text:
Research Strategic Research Areas Oceans
Pilot programme on Arctic marine environment
Pilot programme on Arctic marine environment - NTNU Oceans
Arctic banner
Photo: SAMCot
NTNU Oceans Pilot on Arctic marine environment
- project addressing Risk, Reliability and Ice data in the Arctic marine environment
Data and knowledge about the marine environment in arctic regions are lacking. How then to achieve a sufficient safety level for humans and the environment at an acceptable cost? Through multidisciplinary research, combining expertise in civil, marine, cybernetics, remote sensing, mechanical, statistics, economy, management, and history, science based tools for rational design of risk and safety in the Arctic Marine and Coastal environment are to be developed. Typical end-users are shipping companies, local communities, oil companies, regulatory bodies, consultants and construction companies.
Projects at NTNU:
Reliability methods and long-term data
There are so far four PhD students and two post-docs working here. The main objective is to establish the scientific and engineering basis for methods to estimate structural reliability with respect to available long-term data.
PhD students Ilija Samardzija and Chana Sinsabvarodom both work with probabilistic approach for the design on fixed structures in ice. Ilija addresses a probabilistic model to estimate ice ridge action on fixed structures. He uses a combination of in-situ measurements and knowledge about the relevant physical processes to establish the relevant sea ice ridge statistics. Chana goes deeper in the probabilistic methods and studies the correlation between different sea ice properties and how these can be included in the estimation of ice actions.
Post-Doc Joshua Turner is engaged through the MarTERA project FATICE and tries to answer the questions: Which ice conditions give ice loads and How can we quantify these?
Post-Doc Roghayeh Shamshiri works with satellite-based measurements of ice and tries to understand how Synthetic Aperture Radar (SAR) data, that is sampled roughly every day can be used to estimate long term sea ice statistics of ice thickness, ice drift and ice classification. She will compare statistics derived from SAR with locally measured data from underwater moored Upward Looking Sonars (sampled every second).
The two PhD students Mostafa Hoseini and Mahmoud Rajabi (both at IBM) work in collaboration with the Small Satellite laboratory at NTNU. They both work to support Arctic and Ocean monitoring focusing on the use of hyperspectral camera in small satellites. In addition, the theory and design of a GNSS-R (Global Navigation Satellite System - Reflectometry) sensor is being investigated for ocean observations (sea ice thickness, sea surface height and wave height) and is being qualified for inclusion in NTNU's SmallSat missions. GNSS-R involves making measurements from the reflections from the Earth of GNSS/GPS satellite signals. The designed GNSS-R sensor provides a high spatiotemporal resolution monitoring of the Arctic and Oceans, as the key components of the Earth's climate system.
Supervisors are Knut V. Høyland (IBM), Bernt Leira (IMT), Egil Eide (IE) and Hossein Nahavandchi (IBM).
Arctic transport
When sailing through ice, risk may fluctuate sharply with varying ice conditions, and understanding this risk is important to avoid accidents with a minimal cost.
To support decision making at high latitudes, Sheng Xu, a PhD student at the Department of Marine Technology attempts to link ice conditions, ice loads, and risks. Currently he tries to gain a better understanding of what is happening with ice thickness along the Norther Sea Route as well as to addresses challenges of risk modeling for strategic and tactical planning of Arctic transportation.
Research at the Department of Industrial Economics and Technology Management (IØT) focuses on optimal design and operations of Arctic maritime transportation systems. PhD candidate Mingyu Li uses methodologies such as Robust Optimization and Stochastic Programming to analyze the impact of uncertainty in environmental conditions, such as ice conditions or the duration of the sailing season, on decisions regarding all levels of the transportation system. Examples of relevant problems are fleet-size-and-mix problems, Annual Delivery Plans for LNG, and ice routing problems.
Supervisors are Ekaterina Kim (IMT) and Peter Schütz (IØT).
Coastal infrastructure in a changing climate
The global warming affects the coastal climate in two ways. First, the increasing air temperatures warm up the permafrost and increase the active layer thickness. Secondly, the decreasing sea ice content gives more open water and a harsher wave climate in coastal waters. These two phenomena enhance the coastal erosion rates and by this threatens the stability and operability of coastal infrastructure such as harbours and terminals.
NTNU collaborates closely with Denmark's Technical University (DTU) on these topics and four PhD students are engaged in projects related to them. Two of them through the EU Horizon 2020 project Nunataryuk where DTU and NTNU are responsible for the work package on Coastal Structures.
PhD student Hongtao Li (NTNU-IBM/DTU) works with the interaction between waves and sea ice. He developed advanced data-processing techniques to analyse the laboratory-data on wave-ice interactions, which were gathered during the Hydralab+ Transnational Access project: Loads on Structure and Waves in Ice (LS-WICE) conducted in 2016 at Hamburgische Schiffbau-Versuchsanstalt GmbH (HSVA). At first, Li focused on the phenomenon of wave-induced ice-ice collisions and he provided estimates of the contact durations and the restitution coefficients (a measure of energy dissipation). Currently, Li is working on the wave-induced flexural motion of ice floes and the consequent ice break-up du to wave actions.
PhD student Dennis Monteban (DTU/ NTNU-IBM) works with measurements and modelling of Arctic coastal environments. His primary research interest is to utilise remote sensing to improve our understanding of the physical environment of the Arctic and to enhance our ability to model the Arctic environmental conditions. So far, he succeeded in using Sentinel-1 TOPS mode to provide spatiotemporal observations of wave dispersion within sea ice. He applied his new techniques to a study site in the Barents Sea where in-situ data became recently available from the Barents Sea Metocean and Ice Network (BaSMIN) measurement campaign. This measurement program was performed by Fugro, on behalf of Equinor (former Statoil). Currently, Monteban is working on enhancing iceberg drift modelling in the Barents Sea with estimates of icebergs release rates and size characteristics at the major sources in the Barents Sea using Sentinel-1 and Sentinel-2.
PhD student Akshanul Islam (NTNU-IBM) works with numerical modelling of short- and long-term erosion of permafrost coastal bluffs. The primary research interest for Islam is to build a high-fidelity numerical model capable simulating wave transformation in the surf-zone in addition to the thermal and mechanical interaction between the waves and frozen bluffs. The model should be able to reproduce the know Arctic-specific coastal erosion phenomena (i.e., thermoabrasion and thermodenudation) and thus it should be a generic model valid to all sites. The high-fidelity model will be used as a base to derive long-term prediction models. Islam is making a good progress in building the numerical model and recently he participated in field work at Baydaratskaya bay northwest Russia and he gathered valuable data for the validation of his model.
Chuangxin Lyu (IBM) works with field and laboratory experiments to quantify the effect of thawing permafrost on the bearing capacity and mechanical behavior of frozen soil. The field work will be done in Svalbard in close collaboration with UNIS.
Supervisors are Raed Lubbad (IBM), Jens Olaf Pedersen (DTU Space) and Gustav Grimstad (IBM).
Oil spill in Arctic marine environment
Climate change is leading to decline and thinning of Arctic sea ice, potentially increasing the commercial and shipping activities in Arctic marine waters. In addition, expanding maritime activity in the region will likely result in greater seasonal use by tankers and bulk carriers, fishing fleets and cruise ships. This foreseen rise in activity raises additional concerns about the risk of oil spills. Recent interest in developing the rich oil and gas resources in Arctic marine waters will also contribute to increased likelihood of spill accidents.
PhD student Hosseinreza Abbasi (NTNU-IBM) investigates the different physical processes associated with oil spill fate, spread and transport in the Arctic waters. The objective of Abbasi's PhD study is to develop a novel numerical model to forecast the fate, spread and transport of spilled oil in the marginal ice zone.
Supervisors are Raed Lubbad (IBM), Olga Shipilova (DNVGL)
Charting the Translocal Arctic
This project gathers scientists to address how Arctic transformations might affect human well-being in global environments.The Arctic region is changing rapidly. These changes are not only material, but also system-wide: Because Arctic biophysical and climatic systems are deeply intertwined with the lifeworlds of the planet, dramatic changes in this sensitive region are likely to be felt elsewhere.
Only sparse knowledge exists about the interconnections between the Arctic's changing environment and cultural systems of other regions across the world. Such interconnections may be referred to as sharting the translocal Arctic which draws attention to significant interactions within and outside the Arctic.
Understanding the Arctic translocally opens up new understandings of networked environmental effects and the various infrastructures through which its resources and representations are produced and circulated. Thus, this research advances systems-level understanding of links between the Arctic's rapidly changing environmental, social and cultural systems with other regions and networks across the world.
Arthur Mason project leader
Climate change - its biological and physical consequences
(NTNU in MOSAiC – the world's largest Polar expedition ever)
MOSAiC - Multidisciplinary drifting Observatory for the Study of Arctic Climate could very well be the largest-scale Arctic research expedition of all time: 20 September 2019 the German research icebreaker Polarstern left Tromsø, Norway, and will spend the next year drifting through the Arctic Ocean, trapped in the ice.
A total of 600 people from 19 countries, who will be supplied by other icebreakers and aircraft, will participate in the expedition – and several times that number of researchers will subsequently use the data gathered to take climate and ecosystem research to the next level. The mission will be spearheaded by the Alfred Wegener Institute, Helmholtz Centre for Polar and Marine Research (AWI).
NTNU, the departments of Civil and Environmental Engineering (IBM) and Biology (IBI) participate in MOSAiC in the framework of the NFR-funded project HAVOC, the Norwegian contribution to MOSAiC.
Researchers from HAVOC will join the MOSAiC legs 2, 4 and 5 to study ice ridge consolidation (ice physics) as well as community composition and biodiversity of ice biota associated to ridges.
The responsible WP leaders/supervisors from NTNU are Knut V. Høyland (IBM), Nicole Aberle-Malzahn (IB) and Pedro de la Torre (IMT).
Cooperative marine operations
Ice monitoring and management
Robust structures and processing technology
Safety, risk management, ethics and politics
Biogeochemical processes
person-portlet
Pilot Leader
Knut Vilhelm Høyland Professor
+47-73594632 knut.hoyland@ntnu.no Department of Civil and Environmental Engineering
Professor Høyland's blog post about the pilot
(in Norwegian)


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text:
\section{Introduction}

In earlier papers (\cite{Mazenc:2019cfg,Cardy:2018sdv,Smirnov:2016lqw,Cavaglia:2016oda}), it was shown that the infinitesimal deformation of a CFT by the determinant of its energy-momentum tensor, a ``$T\bar{T}-$ deformation,''  generates a flow in the space of QFTs along the deformation parameter (see \cite{Jiang:2019epa} for a review of $T\bar{T}$-deformations in 2D). This flow follows a hydrodynamic equation (the inviscid Burgers equation) for the energy levels of the deformed QFT \cite{Cavaglia:2016oda}. It has also been shown that these $T\bar{T}$-deformations, widely studied on flat space previously, could be carried out and formalized in curved space as well \cite{Brennan:2020dkw,Dubovsky:2017cnj,Gross:2019ach,Tolley:2019nmm,Mazenc:2019cfg}. Moreover, \cite{Mazenc:2019cfg} have proven that the partition functions of the QFTs along the flow obey a diffusion-like differential equation in the deformation parameter by introducing a kernel relating the partition functions of deformed and undeformed theories. In that paper, the aforementioned kernel was presented as an ansatz that originated in radial wavefunctions in 3D gravity, which had been shown to be related to CFT partition functions. The same year, \cite{Tolley:2019nmm} drew an explicit parallel between $T\bar{T}$-deformations of QFTs and a coupling to massive gravity. This provides some solid motivation as to why gravity is strongly related to $T\bar{T}$-deformations. Furthermore, since gravity is also very closely related to 3D Chern-Simons theory with gauge group $SL_2$ \cite{Witten:1988hc}, Chern-Simons theory seems like a solid ground to build some further intuitive understanding of $T\bar{T}$-deformations.

