# Ginzburg–Landau theory on euclidean backgrounds

• Effective field theory for superconductivity.

• Unknowns: $\left\{ \begin{array}{ll} A = (A_1, \ldots, A_n) &\mbox{: EM vector potential (1-form) in Coulomb gauge$\partial_a A_a = 0$,} \\ \phi &\mbox{: order parameter (complex scalar field).} \end{array} \right.$

• Induced fields: $\left\{ \begin{array}{lllll} \mbox{EM field tensor: } &(F_A)_{ab} &= \quad i \left( \partial_a A_b - \partial_b A_a \right), &\Rightarrow \quad - \partial_a (F_A)_{ab} &= - i \partial_a^2 A_b = i \Delta A_b, \\ \mbox{kinetic term: } &\nabla_a^A \phi &= \quad (\partial_a + i A_a) \phi, &\Rightarrow \quad \Delta_A \phi &= - (\nabla_a^A)^2 \phi, \\ \mbox{supercurrent: } &j (A, \phi)_a &= \quad i \: \mathrm{Im} \left( \overline{(\nabla^A \phi)_a} \phi \right). \end{array} \right.$

• Coupling constants: $\left\{ \begin{array}{ll} \tau \quad \sim &\mbox{ temperature or volume,} \\ \kappa \quad \sim &\mbox{$\tfrac{\mbox{penetration length}}{\mbox{coherence length}}$; (Type I/II SC's).} \end{array} \right.$

• Energy density & functional: $\: e (A, \phi) = |F_A|^2 + |\nabla^A \phi|^2 + \tfrac{\kappa^2}{2} \left( \tau - |\phi|^2 \right)^2, \quad \& \quad \mathcal{E} (A, \phi) = \displaystyle\int\limits_{\mathbb{R}^n} e (A, \phi) \mathrm{d}^n x.$

• The Euler–Lagrange equations of $\mathcal{E}$ are the Ginzburg–Landau equations: $\left\{ \begin{array}{ll} - \partial_a (F_A)_{ab} &= \quad j (A, \phi)_b, \\ \Delta_A \phi &= \quad \kappa^2 \left( \tau - |\phi|^2 \right) \phi. \end{array} \right.$

# Ginzburg–Landau theory on manifolds

• Replace $\mathbb{R}^n$ with a (closed and oriented) Riemannian manifold, $(X, g)$, and fix a Hermitian line bundle, ($\mathcal{L}, h)$.

• Unknowns: $\left\{ \begin{array}{ll} \nabla &\mbox{: connection on$\mathcal{L}$,} \\ \phi &\mbox{: section of$\mathcal{L}$.} \end{array} \right.$

• Induced fields: $\left\{ \begin{array}{ll} \mbox{the curvature of } \nabla &= \quad F_\nabla, \\ \mbox{the covariant derivative of } \phi &= \quad \nabla \phi, \\ \mbox{the supercurrent generated by$\nabla$and$\phi$} &= \quad j (\nabla, \phi) = i \: \mathrm{Im} \left( h (\nabla \phi, \phi) \right). \end{array} \right.$

• Energy density & functional: $\: e (\nabla, \phi) = |F_\nabla|^2 + |\nabla \phi|^2 + \tfrac{\kappa^2}{2} \left( \tau - |\phi|^2 \right)^2, \quad \& \quad \mathcal{E} (\nabla, \phi) = \displaystyle\int\limits_X e (\nabla, \phi) \mathrm{vol}_g.$
• Ginzburg–Landau equations: $\left\{ \begin{array}{ll} \mathrm{d}^* F_\nabla &= \quad j (\nabla, \phi), \\ \nabla^* \nabla \phi &= \quad \kappa^2 \left( \tau - |\phi|^2 \right) \phi. \end{array} \right.$

• If $\nabla$ solves $\mathrm{d}^* F_\nabla = 0$, then the pair $(\nabla, 0)$ solves the Ginzburg–Landau equation for any $\kappa, \tau$.
• The equation $\nabla$ solves $\mathrm{d}^* F_\nabla = 0$ is equivalent to the vector potential solving the vacuum Maxwell equation(s).
• Such a $\nabla$ is called a normal phase solution.

# Existence of absolute minimizers in dimension 2

• Let $X = \Sigma$ be a closed (oriented) surface and $\nabla^0$ a normal phase solution.

• Let $\lambda_1 =$ smallest eigenvalue of $(\nabla^0)^* \nabla^0 = \underbrace{\tfrac{2 \pi |d_{\mathcal{L}}|}{\mathrm{Area} (\Sigma)}}_{Chern–Weil} = |\underbrace{B_{external}}_{physics}|$. In particular, $\lambda_1 > 0$, unless $\mathcal{L}$ is trivial (zero flux).

