Besides the coupling between gravity and the matter fields, in the standard model there are also interactions between the matter field themselves. Since multiple instances of the same interaction occur, it is useful to spell out the details of the interaction between possible couples of types of fields, keeping the discussion as general as possible. Later on, in Sect. 5, we will combine all the results contained in this section and specify the type of scalar, Yang–Mills and spinor fields we are working on. The possible interactions are the following:

1.

A Yang–Mills field and a spinor field;

2.

A Yang–Mills field and a scalar field;

3.

A scalar field and a spinor field.

We will treat these three interactions respectively in Sects. 4.1, 4.2 and 4.3. For each of the aforementioned interaction we describe the reduced phase space using the KT construction, following the scheme already used for the single matter fields coupled to gravity. Namely, starting from the classical action on the bulk, we derive the EL equations and the pre-symplectic form on the boundary fields. Then, if necessary we perform the reduction and find the geometric phase space. Subsequently we find the functions describing the constraints and we check if they form a coisotropic submanifold. Then the reduced phase space of the theory is found to be the quotient of the geometric phase space with respect to the coisotropic submanifold defined by the constraints.

In order to keep the construction simple, we will make use of the notation introduced in Sect. 1.2. In particular all the relevant quantities will be presented as sums (or products) of the quantities introduced above plus an interaction or correction term.

4.1 Yang–Mills Spinor

We consider in this section the interaction of a Yang–Mills field and a spinor field together with gravity. We denote the spinor field by \(\psi \in S(M)\otimes \mathfrak (N)\) and the Yang–Mills field by A. The action on the bulk is given by the sum of the gravity part, the Yang–Mills part (23), the spinor part (32) and an interaction part:

$$\begin S_ = S+ S_ + S_A + S_ \end$$

where

$$\begin S_= \int _M&\frac}\left( \overline \gamma [A, \psi ] - [A, \overline] \gamma \psi \right) . \end$$

where \(\overline \gamma [A, \psi ]= i g_i \overline_ \gamma A^_J \psi ^J\) and \(g_i\) is a coupling constant. The interaction term does not contain derivatives, and hence, the boundary structure is just the direct sum of the YM structure and Spinor structures. In particular the geometric phase space is given by

$$\begin ^_ \rightarrow \Omega _^1(\Sigma , \mathcal _)\oplus \mathcal ^}_ \times S(\Sigma )\times \overline(\Sigma ) \end$$

with fiber \(\mathcal _(\Sigma ) \oplus \Omega ^_(\mathfrak )\) such that (24) and (33) are satisfied. The symplectic form on this space reads

$$\begin \Omega _= \varpi + \varpi _A + \varpi _\psi . \end$$

On this geometric phase space we can then define the following constraints:

$$\begin L^_c&= L_c + l^_; \\ P^_&= P_+ p^_ + p^_+p^_; \\ H^_&= H_+ h^_ + h^_+ h^_;\\ M^_\mu&= M^_\mu + m^_\mu \end$$

where

$$\begin p^_\xi&=\int _- i \frac \left( \overline \gamma [\iota _,\psi ] - [\iota _\xi ,\overline]\gamma \psi \right) \end$$

(38)

$$\begin h^_\lambda&= \int _- \lambda e_n\left[ i\frac\left( \overline\gamma [A, \psi ] - [A,\overline]\gamma \psi \right) \right] , \end$$

(39)

$$\begin m^_\mu&= \int _ - i \frac \left( [\mu ,\overline]\gamma \psi - \overline \gamma [\mu ,\psi ] \right) . \end$$

(40)

Remark 13

Note that for every field \(\mu \) with values in \(\mathfrak \) the following identities hold:

$$\begin [\mu ,\overline\gamma \psi ]&=0 \\ [\mu ,\overline]\gamma \psi - \overline \gamma [\mu ,\psi ]&= 2 [\mu ,\overline]\gamma \psi \end$$

The following theorem proves that these constraints form a coisotropic submanifold.

Theorem 14

Assume that \(g^\partial \) is non-degenerate on \(\Sigma \). Then, the zero locus of the functions \(L^_c\), \(P^_\), \(H^_\) and \(M^_\mu \) is a coisotropic submanifold with respect to the symplectic structure \(\Omega _\). Their mutual Poisson brackets read

$$\begin&\_\mu ,M^_\mu \}_=-\fracM^_&\_\mu ,L^_c\}_&=0\\&\_\mu ,P^_\xi \}_=M^__\xi ^\mu }&\_\mu ,H^_\lambda \}_&= 0\\&\_, P^_\}_ = \fracP^_- \fracL^_\iota _F_}-\fracM^_\iota _F_}&\_c, P^_\}_&= L^__^c}\\&\_c, H^_\}_ \\&\quad = - P^_} + L^_(\omega - \omega _0)_} -H^_} + M^_(A-A_0)_} &\_c, L^_c\}_&= - \fracL^_\\&\_,H^_\}_ = P^_} -L^_ (\omega - \omega _0)_} \\&\quad + H^_} - M^_(A-A_0)_}&\_\lambda ,H^_\lambda \}_&=0 \end$$

with the same notation as in Theorem 7.

Proof

We prove each bracket by using Theorem 2. The first step is to compute the Hamiltonian vector fields of the constraints. Using the notation and the results of Sect. 2, the expressions of \(\mathbb ,\mathbb ^, \mathbb ^, \mathbb \mathbb ^, \mathbb ^, \mathbb ,\mathbb ^, \mathbb ^\) and \( \mathbb ^\) have been computed in [8] and are collected in Appendix B. Hence the only components that we have to compute through (2) are \(\mathbb ^\), \(\mathbb ^\), \(\mathbb ^\) and \(\mathbb ^\). Let us start from \(\mathbb ^\). It must satisfy

$$\begin \iota _^} (\varpi + \varpi _+ \varpi _) + \iota _^} \varpi _+ \iota _^} \varpi _ = 0. \end$$

Since \(\iota _^} \varpi _= \iota _^} \varpi _ = 0\) we conclude \(\mathbb ^=0\). Similarly we have

$$\begin \iota _^} (\varpi + \varpi _+ \varpi _) + \iota _^} \varpi _+ \iota _^} \varpi _ = \delta p^_. \end$$

Since \(\iota _^} \varpi _= \iota _^} \varpi _=0\), the computation is exactly the same as for \(p^_\xi \) with \(A_0\) instead of \(\omega _0\). Hence we get

$$\begin \mathbb ^_e&= 0&\mathbb ^_\omega&= \mathbb _} \\ \mathbb ^_A&= 0&\mathbb ^_\rho&= 0 \\ \mathbb ^_&= - [\iota _,\psi ]&\mathbb ^_}&= - [\iota _,}]. \end$$

