Cuplength estimates for periodic solutions of Hamiltonian particle-field systems

Abstract

We consider a natural class of time-periodic infinite-dimensional nonlinear Hamiltonian systems modelling the interaction of a classical mechanical system of particles with a scalar wave field. When the field is defined on a space torus ${\mathbb {T}}^d={\mathbb {R}}^d/(2\pi {\mathbb {Z}})^d$ and the coordinates of the particles are constrained to a submanifold $Q\subset {\mathbb {T}}^d$, we prove that the number of contractible T-periodic solutions of the coupled Hamiltonian particle-field system is bounded from below by the ${\mathbb {Z}}_2$-cuplength of the space $\Lambda ^{{\text {contr}}} Q$ of contractible loops in Q, provided that the square of the ratio $T/2\pi$ of time period T and space period $X=2\pi$ is a Diophantine irrational number. The latter condition is necessary since for the infinite-dimensional version of Gromov–Floer compactness as well as for the $C^0$-bounds we need to deal with small divisors.

1 A simple particle-field system

As a motivation for the general class of Hamiltonian particle-field systems that we introduce below, we start with the following simple model of a classical particle-field system describing the interaction of a particle with its self-generated scalar wave field. After imposing periodicity conditions in space, we consider a (time-dependent) scalar field $\varphi (t,x)\in {\mathbb {R}}$, $x\in {\mathbb {T}}^d={\mathbb {R}}^d/(2\pi {\mathbb {Z}})^d$, as well as a particle whose center locus q(t) is constrained to a closed submanifold $Q\subset {\mathbb {T}}^d$. While the case $d>3$ only becomes relevant if we also allow for mechanical systems with more than one particle, we will ambiguously speak of a single particle. We assume that q(t) and $\varphi (t,x)$ satisfy the following coupled system of differential equations,

$$\begin{aligned} \nabla _t^2 q(t)= & {} - \nabla V(t,q(t))-\nabla (\varphi *\rho )(t,q(t)),\\ \partial _t^2 \varphi (t,x)= & {} \Delta \varphi (t,x)-\kappa \cdot \varphi (t,x)-\rho (x-q(t)). \end{aligned}$$

Here $\Delta =\partial _{x_1}^2+\cdots +\partial _{x_d}^2$ denotes the Laplacian for functions on ${\mathbb {T}}^d$, while $\nabla$ denotes the gradient for functions on $Q\subset {\mathbb {T}}^d$ as well as the Levi–Civita connection on the tangent bundle TQ of Q with respect to the induced canonical Riemannian metric. Furthermore,

$$\begin{aligned} (\varphi *\rho )(t,q)\,=\,\int _{{\mathbb {T}}^d} \varphi (t,x)\cdot \rho (q-x)\,\textrm{d}x \end{aligned}$$

denotes convolution with respect to the space coordinate $x\in {\mathbb {T}}^d$ with a fixed bump function $\rho \in C^\infty ({\mathbb {T}}^d,{\mathbb {R}})$ which models the particle-field interaction and thereby takes into account the shape and density of the particle, and is symmetric in the sense that $\rho (-x)=\rho (x)$, $x\in {\mathbb {T}}^d$. We emphasize that all studied models for particle-field interaction such as in [3, 13, 18] use a smooth bump function $\rho$ modelling a realistic particle instead of the $\delta$-distribution modelling a point particle to avoid problems with singularities. In the Klein–Gordon equation for $\varphi$ the constant $\kappa$ stands for an arbitrarily fixed positive real number; for notational reasons for the rest of this paper we simply put $\kappa =1$ and claim that everything generalizes to the case of arbitrary $\kappa >0$. Finally $V\in C^\infty ({\mathbb {R}}/(T{\mathbb {Z}})\times {\mathbb {T}}^d,{\mathbb {R}})$, $V_t(x)=V(t,x)$ denotes an external field on $(Q\subset ) {\mathbb {T}}^d$ which is explicitly assumed to be time-dependent with time period T.

Introducing a time-dependent momentum variable $p(t)\in T^*_{q(t)} Q$ and time-dependent momentum field $\pi (t,x)\in {\mathbb {R}}$, $x\in {\mathbb {T}}^d$, it is immediate to see that the above coupled system of differential equations is equivalent to the coupled system of differential equations

$$\begin{aligned} \partial _t q = p,&\nabla _t p= -\nabla V_t(q)-\nabla (\varphi *\rho )(q),\nonumber \\ \partial _t \varphi = \pi ,&\partial _t \pi = \Delta \varphi -\varphi -\rho (q-\cdot ). \end{aligned}$$

(1)

Note that here the induced Riemannian metric on Q is also used to canonically identify $T^*Q\cong TQ$.

It is well-known that the pair of equations

$$\begin{aligned} \partial _t q = p,\quad \nabla _t p= -\nabla V_t(q) \end{aligned}$$

for (q(t), p(t)) is Hamiltonian for the Hamiltonian function

$$\begin{aligned} H_{{\text {part}}}(q,p)=\frac{|p|^2}{2}+V_t(q) \end{aligned}$$

on the phase space $T^*Q$, which is a finite-dimensional symplectic manifold. On the other hand, it is also a classical fact that the pair of equations

$$\begin{aligned} \partial _t \varphi = \pi ,\quad \partial _t \pi = \Delta \varphi -\varphi \end{aligned}$$

for $(\varphi (t,x),\pi (t,x))$ is Hamiltonian for the Hamiltonian function

$$\begin{aligned} H_{{\text {field}}}(\varphi ,\pi )=\int _{{\mathbb {T}}^d} \left( \frac{|\pi (x)|^2}{2}+\frac{|\nabla \varphi (x)|^2}{2}+\frac{|\varphi (x)|^2}{2}\right) \,\textrm{d}x \end{aligned}$$

on the phase space $L^2({\mathbb {T}}^d,{\mathbb {R}})\oplus L^2({\mathbb {T}}^d,{\mathbb {R}})$. Note that $L^2({\mathbb {T}}^d,{\mathbb {R}})\oplus L^2({\mathbb {T}}^d,{\mathbb {R}})$ is an infinite-dimensional symplectic Hilbert space, where the symplectic form is given by the $L^2$ inner product as well as the complex structure $J\cdot (\varphi ,\pi )=(\pi ,-\varphi )$. Taking the particle-field interactions into account, we stress that the coupled system of differential equations is now a Hamiltonian system on the infinite-dimensional symplectic manifold $T^*Q\times L^2({\mathbb {T}}^d,{\mathbb {R}})\oplus L^2({\mathbb {T}}^d,{\mathbb {R}})$ for the Hamiltonian function

$$\begin{aligned} H=H_{{\text {part}}}+H_{{\text {field}}}+H_{{\text {inter}}} \end{aligned}$$

with

$$\begin{aligned} H_{{\text {inter}}}(q, \varphi ,\pi )=(\varphi *\rho )(q)=\int _{{\mathbb {T}}^d}\varphi (x)\rho (q-x)\,\textrm{d}x \end{aligned}$$

denoting the Hamiltonian function modelling the interaction between particle and field.

While this set-up already seems promising, there is an alternative Hamiltonian structure for the Klein–Gordon equation which is more symmetric and turns out to be more suitable for the analysis, see [11, 12]. From now on we will consider the modified coupled system

$$\begin{aligned} \partial _t q = p,&\nabla _t p= -\nabla V_t(q)-\nabla (\varphi *\rho )(q)\\ \partial _t \varphi = B\pi ,&\partial _t \pi = - B\varphi -B^{-1}\rho (q-\cdot ). \end{aligned}$$

with $B=\sqrt{1-\Delta }$. While this modified system is still Hamiltonian, the new phase space is $T^*Q\times {\mathbb {H}}$ with ${\mathbb {H}}=H^{\frac{1}{2}}({\mathbb {T}}^d,{\mathbb {R}})\oplus H^{\frac{1}{2}}({\mathbb {T}}^d,{\mathbb {R}})$, where the symplectic form on ${\mathbb {H}}$ is again given by the standard complex structure $J\cdot (\varphi ,\pi )=(-\pi ,\varphi )$ and the standard inner product $\langle \cdot ,\cdot \rangle$ on $H^{\frac{1}{2}}({\mathbb {T}}^d,{\mathbb {R}})$ given by

$$\begin{aligned} \langle f,g\rangle =\int _{{\mathbb {T}}^d} f(x) (Bg)(x)\,\textrm{d}x. \end{aligned}$$

While it can be observed that the interaction Hamiltonian can remain unchanged,

$$\begin{aligned} H_{{\text {inter}}}(q,\varphi ,\pi )=(\varphi *\rho )(q)=\langle \varphi ,B^{-1}\rho (q-\cdot )\rangle , \end{aligned}$$

the field Hamiltonian is now changed to the more symmetric form

$$\begin{aligned} H_{{\text {field}}}(\varphi ,\pi )=\frac{1}{2}\langle \varphi ,B\varphi \rangle +\frac{1}{2}\langle \pi ,B\pi \rangle . \end{aligned}$$

2 Hamiltonian particle-field systems

For more details and background on Hamiltonian PDEs we refer to [12], see also [1, 11]. We start by recalling that ${\mathbb {H}}=H^{\frac{1}{2}}({\mathbb {T}}^d,{\mathbb {R}})\oplus H^{\frac{1}{2}}({\mathbb {T}}^d,{\mathbb {R}})$ is a separable Hilbert space which is equipped with a (strongly) symplectic form $\omega _{{\mathbb {H}}}=\langle J_{{\mathbb {H}}}\cdot ,\cdot \rangle _{{\mathbb {H}}}:{\mathbb {H}}\times {\mathbb {H}}\rightarrow {\mathbb {R}}$ given by the standard complex structure $J_{{\mathbb {H}}}\cdot (\varphi ,\pi )=(-\pi ,\varphi )$ and the standard inner product $\langle \cdot ,\cdot \rangle _{{\mathbb {H}}}$ on $H^{\frac{1}{2}}({\mathbb {T}}^d,{\mathbb {R}})\oplus H^{\frac{1}{2}}({\mathbb {T}}^d,{\mathbb {R}})$. Here being (strongly) symplectic means that $\omega$ is anti-symmetric and defines an isomorphism between ${\mathbb {H}}$ and its dual ${\mathbb {H}}^*$. There exists a Hilbert basis $(e_n^\pm )_{n\in {\mathbb {Z}}^d}$ of ${\mathbb {H}}$ with $\omega _{{\mathbb {H}}}(e_n^+,e_m^-)=\delta _{n,m}$, $J_{{\mathbb {H}}}e_n^\pm :=\pm e_n^\mp$ of the form $e_n^+=(\xi _n,0)$, $e_n^-=(0,\xi _n)$, where $(\xi _n)_{n\in {\mathbb {Z}}^d}$ is a suitably normalized Hilbert basis of $H^{\frac{1}{2}}({\mathbb {T}}^d,{\mathbb {R}})$ in terms of sine and cosine functions. Furthermore, after identifying ${\mathbb {H}}$ with the subspace of ${\mathbb {H}}\otimes {\mathbb {C}}$ on which $J_{{\mathbb {H}}}=i$, note that there is a complete unitary basis $(z_n)_{n\in {\mathbb {Z}}^d}$ which further allows us to identify ${\mathbb {H}}$ with the Hilbert space $\ell ^2({\mathbb {Z}}^d,{\mathbb {C}})$.

The symplectic Hilbert space ${\mathbb {H}}=H^{\frac{1}{2}}({\mathbb {T}}^d,{\mathbb {R}})\oplus H^{\frac{1}{2}}({\mathbb {T}}^d,{\mathbb {R}})$ naturally comes equipped with a symplectic Hilbert scale $({\mathbb {H}}_h)_{h\in {\mathbb {R}}}$ with

$$\begin{aligned} {\mathbb {H}}_h=H^{\frac{1}{2}+h}({\mathbb {T}}^d,{\mathbb {R}})\oplus H^{\frac{1}{2}+h}({\mathbb {T}}^d,{\mathbb {R}}) \end{aligned}$$

in the sense that the inclusion ${\mathbb {H}}_h\subset {\mathbb {H}}_i$ is compact and dense if $h>i$, and $\omega _{{\mathbb {H}}}$ defines an isomorphism between the Hilbert spaces ${\mathbb {H}}_h$ and ${\mathbb {H}}_{-h}^*$. Furthermore we define

$$\begin{aligned} {\mathbb {H}}_\infty =\bigcap _{h\in {\mathbb {R}}}{\mathbb {H}}_h,\,\, {\mathbb {H}}_{-\infty } = \bigcup _{h\in {\mathbb {R}}}{\mathbb {H}}_h. \end{aligned}$$

In particular, we stress that $\omega _{{\mathbb {H}}}:{\mathbb {H}}_\infty \rightarrow {\mathbb {H}}_\infty ^*$ is only injective and hence $\omega _{{\mathbb {H}}}$ only defines a weakly symplectic form on the Frechet space${\mathbb {H}}_\infty =C^{\infty }({\mathbb {T}}^d,{\mathbb {R}})\oplus C^{\infty }({\mathbb {T}}^d,{\mathbb {R}})$.

