What I'm working on now
Monday, 07-Jan-2019 07:40:18 EST
Most of my research deals with supersymmetry, Yang-Mills, gravity, strings, & Feynman diagrams. Here are a few of the topics I'm looking @ now: Some I just started on recently, some I come back to periodically, on some I've been working almost continuously for quite some time. (Here's an older page.)
New methods for Feynman diagrams
Strings have long been used to find new ways to evaluate Feynman diagrams in particle physics, but in ways that give complete amplitudes, rather than having to sum individual diagrams. Recently relatively simple string-motivated expressions were found for arbitrary dimensions, & given a string-like derivation. Last year I found a derivation of these results directly from standard string theory, by modifying worldsheet boundary conditions. This year we found expressions for the resulting amplitudes in cases not covered previously.
This work was also based on our work in 2013 on worldsheet-quantum corrections to T-theory, where all string excitations were treated as propagating in one direction around closed strings...
T-theory on the worldsheet
Almost all of 2014 we spent working on a version of string theory where T-duality (electromagnetic-style duality of the 2D worldsheet theory) is manifest by including both momenta & winding modes. Although I invented almost everything defining this theory back in 1993, my work has only recently become popular. Most work has concentrated on massless backgrounds, whose properties can be derived most simply by examining the string's current algebra. (Our more recent F-theory work is a generalization of this: see below.)
If string theory is quantized by regularizing the worldsheet on a random (for curvature) 2D lattice, its 1st-quantized path integral becomes a 2nd-quantized particle Feynman diagram. Stringy T-duality then becomes Fourier transformation of this "parton" theory. In 1996 I showed that in 0 dimensions this duality can be performed explicitly on the action, relating equivalent theories of "color" & "flavor". In the AdS/CFT correspondence (see below), the T-duality of the string implies an infinite-dimensional "dual superconformal" invariance of 4D N=4 Yang-Mills. A generalization of my 0-dimensional result is hinted @ by the fact that the Fourier transform of the massless propagator 1/p² is 1/x² only in D=4. Every once in a while I look @ this problem again, but with no success so far.
A related problem is finding 1st-quantized methods for evaluating Feynman diagrams in particle theory, analogous to those already used in string theory. We found explicit string-like results for loops in scalar theories, & how to generalize to Yang-Mills.
AdS/CFT in superspace
The AdS/CFT correspondence between superstrings & 4D N=4 super Yang-Mills (& its generalizations) promises new insights into both theories, but it's rather difficult to do much more than derive properties of the free string theory (i.e., spectrum). 2 notable exceptions are (1) using Feynman diagrams for 10D supergravity on AdS to calculate large-coupling results for Yang-Mills BPS states, & (2) the calculation of the 4-point amplitude in 4D N=4 Yang-Mills to leading order in the inverse of the number of colors but all orders in the coupling. Neither of these has been generalized to superspace; we're working on it.
Feynman graphs in extended superspace
4D N=4 Yang-Mills theory has drawn a lot of renewed interest in recent decades because of its appearance in the AdS/CFT correspondence & indications that it's the simplest quantum field theory in 4 dimensions. Although N=1 supergraphs have shown vast superiority to ordinary Feynman diagrams for N=1 supersymmetric theories, we extended these advantages to N=2 only a couple of years ago. (Previously the N=2 rules had been formally constructed, but not in a way analogous to N=1 that took full advantage of the formalism.) Although a superspace formulation of N=4 supergraphs in unknown, we're now working on the N=3 version of the N=4 theory. (Previously these rules were also constructed, but in a form so unmanageable that no calculations had been done.)
Another technique for Feynman diagrams is twistors: In 1998 we showed how this approach followed directly in the action by use of the "spacecone gauge", a generalization of the lightcone gauge to a null (complex) spatial direction. (2-component) spinor notation is thus found useful even for bosonic theories. Whereas these momentum-space twistors are always on-shell & positive-momentum (making it difficult to deal with infrared divergences & iε prescriptions), there are also similar off-shell coordinate-space twistors: They have 8-4=4 components instead of 4-1=3 components, in a counting related to coset constructions (see below). I showed the advantages of spinor notation in using these to calculate off-shell (massless) Feynman diagrams, & their extension to superspace. (In 1994 I used them to generalize the general Yang-Mills instanton construction to super Yang-Mills.)
New dimensions for particles & strings
F-theory on branes
Last year we started work on a formulation of F-theory as a fundamental theory defined on branes. "F-theory" is a theory in even higher dimensions than M-theory, with respect to both spacetime & worldvolume. Like T-theory, it involves selfduality of fields living on the worldvolume. It reduces to M-theory & all versions of strings without performing additional duality transformations, because all dualities (S,T,U) are manifest. It hasn't been completely defined yet (neither has M-theory), but should describe all massive as well as massless states (supergravity). The massless bosonic states (including the graviton) are constructed on coset spaces (see below).
In a paper this year I discussed reproduction of string amplitudes from F-theory by a kind of "zeroth-quantization": There are then ghosts associated not only with (1) spacetime fields, which are functions of worldvolume fields (spacetime coordinates), & with (2) worldvolume fields, which are functions of worldvolume coordinates, but also with (3) the worldvolume coordinates themselves.
Lorentz coordinates for strings & particles
In general it's often convenient to study groups by introducing coordinates for the group space. A particularly useful realization of a group G is a coset space G/H, where all points in certain subspaces, related by the action of a subgroup H, are identified ("modded out"). This is often used for internal symmetries, but is also useful for spacetime. Well known examples include Minkowski space realizing Poincaré (Poincaré/Lorentz) or conformal (conformal/everything but translations) groups, & superspace. But usually the coordinates for H are ignored, & only those for G/H are kept.
But we've found various advantages to keeping all coordinates: (1) a better treatment of lightcone gauges by including the components of the gauge field in the H directions, (2) a nicer ("1st-quantized") formulation of gravity by associating the vierbein with translation derivatives & Lorentz connection with Lorentz derivatives, (3) introduction of Lorentz connection as a background field into string theory, & (4) constructing the 1st-order formalism for Yang-Mills as a Chern-Simons action by making the field strength the gauge field for derivatives "dual" to Lorentz. (This is a generalization of work I did in 1994 where the spinor superfield strength of super Yang-Mills appeared as the gauge field for derivatives dual to supersymmetry.)