For \(\mathbb ^\) we need some more work. We have \(\iota _^} \varpi _= \iota _^} \varpi _=0\) and

$$\begin \delta h^_\lambda&= \int _- \lambda e_n ie \delta e\left( \overline\gamma [A, \psi ]\right) \\&\quad - \lambda e_n\left[ i\frac\left( \delta \overline\gamma [A, \psi ] +\overline\gamma [\delta A, \psi ]-\overline\gamma [A, \delta \psi ] \right) \right] . \end$$

Hence we get:

$$\begin \mathbb ^_e&= 0&\mathbb ^_\omega&= - \frac\lambda e_n \overline\gamma [A, \psi ]+ \mathbb _}\\ \mathbb ^_A&= 0&(\mathbb ^_\rho )^J_I&=- \fracg_i \lambda e_n e^2 \overline_I\gamma \psi ^J \\ \frac\gamma \mathbb ^_&= \frac\gamma [A, \psi ]&\frac\mathbb ^_}\gamma&= \frac [A, \overline] \gamma . \end$$

As for \(p^_\), the Hamiltonian vector field of \(m^_\) can be obtained by noticing that it is equal to that of \(l^_\) by substituting c with \(\mu \). The result is

$$\begin \mathbb ^_e&= 0&\mathbb ^_\omega&= \mathbb _} \\ \mathbb ^_A&= 0&\mathbb ^_\rho&= 0 \\ \mathbb ^_&= [\mu , \psi ]&\mathbb ^_}&= [\mu , }]. \end$$

We can now compute the constraints using Theorem 2. Before beginning the actual computation we note that for all constraints

$$\begin \iota _+\mathbb ^}\iota _+\mathbb ^}\varpi _+ \iota _+\mathbb ^}\iota _+\mathbb ^}\varpi _=0,\\ \iota _}\iota _}\varpi _=0, \\ \iota _}\iota _}\varpi _=0. \end$$

Furthermore it is also possible to note that

$$\begin \iota _^}\iota _^}\Omega _=0\\ \iota _^}\iota _^}\Omega _=0 \end$$

for all brackets except \(\_\lambda ,H^_\lambda \}\)

$$\begin \iota _^}\iota _^}\Omega _=\iota _^}\iota _^}\varpi = \int _ e \mathbb _e^ \mathbb _\omega ^=0 \end$$

since \(\mathbb _e^ \sim \lambda \), \(\mathbb _\omega ^ \sim \lambda \) and \(\lambda ^2=0\). Hence we conclude that we have

$$\begin \,Y^\}_&=\, Y+y^\}_A+\, Y+y^\}_\psi -\\\&\quad + \iota _^}\delta (X+ x^+x^+x^) + \iota _^}\delta (Y+ y^+y^+y^)\\&\quad - \iota _^}\iota _^}\Omega _. \end$$

Using this formula we can compute the brackets, omitting the terms that are zero.

$$\begin&\^,M_^\}_=\^, M_^\}_A 2\iota _^}\delta ( M_^+m_^) \iota _^}\iota _^}\Omega _ \\&\quad = \frac M_^ + 2\int _ i \frac \left( [\mu ,[\mu ,\overline]]\gamma \psi - \overline \gamma [\mu ,[\mu ,\psi ]] +[\mu ,\overline]\gamma [\mu ,\psi ]\right) \\&\qquad +2\int _ i \frac [\mu ,\overline] \gamma [\mu ,\psi ]-2\int _ i \frac [\mu ,\overline] \gamma [\mu ,\psi ]\\&\quad = \frac M_^ + \frac m_^= \fracM_^. \end$$

Since \(\mathbb ^=0\), we get

$$\begin&\^,L_c^\}_=\^, L_c\}_A + \iota _^}\delta (L_c+l_c^)\\&\quad = \int _ - i \frac \left( -[c,[\mu ,\overline]]\gamma \psi -[c,\overline]\gamma [\mu ,\psi ] - [\mu , \overline] \gamma [c,\psi ] + \overline \gamma [c, [\mu , \psi ]]\right) =0. \end$$

where we used that \([c,[\mu ,\overline]]=[\mu ,[c,\overline]]\) and the properties in Remark 13.

$$\begin&\^,P_^\}_=\^, P_+p_^\}_A+ \iota _^}\delta (M_^+m_^)\\&\qquad + \iota _^}\delta (P_+ p_^+p_^+p_^) - \iota _^}\iota _^}\Omega _\\&\quad = M^__\xi ^\mu }- \int _ i \frac \left( [\mu ,\overline]\gamma [\iota _ , \psi ] + [\iota _, \overline] \gamma [\mu ,\psi ] \right) \\&\qquad -\int _ i \frac \left( [\mu ,\overline] \gamma \textrm_^(\psi ) -\overline \gamma \textrm_^([\mu ,\psi ])+ \textrm_\xi ^([\mu ,\overline])\gamma \psi \right) \\&\qquad + \int _ i \frac \left( \textrm_\xi ^(\overline)\gamma [\mu ,\psi ] +2\left( [\mu ,\overline]\gamma [\iota _ , \psi ] + [\iota _, \overline ]\gamma [\mu ,\psi ] \right) \right) \\&\quad = M^__\xi ^\mu } -\int _ i \frac \left( -\overline \gamma [\textrm_^\mu ,\psi ]+ [\textrm_\xi ^\mu ,\overline]\gamma \psi \right) \\&\quad = M^__\xi ^\mu } -\int _ i \frac \left( -\overline \gamma [\textrm_^\mu ,\psi ]+ [\textrm_\xi ^\mu ,\overline]\gamma \psi \right) \\&\quad = M^__\xi ^\mu }+ m^__\xi ^\mu }= M^__\xi ^\mu }. \end$$

where we used \(\textrm_^\mu =\textrm_^\mu \) and \([\mu ,\overline] \gamma \textrm_^(\psi )=-\overline \gamma [\mu ,\textrm_^(\psi )]\). Similarly we get

$$\begin&\^,H_^\}_\\&\quad =\^, H^A_\}_A + \iota _^}\delta (M_^+m_^) + \iota _^}\delta (H_+ h_^+h_^+h_^)\\&\qquad - \iota _^}\iota _^}\Omega _\\&\quad = -\int _\frac \overline\gamma [d_A \mu , \psi ] +\int _\frac \left( [\mu ,\overline]\gamma [A, \psi ] - [A,\overline]\gamma [\mu , \psi ] \right) \\&\qquad +\int _\frac \left( [\mu ,\overline]\gamma d_ \psi - \overline\gamma d_([\mu , \psi ]) + d_([\mu , \overline])\gamma \psi +d_( \overline)\gamma [\mu , \psi ] \right) \\&\qquad +\int _\frac \left( [\mu ,\overline]\gamma [A, \psi ]- \overline\gamma [A,[\mu , \psi ]] + [A,[\mu , \overline]]\gamma \psi +[A,\overline]\gamma [\mu , \psi ] \right) \\&\qquad -\int _\frac \left( [\mu ,\overline]\gamma [A, \psi ] - [A,\overline]\gamma [\mu , \psi ] \right) \\&\quad = -\int _\frac \overline\gamma [d_A \mu , \psi ]+\int _\frac \left( [\mu ,\overline]\gamma d_ \psi - \overline\gamma d_([\mu , \psi ])\right) \\&\qquad +\int _\frac \left( d_([\mu , \overline])\gamma \psi +d_( \overline)\gamma [\mu , \psi ] \right) \\&\quad = -\int _\frac \overline\gamma [d_A \mu , \psi ]+\int _\frac \overline\gamma [d_ \mu , \psi ]=0 \end$$

where we used \(d_ \mu =d_ \mu .\) Using again that \(\mathbb ^=0\) we get

$$\begin \,L_c^\}_&=\_A+\, L_c+l_c^\}_\psi -\\\&= \frac(L_+l_^)= \fracL^_. \end$$

