Supersymmetric Discrete Light-cone Quantization Research Articles

Numerical evidence for the Maldacena conjecture in two-dimensional super Yang–Mills theory

The N=(8,8) super Yang-Mills theory in 1+1 dimensions is solved at strong coupling to directly confirm the predictions of supergravity at weak coupling. The calculations are done in the large-N_c approximation using Supersymmetric Discrete Light-Cone Quantization. The stress-energy correlator is obtained as a function of the separation r; for intermediate values of r, the correlator behaves in a manner consistent with the 1/r^5 behavior predicted by weak-coupling supergravity.

Direct evidence for the Maldacena conjecture for [formula omitted] super-Yang–Mills theory in [formula omitted] dimensions

We solve N=(8,8) super-Yang–Mills theory in 1+1 dimensions at strong coupling to directly confirm the predictions of supergravity at weak coupling. We do our calculations in the large-Nc approximation using Supersymmetric Discrete Light-Cone Quantization with up to 3×1012 basis states. We calculate the stress-energy correlator 〈T++(r)T++(0)〉 as a function of the separation r and find that at intermediate values of r the correlator behaves as r−5 to within errors as predicted by weak-coupling supergravity. We also present an extension to significantly higher resolution of our earlier results for the same correlator in the N=(2,2) theory and see that in this theory the correlator has very different behavior at intermediate values of r.

N=1super Yang-Mills theory on a(3​+​1)dimensional transverse lattice with one exact supersymmetry

We formulate ${\cal N}$=1 super Yang-Mills theory in 3+1 dimensions on a two dimensional transverse lattice using supersymmetric discrete light cone quantization in the large-$N_c$ limit. This formulation is free of fermion species doubling. We are able to preserve one supersymmetry. We find a rich, non-trivial behavior of the mass spectrum as a function of the coupling $g\sqrt{N_c}$, and see some sort of "transition" in the structure of a bound state as we go from the weak coupling to the strong coupling. Using a toy model we give an interpretation of the rich behavior of the mass spectrum. We present the mass spectrum as a function of the winding number for those states whose color flux winds all the way around in one of the transverse directions. We use two fits to the mass spectrum and the one that has a string theory justification appears preferable. For those states whose color flux is localized we present an extrapolated value for $m^2$ for some low energy bound states in the limit where the numerical resolution goes to infinity.

Solution to the fermion doubling problem for supersymmetric theories on the transverse lattice

Species doubling is a problem that infects most numerical methods that use a spatial lattice. An understanding of species doubling can be found in the Nielsen-Ninomiya theorem which gives a set of conditions that require species doubling. The transverse lattice approach to solving field theories, which has at least one spatial lattice, fails one of the conditions of the Nielsen-Ninomiya theorem nevertheless one still finds species doubling for the standard Lagrangian formulation of the transverse lattice. We will show that the Supersymmetric Discrete Light Cone Quantization (SDLCQ) formulation of the transverse lattice does not have species doubling.

N=(1,1)super Yang-Mills theory in1+1dimensions at finite temperature

We present a formulation of $\mathcal{N}=(1,1)$ super Yang-Mills theory in $1+1$ dimensions at finite-temperature. The partition function is constructed by finding a numerical approximation to the entire spectrum. We solve numerically for the spectrum using supersymmetric discrete light-cone quantization (SDLCQ) in the large-${N}_{c}$ approximation and calculate the density of states. We find that the density of states grows exponentially and the theory has a Hagedorn temperature, which we extract. We find that the Hagedorn temperature at infinite resolution is slightly less than one in units of $\sqrt{{g}^{2}{N}_{\mathrm{c}}/\ensuremath{\pi}}$. We use the density of states to also calculate a standard set of thermodynamic functions below the Hagedorn temperature. In this temperature range, we find that the thermodynamics is dominated by the massless states of the theory.

Improved results forN=(2,2)super Yang-Mills theory using supersymmetric discrete light-cone quantization

We consider the (1+1)-dimensional ${\cal N}=(2,2)$ super Yang--Mills theory which is obtained by dimensionally reducing ${\cal N}=1$ super Yang--Mills theory in four dimension to two dimensions. We do our calculations in the large-$N_c$ approximation using Supersymmetric Discrete Light Cone Quantization. The objective is to calculate quantities that might be investigated by researchers using other numerical methods. We present a precision study of the low-mass spectrum and the stress-energy correlator $<T^{++}(r) T^{++}(0)>$. We find that the mass gap of this theory closes as the numerical resolution goes to infinity and that the correlator in the intermediate $r$ region behaves like $r^{-4.75}$.

