## Lattice Gauge Theory

### Prof. B. Svetitskyfirst semester 5765lecture: Wednesday 12-14 in Dan David 212

This will be an abbreviated version of the course, with the selection of topics to be made according to the wishes of the students.  The complete course is described below.

The aim of the course is to describe non-perturbative approaches to non-Abelian gauge theories. Our main physical problem is quark confinement. The lattice provides a cutoff for field theory which permits many manipulations beyond the reach of Feynman-diagram perturbation theory. We will exploit exact transformations among different field theories and use topological methods and duality to study the physics of these theories.

After introducing the lattice formulation of QCD, we will develop techniques by using simpler field theories as laboratories: the Ising model, the XY (or planar Heisenberg) model, the Ising gauge theory (due to Wegner), the Abelian gauge theory (related to QED). The problem of taking the continuum limit will bring us to the study of critical points and the renormalization group.

The lattice has also provided the easiest approach to numerical calculation in field theory. We will describe the basics of Monte Carlo methods and show how the qualitative ideas presented above find quantitative realization in these calculations.

The foremost practical problem standing in the way of the calculation of hadron properties is also the most interesting open theoretical problem: how to put fermion field theories on the lattice.  We will review this long-standing problem and current attempts to solve it.

### Prerequisites

• Field Theory 1 and 2

### Tentative syllabus

• Gauge theory on a lattice: Ingredients
• The gauge field - Abelian and non-Abelian
• The Wilson loop & q-qbar potential
• Matter fields; fermions
• The transfer matrix and the Hamiltonian
• Finite temperature
• Pure gauge fields - Euclidean & Hamiltonian
• Perturbation theory
• Strong coupling expansion
• 't Hooft's Z(N) duality
• Abelian spin systems and gauge theories: magnetic monopoles and phase structure
• The continuum limit
• Critical phenomena and the renormalization group
• The beta function and asymptotic freedom
• Monte Carlo methods
• Algorithms for simulation of bosonic fields
• Glueballs, string tension, high temperature
• Monte Carlo renormalization group
• Lattice fermions
• Problems in formulation, no-go theorem
• Solution to the formulation problem: Domain-wall (alias overlap)
• Strong coupling methods and chiral symmetry breaking
• Monte Carlo algorithms

### References

• J. M. Drouffe and C. Itzykson, Statistical Field Theory (1989)
• J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (1989)
• G. Parisi, Statistical Field Theory (1988)
• A. M. Polyakov, Gauge Fields and Strings (1987)
• J. J. Binney et al., The Theory of Critical Phenomena (1992)
• N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group (1992)
• proceedings of lattice conferences and the hep-lat archive