# Measuring photophysical transition rates with fluorescence correlation
spectroscopy and antibunching

Damir Sakhapov  [  Ingo Gregor  [  Narain Karedla  The Rosalind Franklin
Institute, Harwell Campus, Didcot, OX11 0FA, UK  Jörg Enderlein
<EMAIL_ADDRESS>[

###### Abstract

We present a new method that combines fluorescence correlation spectroscopy
(FCS) on the microsecond time scale with fluorescence antibunching
measurements on the nanosecond time scale for measuring photophysical rate
constants of fluorescent molecules. The antibunching measurements allow us to
quantify the average excitation rate of fluorescent molecules within the
confocal detection volume of the FCS measurement setup. Knowledge of this
value allows us then to quantify, in an absolute manner, the intersystem
crossing rate and triplet state lifetime from the microsecond temporal decay
of the FCS curves. We present a theoretical analysis of the method and
estimate the maximum bias caused by the averaging of all quantities
(excitation rate, photophysical rates) over the confocal detection volume, and
we show that this bias is smaller than 5% in most cases. We apply the method
for measuring the photophysical rate constants of the widely used dyes
Rhodamine 110 and ATTO 655.

###### keywords:

American Chemical Society, LaTeX

Georg August] III. Institute of Physics – Biophysics, Georg-August-University,
37077 Göttingen, Germany Georg August] III. Institute of Physics – Biophysics,
Georg-August-University, 37077 Göttingen, Germany +49 (0)551 3926908 +49
(0)551 3927720 Georg August] III. Institute of Physics – Biophysics, Georg-
August-University, 37077 Göttingen, Germany [MBExC] Cluster of Excellence
“Multiscale Bioimaging: from Molecular Machines to Networks of Excitable
Cells” (MBExC), Georg August University, 37077 Göttingen, Germany IR,NMR,UV

##  1 Introduction

Fluorescent dyes have become indispensable for a myriad of microscopy and
spectroscopy applications in the life sciences. Although the fundamental
principles of fluorophore photophysics are already known since the start of
quantum mechanics ca. 100 years ago, the quantitative measurement of the
involved photophysical transition rates has become possible only with the
advent flash- and laser photolysis and fast wide-band recording electronics 1.
Newer techniques have used rapid fluid streams for kinetic measurements of dye
photophysics 2, 3, but both photolysis or fluid stream techniques are
experimentally challenging and require large amounts of sample.

A powerful, single-molecule sensitive yet relatively simple technique for
measuring fluorescence-related intensity fluctuations is fluorescence
correlation spectroscopy (FCS), pioneered by Magde, Elson and Webb in 1972 4.
This method allows for studying any dynamic process that modulates the
fluorescence intensity measured within the tiny femtoliter-sized confocal
volume of a confocal microscope. Initially, FCS was mostly used for
determining diffusion coefficients of fluorescent molecules diffusing through
the detection volume. Since the beginning of the nineties it was realized that
FCS can be also used for determining fast photophysical transitions, in
particular the intersystem crossing rate from the first excited state to the
triplet state, and the transition rate from the triplet state to the ground
state (i.e. triplet state lifetime) 5, 6, 7, 8, 9, 10, 11. However, for
obtaining absolute numbers for the photophysical transition rates, these FCS
studies made independent estimates about the excitation intensity in the
detection volume, which can result in considerable uncertainty for the final
values.

Already in 1997, Mets et al. demonstrated that it is possible to directly
determine absolute excitation rates of fluorescent molecules from fluorescence
antibunching measurements 12. At that time, these measurements were quite
demanding, with poor signal-to-noise ratio, and could not be readily combined
with FCS experiments within the micro- to millisecond timescale. This was
probably the reason for why their method was not picked up later by
researchers who wanted to study the photophysics of fluorescent molecules with
single-molecule sensitivity.

In the present paper, we combine fluorescence antibunching measurements with
FCS to measure absolute intersystem crossing rates and triplet state lifetimes
of fluorescent molecules. As we will show below, this can be done within one
single experiment where fluorescence intensity fluctuations are recorded with
sufficiently high temporal resolution. Thus, both the excitation rate and the
photophysical transition rates are simultaneously measured for exactly the
same sample on the same experimental setup, which eliminates additional
sources of error when one measures both these quantities in different
experiments. In the next sections, we will develop the theoretical basis of
our method, present numerical simulations for estimating its performance, and
then experimentally apply it for measuring the photophysical transition rates
of the widely used dyes Rhodamine 110 and ATTO 655.

##  2 Theory

We consider the three-level system shown in fig. 1 consisting of an electronic
singlet ground state $S_{0}$, first excited singlet state $S_{1}$, and a
triplet state $T_{1}$. We will further assume that the transitions rate
$k_{\mathrm{fl}}$ from $S_{1}$ to $S_{0}$ is by orders of magnitude faster
than the intersystem crossing rate $k_{\mathrm{isc}}$ from state $S_{1}$ to
$T_{1}$ or phosphorescence rate $k_{\mathrm{ph}}$ for the transition from
$T_{1}$ to $S_{0}$. This allows us to consider the $S_{0}\rightleftharpoons
S_{1}$ photo-kinetics separately and decoupled from the photo-kinetics
$\\{S_{0},S_{1}\\}\rightleftharpoons T_{1}$.

\begin{overpic}[width=216.81pt]{Fig1.png}\put(27.5,32.0){$k_{\mathrm{ex}}$}\put(46.0,32.0){$\tau^{-1}$}\put(56.0,47.0){$k_{\mathrm{isc}}$}\put(56.0,14.0){$k_{\mathrm{ph}}$}\end{overpic}
Figure 1: Jablonski scheme of the considered photophysics: A fluorescent
molecule has a singlet ground state ($S_{0}$) and first excited singlet state
($S_{1}$), and a triplet state $T_{1}$. The molecule is excited via single-
photon excitation from $S_{0}$ into $S_{1}$, where it relaxes back into the
lowest vibrational level within a few picoseconds. Subsequently, the molecule
returns to the ground state $S_{1}$ with rate $\tau^{-1}$, where $\tau$ is the
fluorescence lifetime (typically few nanoseconds), or it switches to the
triplets state $T_{1}$ with intersystem crossing rate $k_{\mathrm{isc}}$,
where in all relevant cases $k_{\mathrm{isc}}\ll\tau^{-1}$. From the triplet
state, it eventually returns to the ground state with phosphorescence rate
$k_{\mathrm{ph}}$.

Let us first consider the fast kinetics of transitions between the $S_{0}$ and
$S_{1}$ states. If the molecule is in its ground state at time zero, the
chance $s_{1}(t)$ to find it in its excited state $S_{1}$ at time $t$ is given
by solving the kinetic equation

$\displaystyle\frac{ds_{1}(t)}{dt}=k_{\mathrm{exc}}-\left(k_{\mathrm{exc}}+\frac{1}{\tau}\right)s_{1}(t)$
(1)

where $k_{\mathrm{exc}}$ is the excitation rate that depends on the power
density of the excitation laser, and where we have also taken into account
that the chance to have the molecule in its ground state is $1-s_{1}(t)$. The
solution of this equation reads

$\displaystyle
s_{1}(t)=\kappa\,\left\\{1-\exp\left[-\left(k_{\mathrm{exc}}+\frac{1}{\tau}\right)t\right]\right\\}\,\textrm{,}$
(2)

where we have introduced the abbreviation

$\displaystyle\kappa=\frac{k_{\mathrm{exc}}\tau}{1+k_{\mathrm{exc}}\tau}$ (3)

which is the _average excitation rate of a molecule_ when it is _not_ in the
triplet state, i.e. the limit for $t\rightarrow\infty$ of $s_{1}(t)$ in eq.
(2).

Next, let us consider the slow kinetics of intersystem crossing and
phosphorescence, and let us assume that this takes place on such a slow time
scale that at any moment in time, the fast transitions between $S_{0}$ and
$S_{1}$ are in equilibrium. We denote the probability to find the molecule in
any of its two singlet states by $s(t)$. Then, the probability of finding it
in its first excited singlet state $S_{1}$ is given by $\kappa\,s(t)$, see
also asymptotic value of solution (2), and the time evolution of $s(t)$ is
governed by the differential equation

$\begin{split}\frac{ds(t)}{dt}&=-k_{\mathrm{isc}}\,\kappa\,s(t)+k_{\mathrm{ph}}\left[1-s(t)\right]\\\
&=k_{\mathrm{ph}}-(k_{\mathrm{ph}}+\kappa\,k_{\mathrm{isc}})\,s(t)\,\textrm{.}\end{split}$
(4)

With the initial condition $s(t=0)=1$, eq. (4) has the solution

$\begin{split}s(t)=\frac{k_{\mathrm{ph}}+\kappa
k_{\mathrm{isc}}\exp\left[-(k_{\mathrm{ph}}+\kappa
k_{\mathrm{isc}})t\right]}{k_{\mathrm{ph}}+\kappa
k_{\mathrm{isc}}}\end{split}$ (5)

The autocorrelation function $g(t)$ measures the probability to detect a
photon at time $(t_{0}+t)$ if there was a photon detection event at time
$t_{0}$. The probability $p_{0}$ to detect a photon at $t_{0}$ (i.e. at an
arbitrary time) is proportional to the probability that a molecule is _not_ in
its triplet state (given by the asymptotic value of eq. (5) for large time
$t$) times the probability that it is then in its excited state $S_{1}$ (given
by $\kappa$ from eq. (3)). This has to be multiplied by the overall detection
efficiency of the microscopy system $\epsilon$, so that we find

$\displaystyle
p_{0}=\frac{\epsilon\,\kappa\,k_{\mathrm{ph}}}{k_{\mathrm{ph}}+\kappa
k_{\mathrm{isc}}}$ (6)

Directly after such an detection event the molecule had returned to its ground
state, so that the probability to detect the _next_ photon is proportional to
the probability $s_{1}(t)$, i.e. state occupancy of the first excited state.
On a short timescale, this $s_{1}(t)$ is given by eq. (2), and on a longer
timescale, it is given by the product of the probability $s(t)$ to find the
molecule in one of its singlet states, see eq. (5), times the probability
$\kappa$ that it is then in its excited state, see eq. (3). In a FCS
experiment with a confocal microscope, see fig. 2, the situation is
complicated by the fact that both the ratio $\kappa$ as well as the detection
efficiency $\epsilon$ are functions of position $\mathbf{r}$ (the function
$\kappa$ via the position-dependent excitation rate $k_{\mathrm{exc}}$). When
joining all these pieces together, the _lag-time dependent_ part of the
correlation function reads (up to some constant factor)

$\begin{split}g(t)&=\int
d\mathbf{r}p_{0}(\mathbf{r})\epsilon(\mathbf{r})s_{1}(t,\mathbf{r})\\\ &=\int
d\mathbf{r}\frac{k_{\mathrm{ph}}\epsilon(\mathbf{r})^{2}\kappa(\mathbf{r})s_{1}(t,\mathbf{r})}{k_{\mathrm{ph}}+\kappa(\mathbf{r})k_{\mathrm{isc}}}\end{split}$
(7)

where the integration extends over the whole sample volume. Thus, for lag-time
values on the order of the fluorescence lifetime $\tau$, we have to use eq.
(2) for $s_{1}(t)$ and we explicitly find

$\displaystyle g_{a}(t)=\int
d\mathbf{r}\frac{k_{\mathrm{ph}}\epsilon(\mathbf{r})^{2}\kappa(\mathbf{r})^{2}}{k_{\mathrm{ph}}+\kappa(\mathbf{r})k_{\mathrm{isc}}}\left\\{1-e^{-\left[k_{\mathrm{exc}}(\mathbf{r})+\tau^{-1}\right]t}\right\\}$
(8)

which describes the well-known anti-correlation (antibunching) of the
correlation function on very short timescales, and which is indicated by the
index $a$ in $g_{a}$. For longer lag-time values on the timescale of the
triplet-state lifetime $k_{\mathrm{ph}}^{-1}$, we have to use $s_{1}(t)=\kappa
s(t)$ with $s(t)$ from eq. (5) and we find

$\begin{split}g_{p}(t)=\int
d\mathbf{r}&\frac{k_{\mathrm{ph}}\epsilon(\mathbf{r})^{2}\kappa(\mathbf{r})^{2}}{k_{\mathrm{ph}}+\kappa(\mathbf{r})k_{\mathrm{isc}}}\\\
&\cdot\frac{k_{\mathrm{ph}}+\kappa(\mathbf{r})k_{\mathrm{isc}}e^{-(k_{\mathrm{ph}}+\kappa(\mathbf{r})k_{\mathrm{isc}})t}}{k_{\mathrm{ph}}+\kappa(\mathbf{r})k_{\mathrm{isc}}}\end{split}$
(9)

where the index $p$ in $g_{p}$ indicates that this is the part of the
correlation function that reflects the dye’s photophysics. On even longer
timescales, the temporal decay of the correlation function is governed by the
diffusion of the molecules through the detection volume and will be modeled
here by the conventional expression 13

$\begin{split}g_{d}(t)=\frac{1}{(1+4Dt/a^{2})\sqrt{1+4Dt/b^{2}}}\end{split}$
(10)

where $D$ denotes the diffusion coefficient, $a$ and $b$ are the short and
long axes of the assumed three-dimensional axially-symmetric Gaussian
detection volume, and we have again omitted all constant factors. For
sufficiently well-separated timescales of fluorescence lifetime, triplet-state
photophysics, and diffusion (given by the characteristic diffusion time
$a^{2}/4D$), the full correlation curve is well described by the product
$g_{a}(t)g_{p}(t)g_{d}(t)$.

