CUBO A Mathematical Journal

Vol.11, No¯ 01, (21–54). March 2009

Dirac Type Gauge Theories – Motivations and

Perspectives

Jürgen Tolksdorf

TU Bergakademie Freiberg,

Freiberg/Sachsen, Germany.

email: juergen.tolksdorf@math.tu-freiberg.de

ABSTRACT

We summarize the geometrical description of a specific class of gauge theories, called “of

Dirac type”, in terms of Dirac type first order differential operators on twisted Clifford

bundles. We show how these differential operators may be geometrically considered as

being the images of sections of a specific principal fibering naturally associated with

twisted Clifford bundles. Based on the notion of real Hermitian vector bundles, we dis-

cuss the most general real Dirac type operator on “particle-anti-particle” modules over

an arbitrary (orientable) semi-Riemannian manifold of even dimension. This setting

may be appropriate for a common geometrical description of both the Dirac and the

Majorana equation.

RESUMEN

Nosotros resumimos la descripción geométrica de una clase específica de teoría gauge,

llamada "de tipo Dirac", en términos del tipo de Dirac de operadores diferenciales

de primer orden sobre fibrados de Clifford twisted. Mostramos como esos operadores

pueden ser geométricamente considerados como siendo imágenes de secciones de una fi-

bra principal específica naturalmente asociada con el fibrado de Clifford twisted. Basado

en la noción de fibrado vectorial Hermitiano real, discutimos el más general oper-

ador de tipo Dirac real sobre módulos "partícula-anti-partícula" sobre una variedad

semi-Riemanniana (orientable) arbitraria de dimensión par. Este contexto puede ser

apropiado para una descripción geométrica común para las ecuaciones de Dirac y de

Majorana.



22 Jürgen Tolksdorf CUBO
11, 1 (2009)

Key words and phrases: Dirac Type Differential Operators, Real Clifford Modules, General

Relativity, Gauge Theories, Majorana equation

Math. Subj. Class.: 53C05, 53C07, 70S05, 70S15, 83C05.

1 Synopsis

In a nutshell, Dirac type gauge theories are based on the following “universal (Dirac) action func-

tional”:

ID :

∫

M

[〈ψ, /Dψ〉E + trγcurv( /D)] dvolM . (1)

Here, /D ∈ D(E) denotes the most general Dirac type first order differential operator, acting

on the C∞(M)−module of smooth sections ψ ∈ Sec(M,E) on a Hermitian Clifford module bundle

πE : E −→ M over a smooth orientable semi-Riemannian manifold (M,gM) of even dimension

n = 2k ≥ 2. The notation 〈·, ·〉E denotes a chosen Hermitian form on E and

curv( /D) ∈ Ω2(M, End(E)) (2)

is the curvature of /D.

A detailed general discussion of the geometrical background of the functional (1) and how it

is related to the well-known general Lichnerowicz decomposition (c.f. [5] and [3])

/D
2

= −△B + VD (3)

can be found in [22] and [23]. Note that in contrast to what has been stated in the latter Reference,

however, the “Dirac potential” VD actually reads:

VD = γ(curv( /D)) − evg(ω
2
D
) + evg(∂DωD) (4)

where the “Dirac form” ωD ∈ Ω
1
(M, End(E)) is a certain one-form canonically associated with /D

and “evg” means evaluation with respect to the metric gM (c.f. below). In our discussion presented

here we shall omit the quadratic term. The third term in (4) only contributes as a boundary term.

As a consequence, the functional (1) differs from the “universal Dirac action” that is discussed in

Reference [22] by the quadratic term (and the boundary term) in (4). To calculate the curvature

of a general Dirac type operator is generally more involved than for a connection. In contrast,

explicit formulae are available for the corresponding Dirac potential. Hence, (4) may be used to

also obtain the curvature of a general /D via (neglecting the boundary term)

γ(curv( /D)) = VD + evg(ω
2
D
) . (5)

We follow this line of reasoning to calculate the curvature of the most general Dirac type operator

on “particle-anti-particle” modules in section four.



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Dirac Type Gauge Theories – Motivations and Perspectives 23

In the following we focus on a summary of some of the basic features of the universal Dirac

action (1). A detailed discussion of its motivation is presented, whereby we put emphasize to its

“universality” and its relation to various partial differential equations well-known from physics and

geometry. In the particular case of twisted Clifford bundles we discuss how Dirac type first order

differential operators can be geometrically considered as being images of sections of a principal

fibering that is naturally associated with the geometry of Clifford modules. Finally, we discuss a

specific class of real Hermitian Clifford modules. For these we present an explicit formula for the

universal Dirac action.

Our work is organized as follows: The second section is addressed to present some detailed

discussion of the motivation for the Dirac action and how it is related to various well-known “field

equations”, like Yang-Mills and Einstein’s equation of gravity. In the third section we discuss how

the Dirac action may be regarded as a functional of the metric and (endomorphism valued) super

fields. The fourth section is addressed to the Dirac action on the geometrical background of (a

specific class of) real Hermitian Clifford modules which may allow to incorporate the geometrical

description of the Majorana equation in terms of the universal Dirac action. Finally, in the fives

section we present some outlook.

2 Motivation: Four equations and one action

To get started, let us call in mind that the two most profound equations in classical physics are

provided by the Maxwell equations of electrodynamics:

dF = 0 , (6)

d∗F = jelm (7)

and the Einstein equation of gravity:

Ric(gM) −
1

2
scal(gM) = λgravτ . (8)

Here, the “electromagnetic Field strength” is geometrically represented by a (closed) two-form

F ∈ Ω2(M) on a given four-dimensional, orientable semi-Riemannian manifold (M,gM) with index

of gM equals ±2. Accordingly, the two-form ∗F denotes the Hodge-dual of F with respect to gM and

a chosen orientation of M. Moreover, the differential operator d is the usual exterior derivative.

We stress, that in the case of the Maxwell equations (6–7) the metric structure gM on the manifold

M is supposed to be fixed.

In contrast, in the case of Einstein’s theory of gravity the gravitational field is supposed to be

fully described in terms of the metric structure gM on M. However, only those metric structures are

physically admissible which satisfy Einstein’s field equation (8). The tensor Ric ∈ Sec(M, EndTM)

denotes the “Ricci-tensor” and scal ∈ C∞(M) its trace the so-called “Ricci-scalar”. For TM ։ M



24 Jürgen Tolksdorf CUBO
11, 1 (2009)

being the tangent bundle of M, the bundle EndTM ։ M is the associated bundle of endomor-

phisms on TM (over the identity on M).

The right-hand side of the Maxwell equation (7) denotes the “electrically charged matter

current”. Similarly, the right-hand side of the Einstein equation (8) is the “energy-momentum

current” associated with any form of energy and matter. The numerical constant λgrav is called the

“gravitation coupling constant”. It carries a physical dimension in contrast to the electromagnetic

coupling constant which is purely numerical (approximately 1/137).

In classical physics these source terms for non-trivial electromagnetic field strength and gravi-

tational fields are supposed to be given objects, reflecting the physical situation at hand. Of course,

as a special case one may consider the physical situation where (part of) space-time (M,gM) is filled

only with an electromagnetic field that is generated by electrically charged matter whose support

is outside the considered region of space-time. Then, within this region jelm vanishes identically

and τ is a unique function of F such that the pair (gM,F) is physically admissible provided it is a

solution of the coupled Einstein-Maxwell equations.

In a so-called “semi-classical” description of matter (i.e. within a certain approximation of a

full quantum description), the classical Maxwell and Einstein equations are supplemented by the

(gauge covariant) Dirac equation

(i/∂
A
− m)ψ = 0 . (9)

In particular, the electromagnetic current

∗jelm =
3∑

µ,ν=0

qelm〈ψ,γ(e
µ
)ψ〉E gM(eµ,eν)e

ν

≡ qelm〈ψ,γ(e
µ
)ψ〉E gM(eµ,eν)e

ν (10)

becomes a (quadratic) function of ψ ∈ Sec(M,E) such that the triple (gM,F,ψ) is physically

admissible if and only if it fulfills the now coupled Einstein-Maxwell-Dirac equation (6–9). Here,

when appropriate units are used the parameters (m,qelm) ∈ R+ × Z are physically interpreted

as “mass” and “electric charge” of the matter described in terms of the “matter field” ψ (e.g. an

electron).

In (10), e0, . . . ,e3 ∈ Sec(M,TM) is a local orthonormal frame with respect to gM and

e0, . . . ,e3 ∈ Sec(M,T∗M) its (local) dual frame.

Note that henceforth we will make use of Einstein’s summation convention whenever local

expressions come up like in (10).

Geometrically, the matter field ψ is usually considered as a section of a twisted spinor bundle

πE : E = S ⊗ W −→ M (11)

over a semi-Riemannian spin-manifold (M,gM, ΛSpin) with ΛSpin being a chosen spin structure on

M. The Hermitian vector bundle W = P ×ρ V ։ M is an associated vector bundle of a given



CUBO
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Dirac Type Gauge Theories – Motivations and Perspectives 25

principal G-bundle G →֒ P ։ M that represents the so-called “internal gauge degrees of freedom”

of matter. Here, ρ : G → GL(V ) is a unitary representation of G on a Hermitian vector space V

which servers a the typical fiber of the twisting bundle W ։ M.