The above setup immediately generalizes from ${\mathbb {H}}$ to $\widetilde{M}:=M\times {\mathbb {H}}$, where $(M,\omega _M)=(T^*Q,d\lambda _M)$ is the cotangent bundle of the closed manifold Q with its canonical symplectic form given by the Liouville one-form $\lambda _M$. We assume that $\widetilde{M}$ carries the product symplectic form $\omega =\pi _M^*\omega _M+\pi _{{\mathbb {H}}}^*\omega _{{\mathbb {H}}}$, where $\pi _M$, $\pi _{{\mathbb {H}}}$ denote the projection onto the first or second factor of $\widetilde{M}$, respectively. Note that the Riemannian metric on the submanifold Q, obtained from the embedding into ${\mathbb {T}}^d$, leads to a natural choice of Riemannian metric $\langle \cdot ,\cdot \rangle _M$ on $T^*Q$, defined using the Levi–Civita connection on $T^*M\cong TM$, with corresponding $\omega _M$-compatible almost complex structure $J_M$, see [5] for details. It follows that $\omega =\langle J\cdot ,\cdot \rangle$ for the canonical Riemannian metric $\langle \cdot ,\cdot \rangle =\pi _M^*\langle \cdot ,\cdot \rangle _M+\pi _{{\mathbb {H}}}^*\langle \cdot ,\cdot \rangle _{{\mathbb {H}}}$ and the canonical almost complex structure $J={\text {diag}}(J_M,J_{{\mathbb {H}}})$ on $\widetilde{M}=T^*Q\times {\mathbb {H}}$. Furthermore the generalization of the scale structure is given by $\widetilde{M}_h:=M\times {\mathbb {H}}_h$ for $h\in {\mathbb {R}}$.

As in [7, 9] we consider a class of time-dependent Hamiltonians $H_t$ of the form

$$\begin{aligned} H_t(u)=H^A(u)+F_t(u)\,\,\text {with}\,\,H^A(u)=\frac{1}{2}\langle \pi _{{\mathbb {H}}}u,A \pi _{{\mathbb {H}}}u\rangle ,\,\,u=(q,p,\varphi ,\pi ), \end{aligned}$$

where A is a differential operator on ${\mathbb {H}}$ such that $(e_n^\pm )_{n\in {\mathbb {Z}}^d}$ is a basis of eigenvectors for A with real eigenvalues, and $F_t: \widetilde{M}\rightarrow {\mathbb {R}}$ is time-periodic with period T and smoothly depending on the time $t\in {\mathbb {R}}$. Since in this paper we are particularly interested in particle-field systems, we restrict the general class of Hamiltonians to the case where

$$\begin{aligned} H^A(u)= & {} H_{{\text {field}}}(\varphi ,\pi )\,\,\text {with}\,\, H_{{\text {field}}}(\varphi ,\pi )=\frac{1}{2}\langle \varphi ,B\varphi \rangle +\frac{1}{2}\langle \pi ,B\pi \rangle \,\,\text {as before},\\ F_t(u)= & {} F_{{\text {part}},t}(q,p)+F_{{\text {inter}},t}(q,\varphi ,\pi ). \end{aligned}$$

Note that in this case $A={\text {diag}}(B, B)$ is of order 1 and the eigenvalue corresponding to the eigenfunction $e_n^{\pm }$ is $\sqrt{n^2+1}$ with $n^2=n_1^2+\cdots +n_d^2$ for all $n=(n_1,\ldots ,n_d)\in {\mathbb {Z}}^d$. As in [5] we assume that the particle Hamiltonian $F_{{\text {part}},t}\in C^{\infty }(T^*Q,{\mathbb {R}})$ is asymptotically quadratic with respect to the momentum coordinates p in the sense that

• (F1) $dF_{{\text {part}},t}(q,p)\cdot p\frac{\partial }{\partial p} - F_{{\text {part}},t}(q,p)\ge c_0 |p|^2 - c_1$, for some constants $c_0>0$ and $c_1\ge 0$,

• (F2) $\displaystyle \left| \frac{\partial ^2 F_{{\text {part}},t}}{\partial p_i\partial p_j}(q,p)\right| ,\,\, \left| \frac{\partial ^2 F_{{\text {part}},t}}{\partial p_i\partial q_j}(q,p)\right| < c_2$ for some constant $c_2\ge 0$,

while for the interaction Hamiltonian $F_{{\text {inter}},t}(q,\varphi ,\pi )$ we require that

• (F3) $F_{{\text {inter}},t}(q,\varphi ,\pi )=f_t((\varphi *\rho )(q),(\pi *\rho )(q))$ with $f_t\in C^{\infty }({\mathbb {R}}^2,{\mathbb {R}})$, $f_{t+T}=f_t$ having bounded first derivatives.

As in [5] $(q_i,p_i)_i$ are coordinates on $T^*Q$ induced by geodesic normal coordinates $(q_i)$ on Q. Finally we assume without loss of generality that all frequencies are present in $\rho =\sum _{n\in {\mathbb {Z}}^d}\hat{\rho }(n)z_n$ in the sense that $\hat{\rho }(n)\ne 0$ for all $n\in {\mathbb {Z}}^d$.

In analogy with our work in [7, 9], it is the goal of this paper to prove the existence of contractible T-periodic solutions of

$$\begin{aligned} \partial _t u\,=\,X^H_t(u)\,=\,JA\pi _{{\mathbb {H}}}u\;+\;J\nabla F_t(u). \end{aligned}$$

Before we can state the main result, we recall that an irrational number $\sigma$ is called Diophantine if there exists $c>0$ and $r>0$ such that

$$\begin{aligned} \inf _{m\in {\mathbb {Z}}}\Big |\sigma -\frac{m}{n}\Big |\,\ge \, c\cdot n^{-r}\,\,\text {for all}\,\, n\in {\mathbb {N}}. \end{aligned}$$

We stress that the set of Diophantine numbers has full measure. Furthermore the ${\mathbb {Z}}_2$-cuplength $c\ell _{{\mathbb {Z}}_2}(\Lambda ^{{\text {contr}}}Q)$ of the space $\Lambda ^{{\text {contr}}}Q=C^0_{{\text {contr}}}({\mathbb {R}}/(T{\mathbb {Z}}),Q)$ of contractible loops in Q is defined as

$$\begin{aligned} c\ell _{{\mathbb {Z}}_2}(\Lambda ^{{\text {contr}}}Q){} & {} =\sup \{N+1:\exists \theta _1,\ldots \theta _N\in H^{*\ne 0}(\Lambda ^{{\text {contr}}}Q,{\mathbb {Z}}_2):\\{} & {} \quad \theta _1\cup \cdots \cup \theta _N\ne 0\}. \end{aligned}$$

Theorem 2.1

Assume that the squared ratio $T^2/(2\pi )^2$ of time and space period is a Diophantine irrational number, $F_{{\text {part}},t}$ satisfies (F1), (F2), and $F_{{\text {inter}},t}$ satisfies (F3). Then the number of contractible T-periodic solutions

$$\begin{aligned} u=(q,p,\varphi ,\pi ):{\mathbb {R}}/(T{\mathbb {Z}})\rightarrow T^*Q\times H^{\frac{1}{2}}({\mathbb {T}}^d,{\mathbb {R}})\oplus H^{\frac{1}{2}}({\mathbb {T}}^d,{\mathbb {R}}) \end{aligned}$$

of the coupled system of Hamiltonian equations

$$\begin{aligned} \partial _t q= & {} \nabla _p F_{{\text {part}},t},\\ \nabla _t p= & {} -\nabla _q F_{{\text {part}},t} - \partial _1 f_t\cdot \nabla (\varphi *\rho )(q) - \partial _2 f_t\cdot \nabla (\pi *\rho )(q),\\ \partial _t \varphi= & {} B\pi + \partial _2 f_t\cdot B^{-1}\rho (q-\cdot ),\\ \partial _t \pi= & {} - B\varphi - \partial _1 f_t\cdot B^{-1}\rho (q-\cdot ) \end{aligned}$$

is bounded from below by $c\ell _{{\mathbb {Z}}_2}(\Lambda ^{{\text {contr}}} Q)$; in particular, it is infinite if $\pi _1(Q)$ is finite. Furthermore u has image in the weakly symplectic manifold $T^*Q\times C^\infty ({\mathbb {T}}^d,{\mathbb {R}})\oplus C^\infty ({\mathbb {T}}^d,{\mathbb {R}})$.

Here $\nabla _qF_{{\text {part}},t},\;\nabla _pF_{{\text {part}},t}$ denote the components of the gradient$\nabla F_{{\text {part}},t}$ with respect to the splitting $TT^*Q\cong TQ\oplus T^*Q$ obtained using the Levi–Civita connection for the canonical Riemannian metric on $T^*Q$, and $\partial _{1,2}f_t$ denote the first derivatives of $f_t$ with respect to the first and second coordinate. Further recall that $B=\sqrt{1-\Delta }$.

We emphasize that Theorem 2.1 generalizes the celebrated cuplength result for Hamiltonian systems on cotangent bundles in [5] to the case of Hamiltonian particle-field systems: When $F_{{\text {inter}},t}=0$, then our result is equivalent with [5, Main Theorem 1.1], since the linear Klein–Gordon equation admits only the trivial T-periodic solution in the case when $T^2/(2\pi )^2$ is irrational. Apart from the fact that we now consider an infinite-dimensional Hamiltonian system with a densely defined Hamiltonian function and a small divisor problem, it is a nontrivial observation that we can still establish $C^0$-bounds.

For the latter the trick is to replace the original interaction Hamiltonian $F_{{\text {inter}},t}$ by a slightly modified version $\bar{F}_{{\text {inter}},t}$ such that the resulting modified particle-field Hamiltonian system however still has the same set of periodic orbits. This is the content of Sect. 3. As for the Gromov–Floer compactness theorem in infinite dimensions, we need to deal with small divisors and crucially use that the bump function $\rho$ is smooth.

Remark 2.2

We expect that Theorem 2.1 can be generalized in the following directions:

• (1) By inspecting the proof given below it becomes apparent that the statement of Theorem 2.1 continues to hold true for interaction Hamiltonians $F_{{\text {inter}},t}$ which additionally depend on the p-component and the condition (F3) is generalized to (F3’): $F_{{\text {inter}},t}(q,p,\varphi ,\pi )=f_t(q,p,(\varphi *\rho )(q),(\pi *\rho )(q))$ with $f_t\in C^{\infty }(T^*Q\times {\mathbb {R}}^2,{\mathbb {R}})$, $f_{t+T}=f_t$ having bounded first derivatives with respect to the third and fourth entry, and $\nabla _p F_{{\text {inter}},t}\equiv 0$ for $|p|>R$ for some $R>0$. To illustrate the nontriviality of the problem of establishing $C^0$-bounds, note that we currently do not know how to extend our result to the case when the magnetic Lorentz force would be included.

• (2) Since this paper builds on the work of Cieliebak in [5] and not on the work of Abbondandolo-Schwarz in [2], we have chosen our condition (F2) to agree with the original condition (H2) from [5] and not with the more general condition (H2) used in [2]. In the same way as the result in [5] is expected to hold in the generalized setting from [2], we claim that our result generalizes to the case when (F2) agrees with (H2) from [2].

Since the particle-field Hamiltonian of the simple particle-field system that we consider above satisfies (F1), (F2), and (F3), we have the following consequence.

Corollary 2.3

The number of contractible T-periodic solutions $u=(q,\varphi ):{\mathbb {R}}/(T{\mathbb {Z}})\rightarrow Q\times C^\infty ({\mathbb {T}}^d,{\mathbb {R}})$ of the simple particle-field system

$$\begin{aligned} \nabla _t^2 q(t)= & {} - \nabla V(t,q(t))-\nabla (\varphi *\rho )(t,q(t)),\\ \partial _t^2 \varphi (t,x)= & {} \Delta \varphi (t,x)-\varphi (t,x)-\rho (x-q(t)). \end{aligned}$$

is bounded from below by $c\ell _{{\mathbb {Z}}_2}(\Lambda ^{{\text {contr}}} Q)$, provided that the squared ratio $T^2/(2\pi )^2$ of time and space period is a Diophantine irrational number. In particular, it is infinite if $\pi _1(Q)$ is finite.

