Notes on string theory #2: The relativistic point particle (pp. 9-11)

1. Introduction

In Chapter 1 of Polchinski’s textbook, we start with a discussion on the relativistic point particle (pp. 9-11).

String theory proposes that elementary particles are not pointlike, but rather 1-dimensional extended objects (i.e., strings). In fact, string theory (both the bosonic string in Volume 1 of Polchinski and the superstring that comprises much of Volume 2) can be seen as a special generalisation of point particle theory. But the deeper and more modern view is not one that necessarily begins with point particles and then strings, instead the story begins with branes. In that a number of features of string theory are shared by the point particle – as we’ll see in a later note, the point particle can be obtained in the limit the string collapses to a point – the bigger picture is that both of these objects can be considered as special cases of a p-brane.

We refer to p-branes as p-dimensional dynamical objects that have mass and can have other familiar attributes such as charge. As a p-brane moves through spacetime, it sweeps out a latex (p+1)-dimensional volume called its worldvolume. In this notation, a 0-brane corresponds to the case where p = 0. It simply describes a point particle that, as we’ll discuss in this note, traces out a worldline as it propagates through spacetime. A string (whether fundamental or solitonic) corresponds to the p = 1 case, and this turns out to be a very special case of p-branes (for many reasons we’ll learn in following notes). Without getting too bogged down in technical details that extend well beyond the current level of discussion, it is also possible to consider higher-dimensional branes. Important is the case for p = 2, which are 2-dimensional branes called membranes. In fact, the etymology for the word ‘brane’ can be viewed as derivative from `membrane’. As a physical object, a p-brane is actually a generalisation of a membrane such that we may assign arbitrary spatial dimensions. So, for the case {p \geq 2} , these are p-branes that appear in string theory as solitons in the corresponding low energy effective actions of various string theories (in addition to 0-branes and 1-branes).

In Type IIA and Type IIB string theories, which again is a subject of Volume 2, we see that there is entire family of p-brane solutions. From the viewpoint of perturbative string theory, which is the primary focus of Volume 1, solitons as p-branes are strictly non-perturbative objects. (There are also other classes of branes, such as Dp-branes that we’ll come across soon when studying the open string. The more complete picture of D/M-brane physics, including brane dynamics, is anticipated to be captured by M-theory. This is a higher dimensional theory that governs branes and, with good reason, is suspected to represent the non-perturbative completion of string theory).

In some sense, one can think of there being two equivalent ways to approach the idea of p-branes: a top-down higher dimensional view, or from the bottom-up as physical objects that generalise the notion of a point particle to higher dimensions. But given an introductory view of p-branes, perhaps it becomes slightly more intuitive why in approaching the concept of a string in string theory we may start (as Polchinski does) with a review of point particle theory. Indeed, it may at first seem odd to model the fundamental constituents of matter as strings. Indeed, it could seem completely arbitrary and therefore natural to ask, why not something else? But what is often missed, especially in popular and non-technical physics literature, is the natural generalising logic that leads us to study strings in particular. These are remarkable objects with remarkable properties, and what Polchinski does so well in Volume 1 is allow this generalising logic to come out naturally in the study of the simplest string theory: bosonic string theory.

In this note, we will construct the relativistic point particle action as given in p.10 (eqn. 1.2.2) and then work through the proceeding discussion in pages 10-11. The quantisation of the point particle is mentioned several pages later in the textbook, so we’ll address that topic then. In what follows, I originally also wanted to include notes on the superparticle and its superspace formulation (i.e., the inclusion of fermions to the point particle theory of bosons), as well as introduce other advanced topics; but I reasoned it is best to try to keep as close to the textbook as possible. The only exception to this rule is that, at the end of this note, we’ll finish by quickly looking at the p-brane action.

2. Relativistic point particle

Explanation of the action for a relativistic point particle as given in Polchinski (eqn. 1.2.2) is best achieved through its first-principle construction. So let us consider the basics of constructing the theory for a relativistic free point particle.

2.1. Minkowski space

We start with a discussion about the space in which we’ll build our theory [Moh08].

As one may recall from studying Einstein’s theory of relativity, spacetime may be modelled by D-dimensional Minkowski space {\mathbb{M}^D} . In the abstract, the basic idea is to consider two (distinct) sets E and {\vec{E}} , where E is a set of points (with no given structure) and {\vec{E}} is a vector space (of free vectors) acting on the set E. We view the elements of {\vec{E}} as forces acting on points in E, which we in turn think of as physical particles. Applying a force (free vector) {X \in \vec{E}} to a point {P \in E} results in a translation. In other words, the action of a force X is to move every point P to the point {P + X \in E} by translation that corresponds to X viewed as a vector.

In physics, the set E is viewed as the D-dimensional affine space {\mathbb{M}^D} , and then {\vec{E}} is the associated D-dimensional vector space {\mathbb{R}^{1,D-1}} defined over the field of real numbers. The choice to model spacetime as an affine space is quite natural, given that an affine space has no preferred or distinguished origin and, of course, the spacetime of special relativity possesses no preferred origin.

As the vectors {X \in \mathbb{R}^{1,D-1}} do not naturally correspond to points {P \in \mathbb{M}} , but rather as displacements relating a point P to another point Q, we write {X = \vec{PQ}} . The points can be defined to be in one-to-one correspondence with a position vector such that {\vec{X}_P = \vec{OP}} , with displacements then defined by the difference {\vec{PQ} = \vec{OQ} - \vec{OP}} . The associated vector space possesses a zero vector {\vec{0} \in \mathbb{R}^{1,D-1}} , which represents the neutral element of vector addition. We can also use the vector space {\mathbb{R}^{1,D-1}} to introduce linear coordinates on {\mathbb{M}^{D}} by making an arbitrary choice of origin as the point {O \in \mathbb{M}^D} .

The elements or points {P,Q,..., \in \mathbb{M}^D} are events, and they combine a moment of time with a specified position. With the arbitrary choice of origin made, we can refer to these points in Minkowski space in terms of their position vectors such that the components {X^{\mu} = (X^0, X^i) = (t, \vec{X})} , with {\mu = 0,..., D-1, i = 1,...,D-1} of vectors {X \in \mathbb{R}^{1,D-1}} correspond to linear coordinates on {\mathbb{M}^D} . The coordinates {X^{0}} is related to the time t, which is measured by an inertial or free falling observer by {X^0 =ct} , with the c the fundamental velocity. The {X^i} coordinates, which are combined into a (D-1)-component vector, parameterise space (from the perspective of the inertial observer).

It is notable that a vector {X} has contravariant coordinates {X^{\mu}} and covariant coordinates {X_{\mu}} which are related by raising and lowering indices such that {X_{\mu} = \eta_{\mu \nu}X^{\nu}} and {X^{\mu} = \eta^{\mu \nu}x_{\nu}} .

We still need to equip a Lorentzian scalar product. In the spacetime of special relativity, the vector space {\mathbb{R}} is furnished with the scalar product (relativistic distance between events)

\displaystyle  \eta_{\mu \nu} = X^{\mu}X_{\mu} = -t^2 + \vec{X}^2 \begin{cases} <0 \ \text{for timelike disrance} \\ =0 \ \text{for lightlike distance} \\ >0 \ \text{for spacelike distance} \end{cases} \ \ (1)

with matrix

\displaystyle  \eta = (\eta_{\mu \nu}) = \begin{pmatrix} - 1 & 0 \\  0 & 1_{D-1} \end{pmatrix}, \ \ (2)

where we have chosen the mostly plus convention. To make sense of (1), since the Minkowski metric (2) is defined by an indefinite scalar product, the distance-squared between events can be positive, zero or negative. This carries information about the causal structure of spacetime. If {X = \vec{PQ}} is the displacement between two events, then these events are called time-like, light-like or space-like relative to each other, depending on X. The zeroth component of X then carries information about the time of the event P as related to Q relative to a given Lorentz frame: P is after Q ({X^0 > Q} ), or simultaneous with Q ({X^0 = 0} ), or earlier than Q ({X^0 < 0} ).

2.2. Lorentz invariance and the Poincaré group

Let’s talk more about Lorentz invariance and the Poincaré group. As inertial observers are required to use linear coordinates which are orthonormal with respect to the scalar product (1), these orthonormal coordinates are distinguished by the above standard form of the metric. It is of course possible to use other curvilinear coordinate systems, such as spherical or cylindrical coordinates. Given the standard form of the metric (2), the most general class of transformations which preserve its form are the Poincaré group, which represents the group of Minkowski spacetime isometries.

The Poincaré group is a 10-dimensional Lie group. It consists of 4 translations along with the Lorentz group of 3 rotations and 3 boosts. As a general review, let’s start with the Lorentz group. This is the set of linear transformations of spacetime that leave the Lorentz interval unchanged.

From the definitions in the previous section, the line element takes the form

\displaystyle  ds^2 = \eta_{\mu \nu}dX^{\mu}dX^{\nu} = - dt^2 + d\vec{X}^2. \ \ (3)

For spacetime coordinates defined in the previous section, the Lorentz group is then defined to be the group of transformations {X^{\mu} \rightarrow X^{\prime \mu}} leaving the relativistic interval invariant. Assuming linearity (we will not prove linearity here, with many proofs easily accessible), define a Lorentz transformation as any real linear transformation {\Lambda} such that

\displaystyle  X^{\mu} \rightarrow X^{\prime \mu} = \Lambda^{\mu}_{\nu}X^{\nu} \ \ (4)

with

\displaystyle  \eta_{\mu \nu} dX^{\prime \mu} dX^{\prime \nu} = \eta_{\mu \nu} dX^{ \mu} dX^{\nu}, \ \ (5)

ensuring from (1) that

\displaystyle  X^{\prime 2} = X^{2}, \ \ (6)

which, for arbitrary X, requires

\displaystyle  \eta_{\mu \nu} = \eta_{\alpha \beta} \Lambda^{\alpha}_{\mu} \Lambda^{\beta}_{\nu}. \ \ (7)

Note that {\Lambda = (\Lambda^{\mu}_{\nu})} is an invertible {D \times D} matrix. In matrix notation (7) can be expressed as

\displaystyle  \Lambda^T \eta \Lambda = \eta. \ \ (8)

Matrices satisfying (8) contain rotations together with Lorentz boosts, which relate inertial frames travelling a constant velocity relative to each other. The Lorentz transformations form a six-dimensional Lie group, which is the Lorentz group O(1,D-1).

For elements {\Lambda \in O(1, D-1)} taking the determinant of (8) gives

\displaystyle  (\det \Lambda)^2 = 1 \implies \det \Lambda = \pm 1. \ \ (9)

By considering the {\Lambda^0_0} component we also find

\displaystyle  (\Lambda^0_0)^2 = 1 + \Sigma_i (\Lambda^0_i)^2 \geq 1 \Rightarrow \Lambda^0_0 \geq 1 \ \text{or} \ \Lambda^0_0 \leq -1. \ \ (10)

So, the Lorentz group has four components according to the signs of {\det \Lambda} and {\Lambda^0_0} . The matrices with {\det \Lambda = 1} form a subgroup SO(1,D-1) with two connected components as given on the right-hand side of (10). The component containing the unit matrix {1 \in O(1,D-1)} is connected and as {SO_0(1,D-1)} .

