Fermionic Higher-form Symmetries

Yi-Nan Wang$\,{}^{1,2}$, Yi Zhang$\,{}^{2}$

1 School of Physics,
Peking University, Beijing 100871, China

2 Center for High Energy Physics, Peking University,
Beijing 100871, China

<EMAIL_ADDRESS><EMAIL_ADDRESS>
In this paper, we explore a new type of global symmetries—the fermionic
higher-form symmetries. They are generated by topological operators with
fermionic parameter, which act on fermionic extended objects. We present a set
of field theory examples with fermionic higher-form symmetries, which are
constructed from fermionic tensor fields. They include the free fermionic
tensor theories, a new type of fermionic topological quantum field theories,
as well as the exotic 6d (4,0) theory. We also discuss the gauging and
breaking of such global symmetries and the relation to the no global symmetry
swampland conjecture.

###### Contents

  1. 1 Introduction
  2. 2 Fermionic Higher-form symmetries
  3. 3 Examples with fermionic higher-form global symmetries
    1. 3.1 Fermionic “BF” theory
    2. 3.2 6d exotic theories
  4. 4 Gauging of fermionic higher-form symmetries
  5. 5 Examples with broken fermionic symmetries
    1. 5.1 Rarita-Schwinger field coupled to gauge field
    2. 5.2 Swampland implications
  6. 6 Discussions
  7. A Conventions

##  1 Introduction

In recent years, a lot of progress has been made in the rapidly evolving field
of generalized global symmetries, with fruitful applications in both high
energy and condensed matter physics. These include higher-form symmetries
which act on extended objects, higher-group symmetries which combine
symmetries of different form degrees, and more generally, non-invertible and
higher-categorical symmetries [1, 2, 3].

In this paper, we explore the possibilities of fermionic $p$-form global
symmetries in $d$-dimensional quantum field theories, which act on
$p$-dimensional fermionic extended operators. They are generated by
$(d-p-1)$-dimensional topological operators with fermionic parameters (spinors
with Grassmannian components). We will be restricting ourselves to the cases
of invertible fermionic global symmetries.

One simple example is the case of a free Rarita-Schwinger field $\psi_{\mu}$,
as we explain in section 2. Similar to the $U(1)$ Maxwell theory, we can
define a gauge invariant “Wilson loop” operator by integrating $\psi_{\mu}$
along a circle $\mathcal{C}$. The operator is charged under a
$(d-2)$-dimensional topological operator. We also extend the discussion to the
cases of a free fermionic $p$-form gauge field (first defined in [4, 5, 6],
and whose properties were recently discussed in [7]) and a more general type
of actions with fermionic tensor gauge fields and a single derivative in each
term.

Another playground for $p$-form fermionic symmetry is the 6d (4,0) theory,
which is conjectured to be the UV completion of strongly coupled 5d maximal
supergravity [8]. In particular, we consider the free limit of 6d (4,0) theory
with a free kinetic term for the fermionic 2-form gauge field (the exotic
gravitino) $\Psi_{MN}$ ($M,N=0,\dots,5$) in the 6d (4,0) supermultiplet. We
show that $\Psi_{MN}$ possesses a fermionic 2-form symmetry in this limit. We
also discuss the dimension reduction of the free action on $S^{1}$. The
fermionic 2-form symmetry is reduced to a fermionic 2-form symmetry and a
fermionic 1-form symmetry in 5d, however only one of them is preserved after
integrating out a 5d fermionic gauge field.

Analogous to the bosonic higher-form symmetries, we consider the gauging of
fermionic $p$-form global symmetries by coupling to a background fermionic
$(p+1)$-form gauge field, in section 4. Nonetheless, unlike the case of
bosonic $p$-form symmetry, we found that even for the case of a free fermionic
$p$-form gauge field, a ’t Hooft anomaly exists which obstructs the gauging of
fermionic $p$-form global symmetries. On the other hand, we also present a few
cases where the fermionic symmetries can be gauged.

One may also wonder what is the fate of the fermionic $p$-form global symmetry
after the corresponding fermionic $p$-form gauge field is coupled to other
fields. We discuss the case of a charged Rarita-Schwinger field under a
bosonic $U(1)$ gauge group as well as the case of supergravity in section 5.
In these cases, the fermionic 1-form global symmetry is explicitly broken by
the coupling terms with the $U(1)$ gauge fields $A_{\mu}$ or the Christoffel
connection. We also briefly discuss the relation to the no global symmetry
swampland conjecture [9, 10, 11].

The structure of the paper is as follows: in section 2 we introduce the
notions of fermionic $p$-form symmetries and discuss the examples of free
fermionic $p$-form gauge fields. In section 3 we discuss examples with
unbroken fermionic $p$-form symmetries, including a generalized fermionic “BF”
theory and the free limit of 6d (4,0) theory. In section 4 we discuss the
gauging of fermionic $p$-form symmetries. In section 5 we present a few
examples where the fermionic symmetries are explicitly broken, and discussed
the swampland implications. Finally, we put some discussions and future
directions in section 6. The conventions in the formula are organized in
appendix A.

##  2 Fermionic Higher-form symmetries

We introduce the notion of fermionic $p$-form symmetries, which are the
fermionic counterparts of the $p$-form symmetries acting on bosonic
$p$-dimensional operators [1]. We work in $d$-dimensional Minkowski space-
time111The fermionic $p$-form symmetry exists on any flat space-time as well,
such as the torus. We briefly comment on the cases of curved space-time
manifold in section 5.2, but we leave the detailed analysis in future work.
$\mathbb{R}^{1,d-1}$, and the results can be applied to Euclidean cases as
well.

An invertible fermionic $p$-form symmetry is generated by topological
operators $U_{\epsilon}(M^{(d-p-1)})$, defined on a $(d-p-1)$-dimensional
submanifold $M^{(d-p-1)}\subset\mathbb{R}^{1,d-1}$. The symmetry parameter
$\epsilon\in G$ is a fermionic spinor222The symmetry group $G$ can be thought
as a $\mathbb{R}^{n}$ with Grassmannian components, where $n$ is the number of
spinor components., i.e. it has anti-commuting (Grassmannian) components. The
symmetry group $G$ is also fermionic, which can be interpreted as the non-
compact translation group of spinors with Grassmannian components, thus it is
always abelian. $U_{\epsilon}(M^{(d-p-1)})$ acts on $p$-dimensional operators
$V(\mathcal{C}^{(p)})$ as

$\langle
U_{\epsilon}(M^{(d-p-1)})V(\mathcal{C}^{(p)})\rangle=R(\epsilon)^{\langle\mathcal{C}^{(p)},M^{(d-p-1)}\rangle}\langle
V_{j}(\mathcal{C}^{(p)})\rangle\,.$ (2.1)

$\langle\mathcal{C}^{(p)},M^{(d-p-1)}\rangle$ is the linking number (cf.
(A.13)) between $\mathcal{C}^{(p)}$ and $M^{(d-p-1)}$. $R$ is a representation
of the symmetry group $G$.

As usual, when the symmetry parameter $\epsilon$ is space-time independent, we
call it a fermionic $p$-form global symmetry. Otherwise it is a gauge
symmetry.

Now we discuss some examples.

#### Fermionic 0-form symmetry.

A common example of fermionic 0-form symmetry is the _global supersymmetry_ ,
with a spinor $\epsilon$ as the symmetry parameter. The $(d-1)$-dimensional
topological operator is

$\displaystyle U_{\epsilon}(M^{(d-1)})$
$\displaystyle=\exp\left(\int_{M^{(d-1)}}i(\bar{\epsilon}\mathcal{J}+\bar{\mathcal{J}}\epsilon)\right)$
(2.2) $\displaystyle=e^{i(\bar{\epsilon}Q+\bar{Q}\epsilon)}\,.$

$\mathcal{J}$ is the supercurrent $(d-1)$-form and $Q$ is the supercharge,
which acts on the space of local operators.

As another simple example, let us consider a free Dirac spinor $\psi$ with the
action

$S=\int-\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi d^{d}x\,.$ (2.3)

This action admits a _fermionic shift symmetry_

$\psi\rightarrow\psi+\epsilon\,,$ (2.4)

where the spinor parameter $\epsilon$ satisfies $d\epsilon=0$. It is a
fermionic 0-form symmetry generated by the topological operator

$U_{\epsilon}(M^{(d-1)})=\exp\left(i\int_{M^{(d-1)}}\frac{1}{(d-1)!}\varepsilon_{\mu_{1}\dots\mu_{d-1}\nu}\left[\bar{\epsilon}\gamma^{\nu}\psi-\bar{\psi}\gamma^{\nu}\epsilon\right]\,dV^{\mu_{1}\dots\mu_{d-1}}\right)\,,$
(2.5)

where $\varepsilon$ is the Levi-Civita symbol in $d$ dimensions and
$dV^{\mu_{1}\dots\mu_{d-1}}$ is the volume element on the $(d-1)$-dimensional
submanifold $M^{(d-1)}$.
Conveniently, $U_{\epsilon}(M^{(d-1)})$ can be written compactly using
differential form notation

$U_{\epsilon}(M^{(d-1)})=\exp\left(i\int_{M^{(d-1)}}\star\left[\bar{\epsilon}\gamma_{(1)}\psi-\bar{\psi}\gamma_{(1)}\epsilon\right]\right)\,,$
(2.6)

where $\star$ is the Hodge star operator. The combination
$\bar{\epsilon}\gamma_{(1)}\psi$ can be regarded as a 1-form
$\bar{\epsilon}\gamma_{(1)}\psi=\bar{\epsilon}\gamma_{\mu}\psi\,dx^{\mu}$. The
Dirac equation $\not{\partial}\psi=0$ is translated into
$d\star\gamma_{(1)}\psi=0$, and together with Stokes’ theorem we see that
$U_{\epsilon}(M^{(d-1)})$ is on-shell topological.

As another comment, note that we can relax the spinor parameter $\epsilon$ in
(2.4) to only satisfy a weaker constraint, i. e. the Dirac equation
$\not{\partial}\epsilon=0$. Now the parameter $\epsilon$ is space-time
dependent in general, but the action (2.3) is still invariant. In this case,
we say the shift symmetry (2.4) is a “semi-local” symmetry333In an analogous
setup, a free non-compact scalar with action
$S=\int_{M_{d}}\partial_{\mu}\phi\partial^{\mu}\phi$ also has a shift symmetry
$\phi\rightarrow\phi+a$, where the action is invariant if $a$ only satisfies
the weaker condition $\partial_{\mu}\partial^{\mu}a=0$. This is known as the
“Galileon symmetry” [12]..

