Seminar on F-theory
An informal seminar/working group on F-theory at MPI Bonn, during winter 2015. We try to meet each week, starting Nov 20, but there may be interruptions.
We proceed in two parallel threads, alternating each week:
After a few sessions these two strands meet in F-theory. Then we will decide how to continue with the seminar.
We give an introduction to the idea of string theory, M-theory and F-theory.
For some absolute basic orientation, some readers may find it useful to glance over the string theory FAQ.
Readers with a basic idea of string theory but lacking a satisfactory idea of M-theory/F-theory might benefit from looking at the introductions to each chapter of
Readers happy with higher differential geometry may find a mathematical derivation from first principles of the key structures that we need below in these lecture notes:
Here are possible session topics for this thread of the seminar:
worldline formalism for quantum field theory – To understand the principle of perturbative string theory, one needs to know that traditional Feynman diagram rules for the perturbation theory of scattering amplitudes in standard quantum field theory has an equivalent reformulation – called the worldline formalism. Here one observes that the integrand associated with each Feynman diagram is equivalently the correlator of the 1-dimensional sigma-model field theory that describes particles running along a worldline of shape the given Feynman diagram. One may think of this as making precise how quantum field theory on spacetime is, perturbatively at least, the “second quantization” of the quantum mechanics of the particle quanta.
perturbative string theory – From the perspective of the worldline formalism the idea of perturbative string theory is immediate: see what kind of scattering amplitudes one obtaines when the sum over correlators of 1-dimensional field theories describing propagation of 0-dimensional particles running along 1-dimensional trajectories is replaced by a sum over correlators of -dimensional field theories describing propagation of -dimensional “branes” running along -dimensional trajectories.
This is a re-incarnation of old idea of S-matrix theory (which predated modern local field theory on spacetime and then fell out of fashion when QCD celebrated its successes ): produce any consistent formula (the “S-matrix”) for probability amplitudes that certain incoming stuff turns into certain outgoing stuff and don’t worry about what it “means” that happens in between.
However, it turns out that making sense of such an S-matrix for -dimensional objects is in general elusive – except for one special case, namely for and assuming worldsheet supersymmetry. This is the case of the “super 1-brane”, the superstring.
effective field theory – In order to understand what the S-matrix string perturbation series defined just by worldvolume correlators (no explicit spacetime) thus defined means, physically, one needs to resort to local field theory after all: one asks for an ordinary field (of particle quanta) theory whose Feynman diagram perturbation series coincides with the abstractly defined string pertrubation series below some given energy. This is then called the effective field theory description of the perturbative string theory.
Much of what people say about string theory are really statements about these effective field theories approximating the string perturbation series. Notably one finds that these are theories of 10-dimensional supergravity.
But all this is based on comparison of perturbation theories. It is an open problem to identify what string perturbation theory is actually the perturbation of.
black branes and D-branes – Even though an S-matrix made of worldvolume correlators of -branes seems to mathematically exist only for , avatars of other values of emerge from the structure of the string perturbation series. In two ways:
The 10-dimensional supergravity theories that appear as effective field theory descriptions of string scattering amplitudes admit black hole-like solutions of higher dimension. Since a black hole may be thought of as caused by a point mass, hence a “black 0-brane”, these are called black p-branes. (In fact those of main interest are BPS states, hence are extremal black branes, hence have a naked singularity, hence are not as “black” after all, but anyway.)
To define open string correlators, one needs to specify consistent boundary conditions on the worldsheet. The choice is not arbitrary, and one finds that in sigma-model perspective these boundary conditions are given by certain submanifolds of spacetime which the endpoints of the string are constrained to sit on. These submanifolds are called D-branes.
Now a miracle happens: It so happens that the dimension, charges and other properties of the possible black p-branes exactly matches those of the D-branes. Most strikingly, the Bekenstein-Hawking entropy for (near-)supersymmetric black holes in string theory (which a priori was lacking an explanation as an actual state counting of a microscopic ensemble) has been checked to accurately agree with the well-defined entropy of the corresponding perturbative string worldsheet theories.
This fact is regarded to support the point of view that indeed there is a non-perturbative string theory. One argues that starting with the well-defined but pertrubative theory of string scattering amplitudes, then entropy of supersymmetric configurations that its compute will not change as the coupling constant increases, and so even while one does not have the theory to follow as the coupling becomes strong, one imagines that the result must be the corresponding black brane (which due to its strong gravitational coupling heabvily curves spacetime).