We claim that this result is already interesting in the case when there is no exterior time-dependent potential V: While in the case without interaction ($\rho =0$) one only can prove the existence of at least one closed geodesic as all periodic orbits could be iterates of the same geodesic, in the case when $\rho \ne 0$ it can be checked that iterates of $t\mapsto q(t)$ no longer lead to solutions $t\mapsto (q(t),\varphi (t))$ of

$$\begin{aligned} \nabla _t^2 q(t)= & {} - \nabla (\varphi *\rho )(t,q(t)),\\ \partial _t^2 \varphi (t,x)= & {} \Delta \varphi (t,x)-\varphi (t,x)-\rho (x-q(t)). \end{aligned}$$

More precisely, it follows from the proof of Lemma 3.1 that a solution $t\mapsto (q(t),\varphi (t))$ could at most lead to a finite number of solutions by taking iterates of $t\mapsto q(t)$, since $\nabla (\varphi *\rho )(t,q(t))$ is bounded uniformly with respect to $(q,\varphi )$.

Furthermore our result indeed crucially relies on the fact that the underlying space ${\mathbb {T}}^d={\mathbb {R}}^d/(2\pi {\mathbb {Z}})^d$ is periodic: Informally speaking, if no space periodicity was assumed, then the energy that was transferred from the mechanical system to the field would disappear towards infinity and hence would have no chance to come back to the particle, see [13] for a negative result for a closely related problem.

Remark 2.4

While Corollary 2.3 follows from Theorem 2.1, we would like to comment on two alternative approaches to prove Corollary 2.3:

• (1) While no convexity assumptions were made on $F_{{\text {part}},t}$ and $F_{{\text {inter}},t}$ in (F2) and (F3) and hence no Legendre transform is available, the simple particle-field system has a Lagrangian formulation and hence time- and space-periodic solutions can be found by looking for critical points of the Lagrangian action functional

  $$\begin{aligned} \mathcal {L}(q,\varphi ){} & {} = \int _0^T \frac{1}{2}|\partial _t q(t)|^2-V(q(t))\,\textrm{d}t\\ {}{} & {} \quad + \int _0^T \int _{{\mathbb {T}}^d} \frac{1}{2}|\partial _t\varphi (t,x)|^2 - \frac{1}{2}|\nabla \varphi (t,x)|^2 -\frac{1}{2}|\varphi (t,x)|^2\,\textrm{d}x\,\textrm{d}t\\ {}{} & {} \quad - \int _0^T \int _{{\mathbb {T}}^d} \varphi (t,x)\cdot \rho (q(t)-x)\,\textrm{d}x\,\textrm{d}t. \end{aligned}$$

  In the case when the second and the third summand involving $\varphi$ are removed, it was first shown by Benci in [4] that the number of critical points of the remaining mechanical Lagrangian action functional is bounded from below by the ${\mathbb {Z}}_2$-cuplength of $\Lambda ^{{\text {contr}}} Q$. On the other hand, when the first and the third summand involving q are removed and the Klein–Gordon equation is generalized to a nonlinear wave equation with specific growth conditions on the nonlinearity, Rabinowitz established in [15] the existence of a critical point of the resulting Lagrangian action functional for $\varphi$. The small divisor problems that we resolve by imposing the Diophantiness condition on $T^2/(2\pi )^2$ are intendendly avoided by Rabinowitz by restricting to the case when the time period T is a rational multiple of $2\pi$. We claim without proof that the adiabatic limit picture in [16] allows to specialize our Floer-theoretic proof of Theorem 2.1 to a Morse-theoretic proof of Corollary 2.3. Furthermore, in complete analogy with Rabinowitz’ original approach, we make use of a modified Hamiltonian function in order to establish $C^0$-bounds for the Floer/Morse trajectories in finite dimensions.

• (2) While in (F3) we only require that $F_{{\text {inter}},t}$ has bounded first derivatives, the interaction Hamiltonian for the simple particle-field system is indeed linear with respect to the field $\varphi$, which in turn allows for yet another alternative approach in this special case. Assuming that q is time-periodic and given a priori, the Klein–Gordon equation

  $$\begin{aligned}\partial _t^2 \varphi (t,x) = \Delta \varphi (t,x)-\varphi (t,x)-\rho (x-q(t))\end{aligned}$$

  has a unique time- and space-periodic solution $\varphi _q=\varphi$, provided that $T^2/(2\pi )^2$ is Diophantine: Indeed, expanding $\varphi _q$ and $\rho _q$ with $\rho _q(t,x)=\rho (q(t)-x)$ into Fourier series, the corresponding Fourier coefficients $\hat{\varphi }_q$, $\hat{\rho }_q$ satisfy the equation

  $$\begin{aligned} \left( \left( \frac{T}{2\pi }\right) ^2-\frac{m^2}{n^2+1}\right) \hat{\varphi }_q(m,n)=\frac{T^2}{(2\pi )^2 (n^2+1)}\hat{\rho }_q(m,n). \end{aligned}$$

  While the first factor is never zero, there is however a small divisor problem in the sense that there exists a subsequence of tuples (m, n) for which it converges to zero. When $T^2/(2\pi )^2$ is Diophantine, then this however only happens with at most polynomial speed in the frequency variable n for x. Since $\rho$ is assumed to be smooth, it follows that $\varphi _q$ is smooth with respect to x, while the regularity of $\varphi _q$ with respect to t agrees with the regularity of q. This in turn suggests to look at the reduced Lagrangian action functional

  $$\begin{aligned} \widetilde{\mathcal {L}}(q) = \int _0^T \frac{1}{2}|\partial _t q(t)|^2-V_q(q(t))\,\textrm{d}t\,\,\text {with}\,\, V_q = V + \varphi _q*\rho . \end{aligned}$$

  on the Hilbert manifold $H^1({\mathbb {R}}/T{\mathbb {Z}},Q)$ of contractible loops in the finite-dimensional manifold Q, and to give a more direct proof of Corollary 2.3 based on [4]. While it is clear that the critical points q of $\widetilde{\mathcal {L}}$ are in one-to-one correspondence with the critical points $(q,\varphi =\varphi _q)$ of $\mathcal {L}$ and we could already clarify the role of the Diophantiness condition, we expect that the Morse-theoretic approaches using $\widetilde{\mathcal {L}}$ and using $\mathcal {L}$ can again be connected via an adiabatic limit approach, as the gradient flow equations for $\mathcal {L}$ and for $\widetilde{\mathcal {L}}$ are obtained by setting $\epsilon =0$ and $\epsilon =1$ in

  $$\begin{aligned}{} & {} \partial _s q(s,t)+\nabla _t^2 q(s,t) = - \nabla V(t,q(s,t))-\nabla (\varphi *\rho )(s,t,q(s,t)),\\{} & {} \epsilon \cdot \partial _s\varphi (s,t,x)+\partial _t^2 \varphi (s,t,x) = \Delta \varphi (s,t,x)-\varphi (s,t,x)-\rho (x-q(s,t)). \end{aligned}$$

  In the case when the convexity assumption is dropped, we stress that this approach also leads to an alternative proof for the special case of Theorem 2.1 when again the strong additional restriction is imposed that the interaction Hamiltonian is actually linear with respect to the fields. While at first sight we now only need to study Floer curves in $T^*Q$, we stress that the underlying time-dependent Hamiltonian function indeed depends in a nonlocal manner on the q-component of the loop $(q,p):{\mathbb {R}}/T{\mathbb {Z}}\rightarrow T^*Q$. We plan to discuss this nonlocal finite-dimensional Floer-theoretic approach and the adiabatic limit that connects it with the infinite-dimensional Floer-theoretic proof for the general case in future work.

We would like to thank the anonymous referee for the valuable comments and suggestions that helped improving the quality of this paper. Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

3 Properties of the particle-field Hamiltonian

To prove Theorem 2.1, we essentially combine our work in [7] and [9] on pseudoholomorphic curves for infinite-dimensional Hamiltonian systems with the celebrated paper [5] on pseudoholomorphic curves in cotangent bundles to prove an existence result for Floer curves in $T^*Q\times {\mathbb {H}}$.

Apart from the fact that the underlying phase space $\widetilde{M}=T^*Q\times {\mathbb {H}}$ is infinite-dimensional, as in [7, 9] we first need to deal with the fact that the field Hamiltonian

$$\begin{aligned} H^A(u)=\frac{1}{2}\langle u_{{\mathbb {H}}}, Au_{{\mathbb {H}}}\rangle =\frac{1}{2}\langle \varphi ,B\varphi \rangle +\frac{1}{2}\langle \pi ,B\pi \rangle \,\,\text {with}\,\,u_{{\mathbb {H}}}=\pi _{{\mathbb {H}}}u=(\varphi ,\pi ) \end{aligned}$$

is only defined on the dense subspace ${\mathbb {H}}_{\frac{1}{2}}=H^1({\mathbb {T}}^d,{\mathbb {R}})\oplus H^1({\mathbb {T}}^d,{\mathbb {R}})$ of the symplectic Hilbert space ${\mathbb {H}}$.

As in [7, Prop. 2.1], [9, Section 1], one finds that this problem can be resolved by only working with the Hamiltonian flow $\phi ^A_t$ of $H^A$, defined by

$$\begin{aligned} \partial _t|_{t=0}\phi ^A_t(u) = X^A(u) = J A u_{{\mathbb {H}}}, \end{aligned}$$

since it has much better properties: While it is trivial on $T^*Q$, for every fixed time $t\in {\mathbb {R}}$ it extends to a unitary linear map on ${\mathbb {H}}$, since

$$\begin{aligned} \phi ^A_t\cdot z_n = \exp (\sqrt{n^2+1}\cdot it)\cdot z_n. \end{aligned}$$

In particular, for every chosen time period T it follows that the time-T map $\phi ^A_T$ is a smooth symplectomorphism of $(\widetilde{M},\omega )$ which furthermore preserves the complex structure J on $\widetilde{M}$. While the Hamiltonian flow $\phi ^A_t$ is smooth with respect to the time coordinate t on ${\mathbb {H}}_{\infty }=C^\infty ({\mathbb {T}}^d,{\mathbb {R}})\oplus C^\infty ({\mathbb {T}}^d,{\mathbb {R}})$, we stress that on ${\mathbb {H}}=H^{\frac{1}{2}}({\mathbb {T}}^d,{\mathbb {R}})\oplus H^{\frac{1}{2}}({\mathbb {T}}^d,{\mathbb {R}})$ it is only continuous, see also [7, Prop. 2.5].

Defining for each $u:{\mathbb {R}}\rightarrow T^*Q\times {\mathbb {H}}$ a map $\bar{u}:{\mathbb {R}}\rightarrow T^*Q\times {\mathbb {H}}$ via $\bar{u}(t)=\phi ^A_{-t}\cdot u(t)$, it follows as in [7, Prop. 2.5] that u solves $\partial _t u=JA\pi _{{\mathbb {H}}}u+J\nabla F_t(u)$ with $u(t+T)=u(t)$ if and only if $\bar{u}$ solves $\partial _t\bar{u}=J\nabla G_t(\bar{u})$, $\bar{u}(t+T)=\phi ^A_{-T}\cdot \bar{u}(t)$, where we define $G_t=F_{{\text {part}},t}+G_{{\text {inter}},t}$ with $G_{{\text {inter}},t}=F_{{\text {inter}},t}\circ \phi ^A_{-t}$. Writing $u_{{\mathbb {H}}}*\rho =(\varphi *\rho ,\pi *\rho )$, note that since $G_{{\text {inter}},t}(q,\varphi ,\pi )=f_t((\phi ^A_{-t}u_{{\mathbb {H}}}*\rho )(q))=f_t(\phi ^A_{-t}(u_{{\mathbb {H}}}*\rho )(q))$, it follows that $G_{{\text {inter}},t}$ is still smooth with respect to the time coordinate t, using that $u_{{\mathbb {H}}}*\rho \in {\mathbb {H}}_{\infty }$.

Denote by ${\mathcal {P}}(\phi ^A_T,G)$ the set of $\phi ^A_T$-periodic solutions $\bar{u}=(\bar{u}_M,\bar{u}_{{\mathbb {H}}})$ of $\partial _t\bar{u}=J\nabla G_t(\bar{u})$ with contractible image in Q, and for given $R>0$ let $\chi _R:{\mathbb {R}}\rightarrow [0,1]$ denote a smooth cut-off function with $\chi _R(r)=1$ for $r\le R$ and $\chi _R(r)=0$ for $r\ge R+1$.

