0.$$

So the RNS action is a two-dimensional supersymmetric field theory with $D$ free scalar multiplets.

Worldsheet supersymmetry pairs the scalar $X^\mu$ with the two chiral worldsheet fermions $\psi^\mu_+$ and $\psi^\mu_-$. Two transformations close onto translations along $\sigma^+$ or $\sigma^-$.

Notice what this does not yet say. The field $\psi^\mu$ carries a vector index $\mu$, not a spacetime spinor index. The RNS action has worldsheet supersymmetry manifestly, but spacetime supersymmetry remains hidden until the spectrum is projected appropriately.

Coupling to worldsheet supergravity

For the bosonic string, we first wrote a generally covariant worldsheet theory and then fixed conformal gauge. The analogous supersymmetric procedure is to couple the matter multiplet $(X^\mu,\psi^\mu)$ to two-dimensional supergravity.

The worldsheet gravity multiplet contains

$$(e_a{}^m,\chi_a),$$

where $e_a{}^m$ is the zweibein and $\chi_a$ is the worldsheet gravitino. The locally supersymmetric action has the schematic form

$$S=-{1\over4\pi\alpha'}\int d^2\sigma\,e\, \left[ h^{ab}\partial_aX\cdot\partial_bX -i\bar\psi\cdot\rho^aD_a\psi +2\bar\chi_a\rho^b\rho^a\psi\cdot\partial_bX +\cdots \right].$$

Here

$$e=\det(e_a{}^m), \qquad h_{ab}=e_a{}^m e_b{}^n\eta_{mn},$$

and the ellipsis denotes terms quadratic in the gravitino that are required by local supersymmetry. We will rarely need the full expression. What matters conceptually is the role of the gauge fields:

$${\delta S\over\delta h^{ab}}=0 \quad\Longrightarrow\quad T_{ab}=0,$$

and

$${\delta S\over\delta\chi_a}=0 \quad\Longrightarrow\quad J_a=0.$$

The first equation is the Virasoro constraint. The second is its fermionic partner: the vanishing of the supercurrent. These are the classical seeds of the super-Virasoro constraints.

The locally supersymmetric theory has worldsheet diffeomorphisms, Weyl symmetry, local Lorentz symmetry, local supersymmetry, and super-Weyl symmetry. Using these, one can impose superconformal gauge,

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

After this gauge choice, the matter theory again looks free, but the constraints survive:

$$T_{++}=T_{--}=0, \qquad J_+=J_-=0.$$

This is exactly parallel to the bosonic string. Conformal gauge made the bosonic action free but did not remove the Virasoro constraints. Superconformal gauge makes the RNS action free but does not remove the superconformal constraints.

Stress tensor and supercurrent in Lorentzian signature

The holomorphic and antiholomorphic stress-tensor components are, in light-cone coordinates,

$$T_{++} ={1\over\alpha'}\partial_+X\cdot\partial_+X +{i\over2\alpha'}\psi_+\cdot\partial_+\psi_+,$$ $$T_{--} ={1\over\alpha'}\partial_-X\cdot\partial_-X +{i\over2\alpha'}\psi_-\cdot\partial_-\psi_-.$$

The fermionic currents are

$$J_+=\psi_+\cdot\partial_+X, \qquad J_-=\psi_-\cdot\partial_-X.$$

On shell,

$$\partial_-T_{++}=0, \qquad \partial_+T_{--}=0,$$

and

$$\partial_-J_+=0, \qquad \partial_+J_-=0.$$

Thus the left-moving and right-moving sectors decouple classically. The stress tensor generates conformal transformations, while the supercurrent generates the local fermionic transformations that mix $X^\mu$ and $\psi^\mu$. At the quantum level their Laurent modes become the generators $L_n$ and $G_r$ of the superconformal algebra.

