Superconformal Currents, Maxwell Lines, and Worldline Actions

The previous page ended with the fermionic $N=1$ extension of the tricritical Ising model and its weight-$3/2$ chiral current. That current is the first place in the course where a symmetry generator is fermionic rather than bosonic: its modes do not form an ordinary Lie algebra by commutators, but a graded algebra whose anticommutator contains the stress tensor. In slogan form,

$$G\times G \sim T.$$

This page makes that statement precise, then uses it as a bridge. The first half of the page organizes the supercurrent as a chiral field: conservation, transformation under conformal maps, OPE with $T$, and mode algebra. The second half changes gears in a way that is characteristic of Polyakov’s style. Maxwell’s field lines are reinterpreted as line observables, and then a relativistic particle is written as a one-dimensional generally covariant system. This is the worldline version of the Polyakov trick: replace a square-root geometric action by a quadratic action plus an auxiliary metric.

The result is the action

$$S[x,h] ={1\over2}\int_0^1 d\tau\, \left({\dot x^2\over h}+m^2h\right),$$

where $h(\tau)$ is the einbein, the one-dimensional metric density on the particle path. Varying $h$ imposes the mass-shell constraint, while reparametrization invariance explains why one cannot simply set $h=1$ without thinking about the remaining proper-time modulus.

Chiral conservation and holomorphic currents

Supercurrent symbol

We write the holomorphic $N=1$ superconformal current as $G(z)$, the standard notation in most CFT references. Thus $G$ is a fermionic chiral field of conformal weight $3/2$. The stress tensor is $T(z)$, with modes $L_n$.

In two Euclidean dimensions use complex coordinates

$$z=x+iy, \qquad \bar z=x-iy, \qquad \partial={\partial\over\partial z}, \qquad \bar\partial={\partial\over\partial\bar z}.$$

A conserved current can be written in complex components. In schematic notation,

$$\bar\partial A+\partial B=0.$$

This equation by itself does not imply that $A$ is holomorphic. It says that the failure of $A$ to be holomorphic is balanced by the $z$ derivative of the other component $B$. Holomorphy requires extra information: for a genuinely chiral current, or when a justified improvement makes the opposite component vanish, one has

$$B=0,$$

Conservation then becomes

$$\boxed{\bar\partial A=0.}$$

This is an enormous simplification of two-dimensional CFT, but it is not true for every conserved current. For the stress tensor, tracelessness and conservation imply that the chiral component satisfies

$$\bar\partial T(z)=0,$$

away from operator insertions. In an $N=1$ superconformal theory there is also a holomorphic supercurrent

$$\boxed{\bar\partial G(z)=0,}$$

again away from insertions. The stress tensor generates local conformal transformations; the supercurrent generates their fermionic square root.

A two-dimensional conservation equation has two complex components. In a chiral sector the opposite component vanishes, so conservation reduces to holomorphy. Chiral currents then transform naturally under $z\mapsto f(z)$, and the superconformal extension adds a nilpotent coordinate $\theta$.

For an ordinary primary field $O(z)$ of holomorphic weight $h$, a finite conformal map $z\mapsto f(z)$ acts on correlation functions as

$$\langle O_1(z_1)\cdots O_n(z_n)\rangle = \prod_{i=1}^n \bigl(f'(z_i)\bigr)^{h_i} \langle O_1(f(z_1))\cdots O_n(f(z_n))\rangle,$$

with the analogous anti-holomorphic factor for nonchiral fields. The stress tensor is special because of the Schwarzian derivative, but the supercurrent itself is primary under the bosonic conformal group:

$$G(z)\mapsto \bigl(f'(z)\bigr)^{3/2}G(f(z)).$$

The exponent $3/2$ is already a warning that $G$ is not an ordinary bosonic current. The half-integer weight is tied to spin structure and to the choice between Neveu–Schwarz and Ramond sectors.

