Superconformal Currents, Maxwell Lines, and Worldline Actions

The tricritical Ising lesson introduced a fermionic $N=1$ extension and its weight-$3/2$ chiral current. Its modes form a graded algebra whose anticommutator contains the stress tensor. In schematic form,

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

The first half of this page organizes the supercurrent as a chiral field: conservation, conformal transformation, OPE with $T$, and mode algebra. The second half turns to line geometry. Wilson lines describe charged probes on prescribed paths, while a relativistic particle can be written as a one-dimensional generally covariant system. The worldline construction replaces a square-root geometric action by a quadratic action with an auxiliary metric; the normalized quantum propagator then follows from a heat kernel.

For a free particle of mass $m>0$ in flat Euclidean target space, the action is

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

where $h(\tau)>0$ is the einbein and $\dot x^2$ is the Euclidean squared speed. Varying $h$ imposes a classical constraint. Reparametrization invariance leaves a positive modulus that cannot be removed by setting $h=1$ on the fixed parameter interval. The Euclidean classical momentum and the Fourier momentum in the quantum propagator will be distinguished below.

Helpful background. Conformal generators and descendants supply the active primary pullback and origin-mode contours. The Majorana and tricritical theories supply the fermionic chiral extension, its NS/R sectors and supercurrent normalization.

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.

Holomorphy alone does not imply a primary transformation law: the stress tensor is holomorphic but has a Schwarzian term. For chiral primary fields, the normalized identity-vacuum correlator on the plane obeys the following covariance under a global Möbius map $f$, with all displayed insertion coordinates finite, compatible branches of the weight factors, and a consistent spin lift where needed:

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

There is an analogous anti-holomorphic factor for nonchiral fields. A general locally invertible conformal map instead transports the domain, state and boundary data along with the fields; it is not automatically a symmetry of the same-plane vacuum correlator. The local active pullback of the supercurrent is

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

For the factor $(f')^{3/2}$ choose a local square root of $f'$ and a compatible spin lift when continuing around contours. The primary-field and cylinder-map discussion shows how transported states and sector choices enter this local field law.

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 closes on the stress tensor and hence on conformal generators. The constant NS supercharge gives the translation example below, making precise the supersymmetry slogan

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

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. These normalizations and the NS/R sectors are given in Di Francesco, Mathieu and Sénéchal 1997, §7.4.3, pp. 223–225, Eq. 7.87. The third-order pole supplies the central residue derived in the previous lesson; Exercise 1 derives the mixed commutator directly.

A particularly useful case in the Neveu–Schwarz sector 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

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

Use left Grassmann differentiation, with the odd coordinate and parameters in the order written below. The basic superderivative is

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

so that

$$D^2=\partial_z.$$

This is a geometric version of the square-root relation:

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

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 even and $\eta$ is odd, and products of infinitesimal parameters are discarded. For example, both $Dz'$ and $\theta'D\theta'$ equal $\theta+\eta+\theta\epsilon'$ to this order. The functions $\epsilon$ and $\eta$ correspond to the bosonic and fermionic current transformations.

The coordinate vector fields realize the centerless superconformal algebra. The quantum currents can realize its central extension, with the $c$-dependent terms displayed above; those terms do not follow from coordinate changes alone.

Also distinguish the covariant derivative $D$ from a supersymmetry generator. For a constant odd parameter placed on the left, the displayed coordinate transformation is generated by

$$\mathsf q=\partial_\theta-\theta\partial_z, \qquad \mathsf q^2=-\partial_z, \qquad \{\mathsf q,D\}=0.$$

Indeed, $\delta=\eta\mathsf q$ gives $\delta z=-\eta\theta=\theta\eta$ and $\delta\theta=\eta$. With the geometric generator convention $\ell_{-1}=-\partial_z$, this reads $\mathsf q^2=\ell_{-1}$. On fields the course instead uses the positive active action $[L_{-1},O]=\partial O$, with quantum NS relation $G_{-1/2}^2=L_{-1}$. The identity $D^2=\partial_z$ is not an identification of $D$ with that Noether charge.