In the case of electromagnetism the (semi-simple real) Lie group G equals the unitary group

U(1) with Lie-algebra LieG = iR. Accordingly, the gauge covariant Dirac operator

i/∂
A

≡ iγ ◦ (∂A)iγ ◦ (∂
S ⊗ IdW + IdS ⊗ ∂

W
) (12)

is given by

• The covariant derivative with respect to the spin connection on the spinor bundle S ։ M :

∂S
µ

loc.
= ∂µ +

1

4
[γa,γb] ωLC

µab
(13)

with ωLC ∈ Ω1(M,so(p,q)) being the Levi-Civita form that is determined by gM;

• The gauge covariant derivative on the Hermitian vector bundle W ։ M :

∂W
µ

loc.
= ∂µ + ρ

′
(Aµ) (14)

with ρ′(A) ≡ ρ′ ◦ A ∈ Ω1(M, End(W)) being a (local) U(1)−gauge potential represented on

W . The Lie-algebra representation ρ′ : LieG → End(V ) is the derived representation with

respect to the underlying group representation ρ.

Hence, locally the gauge covariant Dirac operator reads:

i/∂
A

loc.
= iγµ

(
∂µ +

1

4
ωLC
µab

[γa,γb] ⊗ IdW + IdS ⊗ ρ
′
(Aµ)

)
. (15)

Here and in the expression (10) the symbol “γ” denotes a Clifford mapping, i.e.

γ : T∗M −→ End(E)

ω 7→ γ(ω) (16)

satisfying γ(ω)2gM(ω,ω) IdE. In the following we will suppress the identity mappings IdE, IdS, IdW

on E,S,W whenever this will not cause any confusion. Also, we will not make a distinction in our

notation with respect to the metric on the tangent and the co-tangent bundle T∗M ։ M. Finally,

γa ≡ γ(ea) are the usual “gamma matrices” and [·, ·] is the ordinary commutator.

Note that the Lie algebra so(p,q) is isomorphic to the Lie algebra spin(p,q) of the spin group

and 1
2
[γa,γb] (0 ≤ a 6= b ≤ 3) are the corresponding generators of the “spinor representation” of

so(p,q). Indeed, the Clifford action γ on a twisted spinor bundle (11) is simply given by the regular

left action of the Clifford bundle

ClM ։ M (17)



26 Jürgen Tolksdorf CUBO
11, 1 (2009)

on S ⊂ ClM. Here, the Clifford bundle is the algebra bundle over (M,gM) whose typical fiber is

given by the Clifford algebra Clp,q that is generated by the Minkowski space R
p,q ≡ (R4,η), where

η(eµ, eν) :





±1 for µ = 0 ,

∓1 for 1 ≤ µ = ν ≤ 3 ,

0 for 0 ≤ µ 6= ν ≤ 3

(18)

and e0, . . . , e3 ∈ R
4 the standard basis.

Therefore, on a twisted spinor bundle the Clifford action γ is uniquely determined by the

metric gM. If γ denotes the Clifford action with respect to the metric gM, then iγ is the Clifford

action with respect to the metric −gM. Likewise, if the Clifford action is “even”, i.e. γ(ω)
t
= −γ(ω)

for all ω ∈ T∗M, then the Clifford action given by iγ is “odd”: iγ(ω)t = iγ(ω) and vice verse.

Apparently, the geometrical background of the equations (6– 9) seems quite different like

the equations themselves. To summarize: The geometrical background of the Maxwell equations

is given by the Grassmann bundle ΛM ։ M over a given orientable semi-Riemannian manifold

(M,gM). In contrast, the geometrical background of the Einstein equation is provided by so-called

SO(p,q)−reductions of the frame bundle FM ։ M. That is, the geometrical background is given

by the fiber bundle

EEH := FM ×Gl(4) GL(4)/SO(p,q) −→ M . (19)

In fact, a section of this associated bundle with typical fiber GL(4)/SO(p,q) is in one-to-one

correspondence to a semi-Riemannian structure gM of signature (p,q). We thus do not make a

distinction between a section of the Einstein-Hilbert bundle (19) and the corresponding metric.

We denote both by the same symbol. Finally, the geometrical background of the Dirac equation

is provided by a Clifford module (E,γ) over a given orientable semi-Riemannian (spin-)manifold

(M,gM).

Apparently, Maxwell’s equations, Einstein’s equation and Dirac’s equation are rather different

equations. Nonetheless, one may ask for a common geometrical root of these three equations which

play such a fundamental role in physics and mathematics.

An appropriate hint is provided by the gauge covariant Dirac equation (9) and the geometrical

interpretation of the Maxwell equations (6–7) in terms of gauge theory. For this one may regard

the electromagnetic field strength F as a section of the complexified Grassmann bundle

ΛM ⊗R C ։ M (20)

which corresponds to the curvature of a U(1)−connection on U(1) →֒ P ։ M. We emphasize that

with respect to this geometrical interpretation of the electromagnetic field strength the Maxwell

equation (6) becomes an identity (the “Bianci-identity”). If A denotes a local gauge potential of

the curvature, then (7) is read as the U(1)−Yang-Mills equation:

dA∗FA = j . (21)



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Dirac Type Gauge Theories – Motivations and Perspectives 27

Here, respectively, FA = iF ∈ Ω
2
(M,iR) is, again, the curvature of a U(1)−connection and dA its

gauge covariant exterior derivative, locally given by the first order differential operator d + A and

A ∈ Ω1(M,iR). Clearly, the adjoint action is trivial, for U(1) is abelian. Hence,

dA∗FA
loc.
= d∗FA + [A,∗FA]

= d∗FA . (22)

If j ≡ ijelm, then the Yang-Mills equation (21) is equivalent to (7).

Note that FA = d
2

A

loc.
= dA. That is, the square of the first order differential operator dA is a

zero order differential operator taking values in Ω2(M,iR).

Let δA be the formal adjoint of dA with respect to the pairing
∫
M
αc ∧ ∗β for all compactly

supported α,β ∈ Ω(M, C) ≡ Sec(M, ΛM ⊗R C). By α
c we denote the complex conjugate of α

with respect to the canonical real structure on ΛM ⊗R C that is given by α
c

:= eµ ⊗ λµ for

α
loc.
= eµ ⊗ λµ ∈ Ω

1
(M, C). It follows that δA = ± ∗ dA∗, where the sign depends on the signature

of gM and the degree of the form the operator acts on. Then, the equation (21) may be rewritten

as

δAFA = ±j (23)

and thus the original Maxwell equations become equivalent to

( dA + δA)FA = ±j . (24)

The point to be stressed here is, that the (complexified) Grassmann bundle serves as a canon-

ical Clifford module with respect to the Clifford action

γ : T∗M −→ End(ΛM ⊗R C)

ω 7→

{
ΛM ⊗R C −→ ΛM ⊗R C

α 7→ −i(extω(α) − intω(α)) .
(25)

Here, extω(α) := ω ∧ α and intω(α) := α(ω
♯, ·) with ω♯ ∈ TM is the metric dual with respect to

gM : β(ω
♯
) := gM(ω,β) for all β ∈ T

∗M.

As consequence,

dA + δAi/∂A , (26)

with

i/∂
A

loc.
= iγµ(∂µ +

1

4
ωLC
µab

[γa,γb] + Aµ) . (27)

The Maxwell equations for purely imaginary FA ∈ Sec(M, ΛM ⊗R C) can thus be brought into

a form analogous to the Dirac equation for ψ ∈ Sec(M,E) :

i/∂
A
FA = ±j . (28)



28 Jürgen Tolksdorf CUBO
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The similarity between the Dirac equation (9) and (28) can be made even more close by noting

that ΛM ⊗R C ≃ ClM ⊗R C ≃ End(SC), where SC ≡ S ⊗R C. Hence, ΛM ⊗ C ≃ SC ⊗ S
∗
C

and the

(complexified) spinor bundle SC ։ M (with respect to a chosen spin structure) can be regarded

as a sub-vector bundle of the Grassmann bundle:

SC →֒ ΛM ⊗ C ։ M . (29)

Geometrically, the complexified Grassmann bundle ΛM⊗R C is but a special twisted Grassmann

bundle

ΛM ⊗ L −→ M (30)

with L := M × C → M being the trivial complex line bundle over M. The Hermitian Clifford

module

πΛ, E : EΛ, E ≡ ΛM ⊗ E −→ M , (31)

with E := L ⊕ W ։ M being the Whitney sum of the two Hermitian vector bundles L ։ M and

E ։ M, actually provides a common geometrical setting for the Dirac and Maxwell equations.

Obviously, all of this can be immediately generalized to arbitrary twisted Grassmann bundles

parameterized by arbitrary Hermitian vector bundles E ։ M over (M,gM). In this case, one only

has to replace the covariant derivative ∂S that corresponds to a chosen spin structure on M by

the covariant derivative ∂Λ of the Levi-Civita connection on ΛM ։ M with respect to the induced

metric gΛM. Then, (12) is replaced by the twisted Gauss-Bonnet like operator

i/∂
A

= iγ ◦ (∂Λ ⊗ IdE + IdΛ ⊗ ∂
E
)

= dA + δA . (32)

Note that locally there is no distinction between the first order operators (32) and (12). This is,

because the bundle of homomorphisms End(E) ։ M of any Clifford module (E,γ) over an even

dimensional semi-Riemannian manifold (M,gM) globally decomposes as

End(E) ≃ (ClM ⊗R C) ⊗ EndCl(E) . (33)

Here, EndCl(E) denote the sub-algebra of γ−invariant endomorphisms on E ։ M. The fundamental

isomorphism (33) can be inferred from the two Wedderburn Theorems about “invariant linear

mappings” (c.f. [9] and [3]). In fact, the use of this global decomposition forces the dimension of

M to be even such that Clp,q is simple.

Finally, nothing basically chances even in the case the Maxwell equations are replaced by

general Yang-Mills equations, i.e. the abelian structure group U(1) is replaced by an arbitrary

(semi-simple, real and compact) Lie group G. In this case, one only has to replace the (trivial)

line bundle L ։ M by the adjoint bundle ad(P) := P ×ad LieG ։ M.