In this paper, we will introduce a new perspective on $T\bar{T}$-deformations on curved space, based on Chern-Simons theory. We will show that it is possible to couple a 2D CFT that factors into chiral and antichiral sectors to a 4D version of Chern-Simons theory (introduced by Costello et al. \cite{Costello:2017dso}). We will then argue that the coupling to 4D CS can be used to engineer $T\bar{T}$-deformations, at least perturbatively. The basic idea is to look at the coupled theory at low energy and identify an interaction term proportional to $T\bar{T}$ that deforms the effective 2D action. The coefficient in front of $T\bar{T}$ then corresponds to the metric of the space on which the CFT that undergoes the deformation lives. We will use this to show that our approach can be used to engineer $T\bar{T}$-deformed theories in a variety of curved backgrounds, within a common framework.

The paper is organized as follows. In Section \ref{sec::ttbar}, we shall first swiftly go over the principle behind $T\bar{T}$-deformations in order to give the basic definition and notation that we will be using in the following sections. We shall then define a 4D Chern-Simons theory framework following Costello's prescription and insert two 2D chiral and antichiral CFTs as surface defects in this 4D bulk, and give an example of how such a framework has already been used by \cite{Costello:2019tri} to recover deformations similar to $T\bar{T}$. Section \ref{sec::integout} will start with a brief reminder of how the 4D Chern-Simons setup that we have relates to gravity, before getting to the heart of the matter. Coupling the two CFTs through their stress-energy tensor to the gauge field of the bulk theory and integrating out the gauge field then allows us to recover a 2D effective field theory with an interaction term proportional to $T\bar{T}$ whose dependence in 2D spacetime corresponds to a metric term. We will follow this procedure with different gauge groups and prescriptions for our 4D Chern-Simons theory. We will see in Section \ref{sec::TTbarCP1} that using gauge group $SL_2(\mathbb{C})$ and a holomorphic 1-form $dz$ in the definition of 4D CS theory one can recover a $T\bar{T}$-deformation on $\mathbb{CP}^{1}$ or on $AdS_{2}$; while using the same gauge group but the form $dz/z$ in Section \ref{sec::TTbarsausage} will yield a deformed theory on a space endowed with the so-called sausage metric. Finally, a quick look at another gauge group with similar presentation, namely that of the isometries of the plane, will provide us with a way of getting a deformed theory on a space equipped with Witten's ``cigar metric,'' as shown in Section \ref{sec::cigar}

These results constitute a proof of concept showing that we can interpret some $T\bar{T}$-deformations of theories on curved space as coming from conformal surface defects in a well-tuned 4D gauge theory. It would be interesting in future work to fully determine on which kinds of geometries $T\bar{T}$-deformations can be recovered with this approach.

\newpage
\section{Brief review of $T\bar{T}$ deformations and 4D Chern-Simons Theory}\label{sec::ttbar}
\subsection{$T\bar{T}$ deformation of conformal field theories}

The principle of $T\bar{T}$-deformations (see \cite{Jiang:2019epa} for further references and developments) is quite simple at first order in perturbation theory: given a 2D conformal field theory with stress tensor $T$ in complex coordinates, one can build an irrelevant operator by taking the determinant of the energy momentum tensor:
\vspace{7mm}
\begin{equation}
    \text{det}\left(\begin{matrix}
    T_{zz} & 0 \\
    0 & T_{\bar{z}\bar{z}}
    \end{matrix}\right)=T_{zz}T_{\bar{z}\bar{z}}\mydef T\bar{T}\ \ ,
\end{equation}

\vspace{4mm}
\noindent and infinitesimally deforming the 2D CFT with this operator:
\begin{equation}
    S[\lambda]=S[0]+\lambda\int_{2D}\sqrt{-g}\,T\bar{T}\ \ ,
\end{equation}
with $\lambda$ a small deformation parameter. It has been shown that such deformations in flat space do not affect integrability (inherited from the seed theory) \cite{Smirnov:2016lqw} and that these deformations can be generalized to curved space \cite{Mazenc:2019cfg}. Of course, determining a non-perturbative formula for the deformed action or the deformed partition function is incomparably more involved and we will not attempt to describe it in this paper.

Previous works \cite{Dubovsky:2018bmo} proved that the energy levels in the spectrum of the deformed theory (when compactifying on a cylinder) satisfy a well-known non-linear differential equation (Bürger's equation) with dependence in the deformation parameter and the radius of the cylinder, while the partition function itself satisfies a close-to-diffusion flow equation (with the deformation parameter playing the role of "time"). Although we will not delve any further into these considerations, we will also mention that \cite{Iliesiu:2020zld,Dubovsky:2017cnj} showed that $T\bar{T}$-deformations can be interpreted - with some relabeling - as a gravitational perturbation of Jackiw-Teitelboim (JT) gravity, and that through AdS/CFT correspondence in the absence of matter fields, a $T\bar{T}$-deformation of the boundary 2D CFT corresponded to a theory of gravity in AdS at a finite cutoff determined by the deformation parameter. Here, however, we will not follow this route, and will endeavor instead to find a simple way of obtaining $T\bar{T}$-deformed theories from higher dimensional gauge theories.

\subsection{4D Chern-Simons theory}\label{sec::4DCST}

4D Chern-Simons is a partially holomorphic extension of 3D Chern-Simons introduced by Costello \cite{Costello:2017dso}. It makes sense in particular on spaces of the form $\Sigma\times \mathcal{C}$, with $\Sigma$ a 2-dimensional oriented surface, and $\mathcal{C}$ a complex manifold endowed with a meromorphic one-form. In this paper, we will choose $\Sigma$ to be various spaces (like $\mathbb{CP}^1$) parametrized with a complex variable for convenience (we will not be explicitly using any complex structure on it), and we will use two different $\mathcal{C}$, namely $\mathbb{C}_{z}$ (where the subscript refers to the complex coordinate we put on that space) endowed with the 1-form $dz$ and $\mathbb{C}^{\times}_{z}$ endowed with the 1-form $dz/z$. The gauge part of the action reads:

\begin{equation}
    S=\int_{\Sigma\times\mathcal{C}}\omega\wedge CS(A)\ \ ,
\end{equation}
where the $\omega$ is the aforementioned 1-form on $\mathcal{C}$, and where $CS(A)$ is the standard Chern-Simons 3-form:
\begin{equation}
    CS(A)=\Tr{A\wedge \text{d}A + \frac{2}{3}A\wedge A\wedge A}\ \ ,
\end{equation}

\noindent where the trace is over the fundamental representation of the gauge group.Moreover, it is worth mentioning that since CS theory is topological in the first place, this 4D version of Chern-Simons depends topologically on $\Sigma$ (and holomorphically on $\mathcal{C}$).

Taking the slightly easier example of $\mathcal{C}=\mathbb{C}_z$, the action reads
\begin{equation}
    S=\int_{\mathbb{C}_z\times\mathbb{C}_w}\text{d}z\wedge \Tr{A\wedge \text{d}A + \frac{2}{3} A\wedge A\wedge A}\ \ ,
\end{equation}
and it is useful to note an extra gauge freedom (on top of the usual gauge symmetry associated to Chern-Simons theory) coming from the $dz\wedge$. The usual gauge symmetry is captured by the transformation

\begin{equation}
    A\to A+d_{A}\chi\ \
\end{equation}
for $\chi$ an infinitesimal parameter, where we define $d_{A}$ to be
\begin{equation}\label{eq::gaugetsfo}
    d_{A}\chi \mydef d\chi + [A,\chi]\ \ ,
\end{equation}
while the outstanding one coming from wedging with the holomorphic one-form reads
\begin{equation}
    A_z\to A_z + \phi(z,\bar{z})\ \
\end{equation}
for any function $\phi$ of $z$ and $\bar{z}$. The latter equation makes the $A_z$ component of the action irrelevant, as the action is then invariant under any shift of that component (including by negative itself)
.

Thus, we might as well fix the gauge by choosing $A_z=0$, making our 1-form $A$ effectively read:
\begin{equation}
    A=A_{\bar{z}}\text{d}\bar{z}+A_w \text{d}w+A_{\bar{w}}\text{d}\bar{w}\ \ .
\end{equation}
Moreover, we can further gauge fix the usual symmetry by choosing a holomorphic gauge:
\begin{equation}
\partial_{z}A_{\bar{z}}=0\ \ ,
\end{equation}
which happens to be related to the Lorentz gauge
\begin{equation}
\partial_{z}A_{\bar{z}}+ \partial_{w}A_{\bar{w}}+ \partial_{\bar{w}}A_w=0
\end{equation}
by using the topological character of Chern-Simons theory to infinitely rescale $\mathbb{C}_w$.
If we now add to this the asymptotic boundary condition
\begin{equation}
    \lim_{|z|\to \infty}A_{\mu}(z)= 0 \ \forall \mu\ \ ,
\end{equation}
we get
\begin{equation}
    A_{\bar{z}}=0\ \ ,
\end{equation}
leaving only two effective components to our gauge field:
\begin{equation}
    A_{GaugeFixed}=A_w \text{d}w + A_{\bar{w}}\text{d}\bar{w}\ \ .
\end{equation}

\subsection{$J\bar{J}$ deformations}

Recently, Costello and Yamazaki showed that such a 4D Chern-Simons setup could be used to recover a kind of deformations that looks very similar to $T\bar{T}$-deformations, namely, $J\bar{J}$ deformations\cite{Costello:2019tri}.

The authors introduce what they call ``order defects", which they define to be (surface) defects on which the degrees of freedom are coupled to the bulk theory. Their 2D theory (on the defects) having a global G-symmetry, they choose a coupling involving its current in order for the symmetry to remain manifest. Schematically,
\begin{equation}
    \int_{w\in \Sigma} A\, J\ \ ,
\end{equation}
where $J$ denotes the current of the G-symmetry and $A$ the 4-dimensional gauge field.

By inserting chiral and anti-chiral order defects at positions $z$ and $z'$ in the holomorphic plane, coupled through the chiral and anti-chiral currents respectively, and by then integrating out the gauge field, they recover a 2D effective theory with an interaction term that reads
\begin{equation}
    \mathcal{L}^{eff}_{int}=\int_{w\in \Sigma} r_{ab}(z-z')J_a (w,\bar{w}) \bar{J}_b (w,\bar{w})\ \ ,
\end{equation}
where $r_{ab}$ is the so-called classical $r$-matrix involved in the scattering process and can just be thought of as the propagator between two components of the 1-form gauge field.

This result inspired us to wonder whether the same can be done for $T\bar{T}$-deformations, so as to recover in a very similar fashion the deformation term (to first order) from a 4D Chern-Simons setup with surface defects involving an $AT$ coupling.

\section{How to get 2D $T\bar{T}$-deformed theories from 4D Chern-Simons}\label{sec::integout}

\subsection{Brief review of the connection to gravity}\label{sec::gravity}

Although this is not the focus of this paper, it is interesting to keep in mind that our 4-dimensional setup is strongly related to theories of gravity. This suggests the existence of a parallel between $T\bar{T}$-deformations of CFT's as studied in our paper and the appearance of the same deformations in a JT gravity context, for instance. Let us therefore give a lightning review on how 4D Chern-Simons relates to gravity.