M-theory and F-theory – Via such a mixture of arguments, combining the string perturbation series at low coupling constant but for high energy with the corresponding effective field theory defined also for large coupling constant but only at low energy, and using a bunch of intricate relations between this data, called duality in string theory, one may make an educated guess as to what the ingredients of non-perturbative string theory ought to be.
The result of this mixture of arguments is called M-theory and F-theory (which are meant to be two pespectives on the same thing). Accordingly, these “theories” are a combination of ordinary (classical) 11-dimensional supergravity “with M-brane-effect added”. While it is a bit of an art to figure out what classical supergravity with such non-perturbative brane effects added is, once this is said then, contrary to a common perception, the result is a perfectly well defined mathematical thing about which mathematicians have proven many non-trivial theorems. One just needs to be aware that M-theory-in-the-grandiose-sense of a complete non-perturbative definition of string theory is still an open problem, what is called M-theory in most of the literature are just facets of it.
Green-Schwarz super p-branes – With the F/M-theory perspective in hand, one may again go back and see if all this structure has an incarnation in worldvolume sigma-model field theory. And it does: The brane bouquet of Green-Schwarz sigma-models. These are classical field theories (or rather: prequantum field theories) on “worldvolumes” that describe the propagation of -dimensional objects on super spacetimes, in direct generalization of the basic worldline field theory that describes the propagation of an electron on a spacetime. It so happens that they exist precisely in all the dimensions in which M-theory/F-theory predicts there to be branes. In particular there are sigma-models for the M-branes: the M2-brane and the M5-brane. Since all the other branes are supposed to derive from these (see at F-branes – table), the open problem of making sense of non-perturbative string theory might well reduce to the open problem: Make sense of quantum M2-brane and M5-brane Green-Schwarz sigma-models.
For lecture notes on this see Structure Theory for Higher WZW Terms.
Outlook: mathematical problems – One may phrase major aspects of the open question What is non-perturbative string theory? in the form of fairly precise open mathematical problems. Here are some:
the cohomology theory hosting F-brane charges – It has been well-understood that the charges of the string and the D-branes are not classes in ordinary cohomology, as originally assumed, but that the string rather has charges in super line 2-bundles and the D-branes in topological K-theory twisted by the string charges. The precise conditions is known in considerable mathematical detail, see (Distler-Freed-Moore 09).
Therefore, for the M-theory/F-theory story to work out, it must be true that there is some twisted generalized cohomology theory for M-branes/F-branes, together with a concept of taking a pertrubative approximation, such that in this limit it reporduces the twisted K-theory charges of D-branes.
This issue has been left wide open. The only exception is Hisham Sati, who has been pointing out various plausibility arguments about what that M-brane cohomology theory could be (_Geometric and topological structures related to M-branes)
Sati’s proposal for the purpose of F-theory is that F-branes should carry charges in parameterized modular equivariant elliptic cohomology (see there) with elliptic curve determined by the F-theory elliptic fibration.
Here is one argument for why this proposal goes in the right direction: by the overview at F-branes – table, it is the neighbourhood of the loci of the nodal curves in the elliptic fibration at which open strings stretching between D-branes (and hence inducing D-brane charge via Chan-Paton gauge fields) appear. So for the proposal to be consistent, elliptic cohomology in the neighbourhood of the nodal curve – called the the Tate curve – needs to reduce to topological K-theory. And it does! Not just that, it reduces to KR-theory with just the right -equivariant needed for general type II backgrounds (“orientifolds”, see there).
If this is right, then the S-duality of F-theory requires that there is some kind of modular group equivariance on elliptic cohomology. This was indeed found to exist by Lawson et al, see at modular equivariant elliptic cohomology.
So this looks good. But there are further subtleties to sort out. For instance from the computation in (Fiorenza-Sati-Schreiber 15) we know that the rationally charges of the M2-braneM5-brane jointly take values in the quaternionic Hopf fibration, hence in twisted cohomotopy with coefficients in . How is this related to elliptic cohomology (if it is)?
gauge enhancement from M-branes – The key hint for what the elusive “microscopic degrees of freedom” of genuinely non-perturbative string theory should be comes from the expected gauge enhancement of M-theory at ADE-singularities? and of F-theory at 2-cycle degenertion loci. This is all supposed to be controled (and in fact to explain) the infamous list of ADE classifications of mathematical structures, hence points to a deep statement in pure mathematics. Combined with the previous point the question is: is there any version of elliptic cohomology such that evaluated at ADE-orbifolds it exhibits the appropriate phenomena of exceptional gauge groups. (…)
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under construction
Something like the following – to be expanded:
Last revised on November 17, 2015 at 16:25:33. See the history of this page for a list of all contributions to it.