Philosophy of Dimensions: M-Theory. Thought of the Day 85.0

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Superstrings provided a perturbatively finite theory of gravity which, after compactification down to 3+1 dimensions, seemed potentially capable of explaining the strong, weak and electromagnetic forces of the Standard Model, including the required chiral representations of quarks and leptons. However, there appeared to be not one but five seemingly different but mathematically consistent superstring theories: the E8 × E8 heterotic string, the SO(32) heterotic string, the SO(32) Type I string, and Types IIA and IIB strings. Each of these theories corresponded to a different way in which fermionic degrees of freedom could be added to the string worldsheet.

Supersymmetry constrains the upper limit on the number of spacetime dimensions to be eleven. Why, then, do superstring theories stop at ten? In fact, before the “first string revolution” of the mid-1980’s, many physicists sought superunification in eleven-dimensional supergravity. Solutions to this most primitive supergravity theory include the elementary supermembrane and its dual partner, the solitonic superfivebrane. These are supersymmetric objects extended over two and five spatial dimensions, respectively. This brings to mind another question: why do superstring theories generalize zero-dimensional point particles only to one-dimensional strings, rather than p-dimensional objects?

During the “second superstring revolution” of the mid-nineties it was found that, in addition to the 1+1-dimensional string solutions, string theory contains soliton-like Dirichlet branes. These Dp-branes have p + 1-dimensional worldvolumes, which are hyperplanes in 9 + 1-dimensional spacetime on which strings are allowed to end. If a closed string collides with a D-brane, it can turn into an open string whose ends move along the D-brane. The end points of such an open string satisfy conventional free boundary conditions along the worldvolume of the D-brane, and fixed (Dirichlet) boundary conditions are obeyed in the 9 − p dimensions transverse to the D-brane.

D-branes make it possible to probe string theories non-perturbatively, i.e., when the interactions are no longer assumed to be weak. This more complete picture makes it evident that the different string theories are actually related via a network of “dualities.” T-dualities relate two different string theories by interchanging winding modes and Kaluza-Klein states, via R → α′/R. For example, Type IIA string theory compactified on a circle of radius R is equivalent to Type IIB string theory compactified on a circle of radius 1/R. We have a similar relation between E8 × E8 and SO(32) heterotic string theories. While T-dualities remain manifest at weak-coupling, S-dualities are less well-established strong/weak-coupling relationships. For example, the SO(32) heterotic string is believed to be S-dual to the SO(32) Type I string, while the Type IIB string is self-S-dual. There is a duality of dualities, in which the T-dual of one theory is the S-dual of another. Compactification on various manifolds often leads to dualities. The heterotic string compactified on a six-dimensional torus T6 is believed to be self-S-dual. Also, the heterotic string on T4 is dual to the type II string on four-dimensional K3. The heterotic string on T6 is dual to the Type II string on a Calabi-Yau manifold. The Type IIA string on a Calabi-Yau manifold is dual to the Type IIB string on the mirror Calabi-Yau manifold.

This led to the discovery that all five string theories are actually different sectors of an eleven-dimensional non-perturbative theory, known as M-theory. When M-theory is compactified on a circle S1 of radius R11, it leads to the Type IIA string, with string coupling constant gs = R3/211. Thus, the illusion that this string theory is ten-dimensional is a remnant of weak-coupling perturbative methods. Similarly, if M-theory is compactified on a line segment S1/Z2, then the E8 × E8 heterotic string is recovered.

Just as a given string theory has a corresponding supergravity in its low-energy limit, eleven-dimensional supergravity is the low-energy limit of M-theory. Since we do not yet know what the full M-theory actually is, many different names have been attributed to the “M,” including Magical, Mystery, Matrix, and Membrane! Whenever we refer to “M-theory,” we mean the theory which subsumes all five string theories and whose low-energy limit is eleven-dimensional supergravity. We now have an adequate framework with which to understand a wealth of non-perturbative phenomena. For example, electric-magnetic duality in D = 4 is a consequence of string-string duality in D = 6, which in turn is the result of membrane-fivebrane duality in D = 11. Furthermore, the exact electric-magnetic duality has been extended to an effective duality of non-conformal N = 2 Seiberg-Witten theory, which can be derived from M-theory. In fact, it seems that all supersymmetric quantum field theories with any gauge group could have a geometrical interpretation through M-theory, as worldvolume fields propagating on a common intersection of stacks of p-branes wrapped around various cycles of compactified manifolds.

In addition, while perturbative string theory has vacuum degeneracy problems due to the billions of Calabi-Yau vacua, the non-perturbative effects of M-theory lead to smooth transitions from one Calabi-Yau manifold to another. Now the question to ask is not why do we live in one topology but rather why do we live in a particular corner of the unique topology. M-theory might offer a dynamical explanation of this. While supersymmetry ensures that the high-energy values of the Standard Model coupling constants meet at a common value, which is consistent with the idea of grand unification, the gravitational coupling constant just misses this meeting point. In fact, M-theory may resolve long-standing cosmological and quantum gravitational problems. For example, M-theory accounts for a microscopic description of black holes by supplying the necessary non-perturbative components, namely p-branes. This solves the problem of counting black hole entropy by internal degrees of freedom.

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