Back to Basics

Well, today was the first lecture of the string theory course (part 2) that I mentioned in the previous post. And I applied the “when they think you’re going to zig, you zag” principle. They have been expecting me to dive into the whole business of open strings and D-branes and so forth (the subject of the book), and I did not. Sure, that will come, and sure, we’ll explore what they mean and what they can tell us about string theory beyond perturbation theory and so on and so forth. But I want first to spend a couple of weeks on getting to the heart of the matter. They made several standard choices along the way in doing their first semester of study of string theory. What did they mean? Why did they work? Were those the only choices? What is underlying a lot of it all, and what, when stripped down to the essence, is at the core of string perturbation theory and beyond? In other words, let’s look more closely at the path integral definition (such as it is) of a string theory (slightly schematically):

[tex] Z=\int [{\cal D}g {\cal D}X] e^{-S(X,g)}\ ,[/tex]

and make sense of all the bits. (Er, for the two of you still reading, [tex]S(X,g)[/tex] is an appropriately chosen action built out of the various quantities describing a relativistic string moving in spacetime, and the full quantity [tex]Z[/tex] describes all the possible splittings and joinings the string can do quantum mechanically, and so on and so forth. The expression is all very short hand, in the language of quantum field theory, for a lot that’s going on.)

Doing path integrals over fields ([tex]X[/tex]) is familiar enough, but how do you do the integral over all world-sheet metrics [tex]g[/tex], and the sum over all world-sheet topologies (that is what encodes all the quantum splittings and joinings)? Is it really true that you can only do string theory in the famous so-called “critical” dimensions, such as 26 and 10 (er… no!), and what is going on really? (And is this really the best starting point anyway, being so wedded to world-sheets and strings at the outset?) So before rushing off to do all the fancy fashoinable stuff, we’re going to pause a bit and linger over this sort of thing. It’ll make the fashionable stuff better motivated, and they’ll be able to read more widely, and ultimately to make (I hope) wiser choices about what problems to work on, as opposed to just doing what is the flavour of the month.

I had fun unpacking the answers to some of the above for two hours, and I think they found it quite interesting as well as unexpected. A win-win. What will we do next? Strip off a lot of the baggage of string theory and cut down to the core, exploring how to do the sum over all world-sheet metrics and topologies exactly in a series of examples, and see what lovely surprises and lessons lay in store once we’ve done that.


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