The stress-tensor OPE with the supercurrent

The defining statement that $G$ has conformal weight $3/2$ is the OPE

$$\boxed{ T(z)G(w) \sim {{3\over2}G(w)\over (z-w)^2} +{\partial G(w)\over z-w}. }$$

This is the same primary-field formula as

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

with $h=3/2$. It implies the local Ward identity

$$\langle T(z)G(w)X\rangle \sim {{3\over2}\over(z-w)^2}\langle G(w)X\rangle +{1\over z-w}\partial_w\langle G(w)X\rangle +\cdots,$$

where $X$ denotes other insertions. The omitted singularities occur when $z$ approaches those other insertions. Equivalently, inserting $T$ in a contour integral around $w$ implements an infinitesimal conformal transformation of $G(w)$.

The second defining OPE is the supercurrent OPE with itself:

$$\boxed{ G(z)G(w) \sim {2c/3\over (z-w)^3} +{2T(w)\over z-w}. }$$

There is no $(z-w)^{-2}$ term. The leading pole is the central term; the simple pole says that the product of two supersymmetry transformations is a translation generated by the stress tensor. This is the local CFT version of the supersymmetry slogan

$$\{Q,Q\}\sim P.$$

The OPE $T(z)G(w)$ says that $G$ has weight $3/2$. The OPE $G(z)G(w)$ says that the square of the supercurrent is the stress tensor, up to the central term. Passing to modes gives the $N=1$ superconformal algebra.

Define the modes by

$$T(z)=\sum_{n\in\mathbb Z}L_n z^{-n-2}, \qquad G(z)=\sum_r G_r z^{-r-3/2}.$$

The allowed values of $r$ depend on the spin structure:

$$r\in\mathbb Z+{1\over2} \quad\text{in the Neveu–Schwarz sector}, \qquad r\in\mathbb Z \quad\text{in the Ramond sector}.$$

The OPEs are equivalent to the $N=1$ super-Virasoro algebra

$$[L_m,L_n] =(m-n)L_{m+n} +{c\over12}m(m^2-1)\delta_{m+n,0},$$ $$\boxed{ [L_n,G_r] =\left({n\over2}-r\right)G_{n+r}, }$$

and

$$\boxed{ \{G_r,G_s\} =2L_{r+s} +{c\over3}\left(r^2-{1\over4}\right)\delta_{r+s,0}. }$$

The anticommutator is essential. Since $G$ is fermionic, two $G$ modes close through the graded bracket. If one writes an ordinary commutator here, the algebra has already lost the spin-statistics information.

A particularly useful case is

$$\{G_{-1/2},G_{-1/2}\}=2L_{-1}.$$

The central term vanishes because $(-1/2)^2-1/4=0$. Since $L_{-1}$ generates translations,

$$[L_{-1},O(z)]=\partial O(z),$$

the mode $G_{-1/2}$ is literally a square root of the holomorphic translation operator.

Supercoordinates and superconformal maps

The compact way to package this square-root structure is to introduce a Grassmann coordinate $\theta$ satisfying

$$\theta^2=0.$$

A holomorphic superspace point is

$$Z=(z,\theta).$$

The basic superderivative is

$$D=\partial_\theta+\theta\partial_z,$$

so that

$$D^2=\partial_z.$$

This equation is the differential-operator version of the same idea:

$$\text{fermionic square root}^2=\text{translation}.$$

A holomorphic superconformal transformation is a change of variables

$$(z,\theta)\mapsto (z',\theta')$$

that preserves the distribution generated by $D$, meaning

$$D=(D\theta')D'$$

or equivalently

$$Dz'=\theta' D\theta'.$$

Infinitesimally one may write

$$z'=z+\epsilon(z)+\theta\eta(z),$$ $$\theta'=\theta+\eta(z)+{1\over2}\theta\epsilon'(z),$$

where $\epsilon$ is bosonic and $\eta$ is fermionic. The function $\epsilon(z)$ is generated by the stress tensor $T(z)$; the function $\eta(z)$ is generated by the supercurrent $G(z)$.

