Fugacity

Fugacity is a quantity in statistical mechanics which allows one to factor conserved quantities into the partition function. In the context of the superconformal index, fugacity is used to take into account symmetries of the system, including gauge invariance.

Fugacity in Supersymmetric Theories

For instance, consider a supersymmetric theory with \(2N\) bosons and \(2N\) fermions having generic Lagrangian

\[ \mathcal{L} = \frac{1}{2} \dot{x}_i^2 - \frac{1}{2} \left\vert \frac{\partial W(x_i)}{\partial x_i} \right\vert^2 - i \bar{\psi}_i \dot{\psi}_i - \frac{\partial^2 W(x)}{\partial x_i \partial x_j} \bar{\psi}_i \psi_j \]

The bosonic and fermionic one-letter partition functions are \(z_B(x) = z_F(x) = 2 N x\), which through plethystic exponentiation gives

\[ Z(x) = \left(\frac{1+x}{1-x}\right)^{2N} \]

However, we should like to keep track of how the states in this system transform, in this case under our global \(SO(2N)\) symmetry. We do this by turning on fugacities with respect to the Cartan generators:

\[ z_B(x, a_i) = \text{Tr}\: x^H a_i^{J_i} = x \left( a_1 + \frac{1}{a_1} + \dots + a_n + \frac{1}{a_n} \right) = x \chi_{\text{fund}}(a_i) \]

where \(\chi_{\text{fund}}(a_i)\) is the [[Character of a Representation|character]] of the fundamental representation, in which these states transform. We have the same for the fermionic single-letter partition function, and so:

\[ Z(x, a_i) = \text{PE}[z_B(x, a_i)]\: \tilde{\text{PE}}[z_F(x, a_i)] = \prod_{i=1}^{N} \frac{(1+x a_i)(1 + x/a_i)}{(1-x a_i)(1 - x/a_i)} \]

Setting the fugacities \(a_i\) to \(1\) recovers the standard partition function \(Z(x)\). However, keeping these fugacities allows us to compute the partition function over only those states which transform under a gives representation of our group of interest, due to the orthogonality of characters of group representations.

Fugacity in Statistical Mechanics

Consider a statistical system with temperature \(T\), pressure \(P\), volume per mole \(V_m\), entropy per mole \(S_m\) , and chemical potential \(\mu\). The differential of chemical potential is \(d\mu = V_m dP\). For an ideal gas, this is:

\[ d\mu = V_m dP = \frac{RT}{P} dP = RT\: d \log P \]

but this is untrue for a real gas. However, we can define the fugacity \(f\) such that

\[ d\mu = RT\: d\log f \]

such that \(\lim_{P \to 0} \frac{f}{P} = 1\). The ratio \(\phi \equiv \frac{f}{P}\) is called the figacity coefficient. This is to say, the fugacity is an effective pressure such that the chemical potential of a real gas varies the same as for an ideal gas with that fugacity as its pressure.