Wick rotation and Euclidean worldsheet notation

For operator methods it is best to Wick rotate the worldsheet and use complex coordinates. Let

$$\tau=-i\tau_E,$$

and set

$$\sigma^1=\tau_E, \qquad \sigma^2=\sigma.$$

The Euclidean cylinder coordinate is

$$w=\sigma^1+i\sigma^2=\tau_E+i\sigma, \qquad \bar w=\sigma^1-i\sigma^2=\tau_E-i\sigma.$$

Locally we may map the cylinder to the complex plane by

$$z=e^w, \qquad \bar z=e^{\bar w}.$$

In the common CFT normalization $\alpha'=2$, the Euclidean RNS matter action in superconformal gauge is

$$S={1\over4\pi}\int d^2z\, \left( \partial X^\mu\bar\partial X_\mu +\psi^\mu\bar\partial\psi_\mu +\tilde\psi^\mu\partial\tilde\psi_\mu \right).$$

The equations of motion are

$$\partial\bar\partial X^\mu=0, \qquad \bar\partial\psi^\mu=0, \qquad \partial\tilde\psi^\mu=0.$$

Therefore

$$X^\mu(z,\bar z)=X_L^\mu(z)+X_R^\mu(\bar z),$$

and

$$\psi^\mu=\psi^\mu(z), \qquad \tilde\psi^\mu=\tilde\psi^\mu(\bar z).$$

The basic short-distance singularities are

$$X^\mu(z,\bar z)X^\nu(0,0) \sim -\eta^{\mu\nu}\ln |z|^2,$$ $$\psi^\mu(z)\psi^\nu(0) \sim {\eta^{\mu\nu}\over z}, \qquad \tilde\psi^\mu(\bar z)\tilde\psi^\nu(0) \sim {\eta^{\mu\nu}\over\bar z}.$$

The holomorphic stress tensor and supercurrent are

$$T(z)=-{1\over2}:\partial X^\mu\partial X_\mu: -{1\over2}:\psi^\mu\partial\psi_\mu:,$$ $$T_F(z)=i:\psi^\mu\partial X_\mu:.$$

Similarly,

$$\tilde T(\bar z)=-{1\over2}:\bar\partial X^\mu\bar\partial X_\mu: -{1\over2}:\tilde\psi^\mu\bar\partial\tilde\psi_\mu:,$$ $$\tilde T_F(\bar z)=i:\tilde\psi^\mu\bar\partial X_\mu:.$$

The weights are

$$X^\mu:(0,0), \qquad \partial X^\mu:(1,0), \qquad \psi^\mu:\left({1\over2},0\right), \qquad T_F:\left({3\over2},0\right), \qquad T:(2,0).$$

The antiholomorphic fields have the analogous barred weights.

Currents as contour generators

A holomorphic stress tensor acts on local operators by contour integration. If $\phi(w,\bar w)$ is a primary field of holomorphic weight $h$, then

$$T(z)\phi(w,\bar w) \sim {h\phi(w,\bar w)\over(z-w)^2} +{\partial_w\phi(w,\bar w)\over z-w} +\text{regular}.$$

Equivalently,

$$\delta_\epsilon\phi(w,\bar w) =\oint_w {dz\over2\pi i}\,\epsilon(z)T(z)\phi(w,\bar w),$$

where $\epsilon(z)$ is the infinitesimal conformal vector field. The supercurrent generates the fermionic transformation,

$$\delta_\eta\phi(w,\bar w) =\oint_w {dz\over2\pi i}\,\eta(z)T_F(z)\phi(w,\bar w),$$

where $\eta(z)$ has conformal weight $-1/2$, so that $\eta T_F$ has weight one and may be integrated around a contour.

The stress tensor $T$ generates conformal transformations. Its fermionic partner $T_F$ generates worldsheet supersymmetry transformations. Their modes form the super-Virasoro algebra.

For example, using

$$\partial X^\mu(z)X^\nu(w,\bar w) \sim -{\eta^{\mu\nu}\over z-w},$$

one obtains

$$T_F(z)X^\mu(w,\bar w) \sim -{i\psi^\mu(w)\over z-w}.$$

Likewise, using $\psi^\nu(z)\psi^\mu(w)\sim\eta^{\nu\mu}/(z-w)$,

$$T_F(z)\psi^\mu(w) \sim {i\partial X^\mu(w)\over z-w}.$$

These two OPEs are the Euclidean CFT version of

$$\delta X^\mu\sim\eta\psi^\mu, \qquad \delta\psi^\mu\sim\eta\partial X^\mu.$$

This is the compact reason the RNS theory is solvable: the matter fields are free, and all the symmetry generators are explicit bilinears in free fields.