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 an NS representation, acting with $G_{-1/2}$ on a superconformal primary gives the candidate superpartner. In state–operator notation,

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

has holomorphic weight

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

provided it is nonzero in the physical quotient. The eigenvalue equation also holds when the vector vanishes, but the zero vector does not define another state or field. Superconformal representations therefore organize surviving states with half-unit weight shifts. Nontrivial minimal models have finitely many irreducible primary families; their descendant modules remain infinite-dimensional after null relations are imposed. Finiteness of primary-family data is not finiteness of the state space.

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

Classical Maxwell theory is often visualized in terms of Faraday lines of force. In the external-source convention, with unit Maxwell kinetic normalization, the electric field satisfies

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

Here $\rho=J_{\rm source}^0$ is the external Maxwell source density, whose Lorentzian coupling is $-A_\mu J_{\rm source}^\mu$. Electric field lines are tangent to the electric field determined by that source and the boundary data. A prescribed probe path in a Wilson line need not follow such a field line; the relation between the two pictures is an analogy, not an identification.

For the Maxwell and worldline discussion we now use Euclidean signature and the weight $e^{-S_E}$. A gauge field couples to a charged particle moving along a path $\gamma$ through

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

Consequently, the path-integral weight 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 $y$.

For an oriented charged path, define the current entering the plus Wilson phase by

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

so

$$-S_{E,\text{int}}=i\int d^d x\,J^\mu A_\mu =iq\int d\tau\,\dot x^\mu A_\mu(x(\tau)) =iq\int_\gamma A.$$

This $J$ belongs to the charge convention with Lorentzian interaction $+A_\mu J^\mu$. In the external-source convention used for Gauss’s law, the corresponding source is $J_{\rm source}=-J$. The source sign must not be inferred by identifying the two current symbols; the plus Wilson phase and its endpoint transformation above are unchanged.

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 flat Euclidean target space. The parameter $\tau$ is arbitrary. The geometric path is unchanged by a smooth orientation-preserving relabeling. On the fixed interval the allowed reparametrizations obey

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

The most direct geometric action is the length action

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

with $m>0$. The length action and endpoint-preserving transformations are the starting point of Polyakov 1987, §9.1, p. 152, Eqs. 9.3–9.5. Reparametrization invariance follows 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 on a smooth regular curve, a classically equivalent quadratic 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 line element $d\ell=h\,d\tau$ should not be confused with 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 dimensionless $\tau$ and target coordinates measured in units of length, $h$ and $L=\int h\,d\tau$ have units of length squared. This $L$ is the worldline proper-time modulus; it is not a target-space travel time or arclength.

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 figure compares two clocks on the dimensionless curve $\gamma(u)=(u,0.3\sin\pi u)$, $0\le u\le1$. If a physical target-space scale is needed, take $x=\ell_0\gamma$ with $\ell_0>0$. Follow the labeled parameter values in the two panels: their positions change along the same oriented curve. The chosen map obeys $f'(\tau)=1+\tfrac12\cos(2\pi\tau)\ge\tfrac12$, so it is smooth, endpoint-preserving and invertible.

The same kinematical worldline is labeled by two smooth, orientation-preserving clocks. Dots mark $\tau=0,1/4,1/2,3/4,1$; equal parameter steps need not give equal arclength steps. With reference $h=L>0$, the transformed density $h_f=(h\circ f)f'=Lf'$ preserves its integral $L$ and the reparametrization-invariant quadratic action. Coordinates and scales agree in both panels. This explicit curve is not asserted to be a free-particle classical solution; $L$ is an einbein modulus, not its arclength.

Editable TikZ source. Original diagram: QFT.org, created with OpenAI Codex; CC BY 4.0.

