# Work fluctuation theorems with initial quantum coherence

Gianluca Francica and Luca Dell’Anna  Dipartimento di Fisica e Astronomia e
Sezione INFN, Università di Padova, via Marzolo 8, 35131 Padova, Italy

###### Abstract

Fluctuation theorems are fundamental results in nonequilibrium thermodynamics
beyond the linear response regime. Among these, the paradigmatic Tasaki-Crooks
fluctuation theorem relates the statistics of the works done in a forward out-
of-equilibrium quantum process and in a corresponding backward one. In
particular, the initial states of the two processes are thermal states and
thus incoherent in the energy basis. Here, we aim to investigate the role of
initial quantum coherence in work fluctuation theorems. To do this, we
formulate and examine the implications of a stronger fluctuation theorem,
which reproduces the Tasaki-Crooks fluctuation theorem in the absence of
initial quantum coherence.

##  I Introduction

Work is a fundamental nonequilibrium stochastic quantity, which plays a
relevant role in out-of-equilibrium processes generated by changing some
parameters of the system in a certain time interval. In classical systems,
when there is equilibrium at the initial time, the work fluctuations satisfy
an integral fluctuation relation, the Jarzynski equality [1]. Furthermore, the
statistics of the work done in the process is related to its time reversal by
a detailed fluctuation relation given by the Crooks fluctuation theorem [2],
when both the forward and backward processes start from equilibrium states. In
particular, the integral fluctuation relation can be obtained from the
detailed one by integrating it. When the quantum effects cannot be neglected,
the detailed fluctuation relation still holds if we describe the statistics of
work with the help of a two-projective-measurement scheme, as it was
originally shown in Ref. [3], after the relevant works in Refs. [4, 5]. This
is known as Tasaki-Crooks fluctuation theorem (see, e.g., Ref. [6] for a
review). However, in the presence of initial quantum coherence in the energy
basis, in this scheme, where two projective measurements of the energy are
performed at the initial and final times to infer the work statistics, the
first measurement destroys the initial quantum coherence in the energy basis,
and the protocol becomes invasive since irreversibly changes the real process.

There are several attempts to describe the work statistics in quantum regime
(see, e.g., Refs. [7, 8, 9, 10, 11, 12, 13, 14]). If we require that the two-
projective-measurement statistics is also reproduced for incoherent initial
states, a no-go theorem [15] suggests that the statistics of work should be
represented by a quasiprobability distribution. Furthermore, if some
conditions are required, the work can be described by a class of
quasiprobability distributions [16, 17], which includes the ones of Refs. [9,
10]. The initial states of the Tasaki-Crooks fluctuation theorem are
incoherent mixtures of the energy basis, and so are incoherent states. Then,
the effects coming from the initial quantum coherence in the energy basis,
e.g., quantum contextuality [18, 19, 20], are absent in this case.

Here, we aim to investigate the effects of the initial quantum coherence in
the work fluctuation theorems. After introducing some preliminary notions in
Sec. (II), we discuss the time reversal of our quasiprobability distribution
in Sec. (III). In particular, although the forward process can have a non-
contextual representation, its time reversal can exhibits contextuality. Thus,
we derive our main result in Sec. IV, a detailed fluctuation theorem which
holds in the presence of initial quantum coherence. In detail, it reproduces
the Tasaki-Crooks fluctuation relation in the absence of quantum coherence and
implies two different integral fluctuation theorems (see Sec. V), one of which
was introduced in Ref. [16]. To do this, we will focus on initial states with
thermal populations and nonzero coherence in the energy basis, such as the
coherent Gibbs state.

##  II Preliminaries

We start our discussion by introducing some preliminary notions, which are the
Tasaki-Crooks fluctuation theorem (see Sec. II.1) and the quasiprobability
distribution of work (see Sec. II.2).

###  II.1 Incoherent initial state: Tasaki-Crooks fluctuation theorem

We focus on a quantum coherent process generated by changing some parameters
of the system in the time interval $[0,\tau]$. Thus, we get the time-dependent
Hamiltonian
$H(t)=\sum\epsilon_{k}(t)|\epsilon_{k}(t)\rangle\langle\epsilon_{k}(t)|$,
where $|\epsilon_{k}(t)\rangle$ is the eigenstate with eigenvalue
$\epsilon_{k}(t)$ at the time $t\in[0,\tau]$, which generates the unitary time
evolution operator $U_{t,0}=\mathcal{T}e^{-i\int_{0}^{t}H(s)ds}$, where
$\mathcal{T}$ is the time order operator. The system is prepared at the
initial time $t=0$ in an initial state $\rho$, which evolves to the final
state $\rho^{\prime}=U_{\tau,0}\rho U_{\tau,0}^{\dagger}$ at the time
$t=\tau$. There is initial quantum coherence in the energy basis if there are
non-zero coherences (i.e., off-diagonal elements of the density matrix) with
respect to the energy basis. However, before discussing the effects of the
initial quantum coherence, in this section we recall some results concerning
an incoherent initial state, such that $\rho=\Delta(\rho)$, where we have
defined the dephasing map

$\Delta(\rho)=\sum_{i}\Pi_{i}\rho\Pi_{i}\,,$ (1)

with the initial projectors $\Pi_{i}=|\epsilon_{i}\rangle\langle\epsilon_{i}|$
and $\epsilon_{i}=\epsilon_{i}(0)$. For an incoherent initial state $\rho$ the
work can be represented by the two-projective-measurement scheme [3, 6], which
has the probability distribution

$p_{\text{TPM}}(w)=\sum_{k,i}\text{Tr}\left\\{\Pi_{i}\rho\Pi_{i}\Pi^{\prime}_{k}\right\\}\delta(w-\epsilon^{\prime}_{k}+\epsilon_{i})\,,$
(2)

where the final projectors are defined as
$\Pi^{\prime}_{k}=U_{\tau,0}^{\dagger}|\epsilon^{\prime}_{k}\rangle\langle\epsilon^{\prime}_{k}|U_{\tau,0}$
and $\epsilon^{\prime}_{k}=\epsilon_{k}(\tau)$.

The backward process is obtained by considering the backward time evolution
from $\tau$ to zero, given by the unitary operator
$U_{\tau-t,\tau}=U^{\dagger}_{\tau,\tau-t}$, where $t$ goes from zero to
$\tau$. This backward process will start from an initial state $\bar{\rho}$.
The two-projective-measurement scheme leads to the probability distribution
for the backward process

$\bar{p}_{\text{TPM}}(w)=\sum_{k,i}\text{Tr}\left\\{\bar{\Pi}^{\prime}_{k}\bar{\rho}\bar{\Pi}^{\prime}_{k}\bar{\Pi}_{i}\right\\}\delta(w-\epsilon_{i}+\epsilon^{\prime}_{k})\,,$
(3)

where
$\bar{\Pi}^{\prime}_{k}=U_{\tau,0}\Pi^{\prime}_{k}U^{\dagger}_{\tau,0}=|\epsilon^{\prime}_{k}\rangle\langle\epsilon^{\prime}_{k}|$
and
$\bar{\Pi}_{i}=U_{\tau,0}\Pi_{i}U^{\dagger}_{\tau,0}=U_{\tau,0}|\epsilon_{i}\rangle\langle\epsilon_{i}|U^{\dagger}_{\tau,0}$.
If the initial state of the forward process is a Gibbs state at a certain
inverse temperature $\beta$, i.e., $\rho=\rho_{\beta}\equiv e^{-\beta
H(0)}/Z$, where $Z=\text{Tr}\left\\{e^{-\beta H(0)}\right\\}$, for the forward
process we get the Jarzynski equality [1, 3, 6]

$\langle e^{-\beta(w-\Delta F)}\rangle=1\,,$ (4)

where the equilibrium free energy difference reads $\Delta
F=-\beta^{-1}\ln(Z^{\prime}/Z)$, where $Z^{\prime}=\text{Tr}\left\\{e^{-\beta
H(\tau)}\right\\}$. In this case, if the initial state $\bar{\rho}$ of the
backward process is the Gibbs state $\bar{\rho}=\rho^{\prime}_{\beta}\equiv
e^{-\beta H(\tau)}/Z^{\prime}$, we get the Tasaki-Crooks fluctuation relation
[3, 6]

$e^{-\beta(w-\Delta F)}p_{\text{TPM}}(w)=\bar{p}_{\text{TPM}}(-w)\,.$ (5)

In particular, by integrating Eq. (5) over $w$ we achieve the integral
fluctuation theorem of Eq. (4). Our main aim is to generalize the detailed
fluctuation theorem of Eq. (5) in the presence of initial quantum coherence.
Since the two-projective-measurement scheme erases the initial quantum
coherence, Eq. (5) still holds for any $\rho$ and $\bar{\rho}$ such that
$\Delta(\rho)=\rho_{\beta}$ and
$\bar{\Delta}(\bar{\rho})=\rho^{\prime}_{\beta}$, where
$\bar{\Delta}(\bar{\rho})=\sum_{k}\bar{\Pi}^{\prime}_{k}\bar{\rho}\bar{\Pi}^{\prime}_{k}$.
However, the two-projective-measurement work does not satisfy the first law of
thermodynamics and in this case the work can be represented by a
quasiprobability distribution.

###  II.2 Quasiprobability distribution of work

Given a quantum observable $W$, having the spectral decomposition $W=\sum
w_{n}P_{n}$, the projective measurements with projectors $\\{P_{n}\\}$ lead to
the probability distribution
$p_{\text{obs}}(w)=\sum_{n}v(P_{n})\delta(w-w_{n})$, where
$v(P_{n})=\text{Tr}\left\\{P_{n}\rho\right\\}$ is a probability given by the
Born rule, for the density matrix $\rho$. In general, for an effect $E$, which
plays the role of event, the probability
$v(E)=\text{Tr}\left\\{E\rho\right\\}$ results from the Gleason’s theorem
(see, e.g., Ref. [21]). Here, we are considering an out-of-equilibrium process
generated by changing some parameters in the time interval $[0,\tau]$. The
first law of thermodynamics leads to the average work

$\langle w\rangle=\text{Tr}\left\\{(H^{(H)}(\tau)-H(0))\rho\right\\}\,,$ (6)

where $\rho$ is the initial density matrix and given an operator $A(t)$ we
define the Heisenberg time evolved operator
$A^{(H)}(t)=U_{t,0}^{\dagger}A(t)U_{t,0}$. Although $W=H^{(H)}(\tau)-H(0)$ is
a quantum observable, in general its statistics does not reproduce the two-
projective-measurement statistics for a Gibbs initial state
$\rho=\rho_{\beta}$, so that the Jarzynski equality in Eq. (4) is not
satisfied (as originally observed in Refs. [22, 23]). Actually, we can have
two noncommuting quantum observables defining the work, which are
$H^{(H)}(\tau)$ and $H(0)$. They give two sets of projectors $\\{\Pi_{i}\\}$
and $\\{\Pi^{\prime}_{k}\\}$. In this case, Gleason’s theorem cannot be used
in order to achieve a distribution of work that is linear in the initial
state. However, we can generalize Gleason’s axioms so that they lead to
quasiprobabilities, e.g., $v(E,F)=\text{Re}\text{Tr}\left\\{EF\rho\right\\}$,
instead of the probabilities $v(E)$ (e.g., see Appendix for details).

From these quasiprobabilities, we can write a quasiprobability distribution.
In general, the work will be represented in terms of the events $\Pi_{i}$,
$\Pi_{j}$, $\cdots$, $\Pi^{\prime}_{k}$, $\cdots$, and has a quasiprobability
distribution of the form [17]

$p(w)=\sum_{i,j,\ldots}v(\Pi_{i},\Pi_{j},\ldots,\Pi^{\prime}_{k},\ldots)\delta(w-w(\epsilon_{i},\ldots))\,,$
(7)

where the support is defined by $w(\epsilon_{i},\ldots)$, which is a function
of the eigenvalues of the initial and final Hamiltonian. The moments of the
work are $\langle w^{n}\rangle=\int w^{n}p(w)dw$, where $n$ is an integer.
Without loss of generality, for the quasiprobability
$v(\Pi_{i},\Pi_{j},\ldots,\Pi^{\prime}_{k},\ldots)$ we can focus on definite
decompositions of the proposition $\Pi_{i}\land\Pi_{j}\land\cdots$, since an
affine combination of these quasiprobabilities gives any $p(w)$ of the form in
Eq. (7). As shown in Ref. [17], by requiring that (W1) the quasiprobability
distribution $p(w)$ reproduces the two-projective-measurement statistics,
i.e., $p(w)=p_{\text{TPM}}(w)$ when the initial state is incoherent in the
energy basis, $\rho=\Delta(\rho)$, (W2) the average work is equal to Eq. (6)
and (W3) the second moment of work is equal to

$\langle
w^{2}\rangle=\text{Tr}\left\\{(H^{(H)}(\tau)-H(0))^{2}\rho\right\\}\,,$ (8)

we get a class of quasiprobability distributions of the form [16, 17]

$p_{q}(w)=\sum_{i,j,k}\text{Re}\text{Tr}\left\\{\Pi_{i}\rho\Pi_{j}\Pi^{\prime}_{k}\right\\}\delta(w-\epsilon^{\prime}_{k}+q\epsilon_{i}+(1-q)\epsilon_{j})\,,$
(9)

with $q$ real number.

