PProgress and prospects of lattice supersymmetry
David Schaich ∗ AEC Institute for Theoretical Physics, University of Bern, 3012 Bern, SwitzerlandE-mail: [email protected]
Supersymmetry plays prominent roles in the study of quantum field theory and in many proposalsfor potential new physics beyond the standard model, while lattice field theory provides a non-perturbative regularization suitable for strongly interacting systems. Lattice investigations ofsupersymmetric field theories are currently making significant progress, though many challengesremain to be overcome. In this brief overview I discuss particularly notable progress in threeareas: supersymmetric Yang–Mills (SYM) theories in fewer than four dimensions, as well as bothminimal N = 1 SYM and maximal N = 4 SYM in four dimensions. I also highlight super-QCDand sign problems as prominent challenges that will be important to address in future work. ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ a r X i v : . [ h e p - l a t ] J un rogress and prospects of lattice supersymmetry David Schaich
0. Introduction, motivation and background
Supersymmetry plays prominent roles in modern theoretical physics, as a tool to improve ourunderstanding of quantum field theory (QFT), as an ingredient in many new physics models, andas a means to study quantum gravity via holographic duality. Lattice field theory provides a non-perturbative regularization for QFTs, and other contributions to these proceedings document theprodigious success of this framework applied to QCD and similar theories. It is natural to consideremploying lattice field theory to investigate supersymmetric QFTs, especially in strongly coupledregimes. In this proceedings I review the recent progress and future prospects of lattice studiesof supersymmetric systems, focusing on four-dimensional gauge theories and their dimensionalreductions to d < . Lattice supersymmetry now has more than four decades of history [1], much of which is re-viewed by Refs. [2 – 7]. Unfortunately, progress in this field has been slower than for QCD, inlarge part because the lattice discretization of space-time breaks supersymmetry. This occurs inthree main ways. First, the anti-commutation relation (cid:8) Q α , Q ˙ α (cid:9) = 2 σ µα ˙ α P µ in the super-Poincaréalgebra connects the spinorial generators of supersymmetry transformations, Q α and Q ˙ α , to thegenerator of infinitesimal space-time translations, P µ . The absence of infinitesimal translations onthe lattice consequently implies broken supersymmetry.Next, bosonic and fermionic fields are typically discretized differently on the lattice (in partdue to the famous fermion doubling problem). In the specific context of supersymmetric gaugetheories, standard discretizations associate the gauginos with lattice sites n (i.e., they transform as G ( n ) λ α ( n ) G † ( n ) under a lattice gauge transformation) while the gauge connections are associatedwith links between nearest-neighbor sites, transforming as G ( n ) U µ ( n ) G † ( n + a (cid:98) µ ) where ‘ a ’ is thelattice spacing. Away from the a → continuum limit, these differences prevent supersymmetrytransformations from correctly interchanging superpartners.Finally, supersymmetry requires a derivative operator that obeys the Leibniz rule [1], viz. ∂ [ φη ] = [ ∂φ ] η + φ∂η , which is violated by standard lattice finite-difference operators. ‘No-gotheorems’ presented by Refs. [8, 9] establish that only non-local derivative and product opera-tors can obey the Leibniz rule (and hence fully preserve supersymmetry) in discrete space-time.Efforts continue to construct and study alternate formulations that may better balance locality andsupersymmetry. For example, Ref. [10] finds that a lattice field product operator obeying a ‘cyclicLeibniz rule’ [11] suffices to preserve partial supersymmetry and establish non-renormalizationfor a quantum-mechanical system. Ref. [12] introduces a different non-local ‘star product’ thatpreserves the Leibniz rule, with the consequence that the lattice spacing no longer acts as a regu-lator. Despite the complicated intricacies of these constructions, so far their applicability appearslimited to systems without gauge invariance, and only in (0+1) dimensions [10, 13] or on infinitelattices [12].As a consequence of broken supersymmetry, quantum effects in the lattice calculation willgenerate supersymmetry-violating operators. These include, in particular, relevant operators forwhich counterterms will have to be fine-tuned in order to recover the supersymmetric QFT of in-terest in the a → continuum limit that corresponds to removing the UV cutoff a − . In theories Theories without gauge invariance, such as Wess–Zumino models and sigma models, are reviewed in Refs. [2, 5].More recent work in this area includes Refs. [171 – 174]. rogress and prospects of lattice supersymmetry David Schaich with scalar fields (either squarks in super-QCD or scalar elements of the gauge supermultiplet in N > theories with extended supersymmetry), these scalars’ mass terms present fine-tuning prob-lems similar to that of the Higgs boson in the standard model. Additional supersymmetry-violatingoperators include fermion (quark and gaugino) mass terms, Yukawa couplings, and quartic (four-scalar) terms. Altogether there are typically O (10) of these operators [3, 14, 15], implying suchhigh-dimensional parameter spaces that there seems to be little hope of effectively navigating themin numerical lattice calculations.The following three sections focus on three different ways to reduce the amount of fine-tuningin lattice studies of supersymmetric Yang–Mills (SYM) theories. We begin in the next sectionby reviewing dimensional reductions to SYM theories in fewer than four space-time dimensions,which has received the most attention from the community so far. We return to four dimensionsin Sec. 2, considering first the special case of minimal ( N = 1 ) SYM, which is vastly simplifiedby the absence of scalar fields. Another special case in four dimensions is maximal ( N = 4 )SYM, the topic of Sec. 3, for which a closed subalgebra of the supersymmetries can be preservedat non-zero lattice spacing, again drastically reducing the necessary fine-tuning. Finally, Sec. 4briefly discusses some prominent challenges to be faced by lattice studies of supersymmetric QFTsin the future, including investigations of supersymmetric QCD (SQCD) and the possibility of signproblems in various theories.