Lemma 3.1

There exists some $R_0>0$ such that $\displaystyle |\pi _{{\mathbb {H}}}\bar{u}(t)|_1\le R_0$ for all $\bar{u}\in {\mathcal {P}}(\phi ^A_T,G)$ and $t\in {\mathbb {R}}$. In particular, we find $R_1>0$ such that ${\mathcal {P}}(\phi ^A_T,G)={\mathcal {P}}(\phi ^A_T,\tilde{G})$ with $\tilde{G}_t=F_{{\text {part}},t}+\tilde{G}_{{\text {inter}},t}$, $\tilde{G}_{{\text {inter}},t}=\tilde{F}_{{\text {inter}},t}\circ \phi ^A_{-t}$ and

$$\begin{aligned} \tilde{F}_{{\text {inter}},t}(u)=\chi _{R_1}(|u_{{\mathbb {H}}}*\rho |_{C^3})\cdot F_{{\text {inter}},t}(u). \end{aligned}$$

Proof

As in the proof of [9, Theorem 9.2], see also [9, Lemma 9.1], we show that, when the ${\mathbb {H}}_1$-norm $|u_{{\mathbb {H}}}|_1$ of $u_{{\mathbb {H}}}=\pi _{{\mathbb {H}}}u$ is too large, the map $\phi ^A_T$ moves the point $u_{{\mathbb {H}}}=(\varphi ,\pi )$ further than $\phi _T^G$ can move points, so that $u_{{\mathbb {H}}}$ cannot be a fixed point of the time-T flow $\phi _T^H=\phi ^A_T\circ \phi _T^G$ of the Hamiltonian vector field of $H_t=H^A+F_t$. As a consequence, multiplying with a cut-off function with cut-off region outside of this region will not alter the set of periodic points. Note that, since the particle Hamiltonian $F_{{\text {part}},t}$ does not depend on $u_{{\mathbb {H}}}$, it suffices to take only the interaction Hamiltonian $F_{{\text {inter}},t}$ into account.

Since $T^2/(2\pi )^2$ is assumed to be a Diophantine irrational number, it follows that there exists $c>0$ and $r>2$ such that

$$\begin{aligned} \inf _{m\in {\mathbb {N}}}\left| \left( \frac{T}{2\pi }\right) ^2-\frac{m^2}{n^2+1}\right| \ge c\cdot (n^2+1)^{-r}. \end{aligned}$$

Since

$$\begin{aligned} \left( \frac{T}{2\pi }\right) ^2-\frac{m^2}{n^2+1}=\left( \frac{T}{2\pi }-\frac{m}{\sqrt{n^2+1}}\right) \cdot \left( \frac{T}{2\pi }+\frac{m}{\sqrt{n^2+1}}\right) , \end{aligned}$$

and the second factor is approximately equal to $2\cdot T/(2\pi )$ whenever the first factor is close to zero, it follows that there also exists some $c'>0$ and $r'=r-\frac{1}{2}$ such that

$$\begin{aligned}{} & {} \inf _{m\in {\mathbb {N}}}\left| T\cdot \sqrt{n^2+1}-m\cdot 2\pi \right| =2\pi \cdot \sqrt{n^2+1}\cdot \\{} & {} \inf _{m\in {\mathbb {N}}}\left| \frac{T}{2\pi }-\frac{m}{\sqrt{n^2+1}}\right| > c'\cdot (n^2+1)^{-r'}. \end{aligned}$$

Together with $\phi ^A_T\cdot z_n=\exp (i\cdot T\cdot \sqrt{n^2+1})\cdot z_n$, it follows with the small angle approximation that

$$\begin{aligned} |\phi ^A_T\cdot z_n - z_n|> c'\cdot (n^2+1)^{-r'}. \end{aligned}$$

It follows that there exists $c''>0$ and $h_0>0$ such that

$$\begin{aligned} |\phi ^A_T(u_{{\mathbb {H}}})-u_{{\mathbb {H}}}|_h>c''\cdot |u_{{\mathbb {H}}}|_{h-h_0} \,\,\text {for every}\,\, h\in {\mathbb {R}}, \end{aligned}$$

where $|\cdot |_h$ denotes the Hilbert space norm on ${\mathbb {H}}_h=H^{h+\frac{1}{2}}({\mathbb {T}}^d,{\mathbb {R}})$. On the other hand, since $F_{{\text {inter}},t}(q,u_{{\mathbb {H}}})=f_t((u_{{\mathbb {H}}}*\rho )(q))$ satisfies (F3) and $\rho \in C^{\infty }({\mathbb {T}}^d,{\mathbb {R}})$, it follows that the ${\mathbb {H}}$-component

$$\begin{aligned} \nabla ^{{\mathbb {H}}}F_{{\text {inter}},t}(q,u_{{\mathbb {H}}})=B^{-1}\rho (q-\cdot )\cdot (\partial _1f_t((u_{{\mathbb {H}}}*\rho )(q)),\partial _2f_t((u_{{\mathbb {H}}}*\rho )(q))) \end{aligned}$$

of the gradient of $F_{{\text {inter}},t}$ is bounded with respect to the ${\mathbb {H}}_h$-norm for every $h\in {\mathbb {R}}$. It follows that the same holds for the ${{\mathbb {H}}}$-component of the gradient of $G_{{\text {inter}},t}$ since $\phi ^A_t$ is a unitary map, and so for every $h\in {\mathbb {R}}$ there exists $c_h'''>0$ with

$$\begin{aligned} |\pi _{{\mathbb {H}}}\phi ^{G_{{\text {inter}}}}_T(u)-\pi _{{\mathbb {H}}}u|_h\le c'''_h. \end{aligned}$$

Choosing $h\in {\mathbb {R}}$ such that $h-h_0=1$, we find that $|\pi _{{\mathbb {H}}}\bar{u}(t)|_1\le R_0$ for every $\bar{u}\in {\mathcal {P}}(\phi ^A_T,G)$ and $t\in {\mathbb {R}}$ with $R_0:=c'''_h/c''$. The second statement follows from the fact that there exists $c''''>0$ such that $|u_{{\mathbb {H}}}*\rho |_{C^3}\le c''''\cdot |u_{{\mathbb {H}}}|_1$ for every $u\in T^*Q\times {\mathbb {H}}$. Hence, when we define $\tilde{G}_{{\text {inter}},t}(u):=\chi _{R_1}(|\pi _{{\mathbb {H}}}u*\rho |_{C^3})\cdot G_{{\text {inter}},t}(u)$ with $R_1=c''''R_0$, and $\tilde{G}_t:=F_{{\text {part}},t}+\tilde{G}_{{\text {inter}},t}$, then we find that ${\mathcal {P}}(\phi ^A_T,\tilde{G})={\mathcal {P}}(\phi ^A_T,G)$. $\square$

Note that we can replace the $C^3$-norm by the $C^\kappa$-norm for any $\kappa$. This would lead to Floer curves of higher regularity in Proposition 4.5. However, since we only require the Floer curves to be of class $C^1$, the $C^3$-norm suffices. It follows that instead of considering the original Hamiltonian $G_t=F_{{\text {part}},t}+G_{{\text {inter}},t}$ from now on we can work with the Hamiltonian $\tilde{G}_t=F_{{\text {part}},t}+\tilde{G}_{{\text {inter}},t}: T^*Q\times {\mathbb {H}}\rightarrow {\mathbb {R}}$ which has bounded support on ${\mathbb {H}}$ in a weak sense. Note that in the language of [9, Definition 3.3] we would call $G_{{\text {inter}},t}$ weakly A-admissible, while $\tilde{G}_{{\text {inter}},t}$ would be called A-admissible. To illustrate the benefit that we gained from passing from $G_t$ to $\tilde{G}_t$, we prove the following

Lemma 3.2

$\tilde{F}_{{\text {inter}},t}$ and hence also $\tilde{G}_{{\text {inter}},t}$ has finite $C^3$-norm.

Proof

We establish the $C^3$-bound for $\tilde{F}_{{\text {inter}},t}(u)=\chi _{R_1}(|u_{{\mathbb {H}}}*\rho |_{C^3})\cdot f_t((u_{{\mathbb {H}}}*\rho )(q))$; the bound for $\tilde{G}_{{\text {inter}},t}$ follows from the fact that $\phi ^A_t$ is a unitary map. Assuming boundedness of the $C^3$-norm of $u_{{\mathbb {H}}}*\rho$, it is clear that the map $(q,u_{{\mathbb {H}}})\mapsto (u_{{\mathbb {H}}}*\rho )(q)$ has bounded q-derivatives up to order 3. Since the ${\mathbb {H}}$-gradient is given by $(B^{-1}\rho (q-\cdot ),B^{-1}\rho (q-\cdot ))$ and in particular does not depend on $u_{{\mathbb {H}}}$, boundedness of all derivatives up to order 3 follows. Since $f_t$ and the cut-off function are smooth and $\tilde{F}_{{\text {inter}},t}$ has bounded support in $\{u\vert |u_{{\mathbb {H}}}*\rho |_{C^3}\le R_1+1\}$, it follows that derivatives of $\tilde{F}_{{\text {inter}},t}$ up to order 3 are bounded. $\square$

Set $\phi :=\phi ^A_T$. The symplectic action ${\mathcal {A}}_{\phi }^{\tilde{G}}$ of a $\phi$-periodic solution $\bar{u}$ of $\partial _t\bar{u}=J\nabla \tilde{G}_t(\bar{u})$ with contractible image in Q is defined as the symplectic action of the corresponding T-periodic solution $u=(u_M,u_{{\mathbb {H}}})$ of $\partial _t u=J\nabla \tilde{H}_t(u)$ with $\tilde{H}_t=H^A+\tilde{F}_t$ given by

$$\begin{aligned} {\mathcal {A}}^{\tilde{H}}(u)=\int \tilde{u}^*\omega -\int _0^T \tilde{H}_t(u)\,\textrm{d}t\,\text {with}\,\,\tilde{u}:D^2\rightarrow T^*Q\times {\mathbb {H}},\,\tilde{u}(e^{2\pi it})=u(t). \end{aligned}$$

Note that this is equal to

$$\begin{aligned} \int _0^T\left( \lambda _M(\partial _t u_M)-\tilde{F}_t(u)\right) \;\textrm{d}t\,+\,\int _0^T\left( \langle \pi ,\partial _t\varphi \rangle -H_{{\text {field}}}(\varphi ,\pi )\right) \;\textrm{d}t. \end{aligned}$$

Denote by ${\mathcal {P}}_{\le a}(\phi ,\tilde{G})$ the set of $\phi$-periodic orbits with symplectic action less than or equal to $a\in {\mathbb {R}}$.

Lemma 3.3

For every $a\in {\mathbb {R}}$ there exists $R_2>0$ such that ${\mathcal {P}}_{\le a}(\phi ,\bar{G})={\mathcal {P}}_{\le a}(\phi ,\tilde{G})$ with $\bar{F}_{{\text {inter}},t}(q,p,u_{{\mathbb {H}}})=\chi _{R_2}(\ln |p|)\cdot \tilde{F}_{{\text {inter}},t}(q,u_{{\mathbb {H}}})$ and $\bar{G}_{{\text {inter}},t}=\bar{F}_{{\text {inter}},t}\circ \phi ^A_{-t}$, $\bar{G}_t=F_{{\text {part}},t}+\bar{G}_{{\text {inter}},t}$.

Proof

Let $\bar{u}$ denote a $\phi$-periodic orbit in ${\mathcal {P}}_{\le a}(\phi ,\bar{G}_t)$ with corresponding T-periodic orbit $u=(q,p,\varphi ,\pi )$ of $\bar{H}_t$. Since the ${\mathbb {H}}_1$-norm of $(\varphi ,\pi )$ is bounded by Lemma 3.1, it follows that there exists $a_0\in {\mathbb {R}}$ such that

$$\begin{aligned} \left| \int _0^T\left( \langle \pi ,\partial _t\varphi \rangle -H_{{\text {field}}}(\varphi ,\pi )\right) \;\textrm{d}t\right| \le a_0. \end{aligned}$$

For this it suffices to observe that $\langle \pi ,\partial _t\varphi \rangle -H_{{\text {field}}}(\varphi ,\pi )$ is given by

$$\begin{aligned} \frac{1}{2}\langle \pi ,B\pi +2\partial _2 f_t\cdot B^{-1}\rho (q-\cdot )\rangle -\frac{1}{2}\langle \varphi ,B\varphi \rangle . \end{aligned}$$

With this we can compute as in [5, Lemma 5.3]

$$\begin{aligned} a+a_0\ge & {} \int _0^T\left( \lambda _M(\partial _t u_M)-\bar{F}_t(u)\right) \;\textrm{d}t\\ {}= & {} \int _0^T \left( \omega _M\left( p\frac{\partial }{\partial p},J\nabla \bar{F}_t(u)\right) -\bar{F}_t(u)\right) \;\textrm{d}t\\ {}= & {} \int _0^T \left( d\bar{F}_t(u)\cdot p\frac{\partial }{\partial p}-\bar{F}_t(u)\right) \;\textrm{d}t\\ {}= & {} \int _0^T \left( dF_{{\text {part}},t}(u_M)\cdot p\frac{\partial }{\partial p}-F_{{\text {part}},t}(u_M) +d\bar{F}_{{\text {inter}},t}(u)\cdot p\frac{\partial }{\partial p}-\bar{F}_{{\text {inter}},t}(u)\right) \;\textrm{d}t\\\ge & {} c_0\int _0^T|p|^2\;\textrm{d}t\,-\,\left( c_1+3\Vert \bar{G}_{{\text {inter}}}\Vert _{C^0}\right) \cdot T, \end{aligned}$$

where we use that $F_{{\text {part}},t}$ satisfies (F1) and $\left| d(\chi _{R_2}\circ \ln \circ |\cdot |)\cdot p\frac{\partial }{\partial p}\right| \le 2$. Furthermore, we clearly crucially use that the $C^0$-norm of $\bar{G}_{{\text {inter}}}$ (which equals the $C^0$-norm of $\tilde{G}_{{\text {inter}}}$) is bounded. As in [5] we see that the $L^2$-norm of $t\mapsto (q(t),p(t))$ is bounded. Since the gradient $\nabla \bar{F}_{{\text {inter}},t}$ is bounded with respect to the $C^0$-norm by Lemma 3.2, it further follows as in [5, Section 5] that there exists $c_3\ge 0$ such that $|\nabla \bar{F}_t(u)|\le 2c_2|p|+c_3$ with $c_2\ge 0$ from (F2). As a consequence we get as in the proof of [5, Lemma 5.3] that $t\mapsto (q(t),p(t))$ is bounded even with respect to the $H^1$-norm; using the Sobolev embedding theorem we can conclude that the same holds true for the $C^0$-norm. Since the p-component of every T-periodic solution of $\partial _t u=J\nabla \bar{H}_t(u)$ and hence of $\partial _t\bar{u}=J\nabla \bar{G}_t(\bar{u})$ of action $\le a$ is hence uniformly bounded, the claim follows.