Similarly,

$$\begin&\,P_^\}_\\&\quad =\+p_^\}_A+\, P_+p_^\}_\psi -\\}+ \iota _^}\delta (L_c+ l_c^)\\&\quad = L^A__^c}+ L^__^c}- L__^c}+ \int _ i \frac \left( [c,[\iota _A_0,\overline]]\gamma \psi + [c,\overline]\gamma [\iota _A_0,\psi ]\right) \\&\qquad + \int _ i \frac \left( - [\iota _A_0,\overline] \gamma [c,\psi ] - \overline \gamma [c,[\iota _A_0,\psi ]]\right) \\&\quad = L^__^c} + \int _ i \frac \left( -[\iota _A_0,[c,\overline]]\gamma \psi + [c,\overline]\gamma [\iota _A_0,\psi ]\right) \\&\qquad + \int _ i \frac \left( - [\iota _A_0,\overline] \gamma [c,\psi ] + \overline \gamma [\iota _A_0,[c,\psi ]]\right) \\&\quad = L^__^c} \end$$

where we used \([c,[\iota _A_0,\overline]] \) \(=-[\iota _A_0,[c,\overline]]\) and \([\iota _A_0,[c,\overline]]\gamma \psi = [c,\overline]\gamma [\iota _A_0,\psi ]\).

$$\begin&\,H_^\}_\\&\quad =\+h_^\}_A+\, H_+h_^\}_\psi \\&\qquad -\\}+ \iota _^}\delta (L_c+ l_c^)\\&\quad = - P^_} + L_(\omega - \omega _0)_} -H^_} + M^_(A-A_0)_}\\&\qquad - P^_} + L^_(\omega - \omega _0)_} - H^_}- P_} + L_(\omega - \omega _0)_} \\&\qquad - H_} - \int _ \frac [\lambda e_n \overline\gamma [A, \psi ], e^2] \\&\qquad - \int _ \frac\left( -[c, \lambda e_n e^2 [A,\overline]]\gamma \psi + [c, \overline]\gamma \lambda e_n e^2 [A,\psi ]\right) \\&\qquad +\int _ \frac\left( - \lambda e_n e^2 [A,\overline] \gamma [c, \psi ] + \overline\gamma [c, \lambda e_n e^2 [A,\psi ]] \right) \\&\quad = - P^_} + L_(\omega - \omega _0)_} -H^_} + M^_(A-A_0)_} \\&\qquad - P^_} + L^_(\omega - \omega _0)_} - H^_}- P_} + L_(\omega - \omega _0)_} - H_}\\&\qquad - \int _[c, \lambda e_n]\left[ i\frac\left( \overline\gamma [A, \psi ] - [A,\overline]\gamma \psi \right) \right] \\&\quad = - P^_} + L_(\omega - \omega _0)_} -H^_} + M^_(A-A_0)_}\\&\qquad - P^_} + L^_(\omega - \omega _0)_} - H^_}- P_} + L_(\omega - \omega _0)_}\\&\quad \quad - H_} - p^_} - h^_} + m^_(A-A_0)_}\\&\quad = - P^_} + L^_(\omega - \omega _0)_} -H^_} + M^_(A-A_0)_} \end$$

where in the second-last passage we used that \([c,\lambda e_n]=X= X^e_\nu + X^e_n\) and that

$$\begin - p^_} + m^_(A-A_0)_}=- \int _[c, \lambda e_n]^e_\nu \left[ i\frac\left( \overline\gamma [A, \psi ] - [A,\overline]\gamma \psi \right) \right] \end$$

Let us now consider

$$\begin \^,P_^\}_&=\+p_^, P_+p_^\}_A+\+p_^, P_+p_^\}_\psi -\,P_\}\\&\quad + 2\iota _^}\delta (P_+ p_^+p_^+p_^) - \iota _^}\iota _^}\Omega _. \end$$

We have

$$\begin&\+p_^, P_+p_^\}_A+\+p_^, P_+p_^\}_\psi -\,P_\}\\&\quad =\frac\left( P^_- L_\iota _F_}-M^_\iota _F_} + P^_- L^_\iota _F_}-P_+ L_\iota _F_}\right) \end$$

and

$$\begin&2\iota _^}\delta (P_+ p_^+p_^+p_^) - \iota _^}\iota _^}\Omega _\\&\quad = -\int _ i \frac \left( [\iota _A_0,\overline] \gamma \textrm_^(\psi )+\overline \gamma \textrm_^([\iota _A_0,\psi ])\right) \\&\qquad -\int _ i \frac \left( - \textrm_\xi ^(\overline)\gamma [\iota _A_0,\psi ]- \textrm_\xi ^([\iota _A_0,\overline])\gamma \psi \right) \\&\quad =-\int _ i \frac \left( -\overline \gamma [\iota _A_0,\textrm_^(\psi )]+\overline \gamma \textrm_^([\iota _A_0,\psi ])\right) \\&\qquad -\int _ i \frac \left( - [\iota _A_0,\textrm_\xi ^(\overline)]\gamma \psi - \textrm_\xi ^([\iota _A_0,\overline])\gamma \psi \right) \\&\quad =\int _- i \frac \left( -\overline \gamma [\textrm_\xi ^(\iota _A_0),\psi ]- [\textrm_\xi ^(\iota _A_0),\overline]\gamma \psi \right) \\&\quad = \int _ i \frac \left( \overline \gamma \left[ \iota _A_0+\iota _\iota _F_,\psi \right] + \left[ \iota _A_0+\iota _\iota _F_,\overline\right] \gamma \psi \right) \\&\quad = \fracp^_- m^_\iota _F_} \end$$

where we used that \(\textrm_\xi ^\iota _A_0=\textrm_\xi ^\iota _A_0= \frac\iota _A_0+\frac\iota _\iota _F_.\) Hence we get

