Cohomology classes associated to anomalies
Authors: Peterson Gregory Lambros 
Published: June 2012

One of the proposed settings for the description of anomalies in the setting of gauge field theories is a local bicomplex associated to a principal fiber bundle

G
→
P
→
M

G \to P \to M

. It is a bigraded algebra with two differentials which is invariantly defined, i.e. independent of local coordinates of

M
M

. I denote it by

S
M

∙
,
∙

S_M^{ \bullet , \bullet }

. Briefly,

S
M

p
,
q

S_M^{p,q}

consists of local

p
p

-multilinear operators from the gauge algebra into

q
q

-forms on

M
M

which depend on a connection

A
A

in a local manner; local means that the operators decrease supports. The gauge algebra is the Lie algebra of the gauge group, which consists of diffeomorphisms of

P
P

that respect the action of

G
G

and cover the identity diffeomorphism of

M
M

. In this setting, the anomalies are described as integrals over

M
M

whose integrands can be shown to be representatives of total cohomology classes in

H
1

(

S
M

∙
,
∙

)

{H^1}(S_M^{ \bullet , \bullet })

. The main reason for restricting to a local bicomplex is due to Peetre’s theorem. It states that local operators are differential operators over open sets

U
⊂
M

U \subset M

. This property is both mathematically natural and required by physical considerations. This paper explores the computation of the total cohomology of the local bicomplex by beginning with the coordinate description of the differential operators and then determining which of these differential operators can be used to construct invariantly defined objects. What is accomplished is the description of the differential operators which are invariant under the action of the local diffeomorphisms of

R

n

{\mathbb {R}^n}

and the computation of their total cohomology over open sets

U
⊂
M

U \subset M

. The main result is that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Subscript d Superscript bullet Baseline left-parenthesis left-parenthesis upper S Subscript upper U Superscript bullet comma bullet Baseline right-parenthesis Superscript upper D i f f Super Subscript loc Superscript left-parenthesis double-struck upper R Super Superscript n Superscript right-parenthesis Baseline right-parenthesis asymptotically-equals upper H Subscript d Superscript bullet Baseline left-parenthesis upper W left-parenthesis German g right-parenthesis Subscript left-bracket StartFraction n Over 2 EndFraction right-bracket Baseline right-parenthesis comma">

H
d
∙

(

(

S
U

∙
,
∙

)

Diff

loc

(

R

n

)

)
≃

H
d
∙

(
W

(

g

)

[

n
2

]

)
,

H_d^ \bullet ({(S_U^{ \bullet , \bullet })^{{\operatorname {Diff}_{{\text {loc}}}}({\mathbb {R}^n})}}) \simeq H_d^ \bullet (W{(\mathfrak {g})_{[\tfrac {n} {2}]}}),

\]
where
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper S Subscript upper U Superscript bullet comma bullet Baseline right-parenthesis Superscript upper D i f f Super Subscript loc Superscript left-parenthesis double-struck upper R Super Superscript n Superscript right-parenthesis">

(

S
U

∙
,
∙

)

Diff

loc

(

R

n

)

{(S_U^{ \bullet , \bullet })^{{{\operatorname {Diff}}_{{\text {loc}}}}({\mathbb {R}^n})}}

denotes the invariant differential operators over the open set

U
U

and

W

(

g

)

[

n
2

]

W{(\mathfrak {g})_{[\tfrac {n} {2}]}}

is the Weil algebra of

g

\mathfrak {g}

, the Lie algebra of

G
G

truncated at

[

n
2

]

[\tfrac {n} {2}]

, the greatest integer less than or equal to half the dimension of

M
M

. This shows that the cohomology groups over open sets are nonzero only in the range

n
≤
q
≤
n
+
r

n \leq q \leq n + r

where

r
r

is the dimension of the Lie algebra

g

\mathfrak {g}

, and in this range they are all finite dimensional. This result is globalized in the special case that the associated fiber bundle

ad
∗

P

\operatorname {ad}^\ast \;P

is trivializable.