This formalism is not needed for every calculation in these notes, but it explains why the pair $(T,G)$ belongs together. The ordinary conformal symmetry of the plane is enlarged by transformations that mix $z$ and a nilpotent coordinate. The algebra of their generators is exactly the super-Virasoro algebra above.

The free fermion as the simplest half-integer primary

The Ising CFT already gave us a chiral Majorana field $\psi(z)$ with

$$h_\psi={1\over2}.$$

Its OPEs are

$$\psi(z)\psi(w) \sim {1\over z-w},$$

and

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

This is structurally similar to the supercurrent OPE with $T$, except that the weight is $1/2$ rather than $3/2$. The free Majorana field is a chiral spinor. Calling it a “spin field” would be misleading in Ising language, where that term normally refers to the twist fields $\sigma$ and $\mu$. The supercurrent, by contrast, is a spin-$3/2$ current that generates a fermionic symmetry.

In a superconformal theory, acting with $G_{-1/2}$ moves a primary into its superpartner. If $\Phi$ is a superconformal primary, then

$$\widehat \Phi=G_{-1/2}\Phi$$

has holomorphic weight

$$h_{\widehat \Phi}=h_\Phi+{1\over2},$$

provided the state is not null. Thus superconformal representations naturally arrange states into pairs separated by half a unit of dimension. In minimal models, null vectors and fusion restrictions make this pairing finite and highly constrained.

This is the end of the purely chiral CFT thread for the moment. The course now pivots from local operator algebras to a more geometric viewpoint: fields, lines, and eventually surfaces.

Maxwell fields and Faraday lines

Classically, Maxwell theory is often visualized in terms of Faraday lines of force. The electric field satisfies

$$\nabla\cdot\mathbf E=\rho,$$

so electric field lines begin and end on charges. This picture is not a substitute for gauge-invariant observables, but it is powerful intuition: the field stores energy in space, and sources are connected to that field by extended lines.

In the quantum theory, the gauge field couples naturally to a charged particle moving along a path $\gamma$:

$$S_{\text{int}} =iq\int_\gamma A_\mu dx^\mu.$$

Equivalently, the path contributes the Wilson-line phase

$$U_\gamma =\exp\left(iq\int_\gamma A_\mu dx^\mu\right).$$

For a closed contour $C$, this becomes the gauge-invariant Wilson loop

$$W(C)=\exp\left(iq\oint_C A_\mu dx^\mu\right).$$

For an open contour, the Wilson line is not gauge invariant by itself; it must end on charged fields. Under

$$A_\mu\mapsto A_\mu+\partial_\mu\alpha,$$

one finds

$$\exp\left(iq\int_x^y A\right) \mapsto \exp\left(iq\alpha(y)\right) \exp\left(iq\int_x^y A\right) \exp\left(-iq\alpha(x)\right),$$

which is exactly the endpoint transformation needed to connect a charge at $x$ to a charge at $y$.

Faraday’s field-line picture becomes precise in the path integral through Wilson-line factors. A charged particle worldline sources the gauge field by $iq\int A_\mu dx^\mu$. In confining systems, field lines can become effective flux tubes, setting the stage for string-like descriptions.

A point charge has current

$$J^\mu(x)=q\int d\tau\,\dot x^\mu(\tau)\,\delta^{(d)}(x-x(\tau)),$$

so

$$i\int d^d x\,J^\mu A_\mu =iq\int d\tau\,\dot x^\mu A_\mu(x(\tau)) =iq\int_\gamma A.$$

Thus line observables and particle worldlines are not decorative additions to gauge theory. They are the natural gauge-covariant language for charged probes.

For a path running from $x_i$ to $x_f$, the current is not conserved by itself:

$$\partial_\mu J^\mu(x) =q\left[\delta^{(d)}(x-x_i)-\delta^{(d)}(x-x_f)\right].$$

It is conserved for a closed worldline. For an open worldline, the endpoint charged operators supply exactly these source and sink terms; this is the current-language version of the endpoint phases of an open Wilson line.