Like in the particular case of a spinor bundle, the Clifford action (25) is uniquely determined

by the metric gM. Actually, both Clifford actions coincide on their common domain. Hence, with



CUBO
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Dirac Type Gauge Theories – Motivations and Perspectives 29

respect to twisted Grassmann bundles we may consider γ and gM as being basically the same.

Accordingly, the Einstein equation is seen to provide a physical constraint on the possible Clifford

module structures to which (31) refers to. Note the change of the meaning of the metric when the

Maxwell equations are written similar to the Dirac equation.

Once we have established a common geometrical setup for the Dirac and the Maxwell (resp.

Yang-Mills) equations we proceed to show that this common setup also provides an appropriate

geometrical background for the Einstein equation. For this we remark that on the one hand side

the Maxwell and Dirac equations make use of a given metric gM (i.e. a fixed Clifford module

structure of the underlying twisted Grassmann bundle). On the other hand, the Einstein equation

are considered as differential equations determining gM. In particular, the (Levi-Civita) connection

which fixes the first order operator ∂Λ is fully determined by gM. In contrast to the Maxwell

equations (resp. Yang-Mills equations), the gravitational gauge potential has thus an underlying

geometrical structure given by the metric gM from which the connection is derived. For this

matter the Einstein-Hilbert functional, from which the Einstein equation can be derived as Euler-

Lagrange equation, is linear in the curvature. In contrast, the Yang-Mills functional, which yields

the (homogeneous) Maxwell equation (7) in the case G = U(1), is quadratic in the curvature:

IEH(gM) := λ
−1
grav

∫

M

scal(gM) dvolM , (34)

IYM(gM; A) := λ
−1
elm

∫

M

gΛM(FA,FA) dvolM . (35)

Note that the variation of IYM(gM; A) with respect to the metric gM gives rise to the energy-

momentum current τ ∈ Sec(M, End(TM)) as a function of FA as mentioned before.

To get a relation between these seemingly different functionals (34) and (35) we notice that

in contrast to the square d2
A

= FA of the first order operator dA, the square of the associated

Dirac operator i/∂
A

has the well-known Lichnerowicz decomposition into a specific second order

differential operator and a specific zero order operator:

i/∂
2

A
= ( dA + δA)

2 G=U(1)
= d ◦ δ + δ ◦ d

= −△B + VD (36)

with △B
loc.
= −gµν(∇µ◦∇ν−Γ

σ

µν
∇σ) being the Bochner-Laplacian and ∇µ ≡ ∂µ+

1

4
ωLC
µab

[γa,γb]+Aµ.

The local functions Γσ
µν

are the usual Christoffel symbols with respect to gM and a chosen coordinate

frame.

The “Dirac potential” has the specific form:

VD =
1

4
scal(gM) + γ(FA) ∈ Sec(M, End(EΛ, E )) (37)

where locally γ(FA)
1

2
γµγν ⊗Fµν. The tensor product refers to the fundamental decomposition (33).

As a consequence, the zero order operator γ(FA) is always a trace-less operator: trE (γ(FA)) ≡ 0,

where the trace is taken in End(EΛ, E ).



30 Jürgen Tolksdorf CUBO
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Therefore, the Einstein-Hilbert functional may be expressed in terms of i/∂
A

as

IEH(gM) = λ
′−1
grav

∫

M

trEVD dvolM . (38)

Note that the Dirac potential is uniquely determined by i/∂
A
.

We notice that the trace-less zero order operator γ(FA) ∈ Sec(M, End(EΛ, E )) is indeed the

“square root” of the Yang-Mills Lagrangian, for

IYM(gM; A) = λ
′−1
elm

∫

M

trE (γ(FA)
2
) dvolM . (39)

However, also in this form the Yang-Mills action is still quadratic in the curvature in contrast to

the Einstein-Hilbert action.

The question then is whether the Yang-Mills Lagrangian can be “linearized” such that it be-

comes most similar to the Einstein-Hilbert Lagrangian without violating the second order character

of the Yang-Mills equations. Note that both the Einstein and the Yang-Mills equations are of sec-

ond order. Hence, one cannot simply try to square the Einstein-Hilbert Lagrangian to bring it into

a form similar to the Yang-Mills Lagrangian without obtaining higher order differential equations

for gM.

In order to appropriately linearize the integrand of (39) one may take into account that i/∂
A

also determines a specific curvature on the bundle (31), denoted by curv(i/∂
A
) ∈ Ω2(M, End(EΛ, E ))

(c.f. [22] and [23]). Explicitly it reads

curv(i/∂
A
) = Rg ⊗ IdE + IdΛ ⊗ FA

≡ Rg + FA . (40)

Again, this is due to the fundamental decomposition (33). Here, Rg ∈ Ω
2
(M, End(EΛ, E )) is

the Riemannian curvature with respect to the induced metric gΛM on the Grassmann bundle

over M. Locally, it reads: Rg
loc.
=

1

2
eµ ∧ eν ⊗ 1

4
[γa,γb]Rabµν, where the local functions Rabµν ≡

gM(ea,ec) e
c
(Riem(eµ,eν)eb) and Riem ∈ Ω

2
(M, End(TM)) denotes the (semi-)Riemann curva-

ture tensor with respect to gM. Note again that Einstein’s summation convention is employed in

local formulae.

Therefore, the Yang-Mills curvature (especially the electromagnetic field strength) may be

expressed in terms of i/∂
A
. In fact, it is but the “relative curvature” of the curvature of i/∂

A
(again,

neglecting the identity mappings):

FA = curv(i/∂A) − Rg ∈ Ω
2
(M, EndCl(EΛ, E )) . (41)

This geometrical interpretation of FA in terms of i/∂A yields a different interpretation of the

Yang-Mills (resp. of the Maxwell) equations. The latter are considered to yield a constraints

for i/∂
A
. Of course, this simply means constraints for ∂E and thus does not yield anything new in



CUBO
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Dirac Type Gauge Theories – Motivations and Perspectives 31

comparison with the usual description of Yang-Mills type gauge theories in terms of G principal

bundles. However, the strength of the presented geometrical viewpoint of the Yang-Mills equations

in terms of i/∂
A

has a powerful potential for a straightforward generalization. This is, because

the geometrical point of view can be immediately generalized to arbitrary Dirac type first order

differential operators. In other word, there are much more general Dirac type operators on (31)

than those given by i/∂
A
. In fact, the latter are only very specific Dirac type operators. They are fully

characterized by the decomposition (37) and the fact that FA ∈ Ω
2
(M, EndCl(EΛ, E )) is γ−invariant.

This may provide a sufficient motivation to consider the form (38) of the Einstein-Hilbert function

as more profound than the form (34). In fact, the former should be considered as a functional of

a specific class of Dirac type operators on (31) and hence as a specific restriction of a much more

general functional (c.f. our discussion in the next section).

As discussed in ([23]), the trace of the Dirac potential (37) can be recast into the geometrical

form (neglecting boundary terms):

trEVD = trγcurv(i/∂A) . (42)

Therefore,

IEH(gM) ≡ IEH(i/∂A) = λ
′−1
grav

∫

M

trγ(curv(i/∂A)) dvolM

≡ λ′
−1
grav

∫

M

trγ(curv( dA + δA)) dvolM (43)

where trγ(curv(i/∂A)) ≡ trE [γ(curv(i/∂A))] ∈ C
∞

(M).

The form (43) of the Einstein-Hilbert action makes it most explicit how the metric gM can be

replaced by (a specific class of) Dirac type operators and hence how the Einstein-Hilbert functional

determines a Clifford action γ on a twisted Grassmann bundle (31). Note that, despite of its

appearance, (43) is actually independent of the connection on the twisting part E ։ M of (31).

In other words, it is independent of the gauge potential A that (locally) determines the first order

operator ∂E. The functional (43) thus yields a constraint only on how the vector bundle (31) can be

actually regarded as a specific Clifford module. It thus determines γ as stated before. In fact, since

i/∂
A

is fully characterized by (37), it is straightforward to prove that these Dirac type operators

provide the biggest class of Dirac type first order differential operators on a twisted Grassmann

bundle such that the universal Dirac action (1) is proportional to the Einstein-Hilbert action and

thus only depends on gM. Note that there is only a canonical choice for ∂
E if the Hermitian vector

bundle E ։ M equals the trivial bundle M × V ։ M. Only in this case, there is a natural choice

for i/∂
A

given the Gauss-Bonnet like operator d + δ. In the general case, however the latter operator

is not gauge covariant. For this matter one has to chose some ∂E to obtain an appropriate gauge

covariant generalization dA +δA of d+δ. Again, the functional (43) is independent of this arbitrary

choice.

We are still left with the question whether it is possible to find a Dirac type operator i/D
A
, say,



32 Jürgen Tolksdorf CUBO
11, 1 (2009)

on a certain twisted Grassmann bundle such that the Yang-Mills functional can be expressed in

terms of the universal Dirac action (1).

The answer to this question turns out to be affirmative, actually, and has been discussed in

some detail in [22] (c.f. also the appropriate references cited therein, in particular [2] in the case

of a closed compact Riemannian manifold). In general, the Yang-Mills action may be written as

IYM(gM; A) = λ
′
YM (ID(i/DA) − ID(i/∂A))

= λ′YM

∫

M

trγ(curv(i/DA) − curv(i/∂A)) dvolM . (44)

where the corresponding Dirac type operator reads

i/D
A

= i/∂
A

+ I ⊗ γ(FA) . (45)

Here, I ≡ off − diag(−1, 1) is an additional complex structure on the doubled twisted Grassmann

bundle

2EΛ, E ≡ EΛ, E ⊕ EΛ, E ΛM ⊗R (E ⊕ E) −→ M . (46)

The thus defined class of first order differential operators (45) are called Dirac operators of

“Pauli-type”. The reason for this chosen terminology is that first order differential operators of the

form

i/∂
A

+ iγ(FA) (47)

have been introduced in physics in order to describe the so-called “magnetic moment” of the proton

long before it has been realized that the proton is a composite of more fundamental elementary

particles (the “quarks”). In this context, the additional term γ(FA) =
i

2
γµγν⊗Fµν, with F ∈ Ω

2
(M)

being the electromagnetic field strength, is named “Pauli-term” after the famous physicist W. Pauli.