1. Lower-dimensional systems
Dimensionally reduced SYM theories can be much easier to analyze numerically. In additionto the smaller number of degrees of freedom for an L d lattice, the resulting lower-dimensional theo-ries tend to be super-renormalizable and in many cases a one-loop counterterm calculation sufficesto restore supersymmetry in the continuum limit [16 – 18]. We will label systems by their numberof supercharges (generators of supersymmetry transformations): Q = 4 , 8 or 16 respectively cor-responding to N = 1 , 2 or 4 SYM in four dimensions (or equivalently to minimal SYM in d = 4 ,6 or 10 dimensions). For d ≤ these theories involve a gauge field, Q fermionic component fields,and − d , − d or − d real scalar fields, respectively, all of which are massless and transformin the adjoint representation of the gauge group. The gauge groups we consider are SU( N ) andU( N ) = SU( N ) ⊗ U(1).
The reduction to ‘SYM quantum mechanics’ (QM) has been the subject ofmany numerical studies over the past decade, starting with Refs. [19, 20]. These systems involvebalanced collections of interacting bosonic and fermionic N × N matrices at a single spatial point.One reflection of the simplicity of SYM QM is that a lattice regularization may not even be re-quired; a gauge-fixed Monte Carlo approach employing a hard momentum cutoff [19] was usedby Refs. [21 – 26]. Another illustration is a recent proposal [27] that ‘ungauging’ Q = 16 SYMQM (to consider a scalar–fermion system with SU( N ) global symmetry) has relatively little effect,in the sense that both the gauged and ungauged models flow to the same theory in the IR. Thisconjecture was quickly tested by lattice calculations that found consistent results [28].Even though these quantum-mechanical systems are much simpler to study on the lattice thantheir d = 4 SYM counterparts, they remain computationally non-trivial. This is demonstrated by2 rogress and prospects of lattice supersymmetry
David Schaich the state-of-the-art results for Q = 16 SYM QM from Refs. [29, 30] shown in Fig. 1. This Q = 16 case has attracted particular interest due to its connections to string theory [31], and especially theconjecture [32] that the large- N limit of this system describes the strong-coupling (‘M-theory’)limit of type-IIA string theory in light-front coordinates. (Refs. [33, 34] are thorough reviews ofthe stringy side.) At finite temperature, this conjecture relates the large- N limit of the (deconfined) Q = 16 system to a dual compactified 11-dimensional black hole geometry in M-theory, and Fig. 1shows the dual black hole internal energy determined from lattice SYM QM computations. Thisquantity was previously investigated numerically by Refs. [21, 23, 26, 35 – 39].Refs. [29, 30] improve upon the earlier work by carrying out controlled extrapolations to thelarge- N continuum limit, allowing for more robust comparisons with dual gravitational predictions.The left plot of Fig. 1 shows one such extrapolation, for a fixed value T = 0 . of the dimension-less temperature T ≡ t dim /λ / dim . (The subscripts highlight dimensionful quantities, including the’t Hooft coupling λ dim = g N with dimension [ λ dim ] = 4 − d .) With fixed T the continuum limitcorresponds to extrapolating the number of lattice sites L → ∞ . At low temperatures the resultsin the right plot convincingly approach the leading-order gravitational prediction from classicalsupergravity (SUGRA), providing non-perturbative first-principles evidence that the holographicduality conjecture is correct. In addition, the growing difference between the lattice results andthe SUGRA curve at higher temperatures can be considered a prediction of higher-order quantumgravitational effects that are enormously difficult to calculate analytically.The computational non-triviality of these investigations comes primarily from the large valuesof N that are needed ( ≤ N ≤ in Refs. [29, 30], large enough to benefit from dividingindividual N × N matrices across multiple MPI processes via the MMMM code). The computationalcost of N × N matrix multiplication scales ∝ N , compared to the ∼ L d/ costs of the rational hybridMonte Carlo (RHMC) algorithm. In addition to improving control over the N → ∞ extrapolations,large values of N (cid:38) are also required to suppress a thermal instability associated with the non-compact quantum moduli space of Q = 16 SYM QM [36]. For sufficiently low temperatures and
Figure 1:
State-of-the-art results for the dual black hole internal energy from Q = 16 SYM QM latticecalculations, from Refs. [29, 30].
