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October 6, 2006

LazyWeb Chiral Symmetry Breaking

This past week, in the Geometry and String Theory Seminar, as part of our series on differential cohomology, Dan gave a talk about his derivation of the Wess-Zumino term in the Chiral Lagrangian. It’s a pretty little application of generalized differential cohomology theories and it has several advantages over, say, Witten’s derivation

  1. For a general 4-manifold, XX, the signature, σ(X)\sigma(X) is an obstruction to finding a 5-manifold, BB, such that X=BX=\partial B. Witten, of course, worked on X=S 4X=S^4, which has vanishing signature, but this construction clearly doesn’t make sense for general XX.
  2. To define integration in the generalized differential cohomology theory in question (which Dan, unimaginatively, dubs differential “E-theory”) requires a choice of spin structure on XX. In particular, the requirement that XX be a spin manifold, which was clearly present in QCD, but not apparent in previous derivations of the Wess-Zumino term from low-energy considerations, re-emerges here.
  3. When quantizing on X=M (3)×X=M_{(3)}\times \mathbb{R}, the class in differential EE-theory, whose integral over XX is the Wess-Zumino term, gives a natural /2\mathbb{Z}/2 grading to Hilbert space (for N cN_c odd). This is what we call baryon number (compare the Skyrme model).

Anyway, that wasn’t what I really wanted to talk about.

In the course of a subsequent discussion, the following point arose. The 't Hooft Anomaly Matching Conditions ensure that there must be some light degrees of freedom in the low energy theory on which the chiral symmetries are realized. If we assume that the theory confines, then there are really only two possibilities1:

1. There are massless composite fermions which transform in appropriate representations to saturate the 't Hooft Anomaly Matching Conditions. 2. The chiral symmetry is spontaneously broken, and is realized nonlinearly on some Goldstone boson fields, whose Wess-Zumino term provides the anomalous variation satisfying the 't Hooft Anomaly Matching constraints.

Clearly, possibility 1 is impossible for N cN_c even. But, for N cN_c odd, there are all sorts of oddball special cases to check. John Terning found some oddball solutions. Surely there are others.

There’s a broad class of theories, however, where one can essentially prove that chiral symmetry breaking must take place (as it’s the only way to satisfy the Anomaly Matching constraints). Does anyone know of a systematic statement of what’s known?


1 I suppose that one might imagine some circumstances in which some mixture of 1 and 2 arose, with part of the chiral symmetry realized linearly on massless fermions, and part spontaneously-broken. But, since QCD-like theories do not break their vector-like symmetries (by a theorem of Vafa and Witten), this mixed possibility does not arise. Either SU(N f) L×SU(N f) RSU(N_f)_L\times SU(N_f)_R is unbroken, or it is broken to the diagonal SU(N f) VSU(N_f)_V.

Posted by distler at October 6, 2006 11:02 PM

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E, F, K, M-Theory

There is Connes’ E-theory, which is a close relative of K-theory. Usually it is done in the context of non-commutative operator algebras, but certainly one could apply it to the commutative algebra of functions and obtain a generalized cohomology theory on topological spaces. This is not what Dan was talking about, right? I’m confused about the naming clash.

I looked at the article and it appears to me that Freed defines the E-spectrum as something completely different. Did we run out of letters in the alphabet already?

Posted by: Volker Braun on October 7, 2006 12:11 PM | Permalink | Reply to this

Re: E, F, K, M-Theory

I think E-theory is generally attributed to Connes and Higson. It’s definitely not what Dan is talking about.

Posted by: Aaron Bergman on October 7, 2006 12:20 PM | Permalink | Reply to this

Re: E, F, K, M-Theory

{E }\{E_\bullet\}” is often used, generically, to denote a spectrum.

I said it was an unoriginal choice…

In Dan’s case, EE-cohomology is a piece of K-theory/KO-theory (giving a refinement of the usual Chern, Pontryagin and Steifel-Whitney classes).

For the sake of those who haven’t looked at the paper, I’ll give the definition here.

Let H(A) n{H(A)}_n be the nnth Eilenberg-Maclane space associated to the Abelian group, AA. π qH(A) n={A q=n 0 otherwise \pi_q {H(A)}_n = \begin{cases}A& q=n\\ 0 & \text{otherwise}\end{cases} Then his EE-spectrum fits into the fibration H() n i E n j H(/2) n2 \begin{matrix} {H(\mathbb{Z})}_n &\overset{i}{\to} & E_n \\ & & \darr j\\ & & {H(\mathbb{Z}/2)}_{n-2} \end{matrix} This yields the long exact sequence in cohomology H n(X,𝕫)iE njH n2(X,/2)βSq 2H n+1(X,) \dots \to \mathrm{H}^n(X,\mathbb{z}) \overset{i}{\to} E^n \overset{j}{\to} \mathrm{H}^{n-2}(X,\mathbb{Z}/2) \overset{\beta\circ Sq^2}{\to} \mathrm{H}^{n+1}(X,\mathbb{Z}) \to \dots where β\beta is the integer Bockstein for the long exact sequence in cohomology, associated to the coefficient sequence, 0×2/200\to \mathbb{Z} \overset{\times 2}{\to}\mathbb{Z}\to \mathbb{Z}/2\to 0, and Sq 2Sq^2 is the second Steenrod square.

Posted by: Jacques Distler on October 7, 2006 1:06 PM | Permalink | PGP Sig | Reply to this

Re: E, F, K, M-Theory

{E }\{E_\bullet\}” is often used, generically, to denote a spectrum.

In some circles, I think, the EE would indicate a commutative (up to higher homotopy) ring spectrum, i.e. it would refer to an E E_\infty ring spectrum.

Posted by: urs on October 23, 2006 7:42 AM | Permalink | Reply to this

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