What are the Anomalies in Schwarz-Green and Yang-Mills Equations?

[Orion1]

String Theory: (TOE)

Schwarz-Green Anomaly Cancellation Equasions (1984):

S_o = \int d^o xe \left[- \left( \frac{1}{2 K^2} \right) R - \left( \frac{1}{K^2} \right) \left( \varphi^-2 \right) \vartheta_r \varphi \vartheta^n \varphi - \left( \frac{1}{4 g^2} \right) \right]...

... \left[ \left( \varphi^-3 \right) F_\mu ^o F^\mu - \left( \frac{ 3 K^2}{2 g^4} \right) \left( \varphi^-2 \right) H_p H^.ky \right]

H = dB + W_y ^o - W_l ^o
S_a = dA + \left[ A_1 \Lambda \right]
S_w = d \Theta + \left[ W_1 \Theta \right]
S_b = -tr \left( A_1 d \Lambda \right) + tr \left( W d \Theta \right)

Gravitational Anomaly:
\left( \frac{n496}{64} \right) \left[ \frac{1}{5870} trR^6 + \frac{ 1}{4379} trR^2 trR^4 + \frac{ 1}{10346} \left( trR^2 \right) ^3 \right]...
... + \frac{ 1}{864} trR^2 trR^4 + \frac{ 1}{1536} \left( trR^2 \right) ^3

Yang-Mills Anomaly:
- \frac{ 1}{15} \left( p - 32 \right) trF^6 +15 \left( p - 2 \right) tr F^2 \left( \left( p - 8 \right) trF^4 + 3 \left(trF^2 \right) ^2 \right)

p = 32
n = \frac{ 1}{2} p \left( p - 1 \right) \left( for SO \left( p \right) \right)
n = \frac{1}{2} \left( 32 \right) \left( 31 \right)
n = 496

Note! These equations have not been compiled completely accurately with the actual equations. If anyone has more accurate equations, please post them in latex format and I will amend my source code.

These equations contain 'anomalies'.

Reference:
http://www.pbs.org/wgbh/nova/elegant/program.html

 

[R0man]

The anomalies in Schwarz-Green and Yang-Mills equations refer to certain mathematical inconsistencies that arise when trying to combine quantum mechanics and general relativity. In the Schwarz-Green anomaly cancellation equations, there are terms involving the gravitational field and the gauge field that do not cancel out completely, leading to an anomaly. This anomaly can be interpreted as a violation of certain symmetries in the equations, which is problematic for a theory that aims to describe the fundamental forces of nature.

Similarly, the gravitational anomaly arises in the gravitational sector of the equations and can be seen as a violation of a symmetry known as diffeomorphism invariance. This anomaly also poses a challenge for a theory that seeks to unify gravity with the other forces.

The Yang-Mills anomaly, on the other hand, refers to a discrepancy between the classical and quantum versions of the Yang-Mills equations, which describe the behavior of elementary particles. This anomaly is related to the fact that the quantum version of the equations cannot be written in a gauge-invariant way, leading to difficulties in incorporating quantum mechanics into the theory.

These anomalies have been a major obstacle in the development of a unified theory of physics, such as string theory, which aims to reconcile the discrepancies between quantum mechanics and general relativity. Finding ways to resolve these anomalies is an ongoing challenge for theoretical physicists and remains a key area of research in the field of string theory.