arXiv:hep-th/1011.1095

1. Introduction to Geometric BRST

2. Axial Extension of Gauge Theories

3. Anti-Ghost and Anti-BRST Symmetry

4. Application and Conclusions

𝐺𝑎𝑢𝑔𝑒 𝑆𝑦𝑚𝑚𝑒𝑡𝑟𝑦𝑸𝒖𝒂𝒏𝒕𝒊𝒛𝒂𝒕𝒊𝒐𝒏

𝐵𝑅𝑆𝑇 𝑆𝑦𝑚𝑚𝑒𝑡𝑟𝑦

𝑬𝒙𝒕𝒆𝒏𝒔𝒊𝒐𝒏 ↓ ↓ 𝑬𝒙𝒕𝒆𝒏𝒔𝒊𝒐𝒏𝐴𝑥𝑖𝑎𝑙 𝑆𝑦𝑚𝑚𝑒𝑡𝑟𝑦

𝑸𝒖𝒂𝒏𝒕𝒊𝒛𝒂𝒕𝒊𝒐𝒏𝐴𝑛𝑡𝑖 − 𝐵𝑅𝑆𝑇 𝑆𝑦𝑚𝑚𝑒𝑡𝑟𝑦

.

1. Introduction to Geometric BRST

Let 𝑀 be a 2𝑛-dimensional spin manifold with spin bundle 𝑆 ↪ 𝑆(𝑀) ↠ 𝑀 and 𝐺 be a semi simple Lie group with a unitary irreducible representation on finite dimensional complex vector space 𝑉.

Suppose that 𝐺 ↪ 𝑃 ↠ 𝑀 is a principal 𝐺 -bundle equipped with a connection structure given by Cartan connection form 𝔞 and curvature Ω = d𝔞 + 𝔞2. Then 𝐸 ≔ 𝑃 ×𝐺 𝑉, the equivalency classes of (𝑝, 𝑣) for;

𝑝, 𝑣 ~ 𝑝 ⊲ 𝑔, 𝑔−1 ⊳ 𝑣 ,

𝑝 ∈ 𝑃 and 𝑣 ∈ 𝑉, produces a vector bundle over 𝑀 with standard fiber 𝑉;

𝑉 ↪ 𝐸 ↠ 𝑀.

Moreover the Cartan connection form 𝔞 induces a parallelism structure over 𝑉 ↪ 𝐸 ↠ 𝑀. This induced connection structure together with the pull back of Levi-Civita connection over ℝ2𝑛 ↪ 𝑇𝑀 ↠ 𝑀 via;

𝐵 𝑆 𝑀 × 𝑆𝑝𝑖𝑛 2𝑛 𝐴𝑐𝑡𝑖𝑜𝑛

𝐵(𝑆 𝑀 ) → 𝑀

↓ × ↓ ↓ = 𝐵 𝑇𝑀 × 𝑆𝑂 2𝑛

𝐴𝑐𝑡𝑖𝑜𝑛 𝐵 𝑇𝑀 → 𝑀

,

fixes a connection over the 𝑮-gauge theory bundle; 𝑉⨂𝑆 ↪ 𝐸⨂𝑆 𝑀 ↠ 𝑀.

Therefore the 𝐺 -gauge theory bundle is equipped with covariant derivative 𝛻 locally given by;

𝛻𝜇𝜓 = 𝜕𝜇 − 𝑖𝐴𝜇𝑎𝑇𝑎 − 𝑆𝜇

𝜎𝜏 𝛾𝜎 , 𝛾𝜏 𝜓,

for anti-Hermitian matrices *−𝑖𝑇𝑎+𝑎 as a representing basis of 𝔤 ≔ Lie𝐺, and for Dirac matrices *𝛾𝜇+𝜇.

Let 𝒢 ≔ 𝐶∞(𝑀, 𝐺) be the gauge transformation group, 𝐺 ↪ 𝑃 ↠ 𝑀 be a trivial bundle (only for simplicity) and 𝑒: 𝑀 → 𝑃 be considered as the global identity section. Also let 𝒜 be the Affine space of 𝑒∗(𝔠) for Cartan connection forms 𝔠 over 𝐺 ↪ 𝑃 ↠ 𝑀. Indeed 𝒜 can be considered as the space of gauge fields 𝐴 = −𝑖𝐴𝜇

𝑎𝑇𝑎d𝑥𝜇. There is a free action of 𝒢 on 𝒜

from right; 𝐴 ⊲ 𝑔 ≔ Ad𝑔−1 𝐴 + 𝑔−1d𝑔,

for 𝐴 ∈ 𝒜, 𝑔 ∈ 𝒢. Therefore we have the following principal 𝒢-bundle; 𝒢 ↪ 𝒜 ↠ 𝒜/𝒢.