Note that the first order operator (47), however, is not a Dirac type first order operator. This is

because the Pauli-term γ(FA) is an even operator in the sense that it commutes with the canonical

Z2−grading provided by the Riemannian volume form: γM = iγ(dvolM) (called “γ5” in the physics

literature). Indeed, a first order differential operator is said to be of Dirac type provided it is odd

with respect to a given Z2−grading of the underlying Clifford module and the principal symbol of

its square is given by the underlying metric. Only the latter feature is shared by the first order

operator (47). In contrast, the first order operator (45) is both odd and its square is a “generalized

Laplacian”. It is thus of Dirac type.

Note that Dirac’s original first order operator (or its gauge covariant generalization)

i/∂
A
− m (48)

is also not of Dirac type for exactly the same reason as (47) is not of Dirac type. We shall come

back to this in our third section where we discuss a specific class of Hermitian Clifford modules

and the most general Dirac type operators thereof.



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Dirac Type Gauge Theories – Motivations and Perspectives 33

Concerning Dirac type operators of the form (45) the “square root” of the Yang-Mills La-

grangian becomes most obvious. It is not simply given by the traceless zero order operator γ(FA)

itself but, instead, by Dirac operators of Pauli type. Basically, this is because of the additional

grading one obtains from the doubling of (31). This additional grading also allows to express the

“fermionic part” of the universal Dirac action (1) as

〈Ψ, i/D
A

Ψ〉2E = 2〈ψ,i/∂A ψ〉E . (49)

At least, this holds true for those sections Ψ ∈ Sec(M, 2EΛ, E ) that are given by Ψ = (ψ,ψ)

and hence are determined by a section ψ ∈ Sec(M,EΛ, E ). In other words, the “Pauli-term” does

not contribute to the fermionic action but only to the bosonic action. This is a very desirable

feature of this class of Pauli type Dirac operators, for it is well-known that the Pauli-term in the

fermionic action yields a generalized Dirac equation that is not compatible with “quantization”.

We shall return to the Pauli type Dirac operators when considering a specific class of Hermitian

Clifford modules and the corresponding most general Dirac type operators thereof. The underlying

structure of this class of Clifford modules is basically motivated by our fourth equation: the

Majorana equation:

i/∂ψ = mψc (50)

where ψc denotes the “charge conjugate” of ψ (c.f. below).

We call in mind that the Einstein-Hilbert functional may be expressed in terms of Dirac type

operators of the form (32) with an arbitrary choice of ∂E. In contrast, when restricted to Pauli type

Dirac operators i/D
A
, the universal Dirac action (38) yields the combined Einstein-Hilbert-Yang-

Mills functional. It reduces to the pure Yang-Mills functional only if (M,gM) is fixed to be (Ricci)

flat. This is consistent with the Einstein equation, however, only with respect to the physical

approximation that the gravitational field produced by the energy-momentum of the Yang-Mills

field can be neglected to some extend. In general, however, (1) yields the coupled Einstein-Yang-

Mills-Weyl equations as the corresponding Euler-Lagrange equations if (1) is restricted to Pauli

type Dirac operators (45). In this case, the right-hand side of the Yang-Mills equation is similar

to (10) and the energy-momentum current is a well-determined function of (gM,FA,ψ).

We stress that the Pauli type Dirac operators are more general than those given by i/∂
A
. In

particular, the relative curvature of i/D
A

:

FD := curv(i/DA) − Rg (51)

is not γ−invariant, i.e.

FD /∈ Ω
2
(M, EndCl(2EΛ, E )) . (52)

For that matter, γ(FD) ∈ Ω
0
(M, End(E)) is not a traceless operator.

We close our motivation for the universal Dirac action (1) with the remark that the underlying

invariance group of this functional is provided by the full diffeomorphism group Diff(EΛ, E ) of (31).



34 Jürgen Tolksdorf CUBO
11, 1 (2009)

This (infinite) gauge group decompose into the semi-direct product (c.f. [22]):

Diff(EΛ, E )AutM(EΛ, E ) ⋉ Diff(M) (53)

with AutM(EΛ, E ) consisting of all (bundle) automorphisms of (31) over the identity mapping on the

base manifold M. Moreover, this group decomposes further into the direct sum of to sub-groups:

AutM(EΛ, E )AutEH(EΛ, E ) × AutYM(EΛ, E ) . (54)

Here, the “Yang-Mills” sub-group AutYM(EΛ, E ) ⊂ AutM(EΛ, E ) consists of all automorphisms of (31)

being isomorphic to the gauge transformations on the frame bundle that is induced by the vector

bundle (31). It is thus a normal sub-group of AutM(EΛ, E ) and

AutEH(EΛ, E ) : AutM(EΛ, E )/AutYM(EΛ, E ) . (55)

Locally, the “Einstein-Hilbert” sub-group AutEH(EΛ, E ) consists of all SO(p,q) rotations of

orthonormal frames of TM ։ M and AutYM(EΛ, E ) consists of all ordinary gauge transformations

encountered in the usual geometrical description of Yang-Mills gauge theories in terms of principal

G-bundles G →֒ P ։ M. Thus, the universal Dirac action (1) contains all the physical symmetries

which are usually imposed on physical field theories. To enlarge this symmetry to “super-symmetry”

transformations, however, is still an open issue.

Having presented a detailed discussion of the motivation for the universal Dirac action (1) and

how it is related to well-known field equations, we may proceed with a discussion in what sense

the Dirac functional is more general than the ordinary Yang-Mills functional. In other words, in

the following section we want to discuss the precise domain of dependence of the universal Dirac

functional.

3 The Dirac action as a functional of “super fields”

In this section we discuss in more detail the domain of dependence of the (bosonic part of the)

universal Dirac action:

ID,bos :

∫

M

trγcurv( /D) dvolM . (56)

In the foregoing section we have shown how this functional covers both the Einstein-Hilbert

and the Yang-Mills functional. In fact, the Dirac functional may be considered as a natural

generalization of the Einstein-Hilbert functional of the form (43). Hence, when restricted to certain

“sub-domains” on the “set of all Dirac type operators” (c.f. below), the universal functional (56)

becomes a functional on (an appropriate subset of) Sec(M,EEH) in the case of the Einstein-Hilbert

action, or a functional on the affine manifold of all linear connections A(E) in the case of the

Yang-Mills action. Accordingly, when restricted to Pauli type Dirac operators the (bosonic part of



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Dirac Type Gauge Theories – Motivations and Perspectives 35

the) universal Dirac action becomes a functional on the smooth manifold Sec(M,EEH) × A(E). In

general, (1) is considered as a functional on the smooth manifold

D(EΛ, E ) × Sec(M,EΛ, E ) (57)

of all Dirac type first order operators on a twisted Grassmann bundle (31) and the module of

smooth sections therein. Note that the “fermionic part” of (1)

ID,ferm :

∫

M

〈ψ, /Dψ〉E dvolM , (58)

is viewed simply as a quadratic form on Sec(M,EΛ, E ). This quadratic form is fully determined by

(symmetric) elements of D(EΛ, E ). For this reason, it suffices to focus on the affine manifold of all

Dirac type operators on a given Grassmann bundle.

The aim of this section is to make this more precise and to show how the above two cases of

the Einstein-Hilbert and the Yang-Mills functional are special cases of the more general functional

(56). Basically, the reason is provided by the following (highly non-canonical) isomorphisms:

D(EΛ, E ) ≃ Ω
0
(M, End(EΛ, E )) ≃ Ω

∗
(M, EndCl(EΛ, E )) , (59)

which holds true for a fixed Clifford module structure on (31) (i.e. metric on M). The second

isomorphism of (59) is implied by (33), where the abbreviation

Ω
∗
(M, EndCl(EΛ, E )) ≡

⊕

p∈Z
Ω
p
(M, EndCl(EΛ, E )) (60)

has been used.

Consequently, any Dirac type operator on a Clifford module is determined by differential forms

of all degrees. This is in strong contrast to connections on a vector bundle which are determined

by one-forms, only.

To make this more precise, let again M be a smooth orientable manifold of even dimension

n = 2k ≥ 2. Also, let again E ։ M be a smooth Hermitian vector bundle over M and EΛ, E ։ M

the corresponding twisted Grassmann bundle. We call the smooth fiber bundle

ED : EEH × End(EΛ, E ) −→ M (61)

the “Dirac bundle” associated to the twisted Grassmann bundle. So far (31) is considered as a

vector bundle over M. There is no given Clifford structure at all, for M is not yet supposed to be

endowed with a metric. We call in mind that a metric on M is in one-to-one correspondence with

a section of the Dirac bundle given by

σD : M −→ ED

x 7→ (gM(x), 0) (62)

where gM ∈ Sec(M,EEH).



36 Jürgen Tolksdorf CUBO
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We consider the following equivalence relation on the manifold of smooth sections Sec(M,ED) :

σ′
D
≡ (g′, Φ′) ∼ σD ≡ (g, Φ) ∈ Sec(M,ED) (63)

iff g′ = g ∈ Sec(M,EEH) and there exists an α ∈ Ω
1
(M, End(E)) →֒ Ω1(M, EndCl(EΛ, E )), such

that Φ′ = Φ + γ(α) ∈ Ω0(M, End(EΛ, E )). We put

SD := Sec(M,ED)/ ∼ . (64)

There are various equivalent definitions available for Dirac type first order differential opera-

tors, depending on the appropriate focus (see, for example, [1], [5], [3], [4]). We present a different

one which is most adopted to our purpose.

