Surface operators in gauge theory

A recent paper of Sergei Gukov and Edward Witten is quite an extensive opus, and it would take 5 hours or more to give a talk about it. That's why Sergei Gukov focused on a physics part of the paper, namely the surface operators.

In quantum field theory, the most popular operators are the local operators associated with points. However, many of the readers also know the Wilson lines and the 't Hooft lines - examples of one-dimensional operators. The Wilson lines are constructed as the traces of the holonomy of your gauge field over a closed contour and over a representation of the gauge group. The 't Hooft lines are, on the contrary, disorder operators that you can either interpret as objects that create a discontinuity for the perturbative fields or as the Wilson lines constructed from the S-dual gauge field.

Can we increase the dimension of the operators more than that? Sergei answers Yes. A simple operator associated with a surface is the flux over this surface. This looks too easy - it is just a combination of the point-like operators of the field strength from the points on the surface. However, when you take the flux of a dual field strength in the N=4 theory, you obtain something that is closer to a disorder operator, Sergei argues, although I don't quite understand why it should be so: such a dual flux still looks to me as a combination of some components of the same field strength from different points.

Let me stop my skepticism that is most likely based on some misunderstandings. The surface operators within the N=4 theory that they talk about can be constructed in string theory with the help of intersecting stacks of D3-branes. The 33' strings deform the intersection into a smooth connected manifold. Alternatively, most of the 33' scalar degrees of freedom may be viewed as the Fayet-Iliopoulos terms in the two-dimensional theory living on the intersection. These scalar fields parameterize a space that can be expressed either as a coset of a hyperbolic space and a U(1) group, or as a tangent bundle over a complex projective space.




Alternatively, they describe the surface operators as fractional Dirac strings. Dirac strings are invisible if the magnetic charge at their endpoint is properly quantized: otherwise they have physical consequences. Another string-theoretical realization of their picture involves the (2,0) theory in six dimensions compactified on a two-torus - in order to get the N=4 d=4 theory at low energies - where you add some infinite M2-branes ending on the fivebranes at a particular point of the two-torus. This makes some other properties of their construction, especially the transformation rules of some quantities under S-duality, manifest.

Let me omit these technicalities. One of the unusual things about Sergei's statements was his conclusion that the string realization only accounts for some of the interesting mathematical features of their operators in gauge theory but it fails to give the right answer to others. We may ask: if you can do something interesting and fully mathematically consistent with the N=4 gauge theory, can you always realize it within string theory? I still believe that the answer should be Yes, although I don't have any proof, while Sergei's answer, when some subtleties are taken into account, is essentially No.

There are problems with the picture involving infinite membranes because they change the asymptotic behavior, but there are other problems if you compatify these directions - e.g. because the total charge in a compact space should vanish.

On the other hand, Sergei claims that all their constructions have a fully coherent description in terms of the holographic dual. So in the most general sense, they're still fully included in string theory. I still don't understand how can they face any difficulties with the intersecting D3-brane picture of their configuration if they have no difficulties with the AdS picture of the setup: shouldn't the AdS geometry be just a near-horizon approximation of some other, asymptotically Minkowski configuration?

You shouldn't think that most things are confusing. I just focused on those that are confusing to me at this moment.