Three-dimensional super-Yang--Mills theory on the lattice and dual black branes
Simon Catterall, Joel Giedt, Raghav G. Jha, David Schaich, Toby Wiseman
TThree-dimensional super-Yang–Mills theory on the lattice and dual black branes
Simon Catterall, ∗ Joel Giedt, † Raghav G. Jha, ‡ David Schaich, § and Toby Wiseman ¶ Department of Physics, Syracuse University, Syracuse, New York 13244, United States Department of Physics, Applied Physics and Astronomy,Rensselaer Polytechnic Institute, 110 8th Street, Troy, New York 12065, United States Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, United Kingdom Theoretical Physics Group, Blackett Laboratory,Imperial College, London SW7 2AZ, United Kingdom
In the large- N and strong-coupling limit, maximally supersymmetric SU( N ) Yang–Mills theoryin (2 + 1) dimensions is conjectured to be dual to the decoupling limit of a stack of N D -branes,which may be described by IIA supergravity. We study this conjecture in the Euclidean settingusing nonperturbative lattice gauge theory calculations. Our supersymmetric lattice constructionnaturally puts the theory on a skewed Euclidean 3-torus. Taking one cycle to have anti-periodicfermion boundary conditions, the large-torus limit is described by certain Euclidean black holes. Wecompute the bosonic action—the variation of the partition function—and compare our numericalresults to the supergravity prediction as the size of the torus is changed, keeping its shape fixed.Our lattice calculations primarily utilize N = 8 with extrapolations to the continuum limit, and ourresults are consistent with the expected gravity behavior in the appropriate large-torus limit. I. INTRODUCTION
It has been conjectured [1–4] that the large- N limits ofmaximally supersymmetric Yang–Mills (SYM) theories,obtained from the dimensional reduction of N = 1 SYMin ten dimensions down to ( p + 1) dimensions, are dualto string theories containing D p -branes. In the large- N and strong-coupling limit this relates properties of gaugetheories to the dual properties of D p -brane solutions insupergravity. The p = 3 case is the AdS/CFT corre-spondence, which has received much attention, in partdue to its additional conformal symmetries. For directnumerical tests of holographic duality the p < casesare more attractive to consider, as they feature moretractable gauge theories [5].For example, the D -brane or p = 0 case isa quantum-mechanical description well-known as theBanks–Fischler–Shenker–Susskind (BFSS) model [6, 7].One of the earliest efforts to understand holographic du-ality in the quantum-mechanical case directly from non-perturbative gauge theory was described in Refs. [8–10].In recent years, good agreement has been obtained forthe case of p = 0 in the Euclidean setting using nu-merical Monte Carlo calculations. These efforts startedwith Refs. [11–16], and more sophisticated recent lat-tice analyses give convincing agreement with dual-gravityblack hole predictions in the large- N low-temperaturelimit [17–20]. In addition to the BFSS quantum mechan-ics, a maximally supersymmetric deformation of it knownas the Berenstein–Maldacena–Nastase (BMN) model [21] ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] which may also be dual to black holes at low tempera-tures [22], is now also starting to be studied on the lat-tice [23–25].This Euclidean lattice approach was extended to thehigher-dimensional D -brane case in Refs. [26–29]. To al-low numerical lattice calculations, one must compactifythe spatial direction. In the continuum this correspondsto placing the dual theory on a Euclidean torus with allbosonic fields subject to periodic boundary conditionsalong all directions. With periodic fermion boundaryconditions along all directions, supersymmetry is unbro-ken and the partition function is independent of the sizeand shape of this torus. In order to study more inter-esting behavior, we take one cycle to be anti-periodic forfermions.As discussed in Refs. [28, 29], a conventional thermo-dynamic interpretation would require the gauge theoryto be on a rectangular torus with anti-periodic fermion(thermal) boundary conditions on the Euclidean time cy-cle. However, often it is more convenient to work with askewed torus in the Euclidean setting, in order to use su-persymmetric lattice actions which employ non-cubicallattices with enhanced point group symmetries. Whileone cannot continue the numerical results to Lorentziansignature due to the skewing, this is not an obstructionto testing