Definition 1. Let D(EΛ, E ) be the set of all first order differential operators acting on Sec(M,EΛ, E ),

such that for /D ∈ D(EΛ, E ) there exists a section gM ∈ Sec(M,EEH) with

T∗M
γ

−→ End(EΛ, E )

df 7→ [ /D,f] . (65)

Here, the gM−induced Clifford action γ is defined by (25).

A first order differential operator /D ∈ D(EΛ, E ) is called a “Dirac type operator" provided it is

odd with respect to the Z2−grading that is given by an involution τE := γM ⊗ τE.

The set of all Dirac type operators on EΛ, E carries a natural action of the translational group

TE ≡ Ω
1
(M, End

+
(E)) →֒ Ω1(M, End+(EΛ, E )) (66)

that is given by

D(EΛ, E ) × TE
µ

−→ D(EΛ, E )

( /D,α) 7→ /D + γ(α) . (67)

Clearly, this action is free and the corresponding orbit space D(EΛ, E )/µ can be identified with

SD. Furthermore, with respect to this identification

πD : D(EΛ, E ) −→ SD
/D 7→ s ≡ [(gM, Φ)] (68)

is a principal fibering with structure group TE.

This principal fibering is actually trivial. However, every bijection

χA : D(EΛ, E )
≃
−→ SD × TE

/D 7→ (s,α) (69)

strongly depends on the choice of ∂E. This holds true unless E ։ M is trivial.



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Dirac Type Gauge Theories – Motivations and Perspectives 37

Indeed, for every choice of ∂E one may define

D(EΛ, E ) ∋ /D ≡ χ
−1
A

(s,α) := /∂
A

+ Φ̂A + γ(α)

≡ /∂
A

+ ΦA . (70)

Here, /∂
A
∈ D(EΛ, E ) is given by (32) and

Φ̂A ∈ Sec(M, End(EΛ, E )) ≃ Sec(M, ΛM ⊗ EndCl(EΛ, E )) , (71)

which does not contain a one-form part. Note that Φ̂A has to have odd total degree. It is uniquely

defined as follows: every /D ∈ D(EΛ, E ) can be decomposed (in a highly non-unique way) as

/D = /∂
A

+ ΦA with ΦA ≡ /D − /∂A . Then, ΦA =: Φ̂A + γ(α) and /∂A + Φ̂A + γ(α) is equivalent to

/∂
A

+ Φ̂A. It follows that πD(χ
−1
A

(s,α)) = pr1(s,α) = s, if and only if [ /D] ∈ D(E)/µ corresponds to

s ∈ SD.

Proposition 1. Let ED ։ M be the Dirac bundle associated with a twisted Grassmann bundle

EΛ, E ։ M. The functional (56) can be considered as a canonical functional on Sec(M,ED) :

Proof: Since the value of the integral

∫

M

trγcurv(i/∂A) dvolM (72)

is independent of the choice of ∂E, it follows that (56) is constant along the fibers of (68). Hence, it

descents to a well-defined functional on SD. For this matter ID,bos can be considered as a functional

of (gM, Φ) that constitutes a general section of the Dirac bundle (61). 2

As a consequence

ID = I (gM, Φ,ψ) (73)

with

Φ ∈ Sec(M, ΛM ⊗ EndCl(EΛ, E ))
⊕

0≤l≤n
Sec(M, ΛlT∗M ⊗ EndCl(EΛ, E )) . (74)

being a “super-field” of odd total degree that takes values in the γ−invariant endomorphisms on

EΛ, EE
+
Λ, E

⊕ E−
Λ, E

։ M.

Especially, for Φ = 0, the action (1) reduces to the sum of the usual (massless) Dirac functional

and the Einstein-Hilbert functional:
∫

M

[〈ψ, i/∂
A
ψ〉E + trγcurv(i/∂A)] dvolM . (75)



38 Jürgen Tolksdorf CUBO
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In this case, the appropriate Euler-Lagrange equations are given by the combined Einstein-Weyl

equations with the energy-momentum current τ being defined by ψ ∈ Sec(M,EΛ, E ).

Likewise, to obtain the combined Einstein-Yang-Mills-Weyl equations one considers Φ = FA,

with the requirement that FA ∈ Sec(M, Λ
2T∗M ⊗ EndCl(2EΛ, E )) being defined by the curvature

FA ∈ Ω
2
(M, End(E)) with respect to the chosen ∂E. In other words, one restrict the right-hand side

of (1) to Pauli type Dirac operators on 2EΛ, E ։ M. In this case, the energy-momentum current τ

is fully determined as a function of (ψ,FA), whereas the electromagnetic current is given by (10)

(or an appropriate generalization thereof if G 6= U(1)). Of course, this reduces to pure Yang-Mills

theory when one restricts to ψ = 0 and gM (Ricci) flat.

Eventually, one can also recover the full action functional of the so-called Standard Model of

elementary particles including the famous Higgs potential. For this one has to consider even more

general super fields Φ for appropriate twisted Grassmann bundles. Interestingly, the structure of

this bundle is determined by the topology of M and the choice of the “ground-state” of the still to

find “Higgs boson” (c.f. [22] and the corresponding References cited therein).

The above mentioned examples may suffice to exhibit the generality of the Dirac action (1)

and how it covers important classes of coupled partial differential equations as Euler-Lagrange

equations of a natural generalization of the Einstein-Hilbert Lagrangian of Einstein’s theory of

gravity (43). Once one has the universal functional (1) one may ask for the corresponding form of

the Euler-Lagrange equations. This, however, depends on the choice of the (twisting part of the)

underlying twisted Grassmann bundle and is still a major challenge to exhibit in full generality. In

the case, where the bundle is fixed and endowed with sufficient structure one may determine the

most general Dirac type operator that is compatible with the endowed structure. Basically, this

amounts to determine the most general super-field that is compatible with the given structure and

then rewriting the universal Dirac action in terms of this super-field. As a specific example, this

will be demonstrated in the next section in terms of a specific class of “real, Hermitian Clifford

modules”, called “particle-anti-particle modules”.

Before, however, we want to briefly comment on “spin versus non-spin manifolds”. So far,

we concentrated on twisted Grassmann bundles and one may ask what does it give more than

twisted spinor bundles. Also, one may ask how the latter fits with the geometrical frame of twisted

Grassmann bundles.

First of all, if M is a spin-manifold (i.e. it has vanishing second Stiefel-Whitney classes) and

S ։ M is the (complexified) spinor bundle with respect to a chosen spin-structure, then

ΛM ⊗ C ≃ S ⊗ S
∗ −→ M (76)

and hence

EΛ, E ≃ S ⊗ W −→ M (77)

where W ≡ S∗ ⊗ E ։ M.



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Dirac Type Gauge Theories – Motivations and Perspectives 39

Moreover, if S ≃ ClMe ≡ {ae ∈ ClM | a ∈ ClM} with e ∈ Sec(M,ClM) being an appropriately

global primitive idempotent and S ≃ S ⊗ C, then

S ⊗ E →֒ EΛ, E

s ⊗ y 7→ s ⊗ e∗ ⊗ y (78)

yields a canonical inclusion of the twisted spinor bundle

EE := S ⊗ E −→ M (79)

into the twisted Grassmann bundle (31). Here, e∗ is the idempotent that yields the dual spinor

module S∗ := e ClM and S
∗ ≃ S∗ ⊗ C. Note that we only consider complex modules.

In this way, the slightly more general situation of a twisted Grassmann bundles also covers

the geometrical situation where M is supposed to be a spin-manifold. On the other hand, by a

famous Theorem due to R. Geroch, a non-compact four-dimensional Lorentzian manifold possesses

a spin-structure if and only if its frame bundle is trivial (c.f. [7]). Apparently, to propose that M is

a spin-manifold is thus a very strong assumption about the topology of M. Note that locally every

Clifford module looks like a twisted spinor bundle according to the fundamental decomposition

(33).

Therefore, the geometrical setup of twisted Grassmann bundles is slightly more general than

twisted spinor modules and much less restrictive (actually, twisted Grassmann bundles always

exist). On the other hand, to consider arbitrary Clifford modules seems far too general. In

particular, the metric gM does not fix the Clifford module structure γ, in general, like (25) does in

the case of a twisted Grassmann bundle. For that matter it becomes difficult to fix the domain

of the universal Dirac action for general Clifford modules. Only in the case of twisted Grassmann

bundles, the Einstein-Hilbert functional may interpret to provide restrictions also with respect to

the module structure of the vector bundle (31).

We close this section by two remarks: First, one obtains for (M,gM) denoting a closed compact

and orientable Riemannian manifold of even dimension that there exists real constants α,β such

that
∫

M

trEVD dvolM = αIEH(i/∂A) + β Wres
(
/D

2−2k
)

(80)

independent of the chosen ∂E. Here, “Wres” is the “Wodzicki residue”, i.e. the trace functional on

the algebra of classical pseudo-differential operators acting on Sec(M,EΛ, E ) (see, for example, [20]

and the given References therein; also see [2] and [21]).

Therefore, in the case of dimM = 4, the universal Dirac action is basically equal (up to a shift

and the quadratic term in (4)) to the trace of the “propagator” (i.e. the Greens operator) of /D
2
.

This may demonstrate once again how natural the functional (1) actually is.