##  III Time reversal

In general, a work quasiprobability distribution $\bar{p}(w)$ for the backward
process is obtained by taking in account that the initial state is a certain
density matrix $\bar{\rho}$ and the events are $\bar{\Pi}_{i}$,
$\bar{\Pi}_{j}$, $\cdots$, $\bar{\Pi}^{\prime}_{k}$, $\cdots$, so that

$\bar{p}(w)=\sum_{i,j,\ldots}\bar{v}(\bar{\Pi}_{i},\bar{\Pi}_{j},\ldots,\bar{\Pi}^{\prime}_{k},\ldots)\delta(w-\bar{w}(\epsilon_{i},\ldots))\,.$
(10)

In detail, the work support is defined by $\bar{w}(\epsilon_{i},\ldots)$ and
the quasiprobability
$\bar{v}(\bar{\Pi_{i}},\bar{\Pi}_{j},\ldots,\bar{\Pi}^{\prime}_{k},\ldots)$ is
calculated with respect to the initial density matrix $\bar{\rho}$, e.g.,
$\bar{v}(\bar{\Pi}_{i},\bar{\Pi}^{\prime}_{k})=\text{Re}\text{Tr}\left\\{\bar{\Pi}_{i}\bar{\Pi}^{\prime}_{k}\bar{\rho}\right\\}$
if there are only two events, which are $\bar{\Pi}_{i}$ and
$\bar{\Pi}^{\prime}_{k}$. A natural choice of the backward initial state
$\bar{\rho}$ is the final state of the forward process, i.e.,
$\bar{\rho}=\rho^{\prime}$. In this case, among all the representations of the
backward process, there is always a quasiprobability distribution $\bar{p}(w)$
such that

$\bar{p}(w)=p(-w)\,,$ (11)

for any quasiprobability distribution $p(w)$ of the forward process. In
particular, given $p(w)$ of the form in Eq. (7), $\bar{p}(w)$ is obtained by
performing a time reversal, i.e., by replacing $\Pi_{i}\mapsto\bar{\Pi}_{i}$,
$\Pi^{\prime}_{k}\mapsto\bar{\Pi}^{\prime}_{k}$, $\rho\mapsto\bar{\rho}$ and
$w(\epsilon_{i},\ldots)\mapsto\bar{w}(\epsilon_{i},\ldots)$. In detail, if
$\bar{\rho}=\rho^{\prime}$, for the quasiprobabilities we will get
$\bar{v}(\bar{\Pi}_{i},\bar{\Pi}_{j},\ldots,\bar{\Pi}^{\prime}_{k},\ldots)=v(\Pi_{i},\Pi_{j},\ldots,\Pi^{\prime}_{k},\ldots)$,
i.e., they are invariant under the time reversal. To prove it, it is enough to
note that the quasiprobability involves the real part of a trace of the
product of the projectors and the initial density matrix, e.g.,
$\bar{v}(\bar{\Pi}_{i},\bar{\Pi}^{\prime}_{k})=\text{Re}\text{Tr}\left\\{\bar{\Pi}_{i}\bar{\Pi}^{\prime}_{k}\bar{\rho}\right\\}=\text{Re}\text{Tr}\left\\{\Pi_{i}\Pi^{\prime}_{k}\rho\right\\}=v(\Pi_{i},\Pi^{\prime}_{k})$.
Furthermore, by requiring that the work is odd under the time reversal, we
have $\bar{w}(\epsilon_{i},\ldots)=-w(\epsilon_{i},\ldots)$, from which we get
the time-reversal symmetry relation for work in Eq. (11).

We note that Eq. (11) is not satisfied for the two-projective-measurement
probability distribution $p_{\text{TPM}}(w)$. Of course the probability
$\text{Tr}\left\\{\Pi_{i}\rho\Pi_{i}\Pi^{\prime}_{k}\right\\}$ in Eq. (2) is
of the form $v_{TPM}(E,F)=\text{Tr}\left\\{EFE\rho\right\\}$, so that does not
satisfy the Gleason axiom in Eq. (52). Let us focus on the forward class of
quasiprobability distributions in Eq. (9), which reproduce the two-projective-
measurement scheme when the initial state $\rho$ is incoherent with respect to
the projectors $\Pi_{i}$, i.e., $p_{q}(w)=p_{\text{TPM}}(w)$ when
$\rho=\Delta(\rho)$ for any $q$. Similarly, the backward class satisfying the
conditions (W1),(W2) and (W3) is formed by the quasiprobability distributions

$\tilde{p}_{q}(w)=\sum_{i,k,l}\text{Re}\text{Tr}\left\\{\bar{\Pi}^{\prime}_{k}\bar{\rho}\bar{\Pi}^{\prime}_{l}\bar{\Pi}_{i}\right\\}\delta(w-\epsilon_{i}+q\epsilon^{\prime}_{k}+(1-q)\epsilon^{\prime}_{l})\,,$
(12)

with $q$ real, such that $\tilde{p}_{q}(w)=\bar{p}_{\text{TPM}}(w)$ when
$\bar{\rho}=\bar{\Delta}(\bar{\rho})$ for any $q$. On the other hand, the time
reversal of the quasiprobability distribution $p_{q}(w)$ such that Eq. (11)
holds, reads

$\bar{p}_{q}(w)=\sum_{i,j,k}\text{Re}\text{Tr}\left\\{\bar{\Pi}_{i}\bar{\rho}\bar{\Pi}_{j}\bar{\Pi}^{\prime}_{k}\right\\}\delta(w-q\epsilon_{i}-(1-q)\epsilon_{j}+\epsilon^{\prime}_{k})\,,$
(13)

which reproduces the two-projective-measurement scheme,
$\bar{p}_{q}(w)=\bar{p}_{\text{TPM}}(w)$ when
$\bar{\rho}=\bar{\Delta}(\bar{\rho})$, for $q=0,1$ but not for all $q$ (in
particular, in Ref. [24] it has been originally observed how $p_{q}(w)$ does
not satisfy the symmetry relation in Eq. (11) for $q\neq 0,1$, if we use
$\tilde{p}_{q}(w)$ instead of $\bar{p}_{q}(w)$ in this relation). Thus, the
forward class with the quasiprobability distributions of Eq. (9) is not mapped
into the backward class with the quasiprobability distributions of Eq. (12) by
performing the time reversal. In detail, we get
$\tilde{p}_{q}(w)=\bar{p}_{q}(w)=p_{q}(-w)$ for $q=0,1$, but in general
$\tilde{p}_{q}(w)\neq\bar{p}_{q}(w)=p_{q}(-w)$ for $q\neq 0,1$. The symmetry
relation in Eq. (11) relates $p_{q}(w)$ with $\bar{p}_{q}(w)$, and not with
$\tilde{p}_{q}(w)$. Then, the negativity of $\tilde{p}_{q}(w)$ is not
constrained by the negativity of $p_{q}(w)$ for $q\neq 0,1$. This result means
that the (non)negativity of the forward class does not imply the
(non)negativity of the backward class, and vice versa. The negativity is
related to genuine quantum features: If there is some nonnegative distribution
in the forward (backward) class, the forward (backward) work statistics can be
reproduced with a noncontextual hidden variable model [18, 19, 20] which
satisfies the conditions (W1), (W2) and (W3) [17]. For instance, the two-
projective-measurement scheme gives a probability distribution that is
noncontextual [20]. Let us give an example of a forward protocol that is
noncontextual and shows contextuality in its time reversal. We consider a one
dimensional system, with initial Hamiltonian $H(0)=x$ and final Hamiltonian
$H(\tau)=p^{2}$, where $x$ is the position and $p$ is the momentum, such that
$[x,p]=i$. In this case we consider the projectors $\Pi_{x}=|x\rangle\langle
x|$ and $\Pi^{\prime}_{p}=|p\rangle\langle p|$ and the sudden time evolution
$U_{0,\tau}=I$. It is easy to show that for $q=1/2$ the quasiprobability
distribution of work can be expressed in terms of the Wigner function $W(x,p)$
as [17]

$p_{1/2}(w)=\int dxdpW(x,p)\delta(w-p^{2}+x)\,.$ (14)

We consider the initial wave function $\langle
x|\psi\rangle=\exp(-ax^{2}+bx+c)$, we get $W(x,p)\geq 0$, and the protocol is
noncontextual. However, for this state we get the backward quasiprobability
distribution

$\tilde{p}_{q}(w)=\int
dpdp^{\prime}\tilde{v}(p,w+qp^{2}+(1-q){p^{\prime}}^{2},p^{\prime})\,,$ (15)

where $\tilde{v}(p,x,p^{\prime})=\text{Re}\langle
p|\psi\rangle\langle\psi|p^{\prime}\rangle\langle p^{\prime}|x\rangle\langle
x|p\rangle$. Then, $\tilde{p}_{q}(w)$ takes also negative values for any $q$,
so that $\nexists q$ such that $\tilde{p}_{q}(w)\geq 0$ for all $w$, and there
is contextuality for the backward process.

Finally, we will aim to generalize the detailed fluctuation theorem of Eq. (5)
in the next section. Then, for a given $p_{q}(w)$ we will consider the time-
reversed $\bar{p}_{q}(w)$ with the same support of $p_{q}(-w)$ and an
appropriate initial state $\bar{\rho}$ for the backward process. To do this,
we must also take into account quantum coherence as a random variable.

##  IV Detailed Fluctuation Theorem with initial quantum coherence

Given a dephasing map $\Delta$, the quantum coherence of a state $\rho$ can be
characterized by using the relative entropy of coherence [25]

$C_{\Delta}(\rho)=S(\Delta(\rho))-S(\rho)\,,$ (16)

where we have introduced the von Neumann entropy
$S(\rho)=-\text{Tr}\left\\{\rho\ln\rho\right\\}$. Let us focus on the forward
process. By considering the eigenvalues $r_{n}$ and the eigenstates
$|r_{n}\rangle$ of the initial state $\rho$, such that $\rho=\sum r_{n}R_{n}$,
where $R_{n}=|r_{n}\rangle\langle r_{n}|$, we define the probability
distribution of coherence [16]

$p_{c}(C)=\sum_{i,n}r_{n}\text{Tr}\left\\{R_{n}\Pi_{i}\right\\}\delta(C+\ln\langle\epsilon_{i}|\rho|\epsilon_{i}\rangle-\ln
r_{n})\,,$ (17)

such that $C_{\Delta}(\rho)=\langle C\rangle=\int Cp_{c}(C)dC$. We note that
$r_{n}\text{Tr}\left\\{R_{n}\Pi_{i}\right\\}=\text{Tr}\left\\{\rho
R_{n}\Pi_{i}\right\\}=v(R_{n},\Pi_{i})$, which is nonnegative since
$[\rho,R_{n}]=0$ for all $n$. Thus, the state is $\rho$ and the events are
$R_{n}$ and $\Pi_{i}$. In the presence of initial quantum coherence, the work
can be represented by the quasiprobability distribution $p_{q}(w)$. To derive
a detailed fluctuation theorem, we consider an initial state $\rho$ such that
its incoherent part (with respect to the energy basis) is thermal,
$\Delta(\rho)=\rho_{\beta}$. In this case, we get the integral fluctuation
relation of Ref. [16],

$\langle e^{-\beta(w-\Delta F)-C}\rangle=1\,,$ (18)

which is our starting point to derive the detailed fluctuation theorem. In
detail, the average in Eq. (18) is calculated with respect to the joint
quasiprobability distribution

$\displaystyle
p_{q,q^{\prime}}(w,C)=\sum_{k,j,i,n}r_{n}\text{Re}\text{Tr}\left\\{R_{n}\Pi_{j}\Pi^{\prime}_{k}\Pi_{i}\right\\}\delta(w-\epsilon^{\prime}_{k}+q\epsilon_{i}$
$\displaystyle+(1-q)\epsilon_{j})\delta(C+q^{\prime}\ln\langle\epsilon_{i}|\rho|\epsilon_{i}\rangle+(1-q^{\prime})\ln\langle\epsilon_{j}|\rho|\epsilon_{j}\rangle$
$\displaystyle-\ln r_{n})\,.$ (19)

We can easily check that the marginal distributions are the quasiprobability
distribution of work $p_{q}(w)=\int p_{q,q^{\prime}}(w,C)dC$ and the
probability distribution of initial quantum coherence $p_{c}(C)=\int
p_{q,q^{\prime}}(w,C)dw$. To formulate a detailed fluctuation theorem, we
focus on the quantity $e^{-\beta(w-\Delta F)-C}p_{q,q^{\prime}}(w,C)$. Only
for $q=q^{\prime}$, by considering $p_{q}(w,C)=p_{q,q}(w,C)$, we get that
$e^{-\beta(w-\Delta F)-C}p_{q}(w,C)$ is a joint quasiprobability distribution,

$e^{-\beta(w-\Delta F)-C}p_{q}(w,C)=\hat{p}_{q}(-w,C)\,,$ (20)

which explicitly reads

$\displaystyle\hat{p}_{q}(w,\hat{C})=\sum_{k,j,i,n}\frac{e^{-\beta\epsilon^{\prime}_{k}}}{Z^{\prime}}\text{Re}\text{Tr}\left\\{\bar{\Pi}^{\prime}_{k}\bar{\Pi}_{i}\bar{R}_{n}\bar{\Pi}_{j}\right\\}\delta(w-q\epsilon_{i}$
$\displaystyle-(1-q)\epsilon_{j}+\epsilon^{\prime}_{k})\delta(\hat{C}+q\ln\langle\epsilon_{i}|\rho|\epsilon_{i}\rangle+(1-q)\ln\langle\epsilon_{j}|\rho|\epsilon_{j}\rangle$
$\displaystyle-\ln r_{n})\,.$ (21)

The quasiprobability distribution $\hat{p}_{q}(w,\hat{C})$ represents a
backward process, where the initial state is
$\bar{\rho}=\rho^{\prime}_{\beta}$ and the events are
$\bar{\Pi}^{\prime}_{k}$,$\bar{\Pi}_{i}$,
$\bar{R}_{n}=U_{\tau,0}R_{n}U_{\tau,0}^{\dagger}$ and $\bar{\Pi}_{j}$. In
particular, the projector $\bar{\Pi}^{\prime}_{k}$ selects the pure state
$\bar{\Pi}^{\prime}_{k}$ with probability
$e^{-\beta\epsilon^{\prime}_{k}}/Z^{\prime}$. We note that one of the marginal
distribution is the two-projective-measurement probability distribution for
the backward process, $\bar{p}_{TPM}(w)=\int\hat{p}_{q}(w,\hat{C})d\hat{C}$,
so that by integrating Eq. (20) over $C$, and noting that $\int
e^{-C}p_{q,q^{\prime}}(w,C)dC=p_{TPM}(w)$, we get the Tasaki-Crooks
fluctuation relation in Eq. (5). Concerning the variable $\hat{C}$ of the
backward process, its average is

$\int\hat{C}\hat{p}_{q}(\hat{C})d\hat{C}=S(U^{\dagger}_{0,\tau}\rho^{\prime}_{\beta}U_{0,\tau}||\rho_{\beta})-S(U^{\dagger}_{0,\tau}\rho^{\prime}_{\beta}U_{0,\tau}||\rho)\,,$
(22)

where $\hat{p}_{q}(\hat{C})=\int\hat{p}_{q}(w,\hat{C})dw$ and the quantum
relative entropy is defined as
$S(\rho||\eta)=-S(\rho)-\text{Tr}\left\\{\rho\ln\eta\right\\}$.