Remarkably, experimentally measured correlation curves can be perfectly fitted
with similar expressions but where all the position-dependent exponentials are
replaced by single constant values, so that the short and long time parts of
the correlation function are proportional to

$\displaystyle g_{a}(t)\propto 1-Ae^{-\left(\langle
k_{\mathrm{exc}}\rangle+\tau^{-1}\right)t}$ (11)

and

$\displaystyle g_{p}(t)\propto 1+Te^{-(k_{\mathrm{ph}}+\langle\kappa\rangle
k_{\mathrm{isc}})t}$ (12)

with constant factors $A$ and $T$, and with effective excitation rate $\langle
k_{\mathrm{exc}}\rangle$ and effective _mean_ excitation rate
$\left<\kappa\right>$.

The core idea of our paper is now: (i) to use a fit of eq. (11) to an
experimentally measured antibunching curve for determining the effective
excitation rate $\langle k_{\mathrm{exc}}\rangle$, provided that we know the
fluorescence decay time $\tau$; (ii) to use this value $\langle
k_{\mathrm{exc}}\rangle$ in eq. (3) for calculating an effective mean
excitation rate $\left<\kappa\right>$; and (iii) to use this
$\left<\kappa\right>$ to fit eq. (12) to an experimentally measured
fluorescence correlation curve for extracting separate values for
$k_{\mathrm{ph}}$ _and_ $k_{\mathrm{isc}}$. What has been done so far in the
literature is to measure and fit correlation curves for different excitation
intensities (i.e. values of $\left<k_{\mathrm{exc}}\right>$), to estimate the
mean excitation rate $\left<\kappa\right>$ in the detection volume by
_independent measurements_ , and then to fit the obtained dependence of
$k_{\mathrm{ph}}+\langle\kappa\rangle k_{\mathrm{isc}}$ on
$\left<\kappa\right>$ for determining the photophysical rate constants.
Whereas the result for $k_{\mathrm{ph}}$ is relatively robust (it is the
intercept of the curve $k_{\mathrm{ph}}+\langle\kappa\rangle k_{\mathrm{isc}}$
with the abscissa in the limit of zero excitation intensity), the value of
$k_{\mathrm{isc}}$ crucially depends on the correct estimate of
$\left<\kappa\right>$. Our idea here is to determine this value
$\left<\kappa\right>$ in an intrinsic way from antibunching measurements.

In the next section, we will check by numerical simulation what the expected
bias and accuracy of this approach is, in the light that eqs. (11) and (12)
are only rough approximations of the exact equations (8) and (7). In the
section after the next section, we then apply our method for measuring the
photophysical rate constants $k_{\mathrm{ph}}$ and $k_{\mathrm{isc}}$ of
several dyes.

##  3 Numerical simulations

For simulating an FCS experiment, we use a simplified scalar theory of
excitation and detection that was successfully used for modeling dual-focus
fluorescence correlation experiments 14. The excitation intensity distribution
can be described as scalar Gaussian laser beam 15, which has been shown
experimentally to be an excellent approximation of the light intensity
distribution in the focus even when using high numerical aperture objectives
16, 17, 18. The three-dimensional excitation rate distribution is given by

$\displaystyle
k_{\mathrm{exc}}(x,y,z)=k_{\mathrm{exc},0}\exp\left[-\frac{2(x^{2}+y^{2})}{w^{2}(z)}\right]$
(13)

where $k_{\mathrm{exc},0}$ is the excitation rate at the center of the focus,
$x$ and $y$ are the two Cartesian coordinates perpendicular to the optical
axis, and $w(z)$ represents the beam diameter along the optical axis $z$ that
evolves as 15

$\displaystyle w(z)=w_{0}\sqrt{1+\left(\frac{\lambda_{\mathrm{ex}}z}{n\pi
w_{0}^{2}}\right)^{2}}.$ (14)

Here, $w_{0}$ is the beam waist in the focal plane $z=0$, $n$ denotes the
refractive index of the sample, and $\lambda_{\mathrm{ex}}$ represents the
vacuum wavelength of the excitation light. The detection efficiency $\epsilon$
can be approximated as 19, 20, 14

$\displaystyle\epsilon(z)=1-\exp\left[-\frac{2a^{2}}{\rho(z)^{2}}\right]$ (15)

where $a$ is the radius of the confocal aperture divided by the magnification,
and $\rho(z)$ is given by:

$\displaystyle\rho(z)=\rho_{0}\sqrt{1+\left(\frac{\lambda_{\mathrm{em}}z}{n\pi\rho_{0}^{2}}\right)^{2}}$
(16)

where $\lambda_{\mathrm{em}}$ is the vacuum wavelength of the emitted light,
and $\rho_{0}=\lambda_{\mathrm{em}}\pi/\tan\Theta$ is the radius of a
diffraction-limited Gaussian-approximated intensity distribution in the focal
plane that is obtained when focusing a plane wave through the microscope’s
objective, where $\Theta=\arcsin(\mathrm{N.A.}/n)$ is the half angle of the
cone of objective’s light collection with a numerical aperture N.A. This model
of the detection function takes into account the attenuation of the light by
the confocal pinhole only when an emitter is moved along the optical axis, but
ignores any pinhole effects on the detection when the emitter is moved in the
lateral plane. This approximation is acceptable as long as the excitation
laser beam waist is smaller than the size of the pinhole. As an example, the
two insets in Fig. 2 show a $xz$-cross-section of the excitation intensity
distribution $k_{\mathrm{exc}}$ (Fig. 2(a)) and the product
$k_{\mathrm{exc}}\epsilon$ (Fig. 2(b)), for excitation wavelength
$\lambda_{\mathrm{exc}}=663$ nm, a beam waist $w_{0}=350$ nm, and a pinhole
radius divided by magnification of $a=25\leavevmode\nobreak\ $µm$/60=417$ nm.

Figure 2: Schematic of the confocal setup used for all measurements. The inset
(a) shows an exemplary excitation intensity distribution for a laser beam with
663 nm wavelength and focus beam waist of 350 nm, calculated by using eq.
(13), and inset (b) shows the full detection function
$k_{\mathrm{exc}}\epsilon$ when detection is done through a pinhole with 25 µm
radius and at 60$\times$ magnification, calculated using eq. (16). Figure 3:
Model calculations. (a) Antibunching curves. (b) FCS curves for lag times
longer than 10 ns. (c) Linear fit of
$k_{\mathrm{ph}}+k_{\mathrm{isc}}\left<\kappa\right>$ against
$\left<\kappa\right>$, where
$k_{\mathrm{ph}}+k_{\mathrm{isc}}\left<\kappa\right>$ was obtained from
fitting the correlation curves in (b), and $\left<\kappa\right>$ was obtained
from fitting the antibunching curves in (a). The shown values for
$k_{\mathrm{isc}}/k_{\mathrm{ph}}$ and $k_{\mathrm{ph}}$ are the estimates of
the actual values obtained from the linear fit. (d) Relative errors for the
estimates of $k_{\mathrm{isc}}$ and $k_{\mathrm{ph}}$ as a function of the
actual value of $k_{\mathrm{isc}}$, with the value $k_{\mathrm{ph}}$ fixed to
1 µs-1.

Using the above model functions for $k_{\mathrm{exc}}(x,y,z)$ and $\kappa(z)$,
we first calculated an antibunching curve using eq. (8), fitted this curve
with the mono-exponential fit function as written in eq. (11) to obtain an
“averaged” value of $\left<k_{\mathrm{exc}}\right>$, and thereafter used this
value to calculate
$\left<\kappa\right>=\left<k_{\mathrm{exc}}\right>\tau/(1+\left<k_{\mathrm{exc}}\right>\tau)$.
Next, we calculated an FCS curve using eq. (7), fitted this curve with the
mono-exponential expression in eq.(12), which yielded the exponential
coefficient $k_{\mathrm{ph}}+k_{\mathrm{isc}}\left<\kappa\right>$. We repeated
this procedure for several excitation intensities $I_{0}$ yielding the linear
relation between $k_{\mathrm{ph}}+k_{\mathrm{isc}}\left<\kappa\right>$ and
$\left<\kappa\right>$. Using this linear relationship, estimates for
$k_{\mathrm{ph}}$ and $k_{\mathrm{isc}}$ were obtained that can be then
compared against the actual set values used for the model calculations. For
all these calculations, we fixed excitation wavelength
$\lambda_{\mathrm{ex}}=663$ nm, emission wavelength
$\lambda_{\mathrm{ex}}=660$ nm, molar extinction coefficient of the molecule
$\sigma=125000$ l$\cdot$M${}^{-1}\cdot$cm-1, fluorescence lifetime $\tau=1.8$
ns, excitation laser beam waist $w_{0}=350$ nm, and intersystem crossing rate
$k_{\mathrm{isc}}=1$ µs-1. The calculations were done for four different
values of 100 µW, 200 µW, 400 µW, and 800 µW of excitation laser power $P$,
where the corresponding excitation rates $k_{\mathrm{exc},0}$ (excitations per
second) were calculated by

$\displaystyle k_{\mathrm{exc},0}=\frac{2P}{\pi
w_{0}^{2}}\frac{10^{3}\ln(10)\sigma}{N_{\mathrm{A}}}\frac{\lambda_{\mathrm{ex}}}{hc}$
(17)

where $h$ is Planck’s constant, $c$ the vacuum speed of light, and
$N_{\mathrm{A}}$ the Avogadro-Loschmidt number. Fig. 3(a) shows the model
results for the antibunching curves, and fig. 3(b) for the corresponding FCS
curves. The integrals in eqs. (8) and (7) where approximated by summation over
a discretized grid covering a cylinder with 2.1 µm radius and $\pm$2.8 µm
axial extent, with grid spacing of 21 nm radially and 28 nm axially. In all
the calculations so far the phosphorescence rate was set to
$k_{\mathrm{ph}}=1$ µs-1. The linear fit of
$k_{\mathrm{ph}}+k_{\mathrm{isc}}\left<\kappa\right>$ as a function of
$\left<\kappa\right>$ is shown in Fig. 3(c). Next, we repeated these
calculations for a range of phosphorescence rates between 0.1 µs-1 and 10
µs-1, and the relative errors $k_{\mathrm{estimate}}/k_{\mathrm{actual}}-1$ of
the estimated rate values for $k_{\mathrm{isc}}$ and $k_{\mathrm{ph}}$ are
shown in Fig. 3(d).

It is noteworthy that the relative errors, for both the intersystem crossing
rate as well as phosphorescence rate, are below 4% across the whole range of
phosphorescence rates considered. It should also be noted that only the ratio
$k_{\mathrm{ph}}/k_{\mathrm{isc}}$ is important, not the absolute rate values,
so that our error estimate will be valid for all ratios
$k_{\mathrm{ph}}/k_{\mathrm{isc}}$ considered here, independent on the
absolute values. Thus, our model calculations show that despite the highly
non-uniform distributions of $k_{\mathrm{exc}}$ and $\epsilon$ across the
detection volume, fitting the resulting antibunching and FCS curves with mono-
exponential functions, and then using these fits for estimating the photo-
physical rates $k_{\mathrm{isc}}$ and $k_{\mathrm{ph}}$, delivers surprisingly
accurate results within a relative error of 4% (within the considered range of
0.1–10 for the ratio $k_{\mathrm{isc}}/k_{\mathrm{ph}}$). In the next
sections, we will apply our rate estimation scheme to determining intersystem
crossing and phosphorescence rates of several common fluorescent dyes.

##  4 Experimental results

We have measured the photophysics of two dyes, Rhodamine 110 in the green
spectral region, and ATTO 655 in the red spectral region. We chose Rhodamine
110 because there exist already excellent publications that report its
intersystem crossing and triplet-to-ground state transition rates 9, against
which we can validate the current method. ATTO 655 because is routinely used
in photo-electron transfer spectroscopy for monitoring conformational dynamics
in biomolecules (peptides, DNA). We benchmark the photophysical properties of
this dye in this work which will is essential for separating the timescales of
intensity fluctuations arising from its own photophysical transitions from the
conformational dynamics of biomolecules.

For each dye, we performed four antibunching/FCS measurements at four
increasing excitation intensity values (for measurement details see sections
below). The results for Rhodamine 110 are shown in Fig. 4. The four
(normalized) antibunching curves and the corresponding microsecond/millisecond
FCS autocorrelation curves are shown in panels (a) and (b), respectively. We
measured the fluorescence lifetime of Rhodamine 110 in a separate Time-
Correlated Single-Photon Counting (TCSPC) measurement and obtained a
fluorescence lifetime value $\tau=4.0$ ns which in perfect agreement with
reported values. By fitting the antibunching curves with eq. (11), we obtain
an estimate for $\left<k_{\mathrm{exc}}\right>+\tau^{-1}$, which allows us to
determine $\left<k_{\mathrm{exc}}\right>$ and then subsequently to estimate
$\left<\kappa\right>=\left<k_{\mathrm{exc}}\right>\tau/(1+\left<k_{\mathrm{exc}}\right>\tau)$.
The microsecond decays of the autocorrelation curves shown in Fig. 4(b) are
fitted with eq. (12), again using a single exponential rise function where the
exponential constant is equal to
$k_{\mathrm{isc}}\left<\kappa\right>+k_{\mathrm{ph}}$.

To obtain error estimates for our fit results, we applied a bootstrap
procedure. We divided each measurement into 10 sub-measurements, and then
repeated the fitting five times using each time a randomly chosen set
containing half of these sub-measurements. The resulting spread of fit results
is visualized by the point clouds shown in Fig. 4(c). Then, we fitted the
abscissa values $k_{\mathrm{isc}}\left<\kappa\right>+k_{\mathrm{ph}}$ linearly
against the ordinate values
$\left<\kappa\right>=\left<k_{\mathrm{exc}}\right>\tau/(1+\left<k_{\mathrm{exc}}\right>\tau)$,
again using a bootstrap approach. A linear fit was performed on a randomly
chosen set of four points, one from each point cloud. This was repeated
$10^{3}$ times, and the resulting mean value and variance are textually shown
in the figure. For Rhodamine 110, our methods yields an intersystem crossing
rate of $k_{\mathrm{isc}}=0.47\pm 0.01$ µs-1, and a triplet-to-ground-state
transition rate of $k_{\mathrm{ph}}=0.4\pm 0.02$ µs-1.