Cylinder-to-plane map and the origin of spin structures

On the Euclidean cylinder the spatial coordinate is periodic:

$$\sigma^2\sim\sigma^2+2\pi.$$

With $z=e^w$, radial quantization on the plane is cylinder time evolution in disguise. A primary field of weight $h$ transforms as

$$\phi_{\rm cyl}(w)=\left({dz\over dw}\right)^h\phi_{\rm plane}(z) =z^h\phi_{\rm plane}(z).$$

For a fermion, $h=1/2$, hence

$$\psi_{\rm cyl}(w)=z^{1/2}\psi_{\rm plane}(z).$$

That square root is the first hint of a new ingredient. Fermions can be periodic or antiperiodic around the spatial circle:

$$\psi(w+2\pi i)=+\psi(w) \quad\text{or}\quad \psi(w+2\pi i)=-\psi(w).$$

These two choices are the Ramond and Neveu—Schwarz sectors. Their mode expansions, zero modes, and ground-state energies are the next step.

Exercises

Exercise 1: derive the RNS equations of motion

Starting from

$$S={1\over2\pi\alpha'}\int d^2\sigma\, \left( \partial_+X\cdot\partial_-X +{i\over2}\psi_+\cdot\partial_-\psi_+ +{i\over2}\psi_-\cdot\partial_+\psi_- \right),$$

derive the equations of motion for $X^\mu$, $\psi_+^\mu$, and $\psi_-^\mu$.

Solution

Varying $X^\mu$ gives

$$\delta S_X={1\over2\pi\alpha'}\int d^2\sigma\, \left( \partial_+\delta X\cdot\partial_-X +\partial_+X\cdot\partial_-\delta X \right).$$

Integrating by parts and dropping boundary terms,

$$\delta S_X =-{1\over2\pi\alpha'}\int d^2\sigma\, \delta X\cdot \left(\partial_+\partial_-X+\partial_-\partial_+X\right).$$

Since the derivatives commute,

$$\partial_+\partial_-X^\mu=0.$$

For the fermion $\psi_+^\mu$, remember that the variation is Grassmann odd. The first-order kinetic term gives

$$\delta S_{\psi_+} ={i\over4\pi\alpha'}\int d^2\sigma\, \left( \delta\psi_+\cdot\partial_-\psi_+ +\psi_+\cdot\partial_-\delta\psi_+ \right).$$

After integrating by parts, the second term equals the first term rather than canceling it, because both $\psi_+$ and $\delta\psi_+$ are anticommuting. Hence

$$\delta S_{\psi_+} ={i\over2\pi\alpha'}\int d^2\sigma\, \delta\psi_+\cdot\partial_-\psi_+,$$

so

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

Similarly,

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

Exercise 2: show that the supercurrent is conserved

Use the equations of motion to show that

$$J_+=\psi_+\cdot\partial_+X, \qquad J_-=\psi_-\cdot\partial_-X$$

obey

$$\partial_-J_+=0, \qquad \partial_+J_-=0.$$

Solution

For $J_+$,

$$\partial_-J_+ =(\partial_-\psi_+)\cdot\partial_+X +\psi_+\cdot\partial_-\partial_+X.$$

The first term vanishes because $\partial_-\psi_+=0$. The second vanishes because $\partial_-\partial_+X=0$. Hence

$$\partial_-J_+=0.$$

The calculation for $J_-$ is identical:

$$\partial_+J_- =(\partial_+\psi_-)\cdot\partial_-X +\psi_-\cdot\partial_+\partial_-X=0.$$

Exercise 3: compute the conformal weight of $\psi^\mu$

Using

$$T_\psi(z)=-{1\over2}:\psi^\nu\partial\psi_\nu:(z)$$

and

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

show that $\psi^\mu$ has holomorphic conformal weight $h=1/2$.

