Physics — General Relativity · higher-dimensional-gauge-theory-forms
Problem
There are a lot of motivational words attached here to what is a very simple problem; don't get too distracted. In ordinary electromagnetism with point particles, the part of the action which represents the coupling of the gauge-potential one-form A(1) to a charged particle can be written S=∫γA(1), where γ is the particle worldline. (The superscript on A(1) is just to remind you that it is a one-form.) For this problem you will consider a theory related to ordinary electromagnetism, but this time in 11 spacetime dimensions, with a three-form gauge potential A(3) and four-form field strength F(4)=dA(3). Note that the field strength is invariant under a gauge transformation A(3)→A(3)+dλ(2) for any two-form λ(2).
(a) What would be the number of spatial dimensions of an object to which this gauge field would naturally couple (for example, ordinary E+M couples to zero-dimensional objects---point particles)?
(b) The electric charge of an ordinary electron is given by the integral of the dual of the two-form gauge field strength over a two-sphere surrounding the particle. How would you define the ``charge'' of the object to which A(3) couples? Argue that it is conserved if d∗F(4)=0.
(c) Imagine there is a ``dual gauge potential'' A that satisfies d(A)=∗F(4). To what dimensionality object does it naturally couple?
(d) The action for the gauge field itself (as opposed to its coupling to other things) will be an integral over the entire 11-dimensional spacetime. What are the terms that would be allowed in such an action that are invariant under ``local'' gauge transformations, for instance, gauge transformations specified by a two-form λ(2) that vanishes at infinity? Restrict yourself to terms of first, second, or third order in A(3) and its first derivatives (no second derivatives, no higher-order terms). You may use the exterior derivative, wedge product, and Hodge dual, but not any explicit appearance of the metric.