$\square$

Summarizing, we find that for every given action bound $a\in {\mathbb {R}}$ we can find $R_1,R_2>0$ such that $G_t=F_{{\text {part}},t}+G_{{\text {inter}},t}$ and $\bar{G}_t=F_{{\text {part}},t}+\bar{G}_{{\text {inter}},t}$ have the same set of $\phi$-periodic orbits of action less than or equal to a.

4 Floer curves in infinite dimensions

In what follows we fix $R_1>0$ as in Lemma 3.1, whereas $R_2>0$ will be determined below. As in finite dimensions, it follows that ${\mathcal {P}}(\phi ,\bar{G})$ with $\phi =\phi ^A_T$ agrees with the set of critical points ${\text {Crit}}({\mathcal {A}}_{\phi }^{\bar{G}})$ of the symplectic action functional ${\mathcal {A}}={\mathcal {A}}_{\phi }^{\bar{G}}$ on $H^1_{\phi }({\mathbb {R}},T^*Q\times {\mathbb {H}})$ for the time-dependent Hamiltonian $\bar{G}_t$, where $H^1_{\phi }({\mathbb {R}},T^*Q\times {\mathbb {H}})$ contains all $H^1$-maps $\bar{u}:{\mathbb {R}}\rightarrow T^*Q\times {\mathbb {H}}$ with $\bar{u}(t+T)=\phi ^A_{-T}\bar{u}(t)$ and contractible image in Q. As in [2, 5, 7, 9, 17] we follow the idea of A. Floer in [10] to study flow lines of the gradient $\nabla {\mathcal {A}}_{\phi }^{\bar{G}}$ of the action functional with respect to the $L^2$-metric on $H^1_{\phi }({\mathbb {R}},T^*Q\times {\mathbb {H}})$ given by the canonical Riemannian metric $\langle \cdot ,\cdot \rangle$ on $\widetilde{M}=T^*Q\times {\mathbb {H}}$. The reason why Floer preferred to choose the $L^2$-gradient over the more natural $H^1$-gradient follows from the observation that the gradient flow equation $\partial _s\widetilde{u}=\nabla {\mathcal {A}}_{\phi }^{\bar{G}}(\widetilde{u})$ for $\widetilde{u}:{\mathbb {R}}\rightarrow H^1_{\phi }({\mathbb {R}},T^*Q\times {\mathbb {H}})$ is equivalent to the perturbed Cauchy–Riemann equation $\bar{\partial }_{\bar{G}}(\widetilde{u})=\partial _s\widetilde{u}+J(\widetilde{u})\partial _t\widetilde{u}+\nabla {\bar{G}}_t(\widetilde{u})=0$, where we now view $\widetilde{u}$ as a map from ${\mathbb {R}}^2$ to $\widetilde{M}$ satisfying $\widetilde{u}(s,t+T)=\phi ^A_{-T}\widetilde{u}(s,t)$ for all $(s,t)\in {\mathbb {R}}^2$. Note that this is an infinite-dimensional analogue of the perturbed Cauchy–Riemann equation used to define Floer homology for general symplectomorphisms in [6]. The following main theorem of this paper is an analogue of [9, Theorem 10.4], see also [7, Theorem 3.4].

Theorem 4.1

Assume that there exist $\theta _1,\ldots ,\theta _N\in H^{*\ne 0}(\Lambda ^{{\text {contr}}} Q,{\mathbb {Z}}_2)$ with $\theta _1\cup \cdots \cup \theta _N\ne 0$. Then there exist N maps $\widetilde{u}=\widetilde{u}_1,\ldots ,\widetilde{u}_N:{\mathbb {R}}^2\rightarrow T^*Q\times {\mathbb {H}}_{\infty }\subset T^*Q\times {\mathbb {H}}$ with $\widetilde{u}(s,\cdot )$ having contractible image in Q and satisfying the Floer equation and $\phi _T^A$-periodicity condition

$$\begin{aligned} \partial _s\widetilde{u}+J(\widetilde{u})\partial _t\widetilde{u}+\nabla \bar{G}_t(\widetilde{u})=0,\qquad \widetilde{u}(s,t+T)=\phi _{-T}^A\widetilde{u}(s,t) \end{aligned}$$

with $\bar{G}_t(q,p,u_{{\mathbb {H}}})=F_{{\text {part}},t}(q,p)+\chi _{R_2}(\ln |p|)\cdot \chi _{R_1}(|u_{{\mathbb {H}}}*\rho |_{C^3})\cdot G_{{\text {inter}},t}(q,u_{{\mathbb {H}}})$ and $R_1>0$ as in Lemma 3.1. When $R_2>0$ is chosen large enough, for every $\alpha =1,\ldots ,N$ the Floer curve $\widetilde{u}_{\alpha }$ connects two different solutions $\bar{u}=\bar{u}^-_{\alpha },\bar{u}^+_{\alpha }: {\mathbb {R}}\rightarrow T^*Q\times {\mathbb {H}}_{\infty }$ of

$$\begin{aligned} \partial _t\bar{u}=X_t^G(\bar{u}),\qquad \bar{u}(t+T)=\phi ^A_{-T}(\bar{u}(t)) \end{aligned}$$

(2)

with $G_t(q,p,u_{{\mathbb {H}}})=F_{{\text {part}},t}(q,p)+G_{{\text {inter}},t}(q,u_{{\mathbb {H}}})$ in the sense that there exist sequences $s_{\alpha ,n}^\pm \in {\mathbb {R}}$ with $s_{\alpha ,n}^\pm \rightarrow \pm \infty$ as $n\rightarrow \infty$ such that

$$\begin{aligned} \lim _{n\rightarrow \infty }\widetilde{u}_\alpha (s_{\alpha ,n}^-,t)=\bar{u}^-_\alpha (t),\qquad \lim _{n\rightarrow \infty }\widetilde{u}_\alpha (s_{\alpha ,n}^+,t)=\bar{u}^+_{\alpha }(t). \end{aligned}$$

Furthermore, since for the symplectic actions we have

$$\begin{aligned} \mathcal {A}(\bar{u}^-_1)<\mathcal {A}(\bar{u}^+_1)\le \mathcal {A}(\bar{u}^-_2)<\cdots<\mathcal {A}(\bar{u}^+_{N-1})\le \mathcal {A}(\bar{u}^-_N)<\mathcal {A}(\bar{u}^+_N), \end{aligned}$$

it follows that there are at least $N+1$ mutually different solutions of (2) with contractible image in Q.

For the proof we follow the strategy to combine the existence of Floer curves in cotangent bundles proven in [5] with the infinite-dimensional Gromov–Floer compactness result from [7, 9].

As a starting point we first approximate our infinite-dimensional Hamiltonian system by finite-dimensional ones.

For every $k\in {\mathbb {N}}$ let ${\mathbb {H}}^k$ denote the finite-dimensional subspace of ${\mathbb {H}}$ which is spanned by all $e_n^{\pm }$ with $|n|=\sqrt{n_1^2+\cdots +n_d^2}\le k$. First note that as in [9], see also [7, Prop. 2.1], the flow $\phi ^A_t$ of the field Hamiltonian $H^A$ restricts to a unitary linear map on each ${\mathbb {H}}^k$ since $\phi ^A_t\cdot z_n=\exp (it\cdot \sqrt{n^2+1})\cdot z_n$. Here we use that $(e_n^{\pm })_{n\in {\mathbb {Z}}^d}$ is a complete eigenbasis for A with real eigenvalues. For every $k\in {\mathbb {N}}$ let $\bar{G}^k_{{\text {inter}},t}=\bar{G}_{{\text {inter}},t}\circ \pi _k:T^*Q\times {\mathbb {H}}\rightarrow {\mathbb {R}}$ denote the Hamiltonian obtained by composing $\bar{G}_{{\text {inter}},t}$ with the projection $\pi _k: T^*Q\times {\mathbb {H}}\rightarrow T^*Q\times {\mathbb {H}}^k$. Furthermore note that for every fixed $t\in {\mathbb {R}}$, $u\in T^*Q\times {\mathbb {H}}$ the ${\mathbb {H}}$-gradient $\nabla ^{{\mathbb {H}}}\bar{G}_{{\text {inter}},t}(u)$ can be expanded into a Fourier series,

$$\begin{aligned} \nabla ^{{\mathbb {H}}}\bar{G}_{{\text {inter}},t}(u)=\sum _{n\in {\mathbb {Z}}^d} \widehat{\nabla ^{{\mathbb {H}}}\bar{G}_{{\text {inter}},t}(u)}(n)\cdot z_n. \end{aligned}$$

Then we have the following analogue of [7, Lemmata 2.3 and 2.4], [9, Lemma 3.2 and Lemma 6.1] about finite-dimensional approximation.

Lemma 4.2

The gradients $\nabla \bar{G}^k_{{\text {inter}},t}$, $k\in {\mathbb {N}}$ converge uniformly with their derivatives up to order 2 to the gradient $\nabla \bar{G}_{{\text {inter}},t}$. Furthermore, for all $\delta \in {\mathbb {N}}$ there exists $C_{\delta }>0$ such that

$$\begin{aligned} |\widehat{\nabla ^{{\mathbb {H}}}\bar{G}_{{\text {inter}},t}(u)}(n)|\le C_{\delta }\cdot |n|^{-\delta } \end{aligned}$$

for all $t\in {\mathbb {R}}$ and $u\in T^*Q\times {\mathbb {H}}$.

Proof

We start by observing that the statement holds if and only if it holds for $\bar{F}_{{\text {inter}},t}$, since $\phi ^A_t$ is a unitary map. Note that $F^k_{{\text {inter}},t}=F_{{\text {inter}},t}\circ \pi _k$ is given by $F^k_{{\text {inter}},t}=f_t((u_{{\mathbb {H}}}*\rho ^k)(q))$ with $\rho ^k=\sum _{|n|\le k}\hat{\rho }(n)\cdot z_n\in {\mathbb {H}}^k$. It follows that the ${\mathbb {H}}$-gradient of $F^k_{{\text {inter}},t}$,

$$\begin{aligned} \nabla ^{{\mathbb {H}}}F^k_{{\text {inter}},t}(u)=(B^{-1}\rho ^k)(q-\cdot )\left( \begin{matrix} \partial _1f_t((u_{{\mathbb {H}}}*\rho ^k)(q))\\ \partial _2f_t((u_{{\mathbb {H}}}*\rho ^k)(q)) \end{matrix} \right) \end{aligned}$$

converges to the ${\mathbb {H}}$-gradient of $F_{{\text {inter}},t}$,

$$\begin{aligned} \nabla ^{{\mathbb {H}}}F_{{\text {inter}},t}(u)=(B^{-1}\rho )(q-\cdot )\left( \begin{matrix} \partial _1f_t((u_{{\mathbb {H}}}*\rho )(q))\\ \partial _2f_t((u_{{\mathbb {H}}}*\rho )(q)) \end{matrix} \right) \end{aligned}$$

uniformly with respect to $u\in T^*Q\times {\mathbb {H}}$, using (F3). Passing from $F_{{\text {inter}},t}$ to $\bar{F}_{{\text {inter}},t}$, this result does not only continue to hold true, but even holds for all derivatives up to order 3, since we may assume that we have a uniform bound on the $C^3$-norm on $u_{{\mathbb {H}}}*\rho$ and hence in particular a uniform bound on $(u_{{\mathbb {H}}}*\rho )(q)$. For the second statement we observe that since $\rho \in C^{\infty }({\mathbb {T}}^d,{\mathbb {R}})$ we know that the Fourier coefficients $\hat{\rho }(n)$ and hence also $\widehat{\rho (\cdot -q)}(n)$ converge to zero with exponential speed. It immediately follows that the Fourier coefficients $\widehat{\nabla ^{{\mathbb {H}}}F_{{\text {inter}},t}(u)}(n)$ converge to zero with exponential speed and this convergence is uniform with respect to $t\in {\mathbb {R}}$ and $u\in T^*Q\times {\mathbb {H}}$ using (F3). $\square$

After restricting to the finite-dimensional symplectic submanifold $T^*Q\times {\mathbb {H}}^k$, note that, since $\bar{G}_{{\text {inter}},t}$ only has support in $\{u\in T^*Q\times {\mathbb {H}}: |u_{{\mathbb {H}}}*\rho |_{C^3}\le R_1\}$, the finite-dimensional Hamiltonian $\bar{G}^k_{{\text {inter}},t}$ now has compact support in $Q\times B_{R_k}(0)\subset Q\times {\mathbb {H}}^k$, where $B_{R_k}(0)$ denotes a ball around 0 in ${\mathbb {H}}^k$. Note here we crucially need the assumption that all frequencies are present in $\rho$ in the sense that $\hat{\rho }(n)\ne 0$ for all $n\in {\mathbb {Z}}^d$.

