In statistical mechanics one dispenses with the idea of set of numbers that describe a system perfectly and instead one must work with the beautiful and mysterious entity know as the 'partition function'. The partition function is just the sum, over all the possible states of the system, of the probabilities of existing in that particular state. If the state of a system is a continuous variable, you do an integral instead of a sum, and if there is more than one degree of freedom you do more than one integral.

The partition function may seem somewhat arbitrary but its power lies in the fact that the value will depend on the external thermodynamic properties (hence 'function' in the name). For example in a gas it will depend on the volume, temperature, and pressure. The logarithm of the partition function is the free energy which is the energy of the system after the thermal energy has been subtracted. This quantity represents the available energy to do work, and the thermal energy is removed because you're not getting that energy back into anything useful. As an example of this, the pressure of a system is actually just the change in free energy with the system's volume. If you attached the gas to a spring, the system would do work on the spring by exerting pressure.

A duality is just a pair of statistical ensembles that have the same partition function. They are said to be dual because the two systems behave the same way under external stimuli.

This is where things get weird. Replace statistical mechanics, which has a large but finite number of degrees of freedom, with statistical field theory, where the field value at each point in space-time is a degree of freedom. Say that the field theory interacts with electromagnetism and set up an experiment where we are modify the electric and magnetic fields, observing the thermodynamic response of the field theory. This is like changing the volume to measure pressure except that was just one variable whereas now we have six variables (three for E-field and three for B-field) at every point in space-time. The partition function is now calculated by doing a path integral (a fancy way of saying an integral at every point in space-time). It is now a 'functional' which means that it takes functions (in this case the field value as a function of space-time) as input rather than any finite number of parameters.

In statistical field theory a duality is a pair of different field theories that have the same partition-function functional with respect to an external, probing, field.

Particle-vortex duality was something discovered in the 70's by people working on condensed matter systems like superconductors. On the one hand you have a regular old scalar bosonic field theory (scalar meaning that that the field has a number at each point in space-time, and bosonic means that the number is just a normal number). This theory is referred to as 'particle' in the duality because the quantum excitations of a field theory are exactly what we interpret as particles. On the other hand we have and Abelian-Higgs field theory which, for the experts, is a U(1) gauge field that interacts strongly with a scalar Higgs field. I won't go into how this works but its interesting to note that a classical field of this type has vortex solutions which are effectively a result of the field getting itself tangled up into knots.

These knots behave like particles in the sense that they are local excitations of the field with integer number, but their existence is purely a consequence of classical physics and topology rather than quantum effects. The marvelous thing is that these two theories, when coupled to an appropriate external field, can be shown to be dual to one another in the sense of partition functions. There is no observable difference, using the external field to probe, between the quantum-like particle excitations and vortex-like particle excitations, and that is the sort of thing that keeps me up at night.

My final report wasn't actually on the particle-vortex duality, I decided to focus on another, more recent duality which is called 3D bosonizaiton. If you remember from earlier that boson field means that the field values are 'just normal numbers'. The reason I make that distinction is because there are fermionic fields which use Grassman numbers instead. The base unit of a Grassman number, usually represented by theta, is like an imaginary number in that it can be multiplied and divided by real numbers except that instead of i x i = -1 we have theta x theta = 0. In general fermionic particles have spin which is quantized magnetic flux. It turns out that for a certian type of theory (Chern-Simons theory) which is purely classical, magnetic flux is quantized in a way that is very similar to the field-tangling of the vortexes. 3D bosonization duality is between fermionic particles on the one hand and normal, scalar, particles with magnetic flux attached through its interaction with a Chern-Simons field on the other.