• Theorem [Existence; N '18]:
If $\tau \kappa^2 > \lambda_1$, then there are nontrivial (irreducible) solutions to the Ginzburg–Landau equations.

• "Proof":
1. Prove that the Ginzburg–Landau energy is (gauged) Palais–Smale, and thus there are global minima.
2. Compute: $\mathrm{Hess} \left( \mathcal{E} \right)_{(\nabla^0, 0)} (0, \psi) = \langle \psi | \left( (\nabla^0)^* \nabla^0 - \kappa^2 \tau \mathbb{I} \right) \psi \rangle$
3. $\Rightarrow \kappa^2 \tau > \lambda_1$ makes normal phases unstable, so it's not them. ■

# Nonexistence of absolute minimizers in dimension 2

• Let $X = \Sigma$ be a closed (oriented) surface and $\nabla^0$ a normal phase solution.

• Let $\lambda_1 =$ smallest eigenvalue of $(\nabla^0)^* \nabla^0 = \underbrace{\tfrac{2 \pi |d_{\mathcal{L}}|}{\mathrm{Area} (\Sigma)}}_{Chern–Weil} = |\underbrace{B_{external}}_{physics}|$. In particular, $\lambda_1 > 0$, unless $\mathcal{L}$ is trivial (zero flux).

• Theorem [Nonexistence; N '18]:
If $\tau \leqslant \lambda_1 \min \left\{ 2, \kappa^{- 2} \right\}$, then there are no nontrivial solutions to the Ginzburg–Landau equations.

• "Proof":
1. Compute $\Delta (\tau - |\phi|^2)$ + Maximum (well... Minimum) Principle, and get $|\phi|^2 < \tau$.
2. Then compute $\Delta |F|$ + Maximum Principle again, and get \begin{equation} 2 |F| + |\phi|^2 \leqslant \tau \max \left\{ 1, 2 \kappa^2 \right\}. \end{equation} (cf. Taubes CMP '80).
3. Finally, integrate + Chern–Weil. ■

• Recall the previous theorem: If $\tau \kappa^2 > \lambda_1$, then there are nontrivial solutions to the Ginzburg–Landau equations.

• Corollary [Critical and Type II SC's; N '18]:
When $\kappa \geqslant \kappa_c$, then there are nontrivial solutions to the Ginzburg–Landau equations exactly when $\tau \kappa^2 > \lambda_1$.

# Solutions on closed manifolds

• When $\kappa = \kappa_c = \tfrac{1}{\sqrt{2}}$ and $X$ is a surface, then absolute minimizers solve the (anti-)vortex equation: \begin{equation} \left( \nabla_x \pm i \nabla_y \right) \phi = 0, \quad \mbox{and} \quad F_{xy} = \pm \tfrac{1}{2} \left( \tau - |\phi|^2 \right). \end{equation} The vortex equations have solutions exactly when $\tau > \tau_{Bradlow}$ (Bradlow CMP '90).

• The vortex equations generalize to higher dimensional Kähler manifolds (but still under $\kappa = \kappa_c = \tfrac{1}{2}$).

• Pigati–Stern; Invent. Math. '20:
Solution on trivial bundles (zero total magnetic flux).
Their method is topological (uses $\pi_2 (\mbox{configuration space}) \neq \{ 0 \}$):
1. They proved that the Ginzburg–Landau energy is (gauged) Palais–Smale in higher dimensions as well.
2. Then used a mountain-pass argument.
3. These solutions are necessarily irreducible and unstable. First such examples that I know of.
4. Note: on $\mathbb{R}^2$ this cannot happen! (Taubes CMP '80)
5. They used these solutions to construct codimension 2 (stationary, integral) subvarieties.

• Back in dimension 2, but on nontrivial bundles:
• Chouchkov, Ercolani, Rayan, and Sigal constructed solutions using the Lyapunov–Schimdt reduction (CERS; Poincaré C '20).
• We generalized their argument to work in more cases (and to higher dimensions).

# Lyapunov–Schimdt reduction for the Ginzburg–Landau equations I.

• Fix a normal phase $\nabla^0$ and $\kappa > 0$, and impose Coulomb gauge ($\mathrm{d}^* A = 0$).

