Latest results from lattice super Yang–Mills
Abstract:
We present some of the latest results from our numerical investigations of supersymmetric Yang–Mills theory formulated on a spacetime lattice. Based on a construction that exactly preserves a single supersymmetry at nonzero lattice spacing, we recently developed an improved lattice action that is now being employed in largescale calculations. Here we update our studies of the static potential using this new action, also applying treelevel lattice perturbation theory to improve the analysis of the potential itself. Considering relatively weak couplings, we obtain results for the Coulomb coefficient that are consistent with continuum perturbation theory.
Nonperturbative investigations of supersymmetric Yang–Mills (SYM) formulated on a spacetime lattice have advanced rapidly in recent years. In addition to playing important roles in holographic approaches to quantum gravity, investigations of the structure of scattering amplitudes, and the conformal bootstrap program, SYM is also the only known fourdimensional theory for which a lattice regularization can exactly preserve a closed subalgebra of the supersymmetries at nonzero lattice spacing [1, 2, 3, 4, 5]. Based on this lattice construction we have been pursuing largescale numerical investigations of SYM that can in principle access nonperturbative couplings for arbitrary numbers of colors . Here we discuss a selection of our latest results from this work in progress.
Last year we introduced a procedure to regulate flat directions in numerical computations by modifying the moduli equations in a way that preserves the single exact supersymmetry at nonzero lattice spacing [6, 7, 8]. This procedure produces a lattice action that exhibits effective improvement, with significantly reduced discretization artifacts that vanish much more rapidly upon approaching the continuum limit. We have implemented this improved action in our parallel software for lattice SYM [9], and are now employing it in the largescale numerical computations discussed in this proceedings. We make our software publicly available to encourage independent investigations and the development of a lattice SYM community.^{1}^{1}1http://github.com/daschaich/susy
In this proceedings, after briefly reviewing the improved action we revisit our lattice investigations of the static potential [10, 11]. In addition to the new lattice action, we also improve the static potential analysis itself by applying treelevel lattice perturbation theory. We observe a coulombic potential and our preliminary results for the Coulomb coefficient are consistent with continuum perturbative predictions. A separate contribution to these proceedings [12] discusses our efforts to investigate S duality on the Coulomb branch of SYM where some of the adjoint scalar fields acquire nonzero vacuum expectation values leading to spontaneous symmetry breaking. These efforts involve measuring the masses of the elementary W boson and the corresponding dual topological ’t Hooft–Polyakov monopole. Ref. [12] also provides an update on our ongoing investigations of the Konishi operator scaling dimension.
Improved lattice action for Sym
Our lattice formulation of SYM is based on the Marcus (or GeometricLanglands) topological twist of the continuum theory [13, 14]. This produces a gauge theory with a fivecomponent complexified gauge field in four spacetime dimensions. We discretize the theory on the lattice, exactly preserving the closed subalgebra involving the single twistedscalar supercharge . The improved lattice action that we use is [6]
where the operator in the first line is and is the oriented plaquette built from the complexified gauge links in the – plane. Repeated indices are summed and the forward/backward finitedifference operators both reduce to the usual covariant derivatives in the continuum limit [3, 4]. All indices run from 1 through 5, corresponding to the five symmetric basis vectors of the fourdimensional lattice [2, 11].
When this action has the same form as the twisted continuum theory [13, 14]. These two tunable couplings are introduced to stabilize numerical calculations by regulating flat directions and exact zero modes. The scalar potential with coupling lifts flat directions in the SU() sector, while the plaquette determinant with coupling does so in the U(1) sector. Although nonzero softly breaks the supersymmetry, the plaquette determinant deformation is exact. This exact deformation results from the general procedure introduced in Ref. [6], which imposes the Ward identity by modifying the equations of motion for the auxiliary field,
(1) 
With this Ward identity gives after averaging over the lattice volume, while is constrained by the scalar potential. Thanks to the reduced soft supersymmetry breaking enabled by this procedure, Ward identity violations vanish in the continuum limit [8]. This is consistent with the improvement expected since and the other lattice symmetries forbid all dimension5 operators [6].
With the moduli space of the lattice theory survives to all orders of lattice perturbation theory [15]. If nonperturbative effects such as instantons also preserve the moduli space, then the most general longdistance effective action contains only the terms in the improved action above [11, 16]. In addition, all but one of the coefficients on the terms in can be absorbed by rescaling the fermions and the auxiliary field, leaving only a single coupling that may need to be tuned to recover the full symmetries of SYM in the continuum limit.
Treelevel improvement for the lattice SYM static potential
We extract the static potential from the exponential temporal decay of rectangular Wilson loops . To easily analyze all possible spatial separations we gauge fix to Coulomb gauge and compute , where is the product of complexified temporal links at spatial location , extending from timeslice to timeslice .
The static potential analysis can be improved by refining the scalar distance associated with the spatial threevector . This is a longestablished idea in lattice gauge theory, dating back at least to Ref. [17]. Previously we identified the scalar distance as the euclidean norm of , where each is a basis vector of the lattice. Because these basis vectors are not orthogonal, is a fourvector in physical spacetime even though is a threevector displacement on a fixed timeslice of the lattice.
