Plethystic logarithm

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, ${\displaystyle f(t_{1},t_{2},\cdots ,t_{n})}$, which must equal one at the origin, and is given by^[1]

${\displaystyle \mathrm {PL} [f(t_{1},t_{2},\cdots ,t_{n})]=\sum _{k=1}^{\infty }{\frac {\mu (k)}{k}}{\text{ln}}(f(t_{1}^{k},t_{2}^{k},\cdots ,t_{n}^{k}))}$

where ${\displaystyle \mu (k)}$ is the Möbius function and is defined by ^[2]

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

and ${\displaystyle {\text{ln}}(f(t_{1}^{k},t_{2}^{k},\cdots ,t_{n}^{k}))}$ is the natural logarithm of the initial function with every argument raised to the power of ${\displaystyle k}$.

Applications in theoretical physics

The plethystic logarithm has a few applications in theoretical physics, particularly within the study of gauge theories. ^[3]