In ordinary weakly coupled Maxwell theory, electric flux spreads. In a confining phase, flux between external charges can become collimated into a tube, and the long-distance dynamics begins to resemble the motion of a string. Before reaching worldsheets, however, one must understand the one-dimensional version: the relativistic worldline.

The relativistic worldline and the einbein

A relativistic particle traces a curve

$$x^\mu=x^\mu(\tau)$$

in spacetime. The parameter $\tau$ is arbitrary. The geometric path is the physical object, not the specific clock used to label points on it. Therefore the action should be invariant under a reparametrization

$$\tau\mapsto \tau'=f(\tau), \qquad f(0)=0, \qquad f(1)=1, \qquad f'(\tau)>0.$$

The most direct geometric action is the length action

$$S_{\text{length}}=m\int ds =m\int_0^1 d\tau\,\sqrt{\dot x^2},$$

in Euclidean signature. It is reparametrization invariant because

$$\sqrt{\dot x^2}\,d\tau$$

is the line element along the curve. The square root, however, makes quantization awkward.

For a massive particle, the equivalent Polyakov-like worldline action introduces an auxiliary field $h(\tau)>0$:

$$\boxed{ S[x,h] ={1\over2}\int_0^1 d\tau\, \left({\dot x^2\over h}+m^2h\right). }$$

Here $h(\tau)$ is the einbein. The intrinsic one-dimensional metric can be written as

$$d\ell^2=h(\tau)^2d\tau^2.$$

The intrinsic coordinate length $d\ell=h\,d\tau$ should not be confused with the target-space arclength $ds=\sqrt{\dot x^2}\,d\tau$. The einbein equation of motion will relate them by $ds=m\,d\ell$ for $m>0$. With this normalization, $L=\int h\,d\tau$ is the worldline proper-time modulus used below.

Under an active reparametrization

$$x_f^\mu(\tau)=x^\mu(f(\tau)),$$

the einbein transforms as

$$\boxed{ h_f(\tau)=f'(\tau)h(f(\tau)). }$$

This is exactly what is needed for invariance of the action.

The parameter $\tau$ is gauge. A reparametrization changes the density of tick marks along the same geometric curve. The einbein $h(\tau)$ transforms so that the quadratic action describes the same particle path.

Let us check the invariance explicitly. Under $x_f(\tau)=x(f(\tau))$,

$$\dot x_f^\mu(\tau)=f'(\tau)\dot x^\mu(f(\tau)).$$

Using $h_f(\tau)=f'(\tau)h(f(\tau))$, the kinetic term transforms as

$${\dot x_f^2(\tau)\over h_f(\tau)}d\tau = {f'(\tau)^2\dot x^2(f(\tau))\over f'(\tau)h(f(\tau))}d\tau = {\dot x^2(\tau')\over h(\tau')}d\tau',$$

where $\tau'=f(\tau)$. Similarly,

$$h_f(\tau)d\tau=h(\tau')d\tau'.$$

Therefore $S[x_f,h_f]=S[x,h]$.

Constraint and equivalence to the length action

The einbein is not a propagating field. It has no derivative term. Varying the action with respect to $h$ gives an algebraic constraint:

$${\delta S\over \delta h}=0 \quad\Rightarrow\quad -{\dot x^2\over h^2}+m^2=0.$$

Thus

$$\boxed{ h={\sqrt{\dot x^2}\over m} }$$

for positive $h$ and $m>0$. Substituting this solution back into the action gives

$$S[x,h_{\text{cl}}] ={1\over2}\int_0^1 d\tau\, \left(m\sqrt{\dot x^2}+m\sqrt{\dot x^2}\right) =m\int_0^1 d\tau\,\sqrt{\dot x^2}.$$

So the quadratic action with the einbein is classically equivalent to the square-root action when $m>0$. The massless theory still has a useful einbein formulation, but $h$ cannot be eliminated by dividing by $m$; in Lorentzian signature its equation of motion instead imposes the null constraint.