We may also briefly consider translations of the form

\displaystyle  X^{\mu} \rightarrow X^{\prime \mu} = X^{\mu} + a^{\mu}, \ \ (11)

where {a = (a^{\mu}) \in \mathbb{R}^{1, D-1}} . Translations form a group that can be parametrised by the components of the translation vector {a^{\mu}} .

As mentioned, the Poincaré group is then the complete spacetime symmetry group that combines translations with Lorentz transformations. For a Lorentz transformation {\Lambda} and a translation {a} the combined transformation {(\Lambda, a)} gives

\displaystyle X^{\mu} \rightarrow X^{\prime \mu} = \Lambda^{\mu}_{\nu} X^{\nu} + a^{\mu}. \ \ (12)

These combined transformations form a group since

\displaystyle (\Lambda_2, a_2)(\Lambda_1, a_1) = (\Lambda_2 \Lambda_2, \Lambda_2 a_1 + a_2), \ (\Lambda, a)^{-1} = (\Lambda^{-1}, -\Lambda^{-1}a). \ \ (13)

Since Lorentz transformations and translations do not commute, the Poincaré group is not a direct product. More precisely, the Poincaré group is the semi-direct product of the Lorentz and translation group, {IO(1,D-1) = O(1,D-1) \propto \mathbb{R}^D} .

2.3. Action principle

We now look to construct an action for the relativistic point particle (initially following the discussion in [Zwie09] as motivation).

The classical motion of a point particle as it propagates through spacetime is described by a geodesic on the spacetime. As Polchinski first notes, we can of course describe the motion of this particle by giving its position in terms of functions of time {X(t) = (X^{\mu}(t)) = (t, \vec{X}(t))} . For now, we may also consider some arbitrary origin and endpoint {(ct_f, \vec{X}_{f})} for the particle’s path or what is also called its worldline. We also know from the principle of least action that there are many possible paths between these points.

Particle worldline

It should be true that for any worldline all Lorentz observers compute the same value for the action. Let {\mathcal{P}} denote one such worldline. Then we may use the proper time as an Lorentz invariant quantity to describe this path. Moreover, from special relativity one may recall that the proper time is a Lorentz invariant measure of time. If different Lorentz observers will record different values for the time interval between the two events along {\mathcal{P}} , then we instead imagine that attached to the particle is a clock. The proper time is therefore the time elapsed between the two events on that clock, according to which all Lorentz observers must agree on the amount of elapsed time. This is the basic idea, and it means we want an action of the worldline {\mathcal{P}} that is proportional to the proper time.

To achieve this, we first recall the invariant interval for the motion of a particle

\displaystyle  - ds^2 = -c^2 dt^2 + (dX^1)^2 + (dX^2)^2 + (dX^3)^2, \ \ (14)

in which, from special relativity, the proper time

\displaystyle  -ds^2 = -c^2 dt_f \rightarrow ds = c dt_f \ \ (15)

tells us that for timelike intervals ds/c is the proper time interval. It follows that the integral of (ds/c) over the worldline {\mathcal{P}} gives the proper time elapsed on {\mathcal{P}} . But, if the proper time gives units of time, we still needs units of energy or units of mass times velocity-squared to ensure we have the full units of action (recall that for any dynamical system the action has units of energy times time, with the Lagrangian possessing units of energy). We also need to ensure that we preserve Lorentz invariance in the process of building our theory. One obvious choice is m for the rest mass of the particle, with c for the fundamental velocity in relativity. Then we have an overall multiplicative factor {mc^2} that represents the the rest energy of the particle. As a result, the action takes the tentative form {mc^2 (ds/c) = mc ds} . This should make some sense in that {ds} is just a Lorentz scalar, and we have the factor of relativity we expect. We also include a minus sign to ensure the follow integrand is real for timelike geodesics.

\displaystyle  S = -mc \int_{\mathcal{P}} ds. \ \ (16)

A good strategy now is to find an integral of our Lagrangian over time – say, {t_i} and {t_f} which are world-events that we’ll take to define our interval – because it will enable use to establish a more satisfactory expression that includes the values of time at the initial and final points of our particle’s path. If we fix a frame – which is to say if we choose the frame of a particular Lorentz observer – we may express the action (16) as the integral of the Lagrangian over time. To achieve this end, we must first return to our interval (14) and relate {ds} to {dt} ,

\displaystyle  -ds^2 = -c^2 dt^2 + (dX^1)^2 + (dX^2)^2 + (dX^3)^2

\displaystyle  ds^2 = c^2 dt^2 - (dX^1)^2 - (dX^2)^2 - (dX^3)^2

\displaystyle  ds^2 = [c^2 - \frac{(dX^1)^2}{dt} - \frac{(dX^2)^2}{dt} - \frac{(dX^3)^2}{dt}] dt^2

\displaystyle  \implies ds^2 = (c^2 - v^2) dt^2

\displaystyle  \therefore ds = \sqrt{c^2 - v^2} dt. \ \ (17)

With this relation between {ds} and {dt} , in the fixed frame the point particle action becomes

\displaystyle  S = -mc^{2} \int_{t_{i}}^{t_{f}} dt \sqrt{1 - \frac{v^{2}}{c^{2}}}, \ \ (18)

with the Lagrangian taking the form

\displaystyle  L = -mc^{2} \sqrt{1 - \frac{v^{2}}{c^{2}}}. \ \ (19)

This Lagrangian gives us a hint that it is correct as its logic breaks down when the velocity exceeds the speed of light {v > c} . This confirms the definition of the proper time from special relativity (i.e., the velocity should not exceed the speed of light for the proper time to be a valid concept). In the small velocity limit {v << c} , on the other hand, when we expand the square root (just use binomial theorem to approximate) we see that it gives

\displaystyle L \simeq -mc^2 (1 - \frac{1}{2}\frac{v^2}{c^2}) = - mc^2 + \frac{1}{2}m v^2. \ \ (20)

returning similar structure for the kinetic part of the free non-relativistic particle, with ({-mc^2} ) just a constant.

2.4. Canonical momentum and Hamiltonian

We will discuss the canonical momentum of the point particle again in a future note on quantisation; but for the present form of the action it is worth highlighting that we can also see the Lagrangian (19) is correct by computing the momentum {\vec{p}} and the Hamiltonian.

For the canonical momentum, we take the derivative of the Lagrangian with respect to the velocity

\displaystyle  \vec{p} = \frac{\partial L}{\partial \vec{v}} = -mc^{2}(-\frac{\vec{v}}{c^{2}})\frac{1}{\sqrt{1 - \frac{v^{2}}{c^{2}}}} = \frac{m\vec{v}}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}. \ \ (21)

Now that we have an expression for the relativistic momentum of the particle, let us consider the Hamiltonian. The Hamiltonian may be written schematically as {H = \vec{p} \cdot \vec{v} - L} . All we need to do is make the appropriate substitutions,

\displaystyle  H = \frac{m\vec{v}^{2}}{\sqrt{1 - \frac{v^{2}}{c^{2}}}} + mc^{2}\sqrt{1 - \frac{v^{2}}{c^{2}}} = \frac{mc^{2}}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}. \ \ (22)

The Hamiltonian should make sense. Notice, if we instead write the result in terms of the particle’s momentum (rather than velocity) by inverting (22), we find an expression in terms of the relativistic energy {\frac{E^{2}}{c^{2}} - \vec{p} \cdot \vec{p} = m^{2}c^{2}} . This is a deep hint that we’re on the right track, as it suggests quite clearly that we’ve recovered basic relativistic physics for a point-like object.

3. Reparameterisation invariance

An important property of the action (16) is that it is invariant under whatever choice of parameterisation we might choose. This makes sense because the invariant length ds between two points on the particle’s worldline does not depend on any parameterisation. We’ve only insisted on integrating the line element, which, if you think about it, is really just a matter of adding up all of the infinitesimal segments along the worldline. But, typically, a particle moving in spacetime is described by a parameterised curve. As Polchinski notes, it is generally best to introduce some parameter and then describe the motion in spacetime by functions of that parameter.

Furthermore, how we parameterise the particle’s path will govern whether, for the classical motion, the path is one that extremises the invariant distance ds as a minimum or maximum. Our choice of {\tau}-parameterisation is such that the invariant length ds is given by

\displaystyle ds^2 = -\eta_{\mu \nu}(X) dX^{\mu} dX^{\nu}, \ \ (23)

then the choice of worldline parameter {\tau} is considered to be increasing between some initial point {X^{\mu} (\tau_i)} and some final point {X^{\mu}(\tau_f)} . So the classical paths are those which maximise the proper time. It also means that the trajectory of the particle worldline is now described by the coordinates {X^{\mu} = X^{\mu}(\tau)} . As a result, the space of the theory can now be updated such that {X^{\mu}(\tau) \in \mathbb{R}^{1, D-1}} with {\mu, \nu = 0,...,D-1} .

In the use of {\tau} parameterisation, an important idea is that time is in a sense being promoted to a dynamical degree of freedom without it actually being a dynamical degree of freedom. We are in many ways leveraging the power of gauge symmetry, with our choice of parameterisation enabling us to treat space and time coordinates on equal footing. The cost by trading a less symmetric description for a more symmetric one is that we pick up redundancies.

Given the previous preference of background spacetime geometry to be Minkowski, recall the metric

\displaystyle  \eta_{\mu \nu} = \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}, \ \ (24)

such that for the integrand ds we now use

\displaystyle -\eta_{\mu \nu}(X) dX^{\mu} dX^{\nu} = -\eta_{\mu \nu}(X) \frac{dX^{\mu}(\tau)}{d\tau} \frac{dX^{\nu}(\tau)}{d\tau} d\tau^2. \ \ (25)

Therefore, the action (16) may be updated to take the form

\displaystyle  S_{pp} = -mc \int_{\tau_i}^{\tau_f} d\tau \ \sqrt{-\eta_{\mu \nu} \dot{X}^{\mu} \dot{X}^{\nu}} \ \ (26)

with {\dot{X}^{\mu} \equiv dX^{\mu}(\tau) / d\tau} .

Setting {c = 1} , notice (26) is precisely the action (eqn. 1.2.2) in Polchinski. This is the simplest action for a relativistic point particle with manifest Poincaré invariance that does not depend on the choice of parameterisation.

How do we interpret this form of the action? In the exercise to obtain (26) we have essentially played the role of a fixed observer, who has calculated the action using some parameter {\tau} . The important question is whether the value of the action depends on this choice of parameter. Polchinski comments that, in fact, it is a completely arbitrary choice of parameterisation. This should make sense because, again, the invariant length ds on the particle worldline {\mathcal{P}} should not depend on how the path is parameterised.

Proposition 1 The action (26) is reparameterisation invariant such that if we replace {\tau} with the parameter {\tau^{\prime} = f(\tau)} , where f is monotonic, we obtain the same value for the action.