#### Fermionic 1-form symmetry.

The first new example is the case of fermionic 1-form symmetry. Let us
consider a free Rarita-Schwinger field $\psi_{\mu}$ in $d\geq 3$ with the
action

$S[\psi_{\mu}]=\int-\bar{\psi}_{\mu}\gamma^{\mu\nu\rho}\partial_{\nu}\psi_{\rho}\,d^{d}x\,.$
(2.7)

For $d\geq 4$, $\psi_{\mu}$ is fermionic due to spin-statistics theorem, while
there is no restriction in $d=3$. The equation of motion for $\psi_{\mu}$ is

$\gamma^{\mu\nu\rho}\partial_{\nu}\psi_{\rho}=0\,,$ (2.8)

which is equivalent to444In $d=3$, this reduces to $H_{\mu\nu}=0$, which means
that $\psi_{\mu}$ has no physical degrees of freedom. We will discuss these
topological fermionic $p$-forms in a more general context in section 3.

$\gamma^{\mu}H_{\mu\nu}=0\,,$ (2.9)

where we define the field strength $H_{\mu\nu}$ as exterior derivative of the
Rarita-Schwinger field: $H_{\mu\nu}=2\partial_{[\mu}\psi_{\nu]}$.
The action has a fermionic 0-form gauge symmetry

$\psi_{\mu}\rightarrow\psi_{\mu}+\partial_{\mu}\lambda\,,$ (2.10)

and the gauge transformation parameter $\lambda$ is a spinor. The field
strength $H_{\mu\nu}$ is gauge invariant.

Now we construct the gauge invariant 1-dimensional topological operator

$V_{\eta}(\mathcal{C})=\exp\left(i\oint_{\mathcal{C}}(\bar{\eta}\psi_{\mu}+\bar{\psi}_{\mu}\eta)\,dx^{\mu}\right)=\exp\left(i\int_{\mathcal{C}}(\bar{\eta}\psi_{(1)}+\bar{\psi}_{(1)}\eta)\right)\,,$
(2.11)

as the fermionic analog of Wilson loop for the 1-form gauge field $A_{\mu}$,
and $\psi_{(1)}=\psi_{\mu}\,dx^{\mu}$ is the fermionic one-form.

We define a fermionic $2$-form
$\mathcal{J}_{(2)}=\frac{1}{2}\,\mathcal{J}_{\mu\nu}\,dx^{\mu}\wedge dx^{\nu}$
whose components are given as

$\mathcal{J}_{\mu\nu}\equiv\gamma_{\mu\nu\rho}\psi^{\rho}\,.$ (2.12)

By the equation of motion (2.8), this two form is conserved

$\partial^{\mu}\mathcal{J}_{\mu\nu}=-\gamma_{\nu\mu\rho}\partial^{\mu}\psi^{\rho}=0\,.$
(2.13)

This fact indicates that we can define a closed $(d-2)$-form current
$(\star\mathcal{J})_{(d-2)}$ (analogous to the free Maxwell theory, cf.
$\partial^{\mu}F_{\mu\nu}=0$) and the closure of this $(d-2)$-form current
ensures the topological feature of the $(d-2)$-dimensional operator
$U_{\epsilon}(M^{(d-2)})$

$U_{\epsilon}(M^{(d-2)})=\exp\left(i\int_{M^{(d-2)}}\left[\bar{\epsilon}\,(\star\mathcal{J})_{(d-2)}+(\star\bar{\mathcal{J}})_{(d-2)}\,\epsilon\right]\right)\,,$
(2.14)

where $\bar{\mathcal{J}}_{\mu\nu}=\bar{\psi}^{\rho}\gamma_{\mu\nu\rho}$.
In components, it reads

$U_{\epsilon}(M^{(d-2)})=\exp\left(i\left[\bar{\epsilon}\int_{M^{(d-2)}}\frac{1}{(d-2)!}\frac{1}{2!}\varepsilon_{\mu_{1}\dots\mu_{d-2}\mu\nu}\gamma^{\mu\nu\rho}\psi_{\rho}\,dV^{\mu_{1}\dots\mu_{d-2}}+\text{c.c.}\right]\right)\,.$
(2.15)

Let us check the action of $U_{\epsilon}(M^{(d-2)})$ on
$V_{\eta}(\mathcal{C})$ and verify

$\langle
U_{\epsilon}(M^{(d-2)})V_{\eta}(\mathcal{C})\rangle=\exp\left(i(\bar{\epsilon}\eta+\bar{\eta}\epsilon)\langle\mathcal{C},M^{(d-2)}\rangle\right)\langle
V_{\eta}(\mathcal{C})\rangle\,,$ (2.16)

where $\langle\mathcal{O}\rangle$ means the vacuum expectation value of an
operator $\mathcal{O}$.
The left-hand side of (2.16) is by definition computed as the path integral

$\langle
U_{\epsilon}(M^{(d-2)})V_{\eta}(\mathcal{C})\rangle=\int\mathcal{D}\psi_{\mu}\,\mathcal{D}\bar{\psi}_{\mu}e^{iS[\psi_{\mu},\bar{\psi}_{\mu}]+i\int_{M^{(d-2)}}\left(\bar{\epsilon}\,(\star\mathcal{J})_{(d-2)}+(\star\bar{\mathcal{J}})_{(d-2)}\,\epsilon\right)+i\int_{\mathcal{C}}(\bar{\eta}\psi_{(1)}+\bar{\psi}_{(1)}\eta)}\,.$
(2.17)

As a starting point, we consider one of the $(d-2)$-form integrals

$\displaystyle\int_{M^{(d-2)}}\bar{\epsilon}\,(\star\mathcal{J})_{(d-2)}=\int_{M^{(d-1)}}\bar{\epsilon}\,(d\,\star\mathcal{J})_{(d-1)}$
$\displaystyle=\int_{M^{(d)}}J_{(1)}(M^{(d-1)})\wedge\bar{\epsilon}\,(d\,\star\mathcal{J})_{(d-1)}$
(2.18)
$\displaystyle=\int\bar{\epsilon}J_{\mu}\gamma^{\mu\nu\rho}\partial_{\nu}\psi_{\rho}\,d^{d}x\,.$

Here in the first step we used Stokes’ theorem and $\partial
M^{(d-1)}=M^{(d-2)}$. In the second step, we introduced a one-form delta-
function current $J_{(1)}$ with support on the $M^{(d-1)}$ submanifold.
Explicitly, it is given in local coordinates as555We are using a slightly
different definition from [13], since for us the Poincaré dual
$J_{(d-p)}(\mathcal{C}^{(p)})$ of a $p$-cycle $\mathcal{C}^{(p)}$ is
characterized by
$\int_{\mathcal{C}^{(p)}}A_{(p)}=\int_{M^{(d)}}J_{(d-p)}(\mathcal{C}^{(p)})\wedge
A_{(p)}$ for any $p$-form $A_{(p)}$ instead of by
$\int_{\mathcal{C}}A=\int_{M}A\wedge J$.

$J_{\mu}(x,M^{(d-1)})=\int_{M^{(d-1)}}\frac{1}{(d-1)!}\varepsilon_{\mu\mu_{1}\ldots\mu_{d-1}}\,\delta^{(d)}(x-y)\,dV^{\mu_{1}\ldots\mu_{d-1}}(y)\,.$
(2.19)

This is quite similar to the delta-function differential forms (cf. (A.12))
used in describing $D$-brane (Membrane) sources in String (M-) theory and
mathematically it is the cohomology class of the Poincaré dual of $M^{(d-1)}$
[14]. We use this trick to write the $(d-2)$-form integral as an integral on
the entire manifold $M^{(d)}$ and this will help us to absorb it into the free
action by a change of variables.

We now take a look back at the path integral and under the change of variable
$\psi_{\mu}\rightarrow\psi^{\prime}_{\mu}=\psi_{\mu}-\epsilon J_{\mu}$ the
action becomes

$\displaystyle S[\psi_{\mu}-\epsilon J_{\mu}]$
$\displaystyle=\int-\left(\bar{\psi}_{\mu}-\bar{\epsilon}J_{\mu}\right)\gamma^{\mu\nu\rho}\partial_{\nu}\left(\psi_{\rho}-\epsilon
J_{\rho}\right)\,d^{d}x$
$\displaystyle=S[\psi_{\mu}]+\int\left(\bar{\psi}_{\mu}\gamma^{\mu\nu\rho}\epsilon\partial_{\nu}J_{\rho}+\bar{\epsilon}J_{\mu}\gamma^{\mu\nu\rho}\partial_{\nu}\psi_{\rho}-J_{\mu}\bar{\epsilon}\gamma^{\mu\nu\rho}\epsilon\partial_{\nu}J_{\rho}\right)\,d^{d}x\,.$

The last term can be regularized by a local counter term [15]. The third term
is identified with the RHS of (2.18), and similarly the second term is
identified with the hermitian conjugate of the third term. (cf. (2.14))

Apply this change of variable in the path integral (2.17), we get

$\displaystyle\langle U_{\epsilon}(M^{(d-2)})V_{\eta}(\mathcal{C})\rangle$
$\displaystyle=e^{i(\bar{\epsilon}\eta+\bar{\eta}\epsilon)\oint_{\mathcal{C}}J_{\mu}dx^{\mu}}\int\mathcal{D}\psi^{\prime}_{\mu}\,\mathcal{D}\bar{\psi^{\prime}}_{\mu}e^{iS[\psi^{\prime}_{\mu},\bar{\psi^{\prime}}_{\mu}]+i\int_{\mathcal{C}}(\bar{\eta}\psi^{\prime}_{(1)}+\bar{\psi}^{\prime}_{(1)}\eta)}$
(2.20)
$\displaystyle=\exp\left(i(\bar{\epsilon}\eta+\bar{\eta}\epsilon)\langle\mathcal{C},M^{(d-2)}\rangle\right)\langle
V_{\eta}(\mathcal{C})\rangle\,.$

Recall that $\partial M^{(d-1)}=M^{(d-2)}$, and
$\langle\mathcal{C},M^{(d-2)}\rangle$ is the linking number of $\mathcal{C}$
and $M^{(d-2)}$

$\langle\mathcal{C},M^{(d-2)}\rangle=\mathcal{I}(\mathcal{C},M^{(d-1)})=\oint_{\mathcal{C}}J_{(1)}(M^{(d-1)})\,.$
(2.21)

The action of the fermionic 1-form symmetry on the original Rarita-Schwinger
field $\psi_{\mu}$ is the shift by a fermionic 1-form $\xi_{\mu}$ 666Just as
we discussed in the 0-form case, one can relax the flatness condition of
$\xi_{\mu}$, i.e. require it to be “flat” under the Rarita-Schwinger
differential operator $\gamma^{\mu\nu\rho}\partial_{\nu}$ instead of the usual
space-time derivative.

