Some Dialectics About Dielectrics

One of my most rewarding activities last spring was serving as the teaching assistant for Mike Hopkins' course on algebraic topology and condensed matter. I learned so much from many wonderful conversations with Mike, and many incisive questions from students. Throughout the course, it became clear that invertible phases are an excellent entry into stable homotopy theory entirely on their own, and breath life into many of the key manipulations used in the subject. In this post, I want to emphasize one physical concept, the notion of a dielectric brane, which is quite familiar in string theory (and condensed matter, under different names). Hopefully, I'll convey the intuitive, almost tactile understanding of dielectric branes that has helped me enormously in understanding stable homotopy theory in general.

To explain dielectric branes, we should first review conventional dielectrics. A dielectric is an insulating material with the following property: when you place it in an electric field, it becomes electrically polarized. The reason for this is clear: the positive charges will be pushed in the direction of the electric field, while the negative charges will be pulled in the opposite direction, leading to an imbalance of charge. This can be seen even at the level of a single atom, as the positively charged nucleus will be pushed in the opposite direction as the negatively charged electrons.

Let me emphasize the geometry of this transition. Before polarizing, the atom is a single point particle. However, after polarizing, the atom has become smeared out along the direction of the electric field. As we increase the strength of the field, the positive and negative charges move further apart, until eventually we may think of them as sitting at two distinct points in space, separated in the direction of the electric field. The set of two points, one of positive charge (orientation) and one of negative charge (orientation), is simply the zero sphere $S^0$! Thus, we see that a single point particle has puffed up into two particles, with different charges than the original particle, though the net charge is of course conserved.

The concept of a dielectric brane is a categorification of the ordinary notion of a dielectric, in that it replaces point particles with dynamical objects with more dimensions, such and strings and membranes. The basic example introduced by Myers occurs in Type IIA string theory, and involves placing $N$ D0-branes at a point in space, in the presence of a constant four-form field. Here, the D0-branes are playing the role of the atom, and the four-form field replaces the ordinary electric field, but "points" along a three-plane in space rather than along a single direction. In this context, Myers showed that the D0-branes will puff up, or polarize, into a D2-brane supported on a two-sphere $S^2$ in this three-plane, surrounding the location of the original D0-branes. Thus, we have successfully replaced the zero sphere $S^0$ in the original dielectric example with the two sphere $S^2$!

How does charge conservation work in this example? The original configuration of $N$ D0-branes has no net D2-brane charge, but has $N$ units of D0-brane charge. The final configuration might naively look like it has D2-brane charge, but in fact the charge cancels, since the positive D2-brane charge on one side of the $S^2$ is canceled by the negative charge on the opposite side, just as the positive charge of the nuclei is canceled by the negative charge of the electrons in an ordinary dielectric. For the D0-brane charge, recall that the D2-brane carries a $U(1)$ gauge field on its worldvolume. The magnetic fluxes of this gauge field are point particles in the D2 worldvolume, which carry D0 brane charge! Thus, as Myers showed, the D2-brane must carry $N$ units of magnetic flux on the spatial $S^2$, so that the final configuration carries $N$ units of D0-brane charge, as required.

Now, what does any of this have to do with stable homotopy theory? Suppose we wanted to keep careful track of the charges of branes of various dimensions. Usually, we label charges under an abelian gauge field by elements of an abelian group, namely the Pontryagin dual of the gauge group.¹ However, when we have branes of various different dimensions, we need to keep track of the different dimensions, and so maybe we could try a graded abelian group, where the grading corresponds to the dimension of the corresponding brane. Though a better approximation, a graded abelian group misses exactly the possibility of dielectric branes, since it doesn't include any relations between charges of different dimension! We thus need a structure like a graded abelian group, but which remembers things like "a D2-brane wrapped on $S^2$ with $N$ units of flux has the same charge as $N$ D0-branes."

The central point of this post is that the correct description of the various charges, including the possibility for dielectric branes, is a module spectrum! I'm not going to try to give a full definition of spectra here, but I can give some quick heuristics that should be enough for the physical point I'm trying to make. A spectrum $E$ is a higher-categorical analogue of an abelian group. Given a spectrum $E$, you can extract its homotopy groups

\[ \pi_k E, \quad k \in \mathbb{Z},\]

including $k$ negative! These homotopy groups are all abelian, and arrange into a graded abelian group $\pi_* E$. Given two spectra $E_1, E_2$, we may form the wedge sum $E_1 \vee E_2$ and the smash product $E_1 \wedge E_2$, which you can think of as the direct sum and tensor product respectively. Finally, given a graded abelian group $A_*$, we can form its Eilenberg-MacLane spectrum $HA$, which has

\[ \pi_k HA = A_k, \]

and represents $A_*$ in the world of spectra.

