Mirror Symmetry and Other Miracles in Superstring Theory

The dominance of string theory in the research landscape of quantum gravity physics (despite any direct experimental evidence) can, I think, be justified in a variety of ways. Here I focus on an argument from mathematical fertility, broadly similar to Hilary Putnam's 'no miracles argument' that, I argue, many string theorists in fact espouse. String theory leads to many surprising, useful, and well-confirmed mathematical 'predictions' - here I focus on mirror symmetry. These predictions are made on the basis of general physical principles entering into string theory. The success of the mathematical predictions are then seen as evidence for framework that generated them. I attempt to defend this argument, but there are nonetheless some serious objections to be faced. These objections can only be evaded at a high (philosophical) price.Comment: For submission to a Foundations of Physics special issue on "Forty Years Of String Theory: Reflecting On the Foundations" (edited by G. `t Hooft, E. Verlinde, D. Dieks and S. de Haro)

Chiral Symmetry Versus the Lattice

After mentioning some of the difficulties arising in lattice gauge theory from chiral symmetry, I discuss one of the recent attempts to resolve these issues using fermionic surface states in an extra space-time dimension. This picture can be understood in terms of end states on a simple ladder molecule.Comment: Talk at the meeting "Computer simulations studies in condensed matter physics XIV" Athens, Georgia, Feb. 19-24, 2001. 14 page

Quasi-Exact Solvability and the direct approach to invariant subspaces

We propose a more direct approach to constructing differential operators that preserve polynomial subspaces than the one based on considering elements of the enveloping algebra of sl(2). This approach is used here to construct new exactly solvable and quasi-exactly solvable quantum Hamiltonians on the line which are not Lie-algebraic. It is also applied to generate potentials with multiple algebraic sectors. We discuss two illustrative examples of these two applications: an interesting generalization of the Lam\'e potential which posses four algebraic sectors, and a quasi-exactly solvable deformation of the Morse potential which is not Lie-algebraic.Comment: 17 pages, 3 figure