Proof: Consider the following reparameterisation of the particle’s worldline {\tau \rightarrow \tau^{\prime} = f(\tau)} . Then we have

\displaystyle d\tau \rightarrow d\tau^{\prime} = \frac{\partial f}{\partial \tau}d\tau, \ \ (27)

implying

\displaystyle  \frac{dX^{\mu}(\tau^{\prime})}{d\tau} = \frac{dX^{\mu}(\tau^{\prime})}{d\tau^{\prime}}\frac{d\tau^{\prime}}{d\tau} = \frac{dX^{\mu}(\tau^{\prime})}{d\tau^{\prime}} \frac{\partial f(\tau)}{\partial \tau}. \ \ (28)

Plugging this into the action (26) we get

\displaystyle S^{\prime} = -mc \int_{\tau_i}^{\tau_f} d\tau^{\prime} \ \sqrt{\frac{dX^{\mu}(\tau^{\prime})}{d\tau^{\prime}} \frac{dX_{\mu}(\tau^{\prime})}{d\tau^{\prime}}}

\displaystyle  = -mc \int_{\tau_i}^{\tau_f} \frac{\partial f}{\partial \tau} \ d\tau \ \sqrt{\frac{dX^{\mu}}{d\tau} \frac{dX_{\mu}}{d\tau} (\frac{\partial f}{\partial tau})^{-2}}

\displaystyle  = -mc \int_{\tau_i}^{\tau_f} (\frac{\partial f}{\partial \tau})(\frac{\partial f}{\partial \tau})^{-1} \ d\tau \ \sqrt{\frac{dX^{\mu}}{d\tau} \frac{dX_{\mu}}{d\tau}}

\displaystyle  = -mc \int_{\tau_i}^{\tau_f} d\tau \ \sqrt{\frac{dX^{\mu}(\tau)}{d\tau} \frac{dX_{\mu}(\tau)}{d\tau}}. \ \ (29)

\Box

This ends the proof. So we see the value of the action does not depend on the choice of parameter; indeed, the choice is arbitrary.

As alluded earlier in this section, reparameterisation invariance is a gauge symmetry. In some sense, this is not even an honest symmetry; because it means that we’ve introduced a redundancy in our description, as not all degrees of freedom {X^{\mu}} are physically meaningful. We’ll discuss this more in the context of the string (an example of such a redundancy appears in the study of the momenta).

4. Equation of motion for {S_{pp}}

To obtain (eqn. 1.2.3), Polchinski varies the action (26) and then integrates by parts. For simplicity, let us temporarily maintain {c = 1} . Varying (26)

\displaystyle  \delta S_{pp} = -m \int d\tau \delta (\sqrt{-\dot{X}^{\mu}\dot{X}_{\mu}}) \ \ (30)

\displaystyle  = -m \int d\tau \frac{1}{2}(-\dot{X}^{\mu}\dot{X}_{\mu})^{-1/2}(-\delta \dot{X}^{\mu}\dot{X}_{\mu}), \ \ (31)

then from the last term we pick up a factor of 2 leaving

\displaystyle  = -m \int d\tau (-\dot{X}^{\mu}\dot{X}_{\mu})^{-1/2} + (-\dot{X}^{\mu}\delta \dot{X}_{\mu}). \ \ (32)

Next, we make the substitution {u^{\mu} = \dot{X}^{\mu}(-\dot{X}^{\nu}\dot{X}_{\nu})^{-1/2}} such that

\displaystyle  \delta S_{pp} = -m \int d\tau (-u_{\mu})\delta \dot{X}^{\mu}. \ \ (33)

And now we integrate by parts, which shifts a derivative onto u using the fact we can commute the variation and the derivative {\delta \dot{X}^{\mu} = \delta d / d\tau X^{\mu} = d/d\tau \delta X^{\mu}} . We also drop the total derivative term that we obtain in the process

\displaystyle  \delta S_{pp} = -m \int d\tau \frac{d}{d\tau} (-u_{\mu}\delta X^{\mu}) - m \int d\tau \dot{u}_{\mu} \delta X^{\mu}, \ \ (34)

which gives the correct result

\displaystyle  \delta S_{pp} = -m \int d\tau \dot{u}_{\mu}\delta X^{\mu}. \ \ (34)

As Polchinski notes, the equation of motion {\dot{u}^{\mu} = 0} describes the free motion of the particle.

With the particle mass m being the normalisation constant, we can also take the non-relativistic limit to find (exercise 1.1). Returning to (26), one way to do this is for {\tau} to be the proper time, then, as before (reinstating c for the purpose of example)

\displaystyle  \dot{X}^{\mu}(\tau) = c \frac{dt}{d\tau} + \frac{d\vec{X}^{\mu}(\tau)}{d\tau} \ \ (35)

so that we may define the quantity {\gamma = (1 - v^2/c^2)^{-1/2}} . Then, in the non-relativistic limit where {v << c} we have {dt/d\tau = \gamma = 1 + \mathcal{O}(v^2/c^2)} . It follows

\displaystyle  \dot{X}^{\mu}\dot{X}_{\mu} = -c^2 + \mid \vec{v} \mid^2 + \mathcal{O}(v^2/c^2), \ \ (36)

with {\vec{v}} a spatial vector and we define the norm {\mid \vec{v} \mid \equiv v} . Now, equivalent as with the choice of static gauge, the action to order {v/c} takes the form

\displaystyle S_{pp} \approx -mc \int dt \sqrt{c^2 -\mid \vec{v} \mid^2}, \ \ (37)

where we now taylor expand to give

\displaystyle  S_{pp} \approx -mc \int (1 - \frac{1}{2}\frac{\mid \vec{v} \mid^2}{c^2}) \ \ (38)

Observe that we now have a time integral of a term with classical kinetic structure minus a potential-like term (actually a total time derivative) that is an artefact of the relative rest energy

\displaystyle  S_{pp} \approx \int dt \ (\frac{1}{2}m\mid \vec{v} \mid^2 - mc^2). \ \ (39)

5. Deriving {S_{pp}^{\prime}} (eqn. 1.2.5)

The main problem with the action (18) and equivalently (26) is that, when we go to quantise this theory, the square root function in the integrand is non-linear. Analogously, we will find a similar issue upon constructing the first-principle string action, namely the Nambu-Goto action. Additionally, in our study of the bosonic string, we will be interested firstly in studying massless particles. But notice that according to the action (26) a massless particle would be zero.

What we want to do is rewrite {S_{PP}} in yet another equivalent form. To do this, we add an auxiliary field so that our new action takes the form

\displaystyle  S_{pp}^{\prime} = \frac{1}{2} \int d \tau (\eta^{-1} \dot{X}^{\mu} \dot{X}_{\mu} - \eta m^2), \ \ (40)

where we define the tetrad {\eta (\tau) = (- \gamma_{\tau \tau} (\tau))^{\frac{1}{2}}} . The independent worldline metric {\gamma_{\tau \tau}(\tau)} that we’ve introduce as an additional field is, in a sense, a generalised Lagrange multiplier. For simplicity we can denote this additional field {e(\tau)} so that we get the action

\displaystyle  S_{pp}^{\prime} = \frac{1}{2} \int d\tau (e^{-1} \dot{X}^{2} - em^{2}), \ \ (41)

where we have simplified the notation by setting {\dot{X}^{2} = \eta_{\mu \nu}\dot{X}^{\mu}\dot{X}^{\nu}} and completely eliminated the square root. This is equivlant to what Polchinski writes in (eqn.1.2.5). The structure of (41) may look familiar, as it reads like a worldline theory coupled to 1-dimensional gravity (worth checking and playing with).

To see that {S_{pp}^{\prime}} is classically equivalent (on-shell) to {S_{pp}} , we first consider its variation with respect to {e(\tau)}

\displaystyle  \delta S_{pp}^{\prime} = \frac{1}{2}\delta \int d\tau (e^{-1} \dot{X}^{2} - m^2 e)

\displaystyle  = \frac{1}{2} \int d\tau (- \delta (\frac{1}{e})\dot{X}^{2} - \delta (m^{2} e))

\displaystyle  = \frac{1}{2} \int d\tau (- \frac{1}{e^{2}}\dot{X}^{2} - m^{2}), \ \ (42)

which results in the following field equations

\displaystyle  e^{2} = \frac{\dot{X}^{2}}{m^{2}}

\displaystyle  \implies e = \sqrt{\frac{-\dot{X}^{2}}{m^{2}}} \ \ (43).

This again aligns with Polchinski’s result (eqn. 1.2.7).

Proposition 2 If we substitute (43) back into (41), we recover the original {S_{pp}} action (26).

Proof:

\displaystyle  S_{pp}^{\prime} = \frac{1}{2} \int d\tau [(-\frac{\dot{X}^2}{m^{2}})^{-1/2} \dot{X}^{2} - m^{2}(-\frac{\dot{X}^{2}}{m^{2}})^{1/2}]

\displaystyle  = \frac{1}{2} \int d\tau [(-\frac{m^{2}}{\dot{X}^{2}})^{1/2} (\dot{X}^{2} - m^{2}(\frac{\dot{X}^{2}}{m^{2}})^{1/2})]

\displaystyle  = \frac{1}{2} \int d\tau [(-\frac{m^{2}}{\dot{X}^{2}})^{1/2} (\dot{X}^{2} - m (- \dot{X}^{2})^{1/2})] \ \ (44)

Recalling {\dot{X}^{2} = \eta_{\mu \nu} \dot{X}^{\mu}\dot{X}^{\nu}} , substitute for {\dot{X}} in the square root on the right-hand side

\displaystyle  = \frac{1}{2} \int d\tau [(-\frac{m^{2}}{\dot{X}^{2}})^{1/2} \dot{X}^{2} - m (- \eta_{\mu \nu} \dot{X}^{\mu}\dot{X}^{\nu})^{1/2}. \ \ (45)

For the first term we clean up with a bit of algebra. From complex variables recall {i^{2} = -1} .

\displaystyle  (-\frac{m^{2}}{\dot{X}^{2}})^{1/2} \dot{X}^{2} = (-1)(-1) -(\frac{m^{2}}{\dot{X}^{2}})^{1/2} \dot{X}^{2}

\displaystyle  = -(-\frac{m^{2}}{\dot{X}^{2}})^{1/2} i^{2} \dot{X}^{2}

\displaystyle = -(-\frac{m^{2}}{\dot{X}^{2}} i^{4} \dot{X}^{2})^{1/2}

\displaystyle  = -(-m^{2}i^{4}\dot{X}^{2})^{1/2} = -m (-i^{4}\dot{X}^{2})^{1/2}. \ \ (46)

As {i^{4} = 1} , it follows {-m(i^{4}\dot{X}^{2})^{1/2} = -m (-\dot{X}^{2})^{1/2}} . Now, substitute for {\dot{X}^{2}} and we find {-m (-\eta_{\mu \nu}\dot{X}^{\mu}\dot{X}^{\nu})^{1/2}} giving

\displaystyle  S_{pp}^{\prime} = \frac{1}{2} \int d\tau [-m(- \eta_{\mu \nu}\dot{X}^{\mu}\dot{X}^{\nu})^{1/2} - m (- \eta_{\mu \nu} \dot{X}^{\mu}\dot{X}^{\nu})^{1/2}

\displaystyle  = -m \int d\tau (- \eta_{\mu \nu}\dot{X}^{\mu}\dot{X}^{\nu})^{1/2} = S_{pp} \ \ (47).

\Box

This ends the proof, demonstrating that {S_{pp}} and {S_{pp}^{\prime}} are classically equivalent.

