[figure8]

Gauge theory is the collection of techniques used to describe forces in quantum field theory (electro-magnetism, weak force, and strong force). In field theory you have many vector spaces (1-d, 2-d, ... n-d) defined for every point in space time. So a specific field requires a distinct vector value for each space-time point. These fields encode certain particle types like electrons, quarks, etc.

One can ask, how can I compare field values at 2 distinct points a and b? They are in distinct vector spaces V_a vs V_b. Well, gauge theory says you must imagine the vector v_a being moved along a path between a and b, and see how the translated vector value compares to the vector v_b over b. This is called "parallel transport." A rule about how all vectors move along all paths during this translation is called a "connection." These connections encode distinct force-carrying particle types like photons and gluons. Sort of like how an object naturally rotates along a curvy surface. It's naively possible to describe this transport, this curvyness, but the natural notation is such that distinct descriptions are actually the same connection. Dealing with this ambiguity is key to properly computing probabilities using path integrals.

By the way, each gauge theory is required to have an underlying "gauge group," meaning a set of possible symmetries. In the standard model these groups are called U(1), SU(2) and SU(3). U(1) is 1-dimensional, so only 1 kind of electromagnetic force particle, the photon. SU(2) is 3-dimensional so 3 types of weak force particles (W-,W+, and Z) and SU(3) is 8 dimensional so 8 types of gluons.

[espeed]

One emerging realization is that the essence of gauge theory [0] has revealed itself as an insight to some whom have learned to apply it beyond the realm of physics.

Eric Weinstein [1], managing director of Thiel Capital, has been advocating upgrading the current economic calculus with a quantitative model based on gauge theory that can better adapt to changing behavior and changes in consumer preferences across time (for example).

Eric has intimated that James Simons [2] discovered a secret (in the Peter Thiel sense of the word) related to gauge theory during the time Simons and C.N. Yang [3] (as in Yang-Mills Theory [4]) discovered the correspondences in what has become known as the "Wu-Yang Dictionary" [5], which later led to Simons founding Renaissance Technologies [6] in 1982 and is part of the secret-sauce in their mathematical models that has enabled them to dominate quantitative trading ever since. A quote from the 1975 paper:

  ...The mathematics of these results is in fact well
  known to the mathematicians in fiber bundle theory. 
  An identification table of terminologies is given in 
  Sec. V. We should emphasize that our interest in this 
  paper does not lie in the beautiful, deep, and general 
  mathematical development in fiber bundle theory. 
  Rather we are concerned with the necessary concepts to 
  describe the physics of gauge theories. It is 
  remarkable that these concepts have already been 
  intensively studied as mathematical constructs.

See "Gauge Theory and Inflation: Enlarging the Wu-Yang Dictionary to a unifying Rosetta Stone for Geometry in Application - Eric Weinstein" [video] https://www.youtube.com/watch?v=h5gnATQMtPg

[0] Gauge Theory https://en.wikipedia.org/wiki/Gauge_theory

[1] Eric Weinstein https://en.wikipedia.org/wiki/Eric_Weinstein

[2] James Simons https://en.wikipedia.org/wiki/James_Harris_Simons

[3] Nobel Laureate C.N. Yang https://en.wikipedia.org/wiki/Yang_Chen-Ning

[4] Yang-Mills Theory https://en.wikipedia.org/wiki/Yang–Mills_theory

[5] Concept of nonintegrable phase factor and global formulation of gauge fields (AKA the Wu-Yang Dictionary) [pdf] http://www.indiana.edu/~jpac/QCDRef/1970s/Concept%20of%20non...

[6] Renaissance Technologies https://en.wikipedia.org/wiki/Renaissance_Technologies

[7] The conceptual origins of Maxwell's equations and gauge theory, by C.N. Yang (2014) [pdf] http://www.physics.umd.edu/grt/taj/675e/OriginsofMaxwellandG...

Previous Discussion: https://news.ycombinator.com/item?id=16325605

[balsam]

Uh Eric told you this himself? Rentech is famously secretive it wouldnt surprise me if they went after anyone who even leaks the existence of specific sauce.