SYM correlators and the maldacena conjecture

We report on progress in evaluating quantum filed theories with supersymmetric discrete light-cone quantization (SDLCQ). We compare the method to lattice gauge theory and point out its relevance for lattice calculations. As an exciting application we present a test of the Maldacena conjecture. We test the conjecture by evaluating the correlator of the stress-energy tensor in the strong coupling field theory and comparing to the string theory prediction of its behavior as a function of the distance. Our numerical results support the Maldacena conjecture and are within 10–15% of the predicted results.

Anomalously light bound-states in SYM Cherns-Simons theory

We apply supersymmetric discrete light-cone quantization (SDLCQ) to the study of supersymmetric YangMills-Chern-Simons (SYM-CS) theory on R × S 1 × 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. The Chern-Simons term is introduced here to provide masses for the particles while remaining totally within a supersymmetric context. We discuss a set of anomalously light states which are a reflection of the exact BPS state of the underlying of supersymmetric SYM theory.

Properties of the bound states of super Yang-Mills–Chern-Simons theory

We apply supersymmetric discrete light-cone quantization (SDLCQ) to the study of supersymmetric Yang-Mills-Chern-Simons (SYM-CS) 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. The Chern-Simons term is introduced here to provide masses for the particles while remaining totally within a supersymmetric context. We examine the free, weak and strong-coupling spectrum. The transverse direction is discussed as a model for universal extra dimensions in the gauge sector. The wave functions are used to calculate the structure functions of the lowest mass states. We discuss the properties of Kaluza-Klein states and focus on how they appear at strong coupling. We also discuss a set of anomalously light states which are reflections of the exact Bogomol'nyi-Prasad-Sommerfield states of the underlying SYM theory.

Numerical simulations ofN=(1,1)1+1-dimensional super Yang-Mills theory with large supersymmetry breaking

We consider the $N=(1,1)$ super Yang-Mills (SYM) theory that is obtained by dimensionally reducing SYM theory in 2+1 dimensions to 1+1 dimensions and discuss soft supersymmetry breaking. We discuss the numerical simulation of this theory using supersymmetric discrete light-cone quantization when either the boson or the fermion has a large mass. We compare our result to the pure adjoint fermion theory and pure adjoint boson discrete light-cone quantization calculations of Klebanov, Demeterfi, Bhanot and Kutasov. With a large boson mass we find that it is necessary to add additional operators to the theory to obtain sensible results. When a large fermion mass is added to the theory we find that it is not necessary to add operators to obtain a sensible theory. The theory of the adjoint boson is a theory that has stringy bound states similar to the full SYM theory. We also discuss another theory of adjoint bosons with a spectrum similar to that obtained by Klebanov, Demeterfi, and Bhanot.

Anomalously light states in super-Yang–Mills–Chern–Simons theory

Inspired by our previous finding that supersymmetric Yang–Mills–Chern–Simons (SYM–CS) theory dimensionally reduced to 1+1 dimensions possesses approximate Bogomol'nyi–Prasad–Sommerfield (BPS) states, we study the analogous phenomenon in the three-dimensional theory. Approximate BPS states in two dimensions have masses which are nearly independent of the Yang–Mills coupling and proportional to their average number of partons. These states are a reflection of the exactly massless BPS states of the underlying pure SYM theory. In three dimensions we find that this mechanism leads to anomalously light bound states. While the mass scale is still proportional to the average number of partons times the square of the CS coupling, the average number of partons in these bound states changes with the Yang–Mills coupling. Therefore, the masses of these states are not independent of the coupling. Our numerical calculations are done using supersymmetric discrete light-cone quantization (SDLCQ).