It is also possible to show that, like with {S_{pp}} , the action {S_{pp}^{\prime}} is both Poincaré invariant and reparameterisation invariant.

6. Generalising to Dp-branes

As an aside, and to conclude this note, we can generalise the action for a point particle (0-brane) to an action for a p-brane. It follows that a p-brane in a {D \geq p} dimensional background spacetime can be described in such a way that the action becomes,

\displaystyle  S_{pb}= -T_p \int d\mu_p \ \ (48).

The term {T_p} is one that will become more familiar moving forward, especially when we begin to discuss the concept of string tension. However, in the above action it denotes the p-brane tension, which has units of mass/volume. The {d\mu_p} term is the {(p + 1)} -dimensional volume measure,

\displaystyle  d\mu_p = \sqrt{- \det G_{ab}} \ d^{p+1} \sigma, \ \ (49)

where {G_{ab}} is the induced metric, which, in the {p = 1} case, we will understand as the worldsheet metric. The induce metric is given by,

\displaystyle  G_{ab} (X) = \frac{\partial X^{\mu}}{\partial \sigma^{a}} \frac{\partial X^{\nu}}{\partial \sigma^{b}} h_{\mu \nu}(X) \ \ \ a, b \equiv 0, 1, ..., p \ \ (50)  p>

A few additional comments may follow. As {\sigma^{0} \equiv \tau} , spacelike coordinates in this theory run as {\sigma^{1}, \sigma^{2}, ... \sigma^{p}} for the surface traced out by the p-brane. Under {\tau} reparameterisation, the above action may also be shown to be invariant.

7. Summary

To summarise, one may recall how in classical (non-relativistic) theory [LINK] the evolution of a system is described by its field equations. One can generalise many of the concepts of the classical non-relativistic theory of a point particle to the case of the relativistic point particle. Indeed, one will likely be familiar with how in the non-relativistic case the path of the particle may be characterised as a path through space. This path is then parameterised by time. On the other hand, in the case of the relativistic point particle, we have briefly reviewed how the path may instead be characterised by a worldline through spacetime. This worldline is parameterised not by time, but by the proper time. And, in relativity, we learn in very succinct terms how freely falling relativistic particles move along geodesics.

It should be understood that the equations of motion for the relativistic point particle are given by the geodesics on the spacetime. This means that one must remain cognisant that whichever path the particle takes also has many possibilities, as noted in an earlier section. That is, there are many possible worldlines between some beginning point and end point. This useful fact will be explicated more thoroughly later on, where, in the case of the string, we will discuss the requirement to sum over all possible worldsheets. Other lessons related to the point particle will also be extended to the string, and will help guide how we construct the elementary string action.

References

[Moh08] T. Mohaupt, Liverpool lectures on string theory [lecture notes].

[Pol07] J. Polchinski, An introduction to the bosonic string. Cambridge, Cambridge University Press. (2007).

[Wray11] K. Wray, An introduction to string theory [lecture notes].

[Zwie09] B. Zwiebach, A first course in string theory. Cambridge, Cambridge University Press. (2009).

Learning M-theory: Gauge theory of membranes, brane intersections, and the self-dual string

I’ve been learning a lot about M-theory. It’s such a broad topic that, when people ask me ‘what is M-theory?’, I continue to struggle to know where to start. Right now, much of my learning is textbook and I have more questions than answers. I naturally take the approach of first wanting as broad and general of a picture as possible. In some sense, it is like starting with the general and working toward the particular. Or, in another way, it’s like when being introduced to a new landscape and wanting, at the outset, a broad orientation to its general geographical features, except in this case we are speaking in conceptual and quantitative terms. I may not ever be smart enough to grasp M-theory in its entirety, but what is certain is that I am working my hardest.

In surveying its geographical features and charting my own map, if I may continue the analogy, obtaining a better sense of the fundamental objects of M-theory is a particular task; but my main research interest has increasingly narrowed to the study and application of gauge theory and higher gauge theory. This can be sliced down further in that I am very interested in the relationship between string and gauge theory, and furthermore in studying the higher dimensional generalisation of gauge theory. This interest naturally follows from the importance of gauge theory in contemporary physics, and then how we may understand it from the generalisation of point particle theory to string theory and then to other higher dimensional extended objects (i.e., branes). We’ve talked a bit in the past about how the dynamics on the D-brane worldvolume is described by a gauge theory. We’ve also touched on categorical descriptions, and how in p-brane language when we study the quantum theory the resemblance of the photon can be seen as a p-dimensional version of the electromagnetic field (by the way, we’re going to start talking about p-branes in my next string note). That is to say, we obtain a p-dimensional analogue of Maxwell’s equations. More advanced perspectives from the gauge theory view, or in this case higher gauge theory view in M-theory, illuminate the existence of new objects like self-dual strings.

There is so much here to write about and explore, I look forward to sharing more as I progress through my own studies and thinking. In this post, though, I want to share some notebook reflections on things I’ve been learning more generally in the context of M-theory: some stuff about membranes, 11-dimensional supergravity, and the self-dual string. This post is not very technical; it’s just me thinking out loud.

11-dimensional supergravity

The field content of 11-dimensional supergravity consists of the metric g_{\mu \nu}  , with 44 degrees of freedom; a rank 3 anti-symmetric tensor field C_{\mu \nu \rho}  , with 84 degrees of freedom; and these are paired off with a 32 component Majorana gravitino \Psi_{\alpha \mu}  , with 128 degrees of freedom. Although much has progressed since originally conceived, the Lagrangian for the bosonic sector is similar to as it was originally written [3]

S_{SUGRA} = \frac{1}{2k_{11}^2} \int_{M_{11}} \sqrt{g} \ (R - \frac{1}{48}F^{2}_{4}) - \frac{1}{6} F_{4} \wedge F_{4} \wedge C_3. \ \ (1)

The field strength is F_4 = dC_3  and k_{11}  is the 11-dimensional coupling constant. The field strength is defined conventionally,

\mid F_n \mid^2 = \frac{1}{n !} G^{M_1 N_1} G^{M_2 N_2} ... G^{M_n N_n}F_{M_{1}M_{2} ... M_{n}}F_{N_1 N_2 ... N_n}. \ \ (2)

The 11-dimensional frame field in the metric combination is G_{MN} = \eta_{AB}E^{A}_{M}E^{B}_{N}  , where we have the elfbeins E^{B}_{N}  , M,N  are indices for curved base-space vectors, and A,B  are indices for tangent space vectors. The last term in (2) is the Cherns-Simons structure. This is a topological dependent term independent of the metric. We see this structure in a lot of different contexts.

Although, from what I presently understand, the total degrees of freedom of M-theory are not yet completely nailed down, we can of course begin to trace a picture in parameter space. As we’ve discussed before on this blog, it can be seen how 10-dimensional type IIA theory in the strong coupling regime behaves as an 11-dimensional theory whose low-energy limit is captured by 11-dimensional supergravity. Reversely, compactify 11-dimensional supergravity on a circle of fixed radius in the x^{10} = z  direction, from the 11-dimensional metric we then obtain the 10-dimensional metric, a vector field and the dilaton. The 3-form potential leads to both a 3-form and a 2-form in 10-dimensions. The mysterious 11-dimensional theory can also be seen to give further clue at its parental status given how supergravity compactified on unit interval {\mathbb{I} = [0,1]}  , for example, leads to the low-energy limit of E8 \times E8  heterotic theory.

Non-renomoralisability of 11-dimensional SUGRA

One thing that I’ve known about for sometime but I have not yet studied in significant detail concerns precisely how 11-dimensional supergravity is non-renormalisable [4,5,6]. Looking at the maths, what I understand is that above two-loops the graviton-graviton scattering is divergent. Moreover, as I still have some questions about this, what I find curious is that in the derivative expansion in 11-dimensional flat spacetime (using a 1PI/quantum effective Lagrangian approach) the generating functional for the graviton S-matrix is non-local. But due to supersymmetry, low order terms in the derivative expansion can be separated into local terms, such as t_8 t_8 R^4  , and non-local (or global) terms that correspond to loop amplitudes. But what happens is that, at 2-loops, a logarithmic divergence that is cut off at the Planck scale mixes with a local term of the schematic form D^{12}R^4  , where R^4  is the supersymmetrised vertex. In the literature, one will find a lot of discussion about this R^4  vertex. But like I said, I really need more time looking at this.

In short, the important mechanism in string theory that allows us to avoid UV divergences is absent, or appears absent, in maximal supergravity. What could the UV regulator be? As in any supergravity, from what I understand, it is not clear that a Lagrangian description is sufficient at the Planck scale.

Membranes, D-branes, and AdS/CFT

The facts of 11-dimensional supergravity and how it relates to 10-dimensional string theory are textbook and well-known. Going beyond dualities relating different string theories, an obvious question concerns what M-theory actually constitutes. One thing that is known is that M-theory reduces to 11-dimensional SUGRA at low-energies, as we touched on, and it is known that fundamental degrees of freedom are 2-dimensional and 5-dimensional objects, known as M2-branes and M5-branes. Study of these non-perturbative states offer several intriguing hints. There are also solutions to classical supergravity known as F1 – the fundamental string – and its magnetic dual, the NS5-brane. As it relates to the story of the five string theories, the M-branes realize all D-branes, and this is why D-branes are considered consistent objects in quantum gravity.

The way that M-theory sees D-branes is via the net of dualities. All of the D-branes and the NS5 brane are solutions to type II theories, both A and B. So, when you reduce M-theory on a circle, in that you get back to Type IIA, the M2-branes and M5-branes reduce to the various D-branes such that under S-duality from the D5-brane you get the NS5.

The worldvolume theory of the M5-brane is always strongly coupled, which can be seen in moduli space (its parameters are simply a point). So there is no Lagrangian for this theory, and it suggests something deep is needed or is missing. It is expected that its worldvolume theory will be a 6-dimensional superconformal field theory, typically known as the 6d(2,0) theory. The worldvolume theory for M2-branes (on an orbifold) has been found to be a 3-dimensional superconformal Chern-Simons theory with classical \mathcal{N} = 6 supersymmetry.

If one considers a single M5-brane, a theory can be formulated in terms of an Abelian (2,0)-tensor multiplet, consisting of a self-dual 2-form gauge field, 5 scalars, and 8 fermions, but it is not known how to generalise the construction to describe multiple M5-branes. To give an example, using AdS/CFT [7] it is described how the worldvolume theory for a stack of N  M5-branes is dual to M-theory on AdS7 \times S4  with N  units of flux through the 4-sphere, which reduces to 11-dimensional SUGRA on this background in the limit large N  limit.

Brane intersections and stacks

The existence of branes is one of the most fascinating things about quantum gravity. There is a lot to unpack when learning about D1-branes, D3-branes, D5-branes, M2-branes, and M5-branes, as well as how they may intersect and what sort of consistent solutions have already been found [8,9, 10, 11, 12].

For example, an M2-brane, or a stack of coincident M2-branes, can end on a D5-brane. This is similar to the more simplified story of how D-branes, coincident D-branes, can intersect in string theory. Typically, D1-D3 systems in Type IIB string theory are studied because this system relates to the M2-M5 system by dimensional reduction and T-duality.