For fixed $a\in {\mathbb {R}}$ let ${\mathcal {M}}^{k,\le a}_{\tau }$ denote the moduli space of Floer curves satisfying the $\tau$-dependent Floer equation with periodicity condition and with bounded symplectic action,

$$\begin{aligned} {\mathcal {M}}^{k,\le a}_{\tau }=\left\{ \widetilde{u}=(\widetilde{u}_M,\widetilde{u}_{{\mathbb {H}}}):{\mathbb {R}}^2\rightarrow T^*Q\times {\mathbb {H}}^k:\,\,(*1),(*2),(*3),(*4)\right\} \end{aligned}$$

with $\widetilde{u}(s,\cdot )$ having contractible image in Q and

$$\begin{aligned}{} & {} (*1): \partial _s\widetilde{u}+J(\widetilde{u})\partial _t\widetilde{u}+\nabla F_{{\text {part}},t}(\widetilde{u})+\sigma _\tau (s)\nabla \bar{G}^k_{{\text {inter}},t}(\widetilde{u})=0,\\{} & {} (*2): \widetilde{u}(s,t+T)=\phi _{-T}^A\widetilde{u}(s,t),\\{} & {} (*3): \tilde{u}_{{\mathbb {H}}}(s,\cdot )\rightarrow 0\,\,\text {as}\,\,s\rightarrow \pm \infty ,\\{} & {} (*4): {\mathcal {A}}^{F_{{\text {part}}}}_{\phi ^A_T}(\tilde{u}(s,\cdot ))\le a\,\,\text {for}\,\,s\ge 2\tau +1, \end{aligned}$$

where $\sigma _{\tau }:{\mathbb {R}}\rightarrow [0,1]$ is a smooth cut-off function with $\sigma _{\tau }(s)=0$ for $s\le -1$, $s\ge 2\tau +1$ and $\sigma _{\tau }(s)=1$ for $0\le s\le 2\tau$. Note that we can define N evaluation maps ${\text {ev}}_{1,k,\tau },\ldots ,{\text {ev}}_{N,k,\tau }$ from ${\mathcal {M}}^{k,\le a}_{\tau }$ to the space $\Lambda ^{{\text {contr}}} Q=C^0_{{\text {contr}}}({\mathbb {R}}/(T{\mathbb {Z}}),Q)$ of contractible loops by

$$\begin{aligned} {\text {ev}}_{\alpha ,k,\tau }:{\mathcal {M}}^{k,\le a}_{\tau }\rightarrow \Lambda ^{{\text {contr}}} Q,\,\,\widetilde{u}\mapsto \pi _Q\circ \widetilde{u}\left( 2\tau \cdot \frac{\alpha }{N+1},\cdot \right) ,\,\,\alpha =1,\ldots ,N, \end{aligned}$$

where $\pi _Q$ denotes the projection from $T^*Q\times {\mathbb {H}}^k$ onto the base manifold Q.

Lemma 4.3

There is a uniform $C^1$-bound for the first derivative $T\widetilde{u}$ of $\widetilde{u}\in {\mathcal {M}}^{k,\le a}_{\tau }$ which is independent of $k\in {\mathbb {N}}$ and $\tau \ge 0$.

Proof

The first thing that needs to be observed is that the energy $E(\widetilde{u})$ of the Floer curves is uniformly bounded not just for all $\widetilde{u}\in {\mathcal {M}}^{k,\le a}_{\tau }$ and all $\tau \ge 0$, but also for all $k\in {\mathbb {N}}$: Since $\Vert \bar{G}^k_{{\text {inter}},t}\Vert _{C^0}\le \Vert \bar{G}_{{\text {inter}},t}\Vert _{C^0}$ for all $k\in {\mathbb {N}}$, we get from [14, Prop. 9.1.4] that

$$\begin{aligned} E(\widetilde{u})\le (a-b)+4T\Vert \bar{G}_{{\text {inter}},t}\Vert _{C^0}, \end{aligned}$$

where we again stress that $\bar{G}_{{\text {inter}},t}$ has finite $C^3$-norm by Lemma 3.2. Now let $\widetilde{u}^k$ be an arbitrary sequence of Floer curves in $\bigcup _k\bigcup _{\tau }{\mathcal {M}}^{k,\le a}_{\tau }$, where we want to assume without loss of generality that $\widetilde{u}^k\in {\mathcal {M}}^{k,\le a}_{\tau }$. As shown in [9, Prop. 6.3], see also [7, Prop. 6.1], it follows precisely along the same lines as for sequences of Floer curves in fixed finite-dimensional exact symplectic manifolds using bubbling-off analysis and elliptic regularity that the $C^1$-norm of the first derivatives of the Floer curves $\widetilde{u}^k$ is uniformly bounded,

$$\begin{aligned} \sup _k \Vert T\widetilde{u}^k\Vert _{C^1}<\infty . \end{aligned}$$

We start by observing that the corresponding statement for the $C^0$-norm of $T\widetilde{u}$ in [9, Lemma 6.2] is established using bubbling-off analysis as in [7, Lemma 6.2]: Indeed, assume without loss of generality that the first derivative is unbounded in the sense that

$$\begin{aligned} C_k:=\max _{z=(s,t)\in {\mathbb {R}}^2}\left\{ |\partial _s \widetilde{u}^k(z)|\right\} =:\left| \partial _s\widetilde{u}^k(z_k)\right| \rightarrow \infty \,\,\text {as}\,\,k\rightarrow \infty . \end{aligned}$$

Now consider the reparametrized map

$$\begin{aligned} \widetilde{v}^k:B_{\sqrt{C_k}}(0)\rightarrow T^*Q\times {\mathbb {H}}^k:z\mapsto \widetilde{u}^k\left( \frac{z}{C_k}+z_k\right) \end{aligned}$$

so that $|\partial _s\widetilde{v}^k(0)|=1$ and $|\partial _s\widetilde{v}^k(z)|\le 1$ for $|z|\le \sqrt{C_k}$. By finiteness of area, it follows that for all k there exists $\frac{\sqrt{C_k}}{2}\le r_k\le \sqrt{C_k}$ such that the length of the circle $\theta \mapsto \widetilde{v}^k(r_k e^{i\theta })$ goes to zero. By the exactness of $\omega$ the area of $\widetilde{v}^k_{r_k}$, which is the restriction of $\widetilde{v}^k$ to the disk of radius $r_k$, goes to zero. Finally one can employ an a priori bound to deduce that $\partial _s\widetilde{v}^k(0)\rightarrow 0$ as $k\rightarrow \infty$, and the proof of $\sup _k \Vert T\widetilde{u}^k\Vert _{C^0}<\infty$ follows by contradiction. For elliptic bootstrapping we can use [14, App. B] to deduce that

$$\begin{aligned} \Vert \nabla G^k(\tilde{u}^k)\Vert _{H^{2,p}}\le c_1\cdot \Vert \nabla G^k\Vert _{C^2}\cdot (1+\Vert \tilde{u}^k\Vert _{L^\infty })(1+\Vert \tilde{u}^k\Vert _{H^{2,p}}) \end{aligned}$$

and hence the $H^{3,p}$-norm of $\widetilde{u}$ is bounded, see also the proof of [7, Prop. 6.1]. As mentioned already after the proof of Lemma 3.1, replacing the $C^3$-norm in the definition of $\bar{G}_{{\text {inter}},t}$ by the $C^{\kappa }$-norm, we can obtain $H^{\kappa ,p}$-bounds for any $\kappa$. $\square$

From now on fix $\theta _1,\ldots ,\theta _N\in H^{*\ne 0}(\Lambda ^{{\text {contr}}} Q,{\mathbb {Z}}_2)$ with $\theta _1\cup \cdots \cup \theta _N\ne 0$.

Proposition 4.4

There exists $a\in {\mathbb {R}}$ such that for every $k\in {\mathbb {N}}$ and $\tau \ge 0$ we have

$$\begin{aligned} {\text {ev}}_{1,k,\tau }^*\theta _1\cup \cdots \cup {\text {ev}}_{N,k,\tau }^*\theta _N\ne 0\in H^*({\mathcal {M}}^{k,\le a}_{\tau },{\mathbb {Z}}_2). \end{aligned}$$

Proof

The key ingredient in the proof is [5, Theorem 7.6]. Note that for $\tau =0$ the corresponding moduli spaces ${\mathcal {M}}^{k,\le a}_0$ consist of Floer curves $\widetilde{u}:{\mathbb {R}}^2\rightarrow T^*Q\times {\mathbb {H}}^k$ satisfying the Floer equation $\partial _s\widetilde{u}+J(\widetilde{u})\partial _t\widetilde{u}+\nabla F_{{\text {part}},t}(\widetilde{u})=0$. Writing $\widetilde{u}$ as a pair $\widetilde{u}=(\widetilde{u}_M,\widetilde{u}_{{\mathbb {H}}})$ with $\widetilde{u}_M:{\mathbb {R}}^2\rightarrow T^*Q$ and $\widetilde{u}_{{\mathbb {H}}}:{\mathbb {R}}^2\rightarrow {\mathbb {H}}^k$, we find that the Floer equation (as well as the periodicity condition) decouples,

$$\begin{aligned} \overline{\partial }_{J_M}\widetilde{u}_M+\nabla F_{{\text {part}},t}(\widetilde{u}_M)=0,{} & {} \widetilde{u}_M(s,t+T)=\widetilde{u}_M(s,t),\\ \bar{\partial }\widetilde{u}_{{\mathbb {H}}}=0,{} & {} \widetilde{u}_{{\mathbb {H}}}(s,t+T)=\phi ^A_{-T}\widetilde{u}_{{\mathbb {H}}}(s,t). \end{aligned}$$

Since we assume that $\widetilde{u}_{{\mathbb {H}}}(s,\cdot )\rightarrow 0$ as $s\rightarrow \pm \infty$, it follows that $\widetilde{u}_{{\mathbb {H}}}\equiv 0$. But this implies that ${\mathcal {M}}^{k,\le a}_0$ precisely agrees with the moduli space of Floer curves for which Cieliebak stated his Theorem 7.6 in [5]. Indeed it follows from [5, Theorem 7.6] that there exists $a\in {\mathbb {R}}$ such that

$$\begin{aligned} {\text {ev}}_{1,k,0}^*\theta _1\cup \cdots \cup {\text {ev}}_{N,k,0}^*\theta _N={\text {ev}}_{k,0}^*(\theta _1\cup \cdots \cup \theta _N)\ne 0\in H^*({\mathcal {M}}^{k,\le a}_0,{\mathbb {Z}}_2), \end{aligned}$$

where ${\text {ev}}_{1,k,0}=\ldots ={\text {ev}}_{N,k,0}=:{\text {ev}}_{k,0}$ for all $k\in {\mathbb {N}}$ by definition.

The corresponding statement for all $\tau \ge 0$ now follows from a standard finite-dimensional homotopy argument, where the Gromov–Floer compactness theorem plays the central role. We start with establishing the required $C^0$-bounds for Floer curves in $T^*Q\times {\mathbb {H}}^k$. First, since $\bar{G}_{{\text {inter}},t}(q,p,u_{{\mathbb {H}}})=\chi _{R_2}(\ln |p|)\cdot \chi _{R_1}(|u_{{\mathbb {H}}}*\rho |_{C^3})\cdot G_{{\text {inter}},t}(q,u_{{\mathbb {H}}})=0$ and hence $\bar{G}_t(q,p,u_{{\mathbb {H}}})=F_{{\text {part}},t}(q,p)$ for |p| sufficiently large, it follows that the $C^0$-bound for Floer curves in $T^*Q$ established in [5] immediately establishes a $C^0$-bound for our Floer curves in $T^*Q\times {\mathbb {H}}^k$ which is even independent of $k\in {\mathbb {N}}$. On the other hand, the compactness problems due to the unboundedness of ${\mathbb {H}}^k$ is taken care of as in [9, Prop. 7.2]: Since $\bar{G}^k_{{\text {inter}},t}$ has bounded support in $Q\times B_{R_k}(0)\subset Q\times {\mathbb {H}}^k$ and $F_{{\text {part}},t}$ by choice does not depend on the ${\mathbb {H}}^k$-coordinates, it follows from the maximum principle for unperturbed holomorphic curves in ${\mathbb {H}}^k$ that the Floer curve has to stay inside the same bounded subset.

