Research Review

We work within the framework of Supersymmetric Discretized Light-Cone Quantization (SDLCQ) [12]. SDLCQ is a special form of Discretized Light-Cone Quantization (DLCQ), which makes use of the usual virtues of DLCQ, like a well-defined vacuum and a frame independent wavefunction, and adds the advantages of supersymmetric formulations of quantum field theory (QFT). In particular, given enough supersymmetry, theories are finite and the notoriously hard problem of non-perturbative renormalization in a Hamiltonian approach in QFTs are avoided. The main observation is that DLCQ preserves supersymmetry at each stage of the calculation, when the supercharge, rather than the Hamiltonian is diagonalized. Our work on non-perturbative methods in low dimensional field theories is motivated strongly by the fact that string theories can be formulated as field theories (Maldacena conjecture). Typically, this involves the field theories in low dimensions at strong coupling, where to date SDLCQ is the only available method. More recently, we have successfully tried to apply our approach to theories in higher dimensions [3][6][10][11].

In Ref. [6] we calculated the correlator of the stress-energy tensor in N=1 SYM(2+1), which plays a central role in string/field theory correspondences. The surprising outcome is that at large r, where the correlator is dominated by the BPS states of the theory, there is a critical value of the coupling where the correlator goes to zero. We found that this critical coupling grows linearly with the square root of the transverse momentum resolution.

In an attempt to rigorously test the Maldacena conjecture, our results [4][5][9] support the conjecture and are within 10--15\% of the predicted numerical results in some regions. Our results are still not sufficient to demonstrate convergence. Over the last year we have however been able to greatly improve our analytic and numerical treatment of the problem, and the prospects for future decisive tests of the conjecture seem very promising.

The rapid convergence of SDLCQ is especially apparent in two-dimensional problems, but we very able to show that this remains an advantage of this framework also in three dimensions [10][11]. In particular, we studied the behavior of the spectrum of N=1 supersymmetric Yang-Mills theory in three dimensions as a function of the coupling and found at strong coupling a stable, well-defined spectrum.

The standard formulation of SDLCQ allows for periodic boundary conditions only. One is therefore prevented from studying BPS states, because the central charge in supersymmetric theories is in general a boundary integral and therefore vanishes when one uses periodic boundary conditions. In Ref. [5] we presented a novel formulation of SDLCQ where the fields satisfy anti-periodic boundary conditions. Theories with non-zero central charges can thus be investigated with SDLCQ. However, the convergence of this method turns out to be unpleasantly slow.

In Ref. [1][7] we calculated the spectrum of two-dimensional QCD. We formulated the theory with SU(N_c) currents rather than with fermionic operators which decomposes the problem into isolated current sectors. This method has the potential of advancing analytic and numeric computations of spectra in quantum field theory: the entire eigenvalue problem can be written as an algebraic function of the numbers of flavors N_f and colors N_c and the dependence of the eigenvalue spectrum on these parameters can be studied explicitly.

Publications (January 1999-October 2001)

Presentations (January 1999-October 2001)

• Invited talks at international conferences

  • ``The two-point stress-energy correlator in N=1 SYM(2+1)'', talk at the 'Particles and Strings' conference, Trento, Italy, September 3-12, 2001.
  • ``Testing the Maldacena conjecture with SDLCQ'', talk at the 'CPT 2001' conference, Bloomington, IN, August 15-18, 2001.
  • ``Correlators in N=1 SYM(2+1)'', talk at the PASCOS 2001 conference, Chapel Hill, NC, April 10--15, 2001.
  • ``Field Theory Correlators and String Theory'', talk at the International Conference on Orbis Scientiae 2000, Ft. Lauderdale, FL, December 14--17, 2000.
  • ``SDLCQ and String/Field Theory Correspondences'', talk at the Conference Thirty Years of Supersymmetry, Minneapolis, MN, October 13--20, 2000.
  • ``Towards testing the Maldacena Conjecture with SDLCQ'', talk at the X.~International Light-Cone Meeting on Nonperturbative QCD and Hadron Phenomenology -- From Hadrons to Strings, Heidelberg, Germany, June 2000.
  • ``Testing SDLCQ in 2+1 dimensions'', talk at the Workshop on Light-Cone QCD and Nonperturbative Hadron Physics, Adelaide, South Australia, December 1999.
  • ``On the bosonic spectrum of massless QCD(1+1)'', talk at the International Workshop: Challenges in QCD, Kfar Giladi, Israel, June 1999.

• Seminars/Colloquia

  • Albert-Einstein-Institut, Potsdam, Germany, September 2001.
  • Universitaet Muenster, Germany, September 2001.
  • DESY, Hamburg, Germany, September 2001.
  • Universitaet Erlangen/Nuernberg, Erlangen, Germany, September 2001.
  • Kyoto University, Kyoto, Japan, March 2001.
  • Tokyo Institute of Technology, Tokyo, Japan, March 2001.
  • The Ohio State University, Columbus, Ohio, USA, March 2001.
  • Meeting of the Division of Particles and Fields of the American Physical Society, Columbus, OH, August 2000.
  • The Ohio State University, Columbus, Ohio, USA, September 1999.
  • Max Planck Institute for Nuclear Physics, Heidelberg, Germany, August 1999.
  • The Ohio State University, Columbus, Ohio, USA, May 1999.
  • University of Florida, Gainesville, Florida, USA, February 1999.
  • University of California, Los Angeles, California, USA, February 1999.
  • SLAC, Stanford, California, USA, January 1999.
  • University of Washington, Seattle, Washington, USA, January 1999.
  • Argonne National Laboratory, Argonne, Illinois, USA, January 1999.

Last modified: 10/10/2001