supergravity predictions. One may also con-sider the dual supergravity in the Euclidean setting witha skewed torus as the asymptotic boundary geometry, inwhich case in the appropriate large- N ’t Hooft limit itpredicts a behavior governed by certain Euclidean blackholes (which also have no Lorentzian analog).The higher-dimensional SYM theories, such as the oneconsidered in this paper, involve more challenging cal-culations than in the quantum-mechanical case, but of-fer the advantage of richer structures. Distinct phasesare associated to center symmetry breaking signaled bythe eigenvalue distributions of the Wilson lines around a r X i v : . [ h e p - t h ] S e p he spatial torus cycles, and are described in the dualgravity by the competition between different black holesolutions. In Refs. [28, 29] these different phases wereindeed seen in two-dimensional lattice calculations, andreasonable agreement was observed for the variation ofthe partition function with torus size for both IIA andIIB supergravity predictions. (See Ref. [30] for an alter-nate approach to the strong-coupling limit of the p = 1 theory in the Lorentzian signature.)The purpose of this paper is to advance these testsof holographic duality to the next higher dimension —the case of D -branes. Again we consider the Euclideantheory compactified on a torus so that it is amenable tolattice calculations. We take one anti-periodic cycle forfermions. In the conventional Euclidean thermal settingon a rectangular torus, the system has an even richerphase structure than the case of p = 1 , with sensitiv-ity to the dimensionless temperature and the various as-pect ratios of the 3-torus [31, 32]. We consider here theskewed torus, as dictated by our supersymmetric latticediscretization. Keeping the shape of the torus fixed, wevary its size relative to the scale set by the ’t Hooft cou-pling and study the bosonic action—the variation of thepartition function. We choose the shape of the torus sothat we can expect the behavior in the large- N stronglycoupled large-torus limit to be governed by the simplestgravitational dual, a homogeneous Euclidean D -braneblack hole in IIA supergravity with boundary given bythe skewed torus. We then numerically analyze this large- N , large-torus limit, to understand how well the gaugetheory matches the predictions of the supergravity solu-tion.We begin in the next section by discussing (2 + 1) -dimensional SYM on a skewed torus and its supergrav-ity dual in the large- N ’t Hooft limit. In Section IIIwe describe our three-dimensional supersymmetric latticeconstruction, which produces the numerical results pre-sented and compared with supergravity expectations inSection IV. The data leading to these results are availableat [33]. We conclude in Section V by looking ahead tofurther lattice SYM studies that can build on this work inthe future, including prospects for exploring phase tran-sitions by changing the shape of the torus. II. SYM ON A SKEWED TORUS AND THESUPERGRAVITY DUAL
We consider three-dimensional maximally supersym-metric Yang–Mills theory, which we take in Euclidean signature to be on a 3-torus denoted hereafter by T .As in the thermal case, we impose anti-periodic fermionboundary conditions only on one cycle corresponding toEuclidean time. Labelling this coordinate as τ , and theothers as x i , we identify τ ∼ τ + β (anti-periodic forfermions), while the others form the ‘spatial’ torus cy-cles after the identifications ( τ, x i ) ∼ ( τ, x i ) + (cid:126)L , (peri-odic for fermions). If (cid:126)L , were orthogonal to each otherand τ , the torus would be rectangular and we wouldhave a Lorentzian interpretation with β being the in-verse temperature. Here we will consider a skewed torus,for which there is no simple Lorentzian interpretation —the Euclidean torus cannot be analytically continued to areal Lorentzian-signature space-time. Nonetheless, holo-graphic duality states that this theory can be describedby a string theory dual which reduces to supergravity inthe large- N ’t Hooft limit.It is convenient to define dimensionless lengths r τ = βλ and r , = | (cid:126)L , | λ in terms of the (dimensionful) ’t Hooftcoupling λ = N g YM . Here we are interested in fixingthe shape of the torus that the SYM is defined on, whilevarying its size. Thus we make the choice (cid:126)L , = β(cid:126)l , ,with (cid:126)l , being vectors that we take to be fixed withunit length, | (cid:126)l , | = 1 , so that each torus cycle has equalproper length β . The partition