Starting with the 4-dimensional setup described in the previous Section, the first thing we can do is to compactify the theory on a circle in the $z$ (holomorphic) direction. In other words, we are doing a Kaluza-Klein reduction in the holomorphic direction. If the circle we compactify on is small enough, the fields can be considered constant over it (we throw away the KK modes). Our holomorphic plane becomes (through the usual conformal transformation $z\mydef e^{h+i\theta},\bar{z}
\mydef e^{h-i\theta}$) a cylinder $\mathcal{C}$ with height coordinate $h$ and angle $\theta$. In our gauge ($A_z = 0$) the pullback of the 1-forms and form components by that map yields
\begin{align}
    dz&=(dh+id\theta)e^{h+i\theta}\nonumber \\
    d\bar{z}&=(dh+id\theta)e^{h+i\theta}\nonumber \\
    A_h&=e^{h+i\theta}A_{\bar{z}}\nonumber\ \ ,
\end{align}
and our action becomes
\begin{align}
    \int_{z,\bar{z}\in \mathbb{C}\, , \, w,\bar{w}\in \Sigma}\frac{dz}{z}\wedge CS\left[A_{\bar{z}},A_w,A_{\bar{w}}\right] = \int_{\theta\in S^{1},h\in \mathbb{R}\, , \, w,\bar{w}\in \Sigma}d\theta\wedge CS\left[A_h,A_w,A_{\bar{w}}\right](h,w,\bar{w})\ \ .
\end{align}

This is a 3D (analytically continued) Chern-Simons for $SL_2(\mathbb{C})$ which, with the choice of a contour, becomes a Chern-Simons theory for $SL_2(\mathbb{R})$. The latter theory is related to (half of) 3D gravity using the standard identification of Chern-Simons gauge field with vielbein and spin connection \cite{Witten:1988hc}.

We can now go one step further and do yet another compactification, this time over a circle in the topological direction. This yields a 2D $BF$ theory, where $F$ is the field strength of the gauge field and where the auxiliary field $B$ is its monodromy around the circle over which we just compactified. It is then well known that 2D JT gravity and BF theory are equivalent \cite{Isler:1989hq,Chamseddine:1989yz}.

Thus, from 4D Chern-Simons theory, one can recover 2D JT gravity using two Kaluza-Klein compactifications. It would be interesting to use this to explicitly relate $T\bar{T}$ deformations in all these formalisms. In this paper, however, we will focus on establishing how to derive $T\bar{T}$-deformations of 2D CFT's using 4D Chern-Simons.

\subsection{Coupling the CFT to the bulk Chern-Simons theory}
After having introduced the 4D version of Chern-Simons theory, we are now ready to add to the mix the 2D CFT of interest. The incorporation of a 2D (chiral + antichiral) CFT will be carried out as follows: at a given point $z_0$ in $\mathcal{C}$, we insert a free 2D chiral CFT, and at a point $z_1$ we insert its free antichiral counterpart. Finally, we couple both of them to the gauge field $A$ (from Chern-Simons) through their stress tensor, as can be expected in order to recover $T\bar{T}$-deformations. As the coupling process is a little complex, we will first write the coupling terms and then explain why such coupling makes physical sense and yields a gauge-invariant action. To perform it, we need a gauge group generated by three elements (which we call $e,f$ and $h$). The interaction part of the action will read:

\begin{equation}
    S_{int}\mydef\int_{\Sigma,\{z=z_0\} \in \mathcal{C}}\left(A^{e}_{\bar{w}}\mathcal{T}^{f}+A^{f}_{\bar{w}}\mathcal{T}^{e}+A^{h}_{\bar{w}}\mathcal{T}^{h}\right)+\int_{\Sigma,\{z=z_1\} \in \mathcal{C}}\left(A^{e}_{w}\mathcal{\bar{T}}^{f}+A^{f}_{w}\mathcal{\bar{T}}^{e}+A^{h}_{w}\mathcal{\bar{T}}^{h}\right)\ \ ,
\end{equation}
with $\mathcal{T}^{a}=c_{a}w^{p_a}T\mydef \mathcal{V}^{a}T$, and  its antichiral counterpart $\bar{\mathcal{T}}^{a}=c'_{a}\bar{w}^{p'_a}T\mydef \bar{\mathcal{V}}^{a}T$. Here $c,c'$ are complex coefficients, $p,p'$ are natural numbers, and all depend on the choice of gauge group. We want to use the property that $T$ (resp. $\bar{T}$) acts as an operator as $\partial_w$ (resp. $\partial_{\bar{w}}$) to build another copy of the Lie algebra (typically resembling a Virasoro algebra) generated by the $\mathcal{T}^{a}$, such that $e^{-S_{int}}$ stays invariant under gauge transformations.

\subsection{Explicit proof of gauge-invariance in the $SL_2$ case}
Let us work out the gauge-invariance for the case where the gauge group is $SL_2$. By construction, the 4D CS action is invariant under the chosen gauge group's transformations, but the coupling terms themselves are not classically gauge invariant. This is however not a problem as long as the variation of $S_{int}$ is equivalent to zero as an operator. To engineer some $\mathfrak{sl}_2$ gauge invariance, as just mentioned, we use the fact that integrating the stress tensor on a closed contour around an operator acts like a derivative with respect to the complex variable. This allows us to construct the following (Virasoro-like) $\mathfrak{sl}_2$ generators from the stress tensor: $\mathcal{T}^{a}\mydef \mathcal{V}^{a}T$ (with $a\in\left\{e,f,h\right\}$) with

\begin{align}\label{eq::virasorocoeffsCP1}
    \mathcal{V}^{e}(w)&=1\nonumber \\
    \mathcal{V}^{f}(w)&=-w^2 \\
    \mathcal{V}^{h}(w)&=-2w\nonumber \ \ ,
\end{align}
and for the anti-holomorphic counterparts:
\begin{align}\label{eq::virasorocoeffsCP1c}
    \bar{\mathcal{V}}^{e}(\bar{w})&=\bar{w}^2\nonumber \\
    \bar{\mathcal{V}}^{f}(\bar{w})&=-1 \\
    \bar{\mathcal{V}}^{h}(\bar{w})&=2\bar{w}\nonumber \ \ .
\end{align}
Note that we could have taken the same expression for the holomorphic and antiholomorphic vertices in principle, but this prescription will yield more interesting results.

Let us now explicitly check this invariance for the chiral part of the action (the antichiral part will behave in the exact same way), and along only one "direction", say `$e$' (such that $\chi(w,\bar{w})\equiv \chi_{e}(w,\bar{w})\,e$) - since everything is linear, it suffices to check invariance along each generator independently. Note that we shall henceforth use interchangeably $\bar{A}\mydef A_{\bar{w}}$ and $A\mydef A_{w}$ for practical reasons.

Consider the resulting transformation of the barred components of the gauge field:
\begin{equation*}
    A_{\bar{w}}^{a}\to A_{\bar{w}}^{a}+\bar{\partial}\chi^{a} + [A_{\bar{w}},\chi]^{a}\ \ .
\end{equation*}
In particular, if $\chi$ is only along the Lie algebra generator $e$:
\begin{align}\label{eq::varact}
    \delta_{\chi^e}A_{\bar{w}}^{e}&=\bar{\partial}\chi+ 2 A_{\bar{w}}^{h} \chi_e \nonumber\\
    \delta_{\chi^e}A_{\bar{w}}^{f}&=0 \\
    \delta_{\chi^e}A_{\bar{w}}^{h}&=- A_{w}^{f}\chi_e \ \ . \nonumber
    \end{align}
This yields:
\begin{equation}
    \delta_{\chi^e}S_{int}=\int \left(\bar{\partial}\chi^e+2\bar{A}^{h}\chi^{e}\right)\mathcal{T}^{f}+0\cdot \mathcal{T}^{h}+\left(-\bar{A}^{f}\chi^{e}\right)\mathcal{T}^{e}\ \ .
\end{equation}
Thus,
\begin{equation}\label{eq::varexp}
    \delta_{\chi^{e}}\left(e^{-S_{int}}\right)=-\delta_{\chi^{e}}S_{int}+\left(\delta_{\chi^{e}}S_{int}\right)S_{int}+ \dots
\end{equation}
Now, the game is to show that part of the second term in the variation of the exponential cancels out the first one. This is possible because one of the ``sub-terms" of the second term (the one containing $\bar{\partial}\chi^{e}$) is linear in $\bar{A}$ (like the terms we want to cancel out) and quadratic in $\mathcal{T}$, which allows us to use the OPE of the stress tensor
\begin{equation}\label{eq::opeTT}
    T(w')T(w)\sim \frac{\partial T(w)}{w'-w}+2\frac{T}{(w'-w)^2}+\frac{c/2}{(w'-w)^4}
\end{equation}
to recover the right kind of terms, up to integration by parts. This will show (up to generalization to the next orders in $\bar{A}$) that one term at order N+1 in $\bar{A}$ cancels all the remaining terms at order N in $\bar{A}$, that is, the terms that were not used to cancel out the order N-1.

Let us work this out. For the sake of simplicity, since $A=A_w$ does not appear in the part of the action we are considering (only $\bar{A}$), we can partly fix the gauge and impose:
\begin{equation}\label{eq::gaugefix}
    A=0 \,\, \implies \partial \chi = 0\ \ ,
\end{equation}
where the implication comes from disallowing gauge transformations to modify $A$. This allows us to effectively set $[A,\bar{A}]=0=\bar{\partial} A$, which yields a modified version of the equations of motion for $A_{\bar{z}}$ (obtained pre-gauge fixing) in our Chern-Simons theory:
\begin{equation}\label{eq::eomazbar}
    \partial \bar{A} + \bar{\partial}A+[A,\bar{A}]=\partial \bar{A} = 0\ \ .
\end{equation}
This equation will be useful later in the proof.

Combining Equations \ref{eq::varact} and \ref{eq::varexp} we get a quadratic variation that can be decomposed into two kinds of terms. Terms quadratic in $A$, which will be cancelled by the cubic variation, and terms in $\bar{\partial}\chi^{e}$ that read:
\begin{equation}\label{eq::proofhalf}
    \int_{w'}\int_{w}\bar{\partial}\chi^{e}(w')\,\mathcal{T}^{f}(w')\left(A^{e}\mathcal{T}^{f}+A^{h}\mathcal{T}^{h}+A^{f}\mathcal{T}^{e}\right)(w)\ \ .
\end{equation}
An integration by part of the antiholomorphic derivative term will pick out the residue of the $\chi\mathcal{T}\mathcal{T}$ OPE and reduce remove one integral sign by virtue of the complex analysis lemma
\begin{equation}
    \partial\frac{1}{\bar{w}}=\delta^{2}(w)\ \ .
\end{equation}
Using Equation \ref{eq::opeTT}, and recalling that our gauge fixing ensures that $\partial \chi^{e}=0$, we can deduce the residues of the $\chi^{e}\mathcal{T}^{f}(w')\mathcal{T}^{a}(w)$ OPEs:
\begin{align}\label{eq::opesttchi}
    Res\left(\chi^{e}(w')\mathcal{T}^{f}(w')\mathcal{T}^{e}(w)\right)&=-(w^2\partial T+4wT)\chi^{e}(w)\\
    Res\left(\chi^{e}(w')\mathcal{T}^{f}(w')\mathcal{T}^{h}(w)\right)&=(2w^3\partial T+8w^2 T)\chi^{e}(w)\\
    Res\left(\chi^{e}(w')\mathcal{T}^{f}(w')\mathcal{T}^{f}(w)\right)&=(w^4\partial T+4w^3 T)\chi^{e}(w)\ \ .
\end{align}
Plugging the equations in \ref{eq::opesttchi} back into \ref{eq::proofhalf} and integrating by part the $\partial T$ dependences using Equations \ref{eq::eomazbar} and \ref{eq::gaugefix} , we get from the originally $\bar{\partial}\chi^{e}$ term:
\begin{align}\label{eq::proof}
    &\,\,\chi^{e}T\left(\bar{A}^{f}\left(-\partial (w^2)T+4w\right)+\bar{A}^{h}\left(-\partial (-2w^3)-8w^2\right)+\bar{A}^{e}\left(-\partial (-w^4)-4w^3\right)\right)\nonumber \\
    &=\chi^{e}\left(2wT\bar{A}^{f}-2w^{2}T\bar{A}^{h}+ 0T \bar{A}^{e}\right)\nonumber \\
    &=\chi^{e}\left(-\bar{A}^{f}\mathcal{T}^{h}+2\bar{A}^{h}\mathcal{T}^{f}\right)\nonumber\\
    &=\delta_{\chi^{e}}S_{int}-\int \bar{\partial}\chi^{e}\mathcal{T}^{f}\nonumber \\
    &=\delta_{\chi^{e}}S_{int}\ \ ,
\end{align}
where we used the fact that $\mathcal{T}^{f}$ is holomorphic to get the last line by integration by part.