$\displaystyle\psi_{\mu}$ $\displaystyle\longrightarrow\psi_{\mu}+\xi_{\mu}$
(2.22) $\displaystyle\partial_{[\nu}\xi_{\rho]}=0\,.$

One may wonder if there is also an analog of $(d-3)$-form magnetic symmetry as
in Maxwell theory. The associated topological operator would be defined by
using the field strength $H$ of the Rarita-Schwinger field

$U_{\theta}(M^{(2)})=\exp\left(i\int_{M^{(2)}}\left[(\bar{\theta}H)_{(2)}+\text{c.c.}\right]\right)\,,$
(2.23)

and $H$ satisfies the Bianchi identity $dH=0$. However, there is no such
symmetry because there is no electromagnetic type dualities between $p$-form
fermionic fields.

#### Fermionic $p$-form symmetry.

The flat spacetime Rarita-Schwinger action (2.7) for a fermionic one-form
$\psi_{\mu}$ can be generalised to a free action for ‘fermionic $p$-forms’
$\psi_{(p)}$ [4, 5, 6, 7]

$S[\psi_{(p)}]=-(-1)^{\frac{p(p-1)}{2}}\,\int\\!d^{d}\\!x\;\bar{\psi}_{\mu_{1}\mu_{2}\dots\mu_{p}}\,\gamma^{\mu_{1}\mu_{2}\dots\mu_{p}\nu\rho_{1}\rho_{2}\dots\rho_{p}}\,\partial_{\nu}\psi_{\rho_{1}\rho_{2}\dots\rho_{p}}\,.$
(2.24)

This action is invariant under the gauge transformation

$\delta\psi_{\mu_{1}\mu_{2}\dots\mu_{p}}=p\,\partial_{[\mu_{1}}\Lambda_{\mu_{2}\dots\mu_{p}]}\,,$
(2.25)

and $\Lambda_{\mu_{2}\dots\mu_{p}}$ is the components of the fermionic
$(p-1)$-form $\Lambda_{(p-1)}$ gauge parameter. The gauge transformation can
be written compactly as $\delta\psi_{(p)}=d\Lambda_{(p-1)}$. We define the
gauge invariant field strength $H_{(p+1)}$ of the field $\psi_{(p)}$ as
$H_{(p+1)}\equiv d\psi_{(p)}$, i. e.

$\quad
H_{\mu_{1}\dots\mu_{p+1}}=(p+1)\,\partial_{[\mu_{1}}\psi_{\mu_{2}\dots\mu_{p+1}]}\,.$
(2.26)

The equations of motion derived from the action (2.24) read

$\gamma^{\mu_{1}\dots\mu_{p}\nu_{1}\dots\nu_{p+1}}\,H_{\nu_{1}\dots\nu_{p+1}}=0\,,$
(2.27)

which after some gamma matrices identities are equivalent to the single-gamma-
trace equations

$\gamma^{\mu_{1}}H_{\mu_{1}\mu_{2}\dots\mu_{p+1}}=0\,.$ (2.28)

The latter cannot be obtained directly from an action. For $2p+1>d$, the
action (2.24) identically vanishes because of the rank of the gamma matrix
exceeds the highest possible value $d$, while for $2p+1=d$, the equations of
motion are equivalent to $H_{(p+1)}=0$, which implies that $\psi_{(p)}$ is a
pure gauge. The only case the theory described by (2.24) has propagating
degrees of freedom is when $2p<d$.
Analogous to the $1$-form case, we define a conserved $(p+1)$-form
$\mathcal{J}_{(p+1)}$ with components

$\mathcal{J}_{\mu_{1}\mu_{2}\dots\mu_{p+1}}\equiv\gamma_{\mu_{1}\dots\mu_{p+1}\nu_{1}\dots\nu_{p}}\psi^{\nu_{1}\dots\nu_{p}}\,,$
(2.29)

and $\partial_{\mu}\mathcal{J}^{\mu\mu_{1}\dots\mu_{p}}=0$ is achieved by
requiring equation of motions (2.27).

We can then build topological operators on $(d-p-1)$-dimensional submanifolds
$M^{(d-p-1)}$

$U_{\epsilon}(M^{(d-p-1)})=\exp\left(i\,C(d,p)\int_{M^{(d-p-1)}}\left[\bar{\epsilon}\,(\star\mathcal{J})_{(d-p-1)}+(\star\bar{\mathcal{J}})_{(d-p-1)}\epsilon\right]\right)\,$
(2.30)

which act on the following $p$-dimensional objects

$V_{\eta}(\mathcal{C}^{(p)})=\exp\left(i\int_{\mathcal{C}^{(p)}}(\bar{\eta}\psi_{(p)}+\bar{\psi}_{(p)}\eta)\right)\,,$
(2.31)

and the numerical constant is computed (A.14) as
$C(d,p)=-p!(-1)^{\frac{(p+1)(2d-p)}{2}}$, it depends on space-time dimension
and the degree of the fermionic differential form. The symmetry transformation
of $V_{\eta}(\mathcal{C}^{(p)})$ is a straight generalisation of (2.16)

$\langle
U_{\epsilon}(M^{(d-p-1)})V_{\eta}(\mathcal{C}^{(p)})\rangle=\exp\left(i(\bar{\epsilon}\eta+\bar{\eta}\epsilon)\langle\mathcal{C}^{(p)},M^{(d-p-1)}\rangle\right)\langle
V_{\eta}(\mathcal{C}^{(p)})\rangle\,.$ (2.32)

##  3 Examples with fermionic higher-form global symmetries

###  3.1 Fermionic “BF” theory

In this section, we consider a generalization of the free $p$-form fermionic
fields, which consists of an arbitrary number of fermionic $p$-form fields
$\psi_{(p),\alpha}$ with different $p$ and label $\alpha$. For the action, it
is different for the cases of even and odd space-time dimension $d$.

For odd $d$, it can be written as a sum

$S=\sum
c_{i}\int_{M^{(d)}}\bar{\psi}_{(p_{i}),\alpha_{i}}\wedge\gamma_{(d-p_{i}-q_{i}-1)}\wedge
d\psi_{(q_{i}),\beta_{i}}+\text{c.c.}\,.$ (3.1)

$\gamma_{(d-p_{i}-q_{i}-1)}$ is the $(d-p_{i}-q_{i}-1)$-form $\gamma$-matrix,
i.e. $\gamma_{(r)}=\frac{1}{r!}\gamma_{\mu_{1}\ldots\mu_{r}}dx^{\mu_{1}}\ldots
dx^{\mu_{r}}$. Hence $S$ is the integration of a general sum of $d$-forms,
each of which consists of two fermionic fields and a single exterior
derivative.
To see how this generalises the free fermionic $p$-form action (2.24), we
consider a single summand

$\displaystyle S$
$\displaystyle=c\int_{M^{(d)}}\bar{\psi}_{(p),\alpha}\wedge\gamma_{(d-p-q-1)}\wedge
d\psi_{(q),\beta}+\text{c.c.}$ (3.2)
$\displaystyle\propto\int\\!d^{d}\\!x\;\bar{\psi}_{\mu_{1}\ldots\mu_{p},\alpha}\,\gamma_{\nu_{1}\ldots\nu_{d-p-q-1}}\varepsilon^{\mu_{1}\ldots\mu_{p}\nu_{1}\ldots\nu_{d-p-q-1}\rho\sigma_{1}\ldots\sigma_{q}}\,\partial_{\rho}\psi_{\sigma_{1}\ldots\sigma_{q},\beta}+\text{c.c.}$
$\displaystyle\propto\int\\!d^{d}\\!x\;\bar{\psi}_{\mu_{1}\ldots\mu_{p},\alpha}\,\gamma^{\mu_{1}\ldots\mu_{p}\rho\sigma_{1}\ldots\sigma_{q}}\,\partial_{\rho}\psi_{\sigma_{1}\ldots\sigma_{q},\beta}+\text{c.c.}\,,$

where in the last step we used the identity of gamma matrices for odd
dimensions

$\gamma^{\mu_{1}\ldots\mu_{p}}=i^{\frac{d+1}{2}}\frac{1}{(d-p)!}\varepsilon^{\mu_{1}\ldots\mu_{p}\nu_{1}\ldots\nu_{d-p}}\gamma_{\nu_{d-p}\ldots\nu_{1}}\,.$
(3.3)

For even $d$, this identity becomes

$\gamma^{\mu_{1}\ldots\mu_{p}}\gamma_{d+1}=-(-i)^{\frac{d}{2}+1}\frac{1}{(d-p)!}\varepsilon^{\mu_{p}\ldots\mu_{1}\nu_{1}\ldots\nu_{d-p}}\gamma_{\nu_{1}\ldots\nu_{d-p}}\,,$
(3.4)

where the chirality matrix $\gamma_{d+1}$ (see (A.2) for definition) appears.
The general action takes the form of

$S=\sum
c_{i}\int_{M^{(d)}}\bar{\psi}_{(p_{i}),\alpha_{i}}\wedge\gamma_{(d-p_{i}-q_{i}-1)}\left(1+d_{i}\gamma_{d+1}\right)\wedge
d\psi_{(q_{i}),\beta_{i}}+\text{c.c.}\,.$ (3.5)

The actions (3.1), (3.5) are gauge invariant under the gauge transformations

$\delta\psi_{(p),\alpha}=d\lambda_{(p-1),\alpha}\,.$ (3.6)

We prove that there is a fermionic $p$-form global symmetry associated to each
fermionic $p$-form field $\psi_{(p),\alpha}$, acting on the $p$-dimensional
object

$V_{\alpha,\eta}(\mathcal{C}^{(p)})=\exp\left(i\int_{\mathcal{C}^{(p)}}(\bar{\eta}\psi_{(p),\alpha}+\bar{\psi}_{(p),\alpha}\eta)\right)\,.$
(3.7)