function of the Euclideantheory is then just a function of the one dimensionlessparameter r τ , and it is convenient to think in terms of t = 1 /r τ , which we may view as a dimensionless ‘gener-alized’ temperature. At large N in the ’t Hooft limit weregard t ∼ O (1) . In this limit, a large numerical value t = 1 /r τ (cid:29) corresponds to the torus being small inunits of the ’t Hooft coupling, and the theory reduces toa 0-dimensional effective theory of the zero modes on thetorus. This small-torus effective theory corresponds tothe bosonic Yang–Mills matrix integral formed from thebosonic truncation of the p = 0 SYM theory, which wenote is not a weakly coupled description [34, 35].Conversely, a small numerical value t = 1 /r τ (cid:28) cor-responds to the torus being large in units of the ’t Hooftcoupling. The behavior in this regime is given by the de-coupling limit of D -branes [2], which may be describedin supergravity by the ten-dimensional Euclidean stringframe metric and dilaton, ds IIA, String = α (cid:48) (cid:32) U / √ π λ (cid:20)(cid:18) − U U (cid:19) dτ + dx i (cid:21) + √ π λU / (cid:34) dU (cid:18) − U U (cid:19) − + U d Ω (cid:35)(cid:33) e φ = λN (cid:114) π λU / . (1)2here is also a 3-form potential carrying the N unitsof D -charge, with τ and x i forming the ‘world-volume’directions that constitute the asymptotic toroidal bound-ary which we may think of the gauge theory living on.Here U is the radial direction, normalized as an en-ergy scale, and U represents the radial position of aEuclidean ‘horizon’ where the Euclidean time circle di-rection, τ , shrinks to zero size. The smoothness of thegeometry relates this to the inverse temperature β , as U / = π √ λβ . We require large N to suppress stringquantum corrections to the supergravity approximation,while the large torus size, t (cid:28) , is required to suppressthe α (cid:48) corrections to the classical supergravity geometrynear the horizon. Both these conditions are satisfied ifwe take (cid:28) r τ (cid:28) N at large N , which is the regimewe focus on in this work. On a large torus with t (cid:28) , stringy winding modesalong the x i cycles may become relevant, associated to aT-dual Gregory–Laflamme instability [31, 32, 34, 35, 37–39], in the case that r / , (cid:46) r τ . However, since we arefixing the shape of the torus to have r , = r τ , we donot expect such phenomena to occur in a regime wherethe dual supergravity describes the system. Since thedual D -brane solution has non-contractible spatial cy-cles on the torus, we expect the angular distribution ofeigenvalues of a Wilson line about such a cycle to be ho-mogeneous at large N [3, 34, 40]. On the other hand, fora small torus where the theory reduces to a bosonic ma-trix integral, we expect a highly localized distribution ofeigenvalue phases for Wilson lines about any torus cycle.Hence one expects a large- N transition as the torus sizeis varied, associated to center symmetry breaking of thespatial Wilson lines.If the x i directions were not compact, so that T =1 /β is a temperature, then noting that the solution istranslation invariant in the τ and x i directions, one maycompute the free energy density f from the dual-gravitysolution, fN λ = − (cid:18) π (cid:19) / t / ≈ − . t / . (2)Compactifying on a torus doesn’t change this density,and for a rectangular torus it yields a partition function log Z = − f V ( T ) , where V ( T ) denotes the volume ofthe 3-torus. Due to the translation invariance of the so-lution, the skewed-torus partition function is given bythese same expressions, although there is no thermal in-terpretation [35].The SYM action is composed of bosonic and fermionic For still-larger tori it is believed the theory flows to a super-conformal IR fixed point given by the Aharony–Bergman–Jafferis–Maldacena (ABJM) model [36] with a dual M -branedescription. parts having the schematic form S SYM = S Bos + S Ferm (3) S Bos = N λ (cid:90) T dτ d x Tr (cid:104) F + 2 ( D Φ I ) − [Φ I , Φ J ] (cid:105) S Ferm = Nλ (cid:90) T dτ d x Tr (cid:2) ψ T (cid:0) /D − [Γ I Φ I , · ] (cid:1) ψ (cid:3) . Rescaling the gauge field A , scalars Φ I , fermions ψ , andthe coordinates ( τ, x i ) by the torus size so they are alldimensionless, ( A, Φ I ) = ( A (cid:48) , Φ (cid:48) I ) /β ψ = √ λψ (cid:48) /β ( τ, x i ) = β ( τ (cid:48) , x (cid:48) i ) , the action may be written as S SYM = 1 βλ S (cid:48)
