This package presents numerical routines used to obtain or check some of the results of the article [1]. We investigate these phenomena, which can be broken into two separate but related problems. The first problem is to count the number of Goldstone modes arising from spontaneous symmetry breaking, which depends directly on $N$, the rank of the global symmetry group. The second problem is to calculate the dimension of some irreducible representations, which include physical states and also depend on $N$. Two separate chapters are devoted to both of these problems, Goldstone modes count and scaling formula: in each of them these problems are considered numerically for large $N$.

Goldstone modes count

This Mathematica package provides some functions for QFT and representation theory that were used for computations in the project [1].

Theory explanation:

$b$-baryon belongs to the second fundamental representation of the SU($N$) group and transforms as $$b'=UbU^T\approx b+\delta b=b+i\alpha^a{T^a b+b (T^a)^T},$$ where $T^a$ are the SU($N$) generators, $U\in \mathrm{SU}(N)$.

The VEV of massless $b$-baryon, $$ \langle b \rangle = b J ,, \quad J \equiv \oplus_{j=1}^{K=N/2}\left( \begin{array}{cc} 0 & 1 \ -1 & 0 \end{array}\right)_j \equiv \left( \begin{array}{cccc} \epsilon & 0 & \ldots & 0 \ 0 & \epsilon & \ldots & 0 \ \vdots & & \ddots & \vdots \ 0 & & \ldots & \epsilon \end{array} \right),\quad \epsilon=\left( \begin{array}{cc} 0 & 1 \ -1 & 0 \end{array}\right)$$ breaks this global $\mathrm{SU}(N)$ down to some subgroup, causing Goldstone bosons (GB) to emerge. We want to count the number of these GB. To do this, we apply the following algorithm:

1. For a given $N$ build the $N\times N$ matrix $\delta b$;
2. Equate all components of $\delta b$ to 0, which will give the system of $N^2-1$ linear equations (SLE) for $\alpha^a$;
3. Calculate the rank of the SLE matrix $A$. This gives the number of GB.

This algorithm is implemented in this Mathematica package

Package contents:

• Tcartan[k,N]: defines k diagonal $\mathrm{SU}(N)$ Cartan generators via relation $$T^k=\frac{1}{\sqrt{2k(k+1)}}\mathrm{diag}\left( \underbrace{1,1,\ldots,1}{k \text{ terms}},-k,\underbrace{0,0,\ldots,0,0}{N-k-1 \text{ terms}} \right).$$
• T1[i,j,N]: defines set of remaining symmetric $\mathrm{SU}(N)$ generators via relation $$T^{(i,j;1)}{a,b}=\frac12 (\delta{a,i}\delta_{b,j}+\delta_{a,j}\delta_{b,i}).$$
• T2[i,j,N]: defines remaining antisymmetric $\mathrm{SU}(N)$ generators via relation $$T^{(i,j;2)}{a,b}=\frac{1}{2i} (\delta{a,i}\delta_{b,j}-\delta_{a,j}\delta_{b,i}).$$
• bvacMatrix[N]: takes T2[i,j,N] and returns $N\times N$ matrix $\frac{\langle b \rangle}{b}$ using T2[i,j,N] via the formula $$\frac{\langle b \rangle}{b}=2i\sum_{i=1}^{N-1}T^{(N,N+1;2)}.$$
• alpha[N]: defines $N^2-1$ parameters $\alpha^a$.
• deltavac[N]: takes bvacMatrix[N] and \alpha[N] and returns $N\times N$ matrix $\delta b/b$ using the formula $$\delta b/b=\frac{1}{b}\alpha^a{T^a \langle b\rangle+\langle b\rangle (T^a)^T}$$(imaginary unit $i$ is dropped from the definition as it does not contribute to the number of GB calculation).
• \alphamatrix[N]: takes deltavac[N] and returns matrix A. This matrix defines the system of $N^2-1$ linear algebraic equations for $\alpha^a$ and has an extra column of zeros, which makes A an $N^2\times N^2$ square matrix and does not change the number of linearly independent rows.
• gscount[N]: takes \alphamatrix[N] and returns the rank of matrix A.

Scaling formula

These C++ and Python codes provide functions for numerical verification of some representation theory formulas from [1].

Theory explanation:

All massive baryonic states (which can be thought of as composites made of $n$ elementary b-baryons) belong to the irreducible representations with highest weights $$(\lambda_1,\lambda_2,\ldots,\lambda_N)$$ with $\lambda_1\geq\lambda_2\geq\ldots\geq\lambda_N$ and $\lambda_{2i-1}=\lambda_{2i}\equiv\tilde{\lambda}i,,i=1,\ldots,K$, i.e. have the form of "double-ladders". Note that $\sum{i=1}^N\lambda_i=2n$.

