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David Tong's vortices

David Tong who is now a professor in the original Cambridge is visiting us - his friends in the new Cambridge (David returned back to home; he spent a few years at the M.I.T.) - and yesterday he spoke about his work about

  • The Vortices in non-Abelian Yang-Mills
He also told me about some additional interesting ideas, but I feel I can't tell you about this private discussion. In the private discussion, he worked with an N=2, d=4 U(N) gauge theory with N flavors in the fundamental representation (yes, the number of flavors equals the number of colors, but the number of supersymmetries "N" is a different number haha). In the talk, there was no supersymmetry (this sentence was added later because David has pointed out my error in the comments, so don't get confused).

Normally, one can find cosmic strings in Abelian Higgs models - i.e. a spontaneously broken U(1) gauge theory - which are vortices: the Higgs field has minima on a circle, and this circle is identified with a circle surrounding a vortex. You might think that if you work with U(N), only the U(1) part will participate in the solution. Well, while it's true that it's only the U(1) part that is responsible for the existence of the vortex solution, the SU(N) part is important for detailed properties of the solutions.

David Tong has constructed a whole zoo of various objects - domain walls and cosmic strings of different types stretched between them; the types are determined by the U(1) that has a non-zero magnetic flux through the tube. On the cosmic strings, one can place a 't Hooft-Polyakov monopole that is able to change the type of the string. Normally, a 't Hooft-Polyakov monopole would carry no "Dirac strings", but because David works with U(2) and not SU(2), there are two different "Dirac strings" attached to this monopole.

The most intense discussions, especially between David Tong, Nima Arkani-Hamed, and partially me and others were about David's statement that the following widely believed statement is not true:
  • The probability that two field-theoretical cosmic strings intercommute is always essentially one.
Note that this is usually used as a criterion to distinguish "ordinary" field-theoretical cosmic strings, whose probability should be one, from "cosmic superstrings" that intercommute with probability proportional to "g_{string}^2".

David Tong argued that he can arrange his field-theoretical cosmic strings such that the probability that two strings intercommute will be "1/N" at low energies or "e^2" at high energies (or did I confuse the energy regimes again?) which are numbers smaller than one. The probability may be smaller than one because the strings must be oriented within the gauge group exactly in the same way for the interaction to occur. The fermionic zero modes living on the string can change the conclusion, and the various twists and turns are slightly confusing, so I can't explain you these things coherently.

At any rate, if David is correct, then it means that even if we observed cosmic strings whose probability of intercommuting were seen to be much smaller than one, it would not prove that the strings were fundamental strings. Field theory is always able to immitate string theory if it is sufficiently complex - a wisdom that we've encountered many times in the recent era.

If you're interested, you may look at David Tong's recent paper with Norisuke Sakai:
You are strongly encouraged to click "Fast comments" under this article because Joe Polchinski and David Tong posted more relevant remarks about the probabilities.

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