Bos + S (cid:48) Ferm , (4)where S (cid:48) Bos = S Bos βλ and S (cid:48) Ferm = S Ferm involve onlythe dimensionless bosonic fields and fermion fields respec-tively, and have no explicit β or λ dependence. Thus wemay explicitly differentiate the partition function withrespect to β to obtain β ∂∂β log Z = (cid:104) S Bos (cid:105) . (5)While the partition function itself cannot be computedthrough the lattice methods we use, the expectation valueof the bosonic action is very convenient to obtain (asreviewed in the Appendix). We find the prediction fromsupergravity that at large N , (cid:104) S Bos (cid:105) N = − (cid:18) π t (cid:19) / (cid:18) V ( T ) β (cid:19) (6)when t is sufficiently small. In the small-volume limit t (cid:29) we may use the effective dimensional reduction tocompute (cid:104) S Bos (cid:105) N = − (7)at large N [28, 41].We will see in the next section that the most naturaltorus geometry for us to consider is formed by periodi-cally identifying R in the three basis directions of an A ∗ lattice. As discussed above, we do so taking the cyclein each direction to have the same length β . Explicitlyin our coordinates x µ = ( τ, x i ) we may achieve this bytaking (cid:126)l = 13 − √ (cid:126)l = 13 − −√ √ , (8)which gives a volume V ( T ) = 4 β / (3 √ .3efining the bosonic action density s Bos = (cid:104) S Bos (cid:105) /V ( T ) , for our torus geometry we see theholographic large-volume behavior and small-volumelimit imply s Bos N λ = (cid:26) − . . . . t / for t (cid:28) − . . . . t for t (cid:29) . (9)It is worth noting that for SYM on an analogous torus in ( p +1) -dimensions we would have parametric dependence s Bos ∝ t (14 − p ) / (5 − p ) for t (cid:28) from the gravity dual,and the t (cid:29) limit would go as s Bos ∝ t p +1 . In the p = 3 conformal case these powers coincide, and we seethe powers in the case of p = 2 we consider here arerather close. This makes the task of distinguishing thetwo behaviors more challenging than for the p = 0 and cases considered previously [14, 15, 19, 20, 27–29], wherethere is greater contrast between the large- vs. small-volume parametric dependence on t . III. THREE-DIMENSIONALSUPERSYMMETRIC LATTICECONSTRUCTION
In recent years, it has become possible to formulatecertain supersymmetric lattice gauge theories using theidea of topological twisting, in which the superchargesare grouped into p -forms and the -form supercharges canbe preserved in discrete space-time. While this construc-tion is not needed for (0 + 1) -dimensional SYM quantummechanics (where one can show perturbatively that norelevant supersymmetry-breaking counter-terms are pos-sible [12, 42]), in higher dimensions it is a key ingredientto minimize issues of fine-tuning [5, 43].The three-dimensional maximally supersymmetricYang–Mills theory considered here can be obtained byclassical dimensional reduction of four-dimensional N =4 SYM. The N = 4 SYM lattice construction [44–52] dis-cretizes a maximal twist of the continuum theory knownas the Marcus or geometric-Langlands twist [53, 54]. Theresulting lattice theory features many symmetries: inaddition to U( N ) lattice gauge invariance and a singlescalar supersymmetry, it is also invariant under a large S point group symmetry arising from the underlying A ∗ lattice. Using these symmetries, it is possible to showin perturbation theory that radiative corrections gener-ate only a small number of log divergences in the latticetheory [48]. On reduction to three dimensions these di-vergences disappear and no fine-tuning is expected to beneeded to take the continuum limit [44]. The resultingthree-dimensional lattice theory naturally lives on an A ∗ (body-centered cubic) lattice, whose four basis vectorscorrespond to vectors drawn out from the center of anequilateral tetrahedron to its vertices.As we did in Refs. [28, 29], here we use the full four-dimensional lattice construction provided by the publicly available parallel software described in Refs. [51, 55], setting N z = 1 to reduce to the A ∗ lattice. The remain-ing lattice directions are taken to have equal numbers oflattice sites, N x = N y = N τ , with anti-periodic fermionboundary conditions only on the N τ cycle. In the con-tinuum limit this generates the skewed torus geometrydescribed in Section II (re-labelling { x , x } as { x, y } ).We relegate the full details of the lattice action S lattice to the Appendix, and here discuss only the two soft-supersymmetry-breaking deformation that need to be in-cluded in order to enable our three-dimensional numeri-cal computations. The first of these is a scalar potentialterm, which