Set a connection over this principal bundle with Cartan connection form Π, fix a connection over 𝐺 ↪ 𝑃 ↠ 𝑀 with Cartan connection form 𝔞0, 𝐴0 = 𝑒∗(𝔞0), and define the fiber map 𝑖𝐴0

: 𝑀 × 𝒢 → 𝑀 × 𝒜, with;

𝑖𝐴0𝑚, 𝑔 ≔ 𝑚, 𝐴0 ⊲ 𝑔 .

Let Θ ∈ Ω1(𝑀 × 𝒜)⨂𝔤 be defined by;

Θ 𝑣, 𝜂 ≔ 𝐴 𝑣 + Π 𝜂 𝑝 ,

for 𝑣, 𝜂 ∈ 𝑇𝑚𝑀 × 𝑇𝐴𝒜, and define; 𝑖𝐴0

∗ d𝒜 = 𝛿 (the exterior derivative

over 𝒢). Then it can be easily shown that; 𝑖𝐴0

∗ Θ (𝑝,𝑔)

= 𝐴 + 𝜔, for

𝐴 = 𝐴0 ⊲ 𝑔 and for 𝜔 ≔ 𝑖𝐴0

∗ Π a left invariant Lie𝒢-valued 1-form over

𝒢.

Moreover we have;

𝛿𝐴 = d𝜔 + 𝐴, 𝜔 , 𝛿𝜔 = 𝜔2 =1

2,𝜔, 𝜔-

𝛿d + d𝛿 = 𝛿, d = 0 , 𝛿2 = 0.

It is seen that by considering 𝐴 as the gauge field, 𝜔 and 𝛿 are respectively in complete agreement with ghost field and BRST derivation.

2. Axial Extension of Gauge Theories

Let 𝑀 be ℝ2𝑛 equipped with the Minkowski (Euclidean) metric and 𝐺 be a simply connected semi simple Lie group with Lie algebra 𝔤. Also suppose that 𝐺 acts unitarily and irreducibly on finite dimensional complex vector space 𝑉 with *−𝑖𝑇𝑎+𝑎 the anti-Hermitian represented basis for 𝔤.

Consider a Yang-Mills 𝐺-gauge theory over ℝ2𝑛 with Lagrangian density;

ℒ = −1

4𝑡𝑟*𝐹𝜇𝜈𝐹𝜇𝜈+ + 𝑖𝜓 𝛾𝜇𝜕𝜇𝜓 + 𝐴𝜇

𝑎𝜓 𝛾𝜇𝑇𝑎𝜓,

for 𝐹𝜇𝜈 = (𝜕𝜇𝐴𝜈 − 𝜕𝜈𝐴𝜇) − 𝑖,𝐴𝜇 , 𝐴𝜈-, 𝐴𝜇 = 𝐴𝜇𝑎𝑇𝑎.

ℒ is invariant under gauge transformations of 𝑒−𝑖𝛼 with 𝛼 ∈ 𝐶∞(𝑀, 𝔤); 𝜓 → 𝑒−𝑖𝛼𝜓,

𝐴𝜇 → 𝑒−𝑖𝛼𝑖𝜕𝜇𝑒𝑖𝛼 + Ad𝑒−𝑖𝛼 𝐴𝜇 .

To impose the local axial symmetry to the theory, one should replace ℒ by;

ℒ𝑒𝑥 = ℒ𝐺𝑎𝑢𝑔𝑒 + 𝑖𝜓 𝛾𝜇𝜕𝜇𝜓 + 𝐴𝜇𝑎𝜓 𝛾𝜇𝑇𝑎𝜓 + 𝐵𝜇

𝑎𝜓 𝛾𝜇𝛾5𝑇𝑎𝜓,

with ℒ𝐺𝑎𝑢𝑔𝑒 ≔ −1

4𝑡𝑟*𝐹𝜇𝜈𝐹𝜇𝜈+, for;

𝐹𝜇𝜈 = 𝜕𝜇 𝐴𝜈 + 𝐵𝜈 − 𝜕𝜈 𝐴𝜇 + 𝐵𝜇 − 𝑖,(𝐴𝜇 + 𝐵𝜇) , (𝐴𝜈 + 𝐵𝜈)-,

𝐴𝜇 = 𝐴𝜇𝑎𝑇𝑎, and 𝐵𝜇 = 𝐵𝜇

𝑎𝑇𝑎𝛾5.

