Earlier this week Zohar Komargodski (who is now at the Simons Center) visited Columbia, and gave a wonderful talk on recent work he has been involved in that provides some new insight into a very old question about QCD. Simplifying the problem by ignoring fermions, QCD is a pure SU(3) Yang-Mills gauge theory, a simple to define QFT which has been highly resistant to decades of effort to better understand it.

One aspect of the theory is that it can be studied as a function of an angular parameter, the so-called $\theta$-angle. Most information about the theory comes from simplifying by taking $\theta=0$, which seems to be the physically relevant value, one at which the theory is time reversal invariant. There is however another value for which the theory is time reversal invariant, $\theta=\pi$, and what happens there has always been rather mysterious.

The new ideas about this question that Komargodski talked about are in the paper Theta, Time Reversal and Temperature from earlier this year, joint work with Gaiotto, Kapustin and Seiberg. Much of the talk was taken up with going over the details of the toy model described in Appendix D of this paper. This is an extremely simple quantum mechanical model, that of a particle moving on a circle, where you add to the Lagrangian a term proportional to the velocity, which is where the angle $\theta$ appears. You can also think of this as a coupling to an electromagnetic field describing flux through the circle.

Even if you’re put off by the difficulty of questions about quantum field theories such as QCD, I strongly recommend reading their Appendix. It’s a simple and straightforward quantum mechanics story, with the new feature of a beautiful interpretation of the model in terms of a projective representation of the group O(2), or equivalently, a representation of Pin(2), a central extension of O(2). In the analogy to SU(N) Yang-Mills, it is the $\mathbf Z_N$ symmetry of the theory that gets realized projectively.

Komargodski himself commented at the beginning of the talk on the reasons that people are returning to look again at old, difficult problems about QCD. The new ideas he described are closely related to ones that are part of the recent hot topic of symmetry protected phases in condensed matter theory. It’s great to see that this QFT research may not just have condensed matter applications, but seems to be leading to a renewal of interest in long-standing problems about QCD itself.

Besides the paper mentioned above, there are now quite a few others. One notable one is very recent work of Komargodski and collaborators, Time-Reversal Breaking in QCD4, Walls and Dualities in 2+1 Dimensions.

> a projective representation of the group O(2), or equivalently, a representation of Pin(2), a central extension of O(2)

There are two nontrivial central extensions of O(2), called Pin^+(2) and Pin^-(2). The paper didn’t seem to mention which Pin group is appearing here; do you know which one it is?

Thanks for posting on this. I tried looking through their paper. Will take more time to read it. But I was wondering if in the talk at Columbia whether the speaker mentioned how their works fits in with the strong CP phase or strong CP problem. Their paper doesn’t explicitly mention the strong CP phase/problem.

Interesting post, thanks.

Do you have any opinion on Arkani-Hamed’s latest paper? Judging from the title it sounds pretty spectacular. Unfortunately I don’t have the technical skill sett to judge the impact of the paper.

https://arxiv.org/abs/1709.04891

Arun Debray,

No, I don’t (since I don’t know the difference between those two versions of Pin), but I’d think it’s quite explicit and should be easy to figure out.

Anon,

No mention of the strong CP problem (other than the standard fact that it disappears for a massless fermion), no claim to have something new to say about that.

P.

Looks quite interesting, and I’ll look at it more closely, especially since I’m quite interested in how much reps of the Poincare group tell you about QFT. Some of the claims in the abstract look much stronger than plausible, so very interesting. But, this really is off-topic, and I don’t want to host a discussion of this paper here and now.

Like the paper discussed here, another quite encouraging example of some of the best theorists around working on new ideas about 4d qft close to the Standard Model.