We aim to get a more symmetric detailed fluctuation relation, which implies
Eq. (20). We are looking for a fluctuation relation of the form $e^{a\cdot
x}p(x)=\bar{p}(\bar{x})$, where $x$ is a set of variables including the work
$w$, e.g., $x=(w,C,\ldots)$, $p(x)$ is the forward distribution and
$\bar{p}(\bar{x})$ is the backward distribution of
$\bar{x}=(-w,\bar{C},\ldots)$. It is worth noting that the positive
multiplying factor $e^{a\cdot x}$ does not change the support of $p(x)$, then
the backward distribution $\bar{p}(\bar{x})$ has the same support of $p(x)$.
This suggests that the time reversal discussed in the previous section, i.e.,
$\bar{p}_{q}(w)$ in Eq. (13) with an appropriate initial state $\bar{\rho}$,
will play some role. For simplicity, we consider $q=0$, so that, from Eq.
(IV), we get

$\displaystyle
p_{0}(w,C)=\sum_{k,i,n}r_{n}\text{Re}\text{Tr}\left\\{R_{n}\Pi_{i}\Pi^{\prime}_{k}\right\\}\delta(w-\epsilon^{\prime}_{k}+\epsilon_{i})$
$\displaystyle\times\delta(C+\ln\langle\epsilon_{i}|\rho|\epsilon_{i}\rangle-\ln
r_{n})\,.$ (23)

Since the backward distribution in the right side of a detailed fluctuation
relation needs to have the same support of the left side, we introduce a
random variable $\bar{C}$, and the joint time-reversed quasiprobability
distribution of the form

$\displaystyle\bar{p}_{0}(w,\bar{C})=\sum_{k,i,n}\bar{r}_{n}\text{Re}\text{Tr}\left\\{\bar{R}_{n}\bar{\Pi}_{i}\bar{\Pi}^{\prime}_{k}\right\\}\delta(w-\epsilon_{i}+\epsilon^{\prime}_{k})$
$\displaystyle\times\delta(\bar{C}+\ln\langle\epsilon_{i}|\rho|\epsilon_{i}\rangle-\ln
r_{n})\,,$ (24)

such that $\bar{p}_{0}(-w,C)$ has the same support of $p_{0}(w,C)$. The
eigenvalues $\bar{r}_{n}$ and the projectors $\bar{R}_{n}$ are such that
$\bar{\rho}=\sum\bar{r}_{n}\bar{R}_{n}$, where the initial state $\bar{\rho}$
of the backward process can be chosen appropriately in order to get a detailed
fluctuation relation. E.g., for $\rho=\rho_{\beta}$ and
$\bar{\rho}=\rho^{\prime}_{\beta}$ we get the detailed fluctuation theorem of
Eq. (5). However, as just seen in Eq. (20),

$e^{-\beta(w-\Delta F)-C}p_{0}(w,C)\neq\bar{p}_{0}(-w,C)$ (25)

for any $\bar{\rho}$ and the two quasiprobability distributions are not
related by a detailed fluctuation theorem. Of course, the equality in Eq. (25)
is achieved when the initial state is incoherent, $\rho=\Delta(\rho)$, so that
$\bar{p}_{0}(w,\bar{C})=\hat{p}_{0}(w,\bar{C})=\bar{p}_{TPM}(w)\delta(\bar{C})$.
To get a fluctuation theorem we start to focus on the variable $\bar{C}$,
which for the backward process has the marginal probability distribution
$\bar{p}(\bar{C})=\int\bar{p}_{0}(w,\bar{C})dw$, which reads

$\bar{p}(\bar{C})=\sum_{i,n}\bar{r}_{n}\text{Tr}\left\\{\bar{R}_{n}\bar{\Pi}_{i}\right\\}\delta(\bar{C}+\ln\langle\epsilon_{i}|\rho|\epsilon_{i}\rangle-\ln
r_{n})\,.$ (26)

We guess that for the forward process, the variable $\bar{C}$ has the
probability distribution

$p(\bar{C})=\sum_{k,n}r_{n}\text{Tr}\left\\{R_{n}\Pi^{\prime}_{k}\right\\}\delta(\bar{C}+\ln\langle\epsilon^{\prime}_{k}|\bar{\rho}|\epsilon^{\prime}_{k}\rangle-\ln\bar{r}_{n})\,.$
(27)

Thus, by introducing this new random variable, we consider the joint
quasiprobability distribution

$\displaystyle
p_{0}(w,C,\bar{C})=\sum_{k,i,n}r_{n}\text{Re}\text{Tr}\left\\{R_{n}\Pi_{i}\Pi^{\prime}_{k}\right\\}\delta(w-\epsilon^{\prime}_{k}+\epsilon_{i})$
$\displaystyle\times\delta(C+\ln\langle\epsilon_{i}|\rho|\epsilon_{i}\rangle-\ln
r_{n})\delta(\bar{C}+\ln\langle\epsilon^{\prime}_{k}|\bar{\rho}|\epsilon^{\prime}_{k}\rangle$
$\displaystyle-\ln\bar{r}_{n})$ (28)

and the time-reversed one

$\displaystyle\bar{p}_{0}(w,C,\bar{C})=\sum_{k,i,n}\bar{r}_{n}\text{Re}\text{Tr}\left\\{\bar{R}_{n}\bar{\Pi}_{i}\bar{\Pi}^{\prime}_{k}\right\\}\delta(w-\epsilon_{i}+\epsilon^{\prime}_{k})$
$\displaystyle\times\delta(C+\ln\langle\epsilon^{\prime}_{k}|\bar{\rho}|\epsilon^{\prime}_{k}\rangle-\ln\bar{r}_{n})\delta(\bar{C}+\ln\langle\epsilon_{i}|\rho|\epsilon_{i}\rangle$
$\displaystyle-\ln r_{n})\,,$ (29)

so that $p_{0}(w,C,\bar{C})$ and $\bar{p}_{0}(-w,\bar{C},C)$ have the same
support. It is easy to generalize Eqs. (IV)-(IV) for $q\neq 0$, and to check
that these two quasiprobability distributions are related by the detailed
fluctuation relation

$e^{-\beta(w-\Delta
F)-C+\bar{C}}p_{q}(w,C,\bar{C})=\bar{p}_{q}(-w,\bar{C},C)\,,$ (30)

if $\Delta(\rho)=\rho_{\beta}$,
$\bar{\Delta}(\bar{\rho})=\rho^{\prime}_{\beta}$ and
$\bar{R}_{n}=U_{0,\tau}R_{n}U_{0,\tau}^{\dagger}$ (i.e., $\rho^{\prime}$ and
$\bar{\rho}$ are diagonal with respect to the same basis). We note that

$\int e^{-C}\bar{p}_{q}(w,C,\bar{C})dC=\hat{p}_{q}(w,\bar{C})\,,$ (31)

then the fluctuation theorem of Eq. (30) implies the relation in Eq. (20) (it
is enough to multiply Eq. (30) by $e^{-\bar{C}}$ and then to integrate over
$\bar{C}$). Given the initial state $\rho$, there are several states
$\bar{\rho}$ such that $\bar{\Delta}(\bar{\rho})=\rho^{\prime}_{\beta}$, but
in general there is a unique $\bar{\rho}$ if we require also that
$\bar{R}_{n}=U_{0,\tau}R_{n}U_{0,\tau}^{\dagger}$, which is
$\bar{\rho}=\sum\bar{r}_{n}U_{0,\tau}R_{n}U_{0,\tau}^{\dagger}$ where the
eigenvalues $\bar{r}_{n}$ are the solutions of the linear equations

$\sum_{n}\bar{r}_{n}\langle\epsilon^{\prime}_{k}|U_{0,\tau}R_{n}U_{0,\tau}^{\dagger}|\epsilon^{\prime}_{k}\rangle=e^{-\beta\epsilon^{\prime}_{k}}/Z^{\prime}$
(32)

with $k=1,2,\ldots$. While the condition
$\bar{\Delta}(\bar{\rho})=\rho^{\prime}_{\beta}$ determines the populations of
$\bar{\rho}$, the conditions $U^{\dagger}\bar{R}_{n}U=R_{n}$ fix the
coherences of $\bar{\rho}$. We note that for an incoherent state
$\rho=\Delta(\rho)$ there is coherence in the state $\bar{\rho}$, namely
$\bar{\Delta}(\bar{\rho})\neq\bar{\rho}$, thus for $\rho=\rho_{\beta}$ the
backward process starts from the initial state
$\bar{\rho}\neq\rho^{\prime}_{\beta}$ and it is different from the Tasaki-
Crooks backward process of Sec. II.1. However, Eq. (30) implies Eq. (20),
which implies the Tasaki-Crooks fluctuation relation in Eq. (5).

##  V Integral fluctuation theorems and bounds

From the detailed fluctuation theorem of Eq. (30) we can derive two integral
fluctuation relations. By multiplying Eq. (30) by $e^{-\bar{C}}$ and
integrating we get Eq. (18). Furthermore, by integrating Eq. (30), we get

$\langle e^{-\beta(w-\Delta F)-C+\bar{C}}\rangle=1\,.$ (33)

The integral fluctuation theorem of Eq. (18) has been formulated and discussed
in Ref. [16]. In particular, it implies the bound

$\beta(\langle w\rangle-\Delta F)+\langle C\rangle\geq 0\,.$ (34)

Concerning Eq. (33), we note that $\beta(w-\Delta F)+C$ can be replaced with a
random variable $\sigma$, i.e., for any function of two variables $f(x,y)$,

$\displaystyle\langle f(\beta(w-\Delta F)+C,\bar{C})\rangle=\int
f(\beta(w-\Delta F)+C,\bar{C})$ $\displaystyle\times
p_{q}(w,C,\bar{C})dwdCd\bar{C}=\int f(\sigma,\bar{C})p(\sigma,\bar{C})d\sigma
d\bar{C}$ $\displaystyle=\langle f(\sigma,\bar{C})\rangle\,,$ (35)

where we have defined the probability distribution

$\displaystyle p(\sigma,\bar{C})$ $\displaystyle=$
$\displaystyle\sum_{n,k}r_{n}\text{Tr}\left\\{\Pi^{\prime}_{k}R_{n}\right\\}\delta(\sigma-\beta\epsilon^{\prime}_{k}-\ln
Z^{\prime}-\ln r_{n})\delta(\bar{C}$ (36)
$\displaystyle+\ln\langle\epsilon^{\prime}_{k}|\bar{\rho}|\epsilon^{\prime}_{k}\rangle-\ln\bar{r}_{n})\,.$

Thus, the fluctuation relation in Eq. (33) is equivalent to $\langle
e^{-\sigma+\bar{C}}\rangle=1$ and by using the Jensen’s theorem we get
$\langle\sigma\rangle-\langle\bar{C}\rangle\geq 0$, which can be expressed as

$\beta(\langle w\rangle-\Delta F)+\langle C\rangle-\langle\bar{C}\rangle\geq
0\,,$ (37)

since $\langle\sigma\rangle=\beta(\langle w\rangle-\Delta F)+\langle
C\rangle$. Eqs. (34) and (37) give two lower bounds for the average work
$\langle w\rangle$. Which of the two is tighter depends on the sign of
$\langle\bar{C}\rangle$. By noting that

$\langle\sigma\rangle-\langle\bar{C}\rangle=S(\rho^{\prime}||\bar{\rho})\geq
0\,,\quad\langle\sigma\rangle=S(\rho^{\prime}||\rho^{\prime}_{\beta})\geq
0\,,$ (38)

$\langle\bar{C}\rangle$ can be expressed as the difference of quantum relative
entropies

$\langle\bar{C}\rangle=S(\rho^{\prime}||\rho^{\prime}_{\beta})-S(\rho^{\prime}||\bar{\rho})\,.$
(39)

We note that $\langle\bar{C}\rangle$ is related to the final quantum coherence
in the energy basis, quantified by
$C_{\bar{\Delta}}(\rho^{\prime})=S(\bar{\Delta}(\rho^{\prime}))-S(\rho^{\prime})$.
In particular, Eq. (39) can be written as

$\langle\bar{C}\rangle=C_{\bar{\Delta}}(\rho^{\prime})+S(\bar{\Delta}(\rho^{\prime})||\bar{\Delta}(\bar{\rho}))-S(\rho^{\prime}||\bar{\rho})\,.$
(40)

To prove it, it is enough to note that

$\displaystyle
S(\rho^{\prime}||\rho^{\prime}_{\beta})=S(\rho^{\prime}||\bar{\Delta}(\bar{\rho}))=-S(\rho^{\prime})-\text{Tr}\left\\{\rho^{\prime}\ln\bar{\Delta}(\bar{\rho})\right\\}$
$\displaystyle=S(\bar{\Delta}(\rho^{\prime}))-S(\rho^{\prime})-S(\bar{\Delta}(\rho^{\prime}))-\text{Tr}\left\\{\bar{\Delta}(\rho^{\prime})\ln\bar{\Delta}(\bar{\rho})\right\\}$
$\displaystyle=C_{\bar{\Delta}}(\rho^{\prime})+S(\bar{\Delta}(\rho^{\prime})||\bar{\Delta}(\bar{\rho}))\,.$
(41)

From Eq. (39), we get the bounds for $\langle\bar{C}\rangle$

$-S(\rho^{\prime}||\bar{\rho})\leq\langle\bar{C}\rangle\leq
S(\rho^{\prime}||\rho^{\prime}_{\beta})\,,$ (42)

whereas, from Eq. (40), we get

$S(\bar{\Delta}(\rho^{\prime})||\bar{\Delta}(\bar{\rho}))-S(\rho^{\prime}||\bar{\rho})\leq\langle\bar{C}\rangle\leq
C_{\bar{\Delta}}(\rho^{\prime})\,,$ (43)

since $\bar{\Delta}$ is a completely positive and trace preserving map, so
that
$S(\bar{\Delta}(\rho^{\prime})||\bar{\Delta}(\bar{\rho}))-S(\rho^{\prime}||\bar{\rho})\leq
0$ and $C_{\bar{\Delta}}(\rho^{\prime})\geq 0$. The bounds in Eq. (42) and in
Eq. (43) can be saturated, depending on the final state $\rho^{\prime}$. If
$\rho^{\prime}=\bar{\rho}$, i.e., $r_{n}=\bar{r}_{n}$, then
$\langle\bar{C}\rangle=S(\rho^{\prime}||\rho^{\prime}_{\beta})=C_{\bar{\Delta}}(\rho^{\prime})$.
In this case, $p(\bar{C})$ reduces to the probability distribution of
coherence of $\rho^{\prime}$ with respect to the final energy basis and
$\bar{p}(\bar{C})=p_{c}(\bar{C})$. In contrast, the lower bound in Eq. (43)
can be saturated only if $C_{\bar{\Delta}}(\rho^{\prime})=0$, i.e.,
$\rho^{\prime}=\Delta(\rho^{\prime})$. However in this case it is zero since,
if $\rho^{\prime}=\bar{\Delta}(\rho^{\prime})$, from Eq. (32) we get
$\bar{\rho}=\bar{\Delta}(\bar{\rho})$, from which
$S(\rho^{\prime}||\bar{\rho})=S(\bar{\Delta}(\rho^{\prime})||\bar{\Delta}(\bar{\rho}))$.
Instead, the lower bound of Eq. (42) is saturated if
$\rho^{\prime}=\rho^{\prime}_{\beta}$, then
$\langle\bar{C}\rangle=-S(\rho^{\prime}||\bar{\rho})=0$ since
$\rho^{\prime}=\bar{\Delta}(\rho^{\prime})$ and thus
$\rho^{\prime}=\bar{\rho}$. From Eq. (43), by noting that both the lower and
upper bounds are zero if $\rho^{\prime}=\bar{\Delta}(\rho^{\prime})$, we get