Left:
Representative (separate and combined) large- N and continuum( L → ∞ ) extrapolations, for fixed dimensionless temperature T = 0 . . Right:
Large- N continuum-limitresults versus T , compared to earlier investigations [23, 37] that did not carry out controlled extrapolations. rogress and prospects of lattice supersymmetry David Schaich sufficiently small N the system is able to run away along these flat directions (holographicallyinterpreted as D0-brane radiation from the dual black hole). Formally a scalar potential should beadded to the lattice action to stabilize the desired vacuum, and then removed in the course of thecontinuum extrapolation, further increasing computational costs [36, 40]. However, Refs. [29, 30]argue that in practice it is possible to carry out Monte Carlo sampling around a metastable vacuumso long as N is sufficiently large. In particular, N must increase in order to reach smaller T .Further numerical investigations of SYM QM systems are underway [41 – 45]. At the sametime, the good control over the necessary extrapolations that has now been achieved for the Q =16 case also motivates pursuing comparable quality in lattice studies of less-simplified systems.One example of such a system is the Berenstein–Maldacena–Nastase (BMN) deformation of Q =16 SYM QM [46], which introduces a non-zero mass for the 9 scalars and 16 fermions whilepreserving all 16 supercharges. This theory has been studied numerically by Refs. [47 – 49]. Themass deformation explicitly breaks the SO(9) global symmetry (corresponding to the compactifiedspatial dimensions of d = 10 SYM) down to SO( ) × SO( ). It also lifts the flat directions mentionedabove, thus serving as a supersymmetric regulator that need not be removed in the continuum limit.In addition, the mass parameter µ provides a second axis for the finite-temperature phasediagram, as shown in the left plot of Fig. 2. As µ → ∞ the theory becomes gaussian, and thedeconfinement temperature T d can be computed perturbatively in /µ . At small µ , Ref. [50] carriedout the numerical construction of the SUGRA black hole geometry dual to the deconfined phase,predicting T d to linear order in µ . Figure 2 shows recent numerical results from Ref. [49] inreasonably good agreement with these predictions, given the fixed N = 8 and L = 24 . In additionto the deconfinement transition signalled by the Polyakov loop, this work observes a transitionbetween an approximately SO(9)-symmetric phase at high temperatures and an SO( ) × SO( )phase at low temperatures. For small µ (cid:46) these transitions occur at the same T d , while at larger µ higher temperatures are needed to recover approximate SO(9) symmetry. It will be interestingto systematize large- N continuum extrapolations in future lattice BMN investigations, since theseturned out to be significant in the µ = 0 limit considered in Fig. 1. Dimensional reductions of SYM to d = 2 and 3 also provide less-simplifiedsystems compared to SYM QM, while still being significantly more tractable than d = 4 . Al-though there has been a lot of work in this area over the years, much of the effort has focusedon constructing clever lattice formulations that minimize fine-tuning in principle, rather than us-ing these constructions in practical numerical calculations. Here we will highlight the numericalcalculations, which leaves little to say about d = 3 : see Refs. [18, 51, 52] for Q = 8 formulations.The main clever constructions that have been applied are based on either ‘twisting’ [53 – 55]or orbifolding [56 – 59], two approaches that actually produce equivalent constructions [60, 61].(See Ref. [2] for a thorough review.) Here we discuss only the twisting approach, which identi-fies at most (cid:98) Q/ d (cid:99) linear combinations of supercharges, Q , that are nilpotent, Q = 0 . Theseare found by organizing the Q supercharges into irreducible representations of a ‘twisted rotationgroup’ SO( d ) tw ≡ diag [ SO( d ) euc ⊗ SO( d ) R ] , where SO( d ) euc is the Wick-rotated Lorentz groupand SO( d ) R is a global R -symmetry. The nilpotent Q are those that transform in the twisted-scalarrepresentation. The requirement Q ≥ d ensures a sufficiently large R -symmetry. This procedureprovides a closed supersymmetry subalgebra {Q , Q} = 0 at non-zero lattice spacing, leading to a4 rogress and prospects of lattice supersymmetry David Schaich Q -invariant lattice action with no need of the Leibniz rule.For some theories there are multiple ways the twisting procedure can be carried out. Oneapproach [60, 62 – 64] combines the gauge and scalar fields into a complexified gauge field, leadingto U( N ) = SU( N ) ⊗ U(1) gauge invariance and non-compact lattice gauge links (cid:8) U , U (cid:9) with aflat measure. The fermion fields are twisted in the same way as the supercharges, obtaining thesame lattice gauge transformations as the bosonic degrees of freedom. Although the U(1) sectordecouples in the continuum, at non-zero lattice spacing it can introduce unwanted artifacts at strongcoupling, and ongoing work is searching for good ways to suppress these [64 – 66]. For d = 2 adifferent approach [53, 54, 67 – 71] works with compact gauge links and gauge group SU( N ), atthe cost of imposing an admissibility condition to resolve a huge degeneracy of vacua (but seeRef. [72]), which becomes more problematic in higher dimensions [73, 2]. This formulation hasbeen used by several numerical studies of the Q = 4 [40, 74 – 81] and Q = 16 [82, 83] theories.The right plot of Fig. 2 shows recent results from Ref. [65] for the phase diagram of two-dimensional Q = 16 SYM, using the non-compact twisted construction described above [84 – 87].The system is formulated on an r L × r β torus, with r β = 1 /T the inverse dimensionless temperaturewhile r L = L dim √ λ dim is the corresponding dimensionless length of the spatial cycle. At hightemperatures (small r β ), the fermions pick up a large thermal mass and the system reduces to a one-dimensional bosonic QM. In this limit (at large N ), Refs. [88 – 91] predict a ‘spatial deconfinement’transition as r L decreases, signalled by a non-zero spatial Wilson line Tr (cid:2)(cid:81) x i U x ( x i , t ) (cid:3) . It iscurrently unclear whether this is a single first-order transition [91] or two nearby second- and third-order transitions [89, 90].In the low-temperature (large- r β ) limit, there is a large- N holographic prediction for a similartransition. Here the large- r L spatially confined phase is conjectured to be dual to a homogeneousblack string with a horizon wrapping around the spatial cycle, while the small- r L spatially decon- μ ( μ ,T )- phase diagram Figure 2: Left:
Phase diagram for the BMN deformation of Q = 16 SYM QM, in the plane of dimensionlessmass µ ≡ µ dim /λ / dim and temperature T , from Ref. [49]. Lattice results for the confinement (lower bluepoints) and SO( ) → SO( ) × SO( ) (upper green points) transitions with fixed N = 8 and L = 24 are compared to a small- µ holographic calculation (red), three-loop large- µ perturbation theory (blue) andan interpolating resummation (purple). Right:
Phase diagram for two-dimensional Q = 16 SYM on an r L × r β = r L × T torus, from Ref. [65]. Lattice results for the ‘spatial deconfinement’ transition with fixed N = 12 and aspect ratios α = r L /r β from 8 ( × ) to / ( × ) are compared to the large- N high- T bosonic QM behavior (blue) and a low- T holographic calculation (red). rogress and prospects of lattice supersymmetry David Schaich fined phase corresponds to a localized black hole. As in the BMN case, the holographic analysesrequire challenging numerical SUGRA constructions [92] of these dual black hole and black stringgeometries. The lattice results for the spatial deconfinement transition (with N = 12 and fixedlattice sizes × , × , × , × , × and × ) reproduce the high-temperature bosonicQM expectations quite well and are consistent with holography at lower temperatures, albeit withrapidly increasing uncertainties. At low temperatures a scalar potential is added to the lattice actionand then extrapolated to zero in order to avoid the thermal instability mentioned above for SYMQM. Ref. [65] also calculates the internal energies of the dual black hole and black string, in bothphases finding consistency with holographic expectations within large uncertainties. It will be in-teresting to see future work improve upon these results, ideally accessing lower temperatures inaddition to gaining control over extrapolations to the large- N continuum limit.Two-dimensional SYM also possesses rich zero-temperature dynamics that are important toexplore non-perturbatively, in addition to studying the thermal behavior discussed above. For ex-ample, Refs. [93, 94] argue that the ‘meson’ spectrum of the Q = 4 theory should include amassless supermultiplet, unlike the d = 4 N = 1 SYM of which this is the dimensional reduc-tion. A recent lattice calculation using straightforward Wilson fermions observes such a masslessmultiplet [95], and also checks for spontaneous supersymmetry breaking, which Ref. [96] suggestsmight occur for this theory. No evidence of spontaneous supersymmetry breaking is seen, consis-tent with another recent lattice study [97] and older work [75, 76, 79] using twisted formulations.
