Index of lattice dirac operators - Reetabrata Har (2014)
The general topic of this thesis is how to define and compute the index of discretised “lattice” versions of the Dirac operator coupled to a topologically nontrivial gauge field. The famous Atiyah-Singer Index Theorem, applied to the usual continuum Dirac operator, says that its index is equal to the topological charge of the gauge field. This fact plays an important role in theoretical particle physics. However, in the physics context one often solves problems by discretizing the physical theory and simulating it on a computer. This raises the question of whether the Index Theorem (and its physical implications) can be maintained in the discretized lattice setting. This is a subtle mathematical issue since the usual definition of the index of the Dirac operator breaks down in the discretized setting due to its finite-dimensional nature. This thesis reviews how the index can be extracted in another, indirect, way in the discretized setting, and then goes on to consider the index of a new discretization of the Dirac operator introduced recently by D. Adams. This new lattice Dirac operator has the virtue of being more computationally efficient for use in computer simulations of particle physics theory, but before it can be used with confidence a secure theoretical foundation needs to be established. One part of this is to show that its index correctly reproduces the continuum index in the continuum limit where the lattice spacing goes to zero. This is the main goal and new research contribution of this thesis. It is done here by adapting the methods and techniques used to establish this result previously for the standard discretization of the Wilson-Dirac operator, and involves dealing with new issues specific to the new discretization.