$\rho^{\prime}=\bar{\Delta}(\rho^{\prime})\Rightarrow\langle\bar{C}\rangle=0\,.$
(44)

However,
$\langle\bar{C}\rangle=0\nRightarrow\rho^{\prime}=\bar{\Delta}(\rho^{\prime})$.
To show it, we note that

$\displaystyle\langle\bar{C}\rangle$ $\displaystyle=$
$\displaystyle\text{Tr}\left\\{\rho^{\prime}\ln\bar{\rho}\right\\}-\text{Tr}\left\\{\bar{\Delta}(\rho^{\prime})\ln\bar{\Delta}(\bar{\rho})\right\\}$
(45) $\displaystyle=$
$\displaystyle\sum_{n}r_{n}\ln\bar{r}_{n}-\sum_{k}\langle\epsilon^{\prime}_{k}|\rho^{\prime}|\epsilon^{\prime}_{k}\rangle\ln
p^{\prime}_{eq,k}\,,$

where $p^{\prime}_{eq,k}=e^{-\beta\epsilon^{\prime}_{k}}/Z^{\prime}$. Let us
consider $U_{\tau,0}$ such that
$U_{\tau,0}|r_{1}\rangle=\sqrt{a}|\epsilon^{\prime}_{1}\rangle+\sqrt{1-a}|\epsilon^{\prime}_{2}\rangle$,
$U_{\tau,0}|r_{2}\rangle=\sqrt{a}|\epsilon^{\prime}_{2}\rangle-\sqrt{1-a}|\epsilon^{\prime}_{1}\rangle$
and $U_{\tau,0}|r_{n}\rangle=|\epsilon^{\prime}_{n}\rangle$ for $n>2$. We
focus on $a=1-\eta$ and $\eta\to 0$. From Eq. (32) we get the solution
$\bar{r}_{1}=p^{\prime}_{eq,1}+\eta$ and $\bar{r}_{2}=p^{\prime}_{eq,2}-\eta$,
furthermore
$\langle\epsilon^{\prime}_{1}|\rho^{\prime}|\epsilon^{\prime}_{1}\rangle=r_{1}+(r_{2}-r_{1})\eta$
and
$\langle\epsilon^{\prime}_{2}|\rho^{\prime}|\epsilon^{\prime}_{2}\rangle=r_{2}-(r_{2}-r_{1})\eta$,
then

$\langle\bar{C}\rangle\simeq\left(\frac{r_{1}}{p^{\prime}_{eq,1}}-\frac{r_{2}}{p^{\prime}_{eq,2}}+(r_{1}-r_{2})\ln\left(\frac{p^{\prime}_{eq,1}}{p^{\prime}_{eq,2}}\right)\right)\eta$
(46)

which can be negative or positive depending on the eigenvalues $r_{n}$ and the
populations $p^{\prime}_{eq,k}$. It is easy to see that
$\langle\bar{C}\rangle<0$ if
$r_{1}<P\equiv(p^{\prime}_{eq,1}+p^{\prime}_{eq,1}p^{\prime}_{eq,2}\ln(p^{\prime}_{eq,1}/p^{\prime}_{eq,2}))/(1+2p^{\prime}_{eq,1}p^{\prime}_{eq,2}\ln(p^{\prime}_{eq,1}/p^{\prime}_{eq,2}))$,
which is satisfied if $r_{2}>r_{1}$ since
$p^{\prime}_{eq,2}<p^{\prime}_{eq,1}$. To show it, it is enough to note that
$r_{1}<P$ iff
$r_{2}-r_{1}>(1-2p^{\prime}_{eq,1})/(1+2p^{\prime}_{eq,1}p^{\prime}_{eq,2}\ln(p^{\prime}_{eq,1}/p^{\prime}_{eq,2}))$,
which is satisfied if $r_{2}>r_{1}$ since
$(1-2p^{\prime}_{eq,1})/(1+2p^{\prime}_{eq,1}p^{\prime}_{eq,2}\ln(p^{\prime}_{eq,1}/p^{\prime}_{eq,2}))$
is negative for $p^{\prime}_{eq,2}<p^{\prime}_{eq,1}$. Furthermore, we note
that from Eq. (46), $\langle\bar{C}\rangle\simeq 0$ for $r_{1}=P$ although
$\eta>0$ and so $\rho^{\prime}\neq\bar{\Delta}(\rho^{\prime})$.

##  VI Conclusions

Tasaki-Crooks fluctuation theorem still holds in the presence of initial
quantum coherence if the populations of the initial states of the forward and
backward processes are thermal. Here, we proved that a stronger detailed
fluctuation theorem can be formulated by appropriately fixing the initial
coherence of the backward process. In particular, it implies two different
integral fluctuation relations for the forward process, giving two different
bounds for the average work. Furthermore, it is worth noting that this
detailed fluctuation relation involves a backward quasiprobability
distribution that does not satisfy the reproduction of the two-projective-
measurement scheme. Thus, the contextuality of the backward protocol, which is
represented by suitable quasiprobability distributions reproducing the two-
projective-measurement statistics for incoherent initial states, is not
constrained by the contextuality of the forward one.

### Acknowledgements

The authors acknowledge financial support from the project BIRD 2021
”Correlations, dynamics and topology in long-range quantum systems” of the
Department of Physics and Astronomy, University of Padova, from the
EuropeanUnion-NextGenerationEU within the National Center for HPC, Big Data
and Quantum Computing (Project No. CN00000013, CN1 Spoke 10 Quantum Computing)
and from the Project ”Frontiere Quantistiche” (Dipartimenti di Eccellenza) of
the Italian Ministry for Universities and Research.

##  Appendix A Quasiprobabilities

We recall our notion of quasiprobability introduced in Ref. [17]. In general,
events are represented as effects, which are the positive operators which can
occur in the range of a positive operator valued measurement, i.e., an effect
is a Hermitian operator $E$ acting on the Hilbert space $\mathcal{H}$ such
that $0\leq E\leq I$. For a single event, the generalized probability measures
on the set of effects are functions $E\mapsto v(E)$ with the properties

$\displaystyle 0\leq v(E)\leq 1\,,$ (47) $\displaystyle v(I)=1\,,$ (48)
$\displaystyle v(E+F+\cdots)=v(E)+v(F)+\cdots$ (49)

where $E+F+\cdots\leq I$. A theorem [21] states that, if Eqs. (47)-(49) are
satisfied, then the probability corresponding to the event represented by $E$
is $v(E)=\text{Tr}\left\\{E\rho\right\\}$ for some density matrix $\rho$. For
two events, we can define a function $v(E,F)$ with the properties [17]

$\displaystyle v(E,F)\in\mathbb{R}\,,$ (50) $\displaystyle
v(I,E)=v(E,I)=v(E)\,,$ (51) $\displaystyle
v(E+F+\cdots,G)=v(E,G)+v(F,G)+\cdots\,,$ $\displaystyle
v(G,E+F+\cdots)=v(G,E)+v(G,F)+\cdots$ (52)

where $E+F+\cdots\leq I$. If Eqs. (50)-(52) are satisfied, and if $v(E,F)$ is
sequentially continuous in its arguments, then the quasiprobability
corresponding to the events $E\land F$ is a bilinear function, in detail it is
$v(E,F)=\text{Re}\text{Tr}\left\\{EF\rho\right\\}$ for some density matrix
$\rho$ (see Ref. [17] for details). Similarly, for three events, we define a
quasiprobability $v(E,F,G)$ with the properties

$\displaystyle v(E,F,G)\in\mathbb{R}\,,$ (53) $\displaystyle
v(I,E,F)=v(E,I,F)=v(E,F,I)=v(E,F)\,,$ (54) $\displaystyle
v(E+F+\cdots,G,H)=v(E,G,H)+v(F,G,H)+\cdots\,,$ $\displaystyle
v(G,E+F+\cdots,H)=v(G,E,H)+v(G,F,H)+\cdots\,,$ $\displaystyle
v(G,H,E+F+\cdots)=v(G,H,E)+v(G,H,F)+\cdots\,,$ (55)

and in general, for an arbitrary number of events, we define a
quasiprobability $v(E,F,\cdots)$ with the properties

$\displaystyle v(E,F,\cdots)\in\mathbb{R}\,,$ (56) $\displaystyle
v(I,E,F,\cdots)=v(E,I,F,\cdots)=\cdots=v(E,F,\cdots)\,,$ (57) $\displaystyle
v(E+F+\cdots,G,\cdots)=v(E,G,\cdots)+v(F,G,\cdots)+\cdots\,,$
$\displaystyle\cdots$ (58)

where $E+F+\cdots\leq I$. Analogously, if Eqs. (56)-(58) are satisfied, and if
$v(E,F,\cdots)$ is sequentially continuous in its arguments, then the joint
quasiprobability corresponding to the events $E\land F\land\cdots$ can be
expressed as an arbitrary affine combination of
$\text{Re}\text{Tr}\left\\{X_{i}\rho\right\\}$ where the operators $X_{i}$ are
all the possible products of the effects, e.g., for two events we can consider
only the product $X_{1}=EF$, since $X_{2}=FE$ gives the same quasiprobability;
for three events, we can consider the three products $X_{1}=EFG$, $X_{2}=FEG$
and $X_{3}=EGF$, and so on. Basically, the quasiprobability is not fixed for
more than two events since the proposition $E\land F\land\cdots$ is not well
defined. In particular, for more than two events, the quasiprobability depends
on how the events are grouped together. For instance, for three events, a
proposition $E\land F\land G$ can be decomposed in three different ways, which
are $E\land F$, $F\land G$, or $F\land E$, $E\land G$, or $E\land G$, $G\land
F$, then there is a one to one correspondence between the quasiprobabilities
$\text{Re}\text{Tr}\left\\{X_{i}\rho\right\\}$ and the different
decompositions. It is straightforward to see that this correspondence holds
also for an arbitrary number of events, thus we can associate the
quasiprobability
$v(E,F,G,\ldots)=\text{Re}\text{Tr}\left\\{EFG\cdots\rho\right\\}$ to the
definite decomposition $E\land F$, $F\land G$, $G\land\cdots$.

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{A}_{a \d{a}} = {B}_{a \d{a}} + {\alpha}_{a} \alpha^{b} \nabla_{b \d{a}} \mathbf{u} \; .
\end{equation}

Localisation to R4.
Writing the action (<ref>) in terms of the new field variables,
one can see the only terms that will contribute to the effective action given our choice of gauge (<ref>) will be

\begin{equation}
\begin{aligned}
S_{\mathrm{ghCS_6}} = \frac{1}{2 \pi i} \int_{\mathbb{PT}} \b{\pd} \Omega \wedge \Tr( \cA' \wedge \b{\pd} \hat{g} \hat{g}^{-1} - \hat{g}^{-1} \cA' \hat{g} \wedge \cB' - {\hat{g}^{-1} \b{\pd} \hat{g} \wedge \cB'} ) - \frac{1}{6\pi i}\int_{\mathbb{PT} \times [0,1]} \b{\pd}\Omega \wedge \Tr\l((\hat{g}^{-1} \dr \hat{g} )^{3} \r) \; .
\end{aligned}
\end{equation}

The localisation calculation of the gauged model is slightly more involved than the ungauged case due to the additional degrees of freedom appearing. However, as seen in the calculations in previous sections we expect $\mathbf{u}^{2}$ and $\mathbf{u}^{3}$ to appear only as Lagrange multipliers, in particular imposing self-duality type constraints for our gauge field $B$. With this tenet in mind, one can show that the 4d theory is given by

\begin{equation}\label{Gauged LMP action}
\begin{aligned}
S_{\text{gLMP}}[\mathbf{u},B] = k \int_{\mathbb{R}^{4}} \vol_{4} \frac{1}{2} \Tr \, (\nabla^{a \d{a}} \mathbf{u} \nabla_{a \d{a}} \mathbf{u}&) + \frac{1}{3} \epsilon^{\d{a} \d{b}} \, \Tr \, ( \mathbf{u}  \l[ \alpha^{a} \nabla_{a \d{a}} \mathbf{u} , \alpha^{b} \nabla_{b \d{b}} \mathbf{u}  \r] ) +  \mathbf{u} \; \epsilon^{\d{a} \d{b}} \hat{\alpha}^{a} \hat{\alpha}^{b} {F}_{a \d{a} b \d{b}} (B) \\
& + \frac{1}{2}\mathbf{u}^{2} \epsilon^{\d{a} \d{b}} \l( {\alpha}^{a} \hat{\alpha}^{b} + \hat{\alpha}^{a} \alpha^{b} \r) {F}_{a \d{a} b \d{b}}(B) + \tilde{\mathbf{u}}^{3} \epsilon^{\d{a} \d{b}} \alpha^{a} \alpha^{b} {F}_{a \d{a} b \d{b}}(B) \; ,
\end{aligned}
\end{equation}

where we have performed a field redefinition $\mathbf{u}^{3} \rightarrow \tilde{\mathbf{u}}^{3}:= \frac{1}{6}( \mathbf{u}^{3} + 2\l[ \mathbf{u} , \mathbf{u}^{2} \r])$. Upon reducing along a particular $\mathbb{R}^{2}$ subgroup, and appropriately performing redefinitions of our fields and parameters, one finds that the gauged LMP action matches the two-dimensional action eq. (<ref>).

Implementing the Lagrange multipliers.
In section <ref> we reviewed how solutions to the ASDYM can be formulated in terms of a Yang's matrix after a partial gauge fixing of the ASD connection. In this section we will look to integrate out our Lagrange multiplier fields present in the action eq. (<ref>) by solving the self duality constraints they impose in a similar vein.
Indeed, one may understand the LMP equations of motion as the remaining ASDYM equation after these two constraints have been solved. This is analogous to the statement that the WZW$_{4}$ equation of motion is the remaining ASDYM equation for Yang's matrix.