The corresponding measurements and fit results for
$\left<\kappa\right>=\left<k_{\mathrm{exc}}\right>\tau/(1+\left<k_{\mathrm{exc}}\right>\tau)$
and $k_{\mathrm{isc}}\left<\kappa\right>+k_{\mathrm{ph}}$ for the dye ATTO 655
are shown in Fig. 5, where we have used the same fitting and bootstrap
procedures for estimating the mean values and fit errors. For ATTO 655, we
observe a clear deviation from linearity for the highest excitation intensity
used, which we attribute to additional photophysical channels (e.g. triplet-
to-singlet state back pumping) that are activated at higher excitation
intensities and are not taken into account by our simple three-state-one-way
model shown in Fig. 1. Thus, for the linear fit of $\left<\kappa\right>$
against $k_{\mathrm{isc}}\left<\kappa\right>+k_{\mathrm{ph}}$, we used only
the three measurements corresponding to the three lowest excitation
intensities. This gave us an intersystem crossing rate of
$k_{\mathrm{isc}}=0.56\pm 0.05$ µs-1, and a triplet-to-ground-state transition
rate of $k_{\mathrm{ph}}=0.28\pm 0.01$ µs-1.

Finally, one has to add to the experimental uncertainties of
$k_{\mathrm{isc}}$ and $k_{\mathrm{ph}}$ as determined with our bootstrap
analysis a potential systematic bias of below 4% as indicated by our
theoretical-numerical analysis in the preceding chapter (see Fig. 3), which
amounts to a potential difference between our fit results and the actual
values of d$\approx 0.01-0.02$ µs-1 for our transition rates of Rhodamine 110
and ATTO 655.

Figure 4: Photophysics measurements of Rhodamine 110. (a) Measured
antibunching curves and fits (solid lines) for four different excitation laser
powers. (b) Measured µs/ms-FCS curves with fits (solid lines) for the same
four different excitation powers as in (a). (c) Linear fit of the rate
$k_{\mathrm{ph}}+k_{\mathrm{isc}}\left<\kappa\right>$ as extracted from the
fits in (b) against $\left<\kappa\right>$ as extracted from the fits in (a).
For each excitation power, the measurement was repeated five times, and the
linear fit was repeated a thousand times, each time randomly choosing one of
the five measurements for the four excitation powers. This yields the shown
uncertainty values for $k_{\mathrm{isc}}$ and $k_{\mathrm{ph}}$ (standard
deviation of the fitted value distributions). Figure 5: Photophysics of ATTO
655. Linear fit of $\left<\kappa\right>$ against
$k_{\mathrm{ph}}+k_{\mathrm{isc}}\left<\kappa\right>$ for the three lowest-
excitation measurements (three left point clouds). The highest excitation
measurement (right-most point cloud) was excluded because we observe an
obvious deviation from a linear relationship that we attribute to the
activation of other photophysical channels at higher excitation power that are
not included in our simple three-state model shown in Fig. 1.

##  5 Methods

###  5.1 Sample preparation

Fresh samples were prepared and measured at room temperature. Both ATTO 655
and Rhodamine 110 dyes were diluted in PBS (pH 7.4). The final concentration
of ATTO 655 was approximately 0.5 nM, while for Rhodamine 110 the
concentration was 2 nM. Experiments were always performed in 8-well chamber
slides containing a 150 µl aliquot of freely diffusing dye solution sealed
with parafilm to prevent evaporation during the measurements.Each intensity
point was measured for 4 hours with the number of photons which depends on the
excitation intensity with the highest value of 1.226 billion photons for ATTO
655 at the highest excitation intensity and 420 million photons at the lowest
intensity point for the same dye. For Rhodamine 110, the number of photons
ranges from 4 to 2 billions depending on the intensity point.

###  5.2 Confocal Setup and measurements

FCS measurements were performed at a commercially available MicroTime 200
(PicoQuant GmbH, Berlin, Germany) equipped with a water immersion objective
(60x UplanSApo, NA 1.2, Olympus, Japan). All measurements were carried out
with continuous-wave excitation. 488 nm laser (PhoxX 488-60, Omicron-Laserage
Laserprodukte GmbH, Rodgau, Germany) was used to excite Rhodamine 110, whereas
for ATTO 655 a 640 nm laser (LDH-D-C-640s, PicoQuant, Berlin, Germany) was
utilized. The lasers were coupled into a single-mode fiber and then re-
collimated using an apochromat lens. Thereafter, the laser beam was guided
into the microscope with the help of a dichroic mirror. The fluorescence
emission was collected by the same excitation objective, focused through a 50
um pinhole and focused onto two single-photon silicon avalanche photodiodes
(SPCM CD 3516 H, Excelitas Technologies, Inc., Dumberry, Canada) after being
split equally using a 50:50 beamsplitter. Additionally, appropriate bandpass
filters were used to block any back-reflected laser (692/40 for ATTO 655
measurements and 525/45 for Rhodamine 110 measurements). In this series of
measurements, every subsequent intensity point was set to be approximately
twice higher than the previous, with a total laser power of up to 1 mW.
Detected fluorescence photons were registered with the high-speed timing
electronics HydraHarp 400 and SymPhoTime software (PicoQuant GmbH, Berlin,
Germany) in time-tagged time-resoled mode. Raw data was correlated and fitted
using custom written MATLAB (The Mathworks, Inc. Nattick, MA, USA) routines .

##  6 Conclusion

In this paper, we combined nanosecond FCS measurements (fluorescence
antibunching) for estimating the absolute excitation intensity with micro- to
millisecond FCS measurements for monitoring photophysical transitions
(intersystem crossing, phosphorescence) to obtain calibration-free values for
the photophysical transition rate constants. As our numerical simulations
show, the systematic bias of the procedure is estimated to be smaller than
$\sim$5% for a wide range of rate constants. For the highest excitation
intensity used int he ATTO 655 experiments, we observe a slight deviation of
the expected linear relation between (effective) excitation rate and
(effective) photophysical rate constants which hint at the onset of other
photophysical processes that can be neglected at lower intensities. Thus,
future generalizations of our method could extend the theoretical model
underlying our data analysis for including such additional transitions and to
determine their corresponding rate constants. An any case, we believe that our
method presented here is a fast and satisfactorily precise method for
determining photophysical rate constants on minute amounts sample.

##  7 Acknowledgments

DS and JE acknowledge financial support by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) via project A02 of the Collective research
Collaboration SFB1456. JE acknowledges financing by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy - EXC 2067/1- 390729940, and financing by the European
Research Council (ERC) via project “smMIET” (Grant agreement No. 884488) under
the European Union’s Horizon 2020 research and innovation programme. N.K.
acknowledges EPSRC funding for the Rosalind Franklin Institute (RFI).

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$\displaystyle\frac{g_{dc}G_{\mu\nu}}{\sqrt{-\text{det}g}}H_{ab}^{\nu}+\frac{g_{dc}}{2\sqrt{-\text{det}g}}x_{a}^{\alpha}x_{b}^{\beta}\displaystyle\frac{\partial{G_{\alpha\beta}}}{\partial{x^{\mu}}}\Bigg{]}\
,$ $\displaystyle 0$ $\displaystyle=$
$\displaystyle\epsilon^{bd}\epsilon^{ac}\frac{G_{\mu\nu}g_{dc}}{\sqrt{-\text{det}g}}\left(B_{a}^{\mu}-x_{a}^{\mu}\right)\
,$ (61) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\epsilon^{bd}\epsilon^{ac}\Bigg{[}\displaystyle\frac{\partial{}}{\partial{x^{\rho}}}\left(\frac{G_{\mu\nu}g_{dc}}{\sqrt{-\text{det}g}}x^{\nu}_{b}\right)x_{a}^{\mu}B_{k}^{\rho}-\frac{g_{dc}}{2\sqrt{-\text{det}g}}x_{a}^{\alpha}x_{b}^{\beta}B_{k}^{\rho}\displaystyle\frac{\partial{G_{\alpha\beta}}}{\partial{x^{\rho}}}+\displaystyle\frac{\partial{}}{\partial{\sigma^{a}}}\left(\frac{G_{\rho\nu}g_{dc}}{\sqrt{-\text{det}g}}x^{\nu}_{b}\right)B_{k}^{\rho}+$
(62)
$\displaystyle\frac{G_{\mu\nu}g_{dc}}{\sqrt{-\text{det}g}}x_{a}^{\mu}H^{\nu}_{kb}-\frac{G_{\mu\nu}g_{dc}}{\sqrt{-\text{det}g}}(B_{a}^{\mu}H^{\nu}_{kb}-B_{k}^{\mu}H^{\nu}_{ab})-\displaystyle\frac{\partial{}}{\partial{x^{\rho}}}\left(\frac{G_{\mu\nu}g_{dc}}{\sqrt{-\text{det}g}}x_{b}^{\nu}\right)(B_{a}^{\mu}B_{k}^{\rho}-B_{k}^{\mu}B_{a}^{\rho})\Bigg{]}.$

Notice that plugging (61) into (62) implies (60), which means that (62) is an
identity as expected. Equation (61) is the sopde condition for
$\textbf{X}_{\mathscr{L}}$, i.e. $B_{a}^{\mu}=x_{a}^{\mu}$. Furthermore, (60)
becomes the Euler–Lagrange equations when working with
$j^{1}\phi=\left(\sigma^{a},x^{\mu}(\sigma),\displaystyle\frac{\partial{x^{\mu}}}{\partial{\sigma^{a}}}\right)$
that are integral sections holonomic $\textbf{X}_{\mathscr{L}}$ which satisfy

$D_{a}^{\mu}=x_{a}^{\mu}=\displaystyle\frac{\partial{x^{\mu}}}{\partial{\sigma^{a}}}\quad,\quad
H_{ab}^{\nu}=\displaystyle\frac{\partial{x_{b}^{\nu}}}{\partial{\sigma^{a}}}=\frac{\partial^{2}x^{\nu}}{\partial\sigma^{a}\partial\sigma^{b}}\quad,\quad
g_{ab}=h_{ab}=G_{\mu\nu}\displaystyle\frac{\partial{x^{\mu}}}{\partial{\sigma^{a}}}\displaystyle\frac{\partial{x^{\nu}}}{\partial{\sigma^{b}}}\
.$ (63)

Working with such sections that satisfy (63) it follows that (60) becomes

$\displaystyle\epsilon^{bd}\epsilon^{ac}\Bigg{[}\displaystyle\frac{\partial{}}{\partial{\sigma^{a}}}\left(\frac{h_{dc}G_{\mu\nu}}{\sqrt{-\text{det}h}}\displaystyle\frac{\partial{x^{\nu}}}{\partial{\sigma^{b}}}\right)+\displaystyle\frac{\partial{}}{\partial{x^{\rho}}}\left(\frac{h_{dc}G_{\mu\nu}}{\sqrt{-\text{det}h}}\displaystyle\frac{\partial{x^{\nu}}}{\partial{\sigma^{b}}}\right)\displaystyle\frac{\partial{x^{\rho}}}{\partial{\sigma^{a}}}$
$\displaystyle+\frac{h_{dc}G_{\mu\nu}}{\sqrt{-\text{det}h}}\frac{\partial^{2}x^{\nu}}{\partial\sigma^{a}\partial\sigma^{b}}-\frac{h_{dc}}{2\sqrt{-\text{det}h}}\displaystyle\frac{\partial{x^{\alpha}}}{\partial{\sigma^{a}}}\displaystyle\frac{\partial{x^{\beta}}}{\partial{\sigma^{b}}}\displaystyle\frac{\partial{G_{\alpha\beta}}}{\partial{x^{\mu}}}\Bigg{]}=0\
,$

which are the Euler–Lagrange equations for this field theory.