regulates the divergences associated with in-tegration over a non-compact moduli space in the parti-tion function. We have used various scalar potentials inour previous investigations, and here employ the single-trace version also used in Refs. [28, 29]: S soft = N λ lat µ (cid:88) n ,a Tr (cid:34)(cid:18) U a ( n ) U a ( n ) − I N (cid:19) (cid:35) , (10)with µ a tunable coefficient and the dimensionless λ lat defined in the appendix. We need to extrapolate µ → in order to recover the continuum SYM theory of interest,in addition to extrapolating to the continuum limit ofvanishing lattice spacing that corresponds to λ lat → infewer than four dimensions. We guarantee that µ → in the λ lat → continuum limit by setting µ = ζλ lat .This also allows us to extrapolate µ → with λ lat fixedby considering the ζ → limit, which we will do inSection IV.Next, for the dimensionally reduced lattice theory tocorrectly reproduce the continuum physics, we need toensure that the trace of the each gauge link U z ( n ) in thereduced z -direction is close to N , so that the effectivescalar field obtained by dimensional reduction is small inlattice units. In other words, this means that the centersymmetry should be completely broken in the reduced di-rection for proper dimensional reduction. We ensure thisby adding a second soft-supersymmetry-breaking defor-mation to the lattice action: S center = N λ lat κ (cid:88) n Tr (cid:20)(cid:18) U z ( n ) − I N (cid:19) † (cid:18) U z ( n ) − I N (cid:19)(cid:21) , (11)with κ another tunable coefficient that we must alsotake to zero in Section IV. This term is gauge invariantsince N z = 1 . It explicitly breaks the center symmetryin the single reduced direction by forcing the trace of thelink in this direction to be close to N .With this lattice action S lattice for three-dimensionalSYM, we stochastically sample field configurations usingthe rational hybrid Monte Carlo (RHMC) algorithm [56]implemented in the software mentioned above [51, 55]. github.com/daschaich/susy e − S lattice as a Boltzmannweight, requiring that we consider a lattice action thatis real and non-negative. However, gaussian integrationover the fermion fields of three-dimensional SYM pro-duces a pfaffian that is potentially complex, (cid:90) [ d Ψ] e − Ψ T D Ψ ∝ pf D = | pf D| e iφ . (12)Here D is the fermion operator and S lattice = S Bos +Ψ T D Ψ , with S Bos the bosonic part of the lattice action.As in our previous work [5, 28, 29, 49, 52, 57, 58], we‘quench’ the phase e iφ → to obtain a positive latticeaction for use in the RHMC algorithm. Reweighting (cid:104)O(cid:105) = (cid:10) O e iφ (cid:11) pq (cid:104) e iφ (cid:105) pq (13)is then required to recover expectation values from thesephase-quenched (‘ pq ’) calculations, where (cid:104)O(cid:105) pq = (cid:82) [ d U ] O e − S Bos | pf D| (cid:82) [ d U ] e − S Bos | pf D| (14) (cid:104)O(cid:105) = (cid:82) [ d U ] O e − S Bos pf D (cid:82) [ d U ] e − S Bos pf D . (15)This procedure breaks down, producing a sign problem,when (cid:10) e iφ (cid:11) pq is consistent with zero. Fortunately, in thisinvestigation we focus on regimes where (cid:10) e iφ (cid:11) pq ≈ and (cid:104)O(cid:105) ≈ (cid:104)O(cid:105) pq . This follows from the fact that the r τ and N τ we analyze correspond to . < λ lat < . , safely inthe range of couplings where we observe (cid:10) e iφ (cid:11) pq ≈ inthe full four-dimensional theory [5, 57, 58]. In addition,we gain further benefit from the dimensional reduction,since the lower-dimensional continuum limit correspondsto λ lat → . Partly for this reason, previous lattice stud-ies of N = (2 , and N = (8 , SYM theories in twodimensions found (cid:10) e iφ (cid:11) pq → rapidly upon approachingthe continuum limit, with negligible pfaffian phase fluctu-ations even at non-zero lattice spacing [59–63]. Similarlysmall pfaffian phase fluctuations were also seen in the p = 0 case [16, 18]. IV. NUMERICAL RESULTS ANDCOMPARISON WITH SUPERGRAVITY