The equation of motion found by varying $\tilde{\mathbf{u}}^{3}$ is an integrability condition along the 2-plane defined by $\alpha^{a}$, and it maybe be solved by

\begin{equation}
\epsilon^{\d{a} \d{b}} \alpha^{a} \alpha^{b} {F}_{a \d{a} b \d{b}}(B) = 0 \qquad \implies \qquad
\alpha^{a}{B}_{a \d{a}} = {h}^{-1} \alpha^{a} \partial_{a \d{a}}{h} \; ,
\end{equation}

where $h \in C^{\infty}(\mathbb{R}^{4}) \otimes H$. It is helpful to parameterise the remaining degrees of freedom in $B_{a \dot{a}}$ in terms of a new field $C_{\dot{a}}$, defined by the relation

\begin{equation}
B_{a \dot{a}} = h^{-1} \pd_{a \dot{a}} h - \alpha_{a} \, h^{-1} C_{\dot{a}} h \ .
\end{equation}

Then, the $\mathbf{u}^{2}$ equation of motion becomes

\begin{equation}
\epsilon^{\d{a} \d{b}} \l( {\alpha}^{a} \hat{\alpha}^{b} + \hat{\alpha}^{a} \alpha^{b} \r) {F}_{a \d{a} b \d{b}}(B) = 0 \qquad \Longleftrightarrow \qquad
\epsilon^{\d{a} \d{b}} \alpha^{a} \pd_{a \dot{a}} C_{\dot{b}} = 0 \ .
\end{equation}

This may be solved explicitly by ${C}_{\d{a}} = \alpha^{a} \partial_{a \d{a}} f$ for $f \in C^{\infty}(\mathbb{R}^{4}) \otimes \tf{h}$, such that the gauge field $B$ is given by

\begin{equation}\label{on-shell gauge field B}
{B}_{a\d{a}} = {h}^{-1} \partial_{a \d{a}} h + {h}^{-1} X_{a \d{a}} h \; , \; \text{where} \; \; {X}_{a \d{a}} = - {\alpha}_{a} {\alpha}^{b} \partial_{{b} \d{a}} f \; .
\end{equation}

Reinserting this expression into the action eq. (<ref>), the resulting theory may be written as a difference of two LMP actions. This can be done by performing a field redefinition $h \mathbf{u} {h}^{-1} = v - f$, for $v \in C^{\infty}(\mathbb{R}^{4}) \otimes \tf{g}$, such that one arrives at the action

\begin{equation}
\begin{aligned}
S_{\text{gLMP}}[\mathbf{u},B] = \mathscr{k} \int_{\mathbb{R}^{4}} \frac{1}{2} \Tr ( \dr v \wedge \star \dr v ) + & \frac{1}{3} \alpha_{a} \alpha_{b} \Sigma^{ab} \wedge \Tr(v \l[ \dr v , \dr v \r] ) \\
& - \mathscr{k} \int_{\mathbb{R}^{4}} \frac{1}{2} \Tr ( \dr f \wedge \star \dr f ) + \frac{1}{3} \alpha_{a} \alpha_{b} \Sigma^{ab} \wedge \Tr(f \l[ \dr f , \dr f \r] ) \; .
\end{aligned}
\end{equation}

This demonstrates the conclusion

\begin{equation}
S_{\text{gLMP}}[\mathbf{u},B] = S_{\text{LMP}}[v] - S_{\text{LMP}}[f] \; .
\end{equation}

§ OUTLOOK

The construction presented in this work has led us to new integrable field theories in both four and two dimensions.
We conclude by highlighting a number of interesting future directions prompted by these results.

Motivated by the observation that the gauged WZW model on the coset $G/H$ in two dimensions can be written as the difference of WZW models for the groups $G$ and $H$, we took the difference of two hCS$_6$ theories as our starting point.
The boundary conditions (<ref>) led us to add a boundary term resulting in the action (<ref>).
It is worth highlighting that the boundary variation vanishes on the boundary conditions (<ref>) whether or not the boundary term is included, and the contribution of the boundary term to the IFT$_4$ vanishes if we invoke all the boundary conditions.
However, while the algebraic boundary conditions, $\mathcal{A}^{\mathfrak{k}}|_{\alpha,\beta}=0$ and $\mathcal{A}^{\mathfrak{h}}|_{\alpha,\beta}=\mathcal{B}_{\alpha,\beta}$ can be straightforwardly solved, this is not the case for the differential one $\partial_0\mathcal{A}^{\mathfrak{h}}|_{\alpha,\beta}=\partial_0\mathcal{B}_{\alpha,\beta}$.
Therefore, we relaxed this condition meaning that the contribution of the boundary term no longer vanishes.
Remarkably, for the specific boundary term added in (<ref>), the constraints implied by the differential boundary condition now follow as on-shell equations of motion, leading to fully consistent IFT$_4$ and IFT$_2$.

There are compelling reasons to follow this strategy, including that the symplectic potential becomes tautological upon including the boundary term.
However, it remains to understand why the differential boundary condition can be consistently dropped for this particular choice of boundary term, and a systematic interpretation of this is an open question.
To address this, it would be appropriate to pursue a more formal study, complementing a homotopic analysis (along the lines done for CS$_4$ in <cit.>) with a symplectic/Hamiltonian study of the 6d holomorphic Chern-Simons theory (similar to <cit.>  in the context of CS$_4$).

A second arena for formal development is the connection between 6d holomorphic Chern-Simons and five-dimensional Kähler Chern-Simons (KCS$_{5}$) theory <cit.>.  This should mirror the relationship between CS$_{4}$ and CS$_{3}$ theories described by Yamazaki <cit.>.  To make this suggestion precise in the present context one may consider a Kaluza-Klein expansion around the $U(1)$ rotation in the $\mathbb{CP}^1$ that leaves fixed the location of the double poles, retaining the transverse coordinate as part of the bulk five-manifold of KCS$_{5}$. The details of this are left for future study.

It would also be interesting to explore the new integrable IFT$_4$ and IFT$_2$ that we have constructed.
$G/H$ coset CFTs in two dimensions have a rich spectrum of paraferminonic operators <cit.>. It would be very interesting to establish the lift or analogue of these objects in the context of the IFT$_4$.
The natural framework for this is likely to involve the study of co-dimension one defects and associated higher-form symmetries.

For abelian $H$ we find IFT$_2$ that, in the $\kk \to 1$ limit, are related to massive integrable perturbations of the $G/H$ gauged WZW models known as homogeneous sine-Gordon models <cit.>.
These include the sine-Gordon and complex sine-Gordon models as special cases, two of the most well-understood IFT$_2$.
There is nothing in our construction that prohibits non-abelian $H$ and it would be instructive to study the resulting models in more detail.

An important class of IFT$_2$ are the symmetric space sigma models.  These can be constructed either by restricting fields to parameterise $G/H$ directly or by gauging a left action of H on the PCM.
These theories have been realised in CS$_{4}$ through branch cut defects <cit.> and recently in hCS$_{6}$ <cit.>.
One might explore the realisation the gauging construction of such models within the current framework, and generalise to $\mathbb{Z}_4$ graded semi-symmetric spaces (relevant for applications of CS$_4$ to string worldsheet theories <cit.>).

When $G/H$ is a symmetric space, an alternative class of massive integrable perturbations of the $G/H$ gauged WZW model are known as the symmetric space sine-Gordon models <cit.>.
In the landscape of IFT$_2$ these are related to the $\lambda\to0$ limit <cit.> of the $\lambda$-deformation of the symmetric space sigma model <cit.>.
Note that $\kk \to 1$ and $\lambda \to 0$ both correspond to conformal limits and it would be instructive to explore the relation between the two constructions.
More generally, it would be interesting to generalise the construction in this work to deformed models, in particular splitting one or both double poles in the meromorphic (3,0)-form $\Omega$ into simple poles, or dual models, for example considering the alternative boundary conditions (<ref>).

Finally, recently novel approaches to constructing IFT$_3$ using higher Chern-Simons theory in 5d has been explored in <cit.>.
Given that there is an overlap between the models that can be obtained from these constructions and from hCS$_6$, or more precisely its reduction to five dimensions, CS$_5$ on the mini-twistor correspondence space $\mathbb{PN}$  <cit.>, it would be exciting to understand the link between the two, and investigate the existence of categorical generalisations of hCS$_6$.

§ ACKNOWLEDGEMENTS

The authors would like to thank Alex Arvanitakis, Roland Bittleston, Falk Hassler, Tim Hollowood, Sylvain Lacroix, Marc Magro and Benoit Vicedo for interesting discussions.
This work was first presented by D.C.T. at the Workshop on Integrable Sigma-Models held in ETH Zurich in May 2024.
B.H and D.C.T. would also like to thank the organisers and participants for numerous stimulating and lively discussions.
B.H. is supported by a UKRI Future Leaders Fellowship (grant number MR/T018909/1). The work of J.L. is supported by CONICET.
D.C.T. is supported by the STFC grant ST/X000648/1, and by the Royal Society through a University Research Fellowship `Generalised Dualities in String Theory and Holography' URF 150185. For the purpose of open access, the authors have applied a Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising. No new data were generated for this manuscript.

§ SPINOR AND DIFFERENTIAL FORM CONVENTIONS

We work on $\mathbb{R}^4$ and define coordinates in bispinor notation by

\begin{equation}
x^{a \dot{a}} = \frac{1}{\sqrt{2}} \begin{pmatrix}
x_0 +\rmi x_1 & x_2 +\rmi x_3 \\ -x_2 + \rmi x_3 & x_0 - \rmi x_1
\end{pmatrix} \, .
\end{equation}

We fix orientation such that $\star 1 = \vol_4= \dr x_0 \wedge \dr x_1 \wedge \dr x_2 \wedge\dr x_3$. For 1-forms $\sigma = \sigma_{a \dot{a}} \dr x^{a \dot{a}} $ and $\tau = \tau_{a \dot{a}} \dr x^{a \dot{a}} $ we have

\begin{equation}
\star^2 \sigma = - \sigma \, , \quad \sigma \wedge \star \tau = - \star \sigma \wedge \tau = \vol_4 \epsilon^{ab} \epsilon^{\dot{a} \dot{b}} \sigma_{a \dot{a}}\tau_{b \dot{b}} \, , \quad \dr \star \sigma = \vol_4 \epsilon^{ab} \epsilon^{\dot{a} \dot{b}} \pd_{a \dot{a}}\sigma_{b \dot{b}} \, .
\end{equation}

Contraction of spinors is given by

\begin{equation}
\langle \alpha \beta \rangle = \alpha_1 \beta_2 - \alpha_2 \beta_1 = \alpha^a \beta_a
\end{equation}

and raising is achieved by

\begin{equation}
\alpha^a = \epsilon^{ab }\alpha_b \, \quad \epsilon^{12} = - \epsilon^{21} =-1 \, .
\end{equation}

We define $\epsilon_{12} = + 1$ such that $\epsilon^{a b} \epsilon_{b c} = \delta^a_{c}$. The (quaternionic) conjugation of a spinor $\alpha_a = (\alpha_1, \alpha_2)$ is defined to be $\hat{\alpha}_a = (- \bar{\alpha}_2 , \bar{\alpha}_1) $. Identical definitions hold for the anti-chiral spinors with dotted indices and contraction denoted with square brackets though these don't enter in this work.

A basis for self dual two forms is given by

\begin{equation}
\Sigma^{ab} = \epsilon_{\dot{a} \dot{b}} \dr x^{a \dot{a}} \wedge \dr x^{b\dot{b}} \, ,
\end{equation}

from which given any two spinors we can define self dual forms

\begin{equation}
\Sigma_{\alpha,\beta} = \alpha_a \beta_b \Sigma^{ab} = \alpha_a \beta_b \epsilon_{\dot{a} \dot{b}} \dr x^{a \dot{a}} \wedge \dr x^{b\dot{b}} \, , \qquad \star \Sigma_{\alpha,\beta} = \Sigma_{\alpha,\beta} \, .
\end{equation}

As they will play key roles let we denote

\begin{equation}
\omega_{\alpha, \beta} = \frac{1}{\langle \alpha \beta \rangle } \Sigma_{\alpha,\beta} \, , \quad \mu_\alpha = \Sigma_{\alpha,\alpha }\, , \quad \mu_\beta = \Sigma_{\beta,\beta} \, .
\end{equation}

$\mathbb{R}^4$ is equipped with a Hyper Kähler structure and has a $\mathbb{CP}^1$'s worth of complex structures. We can compactly express the corresponding complex structure to a spinor $\gamma_a$ as

\begin{equation}\label{eq:complexStruct}
{\mathcal J}_\gamma = - i (\gamma^a \pd_{a\dot{a}} ) \otimes (\hat{\gamma}_b \dr x^{b \dot{a}} )- i (\hat{\gamma}^a \pd_{a\dot{a}} ) \otimes (\gamma_b \dr x^{b \dot{a}} ) \, ,
\end{equation}

to which adapted complex coordinates are given by

\begin{equation}
\begin{aligned}
\dr z = \gamma_a \kappa_{\dot a} \dr x^{a \dot a } ~, \qquad &
\dr\bar{z} = \hat{\gamma}_a \hat{\kappa}_{\dot a} \dr x^{a \dot a } ~, \qquad
\dr w = \gamma_a \hat \kappa_{\dot a} \dr x^{a \dot a } ~, \qquad
\dr \bar{w} = - \hat \gamma_a \kappa_{\dot a} \dr x^{a\dot a } ~.
\end{aligned}
\end{equation}

With these coordinates we have that

\begin{align}
\mu_\alpha = -2 \langle \alpha \gamma\rangle^2 \dr \bar{w} \wedge \dr \bar{z}- 2 \langle \alpha \gamma\rangle \langle \alpha \hat{\gamma}\rangle ( \dr z \wedge \dr \bar{z} +\dr w \wedge \dr \bar{w} )-2 \langle \alpha \hat{\gamma}\rangle^2 \dr w \wedge \dr z \\
\omega_{\alpha, \beta} = -2 \frac{\langle\alpha \hat\gamma \rangle \langle\beta \hat{ \gamma} \rangle }{\langle\alpha \beta \rangle } \dr w \wedge \dr z -2 \frac{\langle\alpha \gamma \rangle \langle\beta \gamma \rangle }{\langle\alpha \beta \rangle } \dr \bar{w} \wedge \dr \bar{z} - \frac{ \langle\alpha \gamma \rangle \langle\beta \hat{ \gamma} \rangle +\langle\alpha \hat\gamma \rangle \langle\beta \gamma \rangle }{\langle\alpha \beta \rangle} ( \dr z \wedge \dr \bar{z} +\dr w \wedge \dr \bar{w} )
\end{align}

Notice that if we align the spinor $\alpha$ to $\gamma$ and $\beta$ to $\hat{\gamma}$ then $\omega_{\gamma, \hat{\gamma}}$ is (proportional to) the corresponding Kähler form $\varpi$ of type $(1,1)$ and $\mu_\gamma$ is a holomorphic $(2,0)$-form and $\mu_{\hat{\gamma}}$ is $(0,2)$-form.