###  4.2 The String Hamiltonian Formulation

The De Donder–Weyl Hamiltonian formalism is developed starting from the
Legendre map $\mathscr{FL}:J^{1}E\rightarrow J^{1}E^{*}$ which satisfies

$\mathscr{FL}^{*}\sigma^{a}=\sigma^{a}\quad,\quad\mathscr{FL}^{*}x^{\mu}=x^{\mu}\quad,\quad\mathscr{FL}^{*}p^{a}_{\mu}=-T\sqrt{-\text{det}g}\
G_{\mu\nu}g^{ba}x_{b}^{\nu}\ .$ (64)

This Legendre map is invertible as expected from the regularity of the
generalized Hessian (57). The De Donder–Weyl Hamiltonian is defined as

$\mathscr{H}(\sigma^{a},x^{\mu},p^{a}_{\mu})\equiv
p^{a}_{\mu}(\mathscr{FL}^{-1})^{*}x_{a}^{\mu}-(\mathscr{FL}^{-1})^{*}\mathscr{L}\in
C^{\infty}(J^{1}E^{*})\ .$

This Hamiltonian can be expressed in terms of the variables on $J^{1}E^{*}$ as
[32]

$\Pi^{ab}\equiv G^{\mu\nu}p^{a}_{\mu}p^{b}_{\nu}\ \Rightarrow\
\mathscr{FL}^{*}\Pi^{ab}=-T^{2}\text{det}g\ g^{ba}\ \Leftrightarrow\
\mathscr{FL}^{*}\text{det}\Pi=(-T^{2}\text{det}g)^{2}\text{det}(g^{-1})=T^{4}\text{det}g\
,$

from which it follows that

$(\mathscr{FL}^{-1})^{*}x_{b}^{\nu}=-\frac{1}{T}\sqrt{-\text{det}\Pi}\
G^{\mu\nu}\Pi_{ab}p^{a}_{\mu}\ \Longrightarrow\
\mathscr{H}(\sigma^{a},x^{\mu},p^{a}_{\mu})=-\frac{1}{T}\sqrt{-\text{det}\Pi}\
,$

where

$\Pi_{ab}=\frac{1}{\text{det}\Pi}\epsilon_{cb}\epsilon_{da}\Pi^{cd}\ .$ (65)

The Hamilton–Cartan $2$–form and the multisymplectic Hamilton–Cartan $3$-form
on $J^{1}E^{*}$ are

$\displaystyle\Theta_{\mathscr{H}}$ $\displaystyle=$ $\displaystyle
p^{a}_{\mu}\text{d}x^{\mu}\wedge\text{d}^{1}\sigma_{a}-\mathscr{H}\wedge\text{d}^{2}\sigma=p^{a}_{\mu}\text{d}x^{\mu}\wedge\text{d}^{1}\sigma_{a}+\frac{1}{T}\sqrt{-\text{det}\Pi}\
\text{d}^{2}\sigma$ $\displaystyle\Omega_{\mathscr{H}}$ $\displaystyle=$
$\displaystyle-\text{d}p^{a}_{\mu}\wedge\text{d}x^{\mu}\wedge\text{d}^{1}\sigma_{a}+\text{d}\mathscr{H}\wedge\text{d}^{2}\sigma$
$\displaystyle=$
$\displaystyle-\text{d}p^{a}_{\mu}\wedge\text{d}x^{\mu}\wedge\text{d}^{1}\sigma_{a}-\frac{\sqrt{-\text{det}\Pi}}{T}\Pi_{ba}\left(\frac{1}{2}\partial_{\mu}G^{\rho\sigma}p^{a}_{\rho}p^{b}_{\sigma}\text{d}x^{\mu}+G^{\mu\nu}p^{b}_{\nu}\text{d}p^{a}_{\mu}\right)\wedge\text{d}^{2}\sigma\
.$

The field equations, $i(\textbf{X}_{\mathscr{H}})\Omega_{\mathscr{H}}=0$, can
be computed using a representative of a class of $\bar{\pi}^{1}$-transverse
locally decomposable multivector fields $\textbf{X}_{\mathscr{H}}=X_{0}\wedge
X_{1}\in\mathfrak{X}^{2}(J^{1}E^{*})$, with components given by the following
local expression

$X_{a}=\frac{\partial}{\partial\sigma^{a}}+D_{a}^{\mu}\frac{\partial}{\partial
x^{\mu}}+H^{c}_{a\mu}\frac{\partial}{\partial p^{c}_{\mu}}\ .$

Then,

$\displaystyle 0=i(\textbf{X}_{\mathscr{H}})\Omega_{\mathscr{H}}$
$\displaystyle=$
$\displaystyle-\left(H^{a}_{a\mu}-\frac{\sqrt{-\text{det}\Pi}}{2T}\Pi_{ba}\displaystyle\frac{\partial{G^{\rho\sigma}}}{\partial{x^{\mu}}}p^{a}_{\rho}p^{b}_{\sigma}\right)\text{d}x^{\mu}+\left(D_{a}^{\mu}+\frac{\sqrt{-\text{det}\Pi}}{T}\Pi_{ba}G^{\mu\nu}p^{b}_{\nu}\right)\text{d}p^{a}_{\mu}$
$\displaystyle-\left[D_{a}^{\mu}H^{a}_{k\mu}-D_{k}^{\mu}H^{a}_{a\mu}+\frac{\sqrt{-\text{det}\Pi}}{T}\Pi_{ba}\left(\frac{1}{2}D_{k}^{\mu}\displaystyle\frac{\partial{G^{\rho\sigma}}}{\partial{x^{\mu}}}p^{a}_{\rho}p^{b}_{\sigma}+H^{a}_{k\mu}G^{\mu\nu}p^{b}_{\nu}\right)\right]\text{d}\sigma^{k}\
.$

Setting differential forms separately equal to zero produces the following
field equations:

$\displaystyle
H^{a}_{a\mu}-\frac{\sqrt{-\text{det}\Pi}}{2T}\Pi_{ba}\displaystyle\frac{\partial{G^{\rho\sigma}}}{\partial{x^{\mu}}}p^{a}_{\rho}p^{b}_{\sigma}=0,$
(67a) $\displaystyle
D_{a}^{\mu}+\frac{\sqrt{-\text{det}\Pi}}{T}\Pi_{ba}G^{\mu\nu}p^{b}_{\nu}=0,$
(67b) $\displaystyle
D_{a}^{\mu}H^{a}_{k\mu}-D_{k}^{\mu}H^{a}_{a\mu}+\frac{\sqrt{-\text{det}\Pi}}{T}\Pi_{ba}\left(\frac{1}{2}D_{k}^{\mu}\displaystyle\frac{\partial{G^{\rho\sigma}}}{\partial{x^{\mu}}}p^{a}_{\rho}p^{b}_{\sigma}+H^{a}_{k\mu}G^{\mu\nu}p^{b}_{\nu}\right)=0.$
(67c)

Note that plugging (67b) into (67c) implies (67a), which means that (67c) is
an identity as expected. Furthermore, the variational problem on $J^{1}E^{*}$
is solved by integral sections
$\psi(\sigma)=(\sigma^{a},x^{\mu}(\sigma),p^{a}_{\mu}(\sigma))$ of
$\textbf{X}_{\mathscr{H}}$ for which
$D_{a}^{\mu}=\displaystyle\frac{\partial{x^{\mu}}}{\partial{\sigma^{a}}},\
H_{a\mu}^{c}=\displaystyle\frac{\partial{p^{c}_{\mu}}}{\partial{\sigma^{a}}}$,
and so the field equations become

$\displaystyle\displaystyle\displaystyle\frac{\partial{p^{a}_{\mu}}}{\partial{\sigma^{a}}}-\frac{\sqrt{-\text{det}\Pi}}{2T}\Pi_{ba}\partial_{\mu}G^{\rho\sigma}p^{a}_{\rho}p^{b}_{\sigma}=0,$
$\displaystyle\displaystyle\displaystyle\frac{\partial{x^{\mu}}}{\partial{\sigma^{a}}}+\frac{\sqrt{-\text{det}\Pi}}{T}\Pi_{ba}G^{\mu\nu}p^{b}_{\nu}=0,$

which are the corresponding Hamilton–De Donder–Weyl equations for the bosonic
string; these equations can be plugged into another one to give the
Euler–Lagrange equations as usual.

###  4.3 Symmetries

####  4.3.1 Worldsheet Diffeomorphisms

Consider worldsheet diffeomorphisms produced by

$\tilde{\sigma}^{a}=\sigma^{a}+\xi^{a}\ \Rightarrow\
\frac{\partial\tilde{\sigma}^{a}}{\partial\sigma^{b}}=\delta^{a}_{b}+\partial_{b}\xi^{a}\
.$ (68)

It follows that

$\displaystyle\begin{split}\text{det}\tilde{g}=\left[\text{det}\left(\frac{\partial\tilde{\sigma}^{a}}{\partial\sigma^{b}}\right)\right]^{-2}\text{det}g\quad,\quad\text{d}^{2}\tilde{\sigma}=\text{det}\left(\frac{\partial\tilde{\sigma}^{a}}{\partial\sigma^{b}}\right)\text{d}^{2}\sigma\
,\end{split}$

and hence

$\textbf{L}=-T\sqrt{-\text{det}\tilde{g}}\
\text{d}^{2}\tilde{\sigma}=-T\sqrt{-\text{det}g}\ \text{d}^{2}\sigma\ .$

The vector field $\xi\in\mathfrak{X}(\Sigma)$ which generates the worldsheet
diffeomorphisms (68) is given by

$\xi=-\xi^{a}\displaystyle\frac{\partial{}}{\partial{\sigma^{a}}}\in\mathfrak{X}(\Sigma).$

The lift of the vector field $\xi\in\mathfrak{X}(\Sigma)$ to $E=\Sigma\times
M$, obtained from (23), is

$\xi_{E}=-\xi^{a}\frac{\partial}{\partial\sigma^{a}}\in\mathfrak{X}(E),$

and the canonical lift of $\xi_{E}$ to $J^{1}E$ gives

$X_{\xi}=-\xi^{a}\frac{\partial}{\partial\sigma^{a}}+x_{b}^{\mu}\displaystyle\frac{\partial{\xi^{b}}}{\partial{\sigma^{a}}}\frac{\partial}{\partial
x_{a}^{\mu}}\ \in\mathfrak{X}(J^{1}E)\ .$

The resulting field variation given by the Lie derivative (23) of the local
sections $\phi:\Sigma\rightarrow E$ by the vector field $\xi$ is

$\delta
X^{\mu}(\sigma)=-\xi^{a}\displaystyle\frac{\partial{X^{\mu}}}{\partial{\sigma^{a}}}\
.$

It follows that

$\displaystyle\mathpzc{L}_{X_{\xi}}\textbf{L}=0\ \Rightarrow\
\mathpzc{L}_{X_{\xi}}\Theta_{\mathscr{L}}=0\ ,$

so the covariant momentum map is given as

$J_{\mathscr{L}}(X_{\xi})=-i(X_{\xi})\Theta_{\mathscr{L}}=-T\sqrt{-\text{det}g}\left(\epsilon_{ac}\xi^{c}G_{\mu\nu}g^{ba}x_{b}^{\nu}\text{d}x^{\mu}+\xi^{a}\text{d}^{1}\sigma_{a}\right)\
\in{\mit\Omega}^{1}(J^{1}E)\ .$

In the Hamiltonian formalism, the vector field
$Y_{\xi}=\mathscr{FL}_{*}X_{\xi}\in\mathfrak{X}(J^{1}E^{*})$ which takes form

$Y_{\xi}=-\xi^{a}\frac{\partial}{\partial\sigma^{a}}-\frac{1}{T}\sqrt{-\text{det}\Pi}\
G^{\mu\nu}\Pi_{bc}p^{b}_{\nu}\displaystyle\frac{\partial{\xi^{c}}}{\partial{\sigma_{a}}}\frac{\partial}{\partial
p^{\mu}_{a}}\ ,$

generates the worldsheet diffeomorphisms on the multimomentum phase space
$J^{1}E^{*}$ and is used to construct the covariant momentum map

$J_{\mathscr{H}}(Y_{\xi})=-i(Y_{\xi})\Theta_{\mathscr{H}}=\epsilon_{ac}\xi^{c}p^{a}_{\mu}\text{d}x^{\mu}-\frac{1}{T}\sqrt{-\text{det}\Pi}\
\xi^{a}\text{d}^{1}\sigma_{a}\in{\mit\Omega}^{1}(J^{1}E^{*})\ .$

####  4.3.2 Spacetime Isometries

Spacetime symmetries, which arise from performing transformations on the
coordinates of $M$, are gauge symmetries in string theory since they are
generated by vector fields on the configuration manifold $E$ which are
vertical with respect to the projection onto the base space $\Sigma$.

Consider infinitesimal spacetime diffeomorphisms $x^{\mu}\rightarrow
x^{\mu}+\zeta^{\mu}(x)$ generated by the vector field

$\zeta=-\zeta^{\mu}\frac{\partial}{\partial x^{\mu}}\in\mathfrak{X}(M)\ ,$

which is a gauge transformation as the corresponding vector field
$\zeta_{E}\in\mathfrak{X}(E)$ is $\pi$-vertical:

$\zeta_{E}=-\zeta^{\mu}\frac{\partial}{\partial x^{\mu}}\ .$

The canonical lift to $J^{1}E$, given by the jet prolongation of $\zeta$, is

$X_{\zeta}=-\zeta^{\mu}\frac{\partial}{\partial
x^{\mu}}-x^{\nu}_{a}\partial_{\nu}\zeta^{\mu}\frac{\partial}{\partial
x^{\mu}_{a}}\in\mathfrak{X}(J^{1}E)\ .$

The Lie derivative of $\textbf{L}=-T\sqrt{-\text{det}g}\ \text{d}^{2}\sigma$
with respect to $X_{\zeta}$ is

$\displaystyle\mathpzc{L}_{X_{\zeta}}\textbf{L}=\frac{T}{2}\sqrt{-\text{det}g}\
g^{ba}x^{\nu}_{b}x^{\mu}_{a}\mathpzc{L}_{\zeta}G_{\mu\nu}\text{d}^{2}\sigma\
,$

where $\mathpzc{L}_{\zeta}G_{\mu\nu}$ is the Lie derivative of $G_{\mu\nu}$ on
$M$ with respect to
$\displaystyle\zeta=-\zeta^{\mu}\displaystyle\frac{\partial{}}{\partial{x^{\mu}}}\in\mathfrak{X}(M)$.
Then, when $\zeta$ is a Killing vector field,

$\displaystyle\mathpzc{L}_{X_{\zeta}}\textbf{L}=0\ \Rightarrow\
\mathpzc{L}_{X_{\zeta}}\Theta_{\mathscr{L}}=0\ ,$

and it therefore follows that the covariant momentum map is given by

$J_{\mathscr{L}}(X_{\zeta})=-i(X_{\zeta})\Theta_{\mathscr{L}}=T\sqrt{-\text{det}g}\
G_{\mu\nu}G^{ba}x^{\nu}_{b}\xi^{\mu}\text{d}^{1}\sigma_{a}\in{\mit\Omega}^{1}(J^{1}E)\
.$