Therefore the Lagrangian density ℒ𝑒𝑥 is invariant under the union of gauge and local axial transformations respectively given by;

𝜓 → 𝑒−𝑖𝛼𝜓,

𝐴𝜇 + 𝐵𝜇 → 𝑒−𝑖𝛼𝑖𝜕𝜇𝑒𝑖𝛼 + Ad𝑒−𝑖𝛼 𝐴𝜇 + 𝐵𝜇 ,

and; 𝜓 → 𝑒−𝑖𝛼𝛾5𝜓,

𝐴𝜇 + 𝐵𝜇 → 𝑒−𝑖𝛼𝛾5𝑖𝜕𝜇𝑒𝑖𝛼𝛾5 + Ad𝑒−𝑖𝛼𝛾5 𝐴𝜇 + 𝐵𝜇 ,

for 𝛼 ∈ 𝐶∞(𝑀, 𝔤).

1. 1. Does ℒ𝑒𝑥 define a gauge theory with semi simple gauge group?

2. 2. Can this theory be quantized due to path-integral or geometrical versions of Faddeev-Popov quantization?

3. 3. If ℒ𝑒𝑥 be a gauge theory, would it preserve nontrivial topological aspects of gauge theory ℒ such as instantonic and anomalous ones?

Let *𝑡𝑎+𝑎=1𝑘 be a basis for Lie algebra 𝔤. By definition the axial extention

of 𝖌 is a 2𝑘 -dimensional Lie algebra 𝔤 generated with elements *𝑡𝑎, 𝑠𝑎+𝑎=1

𝑘 and commutation relations; 𝑠𝑎 , 𝑠𝑏 = 𝑡𝑎, 𝑡𝑏 , 𝑠𝑎 , 𝑡𝑏 = − 𝑡𝑏, 𝑠𝑎 .

It can be easily seen that the process of axial extension produces a natural transformation in the small category of Lie algebras;

𝜎𝔤: 𝔤 → 𝔤 .

It can also be seen that 𝔤 is semi simple iff 𝔤 is semi simple.

Let 𝐺 be the simply connected semi simple Lie group with Lie algebra 𝔤 .

Thus there is a smooth injective homomorphism Σ𝔤: 𝐺 → 𝐺 such that;

dΣ𝔤 = 𝜎𝔤. In fact 𝐺 is a closed Lie subgroup of 𝐺 .

Using the principal 𝐺-bundle 𝐺 ↪ 𝑃 ↠ 𝑀, the Cartan connection form 𝔞 on 𝑃 and the homomorphism Σ𝔤, one can naturally define a principal 𝐺 -

bundle over 𝑀, say 𝐺 ↪ 𝑃 ↠ 𝑀, a principal bundle homomorphism;

𝑃 × 𝐺 𝐴𝑐𝑡𝑖𝑜𝑛

𝑃 → 𝑀 𝜉 ↓ × ↓ 𝐴𝔤 𝜉 ↓ = ,

𝑃 × 𝐺 𝐴𝑐𝑡𝑖𝑜𝑛

𝑃 → 𝑀

and a Cartan connection form over 𝑃 , say 𝔞 , such that; 𝜉∗ 𝔞 = 𝔞.

Moreover the Lie algebra homomorphism; 𝑡𝑎 ↦ −𝑖𝑇𝑎 , 𝑠𝑎 ↦ −𝑖𝑇𝑎𝛾5,

leads to a unitary irreducible representation of Lie group 𝐺 on 𝑉.

It would be clear that this representation together with principal 𝐺 -bundle 𝐺 ↪ 𝑃 ↠ 𝑀, lead to a Yang-Mills 𝐺 -gauge theory which its Lagrangian density is given by ℒ𝑒𝑥.

More precisely 𝐴⨁𝐵 = −𝑖 𝐴𝜇 + 𝐵𝜇 d𝑥𝜇 can be considered as extended

gauge field.

In fact ℒ𝑒𝑥 is a gauge theory with semi simple gauge group 𝐺 and consequently can be quantized by Faddev-Popov quantization.