This final expression, coming from the $\bar{\partial}\chi^{e}$ part of the second term in \ref{eq::varexp}, indeed cancels out the first (linear) term of the variation in \ref{eq::varexp}. Similarly, all remaining (non $\bar{\partial}\chi$) components of the second term in \ref{eq::varexp} will be cancelled out by the $\bar{\partial}\chi$ part of the third term, and so on. Let us show that explicitly.

So far, what we have done can be summed up in the following schematic way. For a variation of the action looking like
\begin{equation}\label{eq::delS}
    \delta S = D + B\ \ ,
\end{equation}
where $D$ (seen as an operator insertion at all possible $w$'s) is the integral term containing $\bar{\partial}\chi$ and $B$ (operator insertion as well) is the rest of the variation, we have the following:
\begin{equation}\label{eq::proofuseful}
    D\cdot S = B\ \ ,
\end{equation}
where the multiplication here denotes the product of operators (carried out later using OPE's between closest insertions). Thus,
\begin{equation}
    -\delta S + \delta S \cdot S = -D - B + D\cdot S + B\cdot S = B\cdot S\ \ ,
\end{equation}
where we used Equation \ref{eq::proofuseful} and the fact that $D$ by itself is zero by integration by part (see Equation\ref{eq::proof}). Here, $B\cdot S$ is quadratic in $\bar{A}$ and we have therefore successfully cancelled out all terms linear in $\bar{A}$.

Let us now show the cancellation to all orders. Using equation \ref{eq::delS}, we have that the variation of $e^{-S}$ reads
\begin{equation}\label{eq::variationact}
    \delta\left(e^{-S}\right)=(D+B)\cdot e^{-S}\ \ .
\end{equation}
Moreover, $D$ acts as a derivation on analytic functions. Indeed, since $D$ contains a $\bar{\partial}\chi$ term, as we saw before, it is going to pick out the residues of all the OPEs coming from all operators insertion (picking out the residue of all possible singular points $w_1=w_i$ for $i>1$ and where the $w_i$ (resp. $w_1$) are the coordinate of the $S$ (resp. $D$) operator insertions), and therefore act as a derivation.
This yields
\begin{align}
    D\cdot e^{-S}&=-(D\cdot S)e^{-S}\nonumber \\
    &=-B\cdot e^{-S}\ \ ,
\end{align}
which immediately shows that the variation in Equation \ref{eq::variationact} is zero. This concludes our argument.

We have therefore established induction and shown the invariance of the action at the quantum level under a gauge transformation along the $e$ direction. The same work can be done for the other generators ($f$ and $h$) of $\mathfrak{sl}_2$ and for the antichiral CFT coupling. Thus, it concludes our proof of the invariance of $e^{-S_{int}}$ under gauge transformations, thereby showing that the proposed coupling between 4D CS theory and 2D CFTs is viable.

\subsection{Integrating out the gauge field}\label{sec::CStoIFT}
Since we are looking to get a 2D $T\bar{T}$-deformed theory, and as the only objects that are not inherently 2D in our theory are the two remaining components of the gauge field, it seems logical at this point to integrate out those components and compute the effective action resulting from this operation. For the sake of simplicity, this section will describe how to integrate out the gauge field (for any chosen gauge group) in the case of a $dw$ holomorphic 1-form. The relevant expressions for the $\frac{dz}{z}$ case (of which the derivation follows the same steps as in this subsection) will be introduced when needed in a later section.

Seeing as there are only two types of vertices of interest: $\propto A_{\bar{w}}\,T$ at $z_0$, and $\propto A_w\, \bar{T}$ at $z_1$ (the vertices we called respectively $A_{\bar{w}}^{a*}\mathcal{T}^{a}$ and $A_{w}^{a}\bar{\mathcal{T}}^{a}$ in the previous subsection):

\vspace{0.5cm}
\begin{figure}[H]
\begin{center}
\begin{tabular}{ll}
\begin{fmffile}{cubic_vertices}
\begin{fmfgraph*}(100,80)
\fmfleft{e1}
\fmfright{g}

\fmfv{decor.shape=circle, decor.filled=shaded,decor.size=0.2w,label.dist=0.18w}{e1
\fmflabel{$\mathcal{V}^{a}T$}{e1}
\fmflabel{$A_{\bar{w}}^{b}$}{g}
\fmf{gluon}{e1,g}
\end{fmfgraph*}
\end{fmffile}\hspace{1.5cm}
&
\begin{fmffile}{three_body2}
\begin{fmfgraph*}(100,80)
\fmfright{e1}
\fmfleft{g}
\fmfv{decor.shape=circle, decor.filled=shaded,decor.size=0.2w,label.dist=0.18w}{e1}
\fmflabel{$\bar{\mathcal{V}}^{b}\bar{T}$}{e1}
\fmflabel{$A_{w}^{a}$}{g}
\fmf{gluon}{g,e1}
\end{fmfgraph*}
\end{fmffile}

\end{tabular}
\end{center}
\caption{Only two types of vertices participating in the effective action. $a$ and $b$ are such that $\Tr\left(t^{a}t^{b}\right)\neq 0$ with $\Tr$ the non-degenerate invariant bilinear form of $\mathfrak{sl}_2$ and $t^{a}$ the generators of $\mathfrak{sl}_2$.}
\label{feynman::verticesTheo}
\end{figure}

\noindent we will get as an effective interaction vertex all binary combinations of vertices of different types, multiplied by the propagator between the two, whose color part will just be the quadratic Casimir of the gauge group.
As to the color-stripped propagator, it is as always the Green's function of the color-stripped kinetic operator in the action. The kinetic term $\text{d}z\wedge A\wedge \text{d}A$ is whittled down to $A_{\bar{w}}\partial_{\bar{z}}A_w\, \Omega$ (where $\Omega$ is the canonical top form), yielding the following differential equation for the propagator:
\begin{equation}
    \partial_{\bar{z}}G(z-z',w-w')=\delta(z-z')\delta(w-w')\ \ ,
\end{equation}
whose solution is simply
\begin{equation}
    G(z-z',w-w')=\frac{1}{2i\pi (z-z')}\delta(w-w')\ \ .
\end{equation}
Putting color and kinematics together, we finally have
\begin{equation}\label{eq::rmatrix}
    \braket{A^{a}_{\bar{w}}(z,w)\,A_w^{b} (z',w')}=\frac{C^{ab}}{2i\pi (z-z')}\delta(w-w')\ \ ,
\end{equation}

\noindent where $C$ is the (second) Casimir operator of the Lie algebra (here, $\frac{1}{2}h\otimes h+ e\otimes f + f\otimes e$). Now, multiplying the propagators by the couplings of the the vertices that they relate, we get the full effective action:
\begin{equation}
    S_{eff}=\int_{\Sigma}\mathcal{L}^{\text{chiral}}_{free}+\mathcal{L}_{free}^{\text{anti-chiral}}+\frac{\left(\mathcal{V}^{e}\mathcal{\bar{V}}^{f}+\mathcal{V}^{f}\mathcal{\bar{V}}^{e}+\frac{1}{2}\mathcal{V}^{h}\mathcal{\bar{V}}^{h}\right)T\bar{T}}{2i\pi(z_0-z_1)}\ \ ,
\end{equation}
which explicitly involves a term proportional to $T\bar{T}$. The coefficient $\frac{1}{2i\pi(z_0-z_1)}$, which is here a parameter of the problem, can be taken to be the $\lambda$ deformation parameter of the seed CFT (that we have to keep perturbative to be consistent with our first order approach). On the other hand, the term coming from the Casimir operator and the powers of $w$ plays the role of a metric on $\Sigma$.

\section{$T\bar{T}$ deformation on curved space}\label{sec::TTbar}

In this section we shall explore the possible $T\bar{T}$-deformations that we can obtain using two different groups ($SL_2$ and $SO(2)\ltimes \mathbb{R}^2$) and two different holomorphic 1-forms of the complex manifold ($dz$ and $\frac{dz}{z}$). The first two subsections will be devoted to $SL_2$ (with both choices of 1-forms), the last two will cover the case $SO(2)\ltimes \mathbb{R}^2$.
\subsection{Projective space and AdS space}\label{sec::TTbarCP1}
\subsubsection{$T\bar{T}$ deformation on $\mathbb{CP}^1$}
Consider a 4D Chern-Simons theory on $\mathbb{C}_z\times \mathbb{CP}^{1}_{w}$ symmetry group $SL_2(\mathbb{R})$. Let $(z,w)\in \mathbb{C}\times \mathbb{CP}^1$ be the coordinates of a point in the 4D space, and let the 4D CS theory be coupled to a chiral (resp. antichiral) 2D fermionic CFT living on $\mathbb{CP}^1$ at a specific points $z_0$ (resp $z_1$) of $\mathbb{C}$ using the $T\coloneqq  T_{ww}$ (resp. $\bar{T}\coloneqq \bar{T}_{\bar{w}\bar{w}}$) component of the stress energy tensor of said CFT.

Let us now apply the procedure explained in the previous section to recover from our 4D theory with surface defects a deformed 2D effective field theory. Certainly, we expect to recover our 2D CFT that currently plays the role of surface defect in the 4D theory, together with an interaction term arising from the coupling between the CS gauge field and the stress tensor CFT. We finally hope for the powers of $w$ to arrange into a metric term that the $\lambda T\bar{T}$ term would be integrated against. Let us carry out explicitly this computation.

