# N=1 Supersymmetric Quantum Mechanics: SU(2) gauge theory.

Here I make available mathematica code and simple Fortran programs. The most important program is Generate.ma, which is used for making the gauge invariant spherical harmonics for (N=1 supersymmetric) SU(2) gauge theory. The other programs deal with the radial sectors and with finding the lowest eigenvalues and other properties of the Hamiltonian.

The non-supersymmetric (F=0) case reproduces old results by M. Lüscher and G. Münster (Nucl. Phys. B232 (1984) 445), and those I obtained in collaboration with Jeff Koller. See also another paper with him, which forms essential background for this study. The very first time I discussed in writing (with little success) the problem of the adiabatic approximation for determining the Witten Index was in September 1991. Writing a recent review on QCD in a finite volume, prompted me to come back to this problem. The background material for the supersymmetric analysis can be found in the paper "The Witten index beyond the adiabatic approximation" [hep-th/0112072], which is dedicated to the memory of Michael Marinov.

Here is the Mathematica code for generating the basis and matrix elements. I did not use Mathematica Notebook features, so that the available code should run on any platform. It has been tested with Mathematica versions 2.0, 3.0 and 4.0. Use each mathematica program in a separate session, such that global definitions do not interfere!

• Generate.ma, creates spherical harmonics and makes tables of reduced matrix elements (Htab0r, HHtab0r, Htab2r, and HHtab2r) in a format suitable for input in Fortran programs.
• Besj.ma, computes radial matrix elements and makes tables (Rtab2 and RRtab2) suitable for input in Fortran programs.
• Bjtab.ma, makes table (Bjtab) of spherical Bessel functions in a format suitable for input in Fortran programs and used to reconstruct the wave function.
• roots.out, roots of the derivative of the spherical Bessel functions needed in Besj.ma, to make radial matrix elements.
The following deals with diagonalization using Mathematica (slow).
• H0t.ma, builds the full matrix for the F=0 (bosonic) case, allows for diagonalization with mathematica and building the wave function (slow but manageable). Also makes a table for the full matrix (Htab0) in a format suitable for input in Fortran programs.
• H2t.ma, builds the full matrix for the F=2 case, allows for diagonalization and Temple's inequality with mathematica (not advised, except for testing). Also makes tables for the full matrix (Htab2 and HHtab2) in a format suitable for input in Fortran (also not advised, except for testing).
J. Wosiek has implemented symbolically the Hamiltonian for supersymmetric quantum mechanics, including SU(2) gauge theory, using the full Fock space defined in terms of harmonic oscillators. See hep-th/0203116 and hep-th/0204243, as well as more recent work with M. Campostrini, in particular hep-th/0407021 (with a link to supplementary material). For F=0, our basis' agree (take w=1) and a direct comparison (for J=0) gives perfect agreement. Small changes in H0t.ma were made to facilitate this comparison for both parities (run EB[1,Ncut,Em] for the first Em eigenvalues). The following Mathematica program implements the harmonic basis for F=2 and J=0, also showing perfect agreement. I modified diag.f slightly to deal with this case as well.
• H2h.ma, builds the full matrix for the F=2 case with harmonic oscillator basis, allows for diagonalization and makes a table (Htab2h) for the full matrix in a format similar to Htab0 and suitable for input in Fortran.
Here is the Fortran program, which uses the IMSL libraries to find eigenvalues (I used an old Fortran77 version).
• diag.f, used for diagonalizing H(r), computing Veff, diagonalizing H and reconstructing the groundstate wave function. It allows for automatic pruning. Read the extensive comments. See also here.

Page initially created on 10 December 2001. Any changes in the programs can be detected from their version date listed in the beginning.

Disclaimer: Users themselves are responsible for the results obtained with these programs.
This is free code. Of course it has been tested extensively, but I can give no guarantees.

Pierre van Baal, last updated 14 September 2004.