We only need to construct the topological operator generating the fermionic
$p$-form symmetry, which takes the form of

$U_{\alpha,\epsilon}(M^{(d-p-1)})=\exp\left(i\int_{M^{(d-p-1)}}\left[\bar{\epsilon}\mathcal{K}_{(d-p-1),\alpha}+\bar{\mathcal{K}}_{(d-p-1),\alpha}\epsilon\right]\right)\,.$
(3.8)

The $(d-p-1)$-form777In the previous sections, we used its dual $(p+1)$-form
$\mathcal{J}_{(d+1)}$. $\mathcal{K}_{(d-p-1),\alpha}$ appears in the equation
of motion of $\psi_{(p),\alpha}$ as

$d\mathcal{K}_{(d-p-1),\alpha}=0\,.$ (3.9)

It is then obvious that $U_{\alpha,\epsilon}(M^{(d-p-1)})$ acts on
$V_{\alpha,\eta}(\mathcal{C}^{(p)})$ in the standard way

$\langle
U_{\alpha,\epsilon}(M^{(d-p-1)})V_{\alpha,\eta}(\mathcal{C}^{(p)})\rangle=\exp\left(i(\bar{\epsilon}\eta+\bar{\eta}\epsilon)\langle\mathcal{C}^{(p)},M^{(d-p-1)}\rangle\right)\langle
V_{\alpha,\eta}(\mathcal{C}^{(p)})\rangle\,.$ (3.10)

Using the general form of the action (3.1), we can expand out for odd $d$

$\mathcal{K}_{(d-p-1,\alpha)}=-\sum_{i|\alpha_{i}=\alpha}c_{i}\gamma_{(d-p_{i}-q_{i}-1)}\wedge\psi_{(q_{i}),\beta_{i}}-\sum_{j|\beta_{j}=\alpha}c_{j}\gamma_{(d-p_{j}-q_{j}-1)}\wedge\psi_{(q_{j}),\beta_{j}}\,,$
(3.11)

and for even $d$

$\mathcal{K}_{(d-p-1,\alpha)}=-\sum_{i|\alpha_{i}=\alpha}c_{i}\gamma_{(d-p_{i}-q_{i}-1)}(1+d_{i}\gamma_{d+1})\wedge\psi_{(q_{i}),\beta_{i}}-\sum_{j|\beta_{j}=\alpha}c_{j}\gamma_{(d-p_{j}-q_{j}-1)}(1+d_{j}\gamma_{d+1})\wedge\psi_{(q_{j}),\beta_{j}}\,.$
(3.12)

When $d=p_{i}+q_{i}+1$ for all $i$, the action (3.1) describes a non-trivial
TQFT (defined on flat manifold) with a rich spectrum of gauge invariant
extended operators, which have non-trivial correlation functions (3.10). Note
that comparing with the bosonic TQFT constructed with $p$-form gauge fields,
we have many more possibilities from the insertion of $\gamma$-matrices in the
action (3.1).

As a more concrete example, we look at the fermionic analog of the bosonic BF
theory:

$S=\int
d^{d}x\left(-\bar{\chi}_{\mu\nu}\gamma^{\mu\nu\rho\sigma}\partial_{\rho}\psi_{\sigma}-\bar{\psi}_{\mu}\gamma^{\mu\nu\rho\sigma}\partial_{\nu}\chi_{\rho\sigma}\right).$
(3.13)

The action is gauge invariant under the 0-form symmetry

$\delta_{\epsilon}\psi_{\mu}=\partial_{\mu}\epsilon$ (3.14)

and the 1-form symmetry

$\delta_{\lambda}\chi_{\mu\nu}=2\partial_{[\mu}\lambda_{\nu]}\,.$ (3.15)

We can write down the following gauge invariant observables

$V_{\eta}(\mathcal{C})=\exp\left(i\oint_{\mathcal{C}}(\bar{\eta}\psi_{\mu}+\bar{\psi}_{\mu}\eta)\,dx^{\mu}\right)$
(3.16)

and

$W_{\xi}(\mathcal{S})=\exp\left(i\oint_{\mathcal{S}}\frac{1}{2}(\bar{\xi}\chi_{\mu\nu}+\bar{\chi}_{\mu\nu}\xi)\,dS^{\mu\nu}\right)\,.$
(3.17)

In $d=4$ dimensions, both of them are topological888In fact, we can also
insert $\gamma_{5}$ in the spinor bilinears in the integrand, i.e.
$\bar{\eta}\psi_{\mu}\rightarrow\bar{\eta}\gamma_{5}\psi_{\mu}$, to define
topological operators. and they have non-trivial actions on each other

$\displaystyle\langle V_{\eta}(\mathcal{C})W_{\xi}(\mathcal{S})\rangle$
$\displaystyle=\exp\left(\frac{1}{2}(\bar{\xi}\gamma_{5}\eta+\bar{\eta}\gamma_{5}\xi)\langle\mathcal{S},\mathcal{C}\rangle\right)\langle
W_{\xi}(\mathcal{S})\rangle\,,$ (3.18) $\displaystyle\langle
W_{\xi}(\mathcal{S})V_{\eta}(\mathcal{C})\rangle$
$\displaystyle=\exp\left(\frac{1}{2}(\bar{\xi}\gamma_{5}\eta+\bar{\eta}\gamma_{5}\xi)\langle\mathcal{C},\mathcal{S}\rangle\right)\langle
V_{\eta}(\mathcal{C})\rangle\,,$
$\displaystyle=\exp\left(\frac{1}{2}(\bar{\xi}\gamma_{5}\eta+\bar{\eta}\gamma_{5}\xi)\langle\mathcal{S},\mathcal{C}\rangle\right)\langle
V_{\eta}(\mathcal{C})\rangle\,.$

Note that here the chirality matrix $\gamma_{5}$ appears, since in 4d the
action (3.13) that we started with is equivalent to

$S=\int
d^{d}x\left(i\bar{\chi}_{\mu\nu}\varepsilon^{\mu\nu\rho\sigma}\gamma_{5}\partial_{\rho}\psi_{\sigma}+\text{c.c.}\right).$
(3.19)

due to the identity
$\gamma_{\mu\nu\rho\sigma}=-i\varepsilon_{\mu\nu\rho\sigma}\gamma_{5}$.
For $d>4$, although $V_{\eta}(\mathcal{C})$ and $W_{\xi}(\mathcal{S})$ have a
trivial correlation function between each other, we can still construct
topological operators acting on them following the general prescriptions
above. For even dimensions $d=2m$

$\displaystyle U_{\epsilon}(M^{(d-2)})$
$\displaystyle=\exp\left(2(-i)^{m}\int_{M^{(d-2)}}(\bar{\epsilon}\gamma_{(d-4)}\gamma_{d+1}\wedge\chi_{(2)}+\text{c.c.})\right)$
(3.20) $\displaystyle U_{\lambda}(M^{(d-3)})$
$\displaystyle=\exp\left(2(-i)^{m}\int_{M^{(d-3)}}(\bar{\lambda}\gamma_{(d-4)}\gamma_{d+1}\wedge\psi_{(1)}+\text{c.c.})\right)\,,$

and for odd dimensions $d=2m+1$

$\displaystyle U_{\epsilon}(M^{(d-2)})$
$\displaystyle=\exp\left(2(-1)^{(m-2)(2m-3)}i^{m}\int_{M^{(d-2)}}(\bar{\epsilon}\gamma_{(d-4)}\wedge\chi_{(2)}+\text{c.c.})\right)$
(3.21) $\displaystyle U_{\lambda}(M^{(d-3)})$
$\displaystyle=\exp\left(2(-1)^{(m-2)(2m-3)}i^{m}\int_{M^{(d-3)}}(\bar{\lambda}\gamma_{(d-4)}\wedge\psi_{(1)}+\text{c.c.})\right)\,,$

such that their actions on $V_{\eta}(\mathcal{C})$ and $W_{\xi}(\mathcal{S})$
have the same simple form

$\displaystyle\langle U_{\epsilon}(M^{(d-2)})V_{\eta}(\mathcal{C})\rangle$
$\displaystyle=\exp\left(i(\bar{\epsilon}\eta+\bar{\eta}\epsilon)\langle\mathcal{C},M^{(d-2)}\rangle\right)\langle
V_{\eta}(\mathcal{C})\rangle$ (3.22) $\displaystyle\langle
U_{\lambda}(M^{(d-3)})W_{\xi}(\mathcal{S})\rangle$
$\displaystyle=\exp\left(i(\bar{\lambda}\xi+\bar{\xi}\lambda)\langle\mathcal{S},M^{(d-3)}\rangle\right)\langle
W_{\xi}(\mathcal{S})\rangle\,.$

We can also construct fermionic higher-group-like gauge theories. We take the
following 2-group example, where the gauge transformation of $\psi$ (1-form
spinor) and $\chi$ (2-form spinor) fields are

$\displaystyle\delta\psi$ $\displaystyle=d\epsilon+\kappa\lambda$ (3.23)
$\displaystyle\delta\chi$ $\displaystyle=d\lambda\,.$

We can construct the following gauge invariant field strengths:

$\displaystyle\mathcal{F}$ $\displaystyle=d\psi-\kappa\chi$ (3.24)
$\displaystyle\mathcal{H}$ $\displaystyle=d\chi\,,$

which are analogues of the fake curvature and (fake) 2-curvatures of the
bosonic 2-group gauge theory [16, 17, 18, 19].

One can construct gauge invariant actions in $d$ space-time dimensions using
$\mathcal{F}$, $\mathcal{H}$ and gamma matrices. These possibilities would be
further explored in future works.

###  3.2 6d exotic theories

Fermionic 2-forms appear in exotic six-dimensional supermultiplets [20, 21].
These are massless representations of extended Poincaré supersymmetry in
parallel to the standard supergravity multiplets. Intriguingly, the highest
spin field in these multiplets is either a spin-2 boson which is not a
graviton999See [22] for a discussion on bosonic “higher-biform symmetries” of
these exotic gravitons. or a “two indexed” exotic gravitino $\Psi_{MN}$
($M,N=0,\ldots,5$) [8, 23, 7]. The $\mathcal{N}=(4,0)$ and $\mathcal{N}=(3,1)$
maximally supersymmetric cases are conjectured by Hull [24, 8, 25, 26] to play
a role in some strongly coupled regimes of $5d$ maximal supergravity and
further studies have been carried out recently [27, 28, 23, 29, 30, 31, 32,
33, 34, 7, 35, 36]. Nonetheless similar chiral multiplets with less
supersymmetry (e.g. $\mathcal{N}=(2,1)$, $\mathcal{N}=(2,0)$ and etc.) are
discussed a bit in [21, 23, 31], and further properties are yet to be studied.

