Monday, September 18, 2006

Puffed field theory

Tonight, Ori Ganor is proposing
What's that? It's a new, Lorentz violating and nonlocal (but hypothetically rotationally symmetric) deformation of N=4 supersymmetric gauge theory in d=4 - and perhaps other theories - that is conjectured to be UV-complete much like the dipole theories and noncommutative gauge theories.

How do you create these puffed fields? Start with D2-branes, compactify one transverse dimensions X1 on a circle, and T-dualize this circle X1. You will clearly obtain D3-branes with N=4 Super-Yang-Mills at low energies.

Now try something more amusing. Before you T-dualize X1 to obtain D3-branes from D2-branes, twist the remaining six coordinates X4...X9 transverse to the D2-branes by an SO(6) rotation. That means that the identification in the space where the D2-branes live won't be
  • X1 is identified with X1 + 2.pi.R
but rather
  • (X1, X4...X9) is identified with (X1+2.pi.R, rotation(X4...X9))
where the "rotation" is taken from SO(6). Obviously, for this rotation taken inside the SU(3) subgroup, we will obtain a supersymmetric theory.

Fine, what does this twist - the added rotation - do with the theory living on the D3-branes? Yes, your guess is right: it puffs it. If I am using the terminology incorrectly, Ori will probably kindly tell us. :-)

Ori argues that this puffing may be represented by a dimension 7 operator in the d=4 theory at low energies. That looks like if we make the theory non-renormalizable, but Ori argues that this conclusion could be too fast.

In the case of noncommutative gauge theory, we get a theory that is widely believed (or known?) to be UV-finite, even though it looks as a deformation by a dimension 6 operator at low energies (recall that when you expand the star product, the first non-trivial term will have two extra derivatives in it, multiplied by theta, the noncommutativity = commutator of "x" and "y"). The finiteness of the theory arises because the higher derivative terms combine into exponentials - phases - that actually make the loop diagrams' integrals more convergent, not less. Non-commutative field theory has various interesting issues with UV-IR mixing, but still, under appropriate definitions, it is UV-finite.

I wonder whether there is a simple Lagrangian description of this puffed field theory with a transparent puffed star-product that would add something like puffed phases to the integrals over momenta. ;-)

Note that in the dipole theories, an object with a large charge naturally becomes linearly extended in a certain direction. In his theory, what the charge objects acquire is arguably a multi-dimensional volume element although I have not yet figured out what the dimensions are.

I don't yet see a proof of decoupling of gravity. When gravity is decoupled, the magic power of string theory more or less guarantees that the resulting field theory would be local and UV-complete.

Disclaimer for the people who are being brainwashed by dozens of greedy crackpots on the web and on the book market and their pseudoscientific journalistic allies in various Economists and similar tabloids: this magic is not a religious one but a real one and one can explicitly check that it is there, and please kindly allow me to call you idiots if you're unable to understand these things. Thank you very much. ;-)

On the other hand, if gravity is not decoupled, then PFT doesn't exist because the rest of string theory can't be ignored. Ori probably knows why the decoupling is trivial.

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