On the other hand ℒ𝑒𝑥 is the most natural extension of ℒ and thus it preserves all the nontrivial topological aspects of ℒ such as instantons and anomalies even when 𝐺 ↪ 𝑃 ↠ 𝑀 is topologically nontrivial.

3. Anti-Ghost and Anti-BRST Symmetry

Consider the assumptions of last section.

Let 𝑝𝑟: 𝔤 ↠ 𝔤 be the projection map; 𝑝𝑟 𝑡𝑎 = 𝑡𝑎 , 𝑝𝑟 𝑠𝑎 = 0,

for 𝑎 = 1, … , 𝑘, and set 𝑝𝑟5 = 1 − 𝑝𝑟.

Suppose that 𝒜 and ℬ be respectively the spaces of 𝑒∗(𝑝𝑟 ∘ 𝔞 ) and 𝑒∗(𝑝𝑟5 ∘ 𝔞 ) for Cartan connection forms 𝔞 over 𝐺 ↪ 𝑃 ↠ 𝑀 and for identity global section 𝑒: 𝑀 → 𝑃 . Indeed 𝒜 and ℬ can respectively be considered as the spaces of vector and axial parts of extended gauge fields.

Finally denote the (extended) gauge transformation group 𝐶∞(𝑀, 𝐺 ) by 𝒢 .

There is a free action of 𝒢 on 𝒜 × ℬ from right; (𝐴⨁𝐵) ⊲ 𝑔 ≔ Ad𝑔−1 𝐴 + 𝐵 + 𝑔−1d𝑔,

for 𝐴⨁𝐵 ≡ (𝐴, 𝐵) ∈ 𝒜 × ℬ, 𝑔 ∈ 𝒢 .

Therefore we have the following principal 𝒢 -bundle; 𝒢 ↪ 𝒜 × ℬ ↠ (𝒜 × ℬ)/𝒢 .

Set a connection over this principal bundle with Cartan connection form Π , fix a connection over 𝐺 ↪ 𝑃 ↠ 𝑀 with Cartan connection form 𝔞 0, for 𝑒∗ 𝔞 0 = (𝐴0, 𝐵0), and define the fiber map 𝑖(𝐴0,𝐵0):𝑀 × 𝒢 → 𝑀 × (𝒜 ×ℬ), with;

𝑖(𝐴0,𝐵0) 𝑝, 𝑔 ≔ 𝑝, (𝐴0, 𝐵0) ⊲ 𝑔 .

𝒜 × ℬ

(𝒜 × ℬ)/𝒢

𝒢

Let Θ ∈ Ω1 𝑀 × 𝒜 × ℬ ⨂𝔤 be given by;

Θ 𝑣, 𝜂 ≔ (𝐴⨁𝐵) 𝑣 + Π 𝜂 𝑝 ,

for 𝑣, 𝜂 ∈ 𝑇𝑚𝑀 × 𝑇(𝐴,𝐵)(𝒜 × ℬ).

Then it can be shown that;

𝑖(𝐴0,𝐵0)∗ Θ

(𝑝,𝑔)= 𝐴⨁𝐵 + 𝜔⨁𝜔∗ ,

for gauge field 𝐴⨁𝐵 ≡ (𝐴, 𝐵) = (𝐴0, 𝐵0) ⊲ 𝑔 and for 𝜔⨁𝜔∗ a left invariant Lie𝒢 -valued 1-form over 𝒢 . Indeed;

𝜔 = 𝑝𝑟 ∘ 𝑖(𝐴0,𝐵0)∗ Π , 𝜔∗ = 𝑝𝑟5 ∘ 𝑖(𝐴0,𝐵0)

∗ Π .

Set 𝑖(𝐴0,𝐵0)∗ d𝒜 = 𝛿, then we have;

𝛿𝐴 = d𝜔 + 𝐴, 𝜔 , 𝛿𝜔 = 𝜔2 =1

2,𝜔, 𝜔-

𝛿𝐵 = 𝐵, 𝜔 , 𝛿𝜔∗ = 𝜔∗, 𝜔 𝛿d + d𝛿 = 𝛿, d = 0 , 𝛿2 = 0.

On the other hand for 𝑖(𝐴0,𝐵0)∗ dℬ = 𝛿∗ we have;

𝛿∗𝐴 = 𝐵, 𝜔∗ , 𝛿∗𝜔 = 0

𝛿∗𝐵 = d𝜔∗ + 𝐴, 𝜔∗ , 𝛿∗𝜔∗ = 𝜔∗2 =1

2𝜔∗, 𝜔∗

𝛿∗d + d𝛿∗ = 0 , 𝛿∗2 = 0.