#### Free exotic theories.

Our interests are focused on the exotic fermionic field $\Psi_{MN}$ and it is
contained in the covariant field content of all these exotic multiplets. To
begin with, we first look at the 6d $\mathcal{N}=(p,q)$ massless little group
$G_{\text{little}}={\rm SU}(2)\times{\rm SU}(2)\times G^{R}_{(p,q)}$, which is
the subgroup of $\operatorname{\mathit{Spin}}(5,1)\times G^{R}_{(p,q)}$
preserving a null-momentum vector. Omitting the R-symmetry part
$G^{R}_{(p,q)}$ for a moment, the spacetime little group representation
$\mathbf{(4,1)}$ of ${\rm SU}(2)\times{\rm SU}(2)$ corresponds to a covariant
_chiral_ fermionic $2$-form-spinor field $\Psi_{MN}$, which we refer to as the
aforementioned exotic gravitino. This spinor field is anti-symmetric and its
field strength is self-dual and gauge invariant as when introduced in [8]

$\displaystyle\Psi_{MN}$ $\displaystyle=-\Psi_{NM}$ (3.25) $\displaystyle
H_{MNP}\equiv 3\partial_{[M}\Psi_{NP]}\,,\qquad H$
$\displaystyle=\star\;H\qquad\text{invariant
under}\qquad\delta\Psi_{MN}=2\partial_{[M}\epsilon_{N]}$

where $\epsilon_{N}$ is an arbitrary vector-spinor. Remarkably, the self-
duality equation of $H$ alone is not strong enough to ensure that $\Psi_{MN}$
propagates degrees of freedom in $\mathbf{(4,1)}$ of the little group after
gauge fixing101010This is the opposite of what happened with the $6d$ bosonic
chiral 2-form $B_{MN}$, where the self-dual condition $\star(dB)=dB$ does
describe the correct little group representation $\mathbf{(3,1)}$ and it
implies the usual second order field equation $d\star(dB)=0$.. One can show
[23, 37] that the Rarita-Schwinger type field equation (2.27) for $p=2$

$\Gamma^{MNPQR}H_{PQR}=0$ (3.26)

together with chirality condition

$\Gamma_{7}\Psi_{MN}=\Psi_{MN}$ (3.27)

describe the correct little group representation $\mathbf{(4,1)}$, and they
imply the self-dual condition $H=\star H$. We use $\Gamma_{M}$ to denote the
6d gamma matrices and $\Gamma_{7}$ is the chirality matrix. This field
equation (3.26) comes from the free action (2.24) for $p=2$, and in
particular, the discussions for fermionic $p$-form symmetries apply here. The
only difference is that the exotic gravitino is in addition a chiral fermionic
field (3.27). We thus claim that for 6d free exotic theories there is chiral
fermionic 2-form symmetry.

#### Dimensional reduction of 6d chiral fermionic 2-form to 5d.

The reduction of free chiral fermionic 2-form $\Psi_{MN}$ to five dimensions
has been studied in [23] at the level of actions. Following their conventions,
$\Gamma_{7}$ matrix is diagonal in a representation that relates the 6d and 5d
gamma matrices block-wisely. The 6d free chiral fermionic 2-form
$\Psi_{MN}=\begin{pmatrix}\hat{\psi}_{MN}\\\ 0\end{pmatrix}$ described by the
Lagrangian

$\mathcal{L}_{\text{6d}}=\bar{\Psi}_{MN}\Gamma^{MNPQR}\partial_{P}\Psi_{QR}$
(3.28)

yields upon reduction to 5d the following Lagrangian

$\mathcal{L}_{\text{5d}}=\bar{\psi}_{\mu\nu}\gamma^{\mu\nu\rho\sigma\tau}\partial_{\rho}\psi_{\sigma\tau}+2i\bar{\psi}_{\mu\nu}\gamma^{\mu\nu\rho\sigma}\partial_{\rho}\psi_{\sigma}-2i\bar{\psi}_{\mu}\gamma^{\mu\nu\rho\sigma}\partial_{\nu}\psi_{\rho\sigma}\,.$
(3.29)

where the two 5d fields are identified as $\psi_{\mu\nu}=\hat{\psi}_{\mu\nu}$
and $\psi_{\mu}=\hat{\psi}_{\mu 5}$ ($\mu,\nu=0,\ldots,4$). There are 5d gauge
variations descending from $\delta\hat{\psi}_{MN}=2\partial_{[M}\epsilon_{N]}$

$\delta\psi_{\mu\nu}=2\partial_{[\mu}\epsilon_{\nu]},\qquad\delta\psi_{\mu}=\partial_{\mu}\epsilon_{5}\,,$
(3.30)

under which the 5d action is invariant. This allows us to define a loop
operator

$V_{\eta}(\mathcal{C})=\exp\left(i\oint_{\mathcal{C}}(\bar{\eta}\psi_{\mu}+\bar{\psi}_{\mu}\eta)\,dx^{\mu}\right)$
(3.31)

and a surface operator

$W_{\xi}(\mathcal{S})=\exp\left(i\oint_{\mathcal{S}}\frac{1}{2}(\bar{\xi}\psi_{\mu\nu}+\bar{\psi}_{\mu\nu}\xi)\,dS^{\mu\nu}\right)\,.$
(3.32)

Varying the action (3.29) with respect to $\bar{\psi}_{\mu}$ and
$\bar{\psi}_{\mu\nu}$ give rise to

$-2i\gamma^{\mu\nu\rho\sigma}\partial_{\nu}\psi_{\rho\sigma}=0\,,\qquad\gamma^{\mu\nu\rho\sigma\tau}\partial_{\rho}(\psi_{\sigma\tau}-2i\gamma_{\sigma}\psi_{\tau})=0\,,$
(3.33)

which lead to a conserved 2-form current

$\mathcal{J}_{\mu\nu}\equiv
2i\gamma_{\mu\nu\rho\sigma}\psi^{\rho\sigma},\qquad\partial^{\nu}\mathcal{J}_{\mu\nu}=0\,.$
(3.34)

and a conserved 3-form current respectively

$\tilde{\mathcal{J}}_{\mu\nu\rho}\equiv-2\gamma_{\mu\nu\rho\sigma\tau}(\psi^{\sigma\tau}-2i\gamma^{\sigma}\psi^{\tau}),\qquad\partial^{\rho}\tilde{\mathcal{J}}_{\mu\nu\rho}=0\,.$
(3.35)

We then construct topological operators

$U_{\epsilon}(\mathcal{V})=\exp\left(i\int_{\mathcal{V}}\left[\bar{\epsilon}\,(\star\mathcal{J})_{(3)}+(\star\bar{\mathcal{J}})_{(3)}\epsilon\right]\right)\,,$
(3.36)

supported on 3d volume $\mathcal{V}$ and

$\tilde{U}_{\tilde{\epsilon}}(\tilde{\mathcal{S}})=\exp\left(i\int_{\tilde{\mathcal{S}}}\left[\bar{\tilde{\epsilon}}\,(\star\tilde{\mathcal{J}})_{(2)}+(\star\bar{\tilde{\mathcal{J}}})_{(2)}\tilde{\epsilon}\right]\right)\,,$
(3.37)

with surface support $\tilde{\mathcal{S}}$. They act on
$V_{\eta}(\mathcal{C})$ and $W_{\xi}(\mathcal{S})$ as

$\displaystyle\langle U_{\epsilon}(\mathcal{V})V_{\eta}(\mathcal{C}))\rangle$
$\displaystyle=\exp\left(i(\bar{\epsilon}\eta+\bar{\eta}\epsilon)\langle\mathcal{C},\mathcal{V}\rangle\right)\langle
V_{\eta}(\mathcal{C}))\rangle\ \,,$ (3.38)
$\displaystyle\langle\tilde{U}_{\tilde{\epsilon}}(\tilde{\mathcal{S}})W_{\xi}(\mathcal{S})\rangle$
$\displaystyle=\exp\left(i(\bar{\tilde{\epsilon}}\xi+\bar{\xi}\tilde{\epsilon})\langle\mathcal{S},\tilde{\mathcal{S}}\rangle\right)\langle
W_{\xi}(\mathcal{S})\rangle\ \,.$

As a conclusion, when compactified on a circle, the six dimensional chiral
fermionic 2-form symmetry gives fermionic 2-form symmetry and fermionic 1-form
symmetry in five dimensions. Similar result also exists in reduction from ten
to nine dimensions: one considers 10d free chiral fermionic 4-form described
by the action (2.24) for $p=4$, which reduces to a system involving 9d
fermionic 4-form and 3-form. For the same reason, 10d fermionic 4-form
symmetry leads to 9d fermionic 4-form and 3-form symmetries. However, these
fermionic fields do not fit into super Poincaré multiplets.

To make contact with five-dimensional (linearised) supergravity, we can use
the second equation of motion (3.33) to eliminate $\psi_{\mu\nu}$ and obtain a
free Rarita-Schwinger Lagrangian for $\psi_{\mu}$ in 5d [23], which exhibits
solely fermionic 1-form symmetry. Alternatively, the same equation of motion
can be used to integrate out $\psi_{\mu}$ and leave alone a fermionic 2-form
action for $\psi_{\mu\nu}$ in five dimensions (it could be an ingredient of
dual description of five-dimensional supergravity studied in [38]), which only
gives rise to fermionic 2-form symmetry. We also see that this switching
between fermionic 1-form and 2-form symmetries has the origin from one
dimension higher.

##  4 Gauging of fermionic higher-form symmetries

In this section, we discuss the gauging of fermionic $p$-form global
symmetries introduced in this paper. In certain examples (e. g. free fermionic
$p$-form field), the gauging of fermionic $p$-form symmetries is obstructed by
a ’t Hooft anomaly, which can be cancelled by a higher-dimensional anomaly
TQFT.