$$\begin&\^,P_^\}_ = \frac\left( P^_- L^_\iota _F_}-M^_\iota _F_}\right) .\\&\^,H_^\}_ =\+p_^, H_+h_^\}_A+\+p_^, H_+h_^\}_\psi -\,H_\}\\&\qquad + \iota _^}\delta (P_+ p_^+p_^+p_^) + \iota _^}\delta (H_+ h_^+h_^+h_^)\\&\qquad - \iota _^}\iota _^}\Omega _\\&\quad = P^_} -L_ (\omega - \omega _0)_} + H^_} - M^_(A-A_0)_} + p^_}\\&\qquad -l^\psi _ (\omega - \omega _0)_} + h^_}- \int _ \frac_\xi ^e^2 }\left( \overline\gamma [A,\psi ]-[A,\overline])\gamma \psi \right) \\&\qquad - \frac\left( \overline\gamma [\iota _F_+\textrm_\xi ^(A-A_0), \psi ]\right. \\&\qquad \left. - [\iota _F_+\textrm_\xi ^(A-A_0),\overline]\gamma \psi \right) \\&\qquad -\int _ \frac\left( [A,\overline]\gamma \textrm_\xi ^\psi - \textrm_\xi ^\overline\gamma [A,\psi ] \right) \\&\qquad - \int _\frac\left( [\iota _\xi ,\overline ]\gamma d_\psi -\overline \gamma d_[\iota _\xi ,\psi ]+d_[\iota _\xi ,\overline ]\gamma \psi \right) \\&\qquad - \int _\frac\left( d_\overline \gamma [\iota _\xi ,\psi ] +\quad 2[A,\overline]\gamma [\iota _\xi A_0,\psi ] + 2[\iota _\xi A_0 ,\overline]\gamma [A,\psi ] \right) \end$$

Hence we get

$$\begin \^,H_^\}_&= P^_} -L_ (\omega - \omega _0)_} + H^_} - M^_(A-A_0)_}\\&\quad + p^_} -l^\psi _ (\omega - \omega _0)_}+ h^_}\\&\quad - \int _ \frac_\xi ^(\lambda e_n) e^2 }\left( \overline\gamma [A,\psi ]-[A,\overline])\gamma \psi \right) \\&= P^_} -L_ (\omega - \omega _0)_} + H^_} - M^_(A-A_0)_}\\&\quad + p^_} -l^\psi _ (\omega - \omega _0)_}\\&\quad + h^_}+ p^_} + h^_} - m^_(A-A_0)_}\\&= P^_} -L^_ (\omega - \omega _0)_} + H^_} - M^_(A-A_0)_}\\ \^,H_^\}_&=\+h_^, H_+h_^\}_A+\+h_^, H_+h_^\}_\psi -\,H_\}\\&\quad + \iota _^}\delta (H_+ h_^+h_^+h_^) + \iota _^}\delta (H_+ h_^+h_^+h_^)\\&\quad - \iota _^}\iota _^}\Omega _ =0 \end$$

because all the components of the Hamiltonian vector field of \(H_^\) are proportional to \(\lambda \) and \(\lambda ^2=0\). \(\square \)

4.2 Yang–Mills–Higgs

We consider the case of an interacting scalar field and a Yang–Mills field both coupled to gravity, with the addition of a Higgs-type potential.

First of all, let \(P_\) be a SU(n)-principal bundle over M, with the fundamental representation \(n:SU(n)\rightarrow \textrm(\mathbb ^n)\) and its conjugate one \(\bar\) with respect to the canonical Hermitian structure on \(\mathbb ^n\).

We define the Higgs field \(\phi \) (a scalar multiplet) to be a section of the associated bundle \(E_n:=P_ \times _n \mathbb ^n\), while \(\phi ^\dagger \) is a section of \(E_}:=P_ \times _} \mathbb ^n\). Working in the first order formalism, we also introduce the associated momentum \(\Pi \in \Gamma (M,\mathcal \otimes E_n)=:\Omega ^(E_n)\) and its conjugate \(\Pi ^\dagger \in \Omega ^( E_})\).

Remark 15

In the remainder of this section, we will identify (sections of) the Lie algebra \( \mathfrak (n)\) with (sections of) the algebra of Hermitian traceless matrices over \(\mathbb ^n\), i.e.,

$$\begin \Gamma (M,\mathfrak (n))\simeq \Gamma (M,(E_n\otimes E_})_})=:\Gamma (M,(E_n\otimes E_})^). \end$$

Furthermore, we will consider \(\phi \) and \(\Pi \) to be such that the total degrees are \(|\phi |=0\) and \(|\Pi |=1\).

Remark 16

The canonical Hermitian product on \(\mathbb ^n\) induces a Hermitian product on \(E_n\) (and hence on \(\Gamma (E_n)\)). We symmetrize it (i.e., add its complex conjugate) to account for the reality requirement

$$\begin<\cdot ,\cdot>:E_n\times E_n&\longrightarrow \mathbb \\ (\hspace\phi \hspace,\hspace\varphi \hspace)&\longmapsto \hspace <\phi ,\varphi >:=\frac(\phi ^\dagger \varphi + (-1)^ \varphi ^\dagger \phi ). \end$$

Furthermore, we denote the full interior product on \(E_n\otimes \mathcal \) by

$$\begin (<\cdot ,\cdot>):(E_n\otimes \mathcal )^2&\longrightarrow \mathbb \\ (\Pi ,\epsilon )&\longmapsto \hspace (<\Pi ,\epsilon >):=\frac\eta _(\Pi ^_i \epsilon ^ + \text ). \end$$

The last ingredient we need to write the YMH action is the covariant derivative. Letting \(\alpha \in \Gamma (M,(E_n\otimes E_})^)\), we set \(d_\alpha \phi :=d\phi + [A,\phi ]\) and \(d_\alpha \phi ^\dagger :=d\phi ^\dagger + [\alpha ,\phi ^\dagger ]\), while in coordinates we haveFootnote 9

$$\begin [\alpha ,\phi ]^i&:=i g_H (\alpha \phi )^i = i g \alpha ^i_j \phi ^j ;\\ [\alpha ,\phi ^\dagger ]_i&=-(-1)^ i g_H \phi ^\dagger _j \alpha ^j_i, \end$$

where \(g_H\) is a coupling constant related to the representation of SU(n).

With those definitions having been established, denoting the YM field and its conjugate momentum by A and B as before, the desired action in dimension N is given by \(S_=S+S_+S_H\) where \(S_A\) is defined in (23) and

$$\begin S_H=\int _M&\frac<\Pi ,d_A\phi> + \frac (<\Pi ,\Pi>)-\frace^4(<\phi ,\phi >-v^2)^2, \end$$

where q is another coupling constant and v represents the Higgs vacuum.

Remark 17

Notice that the first terms of \(S_H\) are formally equivalent to \(S_\), after substituting \(d\phi \rightarrow d_A\phi \) and generalizing \(\phi \) to a SU(n) multiplet. Then one can easily show that in this case the structural constraint reads

$$\begin (e,\Pi )+d_A\phi =0. \end$$

(41)

(The structural constraint for the B field remains unaltered.) As a consequence the interaction between the YM field and the Higgs field is contained in \(S_H\).