• Let $\mathcal{F} (A, \phi, \tau) = \begin{pmatrix} \mathrm{d}^* F_{\nabla^0 + A} - i \: \mathrm{Im} \left( h \left( (\nabla^0 + A) \phi, \phi \right) \right) \\ (\nabla^0 + A)^* (\nabla^0 + A) \phi - \kappa^2 \left( \tau - |\phi|^2 \right) \phi \end{pmatrix}$$\: = \begin{pmatrix} \Delta_{\mathrm{dR}} A \\ (\nabla^0)^* (\nabla^0) \phi - \kappa^2 \tau \phi \end{pmatrix} +$ L. O. T.

• $\mathcal{F} (A, \phi, \tau) = 0$ is equivalent to the Ginzburg–Landau equations, and for all $\tau$, $\mathcal{F} (0, 0, \tau) = 0$ (normal phases).

• Construct "nearby" solutions through the Implicit Function Theorem:
Obstruction: $\mathcal{K}_\tau = \ker \left( D_{A, \phi} \mathcal{F} (0, 0, \tau) \right) = \{ \mbox{ Harmonic 1-forms } \} \oplus \ker \left( (\nabla^0)^* \nabla^0 - \kappa^2 \tau \mathbb{I} \right)$.

• Harmonic 1-forms are just changes in the normal phase
$\Rightarrow$ interesting bifurcations happen when $\kappa^2 \tau \in \mathrm{Spec} \left( (\nabla^0)^* \nabla^0 \right)$.

# Lyapunov–Schimdt reduction for the Ginzburg–Landau equations II.

• Let $\mathcal{F} (A, \phi, \tau) = \begin{pmatrix} \mathrm{d}^* F_{\nabla^0 + A} - i \: \mathrm{Im} \left( h (\nabla \phi, \phi) \right) \\ (\nabla^0 + A)^* (\nabla^0 + A) \phi - \kappa^2 \left( \tau - |\phi|^2 \right) \phi \end{pmatrix} = \begin{pmatrix} \Delta A \\ (\nabla^0)^* (\nabla^0) \phi - \kappa^2 \tau \phi \end{pmatrix} +$ L. O. T.

• $\mathcal{K}_\tau = \ker \left( D_{A, \phi} \mathcal{F} (0, 0, \tau) \right) = \{ \mbox{ Harmonic 1-forms } \} \oplus \ker \left( (\nabla^0)^* \nabla^0 - \kappa^2 \tau \mathbb{I} \right)$.

• Let $\lambda \in \mathrm{Spec} \left( (\nabla^0)^* \nabla^0 \right)$. Now write $\tau = \tfrac{\lambda}{\kappa^2} + \epsilon$, $A = A_{Harm} + b$ and $\phi = \Phi_\lambda + \Psi$, and then \begin{equation} \mathcal{F}^\parallel = \Pi_{\mathcal{K}_{\lambda/\kappa^2}} \mathcal{F}, \quad \& \quad \mathcal{F}^\perp = \left(\mathbb{I} - \Pi_{\mathcal{K}_{\lambda/\kappa^2}} \right) \mathcal{F}. \end{equation}

• Lyapunov–Schimdt reduction:
Implicit Function Theorem can be used for $(b, \Psi)$ and $\mathcal{F}^\perp$:
$\Rightarrow$ for small $\epsilon, A_{Harm}, \Phi_\lambda$, $\exists ! (b, \Psi)$, such that $\mathcal{F}^\perp = 0$.

• Need to solve $\mathcal{F}^\parallel (\epsilon, A_{Harm}, \Phi_\lambda) = 0 \in \mathcal{K}$, with $(\epsilon, A_{Harm}, \Phi_\lambda) \in \mathbb{R} \times \mathcal{K}$. $\leftarrow$ Finite dimensional equation!
• Theorem [N–Oliveira '21]:
Assume that $X$ is an $n$-dimensional, closed, oriented, Riemannian manifold, and either of the following cases hold:

1. $X$ has trivial first de Rham cohomology.

2. $X$ is Kähler, $\nabla^0$ integrable ($F_\nabla^{0, 2} = 0$), $\mathcal{L}$ carries nontrivial holomorphic sections with respect to $\nabla^0$, and $\lambda = \min \left( \mathrm{Spec} \left( \left( \nabla^0 \right)^* \nabla^0 \right) \right) = \tfrac{2 \pi}{\mathrm{Vol} (X, g)} \left| \left( c_1 (\mathcal{L}) \cup [\omega]^{n - 1} \right) [X] \right|$.