#### Free fermionic $p$-form gauge field

We consider the free fermionic $p$-form gauge field written in the
differential form notations (we set the coefficient to be $(-1)$ for
simplicity)

$S[\psi]_{\text{free}}=\int_{M_{d}}-\bar{\psi}_{(p)}\wedge\gamma_{(d-2p-1)}\wedge
d\psi_{(p)}\,\,.$ (4.1)

To gauge the $p$-form shifting symmetry of $\psi_{(p)}$, we introduce a
background $(p+1)$-form fermionic field $\Psi_{(p+1)}$, with the following
gauge transformation

$\displaystyle\delta\psi_{(p)}$ $\displaystyle=\epsilon_{(p)}$ (4.2)
$\displaystyle\delta\Psi_{(p+1)}$ $\displaystyle=d\epsilon_{(p)}\,.$

Now let us rewrite (4.1) in terms of an integration of a $(d+1)$-form on
$M_{d+1}$, where $M_{d}=\partial M_{d+1}$ is the boundary of $M_{d+1}$.

$S[\psi]_{\text{free}}=\int_{M_{d+1}}-d\bar{\psi}_{(p)}\wedge\Gamma_{(d-2p-1)}\wedge
d\psi_{(p)}\,\,.$ (4.3)

$\Gamma_{(d-2p-1)}$ is the antisymmetric product of $\gamma$-matrices
$\Gamma_{i}$ in $(d+1)$-dimensions. When $d$ is even, we can use a set of
$\Gamma_{i}$ $(i=0,\dots,d)$ with the same dimension as the $\gamma$-matrices
in $d$-dimensions. When $d$ is odd, $\Gamma_{i}$ can be chosen as

$\Gamma_{i}=\begin{pmatrix}0&\gamma_{i}\\\
\gamma_{i}&0\end{pmatrix}\quad(i=0,\dots,d-1)\ ,\
\Gamma_{d}=\begin{pmatrix}I&0\\\ 0&-I\end{pmatrix}\,,$ (4.4)

and dimensions of the spinors $\Psi_{(p+1)}$ and $\psi_{(p)}$ after the uplift
are also doubled.

We use the gauge invariant linear combination $\Psi_{(p+1)}-d\psi_{(p)}$ to
write the gauged action on $M_{d+1}$ as

$S[\psi]_{\text{gauged}}=\int_{M_{d+1}}-(\bar{\Psi}_{(p+1)}-d\bar{\psi}_{(p)})\wedge\Gamma_{(d-2p-1)}\wedge(\Psi_{(p+1)}-d\psi_{(p)})\,\,.$
(4.5)

Nonetheless, the extra terms cannot be absorbed into the $d$-dimensional
action, hence they are interpreted as a ’t Hooft anomaly polynomial.

#### Fermionic “BF” theory with gaugeable fermionic symmetry

We discuss the examples of fermionic “BF” theories with a mixed ’t Hooft
anomaly, where one can gauge a part of fermionic global symmetries.

We construct an action in $d$-dimensions, in the form of

$S_{\text{free}}=\int_{M_{d}}-\bar{\chi}_{(k)}\wedge\gamma_{(d-p-k-1)}\wedge
d\psi_{(p)}+\text{c.c.}\,.$ (4.6)

The theory has a fermionic $p$-form global symmetry

$\delta\psi_{(p)}=\lambda_{(p)}\,$ (4.7)

and a fermionic $k$-form global symmetry

$\delta\chi_{(k)}=\rho_{(k)}\,.$ (4.8)

They have background gauge fields $\Psi_{(p+1)}$ and $X_{(k+1)}$ respectively.

Similar to the discussion for a free fermionic $p$-form field, one cannot
gauge the fermionic $p$-form and $k$-form global symmetries simultaneously,
due to a mixed ’t Hooft anomaly in $(d+1)$-dimensions. However, one can gauge
only one of these symmetries.

For instance, the new action after the gauging the $p$-form symmetry is

$S_{\text{gauged}}=\int_{M_{d}}-\bar{\chi}_{(k)}\wedge\gamma_{(d-p-k-1)}\wedge(d\psi_{(p)}-\Psi_{(p+1)})+\text{c.c.}\,.$
(4.9)

It is gauge invariant under

$\displaystyle\delta\psi_{(p)}$ $\displaystyle=\epsilon_{(p)}$ (4.10)
$\displaystyle\delta\Psi_{(p+1)}$ $\displaystyle=d\epsilon_{(p)}\,.$

Nonetheless, the gauge symmetry of the field $\chi_{k}$ is broken after the
gauging process.

In the special case of $k=0$, which is a spinor coupled to a fermionic
$p$-form field, there is no gauge symmetry to break.

#### Other theories with gaugeable fermionic symmetries

We can also construct actions where all the fermionic fields are accompanied
with a derivative, for instance

$S[\psi]_{\rm free}=-\int_{M^{d}}d\bar{\psi}_{(p)}\wedge*d\psi_{(p)}\,.$
(4.11)

In this case, the fermionic $p$-form global symmetry ($d\lambda_{(p)}=0$)

$\delta\psi_{(p)}=\lambda_{(p)}$ (4.12)

can be gauged by coupling to the background gauge field $\Psi_{(p+1)}$:

$S[\psi]_{\rm
gauged}=-\int_{M^{d}}(\bar{\Psi}_{(p+1)}-d\bar{\psi}_{(p)})\wedge*(\Psi_{(p+1)}-d\psi_{(p)})\,.$
(4.13)

##  5 Examples with broken fermionic symmetries

###  5.1 Rarita-Schwinger field coupled to gauge field

In this section, we discuss an interacting system with a Rarita-Schwinger
field $\psi_{\mu}$ coupled to a gauge field $A_{\mu}$ and a spinor $\xi$,
where the fermionic 1-form symmetry associated to $\psi_{\mu}$ is broken by
the coupling term. The gauge field $A_{\mu}$ can be Abelian or non-Abelian,
and the spinor $\xi$ which plays the role of auxiliary field is also coupled
to the gauge field $A_{\mu}$. This model was firstly introduced in [39] as
_extended gauged Rarita-Schwinger theory_ , where an exact off-shell fermionic
gauge invariance is achieved with help of the auxiliary field $\xi$. This off-
shell fermionic gauge invariance is a generalization of the fermionic $0$-form
gauge symmetry (2.10), and we replace $\partial_{\mu}$ by the usual gauge
covariant derivative $D_{\mu}$. In addition, this theory has no superluminal
modes and it is consistent as a non-supersymmetric, classical field theory in
four dimensions. However, whether it can be properly quantized remains a open
question and for more details we refer to [40, 41] and references therein.

We use the following action, which can be defined in general $d$ space-time
dimensions, and for $d>3$:

$\displaystyle S$ $\displaystyle=-\frac{1}{4}\int
d^{d}xF_{\mu\nu}F^{\mu\nu}-\frac{1}{4}\int
d^{d}x\left(\bar{\psi}_{\mu}\gamma^{\mu\nu\rho}\overset{\rightarrow}{D}_{\nu}\psi_{\rho}-\bar{\psi}_{\mu}\overset{\leftarrow}{D}_{\nu}\gamma^{\mu\nu\rho}\psi_{\rho}\right)$
(5.1) $\displaystyle+\frac{ig}{4}\int
d^{d}x\left(\bar{\xi}F_{\mu\nu}\gamma^{\mu\nu\rho}\psi_{\rho}-\bar{\psi}_{\mu}\gamma^{\mu\nu\rho}F_{\nu\rho}\xi\right)-\frac{ig}{8}\int
d^{d}x\left(\bar{\xi}F_{\mu\nu}\gamma^{\mu\nu\rho}\overset{\rightarrow}{D}_{\rho}\xi-\bar{\xi}\overset{\leftarrow}{D}_{\mu}\gamma^{\mu\nu\rho}F_{\nu\rho}\xi\right)\,.$

The covariant derivatives are111111As mentioned, the non-Abelian version of
the model is also valid and for simplicity we demonstrate here computations
only in the Abelian case.

$\displaystyle\overset{\rightarrow}{D}_{\mu}\psi_{\nu}$
$\displaystyle=\partial_{\mu}\psi_{\nu}-igA_{\mu}\psi_{\nu}$ (5.2)
$\displaystyle\bar{\psi}_{\nu}\overset{\leftarrow}{D}_{\mu}$
$\displaystyle=\partial_{\mu}\bar{\psi}_{\nu}+igA_{\mu}\bar{\psi}_{\nu}$
$\displaystyle\overset{\rightarrow}{D}_{\mu}\xi$
$\displaystyle=\partial_{\mu}\xi-igA_{\mu}\xi$
$\displaystyle\bar{\xi}\overset{\leftarrow}{D}_{\mu}$
$\displaystyle=\partial_{\mu}\bar{\xi}+igA_{\mu}\bar{\xi}\,.$

The above action (5.1) has a fermionic 0-form gauge symmetry

$\delta_{\epsilon}A_{\mu}=0\ ,\
\delta_{\epsilon}\bar{\psi}_{\mu}=\bar{\epsilon}\overset{\leftarrow}{D}_{\mu}\
,\ \delta_{\epsilon}\psi_{\mu}=\overset{\rightarrow}{D}_{\mu}\epsilon\ ,\
\delta_{\epsilon}\bar{\xi}=\bar{\epsilon}\ ,\ \delta_{\epsilon}\xi=\epsilon$
(5.3)

and the usual 0-form gauge symmetry

$\delta_{\alpha}A_{\mu}=\frac{1}{g}\partial_{\mu}\alpha\ ,\
\delta_{\alpha}\bar{\psi}_{\mu}=-i\alpha\bar{\psi}_{\mu}\ ,\
\delta_{\alpha}\psi_{\mu}=i\alpha\psi_{\mu}\ ,\
\delta_{\alpha}\bar{\xi}=-i\alpha\bar{\xi}\ ,\
\delta_{\alpha}\xi=i\alpha\xi\,.$ (5.4)

The equations of motion for $\psi_{\mu}$, $\xi$ and $A_{\mu}$ read

$\displaystyle\frac{1}{2}\gamma^{\mu\nu\rho}\overset{\rightarrow}{D}_{\nu}\left(\psi_{\rho}-\overset{\rightarrow}{D}_{\rho}\xi\right)$
$\displaystyle=0$ (5.5)
$\displaystyle-\frac{ig}{4}F_{\mu\nu}\gamma^{\mu\nu\rho}\left(\psi_{\rho}-\overset{\rightarrow}{D}_{\rho}\xi\right)$
$\displaystyle=0$
$\displaystyle-\frac{1}{2}\partial_{\mu}F^{\mu\nu}+\frac{ig}{4}\partial_{\mu}\left(\bar{\xi}\gamma^{\mu\nu\rho}\psi_{\rho}-\bar{\psi}_{\rho}\gamma^{\rho\mu\nu}\xi+\frac{1}{2}\bar{\xi}\overset{\leftarrow}{D}_{\rho}\gamma^{\rho\mu\nu}\xi-\frac{1}{2}\bar{\xi}\gamma^{\mu\nu\rho}\overset{\rightarrow}{D}_{\rho}\xi\right)$
$\displaystyle-\frac{ig}{2}\bar{\psi}_{\mu}\gamma^{\mu\nu\rho}\psi_{\rho}-\frac{g^{2}}{4}\bar{\xi}F_{\mu\rho}\gamma^{\mu\nu\rho}\xi$
$\displaystyle=0\,.$

Applying $\overset{\rightarrow}{D}_{\mu}$ on the first equation, one gets the
second equation by virtue of the antisymmetrization in covariant derivatives,
thus the equation of motions of $\xi$ is a subset of those of $\psi_{\mu}$ and
indicating that $\xi$ is auxiliary.