We obtain the geometric phase space as the bundle

$$\begin ^\partial _\rightarrow \Omega _^1(\Sigma , \mathcal _)\oplus \mathcal _\Sigma ^\oplus \Gamma (\Sigma ,E_n\vert _\Sigma )\times \Gamma (\Sigma ,E_}\vert _\Sigma ), \end$$

with fiber

$$\begin \mathcal _(\Sigma )\oplus \Omega ^_(\mathfrak )\oplus \Omega ^_\partial (E_\vert _\Sigma )\times \Omega ^_\partial (E_}\vert _\Sigma ). \end$$

where \(\omega ,B,\Pi ,\Pi ^\dagger \) satisfy (8), (24) and (41). \(^\partial _\) is symplectic with symplectic form

$$\begin \Omega _=\varpi + \varpi _+\varpi _H \end$$

where

$$\begin \varpi _H = \int _\Sigma <\delta p,\delta \phi > \end$$

(42)

having defined \(p:=\frac\Pi \) to get rid of the unphysical components of \(\Pi \), as we did in the case of the free scalar field.

Before moving on to the constraint analysis, we provide some useful identities. Let \(\phi ,\varphi \in \Gamma (E_n)\), \(\alpha \in \Gamma (\mathfrak (n))\), then:

$$\begin<\alpha \varphi , \phi>&= (-1)^<\varphi , \alpha \phi > \end$$

(43)

$$\begin <\varphi , [\alpha ,\phi ]>&=\frac\text [(-1)^ \varphi \phi ^\dagger - (-1)^\phi \varphi ^\dagger )\alpha ]\end$$

(44)

$$\begin \phi&=L_\xi ^ d_A\phi + d_A L_\xi ^ \phi = [\iota _\xi F_ + L_\xi ^(A-A_0),\phi ] \end$$

(45)

Let us now consider the constraints. The coupling of the scalar field to the YM field produces the expected constraints (i.e., free gravity + free YM + free scalar) plus the expected interaction terms (the one arising from the minimal coupling via the covariant derivative \(d_A\)). Indeed we have

$$\begin&m_\mu ^=\int _\Sigma \frac g_H \text [\mu (\phi p^\dagger - p \phi ^\dagger )]=\int _\Sigma <p,[\mu ,\phi ]> ; \end$$

(46)

$$\begin&p_\xi ^H =\int _\Sigma -<p,\textrm_\xi ^\phi >; \end$$

(47)

$$\begin&p_\xi ^=\int _\Sigma -<p, \iota _\xi [A_0,\phi ]> \end$$

(48)

$$\begin&h_\lambda ^H=\int _\Sigma \lambda e_n \big [ \frac<\Pi ,d\phi> + \frac(<\Pi ,\Pi>) - \frace^3(<\phi ,\phi >-v^2)^2 \big ]; \end$$

(49)

$$\begin&h_\lambda ^=\int _\Sigma \lambda e_n \frac <\Pi ,[A,\phi ]>. \end$$

(50)

Remark 18

Notice that \(h_\lambda ^H=h_\lambda ^\phi + h_\lambda ^\), where \(h_\lambda ^\) is the term containing the Higgs potential term \(V_H:=\fracq_H (<\phi ,\phi >-v^2)^2\)

$$\begin h_\lambda ^= - \int _\Sigma \lambda e_n \frace^3(<\phi ,\phi >-v^2)^2 =\int _\Sigma \lambda e_n \fracV_H \end$$

Obtaining

$$\begin L_c^&=L_c;&P_^&=P_\xi + p_\xi ^A + p_\xi ^H + p_\xi ^;\\ M_\mu ^&=M_\mu ^A + m_\mu ^;&H_^&=H_\lambda + h_\lambda ^A + h_\lambda ^ + h_\lambda ^ \end$$

Theorem 19

Assume that \(g^\partial \) is non-degenerate on \(\Sigma \). Then, the zero locus of the functions \(L^_c\), \(P^_\), \(H^_\) and \(M_\mu ^\) defined above is a coisotropic submanifold with respect to the symplectic structure \(\Omega _\). Their mutual Poisson brackets read

$$\begin \, L_c^\}_&= - \frac L^_&\_c, P^_\}_&= L^__^c}\\ \ \_c, M^_\mu \}_&= 0&\_c, H^_\}_&= - P^_} + L^_(\omega - \omega _0)_\nu }\\ & &- H^_} + M^_(A-A_0)_\nu } \\ \ \_\mu , P^_\}_&= M^__^\mu }&\_,H^_\}_&= P^_} -L^_ (\omega - \omega _0)_\nu }\\ & &+H^_} - M^_ \\ \ \_\mu , H^_\lambda \}_&= 0&\_, P^_\}_&= \fracP^_- \fracL^_\iota _F_}\\ & &- \fracM^_\iota _F_} \\ \ \,M_\mu ^\}_&=-\fracM_^&\_,H^_\}_&=0 \end$$

with the same notation as in Theorem 7.

Proof

We use the results of appendix B for the components of the Hamiltonian vector fields of the non-interacting theories. In particular, we have \(\mathbb ^H=\mathbb ^\phi \). The residual components are computed using the results in Sect. 2. We start with \(M_\mu ^\). One can quite easily see that

$$\begin \mathbb ^_\phi =[\mu ,\phi ], \qquad \mathbb ^_p=[\mu ,p]. \end$$

Now, since \(l_c^=0\) and \(\iota _^\phi }\varpi _A=\iota _^A}\varpi _H=0\), one finds

$$\begin \mathbb ^=0. \end$$

For \(P_\xi ^\) we find

$$\begin \delta p_\xi ^= & \int _\Sigma -<\delta p, \iota _\xi [A_o,\phi ]>\\ & + <[A_0,\iota _\xi p], \delta \phi >= \underbrace^\phi }\varpi _A}_0 + \underbrace^A }\varpi _H}_0 + \iota _^}\Omega _, \end$$

finding

$$\begin \mathbb ^_\phi =-\iota _\xi [A_0,\phi ], \qquad \mathbb ^_p=[A_0,\iota _\xi p]; \end$$

, while all the other components of \(\mathbb ^\) vanish.

