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.)

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.

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.)

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.

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.)