We will be working to first order in perturbation theory under the gauge fixing given in \ref{sec::4DCST}: consequently, the cubic vertex originally present in the CS action vanishes and we only have the two types of vertices coming from the coupling terms. This gives rise to a limited number of Feynman diagrams that should be all accounted for in order to find the 2D effective action.  As given in Section \ref{sec::4DCST}, the color-stripped propagator is just $\frac{1}{z_{0} -z_{1}}$ and the only possible pairings of gauge fields involving a non-zero propagator are given by the quadratic Casimir operator of $\mathfrak{sl}_2$ :
\begin{equation}\label{eq::casimirsl2}
	C_2(\mathfrak{sl}_2)=e\otimes f+f\otimes e+\frac{1}{2}h\otimes h\ \ ,
\end{equation}
which means that we should consider only three ways to pair up the coupling terms, given by the following Feynman diagrams:

\vspace{0.1cm}
\begin{figure}[H]
\begin{center}
\begin{fmffile}{efCas}
\begin{fmfgraph*}(150,40)
\fmfleft{e1}
\fmfright{g}

\fmfv{decor.shape=circle, decor.filled=shaded,decor.size=0.2w,label.dist=0.18w}{e1
\fmfv{decor.shape=circle, decor.filled=shaded,decor.size=0.2w,label.dist=0.17w}{g}
\fmflabel{$T$}{e1}
\fmflabel{$-T$}{g}
\fmf{gluon,label.dist=0.10w,label.side=bottom,label=$\frac{1}{2i\pi(z_0-z_1)}$}{e1,g}
\end{fmfgraph*}
\end{fmffile}
\end{center}

\caption{e-f part of the Casimir}
\label{feynman::interactionD1}
\end{figure}

\vspace{0.1cm}
\begin{figure}[H]
\begin{center}
\begin{fmffile}{feCas}
\begin{fmfgraph*}(150,40)
\fmfleft{e1}
\fmfright{g}

\fmfv{decor.shape=circle, decor.filled=shaded,decor.size=0.2w,label.dist=0.18w}{e1
\fmfv{decor.shape=circle, decor.filled=shaded,decor.size=0.2w,label.dist=0.17w}{g}
\fmflabel{$-w^2\,T$}{e1}
\fmflabel{$-\bar{w}^2\,\bar{T}$}{g}
\fmf{gluon,label.dist=0.10w,label.side=bottom,label=$\frac{1}{2i\pi(z_0-z_1)}$}{e1,g}
\end{fmfgraph*}
\end{fmffile}
\end{center}

\caption{f-e part of the Casimir}
\label{feynman::interactionD2}
\end{figure}

\vspace{0.1cm}
\begin{figure}[H]
\begin{center}
\begin{fmffile}{hhCas}
\begin{fmfgraph*}(150,40)
\fmfleft{e1}
\fmfright{g}

\fmfv{decor.shape=circle, decor.filled=shaded,decor.size=0.2w,label.dist=0.18w}{e1
\fmfv{decor.shape=circle, decor.filled=shaded,decor.size=0.2w,label.dist=0.17w}{g}
\fmflabel{$-2w\,T$}{e1}
\fmflabel{$-2\bar{w}\,\bar{T}$}{g}
\fmf{gluon,label.dist=0.10w,label.side=bottom,label=$\frac{1}{2i\pi(z_0-z_1)}$}{e1,g}
\end{fmfgraph*}
\end{fmffile}
\end{center}

\caption{h-h part of the Casimir}
\label{feynman::interactionD3}
\end{figure}

\noindent By summing the contributions of all these graphs, one gets the following effective quartic interaction:
\begin{equation}\label{eq::effquartintCP}
    I_{eff}=\frac{1+|w|^4 + 2*|w|^2}{2i\pi(z_1-z_0)}*T\bar{T}(w)=\frac{(1+|w|^2)^2}{2i\pi(z_1-z_{0})}* T\bar{T}(w)\ \ ,
\end{equation}
where we can recognize at the numerator of the fraction the inverse of the complex component of the Fubini-Study metric for $\mathbb{CP}^1$:
\begin{equation}\label{eq::FSmetric}
    g_{\mathbb{CP}^1}(w,\bar{w})=\frac{1}{(1+|w|^2)^2}\mathrm{d}w\mathrm{d}\bar{w} \implies g_{\mathbb{CP}^1}^{-1}(w,\bar{w})=(1+|w|^2)^2\ \ .
\end{equation}
We can therefore rewrite our 2D effective action, denoting by $\lambda_{01}=\frac{1}{2i\pi(z_1-z_0)}$ the first-order deformation parameter, as:
\begin{equation}\label{eq::finaleffquartintCP1}
    S_{eff}=S^{kin}_{c}+S^{kin}_{ac}+\int_{\mathbb{CP}^1}\lambda_{01}\cdot T\bar{T}(w)\cdot  g^{-1}(w)\ \mathrm{d}w\mathrm{d}\bar{w}\ \ ,
\end{equation}
where the subscripts $c$ and $ac$ refer respectively to chiral and antichiral theories.
Clearly, we just get a deformation of our theory by a term proportional to $T\bar{T}$ through a coupling constant $\lambda_{01}$ coming from our choice of position of the surface defects in the initial 4D space.

Let us now take this problem backwards. Had we a 2D chiral theory on $\mathbb{CP}^1$ and should we want to $T\bar{T}$-deform it, we would need to integrate $T\bar{T}$ versus a term that makes dimensional sense. Now $T$ has dimension $(2,0)$ ad $\bar{T}$ has dimensions $(0,2)$, and we want to have an integrand with dimensions $(1,1)$ (like $\mathrm{d}w\mathrm{d}\bar{w}$). We therefore need a quantity with dimensions $(-1,-1)$ to integrate against, and the most natural (and only?) candidate for this definitely seems to be the inverse of the metric.

Thus, from our 4D CS picture with surface defects, we were able to recover a general $T\bar{T}$ deformation of the associated 2D chiral CFT with tunable coupling determined by the initial data of the position of the surface defects in 4D space. This process offers a new point of view on $T\bar{T}$ deformations on curved space, proposing an interpretation thereof in terms of surface operators in 4D Chern-Simons.
\subsubsection{$T\bar{T}$ deformation on $AdS_2$}

The $T\bar{T}$ deformation of a CFT on $ADS_2$ spacetime can be obtained in a very similar fashion. Indeed, noting that the metric on $AdS$ space differs from that on $\mathbb{CP}^1$ by one minus sign only:
\begin{equation}\label{eq::metricAdS2}
g_{AdS_2}=\frac{\mathrm{d}w\mathrm{d}\bar{w}}{(1-|w|^2)^2}\ \ ,
\end{equation}
it suffices to find another representation of the Virasoro $\mathfrak{sl}_2$ algebra such that the quadratic Casimir yields the same term with an extra minus sign. To that end, one can just modify the Equations \ref{eq::virasorocoeffsCP1} in the following way:

\begin{align}\label{eq::virasorocoeffsCP1p}
    \mathcal{V}^{e}(w)&=i\nonumber \\
    \mathcal{V}^{f}(w)&=-i w^2 \\
    \mathcal{V}^{h}(w)&=-2w\nonumber \ \ ,
\end{align}
and same for the anti-holomorphic part with the couplings for $e$ and $f$ swapped. Following the same steps it is then straightforward to show that

\begin{equation}\label{eq::finaleffquartintAdS}
     S_{eff}=S^{kin}_{c}+S^{kin}_{ac}-\int_{AdS_2}\lambda_{01}\cdot T\bar{T}(w)\cdot  g^{-1}(w)\ \mathrm{d}w\mathrm{d}\bar{w}\ \ .
\end{equation}

\subsection{Deformed sphere endowed with the "sausage metric"}\label{sec::TTbarsausage}
As explained in Section \ref{sec::4DCST}, we can choose several different 1-forms to supplement the 3D CS theory and make it four-dimensional \cite{Costello:2017dso}. In particular, instead of just considering $\mathrm{d}z$ with $z\in \mathbb{C}$, one can take the holomorphic 1-form $\frac{\mathrm{d}z}{z}$ for $z\in \mathbb{C}$. Based on calculations by Costello et al. \cite{Costello:2018gyb}, this changes the shape of the two-point correlation function of the 4D Gauge field. This propagator now takes the form of what Costello et al. introduce as the trigonometric classical $r$-matrix (Section 9.4) in their paper, thereby changing the weight given to each of the previous Feynman diagrams (alternatively, changing the form of the propagators in said diagrams):

\begin{equation}
    r=\frac{z_1}{z_1-z_0}e\otimes f+\frac{z_0}{z_1-z_0}f\otimes e
+\frac{1}{4}\frac{z_1+z_0}{z_1-z_0}\ \ ,\end{equation}
where each term corresponds to the propagator between two terms in the action.

Combining the expression for the $r$-matrix given in \cite{Costello:2018gyb} with the Feynman diagram approach used for the previous cases, we get the following expression after summing the contributions:

\begin{equation}\label{eq::effquartintSausage}
    I_{eff}(w)=\left(\frac{z_1}{z_1-z_0}*1+\frac{z_0}{z_1-z_0}*|w|^4+\frac{1}{4}\frac{z_1+z_0}{z_1-z_0}*|-2w|^2\right)T\bar{T}(w)\ \ .
\end{equation}
Denoting by $\alpha=\frac{z_1}{z_1-z_0}$ and by $\beta=\frac{z_0}{z_1-z_0}$ (note that $\alpha-\beta=1$), we can rewrite this interaction as follows:
\begin{equation}\label{eq::effquartintSausage2}
    I_{eff}(w)=\left(\alpha+\beta*|w|^4+(\alpha+\beta)*|w|^2\right)T\bar{T}(w)\ \ .
\end{equation}
By analogy with our previous calculation, this would be the $T\bar{T}$ deformation of a CFT on a space with metric proportional to

\begin{equation}\label{eq::metricneededsausage}
g(w)\propto \frac{\frac{1}{\alpha}\mathrm{d}w\mathrm{d}\bar{w}}{1+\frac{\beta}{\alpha}*|w|^4+(1+\frac{\beta}{\alpha})*|w|^2}\ \ .
\end{equation}
$\frac{\beta}{\alpha}=\frac{z_1}{z_0}$, and we know that the 4D theory is $z$-translation invariant and thus only depends on $(z_0-z_1)$.Consequently, given $z_1-z_0$, we can demand that $\frac{z_1}{z_0}$ be finite non-zero and real by moving $z_0$ freely without loss of generality.

Provided $\frac{\beta}{\alpha}$ is real (which we just made sure of), one can consider the scalar dilation
\begin{align}\label{eq::dilationsausage}
w'&=\left(\frac{\beta}{\alpha}\right)^{\frac{1}{4}}*w \nonumber\\
\bar{w}'&=\left(\frac{\beta}{\alpha}\right)^{\frac{1}{4}}*\bar{w}\ \ ,
\end{align}
which allows us to rewrite the expected metric as follows:

\begin{equation}\label{eq::metricneededsausage2}
g(w')\propto \frac{\frac{1}{\sqrt{\beta*\alpha}}\mathrm{d}w'\mathrm{d}\bar{w}'}{1+|w'|^4+(\sqrt{\frac{\beta}{\alpha}}+\sqrt{\frac{\alpha}{\beta}})*|w'|^2}\ \ .
\end{equation}
Now introducing the variable
\begin{equation}\label{eq::tvariablesausage}
    t=-\frac{1}{8}\ln\left(\frac{\beta}{\alpha}\right)\implies e^{4t}=\sqrt{\frac{\alpha}{\beta}}\ \ ,
\end{equation}
and dropping the primes, we can finally rewrite the metric of our unknown space $g(w)$ as the so-called "Sausage metric" \cite{Bykov:2020llx,Lambert:2012tq} with parameter $t$ as introduced by Lambert et al.:

\begin{equation}\label{eq::sausagemetric}
g(w)\propto \frac{2\sinh(4t)\ \mathrm{d}w\mathrm{d}\bar{w}}{1+|w|^4+2\cosh(4t)|w|^2}\ \ .
\end{equation}
Note that to get this, we used that $2\sinh(4t)=\sqrt{\frac{\alpha}{\beta}}-\sqrt{\frac{\beta}{\alpha}}=\frac{\alpha-\beta}{\sqrt{\alpha\beta}}=\frac{1}{\sqrt{\beta\alpha}}$ by definition of $\alpha$ and $\beta$.