Then there exists $\Phi_r \in \ker \left( (\nabla^0)^* \nabla^0 - \lambda \mathbb{I} \right)$ of unit $L^2$-norm and a branch of triples \begin{equation} \left( A_r, \phi_r, \epsilon_r \right) = \left( A_r r^2 + o \left( r^2 \right), \Phi_r r + \Psi_r r^3 + o \left( r^3 \right), \epsilon_0 r^2 + o \left( r^2 \right) \right), \end{equation} such that for each $r \in [0, r_0)$ the pair $\left( \nabla^0 + A_r, \phi_r \right)$ solves the Ginzburg–Landau equations with $\tau = \tfrac{\lambda}{\kappa^2} + \epsilon_r$.
Furthermore, $\Phi_r$ determines $A_0$, $\Psi_0$, and $\epsilon_0$ explicitly.

• "Proof": Only need to solve $\mathcal{F}^\parallel (A_{Harm}, \Phi_\lambda, \epsilon) = 0$.
1. Hypotheses 1. and 2. are there to make sure that we do not need to worry about $A_{Harm}$
$\: \Rightarrow \: \mathcal{F}^\parallel$ can now be viewed as a map from $\mathbb{R}^{n + 1}$ to $\mathbb{R}^n$ where $n = \dim_{\mathbb{R}} \left( \ker \left( (\nabla^0)^* \nabla^0 - \lambda \mathbb{I} \right) \right) = 2d$ is even.
2. Prove: we can pick $\epsilon = \epsilon (\Phi)$ so that $\Phi \perp \mathcal{F}^\parallel$
$\: \Rightarrow \: \mathcal{F}^\parallel$ can now be viewed as a vector field on $S^{n - 1}$ (for each small sphere around the origin).
(CERS: when $n = 2$, then $S^{2 - 1} = S^1 = \simeq U (1)$ is just gauge, and we're done!)
Problem: $n = 2d$ is even, thus this does not guarantee a zero in higher dimensions.
3. Consider $S^{2d - 1} \subset \mathbb{C}^d$. With a little more work we show that $\mathcal{F}^\parallel$ descends to a vector field on $\mathbb{CP}^{d - 1}$.
Poincaré–Hopf: $\chi (\mathbb{CP}^{d - 1}) = d \neq 0 \: \Rightarrow$ we have a zero!!!
4. The rest is straightforward. ■

# Conjecture [N–Oliveira]

Assume that $X$ is an $n$-dimensional, closed, oriented, Riemannian manifold and $\lambda \in \mathrm{Spec} \left( (\nabla^0)^* \nabla^0 \right)$.
Then for all $\Phi_0 \in \ker \left( (\nabla^0)^* \nabla^0 - \lambda \mathbb{I} \right)$ of unit $L^2$-norm, there exists a branch of triples \begin{equation} \left( A_r, \phi_r, \epsilon_r \right) = \left( A_0 r^2 + O \left( r^4 \right), \Phi_0 r + \Psi_0 r^3 + O \left( r^5 \right), \epsilon_0 r^2 + O \left( r^4 \right) \right), \end{equation} such that for each $r \in [0, r_0)$ the pair $\left( \nabla^0 + A_r, \phi_r \right)$ solves the Ginzburg–Landau equations with $\tau = \tfrac{\lambda}{\kappa^2} + \epsilon_r$.
Furthermore, $\Phi_0$ determines $A_0$, $\Psi_0$, and $\epsilon_0$ explicitly.

# ...might also be true:

\begin{equation} \mathcal{M}_{\tau, \kappa} = \{ \mbox{ Ginzburg–Landau fields with } \tau, \kappa \ \} / \mbox{gauge} = \bigcup\limits_{\lambda \in \mathrm{Spec} \left( (\nabla^0)^* \nabla^0 \right) \cap [0, \kappa^2 \tau]} \mathcal{X}_{\lambda, \tau}, \end{equation} where $\mathcal{X}_{\lambda, \tau}$ is the space of solutions that bifurcates when passing the eigenvalue $\lambda$.

# Remarks

• The error terms are a weaker than in CERS.
We hope to improve them soon.

• The zeros of $\mathcal{F}^\parallel$ actually classifies the "nearby" solutions.
We expect that $\mathcal{F}^\parallel$ vanishes identically.

• The hypotheses on the manifolds are not expected to be needed either.

• Surfaces of constant Gauß curvature: can use Almorox–Prieto JG&P '06 to extend the result.

• We also proved:
• The existence of nonminimal solutions even when $\kappa^2 \tau$ is not near an eigenvalue.
(through topology of the configuration space)
• The instability of nonvortex solutions.

• Related direction: (Morse)-index + Nullity estimates, motivated by the work of Kelleher, Gursky, and Streets CMP '20.

• Potential application to codimension 2 minimal subvarieties, à la Pigati and Stern.