Uwe Trittmann

Towards a SDLCQ test of the Maldacena conjecture

Abstract We consider the Maldacena conjecture applied to the near horizon geometry of a D1-brane in the supergravity approximation and present numerical results of a test of the conjecture against the boundary field theory calculation using DLCQ. We previously calculated the two-point function of the stress-energy tensor on the supergravity side; the methods of Gubser, Klebanov, Polyakov, and Witten were used. On the field theory side, we derived an explicit expression for the two-point function in terms of data that may be extracted from the supersymmetric discrete light cone quantization (SDLCQ) calculation at a given harmonic resolution. This yielded a well defined numerical algorithm for computing the two-point function. For the supersymmetric Yang-Mills theory with 16 supercharges that arises in the Maldacena conjecture, the algorithm is perfectly well defined; however, the size of the numerical computation prevented us from obtaining a numerical check of the conjecture. We now present numerical results with approximately 1000 times as many states as we previously considered. These results support the Maldacena conjecture and are within 10–15% of the predicted numerical results in some regions. Our results are still not sufficient to demonstrate convergence, and, therefore, cannot be considered to a numerical proof of the conjecture. We present a method for using a “flavor” symmetry to greatly reduce the size of the basis and discuss a numerical method that we use which is particularly well suited for this type of matrix element calculation.

Antiperiodic boundary conditions in supersymmetric DLCQ

It is of considerable importance to have a numerical method for solving supersymmetric theories that can support a non-zero central charge. The central charge in supersymmetric theories is in general a boundary integral and therefore vanishes when one uses periodic boundary conditions. One is therefore prevented from studying BPS states in the standard supersymmetric formulation of DLCQ (SDLCQ). We present a novel formulation of SDLCQ where the fields satisfy anti-periodic boundary conditions. The Hamiltonian is written as the anti-commutator of two charges, as in SDLCQ. The antiperiodic SDLCQ we consider breaks supersymmetry at finite resolution, but requires no renormalization and becomes supersymmetric in the continuum limit. In principle, this method could be used to study BPS states. However, we find its convergence to be disappointingly slow.

Mass spectrum of supersymmetric Yang-Mills theory in three dimensions

We consider supersymmetric Yang-Mills theory on R x S^1 x S^1. In particular, we choose one of the compact directions to be light-like and another to be space-like. Since the SDLCQ regularization explicitly preserves supersymmetry, this theory is totally finite, and thus we can solve for bound state wave functions and masses numerically without renormalizing. We present the masses as functions of the longitudinal and transverse resolutions and show that the masses converge rapidly in both resolutions. We also study the behavior of the spectrum as a function of the coupling and find that at strong coupling there is a stable, well defined spectrum which we present. We also find several unphysical states that decouple at large transverse resolution. There are two sets of massless states; one set is massless only at zero coupling and the other is massless at all couplings. Together these sets of massless states are in one-to-one correspondence with the full spectrum of the dimensionally reduced theory.

Wave functions and properties of massive states in three-dimensional supersymmetric Yang-Mills theory

We apply supersymmetric discrete light-cone quantization (SDLCQ) to the study of supersymmetric Yang-Mills theory on R x S^1 x S^1. One of the compact directions is chosen to be light-like and the other to be space-like. Since the SDLCQ regularization explicitly preserves supersymmetry, this theory is totally finite, and thus we can solve for bound-state wave functions and masses numerically without renormalizing. We present an overview of all the massive states of this theory, and we see that the spectrum divides into two distinct and disjoint sectors. In one sector the SDLCQ approximation is only valid up to intermediate coupling. There we find a well defined and well behaved set of states, and we present a detailed analysis of these states and their properties. In the other sector, which contains a completely different set of states, we present a much more limited analysis for strong coupling only. We find that, while these state have a well defined spectrum, their masses grow with the transverse momentum cutoff. We present an overview of these states and their properties.

Two-point stress-tensor correlator in N=1, (2+1)-dimensional super Yang-Mills theory

Recent advances in string theory have highlighted the need for reliable numerical methods to calculate correlators at strong coupling in supersymmetric theories. We present a calculation of the correlator<0|T^{++}(r)T^{++}(0)|0>in N=1 SYM theory in 2+1 dimensions. The numerical method we use is supersymmetric discrete light-cone quantization (SDLCQ), which preserves the supersymmetry at every order of the approximation and treats fermions and bosons on the same footing. This calculation is done at large

Anomalously light mesons in a (1+1)-dimensional supersymmetric theory with fundamental matter

N_c

On the spectrum of ( 1 + 1 ) -dimensional QCD with S U ( N c ) currents

. For small and intermediate r the correlator converges rapidly for all couplings. At small r the correlator behaves like 1/r^6, as expected from conformal field theory. At large r the correlator is dominated by the BPS states of the theory. There is, however, a critical value of the coupling where the large-r correlator goes to zero, suggesting that the large-r correlator can only be trusted to some finite coupling which depends on the transverse resolution. We find that this critical coupling grows linearly with the square root of the transverse momentum resolution.

On rotations in front-form dynamics

Abstract We consider N =1 supersymmetric Yang–Mills theory with fundamental matter in the large-Nc approximation in 1+1 dimensions. We add a Chern–Simons term to give the adjoint partons a mass and solve for the meson bound states. Here mesons are color-singlet states with two partons in the fundamental representation but are not necessarily bosons. We find that this theory has anomalously light meson bound states at intermediate and strong coupling. We also examine the structure functions for these states and find that they prefer to have as many partons as possible at low longitudinal momentum fraction.

Towards finding the single-particle content of two-dimensional adjoint QCD

Extending previous work, we calculate the fermionic spectrum of two-dimensional QCD (QCD2) in the formulation with SU(Nc) currents. Together with the results in the bosonic sector this allows to address the as yet unresolved task of finding the singleparticle states of this theory as a function of the ratio of the numbers of flavors and colors, λ = Nf/Nc, anew. We construct the Hamiltonian matrix in DLCQ formulation as an algebraic function of the harmonic resolution K and the continuous parameter λ in the Veneziano limit. We find that the fermion momentum is a function of λ in the discrete approach. A universality, existing only in two dimensions, dictates that the well-known ’t Hooft and large Nf spectra be reproduced in the limits λ → 0 and ∞, which we confirm. We identify their single-particle content which is surprisingly the same as in the bosonic sectors. All multi-particle states are classified in terms of their constituents. These findings allow for an identification of the lowest singleparticles of the adjoint theory. While we do not succeed in interpreting this spectrum completely, evidence is presented for the conjecture that adjoint QCD2 has a bosonic and an independent fermionic Regge trajectory of single-particle states.

SYM correlators and the maldacena conjecture

Abstract Quantum field theories in front-form dynamics are not manifestly rotationally invariant. We study a model bound-state equation in 3+1 dimensional front-form dynamics, which was shown earlier to reproduce the Bohr and hyperfine structure of positronium. We test this model with regard to its rotational symmetry and find that rotational invariance is preserved to a high degree. Also, we find and quantify the expected dependence on the cut-off.