Is there a more fundamental theory? More than just strings?

If different string theories are related, then maybe they represent different limits of a bigger, more fundamental theory.

To understand the presence of objects in string theory that are not strings, but higher-dimensional objects, or even points, it helps to know the formulation of Maxwell’s equations in the language of differential forms, because this is what tells us that the sources of charge in the Maxwell equations are zero-dimensional objects. Gauge field strengths that are p+2-forms turn out to have sources that are p-dimensional objects. We call these p-branes.

In the regular maxwell equations in d=4 spacetime dimension, the electric and magnetic fields are packed together into the field strength F, which satisfies the equation F=dA, d is the exterior derivative, and A is the vector potential, a one-form. The two-form *F is the dual of F relative to the spacetime volume four-form v. (The subscripts on F, etc, below are just to indicate the degree of the differential form.)

Why String theory?

Maxwell equations, d=4

The charge sources enter through the equation d*F=*J, where *J is the three-form dual to the current four-vector J=(r,j). In the rest frame of the charge density r, J=(r,0), so *J is r times the the volume element for three-dimensional space. In a three-dimensional space, a surface that can be localized in three dimensions (has codimension three) must be a zero-dimensional surface, also known as a point.

This is the math that tells us that the Maxwell equations couple electrically to sources that are points, or zero-branes, as zero-dimensional objects are now called in string theory. (For magnetic couplings, the roles of F and *F are interchanged, but that won’t be covered here.) This same math works for two-forms in any spacetime dimension, so we know that Maxwell’s equations couple to point charges in any spacetime dimension.

Superstring theories contain electromagnetism, but they also contain field strengths that are three-forms, four-forms and on up. These field strengths obey equations just like the Maxwell equations, and their sources can be analyzed in the same manner as above.

Suppose we start in d spacetime dimensions with a vector potential A that is a p+1-form. Then F is a p+2-form, v is a d-form (because it’s the volume element of d-dimensional spacetime), *F is a (d-p-2)-form, and d*F is a (d-p-1)-form. (Once again, the subscripts are just to indicate the degree of the differential form.)

Maxwell equation, d,p version

The equations of motion tell us that the source term *J is also a (d-p-1)-form. In the rest frame of an isolated source, *J is proportional to a volume element of a (d-1-p)-dimensional subspace of (d-1)-dimensional space. The codimension of the source is therefore (d-p-1), and since space has dimension d-1, the charges that serve as sources must be objects with p dimensions, known as p-branes. So a (p+2)-form field strength couples to sources that are p-branes. This little fact has turned out to be extremely important in string theory.

Superstring theories are theories with gravity, so these p-dimensional localizations of charge must lead to spacetime curvature. A p-brane spacetime whose metric solves the equations of motion for a (p+2)-form field strength in d spacetime dimensions can be described using p space coordinates {yi} along the p-brane and (d-1-p) space coordinates {xa} orthogonal to the p-brane.

p-brane metric

The isometries of this spacetime consist of translations (shifting the coordinate by a constant) and Lorentz transformations in the (p+1)-dimensional worldvolume, plus spatial rotations in the (d-1-p)-dimensional space orthogonal to the p-brane.

There’s a problem with adding gravity, however. Most p-brane spacetimes turn out to be unstable. Supersymmetry stabilizes p-branes, but only for the certain values of p and d. Two of the most important p-branes in string theory are the two-brane in d=11 and the five-brane in d=10.

Since we’re talking about a spacetime metric, we’re obviously in the low energy limit of string theory. But p-branes can be protected from quantum corrections by supersymmetry, if they satisfy an equality between mass and charge known as the BPS condition. These branes are then known as BPS branes.

Read more: How many String theories are there?

From p-branes to D-branes

A special class of p-branes in string theory are called D branes. Roughly speaking, a D brane is a p-brane where the ends of open strings are localized on the brane.

D-branes were discovered by investigating T-duality for open strings. Open strings don’t have winding modes around compact dimensions, so one might think that open strings behave like particles in the presence of circular dimensions. However, the stringiness of open strings in the presence of compact dimensions exhibits itself in a more subtle manner, and the T-dual of an open string theory is anything but uninteresting.

The normal open string boundary conditions in the string oscillator expansion come from the requirement that there be no momentum exiting or entering through the ends of an open string. This translates into what is called Neumann boundary conditions at the ends of the string at (s=0) and (s=p):

von Neumann boundary conditions

Suppose d-1-p of the space dimensions are compactified on a torus with radius R, and p of the space dimensions are left noncompact as before. In the T-dual of this string theory, the boundary conditions in those d-1-p directions are changed from Neumann to Dirichlet boundary conditions

Dirichlet boundary conditions in 9-p directions

This T-dual theory has strings with ends localized in d-1-p directions. So the T-dual of open strings compactified on a torus of radius R is open strings with their ends fixed to static p-branes, which we then call D-branes.

D branes have been very important in understanding string theory in general (see below) but also of crucial importance in understanding black holes in string theory, especially in counting the quantum states that lead to black hole entropy.

How many dimensions?

Before string theory won the full attention of the theoretical physics community, the most popular unified theory was an eleven-dimensional theory of supergravity, which is supersymmetry combined with gravity. The eleven-dimensional spacetime was to be compactified on a small 7-dimensional sphere, leaving four spacetime dimensions visible to observers at large distances.

This theory didn’t work as a unified theory of particle physics, because an eleven-dimensional quantum field theory based on point particles is not renormalizable. Also, chiral fermions cannot be defined in spacetimes with an odd number of dimensions. But this eleven-dimensional theory would not die. It eventually came back to life in the strong coupling limit of superstring theory in ten dimensions.

Read more: Why String theory?

The theory currently known as M

Technically speaking, M theory is the unknown eleven-dimensional theory whose low energy limit is the supergravity theory in the eleven dimensions discussed above. However, many people have taken to also using M theory to label the unknown theory believed to be the fundamental theory from which the known superstring theories emerge as special limits.

We still don’t know the M fundamental theory, but a lot has been learned about the eleven-dimensional M theory and how it relates to superstrings in ten spacetime dimensions.

Recall that one of the p-brane spacetimes that are stabilized by supersymmetry is a two-brane in eleven spacetime dimensions. This object is called the M2 brane for short.

Type IIA superstring theory has a stable one-brane solution called the fundamental string. If we take M theory with the tenth space dimension compactified into a circle of radius R, and wrap one of the dimensions of the M2 brane around that circle, then the result is the fundamental string of the type IIA theory. When the M2 brane is not around that circle, then the result is the two-dimensional D-brane, the D2 brane, of the type IIA theory.

If the two theories are identified, the type IIA coupling constant turns out to be proportional to the radius R of the compactified tenth dimension in the M theory. So the weakly coupled limit of type IIA superstring theory, which is the usual ten-dimensional theory, is also an expansion around small R. The strong coupling limit of type IIA theory is where R becomes very large, and the extra dimension of spacetime is revealed. So type IIA superstring theory lives in ten spacetime dimensions in the weak coupling limit, but eleven spacetime dimensions in the strongly coupled limit.

We still don’t know what the fundamental theory behind string theory is, but judging from all of these relationships, it must be a very interesting and rich theory, one where distance scales, coupling strengths and even the number of dimensions in spacetime are not fixed concepts but fluid entities that shift with our point of view.

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