After establishing the necessary $C^0$-bounds, we observe that by Lemma 4.3 we know that we even have uniform bounds for the $C^2$-norm of $\widetilde{u}\in {\mathcal {M}}^{k,\le a}$. By the classical elliptic bootstrapping arguments it follows that we hence have compactness with respect to the $C^1$-norm modulo breaking of Floer curves. $\square$

For every $k\in {\mathbb {N}}$ let $\widetilde{u}^k\in {\mathcal {M}}^{k,\le a}_k$ be an arbitrary Floer curve in $T^*Q\times {\mathbb {H}}^k$ for $\tau =k$. As in the proof of [9, Theorem 10.4] the idea is to apply the infinite-dimensional generalization of the Gromov–Floer compactness result from [7, 9] to sequences of N shifted Floer curves $\widetilde{u}^k_{\alpha }$, $\alpha =1,\ldots ,N$ to obtain Floer curves in the infinite-dimensional symplectic manifold $\widetilde{M}=T^*Q\times {\mathbb {H}}$. More precisely, combining the proof of [9, Theorem 10.4], which itself essentially is based on [7, Lemma 8.1], [9, Theorem 8.1], with the $C^0$-bounds from [5, Theorem 5.4] leads to a proof of the following

Proposition 4.5

For every $\alpha =1,\ldots ,N$, a subsequence of the sequence of shifted Floer curves

$$\begin{aligned} \widetilde{u}^k_{\alpha }:{\mathbb {R}}^2\rightarrow T^*Q\times {\mathbb {H}}^k,\,\, \widetilde{u}^k_{\alpha }(s,t)= \widetilde{u}^k\left( s+2k\frac{\alpha }{N+1},t\right) \end{aligned}$$

$C^1$-converges to a solution $\widetilde{u}=\widetilde{u}_{\alpha }:{\mathbb {R}}^2\rightarrow T^*Q\times {\mathbb {H}}$ of the Floer equation

$$\begin{aligned} \partial _s\widetilde{u}+J(\widetilde{u})\partial _t\widetilde{u}+\nabla \bar{G}_t(\widetilde{u})=0,\,\,\widetilde{u}(s,t+T)=\phi _{-T}^A\widetilde{u}(s,t), \end{aligned}$$

which satisfies the following asymptotic conditions: there exists sequences $s_{\alpha ,n}^\pm \in {\mathbb {R}}$ with $s_{\alpha ,n}^\pm \rightarrow \pm \infty$ as $n\rightarrow \infty$ such that

$$\begin{aligned} \lim _{n\rightarrow \infty }\widetilde{u}_{\alpha }(s_{\alpha ,n}^-,t)=\bar{u}^-_{\alpha }(t),\qquad \lim _{n\rightarrow \infty }\widetilde{u}(s_{\alpha ,n}^+,t)=\bar{u}^+_{\alpha }(t) \end{aligned}$$

in the $C^1$-sense where $\bar{u}^-_{\alpha }$ and $\bar{u}^+_{\alpha }$ are two $\phi _T^A$-periodic orbits of $\bar{G}_t$ with contractible image in Q.

Proof

Note that we have $\sigma _k(s+2k\frac{\alpha }{N+1})\rightarrow 1$ for every $(s,t)\in {\mathbb {R}}^2$ as $k\rightarrow \infty$. While Lemma 4.3 is known to be the main ingredient for Gromov–Floer compactness in the case of closed finite-dimensional symplectic manifolds, here we need to deal with the noncompactness of the target manifold. While we have already discussed how $C^0$-bounds can be established, we now turn to the most striking problem, namely the problem that the target manifold $\widetilde{M}=T^*Q\times {\mathbb {H}}$ is indeed infinite-dimensional. As in [7, 9] we start by observing that we can write the finite-dimensional Floer curve as a tuple

$$\begin{aligned} \widetilde{u}^k=(\widetilde{u}^{k,\ell },\widetilde{u}^{k,\ell }_\perp ):{\mathbb {R}}^2\rightarrow (T^*Q\times {\mathbb {H}}^{\ell })\times ({\mathbb {H}}^k/{\mathbb {H}}^{\ell })=T^*Q\times {\mathbb {H}}^k\subset T^*Q\times {\mathbb {H}}, \end{aligned}$$

where $\widetilde{u}^{k,\ell }_\perp$ denotes the normal component of $\widetilde{u}^k$. As in [7, Prop. 7.1], [9, Prop. 7.2] we prove in Lemma 4.6 below that we have

$$\begin{aligned} \sup _{k\ge \ell }\left\| \widetilde{u}^{k,\ell }_\perp \right\| _{C^1}\rightarrow 0\quad \textrm{as}\quad \ell \rightarrow \infty . \end{aligned}$$

On the other hand, as in the proof of [7, Lemma 8.1], [9, Theorem 8.1] it follows from Lemma 4.3 using the standard elliptic bootstrapping argument together with a diagonal subsequence argument that for each $\alpha =1,\ldots ,N$ there is a subsequence of $(\widetilde{u}^k_{\alpha })_k$ such that for all $\ell \in {\mathbb {N}}$ the corresponding sequence $(\widetilde{u}^{k,\ell }_{\alpha })_k$ $C^1$-converges to a map $\widetilde{u}^\ell _{\alpha }:{\mathbb {R}}^2\rightarrow T^*Q\times {\mathbb {H}}^{\ell }$ as $k\rightarrow \infty$. Together with Lemma 4.6 about the normal component this proves that the given subsequence of $(\widetilde{u}^k_{\alpha })_k$ converges in the $C^1$-sense to a map $\widetilde{u}_{\alpha }$ which solves the Floer equation for $\bar{G}_t=F_{{\text {part}},t}+\bar{G}_{{\text {inter}},t}$ using again Lemma 4.2.

The asymptotic condition is proven as in the proof of [7, Theorem 5.1], [9, Theorem 8.2] and hence again crucially relies on Lemma 4.6. For this choose sequences $s_{n,\alpha }^\pm \in {\mathbb {R}}$ with $n\le s_{n,\alpha }^+\le 2n$ and $n\le -s_{n,\alpha }^-\le 2n$ such that

$$\begin{aligned} \int _0^T\left| \partial _s\widetilde{u}_{\alpha }(s_{n,\alpha }^\pm ,t)\right| ^2\textrm{d}t\le \frac{E(\widetilde{u}_{\alpha })}{n}\rightarrow 0\text { as } n\rightarrow \infty , \end{aligned}$$

where we use that the energy of the limiting Floer curve $\widetilde{u}_{\alpha }$ is still bounded by $(a-b)+4T||\bar{G}^k_{{\text {inter}},t}||_{C^0}$. Now we write $\widetilde{u}_{\alpha }=(\widetilde{u}_{\alpha }^{\ell },\widetilde{u}_{\alpha ,\perp }^{\ell }):{\mathbb {R}}^2\rightarrow (T^*Q\times {\mathbb {H}}^{\ell })\times {\mathbb {H}}/{\mathbb {H}}^{\ell }$ for $\ell \in {\mathbb {N}}$. Using the $C^0$-bounds and after passing to a diagonal subsequence we can assume that $\widetilde{u}_{\alpha }^{\ell }(s_{n,\alpha }^\pm ,\cdot )$ $C^1$-converges as $n\rightarrow \infty$ for all $\ell$ simultaneously. Using again Lemma 4.6 we can deduce that $\widetilde{u}_{\alpha }(s_{n,\alpha }^\pm ,\cdot )$ $C^1$-converges as $n\rightarrow \infty$ to $\phi ^A_T$-periodic orbits $\bar{u}_{\alpha }^\pm$ of $\bar{G}_t(p,q,u_{{\mathbb {H}}})=F_{{\text {part}},t}(q,p)+\chi _{R_2}(\ln |p|)\cdot \chi _{R_1}(|u_{{\mathbb {H}}}*\rho |_{C^3})\cdot G_{{\text {inter}},t}(q,u_{{\mathbb {H}}})$. $\square$

The following lemma is just a recast of [7, Prop. 7.1], [9, Prop. 7.2] and added for completeness.

Lemma 4.6

We have

$$\begin{aligned} \sup _{k\ge \ell }\left\| \widetilde{u}^{k,\ell }_\perp \right\| _{C^1}\rightarrow 0\quad \textrm{as}\quad \ell \rightarrow \infty . \end{aligned}$$

Proof

Since each $\tilde{u}^k:{\mathbb {R}}^2\rightarrow T^*Q\times {\mathbb {H}}^k$ satisfies the Floer equation $\partial _s\widetilde{u}^k+J(\widetilde{u}^k)\partial _t\widetilde{u}^k+\nabla \bar{G}^k_{s,t}(\widetilde{u})=0$, it follows that the ${\mathbb {H}}$-component $\widetilde{u}^k_{{\mathbb {H}}}:=\pi _{{\mathbb {H}}}\widetilde{u}^k:{\mathbb {R}}^2\rightarrow {\mathbb {H}}^k$ satisfies the equation $\partial _s\widetilde{u}^k_{{\mathbb {H}}}+i\partial _t\widetilde{u}^k_{{\mathbb {H}}}+\nabla ^{{\mathbb {H}}} \bar{G}^k_{s,t}(\widetilde{u}^k)=0$. Note that here $\nabla ^{{\mathbb {H}}} \bar{G}^k_{s,t}$ denotes the (orthogonal) projection of the gradient $\nabla \bar{G}^k_{s,t}$ onto ${\mathbb {H}}^k\subset {\mathbb {H}}$ and we identify ${\mathbb {H}}$ with the subspace of ${\mathbb {H}}\otimes {\mathbb {C}}$ on which $J_{{\mathbb {H}}}=i$. In what follows we view each ${\mathbb {H}}$-component $\widetilde{u}^k_{{\mathbb {H}}}$ as a map from ${\mathbb {R}}$ into $L^2_{\phi }({\mathbb {R}},{\mathbb {H}})$, where the Hilbert space $L^2_{\phi }({\mathbb {R}},{\mathbb {H}})$ consists of all maps $\bar{u}\in L^2({\mathbb {R}},{\mathbb {H}})$ satisfying the periodicity condition $\bar{u}(t+T)=\phi ^A_{-T}\bar{u}(t)$. Setting $\nabla ^{{\mathbb {H}}} \bar{G}^k(\widetilde{u}^k)(s)(t):=\nabla ^{{\mathbb {H}}} \bar{G}^k_{s,t}(\widetilde{u}^k(s,t))$, we can view $\nabla ^{{\mathbb {H}}} \bar{G}^k(\widetilde{u}^k)$ also as a map from ${\mathbb {R}}$ into $L^2_{\phi }({\mathbb {R}},{\mathbb {H}})$. The operator $-i\partial _t$ on $L^2_{\phi }({\mathbb {R}},{\mathbb {H}})$ has a Hilbert basis of eigenfunctions $u_{m,n}(t)=e^{\lambda _{m,n}it}\cdot z_n$ with corresponding eigenvalues $\lambda _{m,n}=m\frac{2\pi }{T}-\sqrt{n^2+1}$ for $m\in {\mathbb {Z}}$ and $n\in {\mathbb {Z}}^d$. Now we apply the corresponding Fourier transform to $\widetilde{u}^k_{{\mathbb {H}}}:{\mathbb {R}}\rightarrow L^2_{\phi }({\mathbb {R}},{\mathbb {H}})$ and $\nabla ^{{\mathbb {H}}} \bar{G}^k(\widetilde{u}^k):{\mathbb {R}}\rightarrow L^2_{\phi }({\mathbb {R}},{\mathbb {H}})$ to obtain sequences of maps $w^k_{m,n}:=\widehat{\widetilde{u}^k_{{\mathbb {H}}}}(m,n),g^k_{m,n}:=\widehat{\nabla ^{{\mathbb {H}}} \bar{G}^k(\widetilde{u}^k)}(m,n):{\mathbb {R}}\rightarrow {\mathbb {C}}$ satisfying

$$\begin{aligned} (w^k_{m,n})'(s)=\lambda _{m,n}w^k_{m,n}(s)+g^k_{m,n}(s)\,\,\text {and}\,\,w^k_{m,p}(s)\rightarrow 0\,\,\text {as}\,\,s\rightarrow \pm \infty . \end{aligned}$$

