# How Does Topological Action Simplify with Levi-Civita Tensor Contractions?

• jinbaw
In summary, the conversation discusses simplifying an action with the term levicivita_[a,b,c,d]*levicivita^[mu,nu,rho,sigma]*R^[a,b]_[mu,nu]*R^[c,d]_[rho,sigma], where a, b, c, and d are flat indices and mu, nu, rho, and sigma are curved indices. The term 4*e^mu_a*e^nu_b*e^rho_c*e^sigma_d*R^a,b_mu,nu*R^c,d_rho,sigma is obtained and the question is raised about the possibility of getting a zero if certain indices are equal. The response explains that while this may appear to be the case, there
jinbaw
I'm trying to simplify an action that has the term: levicivita_[a,b,c,d]*levicivita^[mu,nu,rho,sigma]*R^[a,b]_[mu,nu]*R^[c,d]_[rho,sigma]

where a,b,c, and d are flat indices and mu nu rho sigma are curved indices

I got the term: 4*e^mu_a*e^nu_b*e^rho_c*e^sigma_d*R^a,b_mu,nu*R^c,d_rho,sigma

My question is if i have for example a=c levicivita_[a,b,c,d] is 0. however if i have a =c and mu=rho in the answer i got... i won't get a zero. is there some wrong in my computations? thank you

I think the magic is that, for instance, $$4{R^{12}}_{12}{R^{12}}_{12}$$ indeed appears in your term, but it will also appear in those five other terms that you call "permutations" - and they will cancel out. The point of your expansion is to get nice contractions in the formulas, but it is not computationally optimal in the sense that there will be many cancellations of terms. In other words: you are adding and subtracting the same terms in order to get certain nice expressions like square of the scalar curvature etc.

Does it make sense?

## 1. What is topological action in relation to veirbein?

Topological action is the mathematical concept used to describe the behavior of veirbein, which is a mathematical representation of the spin of a particle in a topological space.

## 2. How does veirbein contribute to topological action?

Veirbein is a key component in the formulation of topological action, as it represents the local structure of a topological space and its associated spin connections.

## 3. What is the significance of topological action in physics?

Topological action plays a crucial role in modern physics, particularly in theories such as quantum gravity, superstring theory, and supersymmetry. It helps to explain the behavior of particles in a topological space and provides a framework for understanding the fundamental forces of the universe.

## 4. How is topological action related to other mathematical concepts?

Topological action is closely related to other mathematical concepts such as gauge theory, differential geometry, and topology. It draws upon these disciplines to describe the behavior of particles in a topological space and their interactions with each other.

## 5. What are some applications of topological action?

Topological action has a wide range of applications, including in high-energy physics, condensed matter physics, and cosmology. It is also used in engineering and computer science, particularly in the development of quantum computing and topological data analysis methods.

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