Self-dual string

For a membrane to end on a D5-brane, the membrane boundary must carry the charge of the self-dual field B on the five-brane worldvolume. There are different solutions to the field equations of B. For instance, a BPS solution was found [10] by looking at the supersymmetry transformation.

The linearised supersymmetry equation is

\delta_{\epsilon} \Omega^{j}_{\beta} = \epsilon^{\alpha i}(\frac{1}{2} (\gamma^{a})_{\alpha \beta}(\gamma_{b^{\prime}})^{j}_{i}\partial_a X^{b^{\prime}} - \frac{1}{6}(\gamma^{abc})_{\alpha \beta}\delta^{j}_i h_{abc}) = 0. \ \ (3)

Here b^{\prime}  labels transverse scalars, a indices label worldvolume directions, \alpha, \beta  denote spinor indices of spin(1,5), and i,j are spinor indices of USp(4)  . The solution balances the contribution of the 3-form field strength h with a contribution from the scalars. Additionally, the worldvolume of the string soliton can be taken to be in the 0,1 directions with all fields independent of x^0  and x^1  . An illustration of the solution is given below, showing an M2-brane ending on an M5-brane with a cross section S^3 \times \mathbb{R}  .

M2-branes ending on a M5-brane. The endpoint is a string. Courtesy of N. Copland, Aspects of M-Theory Brane Interactions and String Theory Symmetries [https://arxiv.org/abs/0707.1317].

As I am still trying to understand the calculation, I am currently looking at the following string solution

H_{01m} = \pm \frac{1}{4} \partial_m \phi,

H_{mnp} = \pm \frac{1}{4} \epsilon_{emnpq}\delta^{qr}\partial_r \phi,

\phi = \phi_0 + \frac{2Q}{\mid x - x_0 \mid^2}, \ \ (4)

where \phi  may be replaced by a more general superposition of solutions. We denote \pm Q as the magnetic and electric charge. There is a conformal factor in the full equations of motion which guarantees that they are satisfied even at x = x_0  , which means the solution is solitonic. This string soliton is said to possess its own anomalies that require cancellation (I assume Weyl, Lorentz). What is neat is that this string can be dimensionally reduced to get various T-duality configurations, which is something that would be fun to look into at some point down the road.

References

[1] D. Fiorenza, H. Sati, and U. Schreiber, The rational higher structure of m-theory. Fortschritte der Physik, 67(8-9):1910017, May 2019. [arXiv:1903.02834 [hep-th]].

[2] E. Witten, String theory dynamics in various dimensions. Nuclear PhysicsB, 443(1):85 – 126, 1995.

[3] E. Cremmer, B. Julia, and J. Scherk, Supergravity Theory in 11-dimensions. Phys. Lett. B76, No. 4, (409-412) 19 June 1978.

[4] S. Chester, S. Pufu, and X Yin, The M-Theory S-Matrix from ABJM: Beyond 11D supergravity. (2019). [arXiv:1804.00949v3 [hep-th]].

[5] A. Tseytlin, R4 terms in 11 dimensions and conformal anomaly of (2,0) theory. (2005). [arXiv:hep-th/0005072v4 [hep-th]].

[6] G. Russo, and A. Tseytlin, One-loop four-graviton amplitude in eleven-dimensional supergravity. (1997). [arXiv:hep-th/9707134v3 [hep-th]].

[7] P. Heslop, and A. Lipstein, M-theory Beyond The Supergravity Approximation. (2017). [arXiv:1712.08570 [hep-th]].

[8] P.K. Townsend, D-branes from M-branes. (1995). [arXiv:hep-th/9512062 [hep-th]].

[9] A. Strominger, \textit{Open p-branes}. Phys. Lett. B 383 (1996) 44. [arXiv:hep-th/9512059 [hep-th]].

[10] P.S. Howe, N.D. Lambert, and P.C. West, The self-dual string soliton. Nucl. Phys. B 515 (1998) 203. [arXiv:hep-th/9709014 [hep-th]].

[11] M. Perry and J.H. Schwarz, Interacting chiral gauge fields in six dimensions and Born-Infeld theory. Nucl. Phys. B 489 (1997) 47. [arXiv:hep-th/9611065 [hep-th]].

[12] D.S. Berman, Aspects of M-5 brane world volume dynamics. Phys. Lett. B 572 (2003) 101. [arXiv:hep-th/0307040 [hep-th]].

[13] J. Huerta, H. Sati, and U. Schreiber, Real ADE-equivariant (co)homotopy and Super M-branes. (2018). [arXiv:1805.05987 [hep-th]].

[14] N. Copland, Aspects of M-Theory Brane Interactions and String Theory Symmetries. [https://arxiv.org/abs/0707.1317].

[15] S. Palmer, Higher gauge theory and M-theory. [https://arxiv.org/abs/1407.0298].

Stringification as categorisation

In quantum field theory one is typically taught to use perturbation theory when the equations of motion for the fields are nonlinear and weakly interacting. For example, in \phi^4  theory one can use a formal series as described by Rosly and Selivanov [1]. Perturbative theory is about mastering series expansions. The basic idea, upon constructing some correlation function in the full nonlinear model, is to expand in powers of \alpha  , namely the interaction strength. In the language of perturbative physics, Feynman diagrams give a representation of each term in the expansion such that we use them to illustrate linear operators. This ultimately enables us to obtain a good approximation to the exact solution. Needless to say, there is a real power and usefulness about perturbative methods and the sum of Feynman diagrams.

When computing amplitudes with Feynman diagrams, the amplitudes depend on various topological properties (i.e., vertices, loops, and so on). Although not always made explicit in the perturbative view, from the Fenynman diagrams of 0-dimensional points with 1-dimensional graphs (to use the language of p-branes, which we’ll get to in a moment), we have topologies that describe linear operators: i.e., what Feynman diagrams start to make explicit is the deeper role of topology in physics [2]. This was summarised wonderfully in a lovely article by Atiyah, Dijkgraaf, and Hitchin [3]. Mathematically, and from the perspective of geometry, the main idea is that a linear operator behaves very much like an n-dimensional manifold going between manifolds of one dimension less, which we may define as a cobordism (i.e., think of a stringy ‘trousers’ diagram) [2,4].

Now, consider the story of p-branes, in particular the perspective as we pass from standard quantum field theory to string theory. The language of p-branes as first described by Duff et al [5] may be reviewed in any introductory string theory textbook. We can, from first-principles, motivate string theory thusly: in a special, if not unique way, we may generalise the point-like 0-dimensional particle to the 1-dimensional string, which is made explicit when we generalise the action for a relativistic particle to the Nambu-Goto action for the relativistic string. In the language of p-branes, which are p-dimensional objects moving through a D(D \geq p)  dimensional space-time, a 0-brane is a (0-dimensional) point particle that that traces out a (0+1)-dimensional worldline. The generalisation of the point particle action S_0 = -m \int ds to a p-brane action in a D(\geq p) -dimensional space-time background is given by S_p = -T_p \int d\mu_p . Here T_p is the p-brane tension with units mass/vol, and d\mu_p is the (p + 1)-dimensional volume element. For the special case where p=1 such that we have 1-brane, we obtain the string action which sweeps out a (1+1)-dimensional surface that is the string worldsheet propagating through space-time. We can also go on to speak of higher-dimensional objects, such as those that govern M-theory. For instance, a 2-brane is a membrane. Historically, these were considered as 2-dimensional particles. There are also 3-branes, 4-branes, and so on.

This generalising process, if we can describe it that way, is what I like to think of as stringification. For the case where p=1 , Feynman diagrams of ordinary quantum field theory with 2-dimensional cobordisms represent world-sheets traced out by strings. The generalising picture, or stringification, show these 2-dimensional cobordisms equipped with extra structure give a powerful mathematical language (describing the relation between physics and topology, as string diagrams enable us to sum over the various topologies and provide a valuable mathematical tool for thinking about composition). But of course this picture can still be extended. Not only does the important analogy between operators and cobordisms come directly into focus, it is also, in some sense, where stringification meets categorification. That is, from the maths side, we arrive at the logic of higher-dimensional algebra and the arrows of monoidal and higher categories. In each, physical processes are describe by morphisms or functors (functors are like morphisms between categories). This generalising picture toward higher geometry, higher algebra, and, indeed, higher structures is called ‘categorifying’ or ‘homotopifying’ (my notes on which I have started to upload to this blog). In this post, I want to think a bit about this idea of stringification as categorification.

***

There is a view of M-theory, and I suppose of fundamental physics as whole, that I find fascinating and compelling: stringification as the categorisation of physics. The notion of stringification is not formal, but captures if nothing else an intuition about a certain generalising process or abstract story, or at least that is how I presently see it. It is a term I have picked up that used to float around in different contexts a couple of decades ago. As described through the language of p-branes, the story begins with the generalisation or stringification of point particle theory (and all that it implies) toward the existence of the string and eventually other extended objects in fundamental physics. Meanwhile, the notion of categorification is certainly formal, signalling, at its origin, the process of finding category-theoretic analogs of set-theoretic concepts by replacing sets with categories. This process, when iterated, gives definition to the notion of n-category theory, where we also replace functions with functors, and equations between functions by natural isomorphisms between functors [6]. As Schreiber pointed out in 2004, there is a sort of harmony between these two processes – stringification and categorification – which has certainly started to clarify over the last decade or more.

As one example, the observation that Schreiber describes in the linked post refers to boundaries of membranes attached to stacks of 5-branes, which conceptually appear as a higher-dimensional generalisation of how boundaries of strings appear.

To understand this think, firstly, of the simple example of the existence of D-branes (Dirichlet membranes) and how the endpoints of open strings can end on these extended objects. In fact, an introductory string textbook will guide one to see why the equations of motion of string theory require that the endpoints of an open string satisfy one of two types of boundary conditions (Dirichlet or Neumann) ending on a brane. If the endpoint is confined to the condition that it may move within some p-dimensional hyperplane, one then obtains a first description of Dp-branes. (I think this was one of the first things I calculated when learning strings!). For the sake of saving space I won’t go into the arrangement of D-branes or other related topics. The main point that I am driving at, the technicalities of which we could review in another post, is how these branes are dynamic and as such they may influence the dynamics of a string (i.e., how an open string might move and vibrate). Thus, the arrangement of branes (e.g., we can have parallel branes or ‘stacks’) will also impact or control the types of particles in our theory. It is truly a beautiful picture.

In p-brane language, if you take the Nambu–Goto action and for the quantum theory study the spectrum of particles, you will see that it exhibits what we may describe as the photon, which of course is the fundamental quantum of the electromagnetic field. Now, what is nice about this is that, the resemblance of the photon is actually a p-dimensional version of the electromagnetic field, so it is in fact a p-dimensional analogue of Maxwell’s equations.

What Schreiber is highlighting in his post is not just that in string theory, the points of the string ending on a Dp-brane give rise to ordinary gauge theory. (One could even take the view that string theory predicts electromagnitism such that string theory predicts the existence of D-branes. It is by their nature that these extended objects all carry an electromagnetic field on their volume, i.e., what we call the brane volume). The point made is that, given there is reason to extend the picture further – the picture of stringification so to speak – to higher-dimensional generalisations, we can then replace strings with membranes, and so on. From the maths side, it was realised that from the perspective of categories, something analogous is happening: replacing points with arrows (i.e., morphisms) one finds the gauge string may be described by the structure of nonabelian gerbes (a gerbe is just a generalised analogue of a fibre bundle), and so on.