§ TWISTOR SPACE

In this work we will be working on the Euclidean slice of Penrose's twistor space, $\mathbb{PT}_{\mathbb{E}}$. Starting from the twistor space of complexified Minkowski space,

\begin{equation}
\mathbb{PT} = \mathbb{CP}_3 / \mathbb{CP}_1 = \{ Z^\alpha = (\omega^{\dot{a}} , \pi_a ) \} \vert \pi_a \neq 0 \, , Z^\alpha \sim r Z^\alpha ~~ r\in \mathbb{C}^\times \} \, ,
\end{equation}

we obtain $\mathbb{PT}_{\mathbb{E}}$ by making a choice of reality conditions, in particular by selecting the slice of $\mathbb{PT}$ invariant under the anti-holomorphic (quartic)-involution acting on the holomorphic coordinates as ${Z}^{\alpha} \mapsto \hat{Z}^{\alpha} = (\hat{\omega}^{\d{a}} , \hat{\pi}_{a})$. This choice of reality conditions induces a double fibration and we find that in particular Euclidean twistor space can be viewed as the holomorphic vector bundle $\mathbb{PT}_{\mathbb{E}} \cong \mc{O}(1) \oplus \mc{O}(1) \rightarrow \mathbb{CP}^{1}$, where the holomorphic coordinates along the fibre direction are given by the incidence relations $\omega^{\d{a}} = {x}^{a \d{a}} {\pi}_{a}$. With this we choose a basis of $(1,0)$-forms and $(0,1)$-forms

\begin{equation}
\begin{aligned}
& e^0 = \langle \pi \dr \pi \rangle ~, &&
& e^{\dot{a}} = \pi_{a} \dr x^{a \dot{a} } ~, \\
& \bar{e}^0 = \frac{\langle \hat{\pi} \dr \hat{\pi} \rangle}{\langle \pi \hat{\pi} \rangle^2} ~, &&
& \bar{e}^{\dot{a}} = \frac{\hat{\pi}_{a} \dr x^{\dot{a} a }}{\langle \pi \hat{\pi} \rangle} ~.
\end{aligned}
\end{equation}

and their dual vector fields

\begin{equation}
\label{eq:twistorderivatives}
\begin{aligned}
& \pd_0 = \frac{\hat{\pi}_{a}}{\langle \pi \hat{\pi} \rangle} \frac{\pd}{\pd \pi_{a}} ~, &&
& \pd_{\dot{a}} = - \frac{\hat{\pi}^{a} \pd_{a \dot{a} }}{\langle \pi \hat{\pi} \rangle} ~, \\
& \bar{\pd}_0 = - \langle \pi \hat{\pi} \rangle \, \pi_{a} \frac{\pd}{\pd \hat{\pi}_{a}} ~, &&
& \bar{\pd}_{\dot{a}} = \pi^{a} \pd_{a \dot{a} } ~.
\end{aligned}
\end{equation}

It is important to note that this basis of $1$-forms, and their duals, enjoy the structure equations,

\begin{equation}
\begin{aligned}
\bar{\pd} e^{\d{a}} = e^0 \wedge \bar{e}^{\d{a}} ~, \quad
\pd \bar{e}^{\d{a}} = e^{\d{a}} \wedge \bar{e}^0 ~,\\
\comm{\bar{\pd}_0}{\pd_{\d{a}}} = \bar{\pd}_{\d{a}} ~, \quad
\comm{\bar{\pd}_{\d{a}}}{\pd_0} = \pd_{\d{a}} ~.
\end{aligned}
\end{equation}

§.§ Homogeneous and Inhomogeneous Coordinates

Homogeneous coordinates on $\bCP^{1}$ will be denoted by $\pi_{a} = (\pi_{1} , \pi_{2})$ which are defined up to the equivalence relation $\pi_{a} \sim s \, \pi_{a}$ for any non-zero $s \in \bC^{\ast}$.
These have the advantage of being globally defined on $\bCP^{1}$ but can lead to technical challenges in certain calculations.
It can also be useful to work with inhomogeneous coordinates on two patches covering $\bCP^{1} \cong S^{2}$.
Introducing an arbitrary spinor $\gamma_{a}$ which satisfies $\langle \gamma \hat{\gamma} \rangle = 1$, the two patches covering $\bCP^{1}$ will be defined by

\begin{equation}
U_{1} = \{ \pi_{a} \mid \langle \pi \hat{\gamma} \rangle \neq 0 \} \ , \qquad
U_{2} = \{ \pi_{a} \mid \langle \pi \gamma \rangle \neq 0 \} \ .
\end{equation}

Inhomogeneous coordinates may be defined on each patch by

\begin{equation}
\zeta = \frac{
\langle \gamma \pi \rangle
\langle \pi \hat{\gamma} \rangle
} \ , \qquad
\xi = \frac{
\langle \pi \hat{\gamma} \rangle
\langle \gamma \pi \rangle
} \ , \qquad
\xi = \zeta^{-1} \ .
\end{equation}

In this section, we will restrict our attention to $U_{1}$ and the inhomogeneous coordinate $\zeta$, knowing that an analogous discussion may be had on the other patch.
The complex conjugate of the inhomogeneous coordinate $\zeta$ is

\begin{equation}
\bar{\zeta} = - \frac{
\langle \hat{\pi} \hat{\gamma} \rangle
\langle \gamma \hat{\pi} \rangle
} \ .
\end{equation}

Forms and vector fields on $\bCP^{1}$ written in these coordinates are related to one another by

\begin{equation}
\begin{aligned}
\dr \zeta & = \frac{
\langle \pi \hat{\gamma} \rangle^{2}
} \ , \quad & \quad
\dr \bar{\zeta} & = \frac{
\langle \pi \hat{\pi} \rangle^{2}
\langle \gamma \hat{\pi} \rangle^{2}
} \bar{e}^{0} \ , \\
\pd_{\zeta} & = \langle \pi \hat{\gamma} \rangle^{2} \pd_{0} \ , \quad & \quad
\pd_{\bar{\zeta}} & = \frac{
\langle \gamma \hat{\pi} \rangle^{2}
\langle \pi \hat{\pi} \rangle^{2}
} \bar{\pd}_{0} \ .
\end{aligned}
\end{equation}

It is also helpful to defined a weight zero basis of $(1,0)$-forms on $\bR^{4} \subset \bPT$ by

\begin{equation}
\theta^{\dot{a}} = \frac{
\langle \pi \hat{\gamma} \rangle
= \dr x^{a \dot{a}} \, \gamma_{a} + \zeta \, \dr x^{a \dot{a}} \, \hat{\gamma}_{a} \ .
\end{equation}

Likewise the weight zero basis of $(0,1)$-forms on $\mathbb{R}^{4} \subset \mathbb{PT}$ are defined by

\begin{equation}
\b{\theta}^{\d{a}} = \la \pi \hat{\gamma} \ra \, \b{e}^{\d{a}} = \frac{1}{1+\zeta \b{\zeta}}\l( dx^{a \d{a}} \, \hat{\gamma}_{a} - \b{\zeta} \, d x^{a \d{a}} \, \gamma_{a} \r) \; .
\end{equation}

Given a point on $\bCP^{1}$ defined by $\alpha_{a}$ in homogeneous coordinates, we denote the corresponding point in the inhomogeneous coordinate $\zeta$ by

\begin{equation}
\alpha = \frac{
\langle \gamma \alpha \rangle
\langle \alpha \hat{\gamma} \rangle
= \zeta \vert_{\pi_{a} \sim \alpha_{a}} \ .
\end{equation}

We also have the relationship

\begin{equation} \label{eq:hom_inhom_identity}
\frac{
\langle \pi \alpha \rangle
\langle \pi \hat{\gamma} \rangle
\langle \hat{\gamma} \alpha \rangle
= (\zeta - \alpha) \ .
\end{equation}

§ PROJECTOR TECHNOLOGY

We consider the operator on $1$-forms on $\mathbb{R}^4$ given by

\begin{equation}
{\cal J}_{\alpha,\beta} (\sigma ) = - \rmi \star (\omega_{\alpha,\beta } \wedge \sigma) \, , \quad {\cal J}_{\alpha,\beta}^2 = -\id \,,
\end{equation}

which allows us to define projectors

\begin{equation}
P = \frac{1}{2}\left(\id - \rmi{\cal J} \right) \, \qquad \bar{P} = \frac{1}{2}\left(\id + \rmi{\cal J}\right) \ .
\end{equation}

This is suggestive of a complex structure and indeed if we take $\alpha = \gamma$ and $\beta =\hat{\gamma}$ then ${\cal J}_{\gamma,\hat{\gamma}}$ is the complex structure ${\cal J}_\gamma$ (e.q. (<ref>). The projectors $P$ and $\bar{P}$ project onto $(1,0)$ and $(0,1)$ components thus realising the Dolbeault complex.

These projectors enjoy a range of identities that we deploy in calculation:

\begin{equation} \label{eq:projectorid}
\bar{P} ( \star (\mu_\alpha\wedge \sigma) ) = 0 \, , \quad P( \star (\mu_\beta \wedge \sigma) ) = 0 \,,\quad \mu_\beta \wedge \bar{P}(\sigma) = 0 \, , \quad \mu_\alpha \wedge P(\sigma) = 0 \, ,
\end{equation}

\begin{equation} \label{eq:projectoridII}
\omega_{\alpha, \beta }\wedge \bar{P}(\sigma) = -\star \bar{P}(\sigma) \, , \quad \omega_{\alpha, \beta }P(\sigma) = \star P(\sigma) \,
\end{equation}

\begin{equation} \label{eq:projectoridIII}
\omega_{\alpha, \beta }\wedge \bar{P}(\sigma) \wedge \tau = \omega \wedge \sigma \wedge P(\tau) \, , \quad \omega_{\alpha, \beta }\wedge \bar{P}(\sigma) \wedge \bar{P}(\tau) = 0 \, .
\end{equation}

To move between form and component notation is useful to observe that

\begin{equation}\label{eq:projecttocomp}
P(\sigma)_{a \dot{a}} = - \frac{1}{\ab} \alpha_a \beta^b \sigma_{b \dot{a}} \, , \quad \bar{P} (\sigma)_{a \dot{a}} = \frac{1}{\ab} \beta_a \alpha^b \sigma_{b \dot{a}} \, .
\end{equation}

Further relations, used for processing the $\mathbb{CP}_1$ derivative boundary conditions, are

\begin{align}\label{eq:CP1compid}
\alpha^{a}\sigma_{a\dot{a}} \bar{e}^{\dot{a}}\vert_{\alpha} = 2 \star ( \mu_\alpha \wedge a) \, , \quad \beta^{a}\tau_{a\dot{a}} \bar{e}^{\dot{a}}\vert_{\alpha} = -\langle \alpha \beta \rangle P(\tau) \, , \\
\beta^{a}\sigma_{a\dot{a}} \bar{e}^{\dot{a}}\vert_{\beta} = 2 \star ( \mu_\beta\wedge a) \, , \quad \alpha^{a}\tau_{a\dot{a}} \bar{e}^{\dot{a}}\vert_{\beta} = \langle \alpha \beta \rangle \bar{P}(\tau) \, .
\end{align}

As an application of this projector technology let us consider the (ungauged) WZW$_4$ model for which the equation of motion can be cast in terms of the right-invariant Maurer Cartan form $R= \dr g g^{-1}$ that obeys $\dr R = R\wedge R $ as

\begin{equation}
\dr \star \bar{P}(R) = \frac{1}{2} \dr\; (\star - \omega_{\alpha,\beta} \wedge)\; \dr g g^{-1} = 0 \, .
\end{equation}

We consider now a Yang-Mills connection $A = - \bar{P}(X) $. The equations for this to be anti-self dual are

\begin{equation}
\mu_\beta F[A] = 0 \, , \quad \mu_\alpha F[A]= 0 \, , \quad \omega_{\alpha,\beta} \wedge F[A] = 0\, .
\end{equation}

The first of these vanishes identically by virtue of the fact that $\mu_\beta \wedge A = 0 $. Since $ \mu_\alpha \wedge A = - \mu_\alpha \wedge X$, the second provides a Bianchi identity

\begin{equation}
\mu_\alpha F[A]= - \mu_\alpha \wedge \left( \dr X - X \wedge X\right)
\end{equation}

and hence solved with $X=R$. The final equation returns the desired equation of motion as

\begin{equation}
\omega_{\alpha,\beta} \wedge F[A] = -\dr( \omega_{\alpha,\beta}\wedge \bar{P}(R) ) +\omega_{\alpha,\beta}\wedge \bar{P}(R) \wedge \bar{P}(R) = \dr \star \bar{P}(R) \, .
\end{equation}

At the Kähler point $\beta = \hat{\alpha} = \hat{\gamma}$, we can simply cast the ASDYM equations as

\begin{equation}
F^{2,0} = 0 \ , \qquad
F^{0,2} = 0 \ , \qquad
\varpi \wedge F^{1,1} = 0 \ .
\end{equation}

In this case, the connection is given by $A = - \bar{\pd} g g^{-1}$ is of type $(0,1)$, hence $F^{2,0}=0$ automatically, $F^{0,2}=0$ is zero by Bianchi identity and the equation of motion of WZW$_{4}$ is

\begin{equation}
\varpi \wedge \pd ( \bar{\pd} g g^{-1} ) = 0 \, .
\end{equation}

§ DERIVATION OF LOCALISATION FORMULAE

In this work we are required to evaluate integrals of the form

\begin{equation}\label{General form of integral}
I = \frac{1}{2 \pi \rmi} \int_{\bPT} \bar{\pd} \Omega \wedge Q \ , \qquad
Q \in \Omega^{0,2} (\bPT) \ .
\end{equation}

In this appendix, we will derive general formulae for these integrals for the cases in which $\Omega$ has either two double poles or a single fourth order pole as used in the paper. To compute these integrals efficiently we will move to inhomogeneous coordinates and make use of the identity

\begin{equation} \label{eq:inhom_pole_der}
\pd_{\bar{\zeta}} \bigg( \frac{1}{\zeta - \alpha} \bigg) = - 2 \pi \rmi \, \delta^{2}(\zeta - \alpha) \ , \qquad
\int_{\bCP^{1}} \dr \zeta \wedge \dr \bar{\zeta} \, \delta^{2}(\zeta - \alpha) \, f(\zeta) = f(\alpha) \ .
\end{equation}

§.§ Two double poles

We consider the $(3,0)$-form given by

\begin{equation}
\Omega = \frac{1}{2}
\frac{
\langle \alpha \beta \rangle^{2}
\langle \pi \alpha \rangle^{2}
\langle \pi \beta \rangle^{2}
e^0 \wedge e^{\dot{a}} \wedge e_{\dot{a}} = \frac{1}{2}
\frac{
(\alpha - \beta)^{2}
(\zeta - \alpha)^{2}
(\zeta - \beta)^{2}
\dr \zeta \wedge \theta^{\dot{a}} \wedge \theta_{\dot{a}} \ .
\end{equation}

Substituting this into our integral gives

\begin{equation}
I = - \frac{1}{2} \frac{1}{2 \pi \rmi}
\int_{\bPT} \dr \zeta \wedge \dr \bar{\zeta} \, \pd_{\bar{\zeta}}
\bigg( \frac{
(\alpha - \beta)^{2}
(\zeta - \alpha)^{2}
(\zeta - \beta)^{2}
} \bigg)
\wedge \theta^{\dot{a}} \wedge \theta_{\dot{a}} \wedge Q \ .
\end{equation}