On the Hamiltonian side,

$Y_{\zeta}=\mathscr{FL}_{*}X_{\zeta}=-\xi^{\mu}\frac{\partial}{\partial
x^{\mu}}+\frac{1}{T}\sqrt{-\text{det}\Pi}\,G^{\mu\nu}\Pi_{ba}\,p^{b}_{\mu}\,\displaystyle\frac{\partial{\xi_{\mu}}}{\partial{x^{\nu}}}\frac{\partial}{\partial
p^{a}_{\mu}}\in\mathfrak{X}(J^{1}E^{*})\ ,$

so the covariant momentum map is given by

$J_{\mathscr{H}}(Y_{\zeta})=-i(Y_{\zeta})\Theta_{\mathscr{H}}=-p^{a}_{\mu}\,\zeta^{\mu}\,\text{d}^{1}\sigma_{a}\in{\mit\Omega}^{1}(J^{1}E^{*})\
.$

###  4.4 Generalization to p-Branes (p >1)

The generalization to $p$-branes ($m=p+1$) is straightforward to perform by
using the general Lagrangian density (4) for which the corresponding
Lagrangian energy $E_{\mathscr{L}}\in C^{\infty}(J^{1}E)$ is

$E_{\mathscr{L}}\equiv\frac{\partial\mathscr{L}}{\partial
x_{a}^{\mu}}x_{a}^{\mu}-\mathscr{L}=-T_{p}\sqrt{-\text{det}g}\
(g^{ba}g_{ab}-1)=-pT_{p}\sqrt{-\text{det}g}\ ,$

where $g_{ab}$ is now an $m\times m$ matrix, so $g^{ba}g_{ab}=m=p+1$. Then

$\displaystyle\Theta_{\mathscr{L}}$ $\displaystyle=$ $\displaystyle-
T_{p}\sqrt{-\text{det}g}\
\left[G_{\mu\nu}g^{ba}x_{b}^{\nu}\text{d}x^{\mu}\wedge\text{d}^{m-1}\sigma_{a}-p\text{d}^{m}\sigma\right],$
$\displaystyle\Omega_{\mathscr{L}}$ $\displaystyle=$ $\displaystyle\
T_{p}\partial_{\rho}\left(\sqrt{-\text{det}g}G_{\mu\nu}g^{ba}x^{\nu}_{b}\right)\text{d}x^{\rho}\wedge\text{d}x^{\mu}\wedge\text{d}^{m-1}\sigma_{a}+$
$\displaystyle
pT_{p}\sqrt{-\text{det}g}G_{\mu\nu}g^{ba}\text{d}x^{\nu}_{b}\wedge\text{d}x^{\mu}\wedge\text{d}^{m-1}\sigma_{a}+$
$\displaystyle T_{p}\left[\frac{1}{2}\sqrt{-\text{det}g}\
g^{ba}x^{\alpha}_{a}x^{\beta}_{b}\partial_{\mu}G_{\alpha\beta}-\displaystyle\frac{\partial{}}{\partial{\sigma^{a}}}\left(\sqrt{-\text{det}g}G_{\mu\nu}g^{ba}x^{\nu}_{b}\right)\right]\text{d}x^{\mu}\wedge\text{d}^{m}\sigma-$
$\displaystyle
pT_{p}\sqrt{-\text{det}g}G_{\mu\nu}g^{ba}x^{\mu}_{a}\text{d}x^{\nu}_{b}\wedge\text{d}^{m}\sigma\
.$

Proceeding similarly as before, take an $m$-multivector field
$\textbf{X}_{\mathscr{L}}=X_{0}\wedge X_{1}\wedge\dotsb\wedge X_{m-1}$, with
components

$X_{a}=\frac{\partial}{\partial\sigma^{a}}+B_{a}^{\mu}\frac{\partial}{\partial
x^{\mu}}+H_{ab}^{\nu}\frac{\partial}{\partial x_{b}^{\nu}}$

to obtain the field equations, one of which is an identity, another gives the
sopde condition $B^{\mu}_{a}=x^{\mu}_{a}$, and the third equation upon
plugging in the holonomic condition is

$\displaystyle\displaystyle\frac{\partial{}}{\partial{\sigma^{a}}}\left(\sqrt{-\text{det}g}G_{\mu\nu}g^{ba}x^{\nu}_{b}\right)+\displaystyle\frac{\partial{}}{\partial{x^{\rho}}}\left(\sqrt{-\text{det}g}\
G_{\mu\nu}g^{ba}x^{\nu}_{b}\right)x^{\rho}_{a}$
$\displaystyle+p\sqrt{-\text{det}g}\
G_{\mu\nu}g^{ba}H^{\nu}_{ab}-\frac{1}{2}\sqrt{-\text{det}g}\
g^{ba}x^{\alpha}_{a}x^{\beta}_{b}\frac{\partial}{\partial
x^{\mu}}G_{\alpha\beta}=0\ ,$

which becomes the Euler–Lagrange equations when working on integral sections
of $\textbf{X}_{\mathscr{L}}$:

$\displaystyle\displaystyle\frac{\partial{}}{\partial{\sigma^{a}}}\left(\sqrt{-\text{det}g}G_{\mu\nu}h^{ba}\displaystyle\frac{\partial{x^{\nu}}}{\partial{\sigma^{b}}}\right)+\displaystyle\frac{\partial{}}{\partial{x^{\rho}}}\left(\sqrt{-\text{det}h}\
G_{\mu\nu}h^{ba}\displaystyle\frac{\partial{x^{\nu}}}{\partial{\sigma^{b}}}\right)\displaystyle\frac{\partial{x^{\rho}}}{\partial{\sigma^{a}}}$
$\displaystyle+p\sqrt{-\text{det}g}\
G_{\mu\nu}h^{ba}\frac{\partial^{2}x^{\nu}}{\partial\sigma^{a}\partial\sigma^{b}}-\frac{1}{2}\sqrt{-\text{det}h}\
h^{ba}\displaystyle\frac{\partial{x^{\alpha}}}{\partial{\sigma^{a}}}\displaystyle\frac{\partial{x^{\beta}}}{\partial{\sigma^{b}}}\displaystyle\frac{\partial{G_{\alpha\beta}}}{\partial{x^{\mu}}}=0.$

The Hamiltonian formulation is also carried out as in the string case with the
Legendre map $\mathscr{FL}$ which, for $p$-branes, is

$\mathscr{FL}^{*}\sigma^{a}=\sigma^{a}\quad,\quad\mathscr{FL}^{*}x^{\mu}=x^{\mu}\quad,\quad\mathscr{FL}^{*}p^{a}_{\mu}=\frac{\partial\mathscr{L}}{\partial
x^{\mu}_{a}}=-T_{p}\sqrt{-\text{det}g}\ G_{\mu\nu}g^{ba}x^{\nu}_{b}\ .$

Next, define the matrix

$\Pi^{ab}=G^{\mu\nu}p^{a}_{\mu}p^{b}_{\nu}\ \Rightarrow\
\mathscr{FL}^{*}\Pi^{ab}=-T_{p}^{2}\text{det}g\ g^{ba},$

from which it follows that

$\left(\mathscr{FL}^{-1}\right)^{*}\text{det}g=-\frac{1}{T_{p}^{2}}\left(\frac{\text{det}\Pi}{-T_{p}^{2}}\right)^{1/p}\
,$

and hence

$\left(\mathscr{FL}^{-1}\right)^{*}x^{\nu}_{b}=-\sqrt{\left(\frac{\text{det}\Pi}{-T_{p}^{2}}\right)^{1/p}}G^{\mu\nu}\Pi_{ab}p^{a}_{\mu}.$

The inverse matrix $\Pi_{ab}$ is given by the following general version of the
identity (65):

$\Pi_{ab}=-\frac{1}{p!\text{det}\Pi}\epsilon_{aa_{2}\dotsm
a_{m}}\epsilon_{bb_{2}\dotsm b_{m}}\Pi^{b_{2}a_{2}}\dotsm\Pi^{b_{m}a_{m}}\ .$

Then the De Donder–Weyl Hamiltonian for the $p$-brane theory is

$\mathscr{H}(\sigma^{a},x^{\mu},p^{a}_{\mu})=-\sqrt{\left(\frac{\text{det}\Pi}{-T_{p}^{2}}\right)^{1/p}}\in
C^{\infty}(J^{1}E^{*})\ .$

The De Donder–Weyl field equations are:

$\frac{\partial
X^{\mu}}{\partial\sigma^{a}}=\left(\frac{\partial\mathscr{H}}{\partial
p^{a}_{\mu}}\right)\circ\psi\quad,\quad\frac{\partial
p^{a}_{\mu}}{\partial\sigma^{a}}=-\left(\frac{\partial\mathscr{H}}{\partial
x^{\mu}}\right)\circ\psi\ ,$

however, the Hamiltonian is computationally more tedious to deal with than in
the case of the string.

The multisymplectic symmetries are now worldvolume diffeomorphisms and
spacetime isometires. The worldvolume diffeomorphsims are generated on
$J^{1}E$ by the vector field,

$X_{\xi}=-\xi^{a}\frac{\partial}{\partial\sigma^{a}}+x_{b}^{\mu}\displaystyle\frac{\partial{\xi^{b}}}{\partial{\sigma^{a}}}\frac{\partial}{\partial
x_{a}^{\mu}}\ \in\mathfrak{X}(J^{1}E)\ ,$

and the corresponding momentum map
$J_{\mathscr{L}}(X_{\xi})=-i(X_{\xi})\Theta_{\mathscr{L}}\in{\mit\Omega}^{m-1}(J^{1}E)$
is

$J_{\mathscr{L}}(X_{\xi})=T_{p}\sqrt{-\text{det}g}\
\left[G_{\mu\nu}g^{ba}x_{b}^{\nu}\xi^{c}\text{d}x^{\mu}\wedge\text{d}^{m-2}\sigma_{ac}+p\xi^{a}\text{d}^{m-1}\sigma_{a}\right]\
.$

Furthermore, as in the string theory case, the spacetime isometries which
produce symmetries of the multisymplectic form $\Omega_{\mathscr{L}}$ are
generated on $J^{1}E$ by the following vector field:

$X_{\zeta}=-\zeta^{\mu}\frac{\partial}{\partial
x^{\mu}}-x^{\nu}_{a}\partial_{\nu}\zeta^{\mu}\frac{\partial}{\partial
x^{\mu}_{a}}\in\mathfrak{X}(J^{1}E)\ .$

The corresponding momentum map
$J_{\mathscr{L}}=-i(X_{\zeta})\Theta_{\mathscr{L}}\in{\mit\Omega}^{m-1}(J^{1}E)$
is given as,

$J_{\mathscr{L}}(X_{\zeta})=-T_{p}\sqrt{-\text{det}g}\
G_{\mu\nu}g^{ba}x_{b}^{\nu}\zeta^{\mu}\text{d}^{m-1}\sigma_{a}\ .$

##  5 (2+1)-dimensional Gravity

Gravity with cosmological constant $\lambda$ in $2+1$ dimensions is developed
in the tetrad formalism using the vielbein $e^{a}_{\mu}$ given by
$g_{\mu\nu}=e^{a}_{\mu}e^{b}_{\nu}\eta_{ab}$ ($\mu=0,1,2$, $a=0,1,2$), and the
Hodge dual spin connection
$\omega^{a}_{\mu}=\frac{1}{2}\epsilon^{abc}\omega_{\mu bc}$, which are treated
as the variational fields of the theory as in [40].

###  5.1 Lagrangian Formulation

The configuration bundle $E$ has coordinates
$(x^{\mu},e^{a}_{\mu},\omega_{\rho}^{c})$; the induced coordinates on $J^{1}E$
are
$(x^{\mu},e^{a}_{\mu},\omega_{\rho}^{c},e^{a}_{\sigma\mu},\omega^{c}_{\sigma\rho})$
and $j^{1}\phi:M\rightarrow
J^{1}E:x^{\mu}\mapsto\left(x^{\mu},e^{a}_{\mu}(x),\omega_{\rho}^{c}(x),\partial_{\sigma}e^{a}_{\mu}(x),\partial_{\sigma}\omega^{c}_{\rho}(x)\right)$
are the first-order jet prolongations. The Lagrangian density is given by

$\textbf{L}=\mathscr{L}(x^{\mu},e^{a}_{\mu},\omega_{\rho}^{c},e^{a}_{\sigma\mu},\omega^{c}_{\sigma\rho})\text{d}^{3}x=\epsilon^{\mu\nu\rho}\left[2\eta_{ac}e^{a}_{\mu}\omega^{c}_{\nu\rho}+\epsilon_{abc}\left(e^{a}_{\mu}\omega^{b}_{\nu}\omega^{c}_{\rho}+\frac{1}{3}\lambda
e^{a}_{\mu}e^{b}_{\nu}e^{c}_{\rho}\right)\right]\text{d}^{3}x\ .$

It is well-known that this Lagrangian density can be written as a Chern–Simons
theory [40],

$\textbf{L}=\frac{1}{2}\Big{\langle}A\wedge\text{d}A+\frac{2}{3}A\wedge
A\wedge A\Big{\rangle}\ ,$

with gauge field
$A=\left(e^{a}_{\mu}P_{a}+\omega^{a}_{\mu}J_{a}\right)\text{d}x^{\mu}$, where
$J^{a}=\frac{1}{2}\epsilon^{abc}J_{bc}$ are the Lorentz generators in the dual
form and $P^{a}$ are the translation generators which satisfy

$\left[J_{a},J_{b}\right]=\epsilon_{abc}J^{c}\quad,\quad\left[J_{a},P_{b}\right]=\epsilon_{abc}P^{c}\quad,\quad\left[P_{a},P_{b}\right]=\lambda\epsilon_{abc}J^{c}\
.$ (69)

The invariant bilinear form on the Lie algebra of the gauge group is, in
general, given as

$\left<J_{a},P_{b}\right>=c_{0}\eta_{ab}\quad,\quad\left<J_{a},J_{b}\right>=c_{1}\eta_{ab}\quad,\quad\left<P_{a},P_{b}\right>=\lambda
c_{1}\eta_{ab}\ .$

In this work, the choice $c_{0}=1,\ c_{1}=0$ is made and hence,

$\left<J_{a},P_{b}\right>=\eta_{ab}\quad,\quad\left<J_{a},J_{b}\right>=\left<P_{a},P_{b}\right>=0\
.$

This field theory is singular which can be seen directly from the following
components of the Hessian matrix:

$\frac{\partial^{2}\mathscr{L}}{\partial e^{a}_{\mu\nu}\partial
e^{b}_{\rho\sigma}}=0\quad,\quad\frac{\partial^{2}\mathscr{L}}{\partial\omega^{a}_{\mu\nu}\partial\omega^{b}_{\rho\sigma}}=0\
.$

The null vectors of the Hessian matrix above are given as

$_{(1)}\gamma^{a}_{\mu\nu}\ =\ _{(2)}\gamma^{a}_{\mu\nu}\ =\
\begin{pmatrix}\begin{pmatrix}1&1&1\\\ 1&1&1\\\
1&1&1\end{pmatrix}&\begin{pmatrix}1&1&1\\\ 1&1&1\\\
1&1&1\end{pmatrix}&\begin{pmatrix}1&1&1\\\ 1&1&1\\\
1&1&1\end{pmatrix}\end{pmatrix}^{\textbf{T}}\ ,$ (70)

respectively. The null vector above is expressed as a column vector in the
latin (locally Minkowski tangent space) upper index while, at each row of the
column vector, the $3\times 3$ matrices are indexed by the roman lower
indices.