Second, the Dirac-like form (28) has been studied since from the beginning of the last century,

c.f. [19], [13], [15], [14], [16], [10], [12], [11] and [17]. Apparently, this form of the Maxwell equations



40 Jürgen Tolksdorf CUBO
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has a natural generalization:

/DFD = 0 (81)

where, again, FD := curv( /D) − Rg is the relative curvature with respect to /D ∈ D(EΛ, E ). Accord-

ingly, solutions of this generalized Maxwell equation like, for example, (anti-) self dual solutions

may provide interesting restrictions to D(EΛ, E ) and hence to the Dirac action (1). Note that,

when expressed in terms of the super field Φ ∈ Sec(M, ΛM ⊗ EndCl(EΛ, E )) the generalized Maxwell

equation (81) actually becomes a system of nonhomogeneous partial differential equations.

We finally mention that generalizations of the Dirac type operator i/∂
A

also play a fundamental

role in A. Connes’s noncommutative geometry (c.f., for example, [6]) and in the case of the proof

of the family index theorem, (c.f., for example, in [18], [4]).

4 Particle-anti-particle modules and Dirac operators of Pauli

type

In this section we discuss another specific class of Clifford modules. These modules are mainly

motivated by the structure that underpins the Majorana equation (50). These “particle-anti-

particle” modules will also provide us with a better geometrical understanding of Pauli type Dirac

operators. In particular, these modules will yield a geometrical motivation for the restriction of

“diagonal sections” Ψ = (ψ,ψ), such that the Pauli term appears in the bosonic part of the universal

Dirac action (1) but drops out in fermionic part (58) of (1).

To get started let, again, (M,gM) be a given orientable, semi-Riemannian manifold of even

dimension n = 2k ≥ 2. Also, let τCl ≡ (ClM,M) collect the data of the Clifford bundle ClM ։ M

associated with (M,gM).

Definition 2. By a “real Hermitian Clifford module (bundle)” we understand a collection of data

(E,〈·, ·〉E,τE,JE,γE ) (82)

where, respectively, E is the total space of a complex vector bundle ξE ≡ (E,M,πE ) over M, 〈·, ·〉E
a fiber-wise Hermitian product turning ξE into a Hermitian vector bundle over M, τE ∈ End(E)

is an involution giving rise to a Z2−grading of ξE and JE : E → E denotes a real structure, i.e.

an anti-linear involution on E that allows to identify ξE with its conjugate complex vector bundle

ξĒ := ξ̄E ≡ (Ē,M,π̄E ) over M. Finally,

γE : T
∗M −→ End(E)

ν 7→ γE (ν) (83)

is a Clifford mapping such that all mappings are “quasi-Hermitian” (i.e. either Hermitian or skew-

Hermitian) and τE and γE are “quasi real” (i.e. either real or purely imaginary with respect to



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Dirac Type Gauge Theories – Motivations and Perspectives 41

JE ) :

JE ◦ τE ◦ JE = ±τE

JE ◦ γE ◦ JE = ±γE . (84)

Here, a real structure is called “quasi Hermitian” provided it fulfills

〈JE (z),JE (w)〉E ± 〈w,z〉E (85)

for all z,w ∈ E. Similar to complex linear mappings this is denoted by Jt
E

= ±JE, where, in general,

“ t” means Hermitian transpose with respect to 〈·, ·〉E. If J
t
E

= + JE, the real structure is also called

an “anti-unitary involution”.

In the following we are interested in a specific class of real Hermitian Clifford modules, called

“particle modules”.

Definition 3. A real Hermitian Clifford module over τCl is called a “particle module” if

1. The involution is skew-Hermitian and purely imaginary;

2. The Clifford mapping is skew-Hermitian and real.

The corresponding conjugate complex module is called an “anti-particle module” over τCl.

We denote a particle module by

ξP ≡ (P,〈·, ·〉P,τP,JP,γP) . (86)

A particle-anti particle module over M is a real Hermitian Clifford module (bundle) over τCl

ξPP̄ ≡ (PP̄,〈·, ·〉PP̄,τPP̄,JPP̄,γPP̄) (87)

where, respectively

1. PP̄ := P ⊕M P̄ ;

2. 〈(z1,w1), (z2,w2)〉PP̄ :=
1

2
(〈z1,z2〉P + 〈w1,w2〉P) ;

3. τPP̄(z,w) : (τP(z),−τP(w)) ;

4. JPP̄(z,w) : (JP(w),JP(z)) ;

5. γPP̄(z,w) : (γP(z),γP(w))

for all z,w,. . . ,w2 ∈ P.

It follows that for all ν ∈ T∗M :



42 Jürgen Tolksdorf CUBO
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1. Jt
PP̄

= ±JPP̄ ⇔ J
t
P

= ±JP ;

2. τt
PP̄

= ±τPP̄ ⇔ τ
t
P

= ±τP ;

3. γt
PP̄

(ν) = ±γPP̄(ν) ⇔ γ
t
P
(ν) = ±γP(ν) ;

4. JPP̄ ◦ τPP̄ = ±τPP̄ ◦ JPP̄ ⇔ JP ◦ τP = ∓τP ◦ JP

5. JPP̄ ◦ γPP̄(ν) = ±γPP̄(ν) ◦ JPP̄ ⇔ JP ◦ γP(ν) = ±γP(ν) ◦ JP ;

6. τPP̄ ◦ γPP̄(ν) = ±γPP̄(ν) ◦ τPP̄ ⇔ τP ◦ γP(ν) = ±γP(ν) ◦ τP .

Theorem 1. The most general real Dirac type operator on a particle-anti-particle module ξPP̄ reads

/D
PP̄

(
/D

P
JP ◦ ΦP ◦ JP

ΦP JP ◦ /DP ◦ JP

)
≡

(
/D

P
Φ

c
P

ΦP /D
c

P

)
, (88)

with

/D
P

: Sec(M,P) −→ Sec(M,P) (89)

being a general Dirac type operator on the underlying particle module ξP and ΦP ∈ Sec(M, End(P))

being a zero order operator that is even with respect to the Z2−grading on P

Proof: To prove the statement we mention that an odd first order differential operator on a

Z2−graded vector bundle ξW ≡ (W,M,πW) over a (semi-)Riemannian manifold (M,gM) is of Dirac

type if and only if for all f ∈ C∞(M) the mapping

γW : T
∗M −→ End(W)

df 7→ [ /D,f] ≡ /D ◦ f − f ◦ /D (90)

yields a Clifford action on ξW. Here, the ring C
∞

(M) acts multiplicatively on the (sheave) of

sections Sec(M,W).

Likewise, if (ξW,γW) denotes a Clifford module, then an odd first order differential operator

/D : Sec(M,W) −→ Sec(M,W) (91)

is of Dirac type (that is compatible with the given module structure) if and only if

[ /D,f] = γW(df) . (92)

Let ξPP̄ be a particle-anti-particle module and

/D :=

(
D1 D2

D3 D4

)
(93)



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Dirac Type Gauge Theories – Motivations and Perspectives 43

be a general first order differential operator acting on Sec(M,PP̄) :

Dk : Sec(M,P) −→ Sec(M,P) (94)

for k = 1, . . . , 4.

The operator /D is odd with respect to τPP̄ if and only if

τPP̄ ◦ Dk − Dk ◦ τPP̄ (95)

for k = 1, 4 and

τPP̄ ◦ Dk + Dk ◦ τPP̄ (96)

for k = 2, 4.

Then,

[ /D,f] = γPP̄(df) (97)

for all f ∈ C∞(M) if and only if

[Dk,f] = γP(df) (98)

for k = 1, 4 and

[Dk,f] = 0 (99)

for k = 2, 3.

Therefore, the first order differential operators D1 ≡ /D1 and D4 ≡ /D2 are of Dirac type on the

underlying particle module ξP. In contrast, the operators D2 ≡ Φ2 and D3 ≡ Φ1 are of zero order.

Next, we consider the conditions on the Dirac type operator

/D :=

(
/D

1
Φ2

Φ1 /D2

)
(100)

such that /D is real with respect to JPP̄.

It follows that

JPP̄ ◦ /D ◦ JPP̄ /D ⇔

{
/D

2
= JP ◦ /D1 ◦ JP ,

Φ2 = JP ◦ Φ1 ◦ JP .
(101)

This finally proves the statement. 2

Note that neither /D
P
, nor ΦP are supposed to be real, in general.



44 Jürgen Tolksdorf CUBO
11, 1 (2009)

Let

MPP̄ : {(z,z
c
) ∈ PP̄ |z,zc ≡ JP(z) ∈ P} (102)

be the real subspace defined by JPP̄ such that

PP̄ = MPP̄ ⊗ C . (103)

The corresponding real vector bundle is denoted by ξM ≡ (MPP̄,M,πM) with the projection πM

being given by the restriction of πPP̄ to MPP̄ ⊂ PP̄. Note that ξM ⊂ ξPP̄ is a real τCl submodule.

Clearly, the latter itself contains a distinguished real sub-module given by z ∈ P fulfilling zc = z.

That is, it is given by the real sub (bundle) space

MP ⊕ MP : {(z,z) ∈ PP̄ |JP(z) = z ∈ P} ⊂ MPP̄ , (104)

where MP := {z ∈ P |z = JP(z)} ⊂ P, such that P = MP ⊗ C.

On a particle-anti-particle module, the first order differential operator (88) is the most general

real Dirac type operator. Hence, one may restrict the universal Dirac action (1) to this type of

Dirac operators:

ID,real :
1

2

∫

M

[〈ΨPP̄, /DPP̄ΨPP̄〉PP̄ + trγcurv( /DPP̄)] dvolM (105)

with ΨPP̄ = (ΨP, Ψ
c
P
) ∈ Sec(M,MPP̄) and /DPP̄ any real Dirac operator on the particle-anti-particle

module ξPP̄.