Then, using the identity (<ref>) gives

\begin{equation}
I = - \frac{(\alpha - \beta)^{2}}{2}
\int_{\bPT} \dr \zeta \wedge \dr \bar{\zeta} \,
\bigg[ \frac{
\pd_{\zeta} \delta(\zeta - \alpha)
(\zeta - \beta)^{2}
+ \frac{
\pd_{\zeta} \delta(\zeta - \beta)
(\zeta - \alpha)^{2}
} \bigg]
\wedge \theta^{\dot{a}} \wedge \theta_{\dot{a}} \wedge Q \ .
\end{equation}

Since the integral is symmetric under $\alpha \leftrightarrow \beta$, we will only compute the first term explicitly.
Integration by parts and evaluating the integral over $\bCP^{1}$ gives

\begin{equation} \label{eq:checkpoint}
I = \frac{(\alpha - \beta)^{2}}{2}
\int_{\bR^{4}} \pd_{\zeta}
\bigg( \frac{
\theta^{\dot{a}} \wedge \theta_{\dot{a}} \wedge Q
(\zeta - \beta)^{2}
} \bigg) \bigg\vert_{\alpha}
\quad + \quad \alpha \leftrightarrow \beta \ .
\end{equation}

We will first distribute the $\pd_{\zeta}$ derivative, leaving the 2-form $Q$ completely general, resulting in

\begin{equation}
\begin{aligned}
I = \frac{(\alpha - \beta)^{2}}{2}
\int_{\bR^{4}}
\bigg[
& \frac{-2}{(\zeta - \beta)^{3}} \theta^{\dot{a}} \wedge \theta_{\dot{a}} \wedge Q
+ \frac{2}{(\zeta - \beta)^{2}} \hat{\gamma}_{a} \dr x^{a \dot{a}} \wedge \theta_{\dot{a}} \wedge Q \\
& \qquad + \frac{
\theta^{\dot{a}} \wedge \theta_{\dot{a}}
(\zeta - \beta)^{2}
} \wedge \pd_{\zeta} Q
\bigg] \bigg\vert_{\alpha}
\quad + \quad \alpha \leftrightarrow \beta \ .
\end{aligned}
\end{equation}

The overall factor of $(\alpha - \beta)^{2}$ outside the integral cancels with the denominators in the integrand.
We will also make use of (<ref>) to return to spinor notation, and introduce self-dual 2-forms defined by $\Sigma^{ab} = \varepsilon_{\dot{a} \dot{b}} \dr x^{a \dot{a}} \wedge \dr x^{b \dot{b}}$.

\begin{equation}
\begin{aligned}
I = \frac{1}{2}
\int_{\bR^{4}}
\bigg[
& \frac{
-2 \langle \hat{\gamma} \beta \rangle
\langle \alpha \beta \rangle
\langle \alpha \hat{\gamma} \rangle
} \alpha_{a} \alpha_{b} \Sigma^{ab} \wedge Q \vert_{\alpha}
+ \frac{2}{
\langle \alpha \hat{\gamma} \rangle
\hat{\gamma}_{a} \alpha_{b} \Sigma^{ab} \wedge Q \vert_{\alpha} \\
& \qquad + \alpha_{a} \alpha_{b} \Sigma^{ab} \wedge \frac{
\pd_{\zeta} Q
\langle \pi \hat{\gamma} \rangle^{2}
} \bigg\vert_{\alpha} \bigg]
\quad + \quad \alpha \leftrightarrow \beta \ .
\end{aligned}
\end{equation}

Expanding $\alpha_{a}$ in the basis formed by $\hat{\gamma}_{a}$ and $\beta_{a}$, we see that one component of the first term cancels the entire second term, and only a term proportional to $\alpha_{a} \beta_{b} \Sigma^{ab}$ survives.
In the third term of the action, we recognise the combination $\pd_{0}$ acting on $Q$ and make this replacement.
In conclusion, we have the general formula

\begin{equation} \label{eq:localisationformula}
\frac{1}{2 \pi \rmi} \int_{\bPT} \bar{\pd} \Omega \wedge Q = \int_{\bR^{4}}
\bigg[
\frac{
\alpha_{a} \beta_{b} \Sigma^{ab}
\langle \alpha \beta \rangle
} \wedge Q \vert_{\alpha}
+ \frac{1}{2} \alpha_{a} \alpha_{b} \Sigma^{ab} \wedge (\pd_{0} Q) \vert_{\alpha} \bigg]
\quad + \quad \alpha \leftrightarrow \beta \ ,
\end{equation}

or in differential form notation

\begin{equation}
\frac{1}{2 \pi \rmi} \int_{\bPT} \bar{\pd} \Omega \wedge Q = \int_{\bR^{4}}
\bigg[
\omega_{\alpha,\beta} \wedge Q \vert_{\alpha}
+ \frac{1 }{2} \mu_\alpha \wedge (\pd_{0} Q) \vert_{\alpha} \bigg]
\quad + \quad \alpha \leftrightarrow \beta \ .
\end{equation}

It is also helpful to specialise to 2-forms of the form $Q = \pi^{a} \pi^{b} Q_{a \dot{a} b \dot{b}} \bar{e}^{\dot{a}} \wedge \bar{e}^{\dot{b}}$ which we will often encounter in practice.
In this case, we may make use of the identity

\begin{equation} \label{eq:es_to_vol4}
e^{\dot{c}} \wedge e_{\dot{c}} \wedge \bar{e}^{\dot{a}} \wedge \bar{e}^{\dot{b}} = - 2 \, \vol_{4} \, \varepsilon^{\dot{a} \dot{b}} \ .
\end{equation}

and its generalisation valid for any spinors $\alpha_{a}$ and $\beta_{a}$

\begin{equation}
\alpha_{a} \beta_{b} \Sigma^{ab} \wedge \bar{e}^{\dot{a}} \wedge \bar{e}^{\dot{b}} = - 2 \, \vol_{4} \, \frac{\langle \alpha \hat{\pi} \rangle \langle \beta \hat{\pi} \rangle}{\langle \pi \hat{\pi} \rangle^{2}} \, \varepsilon^{\dot{a} \dot{b}} \ .
\end{equation}

Using these identities on the above formula in the case $Q = \pi^{a} \pi^{b} Q_{a \dot{a} b \dot{b}} \bar{e}^{\dot{a}} \wedge \bar{e}^{\dot{b}}$ gives

\begin{equation}
\label{ec:localisationformula}
\frac{1}{2 \pi \rmi} \int_{\bPT} \bar{\pd} \Omega \wedge Q = - \int_{\bR^{4}} \vol_{4} \,
\bigg[
\frac{
\varepsilon^{\dot{a} \dot{b}} (\alpha^{a} \beta^{b} + \beta^{a} \alpha^{b})
\langle \alpha \beta \rangle
} Q_{a \dot{a} b \dot{b}} \vert_{\alpha}
+ \varepsilon^{\dot{a} \dot{b}} \alpha^{a} \alpha^{b}
(\pd_{0} Q_{a \dot{a} b \dot{b}}) \vert_{\alpha} \bigg]
\quad + \quad \alpha \leftrightarrow \beta \ .
\end{equation}

One final specialism is the case when $Q_{a \dot{a} b \dot{b} } = X_{a \dot{a}} Y_{b\dot{b}} $, in which case the answer can be recast again in differential form notation as

\begin{equation}
\label{ec:localisationformula2}
\frac{1}{2 \pi \rmi} \int_{\bPT} \bar{\pd} \Omega \wedge Q = \int_{\bR^{4}}
\bigg[
\omega_{\alpha,\beta} \wedge X\wedge Y \vert_{\alpha}
+ \frac{1 }{2} \mu_\alpha \pd_{0} \wedge (X\wedge Y ) \vert_{\alpha} \bigg]
\quad + \quad \alpha \leftrightarrow \beta \ .
\end{equation}

To apply these formulae we need the following $\mathbb{CP}_1$ derivatives :

\begin{align}
\pd_0 ( \dr \hat{g} \hat{g}^{-1} ) &= \hat{g} \dr \hat{u} \hat{g}^{-1} \, , \\
\pd_0 ( \hat{g}^{-1}\dr \hat{g} ) &= \dr \hat{u} + [ \hat{g}^{-1}\dr \hat{g}, \hat{u} ] \, , \\
\pd_0 ( A ) &= \pd_0 (B) = 0 \, , \\
\pd_0 (\hat{g}^{-1} A \hat{g} ) &= [\hat{g}^{-1} A \hat{g} ,\hat{u} ] \, , \\
\pd_0 \frac{1}{3} \Tr (\hat{g}^{-1} \dr g)^3 &= \dr ~\Tr( \hat{u} (\hat{g}^{-1} \dr g)^2) \, ,
\end{align}

in which used the definition $\hat{u} = \hat{g}^{-1}\pd_0 \hat{g}$.

§.§ Fourth order Pole

In section <ref>, we considered a different $(3,0)$-form given by

\begin{equation}\label{4th order}
\Omega = k \frac{{e}^{0} \wedge {e}^{\d{a}}\wedge {e}_{\d{a}}}{{\langle \pi \alpha \rangle}^{4}} = \frac{k'}{{\la \hat{\gamma} \alpha \ra}^{4}} \frac{d \zeta \wedge \theta^{\d{a}} \wedge \theta_{\d{a}}}{(\zeta - \alpha)^{4}} \, .
\end{equation}

Substituting this into the general integral expression above gives

\begin{equation}
I = - \frac{k}{{\la \hat{\gamma} \alpha \ra}^{4}} \frac{1}{2 \pi \rmi}
\int_{\bPT} \dr \zeta \wedge \dr \bar{\zeta} \, \pd_{\bar{\zeta}}
\bigg( \frac{1}{(\zeta - \alpha)^{4}} \bigg)
\wedge \theta^{\dot{a}} \wedge \theta_{\dot{a}} \wedge Q \ .
\end{equation}

Then, using the identity (<ref>), we find

\begin{equation}
I = - \frac{k}{6 {\la \hat{\gamma} \alpha \ra}^{4}}
\int_{\bPT} \dr \zeta \wedge \dr \bar{\zeta} \,
\Big( \pd_{\zeta}^{3} \delta (\zeta - \alpha) \Big)
\wedge \theta^{\dot{a}} \wedge \theta_{\dot{a}} \wedge Q \ .
\end{equation}

Applying integration by parts and completing the integral over $\bCP^{1}$ gives

\begin{equation}
I = \frac{k}{6 {\la \hat{\gamma} \alpha \ra}^{4}}
\int_{\bR^{4}} \pd_{\zeta}^{3} \Big( \theta^{\dot{a}} \wedge \theta_{\dot{a}} \wedge Q \Big) \Big\vert_{\alpha} \ .
\end{equation}

In order to distribute this over the argument, it is helpful to have the identities

\begin{equation}
\theta^{a} \big\vert_{\alpha} = \frac{\alpha_{a} \dr x^{a \dot{a}}}{\langle \hat{\gamma} \alpha \rangle} \ , \qquad
\pd_{\zeta} \theta^{a} \big\vert_{\alpha} = \hat{\gamma}_{a} \dr x^{a \dot{a}} \ , \qquad
\pd_{\zeta}^{2} \theta^{a} \big\vert_{\alpha} = 0 \ .
\end{equation}

Then, distributing the three $\pd_{\zeta}$-derivatives gives

\begin{equation}
I = \frac{k}{6 {\la \hat{\gamma} \alpha \ra}^{4}}
\int_{\bR^{4}} \bigg[
\frac{\alpha_{a} \alpha_{b} \Sigma^{ab}}{\langle \alpha \hat{\gamma} \rangle^{2}} \wedge \pd_{\zeta}^{3} Q \big\vert_{\alpha}
+ 6 \, \frac{\alpha_{a} \hat{\gamma}_{b} \Sigma^{ab}}{\langle \alpha \hat{\gamma} \rangle} \wedge \pd_{\zeta}^{2} Q \big\vert_{\alpha}
+ 6 \, \hat{\gamma}_{a} \hat{\gamma}_{b} \Sigma^{ab} \wedge \pd_{\zeta} Q \big\vert_{\alpha}
\bigg] \ .
\end{equation}

Converting this expression back into homogeneous coordinates (and making use of the fact that $Q$ was a $(0,2)$-form on twistor space meaning $\hat{\alpha}_{a} \dr x^{a \dot{a}} \wedge Q \vert_{\alpha} = 0$), this integral becomes

\begin{equation}
I = \frac{k}{6}
\int_{\bR^{4}} \alpha_{a} \alpha_{b} \Sigma^{ab}\wedge \pd_{0}^{3} Q \big\vert_{\alpha} \ .
\end{equation}

§ LOCALISATION DERIVATION WITH GENERAL GAUGINGS

In this appendix we describe in more detail the derivation of the gauged WZW$_4$ model from the gauged hCS$_6$ theory and the application of the localisation formulae above. We shall do this in a more general manner, allowing the gauging of an $H$ subgroup that acts as

\begin{equation}\label{eq:asymaction}
g\mapsto \rho_\beta(\ell) g \rho_\alpha(\ell^{-1}) \, , \quad B \mapsto \ell B \ell^{-1} - \dr \ell \ell^{-1} \, , \quad \ell \in H \subset{G} \,,
\end{equation}

in which $\rho_i: H \rightarrow G$ are group homomorphisms (algebra homomorphisms will be denoted with the same symbol). The covariant derivative is then given by
\begin{equation}
\nabla g g^{-1} =\dr g g^{-1} + B_\beta - g B_\alpha g^{-1} \mapsto \rho_\beta(\ell) ( \nabla g g^{-1})\rho_\beta(\ell^{-1}) \, ,
\end{equation}

in which we ease notation by setting $B_i =\rho_i(B)$.

The starting point is the six-dimensional theory

\begin{equation}
S_{\mathrm{ghCS_6}}[\cA , \cB] = S_{\mathrm{hCS_6}}[\cA] - S_{\mathrm{hCS_6}}[\cB] + S_{\mathrm{bdy}}[\cA , \cB] \,,
\end{equation}

in which we specify a boundary interaction term

\begin{equation}
S_{\mathrm{bdy}}[\cA , \cB]= -\frac{q}{2\pi \rmi }\int_{\mathbb{PT} } \bar{\partial} \Omega \wedge \Tr_\mathfrak{g}\left( \cA \wedge \rho(\cB) \right) \, .
\end{equation}

Here we have introduced a parameter $q$, which will ultimately be set to one, to keep track of the contributions from this boundary term.
To specify this term we include an algebra homomorphism $\rho$ which only needs to be defined piecewise on the components of the support of $\pd \Omega$.
We could choose to dispense the higher-dimensional covariance and simply add different boundary terms specified only at the location of the poles but it is convenient to formally consider $\rho$ to be a defined as a piecewise map that takes values $\rho \vert_{\pi = \alpha , \beta } = \rho_{\alpha, \beta } $.