The equation of motion for $x^\mu$ is

$${d\over d\tau}\left({\dot x^\mu\over h}\right)=0.$$

After choosing a gauge in which $h$ is constant, this becomes

$$\ddot x^\mu=0.$$

The classical trajectory is a straight line in flat space, as expected for a free relativistic particle.

The einbein equation of motion imposes the mass-shell constraint and reduces the quadratic action to the target-space length. Gauge fixing can make $h$ constant, but the invariant quantity $L=\int h d\tau$ remains as a proper-time modulus.

The constraint has a direct Hamiltonian interpretation. The momentum conjugate to $x^\mu$ is

$$p_\mu={\partial \mathcal L\over\partial \dot x^\mu} ={\dot x_\mu\over h}.$$

In the Euclidean action written above, writing the momentum as $p_E$, the einbein equation gives

$$p_E^2=m^2.$$

The Lorentzian einbein action written in the site’s $(+---)$ convention has the corresponding mass shell

$$p^2=m^2.$$

The frequently seen formula $p^2+m^2=0$ uses the opposite, mostly-plus signature. The invariant statement is that the einbein imposes the mass-shell constraint appropriate to the chosen metric.

Gauge fixing and the proper-time modulus

A common trap is to say: since $h$ is pure gauge, set

$$h(\tau)=1.$$

On an infinite line this can be harmless after suitable boundary conditions. But on an interval $0\le \tau\le1$ with endpoints fixed, there is a global invariant:

$$\boxed{ L=\int_0^1 h(\tau)d\tau. }$$

Under $h_f(\tau)=f'(\tau)h(f(\tau))$,

$$\int_0^1 h_f(\tau)d\tau = \int_0^1 f'(\tau)h(f(\tau))d\tau = \int_0^1 h(u)du.$$

Thus $L$ cannot be changed by a reparametrization that preserves the endpoints. The correct local gauge choice is

$$\dot h(\tau)=0,$$

so that

$$h(\tau)=L.$$

This fixes the local wiggle freedom but leaves the proper-time modulus $L$. Setting $h=1$ would also set $L=1$, which is not a local gauge choice; it discards the modulus that must be integrated over in the worldline path integral.

With $h=L$ on $0\le\tau\le1$, the action becomes

$$S[x,L] ={1\over2}\int_0^1 d\tau\, \left({\dot x^2\over L}+m^2L\right).$$

Changing variables to proper time $t=L\tau$ gives

$$S[x,L] ={1\over2}\int_0^L dt\, \left(\left({dx\over dt}\right)^2+m^2\right).$$

This is the form that leads to the Schwinger proper-time representation of a scalar propagator. With the present $1/2$ normalization, the conventional heat-kernel time will be $T=L/2$.

Worldline representation of the scalar propagator

For a free Euclidean scalar field, the Green function is

$$G(x,y)=\langle x|{1\over -\partial^2+m^2}|y\rangle.$$

To keep the normalization consistent with the gauge-fixed action above, use

$${1\over A} ={1\over2}\int_0^\infty dL\,e^{-LA/2}$$

for an operator with positive spectrum. Then

$$G(x,y) = {1\over2}\int_0^\infty dL\,e^{-m^2L/2} \langle x|e^{(L/2)\partial^2}|y\rangle.$$

The matrix element of the heat kernel has a path-integral representation,

$$\langle x|e^{(L/2)\partial^2}|y\rangle = \int_{x(0)=y}^{x(L)=x}\mathcal D x(t)\, \exp\left[-{1\over2}\int_0^L dt\,\dot x^2\right],$$

with a conventional normalization of the measure. Therefore

$$\boxed{ G(x,y) = {1\over2}\int_0^\infty dL\,e^{-m^2L/2} \int_{x(0)=y}^{x(L)=x}\mathcal D x(t)\, \exp\left[-{1\over2}\int_0^L dt\,\dot x^2\right]. }$$