To discuss module spectra, we should first discuss ring spectra. A ring spectrum is a spectrum $E$, together with a multiplication map

\[ \mu : E \wedge E \to E, \]

that is associative and unital (in the appropriate sense). The analogue of the ring of integers $\mathbb{Z}$ in the world of spectra is the sphere spectrum $\mathbb{S}$: just as every abelian group is a module over the integers, every spectrum is a module over the sphere spectrum. The homotopy groups $\pi_k \mathbb{S}$ vanish for $k < 0$, while $\pi_0 \mathbb{S} = \mathbb{Z}$, and $\pi_k \mathbb{S}$ is a finite abelian group for $k > 0$. Thus, while the integers are the freest abelian group, $H\mathbb{Z}$ is not the freest spectrum, as it contains a relation: all the higher homotopy groups vanish!

What does it mean to have a module spectrum $M$, given a ring spectrum $E$? This simply means that we have a multiplication

\[ m : E \wedge M \to M, \]

that satisfies the usual laws for a module over a ring (up to homotopy). In particular, given any element $x \in \pi_k E$, we get a map "multiplication by $x$,"

\[ m_x : \pi_n M \to \pi_{n + k} M, \]

and so we already see that module spectra naturally carry relations between their homotopy groups of different degrees! In fact, since every spectrum is already a module over $\mathbb{S}$, every spectrum already has some canonical relations between its various homotopy groups.

So far, we can only see some analogies between dielectric branes and module spectra, but nothing explicit yet. Let's fix this! The concrete example I want to work out is the following: consider the Majorana wire, an invertible phase in one spatial dimension, characterized by the presence of unpaired Majorana modes at either end of an open wire. One way to distinguish the Majorana wire from the trivial wire is to place it on a circle $S^1$ with periodic boundary conditions for the fermions (i.e., the nonbounding spin structure), and measure the fermion number of the ground state. Unlike the trivial wire, which has a bosonic ground state, the Majorana wire has a fermionic ground state! Now, suppose we place a Majorana wire on a circle in a two-plane, and shrink the radius until the wire becomes indistinguishable from a point particle. Since the fermion parity must be conserved, we know that the resulting point particle must be a fermion!

How can we characterize this (de)polarization in terms of a module spectrum? The ring spectrum in question here is given by spin bordism ${\rm MSpin}$, whose homotopy groups $\pi_k {\rm MSpin}$ are given by cobordism classes of $k$-manifolds with spin structure. The addition and multiplication are given by disjoint union and Cartesian product respectively. The module spectrum is the spectrum ${\rm IP}_f$, whose homotopy groups $\pi_{-k} {\rm IP}_f$ are given by the abelian group of fermionic invertible phases in spacetime dimension $k$.²

Now, what would it mean to have a module structure

\[ m : {\rm MSpin} \wedge {\rm IP}_f \to {\rm IP}_f? \]

In particular, given a closed spin $k$-manifold $X \in \pi_k {\rm MSpin}$, we should have a map

\[ m_X : \pi_{-n} {\rm IP}_f \to \pi_{- (n - k)} {\rm IP}_f, \]

i.e., a way to take an invertible fermionic phase in dimension $n$ and produce one in dimension $(n - k)$. At this point, the answer is clear: you simply dimensionally reduce on the manifold $X$! That is to say, you take $k$ of the dimensions of your material, and wrap them in the shape of the manifold $X$, leaving $(n - k)$ dimensions free.

For the case of wrapping the Majorana wire on the non-bounding circle, we can be even more explicit. We have

\[ \pi_{-1} {\rm IP}_f = \mathbb{Z}_2, \quad \pi_{-2} {\rm IP}_f = \mathbb{Z}_2, \]

generated by the phases of the fermion and the Majorana wire, respectively. The map

\[ m_{S^1_p} : \pi_{-2} {\rm IP}_f \to \pi_{-1} {\rm IP}_f, \]

given by wrapping on the non-bounding circle is the identity map $\mathbb{Z}_2 \to \mathbb{Z}_2$, and so the module structure confirms our physical expectation that the point particle obtained by depolarizing a Majorana wire wrapped on a non-bounding circle is indeed a fermion!

This description of brane charge, including dielectric branes, in terms of module spectra is quite general.³ The ordinary dielectric effect can be described by the module structure of $H\mathbb{Z}$ over itself, while the Myers effect in Type IIA is captured by the module structure

\[ {\rm MSpin}^c \wedge {\rm KU} \to {\rm KU},\]

 of complex K-theory as a module over spin$^c$ bordism. Along the lines of my last post, perhaps we should next think a bit harder about the physical meaning of the module structure

\[ {\rm MString} \wedge {\rm TMF} \to {\rm TMF}, \]

of topological modular forms over string bordism, but this post has gone on long enough, so let's save it for another time!

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1 For example, the Pontryagin dual of $U(1)$ is $\mathbb{Z}$.^↩
2 That invertible phases arrange themselves into a spectrum is due to Kitaev. By Freed-Hopkins, we may identify ${\rm IP}_f = \Sigma I_\mathbb{Z} {\rm MSpin}$, a shift of the Anderson dual of spin bordism.^↩
3 We can even describe electromagnetic duality and self-dual fields in terms of spectra, see this paper.^↩