Jun 14, 2022

Urs Frauenfelder

## Introduction

The V-shaped symplectic homology was introduced in a joint effort with Cieliebak and Oancea where it was shown to be isomorphic to Rabinowitz Floer homology. This work allowed computations of Rabinowitz Floer homology of cotangent bundles to be successful. In Tate Hochschild cohomology, Rivera and Wang conjectured that the algebraic structures discovered in V-shaped symplectic homology by Cieliebak and Oancea would coincide with their computations.

In this blog, we will introduce an intermediate action functional, the V-shaped action functional with delay. It shares some features of Rabinowitz action functional and some features of the V-shaped action functional. This intermediate action functional will shed some light on the ongoing scientific debate about algebraic structures in Rabinowitz Floer homology and their connection with Tate Hochschild homology.

## The V-shaped Action Functional with Delay

We shall now discuss the symplectization of a contact manifold. We will fix a V-shaped function deﬁned in a particular way, and abbreviate the free loop space of the symplectization. The V-shaped action functional with respect to the V-shaped function for a loop is deﬁned in a precise manner.

For the V-shaped action functional with delay, We simply interchange the order of integration and applying the function in the second term. We can interpolate between the two functionals by defining them in a certain way. A key lemma tells us that the critical points and their actions do not depend on the parameter and are in natural one-to-one correspondence with generalized periodic Reeb orbits on the contact manifold.

However, although the functionals have the same critical point, their gradient ﬂow lines are different. We will fix a smooth family of SFT-like almost complex structures and take the gradient with respect to the metric obtained from the family. On the other hand, gradient ﬂow lines of the V-shaped action functional with delay are solutions of a different problem, and hence it's named V- shaped action functional with delay.

## Further Discussions

We will then recall the Rabinowitz action functional on a symplectization, which is given by a precise formula. The gradient ﬂow equations of the Rabinowitz action functionals are delay equations as well. All four problems i.e.; equations, are perturbed Cauchy-Riemann equations where the perturbation is in the Reeb direction.

We shall explore a natural projection down the line, and solutions of a problem can be uniquely characterized as solutions of the perturbed Cauchy-Riemann equation. It was shown previously that there is a natural bijection between ﬁnite energy gradient ﬂow lines of the two Rabinowitz action functionals. The following theorem tells us that the same is true for the gradient ﬂow lines of the V-shaped action functional with delay.

In conclusion, there is a natural bijection between the finite energy gradient flow lines of the V-shaped action functional with delay and the ones of the Rabinowitz action functional. This is a significant finding in understanding the V-shaped action functional with delay, potentially paving the path for further research in this area.

## Acknowledgments

The author acknowledges partial support by DFG grant FR 2637/2-2.

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