When I first learned strings, the picture of stringification was in my mind but I didn’t yet have a word for it. I also didn’t possess category theoretic language at the time; it was really only a vague sense of a picture, perhaps emphasised in the way I learned string theory. So when I discovered and read last year about the idea of stringification as categorisation [7] in Schreiber’s thesis, I was excited.

A nice illustration comes from the first pages of this work. Take some ordinary point-particle, which traces out a worldline over time t . The thrust of the idea is that, given some charge, there is a connection in some bundle (yet unspecified) such that, locally, a group element g \in G is associated to the path. Diagrammatically this may be represented as,

Now consider some time t^{\prime} , where t^{\prime} > t . The particle has travelled a bit further,

We can of course compose these paths. The composition is associative and the operation is multiplication. In fact, what we’re doing is multiplying the group elements. We can also define an inverse g^{-1} . The punchline is that, from the theory of fibre bundles with connection, we can consider how this local picture may fit globally. If g is an element in a non-abelian group, the particle we are generalising is non-abelian. Generalise from a point-particle to a string, and the diagrammatic representation of the world-sheet takes the form

Ultimately, we can continue to play this game and develop the theory of non-abelian strings (and on to higher-dimensional branes), which, it turns out, corresponds with a 2-category theory [7,8]. Sparing details, in n-category theory a 2-category is a special type of category wherein, besides morphisms between objects, it possesses morphisms between morphisms. What is interesting about this example is how we can go on to show the idea of SUSY quantum mechanics on loop space relates to ideas in higher gauge theory, particularly in the sense of categorifying standard gauge theory. For example, John Baez’s paper on higher Yang-Mills [9]. But even before all of that, from the view of perturbative string theory being the categorification of supersymmetric quantum mechanics, we can play the same game such that the generalisation of the membranes of M-theory are a categorification of the supersymmetric string, and so on. The intriguing and, perhaps, grand idea, is that this process of stringification as categorification can be utilised to describe the whole of physics, or, so, it is suspected.

***

I’ve been thinking about this picture quite a bit recently, perhaps spurred by all of my ongoing studies in M-theory. The view to be encircled, as the notion of categorisation enters the stringy picture, also marks for me the beginning of the story about higher structures in fundamental physics (in terms of the view of category theory and higher category theory). In a sense, as much as I currently understand it (as I am very much in the process of studying and forming my thoughts on the matter) we are encircling not much more than an abstract story; but it is one in which many tantalising hints exist about a potentially foundational view.

The history of this higher structure view is rich with examples [10, 11], and, for many reasons, it leads us directly to a study of the plausible existence of M-theory. From the use of braided monoidal categories in the context of string diagrams through to knot theory (See Witten’s many famous lectures); the notion of quantum groups; Segal’s famous work on the axioms of conformal field theory (described in terms of monoidal functors and the category 2Cob_{\mathbb{C}} whose morphisms are string world-sheets such that we can compose the morphisms, and so on); and of course the work of Atiyah in topological quantum field theory (TQFT) followed by Dijkgraaf’s thesis on 2d TQFTs in terms of Frobenius algebras – the list is far to big to summarise in a single paragraph. All of this indicates, in some general sense, a very abstract story from basic quantum mechanics through to string theory and, I would say, as a natural consequence M-theory.

It is a fascinating perspective. There is so much to be said about this developing view, including why higher geometry and algebra seem to hold the important clues of M-theory as a fundamental theory of physics. What is also interesting, as I am beginning to understand, is that in the higher structure picture, a striking consequence from a geometric persective is that the geometry of fundamental physics (higher geometry and supergeometry) may not be described by spaces with sets of points. And, in fact, we start to see this for each value of p  . Instead of a traditional notion space associated with the definition of topological spaces or differentiable manifolds, the geometric observation is that what we’re dealing with is functorial geometry of the sort described by Grothendieck, or synthetic differential geometry of the sort described by Lawvere, or a variation of them both.

Anyway, this is just a short note of me thinking aloud.

References

[1] Rosly, A.A., and Selivanov, K.G., On amplitudes in self-dual sector of Yang-Mills theory. [arXiv:9611101 [hep-th]].

[2] Baez, J., and Stay, M., Physics, Topology, Logic and Computation: A Rosetta Stone. [arXiv:0903.0340 [quant-ph]].

[3] Atiyah, M., Dijkgraaf, R., and Hitchin, N., Geometry and physics. Phil. Trans. R. Soc., (2010), A.368, 913–926. [http://doi.org/10.1098/rsta.2009.0227].

[4] Baez, J., and Lauda, A., A Prehistory of n-Categorical Physics. [https://math.ucr.edu/home/baez/history.pdf.].

[5] M. J. Duff, T. Inami, C. N. Pope, E. Sezgin [de], and K. S. Stelle, Semiclassical quantization of the supermembrane. Nucl. Phys. B297 (1988), 515.

[6] Baez, J., and Dolan, J., Categorification. (1998). [arXiv: 9802029 [math.QA]].

[7] Schreiber, U., From Loop Space Mechanics to Nonabelian Strings [thesis]. (2005). [hep-th/0509163].

[8] Baez, J. et al., Categorified Symplectic Geometry and the Classical String. (2008). [math-ph/0808.0246v1].

[9] Baez, J., \textit{Higher Yang–Mills theory}. (2002). [hep-th/0206130].

[10] Baez, J., and Lauda, A., A Prehistory of n-Categorical Physics. [https://math.ucr.edu/home/baez/history.pdf.]

[11] Jurco, B. et al., \textit{Higher structures in M-theory}. (2019). [arXiv:1903.02807v2].

Swampland Conjectures

Notes on the Swampland (3): Testing the Weak Gravity Conjecture – Gauge Fields, Dp-branes, Type II Strings, and F-Theory-Heterotic Duality

The following collection of notes is based on a series of lectures that I attended by Eran Palti at SiftS 2019 at Universidad Autonoma de Madrid. The theme of the lecture series was ‘String Theory and the Swampland’. Palti’s five lectures were supported by his most recent and impressive 200 page review paper on the Swampland, which includes over 600 references [arXiv: 1903.06239 [hep-th]]. The reader is directed to this paper in addition to supplementary references that I also provide at the end of each set of notes.

In the following entry, the notes presented follow the third lecture of Palti’s series.

1. Introduction

In this collection of notes, we look to review some more basic tests of the Weak Gravity Conjecture. In the last entry, recall that we reviewed a basic relation between the WGC and the Distance Conjecture. We then considered a first test of the Distance Conjecture having compactified our theory on a circle. Additionally, we reviewed evidence for the DC where we found that if we have large expectation values for the scalar fields in string theory, we obtain an infinite tower of exponentially light states. In this sense, we also reviewed the extreme parameter regime for weak and strong coupling. Finally, we reviewed a number of lessons about the DC and T-duality, concluding with a brief review of the parameter space of M-theory.

In the present entry – the third in this series of notes – we continue to expand on past discussions, turning particular attention to another basic test of the WGC. In further testing of the WGC we will also focus on a number of related topics ranging from gauge fields to Dp-branes and Type II strings, ending with a few brief comments on F-theory {\longleftrightarrow} Heterotic duality. This will then lead us directly into the fourth and second-last entry of the series, where we will begin to review more advanced tests of the DC and WGC, using for instance arbitrary Calabi-Yau manifolds.

2. Weak Gravity Conjecture

In this section we return to the WGC, which we have already grown to understand as being closely related to the DC. Following Palti’s lecture series, although the WGC is studied quite extensively from the infrared point of view, we shall instead be studying it from the ultraviolet and maximally stringy perspective.

Proceeding directly from the last entry we return to the simple example of string compactification on a circle and consider some of the physics in [3] as discussed in [1]. This time, in compactifying on {S^{1}}, we are going to instead consider a more general solution for the metric. The reason for this is because we want to study in particular the case of compactification with gauge fields. The metric may be written as follows,

\displaystyle  ds^{2} = e^{2 \alpha \phi}g_{\mu \nu}dX^{\mu}dX^{\nu} + e^{2 \beta \phi}(dX^{d} + A_{\mu}dX^{\mu})^{2} \ \ (1)

A few comments are necessary before proceeding. First, remember that we are working in perturbative superstring theory, so this metric is very similar to the one before, where the first term in the equality is a 9-dimensional object. Second, also remember from the last entry that our original metric encoded the parameter {\phi} such that it became a dynamical field in the lower d-dimensional theory. But, as Palti notes, there is also an additional degree of freedom in the metric. What does this mean? This additional degree of freedom becomes a U(1) gauge field {A_{\mu}} in the d-dimensional theory, as opposed to a scalar field, which will also have a coupling {g}. Furthermore, in that we have added another component to the metric, namely the 9-dimensional {A_{\mu}} term on the right-hand side, this is in fact the graviproton. Altogether, it follows that this is the most general solution for stringy compactification on a circle.

Now, what is of present interest is the Ricci scalar. So let’s look at what dimensional reduction now gives for the Ricci scalar,

\displaystyle  \int d^{D}X \sqrt{-G}R^{D} = \int d^{d}X\sqrt{-g} [R^{d} - \frac{1}{2}(\partial \phi)^{2} - \frac{1}{4}e^{-2(d - 1)\alpha \phi}F_{(A), \mu \nu} F^{\\mu \nu}_{(A)}] \ \ (2)

Where {F_{(A), \mu \nu} =\frac{1}{2} \partial_{[\mu}A_{\nu]}} is the gauge field kinetic term or, in other words, the field strength of the gauge field. Recall, also, from before that the {\phi} in the exponential is related to the radius in the extra dimensions. So from (2) we can read off the gauge coupling for the U(1) gauge field as follows,

\displaystyle  g_{(A)} = e^{d - 1}\alpha \phi = \frac{1}{2 \pi R} (\frac{1}{2 \pi R})^{\frac{1}{d - 2}} \ \ (3)

Which is telling us, similar to the last entry, that if we make the circle very large the theory becomes weakly coupled. But what is the symmetry of the U(1) gauge field? How do we know that symmetry of the gauge field? Consider a general U(1) gauge symmetry transformation of the form (i.e., the circle isometry),

\displaystyle  A_{\mu} \rightarrow A_{\mu} - \partial_{\mu} \lambda (X^{\nu}), \ \ X^{d} \rightarrow X^{d} + \lambda (X^{\nu}) \ \ (4)

Where {\lambda (X^{\nu})} is a local gauge parameter. Notice that the metric remains invariant, and from this we can indeed see how lower d-dimensional theory has a U(1) gauge field with the above gauge coupling.