We now present our lattice results for the bosonic ac-tion density in the two different regimes described in Sec-tion II. Recall that the small-volume regime has dimen-sionless ‘generalized’ temperature t (cid:29) , while t (cid:28) forthe more interesting large-volume regime related to the These calculations used a double-trace scalar potential in placeof Eq. (10), which should not noticeably affect pfaffian phasefluctuations. dual supergravity by holography. We have concentratedresources to analyze these two regimes, with a focus on . < t < . . Our key result is Fig. 1 where we displaythe bosonic action density vs. t for N = 8 and the L lattice sizes we consider, with N x = N y = N τ = L = 8 , , and .After briefly discussing t ≥ results in the small-volume regime, which we use to check our lattice cal-culations, we focus on the more challenging large-volumecase with . < t < . . This range of t is chosen tosatisfy the conditions (cid:28) r τ (cid:28) N discussed in Sec-tion II, which for N = 8 correspond to . (cid:28) t (cid:28) .While it would be straightforward to run numerical cal-culations with smaller t (cid:46) . , for our current N = 8 these may exit the regime in which IIA supergravity isa reliable description of the holographically dual gravi-tational system. Moving to larger N > is also pos-sible, but would demand much more substantial com-putational resources due to computational costs increas-ing more rapidly than N [55]. The results presentedhere required ∼ million core-hours provided by multiplecomputing facilities, with costs dominated by the largest L = 16 we consider. Ref. [33] provides a comprehensiverelease of our data, including full accounting of statis-tics, auto-correlation times, extremal eigenvalues of thefermion operator (which must remain within the spec-tral range where the rational approximation used in theRHMC algorithm is reliable), and other observables com-puted in addition to the bosonic action density. A. Small-volume regime, t (cid:29) To check that our lattice calculations reproduce the ex-pected small-volume behavior of three-dimensional SYM,we analyze several large values of t ≥ . Motivated bythe right panel of Fig. 1, which shows no significant de-pendence on L ≥ for t (cid:38) . , we carry out these calcu-lations for a single L lattice size with L = 8 . For theselarge t we are also able to set κ = 0 in Eq. (11) with-out encountering numerical instabilities (i.e., the centersymmetry in the reduced direction breaks dynamically),leaving Eq. (10) the only soft-supersymmetry-breakingdeformation in the lattice action. As discussed in Sec-tion III, we remove this deformation by extrapolating ζ → , here considering ζ = 0 . , . and . foreach value of t . These linear extrapolations produce the t ≥ results in the left panel of Fig. 1, which are ingood agreement with the solid line showing the expectedsmall-volume limit from Eq. (9).In Fig. 2 we show distributions of the phases of theWilson line (spatial holonomy) eigenvalues for three t ≥ lattice ensembles with ζ = 0 . . As reviewed in the Ap-pendix, our lattice construction naturally provides com-plexified Wilson lines that include contributions fromboth the gauge and scalar fields. In this work, we removethe scalar-field contributions by considering instead uni-tarized Wilson lines. The resulting distributions shown in5 . . . . . . . - s Bos N λ t . t / . t . . . . . . . .
24 0 .
28 0 .
32 0 .
36 0 . .
44 0 . - s Bos N λ t . t / . t Figure 1. The ζ → extrapolated bosonic action density for N = 8 with lattice sizes , , and , compared with thelarge-volume (dashed) and small-volume (solid) expectations from Eq. (9). Left:
The full range of dimensionless temperatures t on log–log axes. Right:
Focusing on . < t < . with linear axes to clarify the absolute size of uncertainties. Phase0.00.10.20.30.40.5 R e l a t i v e f r e q u e n c y Phase of unitarized Wilson line eigenvalues, N=8, L=8 t=2t=1.5t=1
Figure 2. Distributions of N = 8 Wilson line eigenvaluephases over the angular range [ − π, π ) , in the small-volumeregime with dimensionless temperatures t ≥ . The distribu-tions become more localized with increasing t , as expected. Fig. 2 are clearly localized, and the width of the supportdecreases as t increases. This lattice result is consistentwith the expectation that the angular eigenvalue distri-bution is highly localized for t → ∞ , providing anothernon-trivial check that our lattice calculations correctlyreproduce the three-dimensional SYM theory. B. Large-volume regime, t (cid:28) Turning now to the more interesting large-volumeregime where we can compare our results with dual su-pergravity predictions, we analyze . < t < . in or-der to satisfy the conditions N − (cid:28) t (cid:28) discussedabove, with N − ≈ . for the N = 8 we consider. Inthis regime, we need to include both soft-supersymmetry-breaking deformations Eqs. 10 and 11 in the lattice ac-tion. To simplify our analysis we set κ = µ , so that . . . . . . . . . . . . . .
01 0 .
02 0 .
03 0 .
04 0 .
05 0 .
06 0 .
07 0 .
08 0 .