Regarding \(H_\lambda ^\), we find that \(\mathbb ^H=\mathbb ^\phi \) as expected, except for the components that inherit the Higgs potential term:

$$\begin \mathbb ^H_p&=d_\omega \left( \frac e^2 \Pi \right) +\fracq_h e^3 (<\phi ,\phi> - v^2 )\phi ;\\ e\mathbb ^H_\omega&=\lambda e_n \left( e<\Pi ,d\phi> + \frac<(\Pi ,\Pi )> + \fracV_H \right) -\frace^2\Pi (\Pi ,e_n) + \mathbb _. \end$$

For \(h_\lambda ^\) we obtain

$$\begin \delta h_\lambda ^&= \int _\Sigma \lambda e_n \left[ e<\Pi ,[A,\phi ]>\delta e + \frac\left(<\delta \Pi , [A,\phi ]> +<[A,\Pi ],\delta \phi > \right) \right] \\&\quad +\int _\Sigma \lambda e_n \frac \text [(\Pi \phi ^\dagger - \phi \Pi ^\dagger )\delta A] \\&= \underbrace^\phi }\varpi _A}_0 + \underbrace^A }\varpi _H}_0 + \iota _^}\Omega _, \end$$

hence finding

$$\begin \mathbb ^_e&=0&e\mathbb _\omega ^&=-\frace<\Pi ,[A,\phi ]>\\ \frac\mathbb ^_\phi&=\frace^2 [A.\phi ]&\frac\mathbb ^_\Pi&= \frace^2 [A,\Pi ]\\ \mathbb ^_A&=0&\mathbb ^_\rho&=\frac\lambda e_n e^2 \text (\Pi \phi ^\dagger - \phi \Pi ^\dagger ) \end$$

For the computations we use Theorem 2 and the results of Theorem 7 and of corresponding results in the presence of a scalar Higgs field and a Yang–Mills field.

As before we note that for all constraints

$$\begin \iota _+\mathbb ^}\iota _+\mathbb ^}\varpi _+ \iota _+\mathbb ^}\iota _+\mathbb ^}\varpi _=0,\\ \iota _}\iota _}\varpi _=0, \\ \iota _}\iota _}\varpi _=0,\\ \iota _^}\iota _^}\Omega _=0,\\ \iota _^}\iota _^}\Omega _=0. \end$$

Hence we can use the simplified formula

$$\begin&\,Y^\}_\\&\quad =\, Y+y^\}_A+\, Y+y^\}_H-\\\&\qquad + \iota _^}\delta (X+ x^+x^+x^) + \iota _^}\delta (Y+ y^+y^+y^)\\&\qquad - \iota _^}\iota _^}\Omega _. \end$$

Applying it we get:

$$\begin \,M_\mu ^\}_&=\_A+ 2\iota _^}\delta ( M_^+m_^) -\iota _^}\iota _^}\Omega _\\&= \frac M_^ + \int _\Sigma<\mu ,p],[\mu ,\phi >+<,[\mu ,[\mu ,\phi ]>\\&\quad -\int _\Sigma<\mu ,p],[\mu ,\phi > \\&=\frac M^A_+\frac\int _\Sigma<p,[\mu ,\mu ],\phi > \\&=\frac M^_;\\ \_c,M^_\mu \}_&=\^, L_c\}_A + \iota _^}\delta L_c=0\\ \, P_\xi ^ \}_&=\^, P_+p_^\}_A+ \iota _^}\delta (M_^+m_^)\\&\quad - \iota _^}\iota _^}\Omega _+ \iota _^}\iota _+\mathbb ^}\varpi _ +\iota _^}\iota _^}\Omega _\\&= \int _\Sigma \left(<A_0, \iota _p],[\mu ,\phi > +<,[\mu ,[\iota _A_0, \phi ]>\right) \\&\quad +M^A_^\mu }+\int _\Sigma \left( -<\mu ,p],\textrm_\xi ^\phi>-<,\textrm_\xi ^[\mu ,\phi > \right) \\&\quad -\int _\Sigma \left(<\mu ,p],[\iota _A_0, \phi >+<A_0, \iota _p],[\mu ,\phi >\right) \\&= M^A__\xi ^\mu }- \int _\Sigma <, [\textrm_\xi ^\mu , \phi > =M^__\xi ^\mu }, \end$$

where we noticed that in the second step the first and third line cancel and used that

$$\begin -<\mu ,p],\textrm_\xi ^\phi><p,[\mu ,\textrm_\xi ^\phi >. \end$$

$$\begin \, H_\lambda ^ \}_&= \^, H^A_\}_A + \iota _^}\delta (M_^+m_^)\\&\quad + \iota _^}\delta (H_+ h_^+h_^+h_^)\\&\quad - \iota _^}\iota _^}\Omega _\\&= \int _\frac\left(<\Pi , [d_A \mu , \phi > +<A,\Pi ], [\mu , \phi > +<\Pi , [\mu ,[A, \phi >\right) \\&\quad -\int _\Sigma \frac\left(< \mu , \Pi ],d\phi>+<\Pi ,d[\mu ,\phi >\right) + \frac<\mu ,\Pi ], \Pi> \\&\quad +\int _\Sigma \frac<\phi ,\phi>-v^2<\mu ,\phi ],\phi>\\&\quad + \int _\Sigma \frac \left(<\mu ,\Pi ],[A,\phi >+<\Pi ,[A,[\mu ,\phi >\right) \\&\quad -\int _\Sigma \frac \left(<\mu ,\Pi ],[A,\phi >+<A,\Pi ], [\mu , \phi >\right) =0, \end$$

having used the fact that \([\mu ,<\phi ,\phi>=[\mu , <\Pi ,\Pi >=0\) and the Jacobi identity for \(A, \mu \) and \(\phi \).

$$\begin&\_c,L^_c\}_=\_+\_-\\\&\quad =-\frac L_=-\frac L^_;\\&\_c,P_\xi ^\}_\\&\quad =\_H + \_A - \+ \iota _^}\delta L_c= L_^c}^ \end$$

We have

$$\begin \iota _^}\delta L_c= \int _\Sigma c \left[ \frac<\Pi , [A,\phi >,e^2\right] =\int _\Sigma \left[ c , \lambda e_n\right] \frac<\Pi , [A,\phi >. \end$$

Hence, using \([c,\lambda e_n]=X= X^e_\nu + X^e_n\) and

$$\begin - p^_} + m^_(A-A_0)_}=\int _\Sigma \left[ c , \lambda e_n\right] ^e_\nu \frac<\Pi , [A,\phi > \end$$

we get

$$\begin \_c,H_\lambda ^\}_&=\+h_^\}_A+\+h_^\}_H-\\}+ \iota _^}\delta L_c\\&=- P^_} + L^_(\omega - \omega _0)_\nu } - H^_} + M^_(A-A_0)_\nu }\\&\quad - P^_} + L^_(\omega - \omega _0)_\nu }- H^_}+ P_} - L_(\omega - \omega _0)_\nu } \\&\quad + H_} - p^_}-h^_} + m^_(A-A_0)_}\\&=- P^_} + L^_(\omega - \omega _0)_\nu } - H^_} + M^_(A-A_0)_\nu } \end$$