To estimate the number of baryonic states at high energy, we need to understand the behavior of the dimension of an irreducible representation in the limit $$n\gg N \gg1.$$ We start with the general Weyl formula for the dimension of the irreducible representation $\lambda$: $$\mathrm{dim},\lambda=\prod_{1\leq i<j\leq N}\frac{\lambda_i-\lambda_j-i+j}{-i+j}.$$

When $n\rightarrow\infty$, the length of each row of these "double-ladder diagrams" grows correspondingly, and we can account for this growth by scaling the row length as $$\tilde{\lambda}_i\rightarrow t \tilde{\lambda}_i,$$ which leads to $n\rightarrow tn$, and the Weyl formula after this rescaling gives $$\dim\lambda\stackrel{t\rightarrow\infty}{\sim} t^{\frac12N^2-N}, \quad t=O(n).$$ We want to perform two tests of this scaling formula.

Test 1

The first test considers a single "double-ladder" diagram and looks for its asymptotics at large $t$ after rescaling. To do this we apply the following algorithm:

1. Consider a "double-ladder" diagram ${\tilde{\lambda}i}{i=1}^K$ with fixed $n,K$ and $s$, where $s\equiv\tilde{\lambda}i-\tilde{\lambda}{i+1}$ is fixed for every $i$ and denotes the difference in row lengths.
2. Apply the rescaling $\tilde{\lambda}_i\rightarrow t \tilde{\lambda}i$ with $t$ ranging from 1 to some $t{max}=O(N)$ to obtain the "double-ladder" diagram with ${t\tilde{\lambda}i}{i=1}^K$, calculate the logarithm of its dimension $\ln\dim\lambda(t)$ for each $t$ using the Weyl formula, and divide it by the expected asymptotic value $2K^2\ln t$.
3. As $t\rightarrow\infty$ this ratio should tend to 1. This algorithm is implemented in single_double_ladder_scaling.cpp, and scaling_plot.py is used to visualize the results.

Test 2

This test considers a special case of "double-ladders" — "box" diagrams : $$\begin{cases} {\lambda}_1={\lambda}_2=\ldots={\lambda}_s=\frac{2n}{s}, \quad &\text{for some $s\in[1,N]$} \ {\lambda}_i=0, \quad &i=s+1,\ldots, N \end{cases} $$ Here $s$ is the height of this box. For this type of diagram, the scaling method gives the following behavior of the dimension: $$\ln\dim\lambda\sim s(N-s)\ln t.$$ This formula has a maximum at $s=\frac{N}{2}$, and we want to verify this numerically, so we apply the following algorithm:

1. For every even $s$ in the range from $2$ to $N$, calculate $\ln\dim\lambda(s)$ using the Weyl formula. The expected maximum should be at $s=\frac{N}{2}$. This algorithm is implemented in box_dim2_v02.cpp, and box_plot.py is used to visualize the results. *Note: in box_dim2_v02.py, calculations are performed at fixed values of $n$; therefore one should expect the maximum at $s=\frac{N}{2}$ only in the limit $\frac{n}{N}\to\infty$. Otherwise, the peak will be shifted to the left.

Package contents

• single_double_ladder_scaling.cpp: provides functions to calculate the logarithmic asymptotics of the dimension of a "double-ladder" diagram after $t$-rescaling. Parameters n, K, s, tmax are set manually as constants in the main body of the program. The result of executing the code is a file with a dataset, which is presented in the form of two columns of length tmax: $\ln t$ and $\frac{\ln\dim\lambda(t)}{\frac12 N^2\ln t}$:
  • build_ladder(int n, int K, int s): takes positive integer $n, K, s$ and returns the set ${\tilde{\lambda}i}{i=1}^K$ corresponding to the $SU(K)$ Young diagram with $n$ cells, in which the difference between the lenghts of the rows is equal to $s$. *Note: $s$ should satisfy the relation $s<\frac{2n}{K(n-1)}$.
  • double tWeyl(double t, const std::vector<double>& ladder, int N): takes $t$,ladder, from build_ladder(int n, int K, int s), and $N$, and returns $\frac{\ln\dim\lambda(t)}{\frac12 N^2\ln t}$, calculated using the Weyl formula.
• scaling_plot.py: takes the data set from single_double_ladder_scaling.cpp and returns its graphical visualisation as a 2D plot.
• scaling_plot_v03.py: the same as scaling_plot.py, but takes only a few points from single_double_ladder_scaling.cpp to reduce the output 2D plot size.
• box_dim2_v02.py: provides functions to calculate the logarithm of the dimension of "box" diagrams with even $s\in[2,N]$. Parameters n, N are set manually as constants in the main body of the program. The result of executing the code is a file with a dataset, presented as two columns of length N: $s$ and $\ln\dim\lambda(s)$:
  • fdim(int n, int s, int N): takes positive integer $n, s, N$ and returns $\ln\dim\lambda(s)$.
• box_plot.py: takes the data set from box_dim2_v02.pyand returns its graphical visualisation as a 2D plot.

References

[1] E. Ievlev, A. Marshakov, G. Sumbatian and A. Yung, Hadrons in $\mathcal{N}=2$ supersymmetric QCD from non-Abelian string and 2D black hole, to appear