- I introduce a Dirac-type equation in 2D, the
*Jackiw–Rossi equation*. - Main Result #1: I generalize the Jackiw–Rossi equation to:

- Higher dimensions $\sim$ representation theory of Clifford algebras.
- Multi-spinors/spinors with
*flavors*$\sim$ spinors with "twists". - (Pseudo)-Riemannian manifolds $\sim$ spin geometry.

- Main Result #2:

- I describe the "low energy spectrum" in 2D.
- Construct solutions in higher dimensions.
- (If time permits, then) explain connections to vortices and braids.

- Describes electronic surface excitations in a topological insulator in contact with an $s$-wave superconductor.
- Let $\psi = \begin{pmatrix} \psi_\uparrow \\ \psi_\downarrow \end{pmatrix} : \mathbb{R}^2 \mapsto \mathbb{C}^2$ be a spinor field.
- Let $\Phi : \mathbb{R}^2 \mapsto \mathbb{C}$ be the superconducting order parameter.
- The (static, massless) Jackiw–Rossi equation is: \begin{equation} \left( D + \mathcal{R}_\Phi \right) \psi := \begin{pmatrix} \left( \partial_x + i \partial_y \right) \psi_\uparrow + \Phi \overline{\psi}_\uparrow \\ \left( - \partial_x + i \partial_y \right) \psi_\downarrow + \Phi \overline{\psi}_\downarrow \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}. \end{equation}
- Simple example (1-vortex): $\Phi = z = x + i y$, $\psi_\downarrow = 0$, and $\psi_\uparrow = e^{- |z|^2/2}$.
- The "large" 1-vortex limit: $\Phi = t z$, with $t \gg 1$, $\psi_\downarrow = 0$, and $\psi_\uparrow = \sqrt{t} \: e^{- t |z|^2/2}$.
**Important note: If $\psi$ is a (nonzero) solution, then $i \psi$ is not.**$\Rightarrow$ Solutions only form a real (as opposed to complex) vector space!**Majorana fermions!**

- Let $CL (s, t)$ be the Clifford algebra of $\mathbb{R}^{s, t}$, and $S$ be the (unique) spinor representation.
- There is always a(n essentially) unique invariant bilinear pairing $B: S \otimes S \rightarrow \mathbb{C}$. Moreover, $(S \otimes S)^* = \Lambda$, where $\Lambda$ is (a subspace of) the vector space of exterior forms on $\mathbb{R}^{s, t}$ and the bilinear pairings corresponds to the 0-forms in $\Lambda$.
- The metric dual of $B$ yields a map $\mathcal{R} : S \rightarrow \overline{S}$, which can also be viewed as
**complex conjugate linear**map of $S$. - Let $\psi : \mathbb{R}^{s, t} \mapsto S$ be a spinor field and $\Phi : \mathbb{R}^{s, t} \mapsto \mathbb{C}$ be the superconducting order parameter.
- The generalized Jackiw–Rossi equation with mass $m \in \mathbb{R}$ is: \begin{equation} \left( D + \mathcal{R}_\Phi \right) \psi := \left( D + \Phi \mathcal{R} \right) \psi = m \psi. \end{equation}
- When $s = 2, t = 0$, and $m = 0$, then the above equation recovers the original Jackiw–Rossi equation.
- When $s = 2, t = 1$, then the above equation recovers the time dependent Jackiw–Rossi equation with mass $m$!

- Let $CL (s, t)$ be the Clifford algebra of $\mathbb{R}^{s, t}$, and $M$ be any Clifford module.
- Then $M = S \otimes E$, where $E$ is a vanilla complex vector space, the
*flavor space*. **Question:**Are there invariant bilinear pairings on $M$?**Answer:**Yes! Let $F := E \otimes E$. Given any $\Phi \in F^*$ and using $M \otimes M = (S \otimes S) \otimes F$, we can define $B_\Phi : B \otimes \Phi$.

Every invariant bilinear pairing on $M$ has this form.- The metric dual of $B_\Phi$ yields a map $\mathcal{R}_\Phi : M \rightarrow \overline{M}$, which can also be viewed as
**complex conjugate linear**map of $M$. - Let $\psi : \mathbb{R}^{s, t} \mapsto M$ be a spinor field with flavors in $E$ and $\Phi : \mathbb{R}^{s, t} \mapsto F^*$ be a superconducting order parameter.
- Given a vector potential (an EM field) $A : \mathbb{R}^{s, t} \mapsto \mathbb{R}^{s, t}$, we can also twist the Dirac operator: $D_A := D + \sum\limits_{i = 1}^{s + t} A_i \ \mathrm{cl} \left( \mathrm{d} x^i \right)$.
- The generalized Jackiw–Rossi equation with flavors in $E$ and with mass $m \in \mathbb{R}$ is: $\left( D_A + \mathcal{R}_\Phi \right) \psi = m \psi$.
- When $F = \mathbb{C}$ and $A = 0$, then we get back the Jackiw–Rossi equation from the previous slide.
- Using $(S \otimes S)^* = \Lambda$, we can consider even more general "form-order parameters" $\Phi \in \Lambda \otimes F^*$.