Proposition 2. When boundary terms are neglected, the Dirac action (105) decomposes as follows:

ID,realID,ferm( /DPP̄) + ID,bos( /DPP̄) (106)

where

2ID,ferm( /DPP̄) ≡

∫

M

[〈ΨP, /DPΨP〉P + 〈Ψ
c
P
, /D

c

P
Ψ

c
P
〉P

+ 〈ΨP, Φ
c
P
Ψ

c
P
〉P + 〈Ψ

c
P
, ΦPΨP〉P] dvolM ; (107)

2ID,bos( /DPP̄) ≡

∫

M

[trγcurv( /DP) + trγcurv( /D
c

P
) + 2 tr(Φ

c
P
◦ ΦP)

+ 8 (tr ◦ evg)(α
c
P
◦ αP) + 2 (tr ◦ evg)(β

c
P
◦ βP)] dvolM (108)

where 2αP(v) : ΦP ◦ γP(v
♭
) ∈ End(P) and v♭(u) = gM(v,u) for all u,v ∈ TM. Accordingly,

2αc
P
(v) : JP ◦ 2αP(v) ◦ J

−1
P

Φ
c

P
◦ γP(v

♭
) ∈ End(P). Furthermore, βP : extΘ(ΦP − 2 /αP) with Θ ∈

Ω
1
(M, End(P)) being the canonical one-form that exists on every Clifford module (c.f. [22]) and

βc
P

: JP ◦ βP ◦ J
−1
P

extΘ(Φ
c

P
− 2 /α

c

P
).



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Dirac Type Gauge Theories – Motivations and Perspectives 45

In the sequel, we make use of the common “dagger” abbreviation: /α ≡ γ(α) ∈ Ω0(M, End(P))

for any α ∈ Ω∗(M, End(P)). For example, for α = ek ⊗ λk one has /α = γ
k ⊗ λk, etc.

Proof: The fermionic part is straightforward to prove. The bosonic part of the Dirac action is

proved in several steps.

First, we prove the following

Lemma 1. Let /D
1

and /D
2

be two Dirac type first order differential operators on an arbitrary

Clifford module (ξE,γE ) ≡ (E,M,πE,γE ) with πE : E = E1⊕E2 → M being a Z2−graded (Hermitian)

vector bundle over (M,gM). The zero-order term VH of the generalized Laplacian

H : Sec(M,E) −→ Sec(M,E)

Ψ 7→ /D
1
( /D

2
Ψ) (109)

has the explicit form:

VH = VD + Φ ◦ /ωD + evg(α
2
H
) + U . (110)

Here, respectively, VD and ωD are the Dirac potential and Dirac form of /D ≡ /D2 (c.f. [23]).

Moreover, Φ := /D
1
− /D

2
∈ Sec(M, End−(E)) and

U := evg
(
∇T

∗M ⊗End(E)

H
αH
)

(111)

with ∇E
H

being the covariant derivative that defines the connection Laplacian of H :

△H : −evg
(
∇T

∗M ⊗E

H
◦ ∇E

H

)
(112)

and αH ∈ Ω
1
(M, EndM(E)) is given by

2 αH(gradgf) := [Φ ◦ /D,f]

= Φ ◦ γE (df) (113)

for all f ∈ C∞(M).

Here and henceforth we make use of the following notation: “evg” means “evaluation/contraction”

with respect to gM. For instance, evg(α
2
)

loc.
= evg(e

µ ⊗ eν ⊗ αµ ◦ αν) : gM(e
µ,eν) αµ ◦ αν ∈ End(E)

for all α ∈ Ω1(M, End(E)) etc.

Proof: With /D
1

= /D
2
+ Φ ≡ /D + Φ it becomes sufficient to consider Laplace type operators of the

form

H = /D
2

+ Φ ◦ /D . (114)



46 Jürgen Tolksdorf CUBO
11, 1 (2009)

Every generalized Laplacian H decomposes as (see, for instance, in [3])

H = −△H + VH (115)

with ∇E
H

being given by

2 ev(f0 gradf1,∇
E

H
Ψ) := f0 ([H,f1] + △gf1) Ψ (116)

for all f0,f1 ∈ C
∞

(M) and Ψ ∈ Sec(M,E). Here, △g denotes the usual Laplace-Beltrami operator

restricted to zero-forms on M.

It follows that

∇E
H
∇E

D
+ αH (117)

with ∇E
D

being the covariant derivative that defines the Bochner-Laplacian of /D
2
.

As a consequence, the connection Laplacian of H may be expressed in terms of the Bochner-

Laplacian1 of /D
2

:

△H = △D + 2 evg (αH ◦ ∇
E

D
) + evg(α

2
H
) + U . (118)

The statement then follows by comparison of the general Lichnerowicz decomposition (115),

taking into account that /D
2

= −△D + VD and

2 evg(αH ◦ ∇
E

D
) = Φ ◦ /∇E

D
. (119)

2

Note that

trEU = divg ξH (120)

with

ξH := (trEαH)
♯
∈ Sec(M,TM) . (121)

Hence,

trEVH dvolM
[
trEVD + trE (Φ ◦ /ωD) + (trE ◦ evg)(α

2
H
)
]
dvolM + £ξHdvolM (122)

which demonstrates that

[∗trEVH][∗trE (VD + Φ ◦ /ωD + evg(α
2
H
))] ∈ Hn

deR
(M) (123)

with “∗” being the Hodge map induced by gM and the orientation defined by dvolM.

1I would like to thank M. Schneider for appropriate comments.



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Dirac Type Gauge Theories – Motivations and Perspectives 47

Clearly, for /D
1

= /D
2

= /D one has

VH = VD . (124)

Next, we present a Bochner-Lichnerowicz-Weizenböck type formula for a slightly more general

Laplace type second order differential operator H′.

Corollary 2. Let again (ξE,γE ) be a Clifford module over (M,gM). Also, let Dk = /Dk + Φk
(k = 1, 2) be two first order differential operators acting on Sec(M,E). The zero-order term VH′ of

the generalized Laplacian

H′ : Sec(M,E) −→ Sec(M,E)

Ψ 7→ D1(D2Ψ) (125)

reads:

VH′ = VH + V (126)

where VH is given by (110) with the replacement

Φ := D1 − D2 + 2Φ2 (127)

and

V := (Φ − Φ2) ◦ Φ2 + [ /D2, Φ2] . (128)

Proof: We put D1 = /D2 + Φ0 + Φ1 ≡ /D + Φ01 and rewrite H
′ as

H′ = /D
2

+ Φ ◦ /D + V

= H + V . (129)

Hence, the connection Laplacian △H′ of H
′ is the same as the connection Laplacian △H of H.

One may thus apply the former Lemma 1 to prove the statement. 2

As a consequence, one obtains explicitly (neglecting boundary terms):

trEVH′ trEVD + trE (Φ ◦ Φ2 − Φ
2
2
) + trE [ /D2, Φ2] + trE (Φ ◦ /ωD) + (trE ◦ evg)(α

2
H
) . (130)

We are now in the position to prove the bosonic part of Proposition 2. In fact, this will be an

immediate consequence of the following more general



48 Jürgen Tolksdorf CUBO
11, 1 (2009)

Proposition 3. Let (ξE,γE ) be a Clifford module over (M,gM). Also, let (ξ2E,γ2E ) be the Clifford

module that is defined by the corresponding Whitney sum:

ξ2E : ξE ⊕ ξE , τ2E : τE ⊕ (−τE ) , γ2E : γE ⊕ γE (131)

with τE being the grading involution on ξE.

The zero order term VD ∈ Sec(M, End(2E)) associated with the (square of the) most general

Dirac type first order differential operator

/D ≡

(
/D

1
Φ2

Φ1 /D2

)
:

Sec(M,E)

⊕

Sec(M,E)

−→

Sec(M,E)

⊕

Sec(M,E)

(132)

reads:

VD =

(
V1 + Φ2 ◦ Φ1 + 4 evg(α2 ◦ α1) [ /D1 , Φ2] + Φ2 ◦ ( /D1 − /D2) + 2 Φ2 ◦ /ω2

[ /D
2
, Φ1] + Φ1 ◦ ( /D2 − /D1) + 2 Φ1 ◦ /ω1 V2 + Φ1 ◦ Φ2 + 4 evg(α1 ◦ α2)

)
(133)

where, respectively, Vk and ωk denote the Dirac potential and the Dirac form of /Dk (k = 1, 2) and

αk is defined in terms of /D
2

k
, similar to αH of Lemma (1).

Proof: We may write

/D = /D + Φ (134)

with

/D :

(
/D

1
0

0 /D
2

)
, Φ :

(
0 Φ2

Φ1 0

)
. (135)

Then, we simply make use of the preceding Corollary 2 and apply the corresponding Bochner-

Lichnerowicz-Weizenböck type formula to H′ := /D
2
. Note that the Bochner-Laplacian of /D is given

by

∇2E
D
∇2E

D
+ βD (136)

with βD ∈ Ω
1
(M, End(2E)) being

βD :

(
0 2α2

2α1 0

)
. (137)

Here, again

2 αk(gradgf) := [Φk ◦ /Dk ,f]

= Φk ◦ γE (df) (138)



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Dirac Type Gauge Theories – Motivations and Perspectives 49

for all f ∈ C∞(M) and k = 1, 2.

