Tesselation

Gauge Invariant Operator

Lowest dimension gauge-invariant operators in SU(N) Yang-Mills theories.

We look at operators of the form

$$\mathcal{L}{\mathrm{int}} \supset \frac{1}{\Lambda^{k}} X Q{i} G^{\otimes k}_{N},$$

where $X \cong \left(p_{1}, p_{2}, \ldots, p_{N-1} \right)$ is some arbitrary representation of $\mathrm{SU}\left(N\right)$, $Q_{i}$ are generalized matter fields in the i^th fundamental representation, and $G_{N} \cong \left(1, 0, \ldots, 0, 1\right)$ is the adjoint representation. For a given $X$ we then seek the minimum number of copies $k_{\mathrm{min}}$ of the adjoint such that this operator is invariant under a global gauge transformation.

Installation

The code is very simple to install. Simply clone this repository and navigate to the root directory and do

pip install .

Code

The code is a simple collection of functions, all located in tesselation/functions.py. An example of their use is in Tesselation.ipynb. There are four primary functions of interest to the user:

l_pair(p, i)

Pair of l that results in the optimal irreducible representation in tensor product of X times Q_i.

Parameters:
     p : array_like
          Dynkin label of X.
     i : int
          Index of only nonzero Dynkin coefficient in Q_i.

Returns:
     l1, l2 : int, int
          Pair of l such that p[l1] !=0, p[l2] != 0, l1 is largest l such that l <= N - i, and is smallest l such that l > l1.

Nality(p)

N-ality of irreducible representation X. The number of boxes in the Young tableau for X mod N.

Parameters:
     p : array_like
          Dynkin label of X.

Returns:
     t : int
          The number of boxes in the Young tableau for X mod N.

fj(p)

Optimum irreducible representation in the tensor product of X times Q_i.

Parameters:
     p : array_like
          Dynkin label of X.

Returns:
     fj : array_like
          Resulting Dynkin label for optimal irreducible representation.

kmin(p)

Minimum number of copies of the adjoint such that tensor product X Q_i G_N ... G_N contains a trivial subspace.

Parameters:
     p : array_like
          Dynkin label of irreducible representation X.

Returns:
     kmin : int
          Minimum number of copies of the adjoint representation such that the operator is gauge invariant.