Equivalently, set $T=L/2$ and rescale the path parameter. The same expression takes the familiar form

$$\boxed{ \begin{aligned} G(x,y) &=\int_0^\infty dT\,e^{-m^2T} \int_{x(0)=y}^{x(T)=x}\mathcal D x(u)\\ &\qquad\times \exp\left[-{1\over4}\int_0^T du\, \left({dx\over du}\right)^2\right]. \end{aligned} }$$

The measure carries the standard heat-kernel normalization. These are exactly the same convention written with two proper-time variables; their conceptual structure is

$$\text{propagator} = \int_{0}^{\infty}\text{proper time}\times\text{sum over paths}.$$

This is the one-dimensional ancestor of the random-surface and string path integrals that appear later. A field propagator can be expanded as a sum over particle paths. A Wilson loop can be treated as a line observable. A confining flux tube suggests a fluctuating surface. The worldline formalism is the cleanest place to see the gauge principle behind all of these statements.

Summary

The first half of the page completed the superconformal-current thread. A chiral supercurrent $G(z)$ has weight $3/2$,

$$T(z)G(w) \sim {{3\over2}G(w)\over(z-w)^2}+{\partial G(w)\over z-w},$$

and its self-OPE closes on the stress tensor,

$$G(z)G(w) \sim {2c/3\over(z-w)^3}+{2T(w)\over z-w}.$$

In modes this gives the $N=1$ super-Virasoro algebra, especially

$$\{G_{-1/2},G_{-1/2}\}=2L_{-1}.$$

The second half shifted from local chiral operators to line geometry. A charged worldline couples to Maxwell theory by the Wilson factor

$$\exp\left(iq\int A\right),$$

and a relativistic particle is described by a reparametrization-invariant action

$$S[x,h] ={1\over2}\int d\tau\left({\dot x^2\over h}+m^2h\right).$$

The einbein $h$ imposes the mass-shell constraint and makes the action quadratic, but gauge fixing leaves the global proper-time modulus

$$L=\int h\,d\tau.$$

Integrating over this modulus gives the worldline representation of the scalar propagator.

Common pitfalls

The supercurrent is fermionic. Its modes close under an anticommutator, not an ordinary commutator:

$$\{G_r,G_s\}=2L_{r+s}+\cdots.$$

The stress tensor is not an ordinary primary field because of the Schwarzian derivative in finite conformal transformations. The supercurrent $G$ is primary under bosonic conformal maps, with weight $3/2$.

An open Wilson line is not gauge invariant by itself. It becomes gauge invariant only when its endpoints are attached to charged operators, or when the path is closed.

The quadratic worldline action is not the same as the nonrelativistic action unless the einbein gauge and mass-shell constraint are handled correctly. The parameter $\tau$ is gauge, while $L=\int h d\tau$ is a proper-time modulus.

Setting $h=1$ on a finite interval generally overfixes the gauge. The safe gauge is $h(\tau)=L$, followed by an integration over $L$ in the path integral.

The normalization of the Schwinger parameter must track the normalization of the quadratic action. For $S=\frac12\int_0^L(\dot x^2+m^2)dt$, the conventional heat-kernel time is $T=L/2$.