To compute the following bracket, we make use of the following identity

$$\begin \frac\iota _A_0= \iota _ d \iota _ A_0 -\frac\iota _\iota _dA_0. \end$$

$$\begin&\,P_\xi ^\}_\\&\quad = \+p_^, P_+p_^\}_A+\+p_^, P_+p_^\}_H-\,P_\}\\&\qquad + 2\iota _^}\delta (P_+ p_^+p_^+p_^) - \iota _^}\iota _^}\Omega _\\&\quad =\fracP^_- \fracL^_\iota _F_} - \fracM^_\iota _F_} + \fracP^_- \fracL^_\iota _F_}-\fracP_\\&\qquad + \fracL_\iota _F_}+ \int _\Sigma< [A_0, \iota _\xi p], \textrm^_\phi> +<p,\textrm^_[\iota _A_0, \phi ]>\\&\qquad -\int _\Sigma< [A_0, \iota _\xi p],[\iota _A_0, \phi ]>\\&\quad =\fracP^_- \fracL^_\iota _F_} - \fracM^_\iota _F_} + \fracP^_- \fracL^_\iota _F_}-\fracP_\\&\qquad + \fracL_\iota _F_}+ \int _\Sigma<p,[\iota _A_0, \phi ]>\\&\quad =\fracP^_- \fracL^_\iota _F_} - \fracM^_\iota _F_};\\&\,H_\lambda ^\}_\\&\quad =\+p_^, H_+h_^\}_A+\+p_^, H_+h_^\}_H-\,H_\}\\&\qquad + \iota _^}\delta (P_+ p_^+p_^+p_^) + \iota _^}\delta (H_+ h_^+h_^+h_^)\\&\qquad - \iota _^}\iota _^}\Omega _\\&\quad = P^_} -L^_ (\omega - \omega _0)_\nu } +H^_} - M^_+ P^_} -L^_ (\omega - \omega _0)_\nu }\\&\qquad +H^_} - P_} +L_ (\omega - \omega _0)_\nu } -H_}+ \int _\Sigma \textrm^_\xi e \frac<\Pi , [A, \phi ]>\\&\qquad + \int _\Sigma \frac<[A,\Pi ], \textrm^_\xi \phi> - \frac<\Pi , \textrm^_\xi [A,\phi ]>\\&\qquad + \int _\Sigma<\Pi , \textrm^_\xi \left( \frac[A,\phi ]\right)> + \frac<[\iota _A_0,\Pi ],d_A\phi>\\&\qquad + \int _\Sigma \frac\left(<\Pi ,d_A[\iota _A_0,\phi ]>\right) \\&\qquad +\int _\Sigma \frac<[A,\Pi ], \Pi> +\frac(<\phi ,\phi>-v^2)<[A,\phi ],\phi>\\&\quad = P^_} -L^_ (\omega - \omega _0)_\nu } +H^_} - M^_+ P^_} -L^_ (\omega - \omega _0)_\nu }\\&\quad \quad +H^_} - P_} +L_ (\omega - \omega _0)_\nu } -H_}\\&\quad \quad +\int _\Sigma \textrm^_\xi (\lambda e_n)\frac <\Pi ,[A,\phi ]>\\&\quad = P^_} -L^_ (\omega - \omega _0)_\nu } +H^_} - M^_, \end$$

having used (45).

$$\begin \,H_\lambda ^\}_&=\,H_\lambda ^\}_H + \,H_\lambda ^\}_A - \ \\&\quad \quad + 2 \iota __\lambda ^}\delta H_\lambda ^ - \iota __\lambda ^}\iota __\lambda ^}\Omega _=0, \end$$

in fact all terms in \(2 \iota __\lambda ^}\delta H_\lambda ^ - \iota __\lambda ^}\iota __\lambda ^}\Omega _\) contain either \((\lambda e_n)^2=0\) or \(\lambda e_n d_\omega (\lambda e_n)=0\). \(\square \)

Remark 20

Notice that, after choosing U(1) as the gauge group and setting \(q_H=0\), we obtain scalar electrodynamics coupled to gravity as a particular case of this theory.

4.3 Yukawa Interaction

The interaction of a scalar field and a spinor field takes the name of Yukawa interaction.

As before, let us denote the spinor field by \(\psi \in S(M)\) and a scalar field by \(\phi \in C^(M)\). Then an action on the bulk for the Yukawa interaction (i.e., an action correctly reproducing the classical Euler–Lagrange equations) is given by

$$\begin S_Y = S+ S_ + S_ + g_Y \int _M \frac e^N \overline\phi \psi \end$$

where \(g_Y\) is a coupling constant, \(S_\) is defined in (18) and \(S_\) in (32). Since the additional interaction term does not have derivatives, the geometric phase space will be just the direct sum of the building blocks composing this theory. In particular we have that the geometric phase space is the bundle

$$\begin ^_ \rightarrow \Omega _^1(\Sigma , \mathcal _)\oplus \mathcal }(\Sigma )\times S(\Sigma )\times \overline(\Sigma ) \end$$

with fiber \(\mathcal _(\Sigma )\oplus \Omega ^_\) such that (19) and (33) are satisfied.

The corresponding symplectic form is again just the sum of the symplectic form of the building blocks:

$$\begin \Omega _ = \varpi + \varpi _+ \varpi _. \end$$

Let us now consider the constraints. By the previous computation we already know that for a spinor field and a scalar field coupled to gravity the functions building up the constraints are those corresponding to the variations \(\delta e \) and \(\delta \omega \), since all the other ones are evolution equations. From the expression of the action of the Yukawa interaction, we deduce that there are no other constraints than the pure gravity ones and that the only one that is modified is \(H_\). Let

$$\begin h^_:= g_Y \int _ \lambda e_n \frac e^ \overline\phi \psi \end$$

(51)

Then the constraints for Yukawa theory are:

$$\begin L^_c= L_c + l^_; \quad P^_= P_+ p^_ + p^_; \quad H^_= H_+ h^_ + h^_+ h^_. \end$$

Once again, these constraints define a coisotropic submanifold with Poisson brackets analogous to the gravity case, as specified by the following theorem.

Theorem 21

Assume that \(g^\partial \) is non-degenerate on \(\Sigma \). Then, the zero locus of the functions \(L^_c\), \(P^_\), \(H^_\) defined above is coisotropic submanifold with respect to the symplectic structure \(\Omega _\). Their mutual Poisson brackets read

$$\begin \_c, L^_c\}_&= - \frac L^_&\_c, P^_\}_&= L^__^c}\\ \_c, H^_\}_&= - P^_} + L^_(\omega - \omega _0)_} - H^_}&\_, P^_\}_&= \fracP^_- \fracL^_\iota _F_}\\ \_,H^_\}_&= P^_} -L^_ (\omega - \omega _0)_} +H^_}&\_,H^_\}_&=0 \end$$

with the same notation as in Theorem 7.