This shows that the 2D action obtained from the 4D CS theory with holomorphic 1-form $\frac{\mathrm{d}z}{z}$ and gauge group $SL_2$ in the presence of fermionic surface defects on a deformed sphere endowed with the Sausage metric yields an effective action that turns out to be the $T\bar{T}$ deformation of the CFT on that deformed sphere:

\begin{equation}\label{eq::finaleffquartintsausage}
    S_{eff}=S^{kin}_{c}+S^{kin}_{ac}+\int_{S^2}T\bar{T}(w)\cdot  g^{-1}(w)\ \mathrm{d}w\mathrm{d}\bar{w}\ \ .
\end{equation}

\subsection{$T\bar{T}$ deformation on a space with a cigar metric}\label{sec::cigar}
So far, we have used only $SL(2,\mathbb{R})$ symmetry for our 4D CS theory supplemented by its Virasoro counterpart to have invariant coupling terms with 2D CFTs. But we could also consider other symmetry groups $G$, as long as we can engineer coupling terms with the energy-momentum tensors that also present such a symmetry and thus allow to consider a diagonal subgroup of $G\times G$. In particular, the Lie algebra isometries of the plane $SO(2)\ltimes \mathbb{R}^2$ has a similar representation to that of $SL(2,\mathbb{R})$, with only one -sizeable- difference in the commutators. Namely, $Lie\left(SO(2)\ltimes \mathbb{R}^2\right)$ is generated by $e,f$ and $h$ such that:
\begin{align}
    [h,e]&=2e\nonumber \\
    [h,f]&=-2f\\
    [e,f]&=0\ \ ,\nonumber
\end{align}
from which we could assert that $sl(2)$ is but a deformation of this algebra to yield a non-trivial $[e,f]$ Lie bracket.
If we assign this symmetry to our Gauge field $A$, we need to adjust the couplings for it to be also respected by the operator $\oint T*$, which is easily done. It suffices indeed to impose $\mathcal{V}^{f}=0$ and $\bar{\mathcal{V}}^{e} =0$. Now, the more subtle issue is to determine the new shape of the propagator when the symmetry group is the isometries of the plane, that is, find a non-degenerate bilinear 2-form (the Killing form is trivially degenerate) from which to extract the quadratic Casimir. Calling this bilinear form $q$, we can construct it by hand by specifying its value on all of the pairs of generators, and check that it respects the Lie algebra structure. We impose the following equations:

\begin{align}
    q(e,f)&=1\nonumber \\
    q(h,h)&=1\\
    q(e,h)&=q(f,h)=0\ \ ,\nonumber \\
\end{align}
extended by symmetry and $\mathbb{R}$-linearity to a symmetric bilinear map. It is non-degenerate by construction, and we can check that
\begin{equation}
    q(x,[y,z])=q([x,y],z)
\end{equation}
very easily.

This construction leads to a Casimir that turns out to be identical to that of $SL(2)$. Combining this finding with our knowledge of the new couplings $(\mathcal{V}^{a},\bar{\mathcal{V}}^{a})$ and using the same Feynman diagrams as in Section \ref{sec::TTbarCP1}, we now get:
\begin{equation}
    S_{eff}=S_{kin}+\frac{1}{z_0-z_1}\int_{\mathcal{C}} T\bar{T}(1+|w|^2)\mathrm{d}^2 w\ \,
\end{equation}
which suggests that we would be able to $T\bar{T}$ deform a space $\mathcal{C}$ that is endowed with the metric
\begin{equation}
    g(w)=\frac{\mathrm{d}w \mathrm{d}\bar{w}}{1+|w|^2}\ \ ,
\end{equation}
which is precisely the so-called cigar metric.

\subsection{$T\bar{T}$ deformation on a deformed cigar}

Once again, we can wonder if we could recover the $T\bar{T}$-deformation of a different kind of curved space using a different holomorphic 1-form in our initial action: $\frac{dz}{z}$. Following the same steps as in Section \ref{sec::TTbarsausage} (and the same notation) but using this time the commutation relations and couplings introduced in Section \ref{sec::cigar}, we get an expression of the metric for the deformed space that reads
\begin{equation}
    g(w')=-\frac{2\sinh(4t)\ \mathrm{d}w'\mathrm{d}\bar{w}'}{1+2\cosh(4t)\ |w'|^2}\ \ ,
\end{equation}
which means that we recover the first-order $T\bar{T}$-deformation of a space endowed with a slightly deformed version of the cigar metric. Although this deformation does not differ much from the one derived in the previous subsection, it is worth noting that one can easily tune the deformation of the cigar metric using the parameters at our disposal ($z_0$ and $z_1$).

\section{Conclusion}

In this paper we endeavored to interpret curved space $T\bar{T}$-deformations of 2D chiral + antichiral CFTs to first order in perturbation theory as coming from a pair of chiral and antichiral surface CFTs in 4D gauge theories with a suitable coupling scheme. This led to a fairly straightforward 4D interpretation of the deformation parameter as the inverse of the 2D ``holomorphic distance'' ($z_1-z_0$) between the two CFT planes. Moreover, by slightly changing the gauge group of our 4D theory and playing with different holomorphic 1-forms of our complex manifold, we could recover $T\bar{T}$-deformations on spaces of various curved geometries, including projective spaces, and spaces endowed with metrics such as the so-called cigar and sausage metrics.

There are many future directions to explore. Our method only allowed us to retrieve first-order deformations of the action. It would be interesting to compute the effective action at higher order (which is significantly more involved and has not been attempted here) and check that the result matches the expected $T\bar{T}$-deformed theory. Moreover, explicit parallels between $T\bar{T}$-deformations and 3D gravity have previously been drawn, and it is well known that 3D gravity is closely related to 3D Chern-Simons theory. This motivates one to wonder if compactifying 4D Chern-Simons theory along the lines described in Section \ref{sec::gravity} could lead to the same gravity theories as the one we would obtain by using the direct bridge between $T\bar{T}$-deformations and gravity.
\\

\section*{Acknowledgements}

I would like to thank Kevin Costello, who suggested this project and helped and guided me through it. I am also very grateful to my advisor Tudor Dimofte for guidance in preparation for this manuscript. Lastly, I am thankful to Leonel Quinta Queimada for many useful discussions.
\bibliographystyle{JHEP}

redpajama_set_name: RedPajamaArXiv


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text:
Famous Fashion Designer Accused Of Sexually Harassing Two Men
He allegedly touched them inappropriately and asked them to strip.
Famous fashion designer, Kim Young Se, has been accused of sexually harassing two men. A male in his 30s sued him for sexual harassment, claiming that Kim Young Se touched him inappropriately when he went to interview for the position of a chauffeur.
The man claims that Kim Young Se invited the man to his house for the interview and began touching him, even going as far as telling him to strip.
"It started with him wanting to shake my hand to putting his hand on my thighs. He asked me to take off my clothes because he wanted to gift me with a shirt."
— Alleged Victim
The man was so traumatized by the events that he has to get ongoing psychological treatment. He says he can no longer enter a location where only one man is present.
"I can't go anywhere that only has one man present because that event traumatized me greatly."
Another chauffeur who was previously employed by Kim Young Se also confessed that the designer made passes at him. He claims that Kim Young Se told him to touch him inappropriately under the guise of massages and applying lotion.
"He told me to sit on the bed because he had something to tell me. He would tell me to massage him or put lotion on his back."
— Former Chauffeur
Despite these two men's claims, Kim Young Se held a video interview to fight these accusations. He emphasizes that these stories are false, and that the accuser never came within 2 meters of him.
He believes that the men are falsely accusing him of sexual harassment because they know that he is gay and want to get money off of the lawsuit.
"There was no way for me [to come in contact]. He never came within 2 meters from me. Is this a play? Is this a drama?
— Kim Young Se
The police have investigated the two men's stories and have passed it onto prosecution.
Kim Young Se is best known for designing gorgeous pieces for Miss Korea, Miss Universe, and top celebrities like Kim Woo Bin.
Source: MBN News

redpajama_set_name: RedPajamaCommonCrawl


text:
Radiation Oncology: A Physicist's-Eye View, by Michael Goitein.
In an earlier post, I mentioned Michael Goitein textbook Radiation Oncology: A Physicist's Eye View. In Intermediate Physics for Medicine and Biology, Russ Hobbie and I cite Goitein's book when we discuss intensity-modulated radiation therapy.
In classical radiotherapy, the beam was either of uniform fluence across the field, or it was shaped by an attenuating wedge placed in the field. Intensity-modulated radiation therapy (IMRT) is achieved by stepping the collimator leaves during exposure so that the fluence varies from square to square in Fig. 16.45 (Goitein 2008; Khan 2010, Ch. 20)
It was originally hoped that CT reconstruction techniques could be used to determine the collimator settings at different angles. This does not work because it is impossible to make the filtered radiation field negative, as the CT reconstruction would demand. IMRT with conventional treatment planning improves the dose pattern (Goitein (2008); Yu et al. (2008)), providing better sparing of adjacent normal tissue and allowing a boost in dose to the tumor.
Goitein's analysis of IMRT is full of insight, and I quote an excerpt below. Enjoy.
INTENSITY-MODULATED RADIATION THERAPY (IMRT)
So far, we have implicitly assumed that each radiation field is near-uniform over its cross section; dose uniformity of a field within the target volume has, in fact, historically been an explicit goal of radiotherapy. However, the radical suggestion to allow the use of non-uniform fields was made some two decades ago, independently by Anders Brahme and Alan Cormack, fresh from co-inventing the CT scanner-and, in the context of π-meson therapy… (Cormack 1987; Brahme 1988; Pedroni 1981). Their idea was based on the judgment that, using mathematical techniques, an irradiation scheme using non-uniform beams could be found which would more closely achieve the ideal of delivering the desired dose to the target volume while limiting the dose to the normal tissues outside the target volume to some predefined value.
Brahme's and Cormack's approaches were motivated by the observation that, in CT reconstruction, one can deduce from the intensity reduction of X-rays traversing an object along a series of straight paths what the internal structure of the object is. By inverting the mathematics, one can deduce the intensities (pencil beam weights) of a series of very small beams (pencil beams) that pass through the object and deliver dose within it. This procedure leads to highly nonuniform individual fields which, in combination, deliver the desired (usually, uniform) dose to the target volume.
There are two very substantial flaws to the original idea. The first is that, when the problem is posed to deliver zero dose outside the target volume as was initially proposed, many of the computed intensities are negative — a highly unphysical result. The second is that there is no a priori way of specifying a physically possible dose distribution to serve as the goal of the optimization.
However, the basic idea of using non-uniform beams has proven enormously fruitful. A workable computational solution is to use optimization algorithms to iteratively adjust the pencil beam weights such that the resulting dose distribution maximizes some score function. The search is computationally intensive and therefore poses interesting technical challenges. However, the still bigger challenge is to find score functions which give a viable measure of clinical goodness. Increasingly, biophysical models of the dose-response of both tumors and normal tissues are being investigated and are beginning to be used as elements of such score functions. These matters are discussed in Chapters 5 and 9.
Intensity-modulated radiation therapy (IMRT), as treatments featuring non-uniform beams are called, has been most intensely developed for X-ray therapy. However, it is equally appropriate for other radiation modalities-including protons. With charged particles one has an extra degree of freedom. One can vary the beam intensity as a function of lateral position and as a function of penetration (energy).
Who was Michael Goitein? Below is an excerpt from his 2017 obituary in the International Journal of Radiation Oncology Biology Physics.
Michael Goitein was a visionary thought provocateur and is rightly judged to have been an exceptionally innovative and creative physicist in radiation oncology (Fig. 1). He was a critical player in the development of proton radiation therapy, with many of his advances widely used in current proton and photon therapy. His highly important and numerous contributions to medical science have been well rewarded with many important awards: Fulbright Fellowship, 1961–65; US Research Career Development Award, 1976–81; Fellow, American Association of Physicists in Medicine, 2000; Gold Medal of the American Society for Radiation Oncology, 2003; and Lifetime Achievement Award of the European Society for Radiotherapy and Oncology, 2014. Significantly, he was a cofounder of the Proton Therapy Co-Operative Oncology Group and served as the second president. Furthermore, he participated in many of its subsequent functions. In addition, Michael was an invited lecturer at a long list of national and international conferences and medical centers. He published 163 articles and 3 books that have positively affected the practice of radiation oncology (1). It must also be mentioned that he has authored 4 books of a personal nature, one of which is a book of his poems (2, 3, 4, 5).
To learn more, read his ASTRO interview.
READ/DOWNLOAD*[ One Renegade Cell: How Cancer Begi
Bob Park's What's New has been Restored!
From 1983–2006, physicist Bob Park was the director of public information in the Washington D. C. office of the American Physical Society…
The Body Electric
Go to www.amazon.com and look up the best selling book in the category " biophysics." You'll often find #1 is The Body Electric…
Art Winfree and Defibrillation
Yesterday I posted an excerpt from Chaos: Making a New Science. Here's a dirty little secret: I haven't read this book yet. I own a copy…
Life Atomic
Chapter 17 of Intermediate Physics for Medicine and Biology discusses nuclear physics and nuclear medicine. To learn more about this topic…
What Happened to NASA's Big Year?
Hopes for American astronauts to make a dramatic flight into space will have to wait until 2020
Astrology can be real…
I've always been a person who is open to new things, even though sometimes these new things seem outside our world or seem magical…
What does it take to understand spiders? False eyelashes, capes and face paint
Don't Shop, Adopt.
The four lovely dogs that I own were all dogs who were being abandoned by the previous owners or didn't want them because they got too…
Cause and Effect: Are you a deflationist or an inflationist?
What the world of travel looks like post Covid
ABA being used with ASD children Findings