Then, similar to the results presented before

VD = VD + U + 2 Φ ◦ /ωD + evg(β
2
D
) (139)

where

/ω
D

: γ2E (ωD) ≡

(
/ω

1
0

0 /ω
2

)
, U := evg

(
∇T

∗M ⊗End(2E)

D
βD
)
. (140)

2

Therefore,

tr2EVDtrEV1 + trEV2 + 2 trE (Φ1 ◦ Φ2) + 8 (trE ◦ evg)(α1 ◦ α2) + divgξD (141)

with

ξD : (tr2EβD)
♯ . (142)

The bosonic part of the Proposition (2) is then implied by (again, omitting all boundary

terms):

trγ(curv( /D)) = tr2EVD + (tr2E ◦ evg)(ω
2
D
)

= trγ(curv( /D1)) + trγ(curv( /D2)) +

+ 2 trE (Φ1 ◦ Φ2) + 8 (trE ◦ evg)(α1 ◦ α2) +

+ 2 (trE ◦ evg)(σ1 ◦ σ2) , (143)

where σk := extΘ(Φk − 2 /αk) ∈ Ω
1
(M, End(E)) and 2αk(v)Φk ◦ γE (v

♭
) for all v ∈ TM and k = 1, 2.

This finally ends the proof of Proposition (2). 2

The functionals (107–108) may look complicated at first glance. However, they yield a straight-

forward generalization of the usual action of the Standard Model of particle physics as discussed

in [22], which allows to also include Majorana mass terms. The latter feature will be discussed in

detail elsewhere.

Proposition 4. Let JP be anti-unitary and ΨPP̄(ΨP, ΨP) ∈ Sec(M,MP ⊕ MP). Also, let /DP be

real with respect to JP and both /DP and the real part of ΦP be formally self-adjoint. Then, the

Dirac action

ID,realID,ferm( /DPP̄) + ID,bos( /DPP̄) (144)



50 Jürgen Tolksdorf CUBO
11, 1 (2009)

reads:

ID,ferm( /DPP̄) =

∫

M

[〈ΨP, ( /DP + YP)ΨP〉P] dvolM ,

ID,bos(/DPP̄) =

∫

M

[
trγcurv( /DP) + (tr ◦ evg′ )(Y

2
P

) − (tr ◦ evg′ )(F
2

P
)
]
dvolM (145)

where YP, FP ∈ Ω
∗
(M, EndCl(P)).

Proof: This is a simple application of Proposition 2 taking into account that the (endomorphism

valued) one-forms αP, βP ∈ Ω
1
(M, End(P)) are linearly determined by the zero order operator

ΦP ∈ Sec(M, End(P)). Furthermore, due to the fundamental decomposition (33) every zero order

operator locally reads:

ΦPγ
I ⊗ φI (146)

where I = (i1, i2, . . . , il) is a multi-index (1 ≤ ik ≤ n for k = 0, 1, . . . ,n), γ
I ≡ γi1γi2 · · ·γil

and φI are local sections of Sec(M, EndCl(P)) which are totally antisymmetric with respect to the

multi index I. To avoid double counting the summation is thus take only for the ordered indices:

i1 < i2 < ... < il , l = 0, 1, , . . . ,n. In other words, the zero order section ΦP ∈ Ω(M, End(P)) is in

one-to-one correspondence with a general section φP ∈ Ω
∗
(M, EndCl(P)). Then, φP = e

I ⊗ φI with

{eI ≡ ei1 ∧ ei2 ∧ · · · ∧ eil | l = 0, 1, . . . ,n} being a local basis of ΛM ։ M.

Hence, the bosonic part of (105) reduces to

2ID,bos( /DPP̄) ≡

∫

M

[trγcurv( /DP) + trγcurv( /D
c

P
) + 2(tr ◦ evg′ )(φ

c

P
◦ φP)] dvolM (147)

where the evaluation map on the right hand side refers to the re-scaled (fiber) metric g′
ΛM

on the

Grassmann bundle ΛM ։ M that is defined by

g′
IJ

≡ λ′ gΛM(e
I,eJ) := trγIγJ + 1

4
gij trγ

IγiγJγj +

+
1

n2
gij tr

(
γiγI − gab γ

iγaγIγb
)(
γjγJ − gcd γ

jγcγJγd
)
. (148)

Indeed, one explicitly has

ΦP ◦ Φ
c

P
= γIγJ ⊗ φI ◦ φ

c

J
,

evg(αP ◦ α
c

P
) =

1

4
gij γ

IγiγJγj ⊗ φI ◦ φ
c

J
,

evg(βP ◦ β
c

P
) =

1

n2
gij
(
γiγI − gab γ

iγaγIγb
)(
γjγJ − gcd γ

jγcγJγd
)
⊗φI◦φ

c

J
(149)

where gij ≡ gM(ei,ej) etc., J = (j1,j2, . . . ,jk) is again a multi-index and Einstein’s summation

convention is applied. Let us call in mind that we do not distinguish between the metric on the

tangent and the co-tangent space of M.



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Dirac Type Gauge Theories – Motivations and Perspectives 51

As a consequence,

tr(Φ
c
P
◦ ΦP) + 4 (tr ◦ evg)(α

c
P
◦ αP) + (tr ◦ evg)(β

c
P
◦ βP)(tr ◦ evg′ )(φ

c

P
◦ φP) . (150)

Furthermore, every real Dirac type first order differential operator on a particle-anti-particle

module may be rewritten as

/D
PP̄

(
/D

P
YP − FP

YP + FP /D
c

P

)
. (151)

Here,

YP :=
1

2
(ΦP + Φ

c
P
) ≡ ReJΦP , (152)

FP :=
1

2
(ΦP − Φ

c
P
) ≡ iImJΦP , (153)

such that YP is the real and −iFP is the imaginary part of the zero order operator ΦP with respect

to the real structure JP.

According to the general case, we put

YP = γ
I ⊗ YI ,

FP = γ
J ⊗ FJ . (154)

The statement then follows from

tr(Φ
c
P
◦ ΦP)tr(Y

2
P
) − tr(F 2

P
) , (155)

which is analogous to the case of complex numbers (remember that Fc
P
− FP). 2

We note that /D
PP̄

leaves the real submodule Sec(M,MP ⊕ MP) ⊂ Sec(M,MPP̄) invariant if

and only if /D
c

P
= /D

P
and FP = 0.

Clearly, for YP = 0 and FP the curvature of /DP := i/∂A we get back the Pauli type Dirac

operators as specific real Dirac type operators on the real Hermitian Clifford module ξPP̄. Moreover,

the “diagonal sections” are motivated by the distinguished real submodule

Sec(M,MP ⊕ MP) ⊂ Sec(M,PP̄) (156)

of ξPP̄. Note that it is the doubling of MP (which we may identify with our former twisted Grass-

mann bundle EΛ, E ) which allows to add the Pauli-term to i/∂A such that the resulting first order

operator is still of Dirac type. Also note that the additional complex structure encountered in

the definition of (45) corresponds to the assumption that the zero order part of (88) is purely

imaginary. For the same matter it has to drop out in the fermionic part of the universal Dirac

action since it would yield a non-real contribution.



52 Jürgen Tolksdorf CUBO
11, 1 (2009)

5 Outlook

We presented a detailed motivation of “Dirac type gauge theories” which are gauge theories that

are based on the universal Dirac action (1). In particular, we have exhibit how the Dirac action

covers well-known differential equations, like the Maxwell and the Einstein equation. Indeed, the

Dirac action turns out to be a natural generalization of the Einstein-Hilbert functional. To also

obtain the Yang-Mills functional, one has to introduce a specific class of Dirac type operators

and we discussed their geometrical origin in terms of real Hermitian Clifford modules. We also

discussed the domain of the Dirac action from a geometrical point of view. We thereby proved

several Lichnerowicz type formulae for decomposable Laplace type operators which generalize the

corresponding result presented in [23].

It is well-known that there is a one-to-one correspondence between Dirac type operators on

a Clifford module and Clifford super-connections (see, for instance, in [3]). For this reason, the

domain of dependence of the Dirac action may not come as a surprise, especially because of the

isomorphisms (59). However, the latter hold true only when the module structure is fixed from the

outset. This, of course, does not permit interpreting the Einstein-Hilbert functional as a constraint

on the module structure. Moreover, our discussion clearly demonstrates that there exists a natural

functional on the Dirac bundle provided by the Dirac action.

The presented results clearly exhibit in what sense Dirac type operators and Clifford modules

provide a more general geometrical setting to describe gauge theories than connections and principal

bundles. Indeed, Dirac type gauge theories allow to describe different types of gauge theories,

like Yang-Mills theory, Einstein’s theory of gravity and spontaneously broken Yang-Mills gauge

theories, in a geometrically unified setting based on the same universal Dirac functional. This is

independent of whether the base manifold (“space-time”) M is supposed to be spin or not.

In order to gain more insight, however, one has to deal with the “moduli space of Dirac

operators”

M(E) ≡ D(E)/Diff(E) (157)

on which the Dirac functional descents. Of course, this set is probably far too wild and thus has

to be restricted to appropriate subsets like the solutions of

∗FD = ±FD , (158)

similar to the moduli space of (anti-) self-dual solutions of the ordinary Yang-Mills equation:

∗FA ±FA. For this, however, the domain of the Dirac functional has to be discussed more seriously,

in particular from an analytical point of view.

In contrast to the ordinary Yang-Mills equations one obtains still another reasonable con-

strains to Dirac type operators, similar to the (anti-) self-duality condition as, for instance, the

“unimodularity” condition:

£ξDdvolM = 0 . (159)



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Dirac Type Gauge Theories – Motivations and Perspectives 53

Finally, one may pose the question to what extent is there a relation between the stationary

points of the Dirac action (1) and the generalized Maxwell equation

/DFD = 0 . (160)

Again, in full generality this seems a hopeless task. However, it might be reasonable to discuss this

question using appropriate simple geometrical settings. This will be done in a forthcoming work.

Received: February 2008. Revised: August 2008.

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