09 0 . - s Bos N λ ζ t = 0 . t = 0 . t = 0 . t = 0 . t = 0 . t = 0 . t = 0 . Figure 3. Representative linear ζ → extrapolations of thebosonic action density for different temperatures on lat-tices with N = 8 . each ζ → extrapolation (here considering ζ = 0 . , . , and . ) simultaneously removes both deforma-tions. Control over these extrapolations is essential toprecisely determine the SYM bosonic action density tobe compared with the supergravity prediction.Representative linear ζ → extrapolations of ourbosonic action density data are shown in Fig. 3 for allour lattice ensembles with N = 8 . The ζ → limitsin this figure correspond exactly to the points shownin both panels of Fig. 1. Clearly the ζ → extrapolatedresults in Fig. 1 have significantly larger relative uncer-tainties than the input data at non-zero ζ in Fig. 3. Thisis a consequence of the steep extrapolations to the muchsmaller SYM bosonic action densities that remain afterremoving the deformations in our lattice action.These larger uncertainties are even more evident inFig. 4, where we zoom in on the six smallest . (cid:46) t (cid:46) . to investigate the dependence of the ζ → extrap-olated bosonic action densities on the L lattice volumewith L = 8 , and . Since we fix the dimensionless6 . . . . / / / . − s Bos N λ / L t = . t = . t = . t = . t = . t = . Figure 4. Continuum extrapolations of the ζ → extrap-olated bosonic action density for various temperatures with N = 8 , where the limit L → ∞ with fixed t correspondsto λ lat → and vanishing lattice spacing. Small horizontaloffsets are added for clarity. All extrapolations for t ≥ . have slopes consistent with zero, indicating no significant dis-cretization artifacts in the corresponding results. lengths of the lattice, r x = r y = r τ , larger values of L correspond to smaller lattice spacings, allowing us tocheck discretization artifacts and extrapolate to the con-tinuum limit, /L → or equivalently L → ∞ . Mostof the linear /L → extrapolations shown in Fig. 4have slopes consistent with zero, indicating that thereare not significant discretization artifacts in the corre-sponding results, and motivating our choice to includeall our L = 8 , and results in Fig. 1. On the whole,these bosonic action density results are reasonably con-sistent with the large- N prediction from supergravity inEq. (9) (the dashed line in Fig. 1), particularly consider-ing the modest N = 8 and t ≈ . that we have used inthis work.The best agreement with the dual supergravity pre-diction comes from the two smallest t ≈ . and . ,which are also the cases where the /L → contin-uum extrapolations are non-trivial. From Fig. 4 we cansee that these non-trivial extrapolations are driven bythe L = 8 results, with the L = 12 and results fullyconsistent with the respective continuum limits withintheir (relatively large) uncertainties. An obvious ques-tion in this context is whether these results really fallin the large-volume regime, or may still be governed bysmall-volume (or intermediate) behavior. As discussedbelow Eq. (9), the expected parametric dependence of thebosonic action density is rather similar in both regimesfor this p = 2 case, making it more difficult to distinguisha clear change in behavior.Stronger evidence that our small- t results are in thelarge-volume regime can be obtained by again consider-ing the eigenvalues of the Wilson line about the spatialtorus cycles. In Fig. 5 we show distributions of the phasesof these eigenvalues for lattice ensembles with t ≈ . and ζ = 0 . , which follow broad distributions in clearcontrast to the small-volume case shown in Fig. 2. Re-call that the D supergravity solution predicts a homoge-neous distribution of these phases at large N . To checkthe dependence on N , we have generated one ensem-ble with N = 4 and another with N = 6 . In the left panelof Fig. 5 we compare the resulting N = 4 , and Wil-son line eigenvalue phase distributions and confirm thatthey become broader as N increases, consistent with theexpected large- N homogeneous distribution. In the rightpanel we check that there is no visible L dependence inour N = 8 results for this same t ≈ . and ζ = 0 . .Thus we confirm that our small- t results do indeed ap-pear to be in the large-volume regime and consistent withthe dual supergravity predictions. Presumably there is alarge- N phase transition separating the small- and large-volume regimes, although such a transition is difficult tosee in our N = 8 data on the lattice sizes we considerhere. V. CONCLUSIONS AND NEXT STEPS
We have presented the first numerical lattice gaugetheory studies of three-dimensional maximally supersym-metric Yang–Mills theory, advancing our program of non-perturbatively testing holography. Such tests provide di-rect first-principles checks of holographic duality at finitetemperatures and in non-conformal settings, where toolssuch as integrability and supersymmetric localization arenot available.Already at modest N = 8 our results indicate that thelarge- N predictions of the dual-gravity black holes canemerge for large tori. We have seen that the bosonicaction density interpolates rather smoothly between thesmall-volume regime and the large-volume supergravityregime, similar to results for lower-dimensional cases [14,15, 19, 20, 27–29]. We are able to see qualitative agree-ment with the supergravity prediction derived from thedual black hole action density, and continuum extrapo-lations indicate no significant discretization artifacts for t ≥ . . We also see that the Wilson lines about thespatial