Now, just like in the past entry, we want to look at the Kaluza-Klein expansion. Moreover, recalling the KK expansion for the higher D-dimensional field {\Psi (X^{\mu}) = \sum_{n = -\infty}^{\infty} \psi_{n} (X^{\mu})e^{2\pi i n X^{d}}}, notice that the gauge transformation (4) reveals that the KK modes {\psi_{n}} obtain a charge under the U(1) gauge field. This charge is quantised, as anticipated, and for the nth KK mode it may be given as,

\displaystyle  q_{n}^{A} = 2\pi n \ \ (5)

But what is the relation between the charge and the KK modes? Note, firstly, that the charge of {\psi_{n}} are just the phases of these objects. Secondly, the emphasis at this point in Palti’s talk is to remember that the mass of the KK states calculated in a past entry in the Einstein frame, {M^{2}_{\text{n kk mode}} = (\frac{n}{R})^{2} (\frac{1}{2 \pi R})^{2 \ d - 2}}, is related to the charge. More pointedly, we are already familiar with how, for the KK modes, there is an infinite tower of states. We see that the mass increases along this tower, and so too does the charge. In other words, it is argued that we have a charge-mass relation for the infinite tower of states. Here it is for arbitrary {n},

\displaystyle  g_{(A)} q_{(n)}^{(A)} = M_{n, 0} \ \ (6)

This relation between the charge, mass, and couping may have already been anticipated. Since all we’ve considered here is really just a reduction of Einstein gravity, let us consider the effective string action from a past set of notes, written below for convenience,

\displaystyle  S_{D} = 2\pi M_{s}^{D - 2} \int d^{D} X \sqrt{-G}e^{-2 \phi} (R - \frac{1}{12} H_{\mu \nu \rho} H^{\mu \nu \rho} + 4\partial_{\mu} \Phi \partial^{\mu} \Phi) \ \ (7)

If we compactify this action on a circle, as we are so inclined, there is a gauge field obtained from the gravitational sector. This is similar to before, and is nothing new. What is new is that we also now obtain a second gauge field, {V_{\mu}}, which comes from the Kalb-Ramond B-field with a single index in the {X^{d}} direction. For this Kalb-Ramond field we may write,

\displaystyle  V_{\mu} \equiv B_{[\mu d]} \ \ (8)

Where we note that, generally, {B_{[mn]}} is an antisymmetric 2-form. If we also reduce {B_{[mn]}}, this also leads to a gauge field. Additionally, look at {V_{\mu}} in (8). The kinetic terms for this additional gauge field are produced by the dimensional reduction of the kinetic terms from the Kalb-Ramond field. In other words, we can compute the kinetic term for the gauge field, {V_{\mu}}, as it comes from the strength of the 2-form in 10-dimensions,

\displaystyle  \int d^{d}X \sqrt{-g} [R^{d} - \frac{1}{4}e^{-2(\alpha + \beta)\phi}F_{(V), \mu \nu}F^{\mu \nu}_{(V)}] \ \ (9)

The factor in front of the kinetic terms is produced when we reduce {\sqrt{-G}H_{\mu \nu \rho}H^{\mu \nu \rho}}. From (9) one can again read off the gauge coupling,

\displaystyle  g (v) = e^{(\alpha + \beta)\phi} = 2\pi R (\frac{1}{2 \pi R})^{\frac{1}{d - 2}} \ \ (10)

What is different here? Notice, if we now make the circle of radius {R} very large, we obtain a strongly coupled theory. So, in taking from what we reviewed in the last entry, we know that charges under this gauge field are the winding modes of the string. That is, we have stringy or indeed quantum gravity states. Moreover, think about how if we take the basic Polyakov action for a string wrapping in the {X^{d}} direction {w} times in the Einstein frame, which means that we can set {\sigma = \frac{2\pi}{w}X^{d}}, then notice we have

\displaystyle S_{P} = -\frac{T}{2} \int_{\sum} d\tau d\sigma [2i V_{\mu} \partial_{\tau} X^{\mu} \partial_{\sigma} (\frac{w\sigma}{2 \pi})]

\displaystyle  = -i\frac{w}{2 \pi \alpha^{\prime}} \int_{\gamma} d\tau (\partial_{\tau} X^{\mu})V_{\mu} \ \ (11)

Which is the worldline action for a charged particle,

\displaystyle  q_{w}^{(V)} = \frac{w}{2 \pi \alpha^{\prime}} (2\pi R)^{\frac{2}{d - 2}} \ \ (12)

Or we can think of this in another way by remembering that if we have some antisymmetric form of rank {n}, there is going to be some object coupling to it. Hence, we may notice that, if we integrate some Kalb-Ramond 2-form on the string worldsheet, where the 2-form has one leg along the 9th direction and one leg along the extra dimension, and if we consider a string winding around the extra dimension, we find the string worldsheet is just a worldline in the 9th direction times a circle. If we then perform the integral along the extra direction, we obtain the coupling {V_{\mu}}. And so, we may write,

\displaystyle  \int_{\sum = C \times S^{1}} B_{[\mu d]} dX^{\mu} \wedge dX^{d} \sim \int_{C} V_{\mu} \ \ (13)

Where a worldline coupled to a gauge field means that, as in (4.11), we have a particle in the lower d-dimensional theory. What this is telling us is that winding modes in the d-dimensional theory produce charged particles that are gauge fields under the Kalb-Ramond field. Consider again (4.12), we find once again a relation between the coupling, charge, and mass, except this time it is for the winding modes. These are interesting relations,

\displaystyle  g_{(V)}q_{w}^{(V)} = M_{0, w} \ \ (14)

Which are strictly stringy – or quantum gravitational – in nature. Moreover, what we are discovering are what appear to be deeply general relations, where there is always some particle with a relation between its charge and its mass. And if these relations are, in fact, deeply general, then this means they are also intrinsic properties of quantum gravity. We will investigate this idea more deeply in the context of the Swampland in a moment.

In the meantime, also notice something else that is interesting. If we send the gauge coupling to zero (either by making the circle small or large), {g \rightarrow 0}, we obtain an infinite tower of light states. But this is just a special case of the DC, emphasising again the relation between the DC and the WGC. Furthermore, notice that the gauge coupling depends on the scalar field. So should we want to go to weak coupling, we must give the scalar a large expectation value that directly implies an infinite tower of states.

Also notice that, in the context of our wider discussion in these notes, there is a noticeable symmetry in the theory, which until now has been left implicit; because we can exchange the two gauge fields and also the KK and winding modes. This is T-duality.

3. Quick Review: Type IIA String Theory

Let us quickly review another example and think about Type IIA string theory (from the last entry). Remember, Type IIA in the strongly coupled regime is just 11-dimensional supergravity reduced on a circle. Also remember, in thinking of the Type IIA string we have a massive Ramond-Ramond 1-form, {C^{(1)}}, which is just a gauge coupling that is the graviphoton. The gauge group is U(1) and, it follows,

\displaystyle  g_{C^{(1)}} \sim \frac{1}{g_{s}^{3/4}} \ \ (15)

The states charged under this gauge field? A D0-brane, with a D6-brane representing the magnetic dual. Again, we find the following mass-charge relation,

\displaystyle  M_{D0} = g_{c^{(1)}} q_{D0} \ \ (16)

So, as Palti summarises, we have another piece of evidence that the mass-charge-coupling relation is indeed general. And, in fact, the more we search the more we become convinced this relationship is a property of quantum gravity.

4. Weak Gravity Conjecture (d-dimensions)

These considerations bring us to a more formal definition of the WGC than what we have so far previously offered. Consider the following: take a theory coupled to gravity with a U(1) gauge coupling, {g},

\displaystyle  S = \int d^{d}X \sqrt{-g} [] (\frac{M_{p}^{d}}{2})^{d-2}R^{d} - \frac{1}{4g_{s}^{2}} F^{2} + ... ] \ \ (17)

For the Electric WGC, there exists a particle with mass {m} and charge {q} satisfying,

\displaystyle  M \leq \sqrt{\frac{d - 2}{d - 3}} gq (M_{p}^{d})^{\frac{d - 2}{2}} \ \ (18)

And for the Magnetic WGC, the cutoff scale of the effective theory is bounded from above by the gauge coupling, such that we have the general statement,

\displaystyle  \Lambda \lesssim g(M_{p}^{d})^{\frac{d - 2}{2}} \ \ (19)

Where the cutoff, as we understand, should correspond to the mass scale of an infinite tower of charged states. It is argued to be completely general.

5. Testing the WGC: The Heterotic String

Following Palti, let’s now consider testing the WGC even more than what we have done previously. For example, a leading question might be: Is the WGC true for the Heterotic string? The first formal test of the WGC was for the Heterotic string on a {T^{6}} [3]. Again, much of the following discussion also echoes [1], where a summary with additional pedagogical references can be found.

One of the first things we must consider is that we have the non-abelian gauge group {SO(32)}. This is important to note because compactifying on a {T^{6}} yields the following 4-dimensional gauge fields: {U(1)^{28}}. To understand why there are 28 U(1) gauge fields, simply remember that a {T^{6}} may be thought of as a product of 6 circles. In 4-dimensions we obtain 12 gauge fields from the metric and the Kalb-Ramond field. We may break these up into 6 {B_{[mn]}} yielding 6 U(1)’s and 6 graviphotons. Additionally, particular to the Heterotic string is a 10-dimensional gauge group. This gauge group may be broken by Wilson lines on a circle to its Cartan subalgebra. That is to say, if we have a circle and take a gauge field on that circle, this will give us a Wilson line to which we can then give an expectation value. The Wilson line will break the non-abelian group to its Cartan subalgebra. For these reasons, one can see what the Cartan subalgebra gives {U(1)^{16}}.

Let us focus on these last 16 U(1) gauge fields that come from breaking the {SO(32)} gauge group. The states charged under these are string oscillators {\underbar{q} = (q_{1}, ..., q_{16})} from which we once again obtain an infinite tower of states. The first massive excitation is the {SO(32)} spinor with mass,

\displaystyle  m^{2} = \frac{4}{\alpha^{\prime}} \ \ (20)

When we compactify on a {T^{6}} we obtain charged states that correspond to the 16-dimensional charge vectors,

\displaystyle  \textbf{q} = (\pm \frac{1}{2}, ..., \pm \frac{1}{2}) \ \ (21)

The idea now is to consider how, in the Einstein frame, and working in Planck units, we have the following gauge coupling for any of the U(1) gauge fields,

\displaystyle  g^{2} = g_{s}^{2} = \frac{2}{\alpha^{\prime}} \ \ (22)

In which the gauge coupling is equal to the string coupling, and where {\alpha^{\prime}} depends on the expectation value of the dilaton. To put it explicitly, we have a dilatonic coupling. And, so, in terms of the bound set by the WGC for the mass the following inequality is satisfied,

\displaystyle  m^{2} \leq g^{2} \mid \textbf{q} \mid^{2} = \frac{8}{\alpha^{\prime}} \ \ (23)

Which is the limit of the expectation values of the small Wilson lines. As Palti notes, an interesting further test would be for arbitrary Wilson lines, but what he focuses on in his presentation is the way in which the entire analysis may be generalised for the complete {U(1)^{28}} gauge fields in which the U(1)’s from the {T^{6}} are included. So now we consider the mass of the higher oscillator modes,

\displaystyle  m^{2} = \frac{2}{\alpha^{\prime}} (\mid \underbar{q} \mid^{2} - 2) \ \ (24)

For which, in his talk, Palti gives the possible charges,

\displaystyle  \textbf{q} = (q_{1} + \frac{c}{2}, ..., q_{16} + \frac{c}{2}) \ \ (25)

Where {q_{i} \in \mathbb{Z}} and {c = 0,1}. In that the charges should be integer, they must satisfy the lattice condition {\mid \underbar{q} \mid^{2} \in 2N}.