Simulation of dimensionally reduced super Yang-Mills-Chern-Simons theory

A supersymmetric formulation of a three-dimensional SYM-Chern-Simons theory using light-cone quantization is presented, and the supercharges are calculated in light-cone gauge. The theory is dimensionally reduced by requiring all fields to be independent of the transverse dimension. The result is a non-trivial two-dimensional supersymmetric theory with an adjoint scalar and an adjoint fermion. We perform a numerical simulation of this SYM-Chern-Simons theory in 1+1 dimensions using SDLCQ (Supersymmetric Discrete Light-Cone Quantization). We find that the character of the bound states of this theory is very different from previously considered two-dimensional supersymmetric gauge theories. The low-energy bound states of this theory are very ``QCD-like.'' The wave functions of some of the low mass states have a striking valence structure. We present the valence and sea parton structure functions of these states. In addition, we identify BPS-like states which are almost independent of the coupling. Their masses are proportional to their parton number in the large-coupling limit.

The spectrum of SY M2+1

Abstract We apply supersymmetric discrete light-cone quantization (SDLCQ) to the study of supersymmetric Yang-Mills theory on R × S 1 × 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 find that while these state have a well defined spectrum their masses grow with the transverse momentum cutoff.

Exploring N = 1 SYM (2+1): the stress-tensor correlator

The evaluation of field theoretic correlators at strong couplings is especially interesting in the light of recently discovered string/field theory correspondences. We present a calculation of the stress-tensor correlator in N = 1 SYM theory in 2+1 dimensions. We calculate this object numerically with the method of supersymmetric discrete light-cone quantization (SDLCQ) at large N c . For small distances we reproduce the conformal field theory result with the correlator behaving like 1/ r 6 . In the large r limit the correlator is determined by the (massless) BPS states of the theory. We find a critical value of the coupling where the correlator goes to zero in this limit. This critical coupling is shown to grow linearly with the square root of the transverse momentum resolution.

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.

SDLCQ and string/Field theory correspondences

String/Field theory correspondences have been discussed heavily in recent years. Here, we describe a testing scenario involving a non-perturbative field theory calculation using the framework of supersymmetric discrete light-cone quantization (SDLCQ). We consider a Maldacena-type conjecture applied to the near horizon geometry of a D1-brane in the supergravity approximation. Numerical results of a test of this conjecture are presented with orders of magnitude more states as we previously considered. These results support the Maldacena conjecture and are within 10–15% of the predicted results. We present a method for using a “flavor” symmetry to greatly reduce the size of the Fock basis and discuss a numerical method that we use which is particularly well suited for this type of matrix element calculation. Our results are still not sufficient to demonstrate convergence, and, therefore, cannot be considered to be a numerical proof of the conjecture. We update our continuous efforts to improve on these results and present some results on the way to higher dimensional scenarios.

Light-cone quantized supersymmetric abelian Chern—Simons theory

In this lecture we present the light-cone quantized, supersymmetric abelian Chern—Simons theory. While this theory has been studied many times, this particular formulation is of interest because it is the starting point for the numerical technique, SDLCQ (supersymmetric discrete light-cone quantization). This theory is of interest from the SDLCQ perspective because the zero modes of the theory play a fundamental role in the theory and because it is a topological field theory. These are two aspects of supersymmetric field theory that have not previously been addressed within the context of SDLCQ. We will also comment on the dependence of the Witten Index on the Chern—Simons coupling.

Renormalizing DLCQ using supersymmetry

Recent string theory developments suggest the necessity to understand supersymmetric gauge theories non-perturbatively, in various dimensions. In this work we show that there is a standard Hamiltonian formulation that generates a finite and supersymmetric result at every order of the approximation scheme known as discrete light-cone quantization (DLCQ). We present this renormalized DLCQ Hamiltonian and find that it has two novel features: it automatically chooses the 't Hooft prescription for renormalizing the singularities, and it introduces irrelevant operators that serve to preserve the supersymmetry and improve the convergence. We solve for the bound states and the wave functions with and without the irrelevant operators and verify that with the irrelevant operator the exact large- N c supersymmetric DLCQ (SDLCQ) results are reproduced. With the irrelevant operator removed, we show that the bound-state mass and wave functions appear to be converging to the SDLCQ results but very slowly. This is a first step in extending the advantages of SDLCQ to non-supersymmetric theories.

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 $N_c$. 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.

Towards testing the Maldacena conjecture with SDLCQ

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 supersymmetric discrete light-cone quantization (SDLCQ). We 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.