The corresponding Lagrangian energy $E_{\mathscr{L}}\in C^{\infty}(J^{1}E)$,
the Poincaré–Cartan $3$-form
$\Theta_{\mathscr{L}}\in{\mit\Omega}^{3}(J^{1}E)$, and the premultisymplectic
Poincaré–Cartan $4$-form $\Omega_{\mathscr{L}}\in{\mit\Omega}^{4}(J^{1}E)$ are
given by are given by

$\displaystyle E_{\mathscr{L}}$ $\displaystyle=$
$\displaystyle\frac{\partial\mathscr{L}}{\partial
e^{a}_{\sigma\mu}}e^{a}_{\sigma\mu}+\frac{\partial\mathscr{L}}{\partial\omega^{c}_{\sigma\rho}}\omega^{c}_{\sigma\rho}-\mathscr{L}(x^{\mu},e^{a}_{\mu},\omega_{\rho}^{c},e^{a}_{\sigma\mu},\omega^{c}_{\sigma\rho})=-\epsilon^{\mu\nu\rho}\epsilon_{abc}\Big{(}e^{a}_{\mu}\omega^{b}_{\nu}\omega^{c}_{\rho}+\frac{1}{3}\lambda
e^{a}_{\mu}e^{b}_{\nu}e^{c}_{\rho}\Big{)}\ .$
$\displaystyle\Theta_{\mathscr{L}}$ $\displaystyle=$
$\displaystyle\frac{\partial\mathscr{L}}{\partial
e^{a}_{\sigma\mu}}\text{d}e^{a}_{\mu}\wedge\text{d}^{2}x_{\sigma}+\frac{\partial\mathscr{L}}{\partial\omega^{c}_{\sigma\rho}}\text{d}\omega^{c}_{\rho}\wedge\text{d}^{2}x_{\sigma}-E_{\mathscr{L}}\text{d}^{3}x=\widetilde{\Theta}_{\mathscr{L}}-E_{\mathscr{L}}\text{d}^{3}x$
$\displaystyle=$ $\displaystyle
2\epsilon^{\mu\nu\rho}\eta_{ac}e^{a}_{\mu}\text{d}\omega^{c}_{\rho}\wedge\text{d}^{2}x_{\nu}+\epsilon^{\mu\nu\rho}\epsilon_{abc}\Big{(}e^{a}_{\mu}\omega^{b}_{\nu}\omega^{c}_{\rho}+\frac{1}{3}\lambda
e^{a}_{\mu}e^{b}_{\nu}e^{c}_{\rho}\Big{)}\text{d}^{3}x\ ,$
$\displaystyle\Omega_{\mathscr{L}}$ $\displaystyle=$
$\displaystyle-\text{d}\Big{(}\frac{\partial\mathscr{L}}{\partial
e^{a}_{\sigma\mu}}\Big{)}\wedge\text{d}e^{a}_{\mu}\wedge\text{d}^{2}x_{\sigma}-\text{d}\Big{(}\frac{\partial\mathscr{L}}{\partial\omega^{c}_{\sigma\rho}}\Big{)}\wedge\text{d}\omega^{c}_{\rho}\wedge\text{d}^{2}x_{\sigma}+\text{d}E_{\mathscr{L}}\wedge\text{d}^{3}x$
$\displaystyle=$
$\displaystyle-2\epsilon^{\mu\nu\rho}\eta_{ac}\text{d}e^{a}_{\mu}\wedge\text{d}\omega^{c}_{\rho}\wedge\text{d}^{2}x_{\nu}-\epsilon^{\mu\nu\rho}\epsilon_{abc}\left(\omega^{b}_{\nu}\omega^{c}_{\rho}\text{d}e^{a}_{\mu}+2e^{a}_{\mu}\omega^{b}_{\nu}\text{d}\omega^{c}_{\rho}+\lambda
e^{b}_{\nu}e^{c}_{\rho}\text{d}e^{a}_{\mu}\right)\wedge\text{d}^{3}x$
$\displaystyle=$
$\displaystyle\widetilde{\Omega}_{\mathscr{L}}+\text{d}E_{\mathscr{L}}\wedge\text{d}^{3}x\
.$

The field equations are obtained using multivector fields written as
$\textbf{X}_{\mathscr{L}}=X_{0}\wedge X_{1}\wedge
X_{2}\in\mathfrak{X}^{3}(J^{1}E)$ with components given by

$X_{\nu}=\frac{\partial}{\partial
x^{\nu}}+B^{a}_{\nu\mu}\frac{\partial}{\partial
e^{a}_{\mu}}+C^{c}_{\nu\rho}\frac{\partial}{\partial\omega^{c}_{\rho}}+D^{a}_{\nu\sigma\mu}\frac{\partial}{\partial
e^{a}_{\sigma\mu}}+H^{c}_{\nu\sigma\rho}\frac{\partial}{\partial\omega^{c}_{\sigma\rho}}\
.$ (71)

The geometric field equation (3) gives,

$\displaystyle\begin{split}i(\textbf{X}_{\mathscr{L}})\Omega_{\mathscr{L}}&=-2\epsilon^{\mu\nu\rho}\eta_{ac}\left(B^{a}_{\nu\mu}\text{d}\omega^{c}_{\rho}-C^{c}_{\nu\rho}\text{d}e^{a}_{\mu}\right)+\epsilon^{\mu\nu\rho}\epsilon_{abc}\left(\omega^{b}_{\nu}\omega^{c}_{\rho}\text{d}e^{a}_{\mu}+2e^{a}_{\mu}\omega^{b}_{\nu}\text{d}\omega^{c}_{\rho}+\lambda
e^{b}_{\nu}e^{c}_{\rho}\text{d}e^{a}_{\mu}\right)\\\ &\ \
-\epsilon^{\mu\nu\rho}\left[2\eta_{ac}\left(B^{a}_{\lambda\mu}C^{c}_{\nu\rho}-B^{a}_{\nu\mu}C^{c}_{\lambda\rho}\right)-\epsilon_{abc}\left(\omega^{b}_{\nu}\omega^{c}_{\rho}B^{a}_{\lambda\mu}+2e^{a}_{\mu}\omega^{b}_{\nu}C^{c}_{\lambda\rho}+\lambda
e^{b}_{\nu}e^{c}_{\rho}B^{a}_{\lambda\mu}\right)\right]\text{d}x^{\lambda}=0\
.\end{split}$

Setting differential forms separately equal to zero produces the following two
independent field equations:

$\displaystyle\epsilon^{\mu\nu\rho}\left[\eta_{ac}C^{c}_{\nu\rho}+\frac{1}{2}\epsilon_{abc}\left(\omega^{b}_{\nu}\omega^{c}_{\rho}+\lambda
e^{b}_{\nu}e^{c}_{\rho}\right)\right]=0\ ,$ (72a)
$\displaystyle\epsilon^{\mu\nu\rho}\left[\eta_{ac}B^{a}_{\nu\mu}-\epsilon_{abc}e^{a}_{\mu}\omega^{b}_{\nu}\right]=0\
.$ (72b)

These equations are compatible on all of $J^{1}E$ as no compatibility
constraints arise. Note that, as

$\text{ker}\ \widetilde{\Omega}_{\mathscr{L}}\cap\text{ker}\
\text{d}^{3}x=\left\langle\frac{\partial}{\partial
e^{a}_{\rho\mu}},\frac{\partial}{\partial\omega^{a}_{\rho\mu}}\right\rangle\
,$

it follows from Proposition 1 that $\displaystyle
i\left(\frac{\partial}{\partial
e^{a}_{\rho\mu}}\right)\text{d}E_{\mathscr{L}}=i\left(\frac{\partial}{\partial\omega^{a}_{\rho\mu}}\right)\text{d}E_{\mathscr{L}}=0$
is satisfied automatically. Observe also that, as
$\displaystyle\frac{\partial}{\partial
e^{a}_{\rho\mu}},\frac{\partial}{\partial\omega^{a}_{\rho\mu}}\in\ker{\Omega}_{\mathscr{L}}$,
the coefficients $D^{a}_{\nu\sigma\mu},H^{c}_{\nu\sigma\rho}$ in (71) remain
undetermined.

Nevertheless, solutions to the field equations must be holonomic multivector
fields $\textbf{X}_{\mathscr{L}}$ which thereby have components (71) for which
$B^{a}_{\nu\mu}=e^{a}_{\nu\mu}$ and $C^{c}_{\nu\rho}=\omega^{c}_{\nu\rho}$; it
follows that the field equations (72a) and (72b) become:

$\displaystyle\epsilon^{\mu\nu\rho}\Big{[}\eta_{ac}\omega^{c}_{\nu\rho}+\frac{1}{2}\epsilon_{abc}\left(\omega^{b}_{\nu}\omega^{c}_{\rho}+\lambda
e^{b}_{\nu}e^{c}_{\rho}\right)\Big{]}=0\ ,$
$\displaystyle\epsilon^{\mu\nu\rho}\Big{[}\eta_{ac}e^{a}_{\nu\mu}-\epsilon_{abc}e^{a}_{\mu}\omega^{b}_{\nu}\Big{]}=0\
.$

Both of these equations are sopde constraints and define the submanifold
$S_{1}\hookrightarrow J^{1}E$.

Furthermore, the integral sections
$(x^{\mu},e^{a}_{\mu}(x),\omega_{\rho}^{c}(x),e^{a}_{\sigma\mu}(x),\omega^{c}_{\sigma\rho}(x))$
of these holonomic multivector fields satisfy that
$B^{a}_{\nu\mu}=\partial_{\nu}e^{a}_{\mu}$ and
$C^{c}_{\nu\rho}=\partial_{\nu}\omega^{c}_{\rho}$, and therefore the equations
(72a) and (72b) on $S_{1}$ become:

$\displaystyle\epsilon^{\mu\nu\rho}\left[\eta_{ac}\left(\displaystyle\frac{\partial{\omega^{c}_{\rho}}}{\partial{x^{\nu}}}-\displaystyle\frac{\partial{\omega^{c}_{\nu}}}{\partial{x^{\rho}}}\right)+\epsilon_{abc}\left(\omega^{b}_{\nu}\omega^{c}_{\rho}+\lambda
e^{b}_{\nu}e^{c}_{\rho}\right)\right]=0\ ,$
$\displaystyle\epsilon^{\mu\nu\rho}\left[\eta_{ac}\left(\displaystyle\frac{\partial{e^{a}_{\nu}}}{\partial{x^{\mu}}}-\displaystyle\frac{\partial{e^{a}_{\mu}}}{\partial{x^{\nu}}}\right)+\epsilon_{abc}\left(e^{a}_{\mu}\omega^{b}_{\nu}+\omega^{a}_{\mu}e^{b}_{\nu}\right)\right]=0\
.$

These are the well-known field equations for gravity in $2+1$ dimensions; the
first equation is Einstein’s equation with cosmological constant while the
second guarantees that the spin connection is torsion free, thereby fully
specifying the spin connection in terms of the dreibein $e^{a}_{\mu}$.