Peter, Arun Debray is referring to the fact that whereas the (linear algebraic) orthogonal group O(q) associated to a non-degenerate quadratic space (V,q) over a field k is insensitive (say as a subgroup of GL(V)) to replacing q with a scalar multiple cq for nonzero c, the isomorphism class of the (linear algebraic) group Pin(q) considered as a central extension of O(q) by \mu_2 is *very* sensitive to such change in q when c is a non-square in k. This sensitivity to such change in q at the level of the Pin group is a shadow of what occurs at the level of Clifford algebras: for non-square c, one doesn’t see any direct link between the Clifford algebras C(V,q) and C(V,cq). For your situation, the context of interest is (V,q) positive-definite of dimension n>1 over k=R (even n=2) and c=-1.

Staying in the general setting for conceptual clarity, an obstruction to finding an isomorphism between Pin(q) and Pin(cq) as central extensions is encoded in the spinor norm O(q)(k) –> H^1(k,\mu_2) = k*/(k*)^2 that is most efficiently defined as the connecting map associated to the short exact sequence 1 –> \mu_2 –> Pin(q) –> O(q) –> 1 of (linear algebraic) groups, and is characterized most concretely by the condition that it carries the reflection r_v in any non-isotropic v (i.e., v for which q(v) is nonzero, such as any nonzero v when k=R with q definite) to q(v) mod (k*)^2. The point is that if Pin(q) and Pin(cq) are isomorphic as central extensions then the associated connecting maps (i.e., spinor norms) coincide, so then *necessarily* q(v) and c(q(v)) coincide mod (k*)^2 for any non-isotropic v. But that forces c to be a square in k! So when c is a non-square in k, we really get non-isomorphic central extensions.

Working over the real numbers, there is just one isomorphism class of positive-definite quadratic spaces of a given dimension n > 1, so one may refer to its associated Pin group as “Pin^+(n)” (considered as a central extension of O(n) by \mu_2), and refer to the one for the negative-definite variant as “Pin^{-}(n)” (also considered as a central extension of O(n) by \mu_2). Since c=-1 is not a square in R, these two central extensions of O(n) by \mu_2 are not isomorphic. (For expository simplicity I am sweeping under the rug the distinction between linear algebraic R-groups and compact Lie groups because it turns out not to be a problem in this case: see the theorem of Chevalley stated in my answer to https://mathoverflow.net/questions/6079/classification-of-compact-lie-groups/16269#16269 for a precise statement about that.)

Will the Komargodski paper shed some light on confinement?

Thank you!

BCnrd,

Thanks! I see, this is the same phenomenon that shows up in four dimensions as the fact that while Spin(3,1)=Spin(1,3), Pin(3,1) is different than Pin(1,3). This has led to some debate in the physics literature about physics being sensitive to what is usually thought of as a choice of sign convention.

Petite Kabylie,

The paper has some claims about implications for the phase diagram as a function of theta and the temperature, but not I think for what happens at theta=0, which seems to be the physically relevant value.

Peter, that’s right. I should have also noted (for the purposes of the comparisob of spinor-norm calculations upon replacing q with cq) that the reflection r_v \in O(q) in a non-isotropic vector v is *insensitive* to replacing q with cq since by definition r_v(x) = x – (B_q(v,x)/q(v))v where B_q(v,w) = q(v+w)-q(v)-q(w) is the symmetric bilinear form associated to q. (Note this definition of B_q omits the factor of 1/2 that is sometimes used to define B_q, so B_q(v,v)=2q(v) and in particular the factor of 2 one usually sees in the definition of r_v is really lurking inside B_q).

Petite Kabylie,

page 23 onward has the details

P

Peter Woit

New idesas about QFT and Poincare group can be found in the ideas introduced by Mund, Schroer and Yngvason: string localised field that allow getting rid of gauge theory (work in positive definite Hilbert space) and the Higgs mechanism (the Higgs is still there but for other reasons) . Rehren has a preprint the same day as Arkani-Hamed’s:

Pauli-Lubanski limit and stress-energy tensor for infinite-spin fields

https://arxiv.org/abs/1709.04858

(and reference within)

Can a mathematician put in a plug for

https://arxiv.org/abs/1707.05448

The Sum Over Topological Sectors and θ in the 2+1-Dimensional CP1 σ-Model

Daniel S. Freed, Zohar Komargodski, Nathan Seiberg

as well?