Theory on domain walls

Question

In Baryons in Quantum Chromodynamics, Zohar Komargodski have slide:

I wanna understand:

Why domein wall can have nontrivial worldvolume theory?

When such solitonic objects have interior degrees of freedom?

Answer 1

I'm not familiar with the specific example cited in the question, but this answer mentions some other examples illustrating that domain walls generally can have interior degrees of freedom.

A simple example

A simple example is given section 2.1 in Domain Wall Fermions and Chiral Gauge Theories. In that example, the "bulk" theory is a free Dirac spinor field, but with a mass parameter $m$ that smoothly changes sign in a neighborhood of $x=0$. This gives a domain wall with its own internal degrees of freedom.

The one-paragraph derivation in the cited paper is already short and clear, so I'll focus on the idea. The idea is that the smoothed-step-function shape of the mass parameter leads to the Dirac equation having some solutions that are mostly supported only near the step, falling off exponentially with distance orthogonal to the step. These modes are "bound" to the step, but they can still propagate freely as massless particles in the tangential directions. They are the "internal degrees of freedom" associated with the domain wall.

Natural domain walls

The preceding example is "artificial" in the sense that we forced the domain wall to exist by making the mass parameter $x$-dependent, but it still illustrates how a domain wall can have its own internal degrees of freedom. A similar thing can happen on a "natural" domain wall, such as one that separates two different vacuum states in a theory with a spontaneously broken discrete symmetry.

This paper co-authored by Komargodski (the author of the slides referenced in the OP) considers several examples and says this about some of them on page 2:

whenever the theory has more than one vacuum ..., there can be dynamical domain walls separating between the two vacua. These are dynamical excitations of the system. In all our examples the domain wall separates between two gapped ground states — the lowest excitation in the bulk has nonzero energy $M$. It is often the case that there are nontrivial excitations with energy much lower than $M$ living on the domain wall. They are described by a $3d$ quantum field theory. It may also be that the domain wall does not have excitations with energy much smaller than $M$, but it supports a $3d$ Topological QFT (TQFT). ... These 3d QFTs are valid only up to energies of order $M$. At higher energies the bulk cannot be ignored and the theory is no longer a purely 3d QFT.

Just for fun, here's another paper in which the same person (Komargodski) is a co-author with a Physics SE user (Ryan Thorngren):

• Anomaly Matching in the Symmetry Broken Phase: Domain Walls, CPT, and the Smith Isomorphism, https://arxiv.org/abs/1910.14039

Boundary degrees of freedom

A related and experimentally-accessible example with interior degrees of freedom is the integer quantum Hall effect. The (physical) boundary of the condensed-matter system has its own propagating degrees of freedom, confined to the boundary. The introduction to http://arxiv.org/abs/1909.08775 reviews this phenomenon in the light of a broader theoretical context.

That example also illustrates the fact that, despite the language, the domain wall "theory" is not necessarily a well-defined quantum field theory on its own. The "theory" language is often used even when the theory has an anomaly — when it fails to be gauge-invariant by itself — as in the integer quantum Hall effect example. The gauge non-invariance of the boundary (or domain-wall) "theory" is compensated by the gauge non-invariance of the bulk "theory" so that only the combined system is gauge invariant. This is called anomaly inflow.

Application to the Standard Model

The domain-wall theory idea has played a prominent role in the search for mathematically rigorous definitions of chiral gauge theories, such as the Standard Model. Searching for the keywords "domain wall fermions" should lead you to some papers on that subject, like this one. The domain wall's internal degrees of freedom are of the utmost importance in that context, because the (hope is that the) domain wall theory is the theory of interest — such as the Standard Model itself.