Exercises

The stress tensor acting on the supercurrent

Starting from

$$T(z)G(w) \sim {{3\over2}G(w)\over(z-w)^2}+{\partial G(w)\over z-w},$$

show that

$$[L_n,G_r]=\left({n\over2}-r\right)G_{n+r}.$$

Solution

The modes are

$$L_n=\oint_0 {dz\over2\pi i}\,z^{n+1}T(z), \qquad G_r=\oint_0 {dw\over2\pi i}\,w^{r+1/2}G(w).$$

The commutator is obtained by moving the $z$ contour around $w$:

$$[L_n,G(w)] =\oint_w {dz\over2\pi i}\,z^{n+1}T(z)G(w).$$

Using the OPE,

$$[L_n,G(w)] =\oint_w {dz\over2\pi i}\,z^{n+1} \left({{3\over2}G(w)\over(z-w)^2}+{\partial G(w)\over z-w}\right).$$

The residues are

$$\oint_w {dz\over2\pi i}{z^{n+1}\over(z-w)^2} =(n+1)w^n,$$

and

$$\oint_w {dz\over2\pi i}{z^{n+1}\over z-w} =w^{n+1}.$$

Therefore

$$[L_n,G(w)] =w^{n+1}\partial G(w)+{3\over2}(n+1)w^nG(w).$$

Now insert this into the contour defining $G_r$:

$$[L_n,G_r] =\oint_0 {dw\over2\pi i}\,w^{r+1/2} \left(w^{n+1}\partial G+{3\over2}(n+1)w^nG\right).$$

Integrate the first term by parts:

$$\oint w^{n+r+3/2}\partial G =-(n+r+3/2)\oint w^{n+r+1/2}G.$$

Thus

$$[L_n,G_r] =\left[-(n+r+3/2)+{3\over2}(n+1)\right]G_{n+r} =\left({n\over2}-r\right)G_{n+r}.$$

Weight of a superdescendant

Use the supercurrent algebra to show that the mode $G_{-1/2}$ increases the conformal weight of a primary state by $1/2$.

Solution

Set $n=0$ in

$$[L_n,G_r]=\left({n\over2}-r\right)G_{n+r}.$$

This gives

$$[L_0,G_r]=-rG_r.$$

Let $|h\rangle$ be an $L_0$ eigenstate:

$$L_0|h\rangle=h|h\rangle.$$

Then

$$L_0G_r|h\rangle =G_rL_0|h\rangle+[L_0,G_r]|h\rangle =(h-r)G_r|h\rangle.$$

For $r=-1/2$,

$$L_0G_{-1/2}|h\rangle =\left(h+{1\over2}\right)G_{-1/2}|h\rangle.$$

So $G_{-1/2}$ creates a superdescendant with conformal weight $h+1/2$, unless that descendant is null.

Reparametrization invariance

Verify explicitly that the worldline action

$$S[x,h]={1\over2}\int_0^1d\tau\left({\dot x^2\over h}+m^2h\right)$$

is invariant under the active reparametrization

$$x_f(\tau)=x(f(\tau)), \qquad h_f(\tau)=f'(\tau)h(f(\tau)),$$

with $f'(\tau)>0$.

Solution

First compute

$$\dot x_f^\mu(\tau)=f'(\tau)\dot x^\mu(f(\tau)).$$

Then

$${\dot x_f^2\over h_f}d\tau ={f'(\tau)^2\dot x^2(f(\tau))\over f'(\tau)h(f(\tau))}d\tau ={f'(\tau)\dot x^2(f(\tau))\over h(f(\tau))}d\tau.$$

Let $u=f(\tau)$, so $du=f'(\tau)d\tau$. Then

$${\dot x_f^2\over h_f}d\tau ={\dot x^2(u)\over h(u)}du.$$

Similarly,

$$h_f(\tau)d\tau=f'(\tau)h(f(\tau))d\tau=h(u)du.$$

Both terms in the action are invariant after changing variables from $\tau$ to $u$. Hence

$$S[x_f,h_f]=S[x,h].$$

Eliminating the einbein

Eliminate $h$ from the worldline action by its equation of motion and show that the length action is recovered.