Proof

We use the results proved in Sect. 2. The expressions of \(\mathbb \),\(\mathbb ^\), \(\mathbb ^\), \(\mathbb \), \(\mathbb ^\), \(\mathbb ^\), \(\mathbb \), \(\mathbb ^\) and \(\mathbb ^\) have been computed in [8] and are collected in Appendix B. Let us start with the constraint \(L^_c\). We first notice that \(\iota _^} \varpi _= \iota _^} \varpi _=0\). The variation of the interaction term is \(l^_=0\), and hence, we also conclude

$$\begin \mathbb ^=0. \end$$

For \(P^_\), we work in the same way and find

$$\begin \mathbb ^=0. \end$$

On the other hand, for \(H^_\) we get \(\iota _^}\varpi _=0\) and

$$\begin \iota _^}\varpi _= \int _\frace^2\mathbb _e^ \Pi \delta \phi . \end$$

Note that we will not need the explicit expression. The variation of the interaction term reads

$$\begin \delta h^_:= g_Y \int _ \lambda e_n \left( \frace^ \delta e \overline\phi \psi +\frac e^3 \delta \overline\phi \psi -\frac e^3 \overline\delta \phi \psi -\frac e^3 \overline\phi \delta \psi \right) . \end$$

Hence we get:

$$\begin \mathbb ^_e&= 0&e \mathbb ^_\omega&= 0 \\ \mathbb ^_\phi&= 0&\frace^3\mathbb ^_\Pi&=-\frace^2\mathbb _e^ \Pi +\fracg_Y \lambda e_n e^3 \overline \psi \\ \gamma \mathbb ^_&= -\fracg_Y \lambda e_n\phi \psi&\mathbb ^_} \gamma&= \fracg_Y \lambda e_n\overline\phi \end$$

In order to compute the Poisson brackets of the constraints we use Theorem 2. We first note that for all constraints we have

$$\begin \iota _+\mathbb ^}\iota _+\mathbb ^}\varpi _+ \iota _+\mathbb ^}\iota _+\mathbb ^}\varpi _=0. \end$$

For the computations we use repeatedly the results of Theorem7 and the corresponding results in the presence of a scalar field and spinor field. Not writing the zero terms we have

$$\begin \_c,L^_c\}_Y&=\, L_c+l_c^\}_+\, L_c+l_c^\}_-\\\&\quad +\iota _^}\iota _^}\Omega _+\iota _^}\iota _^}\Omega _\\&=-\frac L_ -\frac l_^ = -\frac L^_ \end$$

where we used for the second line \(\mathbb _e^=\mathbb _e^=0\). Similarly

$$\begin \_c,P^_\xi \}_Y&=\, P_\xi +p_\xi ^\}_+\, P_\xi +p_\xi ^\}_-\\\&\quad +\iota _^}\iota _^}\Omega _+\iota _^}\iota _^}\Omega _\\&= L__^c}+ l__^c}^ = L^__^c} \end$$

where we used for the second line \(\mathbb _e^=\mathbb _e^=\mathbb _e^=\mathbb _e^=0\), and

$$\begin \_, P^_\}_Y&=\, P_\xi +p_\xi ^\}_+\, P_\xi +p_\xi ^\}_-\\\&\quad +\iota _^}\iota _^}\Omega _+\iota _^}\iota _^}\Omega _\\&= \fracP_- \fracL_\iota _F_}+ \fracp^_+ \fracp^_- \fracl^_\iota _F_}\\&=\fracP^_- \fracL^_\iota _F_} \end$$

For the brackets with \(H^_\) we have to take into account more terms:

$$\begin \_c, H^_\}_Y&=\, H_+h_^\}_+\, H_+h_^\}_-\\}\\&\quad +\iota _^}\delta (L_c+l_c^) \end$$

Let us compute the last term:

$$\begin&\iota _^}\delta (L_c+l_c^)\\&\quad = \int _ - c e [\mathbb ^_, e] - i \frac \left( -[c,\mathbb ^_}]\gamma \psi - \mathbb ^_} \gamma [c,\psi ] \right) \\&\qquad - i \frac \left( - [c,\overline]\gamma \mathbb ^_\psi + \overline \gamma [c,\mathbb ^_\psi ] \right) \\&\quad = \int _ - \frac g_Y c e [ \lambda e_n e \overline\phi \psi , e] - i \frac \left( -[c,\mathbb ^_}]\gamma \psi - \mathbb ^_} \gamma [c,\psi ] \right) \\&\qquad - i \frac \left( - [c,\overline]\gamma \mathbb ^_\psi + \overline \gamma [c,\mathbb ^_\psi ] \right) \\&\quad = -\int _\frac g_Y [c,\lambda e_n] \overline\phi \psi \end$$

where we used the properties listed in [8, Appendix B.5] for the bracket \([\cdot , \cdot ]\) on spinors. The result is easily recognized as \(h^_}\). Hence we get

$$\begin \_c, H^_\}_Y&= - P_} + L_(\omega - \omega _0)_} - H_} - p^_} - h^_} - p^_} \\&\quad + l^_(\omega - \omega _0)_} - h^_} -h^_}\\&=- P^_} + L^_(\omega - \omega _0)_} - H^_} \end$$

Analogously we compute

$$\begin \_\xi , H^_\}_Y&=\, H_+h_^\}_+\, H_+h_^\}_-\\}\\&\quad + \iota _^}\delta (P_\xi +p_\xi ^+p_\xi ^)+\iota _^}\iota _^}\Omega _ \end$$

Let us consider the terms in the second row. We have

$$\begin \iota _^}\iota _^}\Omega _= \int _\frace^2 \mathbb ^_e \Pi \mathbb ^_= -\int _\frace^2 \mathbb ^_e \Pi \textrm_\xi \phi . \end$$

On the other hand we have

$$\begin&\iota _^}\delta (P_\xi +p_\xi ^+p_\xi ^) \\&\quad = \int _ - \frace^3 \mathbb ^_\textrm_\phi - i \frac \left( \mathbb ^_} \gamma \textrm_^(\psi ) + \textrm_\xi ^(\overline)\gamma \mathbb ^_ \right) \\&\qquad - \frac \left( \textrm_^( e^3 \overline) \gamma \mathbb ^_ + \mathbb ^_}\gamma \textrm_\xi ^(e^3 \psi ) \right) \\&\quad = \int _ \frace^2\mathbb _e^ \Pi \textrm_\phi -\fracg_Y \lambda e_n e^3 \overline \psi \textrm_\phi \\&\qquad + \frac \left( \fracg_Y \lambda e_n\overline\phi \textrm_^(\psi ) - \fracg_Y \lambda e_n\textrm_\xi ^(\overline)\phi \psi \right) \\&\qquad - \frac \left( \fracg_Y \lambda e_n\textrm_^( e^3 \overline) \phi \psi -\fracg_Y \lambda e_n\overline\phi \textrm_\xi ^(e^3\psi ) \right) \\&\quad = \int _ - g_Y \frac \lambda e_n \textrm_^( e^3 \overline\phi \psi )+\frace^2\mathbb _e^ \Pi \textrm_\phi \\&\quad = \int _ g_Y \frac \textrm_^ (\lambda e_n ) e^3 \overline\phi \psi +\frace^2\mathbb _e^ \Pi \textrm_\ph