One can construct the following Wilson loop observable that is gauge invariant
under both (5.3) and (5.4) gauge symmetries:

$\mathcal{L}_{\eta,\zeta}(\mathcal{C})=\mathop{\mathrm{Tr}}\nolimits\exp\mathcal{P}\oint_{\mathcal{C}}\begin{pmatrix}0&(\bar{\psi}_{\mu}-\bar{\xi}\overset{\leftarrow}{D}_{\mu})\eta\\\
\bar{\zeta}(\psi_{\mu}-\overset{\rightarrow}{D}_{\mu}\xi)&0\end{pmatrix}dx^{\mu}\,.$
(5.6)

We used an off-diagonal $2\times 2$ matrix form, similar to the fermionic part
of Wilson loops in the literature [42, 43]. Nonetheless, the loop operator
(5.6) is explicitly Lorentz covariant.

However, one cannot construct a topological generator for fermionic global
1-form symmetry, as the equations of motion (5.5) are not in form of a total
derivative of $d$. Suppose that we write down the operator

$U_{\epsilon}(M^{(d-2)})=\exp\left(i\int_{M^{(d-2)}}\left[\bar{\epsilon}(\star\mathcal{J})_{(d-2)}+(\star\bar{\mathcal{J}})_{(d-2)}\epsilon\right]\right)\,,$
(5.7)
$\mathcal{J}^{\mu\nu}=\gamma^{\mu\nu\rho}\left(\psi_{\rho}-\overset{\rightarrow}{D}_{\rho}\xi\right)\,,$
(5.8)

$\mathcal{J}^{\mu\nu}$ does not satisfy the on-shell condition
$d\star\mathcal{J}=0$. $\int_{M^{(d-1)}}d\star\mathcal{J}$ is not a quantized
quantity either.

###  5.2 Swampland implications

We briefly discuss how to relate the fermionic global symmetries we discussed
in this paper with the no global symmetry swampland conjectures.

Analogous to the bosonic global symmetries, a quantum gravity theory with an
exact fermionic global symmetry should live in the swampland. Since one cannot
consistently gauge the fermionic shifting symmetry in a fermionic $p$-form
field (see section 4), such symmetries should be explicitly broken.

For instance, in the usual formulations of supergravity, the local
supersymmetry transformation of $\psi_{\mu}$ is

$\displaystyle\delta_{\epsilon}\psi_{\mu}$
$\displaystyle=\nabla_{\mu}\epsilon$ (5.9)
$\displaystyle=\partial_{\mu}\epsilon+\frac{1}{4}w_{\mu}^{ab}\gamma_{ab}\epsilon\,,$

where $w_{\mu}^{ab}$ is the spin-connection written in matrix form.

If we write down the loop operator for the Rarita-Schwinger field $\psi_{\mu}$

$V_{\eta}(\mathcal{C})=\exp\left(i\oint_{\mathcal{C}}(\bar{\eta}\psi_{\mu}+\bar{\psi}_{\mu}\eta)\,dx^{\mu}\right)\,,$
(5.10)

it is not gauge invariant on a general curved background. The fermionic 1-form
global symmetries are explicitly broken in supergravity theories, which is
consistent with the no global symmetry swampland conjectures.

We can also consider the fate of fermionic 0-form shift symmetry
$\delta\psi=\epsilon$ for a spinor field ($d\epsilon=0$). In a usual
supergravity theory, the free action for $\psi$ becomes

$S[\psi]_{\text{sugra}}=-\int\,d^{d}x\sqrt{|\det{g}|}\,\bar{\psi}\gamma^{\mu}\nabla_{\mu}\psi\,,$
(5.11)

which breaks the shift symmetry of $\psi$. Of course, one can still ask about
the possibilities to take $\epsilon$ to be covariantly constant, i.e.
$\nabla_{\mu}\epsilon=0$. If the curved space-time admits such spinors then
this shift symmetry (with $d\epsilon=0$ replaced by $\nabla_{\mu}\epsilon=0$)
is unbroken. Such a global symmetry exists on a space-time with covariantly
constant spinors, but it is not preserved in general curved space-time. Hence
we do not have such fermionic global symmetries in a gravity theory.

For a fermionic $p$-form field in curved space-time with action

$S[\psi]=\int_{M_{d}}-\bar{\psi}_{(p)}\wedge\gamma_{(d-2p-1)}\wedge\nabla\psi_{(p)}\,,$
(5.12)

one can also introduce the shifting symmetry
$\delta\psi_{(p)}=\epsilon_{(p)}$, where the parameter $\epsilon_{(p)}$
satisfies the covariantly flat condition

$\nabla\epsilon_{(p)}=0\,.$ (5.13)

Nonetheless, such a symmetry is broken on a general space-time manifold as
well.

##  6 Discussions

In this paper, we introduced and explored the concept of fermionic $p$-form
global symmetries. We discussed physical examples with the symmetry and their
gaugings, as well as examples where the fermionic $p$-form global symmetries
are broken. Here we clarify some points and discuss the future directions.

  * •

Non-compactness of the fermionic symmetry group

In the cases of fermionic $p$-form global symmetries, the symmetry parameter
is a spinor with Grassmannian components, thus the symmetry group is always a
non-compact fermionic translation group. This feature obstructs the partial
breaking of such symmetries, since the fermionic charge associated to the
symmetry is not quantized. One may need to compactify the space of spinors, in
order to construct more non-trivial models with fermionic $p$-form global
symmetries.

  * •

SUSY transformation of Wilson loops in SUSY gauge theories

In SUSY gauge theories, there is a natural fermionic symmetry acting on the
Wilson loop objects, which is the SUSY transformation, see e. g. [44]. In our
language, such a symmetry is still a fermionic 0-form symmetry that acts on
local fields, e. g. $A_{\mu}$. Since the action of symmetry is not related to
the linking of a $(d-2)$-dimensional topological operator with the Wilson
loop.

  * •

Non-invertible fermionic higher-form symmetries

The fermionic topological defect lines (TDL) were recently studied in 2d
fermionic CFTs [45], which generate non-invertible 0-form fermionic
symmetries. It is tentative to discuss the analogue of fermionic topological
defects in 3d or higher dimensional fermionic CFTs, where non-invertible
fermionic higher-form symmetries potentially exist.

  * •

String theory realizations

One may attempt to realize the fermionic $p$-form global symmetries in string
theory. Nonetheless, one needs a massless fermionic $p$-form gauge field in
the space-time, that is almost free. Such a scenario only happens in the free,
tensionless limit of superstring theory, where we only have an infinite tower
of free, massless higher-spin fields. It would be interesting if one can
construct a setup that is not a completely free system.

  * •

6d (4,0) theory

A free theory based on the 6d $\mathcal{N}=(4,0)$ supermultiplet exists, and
its circle reduction yields linearised 5d $\mathcal{N}=8$ supergravity [8].
The interacting $(4,0)$ theory is also conjectured to exist, and as an
extension of the free $(4,0)$ tensor theory the interacting one could be the
strong coupling limit of of 5d non-linear maximal supergravity (see [36] for a
recent review). In this context, the presumed interacting $(4,0)$ theory would
be a new superconformal phase [46] of M-theory at six dimensions with maximal
supersymmetry. On the other hand, standard folklore states that there is no
global symmetries in quantum gravity. According to this criterion, fermionic
2-form symmetries of the free limit must be either gauged or broken in the UV.
Naive gauging can not be applied to free fermionic $p$-form symmetries as
shown in section 4. It would be interesting to ask how the gauging/breaking
mechanism goes when we moving from the free theory to the interacting version.
For instance, given the relation between 5d linearised SUGRA and free (4,0)
theory, one can argue that couplings between $\Psi_{MN}$ and exotic graviton
$C_{MNPQ}$ break the fermionic 2-form symmetry explicitly, while the couplings
between linearised graviton $h_{\mu\nu}$ and gravitino $\psi_{\mu}$ break
fermionic 1-form symmetry in 5d. Moreover, as already pointed out in [8],
extended objects exist and carrying various charges according to the (4,0)
superalgebra. We can ask how to break the fermionic 2-form symmetry by these
localized UV input.

  * •

Fermionic TQFTs

We have also constructed a novel type of topological quantum field theories
(on flat manifolds) using fermionic tensor fields, with an action of the form

$S=\left\\{\begin{array}[]{rl}\sum_{p_{i}+q_{i}=d-1}c_{i}\int_{M_{d}}\bar{\psi}_{(p_{i})}\wedge
d\psi_{(q_{i})}&(\text{odd}\ d)\\\ \\\
\sum_{p_{i}+q_{i}=d-1}c_{i}\int_{M_{d}}\bar{\psi}_{(p_{i})}\wedge(1+\gamma_{d+1})d\psi_{(q_{i})}&(\text{even}\
d)\,.\end{array}\right.$ (6.1)

This is beyond the scope of usual spin TQFTs, which only contain 0-form
spinors, see for example [47]. It would be interesting to further investigate
the properties of the fermionic tensor TQFTs and their relations with the
known models.