And eventually; 𝛿𝛿∗ + 𝛿∗𝛿 = 𝛿, 𝛿∗ = 0.

It is seen that by considering 𝐴 as the vector part of extended gauge field and noting that 𝐵 is a pure gauge field, 𝜔∗ and 𝛿∗ are respectively in complete agreement with anti-ghost field and anti-BRST derivation.

To see this more precisely it is enough to replace the Nakanishi-Lautrup (auxiliary) field in the standard formulation of BRST/anti-BRST derivation with 𝜔∗, 𝜔 .

More precisely in the standard formulation of Yang-Mills theories and Faddeev-Popov Quantization, when local axial symmetry is broken, the ghost field 𝜔 is a differential 1-form along the directions of local symmetry (gauge transformations), but the anti-ghost field 𝜔∗ is a differential 1-form along the directions of global symmetry (axial transformations).

4. Application and Conclusions

Consider the assumptions of previous sections.

One of the most important application of BRST/anti-BRST correlation for axially extended gauge theories is to work out an extended counterpart of consistent anomaly called modified anomaly and to give a new classification of anomalous behaviors in the setting of an extended version of equivariant BRST cohomology called extended BRST cohomology.

To extract the modified anomaly according to the Stora-Zumino procedure, one should use d, 𝛿 and 𝛿∗ alternatively to provide a generalized formulation of descent equations.

Initially, by Bianchi identity we have;

𝛿𝑅 = 𝜔𝑅 − 𝑅𝜔 = ,𝜔, 𝑅-, 𝛿∗𝑅 = 𝜔∗𝑅 − 𝑅𝜔∗ = ,𝜔∗, 𝑅-,

d𝑅 = 𝑅 𝐴⨁𝐵 − 𝐴⨁𝐵 𝑅 = ,𝑅, 𝐴⨁𝐵-,

for 𝑅 = d𝑒∗(𝔞 ) + 𝑒∗(𝔞 )2 = d(𝐴⨁𝐵) + (𝐴⨁𝐵)2 the pull back of the curvature. Indeed 𝐹𝜇𝜈d𝑥𝜇 ∧ d𝑥𝜈 = 2𝑖𝑅.

Thus 𝑡𝑟*𝑅𝑛+1+ , the (𝑛 + 1) th Chern character, is simultaneously a deRham and (anti-) BRST closed form.

Consider 𝑡𝑟*𝑅𝑛+1+ as a (2𝑛 + 2)-form over ℝ2𝑛+2. Thus, the Poincare lemma leads to;

𝑡𝑟 𝑅𝑛+1 = dΩ2𝑛+10,0 ,

𝛿Ω2𝑛+10,0 = dΩ2𝑛

1,0, 𝛿∗Ω2𝑛+10,0 = dΩ2𝑛

0,1,

𝛿Ω2𝑛1,0 = dΩ2𝑛−1

2,0 , 𝛿∗Ω2𝑛0,1 = dΩ2𝑛−1

0,2 ,

𝛿∗𝛿Ω2𝑛+10,0 = dΩ2𝑛

1,1,

where Ω𝑖𝑗,𝑘

is a deRham differential 𝑖-form with ghost number 𝑗 − 𝑘,

while 𝑖 + 𝑗 + 𝑘 = 2𝑛 + 1. Actually, Ω𝑖𝑗,𝑘

is simultaneously a differential 𝑗-form over 𝒜 and a differential 𝑘-form over ℬ.

It is seen that;

𝛿 + 𝛿∗ Ω2𝑛1,0 + Ω2𝑛

0,1 = d(Ω2𝑛−12,0 + Ω2𝑛−1

0,2 + Ω2𝑛−11,1 ),

and hence;

𝛿 + 𝛿∗ Ω2𝑛1,0 + Ω2𝑛

0,1

ℝ2𝑛

= 0.

Thus, up to a constant factor Ω2𝑛1,0 + Ω2𝑛

0,1 can be considered as the modified nonintegrated anomaly.

In the other words, Ω2𝑛1,0 + Ω2𝑛

0,1

ℝ2𝑛 is a candidate for (𝛿 + 𝛿∗)𝑊 for quantum action 𝑊. A direct calculation shows that when 𝑛 = 2 then;

Ω41,0 + Ω4

0,1 = 𝑡𝑟*d(𝜔⨁𝜔∗)( 𝐴⨁𝐵 d 𝐴⨁𝐵 −1

2𝐴⨁𝐵 3)+,

which is the modified consistent anomaly up to a factor of 𝑐2 =1

24𝜋2.

