Draft:Plethystic logarithm

Template:Short description Template:Draft topics Template:AfC topic Template:AfC submission Template:AfC submission In mathematics, the plethystic logarithm is an operator which is the inverse of the plethystic exponential.

Definition

The plethystic logarithm takes in a function with n complex arguments, $f(t_1,t_2,\cdots ,t_n)$, which must equal one at the origin, and is given by

$\mathrm{P}\mathrm{L}[f(t_1,t_2,\cdots ,t_n)]=\sum _{k=1}^{\mathrm{\infty }}\frac{\mu (k)}{k}\text{ln}(f(t_1^k,t_2^k,\cdots ,t_n^k))$

where $\mu (k)$ is the Möbius function and is defined by Template:Sfn

$$\mu (k)=\{\begin{array}{cc}1& \text{if }k=1\\ (-1{)}^n& \text{if }k\text{ is the product of }n\text{ distinct primes}\\ 0& \text{otherwise}\end{array}$$

and $\text{ln}(f(t_1^k,t_2^k,\cdots ,t_n^k))$ is the natural logarithm of the initial function with every argument raised to the power of $k$.

Applications in Theoretical Physics

The plethystic logarithm has a few applications in theoretical physics, particularly within the study of gauge theories.

Sources

• Template:Cite book