## Acknowledgements

We thank Victor Lekeu, Ran Luo, Ruben Minasian, Sakura Schafer-Nameki, Kaiwen
Sun, Qing-Rui Wang, Junbao Wu, Fengjun Xu, Zhi-Cheng Yang for discussions. YNW
and YZ are supported by National Science Foundation of China under Grant No.
12175004 and by Peking University under startup Grant No. 7100603667. YNW is
supported by Young Elite Scientists Sponsorship Program by CAST (2022QNRC001).
YZ is supported by the International Postdoctoral Exchange Program of the
Office of China Postdoc Council (OCPC) and Peking University under Grant No.
YJ20220018.

##  Appendix A Conventions

We use the “mostly plus” signature for $d$-dimensional Minkowski metric:
$\eta=\text{diag}(-,+,\ldots,+)$. Gamma matrices $\gamma_{\mu}$
($\mu=0,\ldots,d-1$) satisfy the anti-commutation relation

$\\{\gamma_{\mu},\gamma_{\nu}\\}=2\eta_{\mu\nu}\,.$ (A.1)

Hermitian property of gammas is
$(\gamma^{\mu})^{\dagger}=\gamma^{0}\gamma^{\mu}\gamma^{0}$. In $d=2m$
dimensions the chirality matrix is defined as

$\gamma_{d+1}=(-i)^{m+1}\gamma_{0}\gamma_{1}\ldots\gamma_{d-1}\,.$ (A.2)

For a spinor $\psi$, its Dirac conjugate is
$\bar{\psi}=i\psi^{\dagger}\gamma^{0}$.
A differential $p$-form $\omega_{(p)}$ is expressed in components as

$\omega_{(p)}=\frac{1}{p!}\omega_{\mu_{1}\ldots\mu_{p}}dx^{\mu_{1}}\wedge\ldots\wedge
dx^{\mu_{p}}\,,$ (A.3)

and its exterior derivative $(d\omega)_{(p+1)}$ is a $(p+1)$-form with
components

$(d\omega)_{\mu_{1}\ldots\mu_{p+1}}=(p+1)\partial_{[\mu_{1}}\omega_{\mu_{2}\ldots\mu_{p+1]}}\,.$
(A.4)

The components of the wedge product of a $p$-form $\omega_{(p)}$ and a
$q$-form $\eta_{(q)}$ are

$(\omega\wedge\eta)_{\mu_{1}\ldots\mu_{p}\nu_{1}\ldots\nu_{q}}=\frac{(p+q)!}{p!q!}\omega_{[\mu_{1}\ldots\mu_{p}}\eta_{\nu_{1}\ldots\nu_{q}]}\,.$
(A.5)

The Hodge star operator $\star$ maps $p$-forms to $(d-p)$-forms and our
convention is

$(\star\omega)_{\mu_{1}\ldots\mu_{d-p}}=\frac{1}{p!}\varepsilon_{\mu_{1}\ldots\mu_{d-p}}{}^{\nu_{1}\ldots\nu_{p}}\omega_{\nu_{1}\ldots\nu_{p}}\,,$
(A.6)

where $\varepsilon_{\mu_{1}\ldots\mu_{d}}$ is the Levi-Civita symbol and
$\varepsilon_{01\ldots d-1}=1$, $\varepsilon^{01\ldots d-1}=-1$.
On a curved manifold with metric $g_{\mu\nu}$, $\varepsilon$ generalises to a
tensor according to the normalisation

$\varepsilon_{01\ldots d-1}=\sqrt{|\det{g}|}\,,\qquad\varepsilon^{01\ldots
d-1}=\frac{-1}{\sqrt{|\det{g}|}}\,.$ (A.7)

The invariant volume form is

$\sqrt{|\det{g}|}\,d^{d}x\equiv\sqrt{|\det{g}|}\,dx^{0}\wedge\ldots\wedge
dx^{d-1}=\frac{1}{d!}\varepsilon_{\mu_{1}\ldots\mu_{d}}\,dx^{\mu_{1}}\wedge\ldots\wedge
dx^{\mu_{d}}\,,$ (A.8)

here we would also write $dV^{\mu_{1}\ldots\mu_{d}}\equiv
dx^{\mu_{1}}\wedge\ldots\wedge dx^{\mu_{d}}$ for short.
Integration of $d$-forms over the manifold $M^{(d)}$ is given as

$\displaystyle\int_{M^{(d)}}\upsilon_{(d)}$
$\displaystyle=\int_{M^{(d)}}\frac{1}{d!}\upsilon_{\mu_{1}\ldots\mu_{d}}\,dx^{\mu_{1}}\wedge\ldots\wedge
dx^{\mu_{d}}=\int_{M^{(d)}}\frac{1}{d!}\upsilon_{\mu_{1}\ldots\mu_{d}}\,dV^{\mu_{1}\ldots\mu_{d}}$
(A.9) $\displaystyle=\int_{M^{(d)}}\upsilon_{01\ldots d-1}\,d^{d}x$
$\displaystyle\equiv\int_{M^{(d)}}\upsilon(x)_{01\ldots d-1}dx^{0}dx^{1}\ldots
dx^{d-1}\,,$

and integration of a scalar (0-form) $\phi$ is defined as the integral of its
Hodge dual

$\int_{M^{(d)}}\star\phi=\int_{M^{(d)}}\phi\,\sqrt{|\det{g}|}\,d^{d}x\,.$
(A.10)

Useful formulae:

$\displaystyle\star\omega\wedge\eta$
$\displaystyle=\star\eta\wedge\omega=\frac{1}{p!}\omega_{\mu_{1}\ldots\mu_{p}}\eta^{\mu_{1}\ldots\mu_{p}}\sqrt{|\det{g}|}\,d^{d}x\,,$
(A.11) $\displaystyle d\star\upsilon\wedge\omega$
$\displaystyle=(-1)^{d-p-1}\frac{1}{p!}\partial_{\mu}\upsilon^{\mu\nu_{1}\ldots\nu_{p}}\omega_{\nu_{1}\ldots\nu_{p}}\sqrt{|\det{g}|}\,d^{d}x\,,$

where $\omega$ and $\eta$ are both $p$-forms and $\upsilon$ is a $(p+1)$-form.
Let $\mathcal{C}^{(p)}$ denote a $p$-cycle (closed oriented submanifold of
dimension $p$), its Poincaré dual is the cohomology class of a $(d-p)$-form
$J_{(d-p)}(\mathcal{C}^{(p)})$ such that for any $p$-form $A_{(p)}$ the
following relation holds

$\int_{\mathcal{C}^{(p)}}A_{(p)}=\int_{M^{(d)}}J_{(d-p)}(\mathcal{C}^{(p)})\wedge
A_{(p)}\,.$ (A.12)

For submanifolds $U$ and $V$ with dimension $p$ and $d-p-1$ and such that $V$
is the boundary of a $(d-p)$-dimensional submanifold $W$, i.e. $\partial W=V$
(in fact, both $U$ and $V$ should be boundaries of some other submanifolds in
order to define the linking number [14]). The linking number $\langle
U,V\rangle$ is given as the intersection number $\mathcal{I}(U,W)$ of $U$ and
$W$

$\langle U,V\rangle=\mathcal{I}(U,W)=\int_{M^{(d)}}J_{(d-p)}(U)\wedge
J_{(p)}(W)=\int_{U}J_{(p)}(W)$ (A.13)

This agrees with the definition of [48].
The free fermionic $p$-form action (2.24) is

$\displaystyle S[\psi_{(p)}]$
$\displaystyle=-(-1)^{\frac{p(p-1)}{2}}\,\int\\!d^{d}\\!x\;\bar{\psi}_{\mu_{1}\mu_{2}\dots\mu_{p}}\,\gamma^{\mu_{1}\mu_{2}\dots\mu_{p}\nu\rho_{1}\rho_{2}\dots\rho_{p}}\,\partial_{\nu}\psi_{\rho_{1}\rho_{2}\dots\rho_{p}}$
$\displaystyle=-(-1)^{\frac{p(p-1)}{2}}(-1)^{p}\,\int\\!d^{d}\\!x\;\bar{\psi}_{\mu_{1}\mu_{2}\dots\mu_{p}}\,\partial_{\nu}\mathcal{J}^{\nu\mu_{1}\mu_{2}\dots\mu_{p}}$
$\displaystyle=-p!(-1)^{\frac{p(p-1)}{2}}(-1)^{p}(-1)^{p(d-p)}(-1)^{d-p-1}\,\int\bar{\psi}\wedge(d\star\mathcal{J})$
$\displaystyle=-C(d,p)\,\int\bar{\psi}\wedge(d\star\mathcal{J})\,,$

with

$C(d,p)=-p!(-1)^{\frac{(p+1)(2d-p)}{2}}\,.$ (A.14)

The shifted $p$-form action used in section 2
($\bar{\mathcal{J}}_{\mu_{1}\dots\mu_{p}\nu}=-\bar{\psi}^{\rho_{1}\dots\rho_{p}}\gamma_{\rho_{p}\dots\rho_{1}\nu\mu_{p}\dots\mu_{1}}$)

$\displaystyle S[\psi_{(p)}-\epsilon J_{(p)}]$
$\displaystyle=S[\psi_{(p)}]+(-1)^{\frac{p(p-1)}{2}}\,\int\\!d^{d}\\!x\;\left((-1)^{p}\bar{\epsilon}J_{\mu_{1}\dots\mu_{p}}\partial_{\nu}\mathcal{J}^{\nu\mu_{1}\mu_{2}\dots\mu_{p}}+(-1)^{p}\partial_{\nu}\bar{\mathcal{J}}^{\nu\rho_{1}\dots\rho_{p}}\epsilon
J_{\rho_{1}\dots\rho_{p}}\right)$
$\displaystyle=S[\psi_{(p)}]+(-1)^{\frac{p(p-1)}{2}}\,\int
p!(-1)^{p}(-1)^{p(d-p)}(-1)^{d-p-1}J\wedge\left(d\star[\bar{\mathcal{J}}\epsilon+\bar{\epsilon}\mathcal{J}]\right)\,,$
$\displaystyle=S[\psi_{(p)}]+C(d,p)\,\int
J\wedge\left(d\star[\bar{\mathcal{J}}\epsilon+\bar{\epsilon}\mathcal{J}]\right)\,,$

suggesting that $U_{\epsilon}(M^{(d-p-1)})$ should be

$U_{\epsilon}(M^{(d-p-1)})=\exp\left(i\,C(d,p)\int_{M^{(d-p-1)}}\left(\star[\bar{\mathcal{J}}\epsilon+\bar{\epsilon}\mathcal{J}]\right)\right)\,.$

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