directions of the torus are consistent with a tran-sition from a localized angular eigenvalue distribution atsmall volumes to the expected homogeneous distributionat large volumes, presumably with a large- N phase tran-sition at an intermediate torus size.In the future, we plan to look at the Maldacena–Wilsonloop and compare it to the results obtained from the dual-gravity computations. In addition, similar to our previ-ous study [28, 29], we can also change the aspect ratios ofthe torus cycle sizes to study phase transitions from thehomogeneous D -phase we consider here to D -phases oreven localized D -phases. It will also be interesting tounderstand the nature of the large- N phase transition atintermediate volumes, although this has proved difficultto study even in simpler settings [64].Though our results approach the supergravity predic-7 Phase0.000.050.100.150.200.250.30 R e l a t i v e f r e q u e n c y Phase of unitarized Wilson line eigenvalues, L=12, t=0.31
N=4N=6N=8 Phase0.000.050.100.150.200.250.30 R e l a t i v e f r e q u e n c y Phase of unitarized Wilson line eigenvalues, N=8, t=0.31
L=8L=12L=16
Figure 5. Distributions of Wilson line eigenvalue phases over the angular range [ − π, π ) for a small t ≈ . . Left:
The L = 12 distributions become broader as N increases, consistent with the homogeneous distribution expected for the large-volume regimein the large- N limit. Right:
The N = 8 distributions are independent of the lattice size L . tions in the appropriate regime, even larger N would helpto better satisfy the conditions on the validity of the clas-sical supergravity description. Numerical calculations atlarger N are certainly possible, but would require muchmore substantial computational resources due to com-putational costs increasing more rapidly than N [55].Our current results in this paper nevertheless show theapproach to this regime in detail and are certainly con-sistent with the supergravity results. ACKNOWLEDGEMENTS
This work was supported by the US Department ofEnergy (DOE), Office of Science, Office of High En-ergy Physics, under Award Numbers DE-SC0009998 (SC)and DE-SC0013496 (JG). RGJ’s research is supportedby postdoctoral fellowship at the Perimeter Institute forTheoretical Physics. Research at Perimeter Institute issupported in part by the Government of Canada throughthe Department of Innovation, Science and Economic Development Canada and by the Province of Ontariothrough the Ministry of Colleges and Universities. DSwas supported by UK Research and Innovation FutureLeader Fellowship MR/S015418/1. Numerical calcula-tions were carried out at the University of Liverpool,on DOE-funded USQCD facilities at Fermilab, and atthe San Diego Computing Center through XSEDE sup-ported by National Science Foundation grant numberACI-1548562.
APPENDIX: LATTICE ACTION ANDCOMPUTATION OF THE BOSONIC ACTION
Our lattice formulation of maximally supersymmetricYang–Mills theory in d < dimensions discretized onthe A ∗ d lattice is obtained by classical dimensional re-duction from the parent four-dimensional theory. Thelattice action for topologically twisted N = 4 SYM in d = 4 dimensions is the sum of the following Q -exactand Q -closed terms [44–52]: S exact = N λ lat (cid:88) n Tr (cid:20) −F ab ( n ) F ab ( n ) − χ ab ( n ) D (+)[ a ψ b ] ( n ) − η ( n ) D ( − ) a ψ a ( n ) + 12 (cid:16) D ( − ) a U a ( n ) (cid:17) (cid:21) , (16) S closed = − N λ lat (cid:88) n Tr (cid:104) (cid:15) abcde χ de ( n + (cid:98) µ a + (cid:98) µ b + (cid:98) µ c ) D ( − ) c χ ab ( n ) (cid:105) , (17)where λ lat is the dimensionless ’t Hooft coupling definedby r τ, lattice = λ lat N − dτ . The indices run from , · · · , ,spanning the basis vectors of the A ∗ lattice, and (cid:80) n isover all lattice sites. The fermion fields η , ψ a and χ ab = − χ ba transform in representations of the S point group symmetry, as do the five complexified gauge links U a and U a that combine the gauge and scalarfield components. These gauge links are used to form thecomplexified field strengths F ab and F ab , as well as thefinite difference operators D (+) a and D ( − ) a .In addition to these terms, we also include the twosoft-supersymmetry-breaking deformations discussed in8ection III. S soft from Eq. (10) is present to regulateflat directions even in four dimensions, while S center fromEq. (11) needs to be added once we specialize to thethree-dimensional theory by setting N z = 1 . The fullthree-dimensional lattice action is then S lattice = S exact + S closed + S soft + S center . (18)As mentioned in Section IV A, we can omit S center (bysetting its coefficient κ = 0 ) in the small-volume regimewhere the center symmetry in the reduced directionbreaks dynamically.Another detail mentioned in Section IV A is the needto remove the scalar-field contributions from the Wil-son lines (spatial holonomies) that we analyze to distin-guish between the small- and large-volume regimes. Asin Ref. [28, 49, 57], we accomplish this by using a po-lar decomposition U a = H a · U a to separate each N × N complexified gauge link into a positive-semidefinite her-mitian matrix H a (containing the scalar fields) and aunitary matrix U a corresponding to the gauge field. Theresulting unitarized Wilson lines are simply the prod-ucts (cid:81) N x i =1 U x ( x i , y, τ ) wrapping around the lattice, andsimilarly in the y -direction. The distributions shownin Figs. 2 and 5 come from the Wilson lines in the x -direction, while the data released in Ref. 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