- Let $X$ be a (pseudo)-Riemannian manifold and $M \rightarrow X$ be a bundle of Clifford modules.
- Locally, $M = S \otimes E$, where $E$ is a complex vector bundle. This picture need not hold globally! In particular, $X$ need not be spin, or even spin$^c$!
- Nonetheless, $(M \otimes M)^*$ is still (canonically) isomorphic to $\Lambda \otimes F^*$, where $\Lambda$ is (a subbundle of) the bundle of differential forms on $X$.
- Given $\nabla$, a compatible connection on $M$, and $\Phi$, a section of $F^*$ (or, more generally $\Lambda \otimes F^*$), we can define the Jackiw–Rossi operator on the sections of $M$ as \begin{equation} D_\nabla + \mathcal{R}_\Phi. \end{equation}

- Let $X$ be a (pseudo)-Riemannian manifold and $M \rightarrow X$ be a bundle of Clifford modules, as before.
- Given $\nabla$, a compatible connection on $M$, and $\Phi$, a section of $F^*$ (or, more generally $\Lambda \otimes F^*$), we get the operators $D_\nabla$ and $\mathcal{R}_\Phi$.
- While $\mathcal{R}_\Phi$ is not complex linear, by doubling the spinors, we can still associate a complex linear operator to this datum, the (generalized)
*Bogoliubov--de Gennes operator*: \begin{equation} \begin{pmatrix} D_\nabla & \mathcal{R}_\Phi \\ \mathcal{R}_\Phi^* & \pm D_\nabla \end{pmatrix} \end{equation} considered as an operator on $S \oplus \overline{S}$. - This is a story for another time though...

- First, let us return to the original Jackiw–Rossi equation (no gauging, flavors, or manifolds).
**Theorem (Rauch '04):**

If $\Phi = t z^p \overline{z}^q$, then the space of all $L^2$ solutions is $|p - q|$ dimensional and purely (anti-)chiral if $p > q$ ($p < q$).**Theorem (N. '21):**

If $\Phi = t z^p \overline{z}^q + \mbox{"remainder terms"}$ and $A$ is smooth and bounded, then the space of all $L^2$ solutions to \begin{equation} \left( D_A + \mathcal{R}_\Phi \right) \psi = 0. \end{equation} is $|p - q|$ dimensional and purely (anti-)chiral if $p > q$ ($p < q$), all of which concentrate around the origin.- Furthermore, I studied the spectral properties of the operator $D_A + \mathcal{R}_\Phi$. In particular, I proved a gap result.

**Theorem (N. '21):**

Let $\Sigma$ be a closed surface, $S \rightarrow \Sigma$ be a spin$^c$ bundle, and $\left( \nabla, \Phi \right)$ as before.

Assume that the zero locus of $\Phi$ is discrete and at each zero $z \in \Sigma$ it has the form $\Phi = z^{p_z} \overline{z}^{q_z} + \mbox{"remainder terms"}$.

Then for $t \gg 1$ there are exactly $\max (\{ p_z - q_z, 0 \})$ linearly independent, chiral eigenspinors concentrating at $z$ with mass $o (1)$. All other eigenvalues are $O (t^{1/(m + 1)})$.

(Similar statement holds for anti-chiral solutions.)- Proof: Glue-in $\oplus$ gap result.
- Furthermore, if for all zeros, $z$, we have $q_z = 0$, then the above "low energy" solutions are in fact massless.

**This gives an independent proof of the Riemann–Roch Theorem!** - One can pull back solutions to $X = \Sigma \times Y$, where $Y$ is any (pseudo)-Riemannian manifold, using harmonic spinors on $Y$.
**Conjecture (vaguely):**Solution to $D_\nabla + t \mathcal{R}_\Phi$ in higher dimensions are classified by harmonic spinors on $Y := \Phi^{- 1} (0)$, which is generically a codimension 2 subset.

- The input for the (generalized) Jackiw–Rossi theories is a pair $\left( \nabla, \Phi \right)$.
- Canonical families of such pairs come from (for example) the Ginzburg–Landau theory.
- The "nicest" Ginzburg–Landau fields are called $\tau$-vortices in 2D. (See Taubes, Bradlow, ...)
**Theorem (Bradlow '90):**For $\tau$ large enough, the moduli space of $\tau$-vortices of degree $d$ is $\mathrm{Sym}^d (\Sigma)$.- It is easy to see that each degree $2k$ vortex field, $(\nabla, \Phi)$, yields a Jackiw–Rossi operator $D_\nabla + \mathcal{R}_\Phi$ whose kernel has real dimension $2k$.
- Problem: Due to the breaking of the gauge symmetry, these kernels only descend to the vortex moduli as a bundle of real projective spaces.
- It is unclear whether this bundle comes from a vector bundle, because
- I don't really understand real Atiyah–Segal class (that would classify bundles of real projective spaces).
- No one else seem to know or care about them either.

- In any case, away from the "diagonal" in $\mathrm{Sym}^d (\Sigma)$, the bundle "flattens" as $\tau \rightarrow \infty$, thus we get a (nontrivial) projective representation of surface braid group of $\Sigma$ with $2k$ braids.

- Understanding solutions in higher dimensions.
- New proofs of Atiyah–Singer type theorems.
- Nonlinear generalizations: $\Phi = (\psi \psi)_0^\flat$.
- Understanding the braid representation and the connections to vortices.
- Understanding the generalized Bogoliubov--de Gennes equation.