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text:
The Kawasaki Z750 Twin
Nobody loved it… till now.
It's a 745cc, 360degree air-cooled parallel twin, made from 1976 – 1984
What sort of bike?
It had no pretensions of being anything in particular, just a bike that goes. It didn't try to appeal to any single group, which inevitably led to its not appealing to any group at all, and sales were dead on the vine. Because of the engine type, it was compared to both the Triumph Bonneville and Yamaha XS650, though not to Yamaha's XS500, although I'm sure the reviewers had this bike in mind at the time (high-tech[!] twin, DOHC, balancer shafts) which was famous only for its unreliability, and was a similar sales disaster.
With hindsight, it was the first Japanese cruiser. Caught on a cusp between engineers who wanted their bike to go round corners, a modern Bonnie, and marketing, who needed something to compete with the Harley, they came up with the Z750 Twin. Chopping bits off the Z1 and rescaling was simple, after all. And so the bike fell between two stools, and boy, did it fall. These days, the genre has mostly been defined as slow, heavy and V-Twin powered. 2 out of 3 ain't bad!
What'll it do mister?
It'll start, probably off the kick-start as the electric foot isn't even an electric inch.
It'll go faster than the traffic, except on motorways, whilst making a gorgeous burble through any non-standard exhaust. It will do this for some considerable time, since it's not prone to chewing chains/tyres, and will actively discourage thrashing (more later). Getting you there and back is all it's for, and it's very, very good at that. A properly set up one will have less vibration than a four, and the handling is way better than any Harley-rep cruiser.
It will also have old men at traffic lights telling you that it sounds just like their old Triumph (which it does). Depending on the mood, you can either engage in a long and pointless conversation or just stare fixedly into the distance. Any more would be rude, and we can't have that can we? 🙂
It will keep your pillion happy. Well mine anyway. Her bum is still soft and pliable after milesandmilesandmiles. She still complains, but that's what other halves are for huh?
It will spray your right leg with oil. This is a problem with the tacho drive seal, and is a 'feature' not a fault!
It will look good. Beauty is in the eye of the beholder, but to my personal eye this is exactly what a bike should look like, and looks are the starting point of any proper love affair. Flowing, curvy lines with a huge chunk of engine in the middle. No plastic, no pretentions. They really don't make them like this any more. Zephyr? Don't make me laugh. W650? OK, a good bike, but not exactly unpretentious, is it?
It will sound good. This type of engine fires once per revolution, which is just inherently right, and will sooth the soul. A 2:2 exhaust is mandatory, otherwise it will sound like Superdreams big brother, and you don't want that.
Oh, speed. About 105mph. But not for long. Too many 14.2 standing quarters will destroy the clutch.
What won't it do?
It won't keep up with anything bigger than a 500, and the only thing less likely to be found at a track day would be a Cossack Urine. It doesn't do fast. It will whinge at sustained revs over 6000 (90mph), and will ultimately overheat. Enjoy the relaxed burble and watch the world go by peacefully, safe in the knowledge that your licence will be yours forever. It's touring range is limited only by fuel stops, since reserve will kick in at an extremely irritating 100miles, when your bum only goes numb after about 400. The fuel tank size is the only thing that Big K got really wrong with this bike. That and the tacho drive seal. There were TWO things that Big K…
It won't do stoppies. The single front disk was a standard fitment to Kwaks throughout the 70's, and was always weak, and positively lethal in the wet. The bike came with mounts and fittings for a second disk as standard, although I've never seen a twin-disk model.
It won't do street-cred, but small boys do seem to like the noise and presence of a near-on 500lb(230kg) lump of chromed metal. The commonest colour (shit brown) doesn't exactly do it a lot of favours either.
It's old, it must be knackered.
Probably. It comes complete with some favourite Japanese stereotypes, the main one being a camchain tensioner which is prone to doing MUCH more harm than good. The electrics will almost certainly have been bodged by some nerk armed only with insulating tape. The shocks will be rusty, and usually empty. Neutral on the early models was as elusive as an honest politician, the late ones have a positive stop to fix this.
The electric starter WILL NOT WORK. Working starters are rare enough to elevate a machine to concours condition on this one point alone. Two reasons for this: the first is that they're really not up to the task of turning 2 big cylinders (the starter was also used on the Z650, a much easier proposition) and second that the starter clutch often fails, either spinning the starter into oblivion or making a sullen schhh noise as it fails to engage on the engine. NOS starters are, amazingly, available on eBay from the States at sensible prices, even including shipping.
The inlet manifolds (carb holders in the US) wear out enough to be considered consumable items. If your bike is blowing and spitting, these are prime suspects. Get NOS ones from anywhere but the UK. 16065-036. I know that number by heart, and have an ebay search on it. The problem seems to be overtightening of the jubilee clips causing the rubber to split at a shear point. So don't do it.
If it still spits and farts, look at the float height. V. difficult to check, and a pain to adjust and get right, but these 38mm Mikunis seem to be critically affected by it. Took me ages to suss that one, but could well be borne in mind when haggling with the owner of an old dog that sounds like a drag bike at idle. Get it home, do the carbs and manifolds, a quick balance by the piss-easy technique of putting your hand over the pipes [not the 2:1 :)] and you'll be smirking everytime you get on. Make sure you put the throttle cables on the right way round though…
These bikes don't vibrate. If you're looking at one that does, walk away, very quickly. You don't want to be messing with those balancers.
It's saving grace is probably its nature and a 4 litre sump. Together, they combine to produce a bike that isn't often thrashed and can survive the inevitable oil-change neglect. Underuse will cause more problems than a daily hack to work.
It's common enough stuff though, the engine and cycle parts are pretty damn simple really. Spares are non-existent in the UK, but quite common in the US, which bought quite a few coz there was a Kawasaki plant there. Handy if you need something really complicated like, say, a clutch cable. The workshop manual is a total gem, and gives good stuff that relates to most bikes, especially about the electrical system. It is to Haynes as John Robinson was to Steve Berry.
What's so good about it then?
As I said, the looks are the starting point. The bike is effectively a Z1 with the engine cut in two, and it shares most of the cycle parts with its bigger brothers. It's heavy and it's planted. It handles. No, really! The road bends, not the frame. OK, so she needs a bit of man-handling, but if a thing is easy, it's not worth doing.
Then there's the sound. When it echoes between the claustrophobic walls of the terraced houses round here it is truly evocative. The point about the engine is that the sound is even, not like any V-twin, and an inline 4 can't come close, no matter how loud. Pobbling around town at 1500 in top, it's happy. It's a biking St. Bernard; it's big, it's strong, and it makes its point in an unconcerned take-it-or-leave-it way. Wanna go faster? Just open the throttle, no gear-changes necessary. Which brings us to…
Bit of a wuss isn't it?
Yep, can't argue really. Your CBZFX600 can blow several shades of shit out of it, given enough revs. But's that's it you see. We don't need revs in Big Twin country. With roads infested by speed cameras, a machine that chills you out the moment it starts is a positive benefit. The engines only rated at 55chevals, but the torque is there from tickover, and never changes. Besides, those 55 nags will be with the stock exhaust system, and anything even moderately fruity will open a few more stable doors. After all, the legendary Z1 alleged 20bhp per 225cc cylinder, so 30bhp from a 375cc version of the same cylinder is hardly too much to ask. Still only sixty ponies, but they all pull together, all the time.
Yebbut it's still SLOOOOOOOW!
In biking terms, it's a dinosaur and behaves like one. But tell that to the Ford Focus which hasn't crossed the lights by the time you're at the next roundabout. Relativity my son, relativity.
OK, if it's so good, why does it rate a specific mention in the uk.rec.motorcycles.classic 'Bikes to Avoid'?
Two reasons:
First, not many people have ridden one, and some of those will have come across the SOB version which has completely knackered every chain in the machine. If it's not fettled, the bike will vibrate like a 500lb dildo, which might be fun for a while, but quickly gets a bit sore!
Second, when it came out, the bike got panned by a bunch of overpaid, overindulged reptiles who said that it lacked 'character'. Saddled with EPA fuel emissions bollox and silenced to buggery, character was always going to be at a bit of a premium on a bike that had no past to nostalge on, when it was in competition with the (equally ruined by that time) Bonnevile and that flat-track favourite, the XS650. All you could hear was the chains rustling, so they fretted about them. Damned with faint praise as it was, the poor bugger never stood a chance. The reputation has survived long after the bike it killed, like fungus on a tree.
And suppose I actually want to buy one?
Genuine UK models aren't common. They didn't sell many, so there's not many left, see? What there are have mostly been left to rot in the dungeon of Chopper-dom, or Bob-ness or some such travesty where Custom usually means 'exactly the same as everyone else who wants to do a bike up'. There are however a lot of US imports. I don't really understand this, as it would've meant paying a huge freight cost to ship a large chunk of metal that no-one here wanted in the first place. A US model is most easily sussed by the rear mudguard (remember them?). Living in a wet and dreary land, our mudguard is longer than theirs. If it ends level with the rear spindle, it's a UK one. Obviously US versions also had higher bars, but they are easily changed.
Check the brakes carefully. As well as being prone to seizing, I've had problems where the pressure operated front brake light switch started to 'give', causing a spongy and completely inadequate brake. Later models had a flat rectangular master cylinder, probably to accommodate twin-disc setups (which didn't happen until the dedicated cruiser models). It's more powerful but has less feel than the classic circular one.
Check the swing-arm. A loose swing arm will give er.. interesting handling, and will probably need an engineering shop to make new bushes.
There will be a grinding noise from the clutch when you start off from cold. There will be, and it should go away. The clutch is not this bikes strong point.
As I've said, if it vibrates in neutral, leave it. If it vibrates only on the move, it might well just be the chain. As with all bikes, a chain and sprockets are real money, so haggle, bearing in mind that the bike has had the sort of owner that not only runs, but sells a bike in this condition.
Author Kaptainkwak
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How exercise can raise self esteem!
A Workout in a gym is a word that the Bible didn't use nor understood. The word for a workout in a gym "gumnatio" (in Greek) that we use today was used in a very different manner in ancient times! Then the word was used for exercising power over others not physical exercise. A "gumnatio" was an authority not a place full of weights and training equipment.
So why then does the Bible not have a word for working out in a gym. They had enough exercise. Their whole daily life was a workout.
The ancient people, were always busy doing physical things Gen 24 : 19 Rebecca quickly emptied the jar from her shoulder, and ran to the well to get more water, … until she watered all the animals. Rebecca's jar probably held 10 litres of water and weighed close to 18 kg's when full. She walked 2 km from her house to the well and to water the animals walked 50m up and down. And she did all of this running. To me this sounds like a very good workout! Rebecca was probably 16 years old when she carried water to her house every day.
The Biblical people danced a lot! even in Church, David danced when they brought the ark of the covenant into Jerusalem, 2 Samuel 6 : 14 David wearing only a linen cloth round his waist, danced with all his might to honour the Lord.
They danced in National festivities, when David defeated the Philistine Goliath, women came out in the streets and danced! 1 Sam 18: 6. Women from every town in <?xml:namespace prefix = st1> Israel came out to meet king Saul … dancing and playing tambourines. Women from all over Israel had the ability to dance and play joyful songs. It wasn't something they just taught themselves after the battle with Goliath, they did that many times before as well. Part of their social get together as women included enhancing their skill as dancers.
Battles in ancient times were fought by muscle power not by high tech missiles and canons. They did not even have assault rifles they fought with heavy clothes, body armour and swords. David a non trained soldier could not walk with the heavy armour because he wasn't trained in that. 1 Samuel 17 : 38 he gave his own armour to David for him to wear, a bronze helmet, which he put on David's head, and a coat of armour. David strapped Saul's sword over the armour and tried to walk, but he couldn't. The armour was too heavy for the untrained soldier. It is unimaginable how strong and fit a soldier had to be to walk in his own heavy armour and be able to fight with a heavy sword and kill somebody else! All young boys had to learn the art of sword fighting to be able to stand up to enemy raids now and then! They had to have a daily workout to be able to serve in the army!
Because their future depended on it, they fought very hard! Physically it was a grueling workout, almost like an iron man race. It could carry on for more than twelve hours of physical activity. Many of them could be running out of fuel very fast. Even in fighting they were sometimes not fed adequately! 1 Sam 4: 9, Be brave Philistines! Fight like men, or we will become slaves of the Israelites just as they were our slaves. So fight like men!
The only mode of transport was donkeys but when you need really fast speed-services there was nothing better than human runners. 2 Samuel 18: 9, Let me run to the King with the good news that the Lord saved him from his enemies. It is true that not every human being at that stage was running for a livelihood but many young people probably ran as messengers.
There were no cranes or mechanical arms in those days. Building work had to be done by hand. They had to work themselves. All buildings were built manually. Nehemiah 3: 5, The Men of Tekoa built the next section… (Then there were fat and important guys who didn't want to work. They are the prototype of modern man). … Leading men of the town refused to do the manual labour assigned to them by the supervisors.
To conclude, the life of an ordinary person 3000 to 5000 years ago was mostly active. They had a moderate calorie intake with little sugar and fat, they must have been gorgeous people! Beautifully built and strong as beasts!
The Bible will take it for granted that you are physically active and healthy as far as possible. Extreme obesity was scarce and you could find it probably only in noblemen and kings. To be physically active is hence a no-brainer!
Make time for exercise, a workout at least every second day is one of the most important spiritual things you can do! God needs you in his kingdom to work for him. You must be fit and strong to endure! Why would you put your life in jeopardy and waste yourself? You are important your skills are needed by Him therefore look after yourself. Be fit and healthy!
Hold on to what is yours.
Things acquired by inheritance are particularly prone to misuse and squandering. It is one of our highest obligations to teach our children to have respect for money and wealth. The values that created abundance in the first place should be instilled in descendants of wealthy parents.
Staan jou plek vol waar jy is.
How humbleness is part of selflove.
Building a good self-esteem by discipline.
Building a good self esteem by understanding your importance.
Understand how special you are!