Indeed, since 𝐵 is a pure gauge then one can set 𝐵 = 0 to achieve the well-known consistent anomaly (the anti-ghost will be killed automatically after taking the trace).

On the other hand, ghost number counting leads to;

𝑐2Ω41,0 =

1

24𝜋2𝑡𝑟*d𝜔( 𝐴⨁𝐵 d 𝐴⨁𝐵 −

1

2𝐴⨁𝐵 3)+,

which is called ghost consistent anomaly and is the anomaly of (vector) gauge current.

Moreover, the other term;

𝑐2Ω40,1 =

1

24𝜋2𝑇𝑟*d𝜔∗( 𝐴⨁𝐵 d 𝐴⨁𝐵 −

1

2𝐴⨁𝐵 3)+,

is called anti-ghost consistent anomaly and is the anomaly of axial current.

Moreover;

𝛿 + 𝛿∗ Ω2𝑛−12,0 + Ω2𝑛−1

0,2 + Ω2𝑛−11,1 = d(Ω2𝑛−2

3,0 + Ω2𝑛−22,1 + Ω2𝑛−2

1,2 + Ω2𝑛−20,3 ).

Therefore,

Ω2𝑛−12,0 + Ω2𝑛−1

0,2 + Ω2𝑛−11,1 ,

is a candidate for the modified Schwinger term up to the factor 𝑐𝑛.

For 𝑛 = 2, the modified Schwinger term is given by;

𝑐2(Ω32,0 + Ω3

0,2 + Ω31,1) =

1

24𝜋2𝑡𝑟*(d(𝜔⨁𝜔∗))2(𝐴⨁𝐵)+.

where;

𝑐2Ω32,0 =

1

24𝜋2𝑡𝑟 d𝜔 2𝐴 ,

𝑐2Ω30,2 =

1

24𝜋2𝑡𝑟 d𝜔∗ 2𝐴 ,

𝑐2Ω31,1 =

1

24𝜋2𝑡𝑟 d𝜔d𝜔∗ + d𝜔∗d𝜔 𝐵 ,

are respectively called ghost/ghost, anti-ghost/anti-ghost and ghost/anti-ghost consistent Schwinger term and are respectively the anomalous terms of vector/vector, axial/axial and vector/axial currents commutation relations.

Consequently the extended descent equations give rise to a bi-complex which commutes up to exact deRham forms;

Ω2𝑛+10,0

d,𝛿 Ω2𝑛

1,0 d,𝛿

Ω2𝑛−12,0

d,𝛿 Ω2𝑛−2

3,0 d,𝛿

. . . .

↕ d, 𝛿∗ ↓ −𝛿∗ ↓ −𝛿∗ ↓ −𝛿∗ . . . .

Ω2𝑛 0,1 𝛿

→Ω2𝑛 2𝑛−1 1,1 𝛿

→Ω2𝑛−1 2𝑛−2 2,1 𝛿

→Ω2𝑛−2 2𝑛−3 3,1 𝛿

→ . . . .

↕ d, 𝛿∗ ↓ −𝛿∗ ↓ −𝛿∗ ↓ −𝛿∗ . . . .

Ω2𝑛−10,2 𝛿

→Ω2𝑛−1 2𝑛−2 1,2 𝛿

→Ω2𝑛−2 2𝑛−3 2,2 𝛿

→Ω2𝑛−3 2𝑛−4 3,2 𝛿

→ . . . .

↕ d, 𝛿∗ ↓ −𝛿∗ ↓ −𝛿∗ ↓ −𝛿∗ . . . .

Ω2𝑛−20,3 𝛿

→Ω2𝑛−2 2𝑛−3 1,3 𝛿

→Ω2𝑛−3 2𝑛−4 2,3 𝛿

→Ω2𝑛−4 2𝑛−5 3,3 𝛿

→ . . . .

↕ d, 𝛿∗ ↓ −𝛿∗ ↓ −𝛿∗ ↓ −𝛿∗ . . . .

⋮ ⋮ ⋮ ⋮ . . . .

In fact modified anomalies and extended descent equations produce a generalized formulation of BRST cohomology, called extended BRST cohomology, which is the cohomology of total complex of the given bi-complex.