Solution

Varying with respect to $h$ gives

$$0={\delta S\over\delta h} ={1\over2}\left(-{\dot x^2\over h^2}+m^2\right).$$

For positive $h$ and $m>0$,

$$h={\sqrt{\dot x^2}\over m}.$$

Substitute this back into

$$S={1\over2}\int d\tau\left({\dot x^2\over h}+m^2h\right).$$

The first term becomes

$${\dot x^2\over h}=m\sqrt{\dot x^2},$$

and the second term becomes

$$m^2h=m\sqrt{\dot x^2}.$$

Therefore

$$S=m\int d\tau\sqrt{\dot x^2}=m\int ds.$$

The surviving proper-time modulus

Show that

$$L=\int_0^1 h(\tau)d\tau$$

is invariant under endpoint-preserving reparametrizations.

Solution

Using the active transformation law,

$$h_f(\tau)=f'(\tau)h(f(\tau)),$$

we have

$$\int_0^1 h_f(\tau)d\tau =\int_0^1 f'(\tau)h(f(\tau))d\tau.$$

Let $u=f(\tau)$. Since $f(0)=0$ and $f(1)=1$, the integration limits remain $0$ and $1$. Thus

$$\int_0^1 h_f(\tau)d\tau =\int_0^1 h(u)du.$$

So $L$ is invariant. This is why gauge fixing can make $h$ constant but cannot fix the value of that constant.

Matching the Schwinger parameter

Starting from

$${1\over2}\int_0^L dt\, \left[\left({dx\over dt}\right)^2+m^2\right],$$

set $T=L/2$ and show that the gauge-fixed path integral has the standard heat-kernel exponent $-\frac14\int_0^T du\,(dx/du)^2-m^2T$.

Solution

Let $t=2u$, so $0\le u\le T=L/2$, $dt=2du$, and

$${dx\over dt}={1\over2}{dx\over du}.$$

The kinetic term becomes

$${1\over2}\int_0^L dt\left({dx\over dt}\right)^2 ={1\over4}\int_0^T du\left({dx\over du}\right)^2,$$

while the mass term becomes

$${1\over2}\int_0^L dt\,m^2=m^2T.$$

Thus the Euclidean weight is

$$\exp\left[-m^2T-{1\over4}\int_0^Tdu \left({dx\over du}\right)^2\right],$$

which is the standard heat-kernel normalization.

Endpoint phases of an open Wilson line

An open Wilson line from $x$ to $y$ is

$$U(x,y)=\exp\left(iq\int_x^y A_\mu dx^\mu\right).$$

Under $A_\mu\mapsto A_\mu+\partial_\mu\alpha$, find its transformation law and explain how it can be made gauge invariant.

Solution

The exponent changes by

$$iq\int_x^y \partial_\mu\alpha\,dx^\mu =iq(\alpha(y)-\alpha(x)).$$

Therefore

$$U(x,y) \mapsto \exp(iq\alpha(y))U(x,y)\exp(-iq\alpha(x)).$$

For Abelian gauge theory the factors commute, but it is useful to keep the endpoint structure visible. If a charged field transforms as

$$\phi(x)\mapsto e^{iq\alpha(x)}\phi(x),$$

then the bilocal dressed operator

$$\phi^\dagger(y)U(x,y)\phi(x)$$

is gauge invariant, with the phases canceling at the endpoints. A closed Wilson loop has $x=y$, so the endpoint phases cancel automatically.

References

• L. Brink, P. Di Vecchia, and P. Howe, “A locally supersymmetric and reparametrization invariant action for the spinning string,” Physics Letters B 65 (1976), 471–474.
• P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, Springer, 1997, Chapters 4–6 and 10.
• P. Ginsparg, “Applied Conformal Field Theory,” in Fields, Strings and Critical Phenomena, Les Houches Session XLIX, Elsevier, 1989, 1–168.
• A. M. Polyakov, Gauge Fields and Strings, Harwood Academic Publishers, 1987.
• J. Schwinger, “On gauge invariance and vacuum polarization,” Physical Review 82 (1951), 664–679.
• S. Weinberg, The Quantum Theory of Fields, Volume III: Supersymmetry, Cambridge University Press, 2000.