Even though Need for Speed: Most Wanted from Criterion Games has yet to release, EA is already hiring for next year's iteration, which will be powered by DICE's Frostbite 2.0. The "help wanted" signal is flashing outside of EA's Gothenburg studio in Sweden.
Gamerzines picked up on the news, but I have no idea how they managed to find that out given that there's no way they could have just known...maybe they were looking for new employment and spotted the opening on EA's page?
Would you like to be a part of EA? Would you like to work on a game in the Need for Speed franchise? You now have the opportunity to become a part of EA Gothenburg and fulfill those two wishes! We are hiring, and we are hiring a lot!
And by "hiring a lot!" they really do mean it. There's a huge list of open positions, from VFX artist to audio engineer, to a producer and even a QA analyst. Just about every position you could imagine required to make a full game is looking to be filled.
Of course, EA has yet to release Most Wanted so there's no telling what the newest game will be called, but if we're sticking to rehashes and re-releases, I'm guessing we're either do up for another entry into Underground or a redo of Carbon. I'd love to play a new NFS: Underground but EA has been so crazy lately with content restriction and DLC I'm almost afraid it'll turn out like the pay-to-win, pay-for-every-customization-part Need for Speed World.
You can check out the jobs page over at the official EA website.


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FTC warns cash option may be small for Equifax settlement
The Federal Trade Commission told consumers affected by the Equifax data breach they are unlikely to get the full $125 they sought.
Equifax last week reached a $700 million settlement over the 2017 data breach that exposed the personal information of 147 million people. The bulk of that money is intended for consumers impacted by the breach.
The company offered credit monitoring or up to $125 cash.
But the FTC said Wednesday that "public response to the settlement has been overwhelming" and millions of people visited the claims site in the first week alone.
Because those payments come from a fixed pot of $31 million, if more people apply than there are full funds for, then the amount each person gets will drop as the pool of money is distributed proportionally.
More Your Money Matters Stories
AMSTERDAM (AP) — Winter blues 0, joyful color 1.
Tulip growers in the Netherlands beat back winter — if only for a day — with a riotous explosion of color Saturday as they turned an Amsterdam square into a multi-colored feast for cold-dulled senses to mark National Tulip Day.
LifeScape's newly proposed building helps solve ongoing space and workforce issues
by Bridget Bennett / Jan 16, 2020
LifeScape announced plans for a major new facility at the University of South Dakota Discovery District. Part of the partnership includes training more therapists to meet the growing demand of the services LifeScape provides for children and adults with disabilities.
"We have a space constraint, but we also have a workforce constraint, so part of our vision was, how can we solve that as we go forward?" LifeScape CEO Steve Watkins said.
Rapid City Regional Airport breaks passenger record 3 years in a row
by Sydney Thorson / Jan 16, 2020
RAPID CITY, S.D (KELO) --- Rapid City Regional Airport is celebrating a record number of passengers for the third year in a row. In 2019 the airport added three new non-stop flights.
Marty Ertman's home is in Wyoming, but Thursday she's headed to Las Vegas. In the past she's flown out of Denver, but she prefers the shorter drive to Rapid City, cheaper parking and that's not all.


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Family Foundations helps support military families expecting their first child
Image: Shutterstock/Felix Mizioznikov
Online training developed at Penn State promotes warmth, nurturing and stability in the family
The arrival of a newborn can be stressful for all new parents, but may be especially so for those in the military. On top of sleep deprivation and being responsible for an infant 24 hours a day, military parents can experience the additional stress of deployment, frequent moves and uncertainty about the future.
A new study found that an online class for military families expecting their first child, developed by a Penn State professor, helped improve new parents' stress and mental health, couples' teamwork and support around parenting, and infant well-being.
"We are proud to support military and civilian families," said Mark Feinberg, senior researcher at Penn State's Edna Bennett Pierce Prevention Research Center and developer of the Family Foundations program. "I grew up on an Air Force base in Ohio, and so I wanted to give something back to the military community. And this has broader implications for all of us, as promoting military readiness depends in part on enhancing military family resilience. The military itself views this as a matter of national security."
According to the researchers, Family Foundations is a training program for expectant couples that is designed to reduce parental stress and mental health problems by helping couples work together smoothly as parents.
"In order to support parenting teamwork — also called coparenting — in military families, it was important for us to create an online version of the program," Feinberg said, "which features interactive lessons designed to promote warmth, nurturing and stability in the family."
Feinberg led a pilot randomized trial of the online program with 56 couples in the U.S. Army, Air Force, Marine Corps and Navy, with 29 of the couples participating in the program. Using pre-test data from pregnancy and post-test data from six months after birth, positive program effects were found for parent depression, mothers' report of coparenting support, and infant mood and soothability, as reported in the journal Family Relations.
The original Family Foundations program, developed by Feinberg with funding from the National Institutes for Health, is a series of prenatal and postnatal classes. The program provides expectant parents with the opportunity to practice self-calming and relationship problem-solving tools and strategies before the baby arrives, and supports parents in refining these skills after the baby has come home.
Previously, the intervention was evaluated in two randomized trials with civilian couples, and found to be effective in reducing postpartum depression, enhancing parent teamwork, better parenting quality, less family aggression, and better child outcomes from infancy through at least age seven.
In addition to the original class series and the online version, the program has been adapted for and is being tested with federal funding for teen parents in schools, poor parents in home visits, parents who use high levels of alcohol, and parents who have a young child just diagnosed with autism spectrum disorder. The Coparenting Relationship Scale, a tool developed by Feinberg and colleagues and used in the Family Foundations study to measure coparenting quality, has been adapted for use in 18 countries.
Jesse Boring of SUNY Broome Community College, and Jennifer Karre, Jamie Irvin, Yunying Le, Michelle Hostetler and Damon Jones, all of Penn State, also participated in this work.
The Eunice Kennedy Shriver National Institute of Child Health and Development helped support this work, as well as the Edna Bennett Pierce Prevention Research Center.
Projects Mentioned in this Article
Perinatal Intervention and Long-Term Outcomes (Family Foundations)
Mark E. Feinberg Michelle Hostetler Damon Jones


Today, I am excited to reveal the new cover for Satchfield Hall. I felt it needed a new look to reflect the emotional story told in those pages. Produced by Cathy Helms at Avalon Graphics, I love it. What do you think?
Celia Bryant-Smythe could never have imagined the high price she would have to pay for her one mistake.
As her father, Henry Bryant-Smythe learns of her indiscretion, he not only deals with it, but stamps on it with a resounding thud that will ricochet through the years. He cares nothing for the consequences of his actions that will bring heartbreak, pain and loss.
Set between 1943 and 1986, Satchfield Hall is a sweeping saga of deep love, evil lies and ruthless power.