To define a six-dimensional theory requires imposing conditions that ensure the vanishing of the boundary term

\begin{equation}
\int_{\mathbb{PT} } \bar{\pd} \Omega \wedge \left( \Tr_{\mathfrak{g}}\left(\delta\cA \wedge (\cA - q \rho (\cB) )+q \rho(\delta \cB) \wedge \cA \right) - \Tr_{\mathfrak{h}}(\delta \cB \wedge \cB) \right) \, .
\end{equation}

We are required to cancel a term involving the inner product on the algebra $\mathfrak{h}$ with one on $\mathfrak{g}$, which can be achieved demanding

\begin{equation}
\Tr_\mathfrak{g} ( \rho(x) \rho(y) )\vert_{\alpha,\beta} = \Tr_\mathfrak{h} ( x y) \quad \forall x,y \in \mathfrak{h} \, .
\end{equation}

Note that as a consequence this implies

\begin{equation}\label{eq:anomv2}
\Tr_\mathfrak{g} ( \rho_\alpha (x) \rho_\alpha (y) ) = \Tr_\mathfrak{h} ( x y) = \Tr_\mathfrak{g} ( \rho_\beta (x) \rho_\beta (y) ) \, ,
\end{equation}

which is the anomaly-free condition allowing for the construction of a gauge-invariant extension to the WZW model for the gauge symmetry (<ref>). With this condition the boundary term produced by variation is given by

\begin{equation}
\int_{\mathbb{PT} } \bar{\pd} \Omega \wedge \left( \Tr_{\mathfrak{g}}\left(\delta\cA \wedge (\cA - q \rho (\cB) )+q \rho(\delta \cB) \wedge (\cA - q^{-1} \wedge \rho(\cB))\right) \right) \, ,
\end{equation}

and is set to zero by the conditions

\begin{equation} \label{eq:ABboundarycond2}
\cA^{\fk} \big\vert_{\alpha , \beta} = 0 \ , \qquad
\cA^{\fh} \big\vert_{\alpha , \beta} =\rho(\cB) \big\vert_{\alpha , \beta} \ , \qquad
\pd_{0} \cA^{\fh} \big\vert_{\alpha , \beta} = \rho( \pd_{0}\cB) \big\vert_{\alpha , \beta} \ .
\end{equation}

It is noteworthy that if we impose all of these conditions from the outset, the contribution from the explicit boundary term $S_{\mathrm{bdy}}[\cA , \cB]$ would vanish.
However, from a four-dimensional perspective the boundary conditions on $\pd_0(\cA)$ lead to differential constraints on the fundamental fields and it is not clear that one should, or could, naively invoke them to produce a Lagrangian description.
Instead what we shall do is only impose the conditions $\cA^{\fk} \big\vert_{\alpha , \beta} = 0$ and $\cA^{\fh} \big\vert_{\alpha , \beta} =\rho(\cB) \big\vert_{\alpha , \beta}$, which can be solved algebraically and substituted into the Lagrangian without concern. Doing this one finds that $S_{\mathrm{bdy}}[\cA , \cB]$ does contribute, and when $q=1$ in particular, it provides a gauge invariant completion of the action. The boundary conditions that we have not imposed have not been forgotten, instead when $q=1$ they are recovered as on-shell equations in this four-dimensional theory. This provides an alternative view of the procedure; the explicit boundary term is serving to implement the constraints arising from $\pd_{0} \cA^{\fh} \big\vert_{\alpha , \beta} = \rho( \pd_{0}\cB) \big\vert_{\alpha , \beta}$ at the Lagrangian level.
We can see this explicitly by observing that if we just impose $\cA^{\fk} \big\vert_{\alpha , \beta} = 0$ and $\cA^{\fh} \big\vert_{\alpha , \beta} =\rho(\cB) \big\vert_{\alpha , \beta}$ then

\begin{equation}\begin{split}
& \Big(\delta\cA \wedge (\cA - q \rho (\cB) )+q \rho(\delta \cB) \wedge \cA - \rho(\delta \cB) \wedge \rho(\cB)\Big)|_{\alpha,\beta} = 0 \,,
\\
& \partial_0 \Big(\delta\cA \wedge (\cA - q \rho (\cB) )+q \rho(\delta \cB) \wedge \cA - \rho(\delta \cB) \wedge \rho(\cB)\Big)|_{\alpha,\beta}
\\ & \qquad =  (1-q)\delta(\partial_0\cA - \rho( \partial_0\cB) )\wedge \rho(\cB) |_{\alpha,\beta}  + (1+q)\rho(\delta\cB) \wedge (\partial_0 \cA - \rho(\partial_0 \cB))|_{\alpha,\beta} \,.
\end{split}\end{equation}

Therefore, for $q=1$ we see that the boundary equation of motion for $\mathcal{B}$ is precisely $\pd_{0} \cA^{\fh} \big\vert_{\alpha , \beta} = \rho( \pd_{0}\cB) \big\vert_{\alpha , \beta}$.

The localisation proceeds as follows. First, we change parametrisation $\cA = \cA'{}^{\hat{g}}$ and $\cB =\cB'{}^{\hat{h} } $ fixing some redundancy by demanding $\cA'$ and $\cB'$ have no $\mathbb{CP}_1$ legs. Second, we fix some of the residual symmetry preserved by the boundary conditions to set $\hat{g} \vert_{\beta} = \hat{h} \vert_{\alpha,\beta} = \id $ and $ \pd_0 \hat{h } \vert_{\alpha,\beta} = 0 $. The remaining fields are $\hat{g}\vert_{\alpha} =g $, $\hat{g}^{-1} \pd_0 \hat{g}\vert_{\alpha} = u $, $\hat{g}^{-1} \pd_0 \hat{g}\vert_{\beta} = \tilde{u} $ and the four-dimensional gauge fields $A$ and $B$ that arise from $\cA'$ and $\cB'$ once their holomorphicity is imposed.

We may now directly apply the localisation formulae (<ref>) to show that the hCS terms localise (without imposing any boundary conditions) to give

\begin{equation}
\begin{split}
S_{\mathrm{hCS}}[\mathcal{A}] \simeq & \int_{\mathbb{R}^4} \omega_{\alpha,\beta} \wedge \Tr_\fg ( A^g \wedge g^{-1} \dr g ) - \omega_{\alpha,\beta}\wedge \mathcal{L}_{\textrm{WZ}}[g] \\
& \qquad + \frac{1 }{2} \mu_\alpha\wedge \Tr_\fg ( A^g \wedge \dr u) + \frac{1 }{2} \mu_\beta \wedge \Tr_\fg ( A \wedge \dr \tilde{u}) \,,
\end{split}
\end{equation}

while $ S_{\mathrm{hCS}}[\cB]$ yields zero in this gauge. Let us first consider the terms involving $\omega_{\alpha,\beta}$. Since the gauge completion of the WZ term is

\begin{equation}
\label{eq:generalgwzw1}
\mathcal{L}_{\textrm{gWZ}}[g,B] = \mathcal{L}_{\textrm{WZ}}[g] + \Tr_\mathfrak{g} \left( g^{-1} \dr g \wedge
B_\alpha + \dr g g^{-1} \wedge B_\beta + g^{-1} B_\beta g B_\alpha \right) \,,
\end{equation}

we may express them (trace left implicit) as

\begin{equation}\begin{split}
& \omega_{\alpha,\beta} \wedge \left( A^g \wedge g^{-1} \dr g - \mathcal{L}_{\textrm{WZ}}[g] \right) \\
= \,& \omega_{\alpha,\beta} \wedge \left( A^g \wedge g^{-1} \dr g - \mathcal{L}_{\textrm{gWZ}}[g,B] + g^{-1} \dr g \wedge B_\alpha + \dr g g^{-1} B_\beta +g^{-1} B_\beta g B_\alpha \right) \\
= \, & \omega_{\alpha,\beta} \wedge \left( (A^g- B_\alpha) \wedge g^{-1} \nabla g - \mathcal{L}_{\textrm{gWZ}}[g,B] + A^g
\wedge B_\alpha - A \wedge B_\beta \right) \, .
\end{split}\end{equation}

To proceed we invoke the algebraic boundary conditions of eq. (<ref>), which in differential form notation become

\begin{align}\label{eq:BCsol}
A = B_\beta - \bar{P}( \nabla g g^{-1} ) \Leftrightarrow A^g= P(g^{-1}\nabla g) + B_\alpha \, .
\end{align}

such that

\begin{equation}\begin{split}
& \omega_{\alpha,\beta} \wedge \left( A^g \wedge g^{-1} \dr g -\mathcal{L}_{\textrm{WZ}}[g] \right) \\
=\, & \omega_{\alpha,\beta} \wedge \left( P( g^{-1} \nabla g) \wedge g^{-1} \nabla g - \mathcal{L}_{\textrm{gWZ}}[g,B] + A^g
\wedge B_\alpha - A \wedge B_\beta \right) \\
=\, & -\frac{1}{2} g^{-1} \nabla g \wedge\star( g^{-1} \nabla g) -\omega_{\alpha,\beta} \wedge \left( \mathcal{L}_{\textrm{gWZ}}[g,B] - A^g
\wedge B_\alpha + A \wedge B_\beta \right)
\, .
\end{split}\end{equation}

Here in the last line we made use of the identity $
\omega \wedge P(\sigma)\wedge \sigma = - \frac{1}{2}\sigma\wedge \star \sigma $ for a 1-form $\sigma$. To treat the terms involving $\mu_{\alpha}$ and $\mu_\beta$ we may combine the algebraic boundary conditions with the properties $\mu_\alpha \wedge P(X) = \mu_\beta \wedge \bar{P}(X) = 0 $ such that $\mu_a\wedge A^g= \mu_\alpha B_\alpha$ and $\mu_\beta \wedge A= \mu_\beta B_\beta$. In summary we find

\begin{equation}
\begin{split}
S_{\mathrm{hCS}}[\mathcal{A}] &\simeq \int_{\mathbb{R}^4} -\frac{1}{2} \Tr_\fg\left(g^{-1}\nabla g \wedge \star g^{-1}\nabla g \right) - \omega_{\alpha,\beta}\wedge\left(\mathcal{L}_{\textrm{gWZ}}[g,B] + \Tr_\fg( A \wedge B_\beta - A^g B_\alpha ) \right) \\
& \qquad + \frac{1}{2} \mu_\alpha\wedge \Tr( B_\alpha \wedge \dr u) + \frac{1 }{2} \mu_\beta \wedge \Tr( B_\beta \wedge \dr \tilde{u}) \,.
\end{split}
\end{equation}

The localisation of the explicit boundary term yields, after using $\mu_a\wedge A^g= \mu_\alpha B_\alpha$,

\begin{equation}
\begin{split}
S_{\mathrm{bdy}}[\cA , \cB] \simeq & - q \int_{\mathbb{R}^4}\omega_{\alpha,\beta} \wedge \Tr_\fg ( A^g B_\alpha - A B_\beta )\\
& \qquad + \frac{1}{2} \mu_\alpha\wedge \Tr_\fg ( (\dr u + [B_\alpha , u]) B_\alpha ) + \frac{1 }{2} \mu_\beta\wedge \Tr_\fg ( (\dr \tilde{u} + [B_\beta , \tilde{u}]) B_\beta ) \, .
\end{split}
\end{equation}

The significance of the boundary term now becomes clear, as it serves to ensure manifest gauge invariance. When $q=1$ the terms $\omega_{\alpha,\beta} \wedge \Tr( A^g B_\alpha - A B_\beta )$ directly cancel. The contributions of the entire localised action that are wedged against $\mu_\alpha$ sum to

\begin{equation}
\mu_\alpha \wedge \Tr_\fg \left( (1- q) ~\dr u\wedge B_\alpha + 2 q ~ u F[B]_\alpha -2q ~\dr ( B_\alpha u) \right) \, .
\end{equation}

We can see that for $q=1$ we find a gauge-invariant field strength together with a total derivative term that we discard.
The terms wedged against $\mu_\beta$ give a similar contribution.
Hence the fully localised action becomes

\begin{equation}
\begin{split}
S \simeq & \int_{\mathbb{R}^4}
-\frac{1}{2} \Tr_\fg\left(g^{-1}\nabla g \wedge \star g^{-1}\nabla g \right) - \omega_{\alpha,\beta}\wedge \mathcal{L}_{\textrm{gWZ}}[g,B] \\
& \qquad + \mu_\alpha\wedge \Tr_\fg( u F[B]_\alpha ) + \mu_\beta \wedge \Tr_\fg( \tilde{u} F[B]_\beta) \, .
\end{split}
\end{equation}

Noting that the components of $u$ and $\tilde{u}$ in complement of $\fh$ decouple we can view $u$ and $\tilde{u}$ as $\fh$-valued and write

\begin{equation}
\begin{split}
S \simeq & \int_{\mathbb{R}^4}
- \frac{1}{2} \Tr_\fg\left(g^{-1}\nabla g \wedge \star g^{-1}\nabla g \right) - \omega_{\alpha,\beta}\wedge \mathcal{L}_{\textrm{gWZ}}[g,B] \\
& \qquad + \mu_\alpha\wedge \Tr_\fh( u F[B] ) + \mu_\beta \wedge \Tr_\fh( \tilde{u} F[B]) \, .
\end{split}
\end{equation}

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titleFour-dimensional avatars of two-dimensional RCFT

titleIntegrable $\mathcal{E}$-Models, 4d Chern-Simons Theory and Affine Gaudin Models. I. Lagrangian Aspects

journaltitleAnnales Henri Poincare
titleIntegrable Degenerate $\mathcal {E}$-Models from 4d Chern–Simons Theory

journaltitlePhys. Lett. B
titleA Kahler-Chern-Simons Theory and Quantization of Instanton Moduli Spaces

journaltitleNucl. Phys. B
titleKahler Chern-Simons theory and symmetries of antiselfdual gauge fields

journaltitlePhys. Lett. B
titleA Cubic action for selfdual Yang-Mills

journaltitlePhys. Lett. B
titleDeformed coset models from gauged WZW actions

titleTwistor Actions for Integrable Systems

journaltitleJ. Math. Phys.
titleTwistor algebra

journaltitlePhys. Lett. B
titleTheory of Nonabelian Goldstone Bosons

journaltitleNucl. Phys. B
titleIntegrable interpolations: From exact CFTs to non-Abelian T-duals

titleFour-Dimensional Chern-Simons and Gauged Sigma Models

title5d 2-Chern-Simons theory and 3d integrable field theories

journaltitleLett. Math. Phys.
title4D Chern–Simons theory and affine Gaudin models

journaltitlePhil. Trans. Roy. Soc. Lond. A
titleIntegrable and solvable systems, and relations among them

journaltitleNucl. Phys. B
titleGauge Theories and Integrable Lattice Models

journaltitleCommun. Math. Phys.
titleOn Holomorphic factorization of WZW and coset models

titleNew T-duality for Chern-Simons Theory

journaltitleSov. Phys. JETP
titleRelativistically Invariant Two-Dimensional Models in Field Theory Integrable by the Inverse Problem Technique. (In Russian)