Solution

We compute the singular terms in the OPE $T_\psi(z)\psi^\mu(w)$. There are two possible contractions. The contraction of $\partial\psi_\nu(z)$ with $\psi^\mu(w)$ gives

$$\partial\psi_\nu(z)\psi^\mu(w) \sim \partial_z\left({\delta_\nu{}^\mu\over z-w}\right) =-{\delta_\nu{}^\mu\over(z-w)^2}.$$

The contraction of $\psi^\nu(z)$ with $\psi^\mu(w)$ gives

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

Carefully keeping the fermion signs in the normal-ordered product, the result is

$$T_\psi(z)\psi^\mu(w) \sim {1\over2}{\psi^\mu(w)\over(z-w)^2} +{\partial\psi^\mu(w)\over z-w}.$$

Comparing with the primary-field OPE

$$T(z)\phi(w)\sim {h\phi(w)\over(z-w)^2}+{\partial\phi(w)\over z-w},$$

we read off

$$h={1\over2}.$$

Exercise 4: recover supersymmetry from the supercurrent OPE

Use

$$T_F(z)=i:\psi^\nu\partial X_\nu:(z)$$

and the free-field OPEs to show that

$$T_F(z)X^\mu(w,\bar w)\sim -{i\psi^\mu(w)\over z-w},$$

and

$$T_F(z)\psi^\mu(w)\sim {i\partial X^\mu(w)\over z-w}.$$

Solution

The only singular contraction in $T_F(z)X^\mu(w,\bar w)$ is between $\partial X_\nu(z)$ and $X^\mu(w,\bar w)$. Since

$$X_\nu(z,\bar z)X^\mu(w,\bar w) \sim -\delta_\nu{}^\mu\ln |z-w|^2,$$

we have

$$\partial X_\nu(z)X^\mu(w,\bar w) \sim -{\delta_\nu{}^\mu\over z-w}.$$

Therefore

$$T_F(z)X^\mu(w,\bar w) \sim i\psi^\nu(z)\left(-{\delta_\nu{}^\mu\over z-w}\right) =-{i\psi^\mu(w)\over z-w}.$$

For $T_F(z)\psi^\mu(w)$, the singular contraction is between $\psi^\nu(z)$ and $\psi^\mu(w)$:

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

Thus

$$T_F(z)\psi^\mu(w) \sim i{\eta^{\nu\mu}\over z-w}\partial X_\nu(z) ={i\partial X^\mu(w)\over z-w}.$$

These are precisely the infinitesimal transformations that exchange $X^\mu$ and $\psi^\mu$.

Exercise 5: map a fermion from the cylinder to the plane

Let $z=e^w$ and let $\psi$ be a primary field of weight $h=1/2$. Show that

$$\psi_{\rm cyl}(w)=z^{1/2}\psi_{\rm plane}(z).$$

Then explain why the cylinder mode expansion is naturally

$$\psi_{\rm cyl}(w)=\sum_r\psi_r e^{-rw},$$

with $r\in\mathbb Z$ for periodic fermions and $r\in\mathbb Z+1/2$ for antiperiodic fermions.

Solution

A primary field of holomorphic weight $h$ transforms under a coordinate change $z=z(w)$ as

$$\psi_{\rm cyl}(w)=\left({dz\over dw}\right)^h\psi_{\rm plane}(z).$$

For $z=e^w$,

$${dz\over dw}=z.$$

With $h=1/2$,

$$\psi_{\rm cyl}(w)=z^{1/2}\psi_{\rm plane}(z).$$

Expanding on the cylinder gives

$$\psi_{\rm cyl}(w)=\sum_r\psi_r e^{-rw}.$$

Under $w\to w+2\pi i$,

$$e^{-r(w+2\pi i)}=e^{-rw}e^{-2\pi i r}.$$

If the fermion is periodic, we need $e^{-2\pi i r}=1$, so

$$r\in\mathbb Z.$$

If the fermion is antiperiodic, we need $e^{-2\pi i r}=-1$, so

$$r\in\mathbb Z+{1\over2}.$$

These are the Ramond and Neveu—Schwarz modings, respectively.