###  5.2 Hamiltonian Formulation

The Legendre map $\mathscr{FL}:J^{1}E\rightarrow J^{1}E^{*}$ associated to
$\mathscr{L}$ gives the multimomenta

$\mathscr{FL}^{*}p_{a}^{\mu\nu}=\frac{\partial\mathscr{L}}{\partial
e^{a}_{\mu\nu}}=0\quad,\quad\mathscr{FL}^{*}\pi_{a}^{\mu\nu}=\frac{\partial\mathscr{L}}{\partial\omega^{a}_{\mu\nu}}=2\epsilon^{\mu\nu\rho}\eta_{ab}e^{b}_{\rho}\
.$

The relations above give the constraints $p_{a}^{\mu\nu}=0$ and
$\pi_{c}^{\mu\nu}-2\epsilon^{\mu\nu\rho}\eta_{ab}e^{b}_{\rho}=0$, which define
the primary constraint submanifold $\jmath_{0}:P_{0}\hookrightarrow
J^{1}E^{*}$ on which the Hamiltonian formulation takes place. As expected from
Proposition 2, the null vectors (70) are given by the partial derivatives of
the primary constraints with respect to the multimomenta according to (14):

${}_{(1)}\gamma^{a}_{\mu\nu}=\mathscr{FL}^{*}\displaystyle\frac{\partial{}}{\partial{p^{\mu\nu}_{a}}}\left(p^{\mu\nu}_{a}\right)\quad,\quad_{(2)}\gamma^{a}_{\mu\nu}=\mathscr{FL}^{*}\displaystyle\frac{\partial{}}{\partial{\pi^{\mu\nu}_{a}}}\left(\pi^{\mu\nu}_{a}-2\epsilon^{\mu\nu\rho}\eta_{ab}e^{b}_{\rho}\right)\
.$

The restricted Legendre transform $\mathscr{FL}_{0}$ is given as
$\mathscr{FL}=\jmath_{0}\circ\mathscr{FL}_{0}$ and the primary constraint
submanifold $P_{0}$ is locally diffeomorphic to $E$ and the system is almost-
regular. Then, the local coordinates on $P_{0}$ are
$(x^{\mu},e^{a}_{\mu},\omega_{\rho}^{c})$.

The De Donder–Weyl Hamiltonian on $P_{0}$, which satisfies
$E_{\mathscr{L}}=\mathscr{FL}_{0}^{*}\mathscr{H}_{0}$, is given by

$\mathscr{H}_{0}=-\epsilon^{\mu\nu\rho}\epsilon_{abc}\Big{(}e^{a}_{\mu}\omega^{b}_{\nu}\omega^{c}_{\rho}+\frac{1}{3}\lambda
e^{a}_{\mu}e^{b}_{\nu}e^{c}_{\rho}\Big{)}\ .$

The Hamilton–Cartan $3$-form
$\Theta^{0}_{\mathscr{H}}\in{\mit\Omega}^{3}(P_{0})$ and the
premultisymplectic Hamilton–Cartan $4$-form
$\Omega^{0}_{\mathscr{H}}\in{\mit\Omega}^{4}({P_{0}})$ are given by

$\displaystyle\Theta^{0}_{\mathscr{H}}$ $\displaystyle=$
$\displaystyle\pi^{c}_{\sigma\rho}\text{d}\omega^{c}_{\rho}\wedge\text{d}^{2}x_{\sigma}-\mathscr{H}_{0}\text{d}^{3}x$
$\displaystyle=$ $\displaystyle
2\epsilon^{\mu\nu\rho}\eta_{ac}e^{a}_{\mu}\text{d}\omega^{c}_{\rho}\wedge\text{d}^{2}x_{\nu}+\epsilon^{\mu\nu\rho}\epsilon_{abc}\Big{(}e^{a}_{\mu}\omega^{b}_{\nu}\omega^{c}_{\rho}+\frac{1}{3}\lambda
e^{a}_{\mu}e^{b}_{\nu}e^{c}_{\rho}\Big{)}\text{d}^{3}x\ ,$
$\displaystyle\Omega^{0}_{\mathscr{H}}$ $\displaystyle=$
$\displaystyle-\text{d}\pi^{c}_{\sigma\rho}\wedge\text{d}\omega^{c}_{\rho}\wedge\text{d}^{2}x_{\sigma}+\text{d}\mathscr{H}_{0}\wedge\text{d}^{3}x$
$\displaystyle=$
$\displaystyle-2\epsilon^{\mu\nu\rho}\eta_{ac}\text{d}e^{a}_{\mu}\wedge\text{d}\omega^{c}_{\rho}\wedge\text{d}^{2}x_{\nu}-\epsilon^{\mu\nu\rho}\epsilon_{abc}\left(\omega^{b}_{\nu}\omega^{c}_{\rho}\text{d}e^{a}_{\mu}+2e^{a}_{\mu}\omega^{b}_{\nu}\text{d}\omega^{c}_{\rho}+\lambda
e^{b}_{\nu}e^{c}_{\rho}\text{d}e^{a}_{\mu}\right)\wedge\text{d}^{3}x\ .$

As it is usual, the field equations are obtained using a multivector field
$\textbf{X}_{\mathscr{H}}=X_{0}\wedge X_{1}\wedge
X_{2}\in\mathfrak{X}^{3}(P_{0})$ with components given by

$X_{\nu}=\frac{\partial}{\partial
x^{\nu}}+B^{a}_{\nu\mu}\frac{\partial}{\partial
e^{a}_{\mu}}+C^{c}_{\nu\rho}\frac{\partial}{\partial\omega^{c}_{\rho}}\ .$

Now the geometric field equation (8) gives

$\displaystyle i(\textbf{X}_{\mathscr{H}})\Omega_{\mathscr{H}}=$
$\displaystyle
2\epsilon^{\mu\nu\rho}\eta_{ac}\left(B^{a}_{\nu\mu}\text{d}\omega^{c}_{\rho}-C^{c}_{\nu\rho}\text{d}e^{a}_{\mu}\right)-\epsilon^{\mu\nu\rho}\epsilon_{abc}\Big{(}\omega^{b}_{\nu}\omega^{c}_{\rho}\text{d}e^{a}_{\mu}+2e^{a}_{\mu}\omega^{b}_{\nu}\text{d}\omega^{c}_{\rho}+\lambda
e^{b}_{\nu}e^{c}_{\rho}\text{d}e^{a}_{\mu}\Big{)}+$
$\displaystyle\epsilon^{\mu\nu\rho}\left[2\eta_{ac}\left(B^{a}_{\lambda\mu}C^{c}_{\nu\rho}-B^{a}_{\nu\mu}C^{c}_{\lambda\rho}\right)+\epsilon_{abc}\Big{(}\omega^{b}_{\nu}\omega^{c}_{\rho}B^{a}_{\lambda\mu}+2e^{a}_{\mu}\omega^{b}_{\nu}C^{c}_{\lambda\rho}+\lambda
e^{b}_{\nu}e^{c}_{\rho}B^{a}_{\lambda\mu}\Big{)}\right]\text{d}x^{\lambda}=0\
.$

Setting differential forms separately equal to zero yields

$\displaystyle\epsilon^{\mu\nu\rho}\left[\eta_{ac}C^{c}_{\nu\rho}+\frac{1}{2}\epsilon_{abc}\left(\omega^{b}_{\nu}\omega^{c}_{\rho}+\lambda
e^{b}_{\nu}e^{c}_{\rho}\right)\right]=0\ ,$
$\displaystyle\epsilon^{\mu\nu\rho}\left[\eta_{ac}B^{a}_{\nu\mu}-\epsilon_{abc}e^{a}_{\mu}\omega^{b}_{\nu}\right]=0\
.$

Similarly to the Lagrangian formalism, no compatibility constraints arise.
Again, observe that

$\text{ker}\ \widetilde{\Omega}^{0}_{\mathscr{H}}\cap\text{ker}\
\text{d}^{3}x=\left\langle\frac{\partial}{\partial
p_{a}^{\rho\mu}},\frac{\partial}{\partial\pi_{a}^{\rho\mu}}\right\rangle\ ,$

from which it follows that $\displaystyle i\left(\frac{\partial}{\partial
p_{a}^{\rho\mu}}\right)\text{d}\mathscr{H}_{0}=i\left(\frac{\partial}{\partial\pi_{a}^{\rho\mu}}\right)\text{d}\mathscr{H}_{0}=0$
automatically. Then, by taking
$(x^{\mu},e^{a}_{\mu}(x),\omega_{\rho}^{bc}(x))$ to be integral sections of
$\textbf{X}_{\mathscr{H}}$, for which
$B^{a}_{\nu\mu}=\partial_{\nu}e^{a}_{\mu}$ and
$C^{c}_{\nu\rho}=\partial_{\nu}\omega^{c}_{\rho}$, the field equations become
again

$\displaystyle\epsilon^{\mu\nu\rho}\left[\eta_{ac}\left(\displaystyle\frac{\partial{\omega^{c}_{\rho}}}{\partial{x^{\nu}}}-\displaystyle\frac{\partial{\omega^{c}_{\nu}}}{\partial{x^{\rho}}}\right)+\epsilon_{abc}\left(\omega^{b}_{\nu}\omega^{c}_{\rho}+\lambda
e^{b}_{\nu}e^{c}_{\rho}\right)\right]=0\ ,$
$\displaystyle\epsilon^{\mu\nu\rho}\left[\eta_{ac}\left(\displaystyle\frac{\partial{e^{a}_{\nu}}}{\partial{x^{\mu}}}-\displaystyle\frac{\partial{e^{a}_{\mu}}}{\partial{x^{\nu}}}\right)+\epsilon_{abc}\left(e^{a}_{\mu}\omega^{b}_{\nu}+\omega^{a}_{\mu}e^{b}_{\nu}\right)\right]=0\
,$

which are the same field equation obtained in the Lagrangian formulation.

###  5.3 Symmetries

####  5.3.1 Spacetime Diffeomorphisms

Spacetime diffeomorphisms are generated by the infinitesimal parameter
$\xi^{\mu}$ via the following transformations:

$x^{\mu}\rightarrow x^{\prime\mu}=x^{\mu}+\xi^{\mu}(x)\ \Rightarrow\
e^{a}_{\mu}(x)\Longrightarrow e^{\prime a}_{\mu}(x^{\prime})=\frac{\partial
x^{\mu}}{\partial x^{\prime\mu}}e^{a}_{\mu}(x),\
\omega^{a}_{\mu}(x)\rightarrow\omega^{\prime
a}_{\mu}(x^{\prime})=\frac{\partial x^{\mu}}{\partial
x^{\prime\mu}}\omega^{a}_{\mu}(x)\ ,$

hence,

$\displaystyle\delta e^{a}_{\mu}(x)=e^{\prime
a}_{\mu}(x)-e^{a}_{\mu}(x)=-\xi^{\nu}\displaystyle\frac{\partial{e^{a}_{\mu}}}{\partial{x^{\nu}}}-e^{a}_{\nu}\displaystyle\frac{\partial{\xi^{\nu}}}{\partial{x^{\mu}}}\
,$ $\displaystyle\delta\omega^{a}_{\mu}(x)=\omega^{\prime
a}_{\mu}(x)-\omega^{a}_{\mu}(x)=-\xi^{\nu}\displaystyle\frac{\partial{\omega^{a}_{\mu}}}{\partial{x^{\nu}}}-\omega^{a}_{\nu}\displaystyle\frac{\partial{\xi^{\nu}}}{\partial{x^{\mu}}}\
.$

Once again, these field variations are given by the Lie derivative of the
local sections $\phi:M\rightarrow E$ generated by the vector field
$\xi=-\xi^{\mu}\displaystyle\frac{\partial{}}{\partial{x^{\mu}}}\in\mathfrak{X}(M)$
as defined in (23). The lift of the vector field $\xi$ to the configuration
bundle $E$ [39, 56] is given by

$\xi_{E}=-\xi^{\mu}\displaystyle\frac{\partial{}}{\partial{x^{\mu}}}+e^{a}_{\nu}\partial_{\mu}\xi^{\nu}\frac{\partial}{\partial
e^{a}_{\mu}}+\omega^{a}_{\nu}\partial_{\mu}\xi^{\nu}\frac{\partial}{\partial\omega^{a}_{\mu}}\in\mathfrak{X}(E)\
.$

The canonical lift $X_{\xi}\in\mathfrak{X}(J^{1}E)$ of $\xi_{E}$ to $J^{1}E$
is given as

$X_{\xi}=-\xi^{\mu}\displaystyle\frac{\partial{}}{\partial{x^{\mu}}}+e^{a}_{\nu}\partial_{\mu}\xi^{\nu}\frac{\partial}{\partial
e^{a}_{\mu}}+\omega^{a}_{\nu}\partial_{\mu}\xi^{\nu}\frac{\partial}{\partial\omega^{a}_{\mu}}+\left(e^{a}_{\sigma\rho}\displaystyle\frac{\partial{\xi^{\rho}}}{\partial{x^{\mu}}}-e^{a}_{\rho\mu}\displaystyle\frac{\partial{\xi^{\rho}}}{\partial{x^{\sigma}}}\right)\frac{\partial}{\partial
e^{a}_{\sigma\mu}}+\left(\omega^{a}_{\sigma\rho}\displaystyle\frac{\partial{\xi^{\rho}}}{\partial{x^{\mu}}}-\omega^{a}_{\rho\mu}\displaystyle\frac{\partial{\xi^{\rho}}}{\partial{x^{\sigma}}}\right)\frac{\partial}{\partial\omega^{a}_{\sigma\mu}}\
.$

It follows that $\mathpzc{L}_{X_{\xi}}\Theta_{\mathscr{L}}=0$ so the momentum
map $J_{\mathscr{L}}(X_{\xi})\in{\mit\Omega}^{2}(J^{1}E)$ is given by

$\displaystyle\begin{split}J_{\mathscr{L}}(X_{\xi})&=-i(X_{\xi})\Theta_{\mathscr{L}}\\\
&=-2\epsilon^{\mu\nu\rho}\eta_{ac}e^{a}_{\mu}\left(\xi^{\sigma}\text{d}\omega^{c}_{\rho}\wedge\text{d}^{1}x_{\nu\sigma}+\omega^{c}_{\sigma}\partial_{\rho}\xi^{\sigma}\text{d}^{2}x_{\nu}\right)+\epsilon^{\mu\nu\rho}\epsilon_{abc}\left(e^{a}_{\mu}\omega^{b}_{\nu}\omega^{c}_{\rho}+\frac{1}{3}\lambda
e^{a}_{\mu}e^{b}_{\nu}e^{c}_{\rho}\right)\xi^{\sigma}\text{d}^{2}x_{\sigma}\
.\end{split}$