To obtain treelevel improvement we instead extract the scalar distance from the Fourier transform of the bosonic propagator computed at tree level in lattice perturbation theory. Then to this order in lattice perturbation theory. Using the treelevel lattice propagator computed in Ref. [15], we have
(2) 
In this expression is the same fourvector discussed above while and the dual basis vectors are defined by . The last identity allows us to replace and , more directly relating to the threevector displacement .
On a finite lattice, the continuous integral in Eq. 2 would reduce to a discrete sum over integer . Since we have not yet computed the zeromode () contribution to the discrete sum, here we determine by numerically evaluating the continuous integral that corresponds to the infinitevolume limit. Ref. [17] argues that infinitevolume can safely be used in finitevolume lattice calculations, without affecting either the Coulomb coefficient or the string tension. In agreement with this argument, we checked that both approaches give us similar results even though we currently omit the zeromode contribution from the finitevolume computation.
We experimented with three integrators to numerically evaluate the fourdimensional integral in Eq. 2, obtaining consistent results but significantly different performance.
For our problem the most efficient integrator we were able to find was the Divonne
algorithm implemented in the Cuba
library [18].
This is a stratified sampling algorithm based on CERNLIB
routine D151 [19].
Especially for large Divonne
’s evaluation of Eq. 2 converged several orders of magnitude more rapidly than two versions of the vegas
algorithm [20] that we tested.
These two versions of vegas
both provide some improvements over the original algorithm, and are implemented in Cuba
and at http://github.com/gplepage/vegas.
Latest results for the static potential
In Fig. 1 we demonstrate the effects of treelevel improvement for lattice SYM computations of the static potential. All four plots in this figure consider lattices generated using the improved action at ’t Hooft coupling . The top row of plots analyze with the scalar distance defined by the naive euclidean norm of . In the topleft plot we show the potential itself for gauge groups U() with , 3 and 4, including fits to the Coulomb form . It is possible to see that the first points at are consistently below the fit curves, while the next points at are well above them. This scatter of the points around the fit is isolated in the topright plot where we show .
It is precisely this scatter at short distances that treelevel improvement ameliorates, as shown in the bottom row of plots.
These results come from the same gauge configurations and measurements as those in the top row, with the only change in the analysis being the use of obtained from Eq. 2 via the Divonne
integrator in Cuba
.
There is not a onetoone correspondence between the points in the two rows of plots.
Several that produce the same euclidean norm (and are therefore combined in our original analyses) lead to distinct .
At the same time, the finitevolume effects also change.
We drop any displacements that extend at least halfway across the spatial volume of the lattice.
When working with euclidean norms for , this imposes , whereas .
In Fig. 2 we collect preliminary results from treelevel improved static potential analyses employing our new ensembles of gauge configurations generated using the improved action. On lattices we consider three U() gauge groups with , 3 and 4, while to explore finitevolume effects we also carry out and calculations for . (Because the larger volumes also help to control discretization artifacts at stronger couplings, so far we have only generated lattices at the strongest included in this analysis.) A notable finitevolume effect that we observe is a small negative value for the string tension when we fit the static potential to the confining form . We can see in Fig. 1 that such a negative string tension would improve the fit for distances near the finitevolume cutoff. As increases we gain data at larger distances, which more effectively constrain . In the right plot of Fig. 2 we see that the string tension moves toward zero as increases, confirming that the static potential is coulombic at all couplings we consider.
We therefore fit the static potential to the Coulomb form to obtain the results for the Coulomb coefficient in the left plot of Fig. 2. For the same gauge groups and lattice volumes discussed above our results are consistent with the nexttonexttoleadingorder (NNLO) perturbative prediction from Refs. [21, 22, 23]. The agreement with perturbation theory tends to improve as and increase, especially at the strongest ’t Hooft coupling where the larger volume helps control discretization artifacts.
Next steps for lattice Sym
We are near to finalizing and publishing our treelevel improved analyses of the lattice SYM static potential based on the improved lattice action introduced last year and summarized above. In addition we are making progress analyzing the anomalous dimension of the Konishi operator, developing a variational method to disentangle the Konishi and supergravity () operators as described in Ref. [12]. We continue to investigate the possible sign problem of the lattice theory, as well as the restoration of the other supersymmetries and in the continuum limit. Finally, Ref. [12] also presents a new project to study S duality on the Coulomb branch of the theory, by measuring the masses of the W boson and the corresponding dual topological ’t Hooft–Polyakov monopole. Ideally this Coulomb branch investigation will allow nonperturbative lattice tests of S duality even at ’t Hooft couplings relatively far from the selfdual point .
Acknowledgments: We thank Tom DeGrand, Julius Kuti and Rainer Sommer for helpful discussions of perturbative improvement for the static potential, and Rudi Rahn for advice on numerical integration. This work was supported by the U.S. Department of Energy (DOE), Office of Science, Office of High Energy Physics, under Award Numbers DESC0009998 (DS, SC) and DESC0013496 (JG). Numerical calculations were carried out on the HEPTH cluster at the University of Colorado, the DOEfunded USQCD facilities at Fermilab, and the Comet cluster at the San Diego Computing Center through the Extreme Science and Engineering Discovery Environment (XSEDE) supported by U.S. National Science Foundation grant number ACI1053575.
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