Now, the whole point of the analysis up to the present is to consider the mass-charge relation. And, in fact, what we find is the following mass-to-charge ratio,

\displaystyle  \mid \textbf{z} \mid^{2} = \frac{\mid \textbf{q} \mid^{2}}{\mid \textbf{q} \mid^{2} - 2} \ \ (26)

Or, to put the matter differently, notice in (24) the {\frac{2}{\alpha^{\prime}}} factor is just {g_{s}^{2}}, and {g_{s}^{2} = \frac{m^{2}}{M_{P}^{2}}}. And so,

\displaystyle  \frac{m^{2}}{g^{2} \mid \textbf{q} \mid^{2}} = \frac{\mid \underbar{q} \mid^{2} - 2}{\mid \underbar{q} \mid^{2}} < 1 \ \ (27)

Where we find quite explicitly that the mass is bounded by the charge for all of the states. This again satisfies the WGC, where, for all the U(1)’s, the mass is less than the charge. We also find that there is an infinite tower of states charging at {g}, and as we go further up the tower (so to speak) the bound in (27) becomes saturated but never violated. So all of our results so far are consistent, and the WGC indeed proves true for the Heterotic string.

6. What About Other Gauge Fields?

The following question we might now ask, as Palti motivates it: what other gauge fields might we consider? So far we have consider some fairly straightforward or simple examples. Can we continue to generalise?

6.1. Testing the Electric WGC: Open String U(1)’s

Another U(1) we get in string theory is an open string U(1), which, considering again Dp-branes, it is a U(1) gauge field on the world-volume. D-branes of course live in Type II string theory, so we could in general consider Type IIA/IIB on {\mathbb{R}^{1, (q - n)} \times T^{6}}, where there is equal radius for the torus. The D-brane can be thought of as filling the non-compact spacetime. In considering string theory on this background, take in particular a Type IIB on a {T^{6}} with 6 circles of radius R as an example. We therefore have some 4-dimensional {M_{1,3} \times T^{6}}, and what we want to do is specifically put a D3-brane with its 4-dimensional world-volume completely in the {M_{1,3}} external spacetime. The D3-brane of course carries U(1), so we therefore now have a U(1) gauge symmetry in our 4-dimensional theory.

Now, with the scenario partly constructed, notice we only have one spacetime filling D-brane, which, impliedly, means that we have some fundamental open string with its endpoints ending on this brane. But this is not consistent. Why? The gauge symmetry we have included is an open string gauge symmetry, and so it is a gauge symmetry being carried by the non-perturbative D3-brane. But if we have just the single D3-brane, it will source the charge inside the 6-dimensional torus, and, one way to put it is that this scenario is akin to inserting a charged particle in a confined space in which there is nowhere for the field lines to propagate. In other words, we have a U(1) neutral state; but D-branes also source R-R fields. This is one of the great facts about D-branes, because insofar that they carry R-R charges, this gives string theory its power of being able to have a source for every gauge field [8]. In our current construction, however, the presence of the D-brane means that it will provide a source in the compact {T^{6}} whilst we lack an appropriate sink for the R-R field lines. This is obviously a problem because the field lines must end somewhere. This is why Palti points in another direction in his talk.

One option is that we could add an anti-brane; but means that the branes will then annihilate one another and, as this is an unstable configuration, it doesn’t really remedy the situation. Instead, the solution is based on a well known fact that orientifold planes are sinks for R-R charge. We might therefore instead introduce the needed negative charges by way of invoking orientifold planes. In doing so, this implies that the spectrum now also contains unoriented strings. These unoriented strings have charge 2 under the U(1), as, under orientifold involution, they stretch between the D-brane and its image. With this configuration, we have a consistent construction, which, with the presence of the orientifold, then means we have a second D3-brane as illustrated below.

In considering the scenario we have constructed, the actual states being charged under the U(1) are open strings whose endpoints end on the D3-branes with a charge {+1}.

Now let us think more deeply about the scenario in relation to the WGC. Is it not possible to violate the WGC? For instance, if the state has charge {+1}, what if we pull the D3-branes apart (i.e., moving away from the orientfold)? The string that is already stretched between the D3-branes would stretch even more over some spatial distance. This would make it massive. But what of the charge? Well, the charge would remain constant. On first inspection, this would seem to violate the WGC. Let us quantify these ideas as follows.

In {D=10}, the relation between the string scale and the Planck scale can be found as (from dimensional reduction and re-writing everything in Planck units),

\displaystyle  M_{s}^{2}g_{s}^{-2} (RM_{s})^{6} \sim M_{P}^{2} \ \ (28)

And the gauge coupling on the D3-brane is simply,

\displaystyle  g \equiv \sqrt{g_{s}} \ \ (29)

Now, for the stretched string, the mass is given as

\displaystyle  m^{2} \sim (RM_{s})^{2}M_{s}^{2} \sim \frac{g_{s}^{2}M_{P}^{2}}{(RM_{s})^{4}} \ \ (30)

Rearranging (30) it can be found that,

\displaystyle  \frac{m^{2}}{g^{2}_{s} M_{P}^{2}} \sim \frac{g_{s}}{(RM_{s})^{4}} \ \ (31)

If the main task was to try and violate the WGC by stretching the string to great length, as we pull the D3-branes away from the orientfold, the question is: have we succeeded? More precisely, to violate the WGC (31) would have to be greater than 1. Is this the case? No, it is not! The reason is because, if we’re working in the perturbative string description – i.e., the controlled weak-coupling regime – than the coupling {g_{s}  1}. So, in fact, the WGC is satisfied. That is,

\displaystyle  \frac{m^{2}}{g^{2}_{s} M_{P}^{2}} \sim \frac{g_{s}}{(RM_{s})^{4}} < 1 \ \ (32)

As we stretch the string and make it massive, with the orientfold growing very large, the gauge coupling does not change. What we are doing, in effect, is diluting gravity. What’s more, we are diluting gravity faster than the mass can increase. And, it turns out, when {M_{P} \rightarrow \infty} we obtain a weakly coupled theory.

6.2. In General for different cases of {n}

Notice that, in general, the scenario constructed above may be considered in terms of compactification of Type IIA/B string theory on {4\mathcal{R}^{1, 9-1} \times T^{n}}. We considered the case for {n >2} when we compactified on a {T^{6}}. But other subtleties arise when considering the case of {n = 2} and especially {n < 2}, particularly due to backreaction on the space. In all cases, it can be seen that the Electric WGC holds for open string U(1)s [1].

7. Testing the Magnetic Weak Gravity Conjecture: Type IIB String Theory in 6d F-theory

In the last example we considered a test of the Electric WGC for open string U(1)s. What about the Magnetic WGC? Does the MWGC likewise hold for open string U(1)s? Recall from earlier in our discussion the MWGC is not making a statement about a single charged state but about an infinite tower of charged states. Where is the infinite tower of charged states in our scenario? The answer is rather non-trivial and can be reviewed in a series of incredibly interesting and mathematically rich papers [5, 6, 7], which display some lovely stringy physics.

We will save a detailed review of these papers for a separate entry (following the formal conclusion of this series of notes on Palti’s lectures). In the meantime, looking at [5] in particular, a brief if not altogether terse description may be considered. What the authors find is that, for the infinite tower of states, they turn out to be non-perturbative states of the theory.

To see these non-perturbative states is difficult. The set-up is this: consider Type IIB string theory on a 4-dimensional manifold, meaning compactification down to 6-dimensions. A powerful method to study non-perturbative type IIB string theory is by way of uplifting to F-theory (or, for Type IIA, uplifting to M-theory). So the framework is 6-dimensional F-theory. The 6-dimensional Planck mass is defined by the volume of the F-theory compactification space, which is a complex Kähler surface {B_{2}} at the base of a Calabi-Yau 3-fold. In these notes, we have not yet considered such complex extended objects. But the idea is that we then consider a D7-brane filling the 6 external dimensions and wrapping a holomorphic curves on the Kähler surface in the 4-dimensional space. In the uncompactified 6 dimensions, the D3-brane wrapping the 2-cycle produces a solitonic ring. Associated strings on the curve {C_{0}} contained in {B_{2}} are sourced under the D7-brane gauge group.

From this construction, however roughly described, the idea is to uplift to a strong coupling (using F-theory). From this, if the goal is {g_{D7} \rightarrow 0}, where the tower of states become light according the WGC, then the 2-cycle must become very large. But, if the 2-cycle becomes big, the volume of the 4-dimensional manifold changes and, impliedly, the values of {M_{P}} and the string scale also change. So one approach is to keep the volume fixed. However, fixing the volume while making the 2-cycle big means that another 2-cycle needs to be small!

\displaystyle  volume \ fixed \rightarrow small \ 2-cycle

Now consider the following. If a D3-brane wrapped in internal dimensions gives a string in external dimensions, impliedly, in the above construction, it seems a D3-brane wrapped on the small 2-cycle is found to produce a string in the 6 external dimensions. But this string propagating in the 6-dimensions is tensionless as the volume of the curve {C_{0}} contained in B_{2} goes to zero, \text{vol}_{j}(C_{0}) \rightarrow 0 . Moreover, as the tension of the string is actually the size of the cycle, the string itself asymptotically describes an open Heterotic string. And so we observe,

\displaystyle  F-theory \longleftrightarrow Heterotic \ duality

And, as it is found that the string is charged under U(1), to finalise what is an incredible piece of evidence, the oscillator modes become massless and again what is found is an infinite tower of light states.

This concludes the summary. In a separate future entry we will study the technicalities in detail.

In the next collection of notes from Palti’s lecture series, we will continue our study by considering more complex manifolds – that is, arbitrary Calabi-Yau manifolds – to see if the WGC still holds! We will also looks to some more advanced tests of the DC, particularly in the context of Type IIB string theory.

Reference

[1] E. Palti, `The Swampland: Introduction and Review’, [arXiv:1903.06239v3[hep-th]]

[2] B. Heidenreich, M. Reece, and T. Rudelius, Sharpening the Weak Gravity Conjecture with Dimensional Reduction, JHEP 02 (2016) 140, [arXiv:1509.06374 [hep-th]].

[3] N. Arkani-Hamed, L. Motl, A. Nicolis, and C. Vafa, The String landscape, black holes and gravity as the weakest force, JHEP 06 (2007) 060, [hep-th/0601001].

[4] B. Heidenreich, M. Reece, and T. Rudelius, Evidence for a sublattice weak gravity conjecture, JHEP 08 (2017) 025, [arXiv:1606.08437].

[5] S.-J. Lee, W. Lerche, and T. Weigand, Tensionless Strings and the Weak Gravity Conjecture, JHEP 10 (2018) 164, [arXiv:1808.05958].

[6] S.-J. Lee, W. Lerche, and T. Weigand, Modular Fluxes, Elliptic Genera, and Weak Gravity Conjectures in Four Dimensions, [arXiv:1901.08065].

[7] S.-J. Lee, W. Lerche, and T. Weigand, A Stringy Test of the Scalar Weak Gravity Conjecture, Nucl. Phys. B938 (2019) 321–350, [arXiv:1810.05169].

[8] J. Polchinski, String theory. Vol. 2: Superstring theory and beyond. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2007.