Furthermore, since $P_{0}$ is locally diffeomorphic to $E$, the vector field
$Y_{\xi}=\mathscr{FL}_{0*}X_{\xi}$ which generates the spacetime
diffeomorphisms on $P_{0}$ is written as

$Y_{\xi}=-\xi^{\mu}\displaystyle\frac{\partial{}}{\partial{x^{\mu}}}+e^{a}_{\nu}\partial_{\mu}\xi^{\nu}\frac{\partial}{\partial
e^{a}_{\mu}}+\omega^{a}_{\nu}\displaystyle\frac{\partial{\xi^{\nu}}}{\partial{x^{\mu}}}\frac{\partial}{\partial\omega^{a}_{\mu}}\in\mathfrak{X}(P_{0})\
,$

and since $\mathpzc{L}_{Y_{\xi}}\Theta^{0}_{\mathscr{H}}=0$, the momentum map
$J_{\mathscr{H}}(Y_{\xi})\in{\mit\Omega}^{2}(P_{0})$ is given by

$\displaystyle\begin{split}J_{\mathscr{H}}(\xi)&=-i(Y_{\xi})\Theta^{0}_{\mathscr{H}}\\\
&=-2\epsilon^{\mu\nu\rho}\eta_{ac}e^{a}_{\mu}\left(\xi^{\sigma}\text{d}\omega^{c}_{\rho}\wedge\text{d}^{1}x_{\nu\sigma}+\omega^{c}_{\sigma}\partial_{\rho}\xi^{\sigma}\text{d}^{2}x_{\nu}\right)+\epsilon^{\mu\nu\rho}\epsilon_{abc}\left(e^{a}_{\mu}\omega^{b}_{\nu}\omega^{c}_{\rho}+\frac{1}{3}\lambda
e^{a}_{\mu}e^{b}_{\nu}e^{c}_{\rho}\right)\xi^{\sigma}\text{d}^{2}x_{\sigma}\
.\end{split}$

####  5.3.2 Local Poincaré Transformations

Local Poincaré transformations generated by the infinitesimal parameter
$\Lambda(x)=P_{a}\rho^{a}(x)+J_{a}\tau^{a}(x)$ induce transformations of the
gauge field given as

$\delta
A_{\mu}(x)=-D_{\mu}\Lambda(x)=-\partial_{\mu}\Lambda-[A_{\mu},\Lambda],$

where $\tau^{a}(x)$ are the infinitesimal generators of local Lorentz
transformations and $\rho^{a}(x)$ are the infinitesimal generators of local
translations. It follows from the Lie algebra (69) that

$\delta
e^{a}_{\mu}(x)=-\displaystyle\frac{\partial{\rho^{a}}}{\partial{x^{\mu}}}-\epsilon_{abc}e^{b}_{\mu}\tau^{c}-\epsilon_{abc}\omega^{b}_{\mu}\rho^{c}\quad,\quad\delta\omega^{a}_{\mu}(x)=-\displaystyle\frac{\partial{\tau^{a}}}{\partial{x^{\mu}}}-\epsilon_{abc}\omega^{b}_{\mu}\tau^{c}-\lambda\epsilon_{abc}e^{b}_{\mu}\rho^{c}\
.$

The vector field on the configuration bundle $E$ which generates the local
Poincaré transformations is

$\zeta_{E}=\left(\displaystyle\frac{\partial{\rho^{a}}}{\partial{x^{\mu}}}+\epsilon^{a}_{\
bc}e^{b}_{\mu}\tau^{c}+\epsilon^{a}_{\
bc}\omega^{b}_{\mu}\rho^{c}\right)\frac{\partial}{\partial
e^{a}_{\mu}}+\left(\displaystyle\frac{\partial{\tau^{a}}}{\partial{x^{\mu}}}+\epsilon^{a}_{\
bc}\omega^{b}_{\mu}\tau^{c}+\lambda\epsilon^{a}_{\
bc}e^{b}_{\mu}\rho^{c}\right)\frac{\partial}{\partial\omega^{a}_{\mu}}\ .$

The canonical lift of $\zeta_{E}$ to $J^{1}E$ is

$\displaystyle\begin{split}X_{\zeta}&=\left(\displaystyle\frac{\partial{\rho^{a}}}{\partial{x^{\mu}}}+\epsilon^{a}_{\
bc}e^{b}_{\mu}\tau^{c}+\epsilon^{a}_{\
bc}\omega^{b}_{\mu}\rho^{c}\right)\frac{\partial}{\partial
e^{a}_{\mu}}+\left(\displaystyle\frac{\partial{\tau^{a}}}{\partial{x^{\mu}}}+\epsilon^{a}_{\
bc}\omega^{b}_{\mu}\tau^{c}+\lambda\epsilon^{a}_{\
bc}e^{b}_{\mu}\rho^{c}\right)\frac{\partial}{\partial\omega^{a}_{\mu}}\\\
&+\epsilon^{a}_{\
bc}\left(e^{b}_{\mu}\displaystyle\frac{\partial{\tau^{c}}}{\partial{x^{\sigma}}}+\omega^{b}_{\mu}\displaystyle\frac{\partial{\rho^{c}}}{\partial{x^{\sigma}}}+e^{b}_{\sigma\mu}\tau^{c}\right)\frac{\partial}{\partial
e^{a}_{\sigma\mu}}+\epsilon^{a}_{\
bc}\left(\omega^{b}_{\mu}\displaystyle\frac{\partial{\tau^{c}}}{\partial{x^{\sigma}}}+\lambda
e^{b}_{\mu}\displaystyle\frac{\partial{\rho^{c}}}{\partial{x^{\sigma}}}+\omega^{b}_{\sigma\mu}\tau^{c}\right)\frac{\partial}{\partial\omega^{a}_{\sigma\mu}}\
,\end{split}$

while the projection $\mathscr{FL}_{0*}X_{\zeta}\equiv Y_{\zeta}$ onto $P_{0}$
is

$Y_{\zeta}=\left(\displaystyle\frac{\partial{\rho^{a}}}{\partial{x^{\mu}}}+\epsilon^{a}_{\
bc}e^{b}_{\mu}\tau^{c}+\epsilon^{a}_{\
bc}\omega^{b}_{\mu}\rho^{c}\right)\frac{\partial}{\partial
e^{a}_{\mu}}+\left(\displaystyle\frac{\partial{\tau^{a}}}{\partial{x^{\mu}}}+\epsilon^{a}_{\
bc}\omega^{b}_{\mu}\tau^{c}+\lambda\epsilon^{a}_{\
bc}e^{b}_{\mu}\rho^{c}\right)\frac{\partial}{\partial\omega^{a}_{\mu}}\ .$

The momentum map $J_{\mathscr{L}}(X_{\zeta})\in{\mit\Omega}^{2}(J^{1}E)$ on
the multivelocity phase space is given by,

$J_{\mathscr{L}}(X_{\zeta})=-i(Y_{\zeta})\Theta_{\mathscr{L}}=-2\epsilon^{\mu\nu\rho}\eta_{ac}e^{a}_{\mu}\left(\displaystyle\frac{\partial{\tau^{c}}}{\partial{x^{\mu}}}+\epsilon^{c}_{\
db}\omega^{d}_{\mu}\tau^{b}+\lambda\epsilon^{c}_{\
db}e^{d}_{\mu}\rho^{b}\right)\text{d}^{2}x_{\nu}\ .$

and the momentum map $J_{\mathscr{H}}(Y_{\zeta})\in{\mit\Omega}^{2}(P_{0})$ on
the multimomentum phase space is given by the same expression

$J_{\mathscr{H}}(Y_{\zeta})=-i(Y_{\zeta})\Theta^{0}_{\mathscr{H}}=-2\epsilon^{\mu\nu\rho}\eta_{ac}e^{a}_{\mu}\left(\displaystyle\frac{\partial{\tau^{c}}}{\partial{x^{\mu}}}+\epsilon^{c}_{\
db}\omega^{d}_{\mu}\tau^{b}+\lambda\epsilon^{c}_{\
db}e^{d}_{\mu}\rho^{b}\right)\text{d}^{2}x_{\nu}\ .$

##  6 Conclusions and Outlook

In this paper, a geometric approach has been used for performing a full
constraint analysis of field theories on both the multivelocity and
multimomentum phase spaces where the Lagrangian and De Donder–Weyl Hamiltonian
formalisms take place respectively. The multivelocity and multimomentum phase
spaces are, respectively, the jet and dual jet bundles of the field
configuration manifold $E$ which is regarded as a fiber bundle over spacetime
(or over the string worldsheet in the case of string theory). These phase
spaces are equipped with multisymplectic or premultisymplectic forms which are
then used to produce the field equations of the theory under investigation,
reveal the symmetries of the field theory, and carry out a full constraint
analysis of the singular field theories.

The approach for carrying out the premultisymplectic constraint analysis of
field theories involves the use of multivector fields to represent
geometrically the solutions of field equations. So, an easy procedural
geometric technique for finding the constraints locally has been described,
and some new properties of the constraints are also exposed (Propositions 1
and 2). The converse of Proposition 1, which states that the compatibility
constraint submanifold $C_{1}$ is completely determined by (13), is left for
further research along with other results and features regarding the geometric
constraint algorithm.

Furthermore, it is shown how further constraint submanifolds (defined by the
so-called sopde constraints) may be found in the Lagrangian formalism when
imposing the holonomic condition which guarantees that the Lagrangian field
equations are second-order partial differential equations. The field
equations, i.e. the well-known Euler–Lagrange equations in the Lagrangian case
and the Hamilton–De Donder–Weyl equations in the Hamiltonian case, are
obtained from the (pre)multisymplectic variational principle in which
multivector fields of a particular type serve as solutions to the field
equations. By example, in the field theories analyzed in Sections 3.1 and 3.3,
it is shown that when sopde constraints exist, the vector fields which
generate symmetries of the field theory under investigation may be projectable
via the Legendre transform only on the sopde constraint submanifold. The
geometric constraint analysis finalizes by imposing stability of the solutions
to the field equations; this is done by demanding tangency of the solution
multivector fields to all constraint submanifolds present in the system until
no new constraint submanifolds are produced. It is also worth noting that this
work presents a new proven proposition, Proposition 2, which states that the
multi-Hessian (2) (which characterizes the De Donder–Weyl Legendre transform
to the multimomentum phase space) has null vectors which are given by the
partial derivative of the primary constraints with respect to the
multimomenta.

The technique used in this paper for performing the geometric constraint
analysis described above is illustrated in various field theory examples, each
of which highlight different aspects of the geometric constraint structure.
The scalar field theories presented in this work include a new approach to the
study of the canonical Klein–Gordon Lagrangian as well as the first
premultisymplectic analysis of Carrollian scalar field theories. The
multisymplectic treatment of bosonic string theory from the Nambu–Goto action
is given along with the generalization to the theory of bosonic $p$-branes.
The regularity of the Nambu–Goto action with respect to the De Donder–Weyl
Legendre transform is shown geometrically by inspection of the multisymplectic
forms that characterize the theory’s multivelocity and multimomentum phase
spaces. The connection between the regularity of bosonic string theory in the
De Donder–Weyl formalism and its singular behavior in the canonical
formulation remains unexplored as, in general, no direct relationship has yet
been established between the premultisymplectic constraint analysis presented
in this work and the canonical constraint structure arising from the canonical
Legendre transform. Understanding the connection between the constraint
structures in these two formalisms is left for future research. Finally, the
premultisymplectic construction of Chern–Simons gravity in $2+1$ dimensions is
given and it is shown that in the Lagrangian formalism there are only sopde
constraints present, while in the De Donder–Weyl Hamiltonian formalism there
are only primary constraints which are produced by the singular Legendre map.

Given that the constraint structure of field theories in the De Donder–Weyl
formalism is only understood geometrically at this point in time, it would be
interesting to develop an algebraic understanding of the constraints a la
Dirac. Such a treatment of constraints requires the construction of Poisson
brackets on some phase space. In the canonical formalism for field theories,
Poisson brackets are well understood in the context of symmetries and
constraints on the covariant phase space by Lee and Wald [9] for example; for
further insight into such covariant phase spaces (and related ones) see [57].
In the De Donder–Weyl formalism however, the construction of Poisson brackets
it still a topic under investigation with many papers published on the matter
(see, for example, [22, 58, 59, 60, 61, 62] and references therein). Covariant
Poisson brackets on jet bundles (where the Lagrangian formulation of field
theories takes place) has also been discussed extensively in the literature
(see for example [63] and references therein) along with the development of
the BRST-BV formalism [64]. However, the understanding of the BRST-BV
formalism in terms of the (pre)multisympletic structures on jet bundles is not
yet fully understood and is left for further research; we also leave gauge
fixing in the De Donder–Weyl formulation of the field theories studied in this
paper for future work.

## Acknowledgments

We acknowledge conversations on non-Lorentzian topics with Eric Bergshoeff,
Roberto Casalbuoni, José Figueroa-O’Farrill, Axel Kleinschmidt, and Alfredo
Peréz. We acknowledge the financial support of the Ministerio de Ciencia,
Innovación y Universidades (Spain), project PGC2018-098265-B-C33 and the
MINECO Grant PID2021-125515NB-C21. The work of JG has been supported in part
by MINECO FPA2016-76005-C2-1-P and PID2019- 105614GB-C21 and from the State
Agency for Research of the Spanish Ministry of Science and Innovation through
the Unit of Excellence Maria de Maeztu 2020-203 award to the Institute of
Cosmos Sciences (CEX2019-000918-M).

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