Since $\bar{G}_t$ is smooth with respect to t and the $C^2$-norms of $\widetilde{u}^k$ are uniformly bounded in $k\in {\mathbb {N}}$, it follows from Lemma 4.2 that for all $\delta \in {\mathbb {N}}$ there exists $C_{\delta }>0$ such that

$$\begin{aligned} |g_{m,n}^k(s)|\le C_{\delta }\cdot |m|^{-2}\cdot |n|^{-\delta } \end{aligned}$$

independent of $k\in {\mathbb {N}}$, $s\in {\mathbb {R}}$. On the other hand we know from the proof of Lemma 3.1 that there exist $c>0$ and $r>0$ such that

$$\begin{aligned} |\lambda _{m,n}|=\frac{2\pi }{T}\cdot \sqrt{n^2+1}\cdot \Big |\frac{T}{2\pi }-\frac{m}{\sqrt{n^2+1}}\Big |\ge c\cdot |n|^{-r}. \end{aligned}$$

Using the elementary estimate $\Vert w^k_{m,n}\Vert _{C^0}\le \Vert g^k_{m,n}\Vert _{C^0}/|\lambda _{m,n}|$ from [7, Lemma 7.2], [9, Lemma 7.1] we get that

$$\begin{aligned} |\widehat{\widetilde{u}^k(s)}(m,n)|=|w_{m,n}^k(s)|\le C_{\delta }/c\cdot |m|^{-2}\cdot |n|^{-\delta +r} \end{aligned}$$

for all $\delta \in {\mathbb {N}}$ independent of $k\in {\mathbb {N}}$, $s\in {\mathbb {R}}$. From

$$\begin{aligned} |\partial _t^j\widetilde{u}^{k,\ell }_\perp (s,t)|^2\le \sum _{|n|=\ell +1}^\infty \left( \sum _{m\in {\mathbb {Z}}}|\widehat{\widetilde{u}^k(s)}(m,n)||m|^j\right) ^2, \end{aligned}$$

we can conclude that

$$\begin{aligned} \sup _{k\ge \ell }\left\| \widetilde{u}^{k,\ell }_\perp \right\| _{C^0},\,\sup _{k\ge \ell }\left\| \partial _t\widetilde{u}^{k,\ell }_\perp \right\| _{C^0}\rightarrow 0\quad \textrm{as}\quad \ell \rightarrow \infty . \end{aligned}$$

On the other hand, using the Floer equation $\partial _s\widetilde{u}^k_{{\mathbb {H}}}+i\partial _t\widetilde{u}^k_{{\mathbb {H}}}+\nabla ^{{\mathbb {H}}} \bar{G}^k_{s,t}(\widetilde{u}^k)=0$ and Lemma 4.2, we also get $\displaystyle \sup _{k\ge \ell }\left\| \partial _s\widetilde{u}^{k,\ell }_\perp \right\| _{C^0}\rightarrow 0$ as $\ell \rightarrow \infty$, and the claim follows. $\square$

Proof

With this we can now finish the proof of Theorem 4.1 (which in turn implies Theorem 2.1). From Proposition 4.5 we know that, given a sequence $\widetilde{u}^k\in {\mathcal {M}}^{k,\le a}_k$, $k\in {\mathbb {N}}$, for every $\alpha =1,\ldots ,N$ a subsequence of the sequence of shifted Floer curves $\widetilde{u}^k_{\alpha }(s,t)= \widetilde{u}^k(s+2k\frac{\alpha }{N+1},t)$ $C^1$-converges to a solution $\widetilde{u}_{\alpha }:{\mathbb {R}}^2\rightarrow T^*Q\times {\mathbb {H}}$ of the Floer equation

$$\begin{aligned} \partial _s\widetilde{u}+J(\widetilde{u})\partial _t\widetilde{u}+\nabla \bar{G}_t(\widetilde{u})=0,\,\,\widetilde{u}(s,t+T)=\phi _{-T}^A\widetilde{u}(s,t) \end{aligned}$$

connecting two $\phi _T^A$-periodic orbits $\bar{u}^-_{\alpha }$ and $\bar{u}^+_{\alpha }$ of $\bar{G}_t$, where we stress that the orbits $\bar{u}^{\pm }_{\alpha }$, $\alpha =1,\ldots ,N$ are partially ordered by their symplectic action,

$$\begin{aligned} \mathcal {A}(\bar{u}^-_1)\le \mathcal {A}(\bar{u}^+_1)\le \mathcal {A}(\bar{u}^-_2)\le \ldots \le \mathcal {A}(\bar{u}^+_{N-1})\le \mathcal {A}(\bar{u}^-_N)\le \mathcal {A}(\bar{u}^+_N). \end{aligned}$$

Assume to the contrary that the number of $\phi ^A_T$-periodic orbits is strictly less than $N+1$. Then there must exist $\alpha \in \{1,\ldots , N\}$ such that

$$\begin{aligned} 0=\mathcal {A}(\bar{u}^+_{\alpha })-\mathcal {A}(\bar{u}^-_{\alpha })=E(\widetilde{u}_{\alpha })=\int _{-\infty }^{+\infty }\int _0^T\left| \partial _s\widetilde{u}_{\alpha }\right| ^2\textrm{d}t\;\textrm{d}s. \end{aligned}$$

But then it follows that $\widetilde{u}_{\alpha }(s,\cdot )$ is independent of $s\in {\mathbb {R}}$ and belongs to the finite set of $\phi ^A_T$-periodic orbits of $\bar{G}_t$. We find that in the limit $k\rightarrow \infty$ the image of the evaluation map ${\text {ev}}_{k,k}=({\text {ev}}_{1,k,k},\ldots ,{\text {ev}}_{N,k,k}): {\mathcal {M}}^{k,\le a}_k\rightarrow (\Lambda ^{{\text {contr}}} Q)^N$ is degenerate in the sense that it factors through a map from $\bigcup _{i=1}^N (\Lambda ^{{\text {contr}}}Q)^{i-1}\times \{\text {point}\}\times (\Lambda ^{{\text {contr}}}Q)^{N-i}$ to $(\Lambda ^{{\text {contr}}}Q)^N$. But this contradicts the fact that we have chosen $a\in {\mathbb {R}}$ as in Proposition 4.4, that is, we have ${\text {ev}}_{1,k,k}^*\theta _1\cup \cdots \cup {\text {ev}}_{N,k,k}^*\theta _N\ne 0\in H^*({\mathcal {M}}^{k,\le a}_k,{\mathbb {Z}}_2)$ for all $k\in {\mathbb {N}}$.

Further, as described in [5, Section 8] it follows from work of D. Sullivan that the finiteness of the fundamental group of Q implies that the ${\mathbb {Z}}_2$-cuplength of the space of contractible loops in Q is infinite.

Analogous to the proof of [7, Prop. 3.5], [9, Theorem 8.3] we note that the limit Floer curve $\widetilde{u}$ has image in $T^*Q\times {\mathbb {H}}_{\infty }$, since for the Fourier coefficients of the Floer curve $\widetilde{u}$ we know that for every $\delta \in {\mathbb {N}}$ there exists $C_{\delta }>0$ such that

$$\begin{aligned} |\widehat{\widetilde{u}(s)}(m,n)|\le C_{\delta }/c \cdot |m|^{-2 +\frac{1}{2}}\cdot |n|^{-\delta +r}. \end{aligned}$$

To see that we actually obtain $\phi ^A_T$-periodic orbits of the original Hamiltonian $G_t(q,p,u_{{\mathbb {H}}})=F_{{\text {part}},t}(q,p)+G_{{\text {inter}},t}(q,u_{{\mathbb {H}}})$ as long as $R_2>0$ is chosen sufficiently large, note that the $\phi ^A_T$-periodic orbits $\bar{u}_{\alpha }^\pm$ of $\bar{G}_t$ that we have found have action less than or equal to $a+4T\Vert \bar{G}_{{\text {inter}},t}\Vert _{C^0}$, see [14, Theorem 9.1.13] for a similar result. Now use Lemma 3.3 to find $R_2>0$ such that the $\phi ^A_T$-periodic orbits of $G_t=F_{{\text {part}},t}+G_{{\text {inter}},t}$ agree with those of $\bar{G}_t=F_{{\text {part}},t}+\bar{G}_{{\text {inter}},t}$ with $\bar{G}_{{\text {inter}},t}(q,p,u_{{\mathbb {H}}})=\chi _{R_2}(\ln |p|)\cdot \chi _{R_1}(|u_{{\mathbb {H}}}*\rho |_{C^3})\cdot G_{{\text {inter}},t}(q,u_{{\mathbb {H}}})$ as long as the symplectic action is less than or equal to $a+4T\Vert \bar{G}_{{\text {inter}},t}\Vert _{C^0}$. On the other hand, working with the modified Hamiltonian $\bar{G}_t$ instead of $G_t$ allowed us to employ a maximum principle as well as the $C^0$-bounds for the p-component of Floer curves established in [5]. $\square$

Remark 4.7

We end this paper with the following remarks.

• (1) By replacing for every $\ell \in {\mathbb {N}}$ the interaction Hamiltonian $F_{{\text {inter}},t}$ by its finite-dimensional approximation $F^{\ell }_{{\text {inter}},t}$ one obtains $N+1$ different $\phi ^A_T$-periodic orbits $\bar{u}^{\ell ,-}_1,\ldots ,\bar{u}^{\ell ,-}_N,\bar{u}^{\ell ,+}_N$ of $\bar{G}^{\ell }_t$ (and hence of $G^{\ell }_t$). While in this finite-dimensional case the use of Lemma 4.6 can be avoided, one might ask whether Theorem 2.1 already follows after proving that for each $\alpha =1,\ldots ,N$ a subsequence of $\bar{u}^{\ell ,\pm }_{\alpha }$, $\ell \in {\mathbb {N}}$ converges to a $\phi ^A_T$-periodic orbit $\bar{u}^{\pm }_{\alpha }$ of $\bar{G}_t$ ($G_t$) with contractible image in Q as $\ell \rightarrow \infty$. Apart from the fact the proof of the necessary analogue of 4.6 for $\phi ^A_T$-periodic orbits still encounters the same small divisor problem, this is not enough to deduce that $\bar{u}^-_1,\ldots ,\bar{u}^-_N,\bar{u}^+_N$ are pairwise different, as for the general cuplength result (we do not assume that $\pi _1(Q)$ is finite and hence the cuplength of $\Lambda ^{{\text {contr}}}Q$ can be finite) the limiting orbits cannot be distinguished by their symplectic action a priori. On the other hand, the argument with the limiting behaviour of the evaluation maps to $\Lambda ^{{\text {contr}}} Q$ that we use in our proof of Theorem 4.1 crucially relies on Gromov–Floer compactness.

• (2) In a different direction, using the Fredholm theory results of our recent paper [8], we claim that one can give a proof of Theorem 2.1 (and hence of Corollary 2.3) which completely avoids finite-dimensional approximations. Indeed one can directly establish the existence of the moduli space

  $$\begin{aligned}{\mathcal {M}}^{\infty ,\le a}_{\tau }=\left\{ \widetilde{u}=(\widetilde{u}_M,\widetilde{u}_{{\mathbb {H}}}):{\mathbb {R}}^2\rightarrow T^*Q\times {\mathbb {H}}:\,\,(*1),(*2),(*3),(*4)\right\} \end{aligned}$$

  with $\widetilde{u}(s,\cdot )$ having contractible image in Q and

  $$\begin{aligned}(*1):\, \partial _s\widetilde{u}+J(\widetilde{u})\partial _t\widetilde{u}+\nabla F_{{\text {part}},t}(\widetilde{u})+\sigma _\tau (s)\nabla \bar{G}_{{\text {inter}},t}(\widetilde{u})=0,\end{aligned}$$

  and $(*2)$, $(*3)$, $(*4)$ as before. In analogy with Proposition 4.4 one can prove that for every $\tau \ge 0$ we have

  $$\begin{aligned} {\text {ev}}_{1,\infty ,\tau }^*\theta _1\cup \cdots \cup {\text {ev}}_{N,\infty ,\tau }^*\theta _N\ne 0\in H^*({\mathcal {M}}^{\infty ,\le a}_{\tau },{\mathbb {Z}}_2), \end{aligned}$$

  again as long as $a\in {\mathbb {R}}$ is large enough, where we now use the natural evaluation maps ${\text {ev}}_{1,\infty ,\tau },\ldots ,{\text {ev}}_{N,\infty ,\tau }$ from ${\mathcal {M}}^{\infty ,\le a}_{\tau }$ to the space $\Lambda ^{{\text {contr}}} Q$. For the latter the key ingredient is that for $\tau =0$ the moduli space ${\mathcal {M}}^{\infty ,\le a}_0$ can still be identified with the moduli space of Floer curves for which Cieliebak stated his Theorem 7.6 in [5].