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 $m>0$ and a regular curve with $\dot x^2>0$. Substituting this positive solution back into the action gives

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

Thus the actions agree after classical elimination of $h$ in this regular massive domain. This does not by itself define an equality of quantum functional measures; the propagator below will be obtained from a normalized heat kernel. 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 constraint also has a classical Legendre-transform interpretation. The real Euclidean conjugate variable is

$$\pi_{E\mu}={\partial \mathcal L_E\over\partial \dot x^\mu} ={\dot x_\mu\over h}.$$

At a smooth saddle, the Euclidean einbein equation gives

$$\pi_E^2=m^2.$$

This real Legendre variable is not the real Fourier variable $k_E$ in the inverse operator $1/(k_E^2+m^2)$. That inverse has no pole on the real Euclidean momentum axis for $m>0$. After continuation to Lorentzian signature, the physical mass shell in the inherited metric convention is

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

The Euclidean saddle constraint and the Lorentzian mass shell serve different roles from the spectral denominator used in the heat-kernel construction.

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 reparametrization freedom but leaves the modulus $L$. Setting $h=1$ would also fix $L=1$, which discards an inequivalent value of the modulus. The constant-metric decomposition is described in Polyakov 1987, §9.2, p. 157, Eq. 9.24. His metric component is the square of the einbein used here; the invariant in his notation is the integral of its square root.

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 scalar on $\mathbb R^d$ with $m>0$, the Euclidean operator $-\partial^2+m^2$ is positive. Its 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 Gaussian Fourier integral fixes the normalization of its massless heat kernel:

$$\begin{aligned} K_L(x,y) &=\int\frac{d^d k_E}{(2\pi)^d} e^{ik_E\cdot(x-y)-Lk_E^2/2}\\ &=\frac1{(2\pi L)^{d/2}} \exp\left[-\frac{|x-y|^2}{2L}\right]. \end{aligned}$$

It integrates to one in $x$ and tends to $\delta^{(d)}(x-y)$ as $L\downarrow0$. Its path-integral representation is

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

where the measure is defined by time slicing: each free step of duration $\Delta t$ uses the factor $(2\pi\Delta t)^{-d/2}$ multiplying its Gaussian. Convolution of these normalized steps gives $K_L$. 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

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

In the $T$ variable the free kernel is $(4\pi T)^{-d/2}\exp[-|x-y|^2/(4T)]$. As a momentum-space check, the remaining integral is $\frac12\int_0^\infty dL\,e^{-L(k_E^2+m^2)/2}=1/(k_E^2+m^2)$. The position-space Green function is understood as a distribution, or pointwise for $x\ne y$; its coincident limit may require a regulator.

Polyakov’s regulated random-path calculation reaches the Euclidean inverse denominator in Polyakov 1987, §9.2, p. 163, Eq. 9.46. His bare length parameter is tuned to obtain a finite physical mass. The explicit Gaussian integral here fixes our free-field normalization and $T=L/2$ without equating that bare parameter to $m$.

The two proper-time variables describe the same representation; its 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, with the NS relation

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

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 the free massive Euclidean particle has the reparametrization-invariant action

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

The einbein $h$ imposes a classical constraint and makes the action quadratic, but endpoint-preserving gauge fixing leaves the positive modulus

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

Integrating the normalized heat kernel over this modulus, with measure $dL/2$, gives 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

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

using origin-centered counterclockwise mode contours in a fixed NS or R sector, show that

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

For the inner residue take $w\ne0$ and a local branch on which the weight is regular. The full weighted outer integrand is single-valued in the chosen sector, so integration by parts has no branch-endpoint term.

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

In the NS sector, use the supercurrent algebra to show that $G_{-1/2}$ increases a primary state’s conformal weight by $1/2$, assuming the descendant is nonzero in the physical quotient.

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

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

For $m>0$, $h>0$ and a smooth Euclidean curve with $\dot x^2>0$, eliminate $h$ by its equation of motion and show that the classical 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

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

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

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

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

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

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

References

• Philippe Di Francesco, Pierre Mathieu, and David Sénéchal, Conformal Field Theory, Springer, 1997. DOI.
• Alexander M. Polyakov, Gauge Fields and Strings, Contemporary Concepts in Physics, Vol. 3, Harwood Academic Publishers, 1987. DOI.