2. Minimally supersymmetric Yang–Mills ( N = 1 SYM) in four dimensions
Returning to four dimensions, we can note that most of the supersymmetry-violating operatorsdiscussed in Sec. 0 involve scalar fields, viz. the scalar mass terms, Yukawa couplings, and quarticoperators. This implies a vast reduction of fine-tuning for N = 1 SYM, the only d = 4 super-symmetric gauge theory with no scalar fields. This theory consists of a SU( N ) gauge field andits superpartner gaugino, a massless Majorana fermion transforming in the adjoint representationof SU( N ). The only relevant (or marginal) operator that may need to be fine-tuned to obtain thecorrect continuum limit is the gaugino mass [98, 99]. We can even avoid this single fine-tuningby working with Ginsparg–Wilson (overlap or domain-wall) lattice fermions that preserve chiralsymmetry and protect the gaugino mass against large additive renormalization. Although the axialanomaly breaks the classical U(1) R -symmetry of N = 1 SYM to its Z N subgroup, this discreteglobal symmetry suffices to forbid a gaugino mass. Gaugino condensation, (cid:104) λλ (cid:105) (cid:54) = 0 , sponta-neously breaks Z N → Z .However, in large part due to their computational expense, there have been no Ginsparg–Wilson studies of N = 1 SYM for most of the past decade [100 – 102]. Instead, current workuses improved Wilson fermions and fine-tunes the gaugino mass to recover both chiral symmetryand supersymmetry in the continuum limit. One major effort by the DESY–Münster–Regensburg–Jena Collaboration, currently using clover improvement, has made significant progress in recentyears [103 – 110]. A second group recently began exploring a SYM analogue of the twisted-massfermion action [111, 112], aiming to improve the formation of composite supermultiplets at non-zero gaugino masses and lattice spacings, and thereby gain better control over the chiral and con-tinuum extrapolations. 6 rogress and prospects of lattice supersymmetry
David Schaich
The larger number of dimensions requires considering much smaller N (cid:28) compared tothe lower-dimensional work discussed above, to keep computational costs under control. Currentefforts study only gauge groups SU(2) [103 – 107] and SU(3) [108 – 113]. The left plot of Fig. 3shows recent SU(3) results from Ref. [108] for the masses of two composite states expected toform (part of) a degenerate multiplet in the supersymmetric continuum chiral limit [114, 115]: the ++ ‘glueball’ and the fermionic ‘gluino–glue’ particle. Even at a fixed lattice spacing the chiralextrapolations of these masses agree within uncertainties. These signs of supermultiplet formationappear much clearer compared to earlier SU(2) results [106], presumably due to either or both thelarger N and the use of clover improvement instead of stout smearing.The chiral extrapolations in Fig. 3 are carried out by computing the mass of an ‘adjoint pion’defined in partially quenched chiral perturbation theory [116] and taking the limit m π → . Whiletwo-point functions for the physical composite states of N = 1 SYM all involve fermion-line-disconnected diagrams, m π is measured from just the connected part of the correlator for the η (cid:48) -like‘gluinoball’. Supersymmetric Ward identities provide an alternative means to determine the critical κ c corresponding to the chiral limit. The difference between these two determinations of κ c can beconsidered a measure of the supersymmetry-breaking discretization artifacts, which is shown fortwo lattice spacings in right plot of Fig. 3. The two available points are consistent with the artifactsvanishing ∝ a as expected for clover fermions, supporting the restoration of supersymmetry in thechiral continuum limit.As for QCD, many other lattice N = 1 SYM investigations may be carried out in addition tocalculations of the spectrum, Ward identities, and the gaugino condensate (cid:104) λλ (cid:105) [100 – 102]. Theseinclude explorations of the finite-temperature phase diagram, with Refs. [103, 117] reporting thatdeconfinement (spontaneous center symmetry breaking) and chiral symmetry restoration appearto occur at the same temperature, which was not known a priori. Refs. [104, 110] investigate thephase diagram on R × S with a small radius for the compactified temporal direction. Comparing . . . . . . . . . . . . . . . . a m ( am π ) glueball ++ gluino-glue . . . .
16 0 0 . . . . . a/w ) △ ( w m S Z − S ) Figure 3:
Recent results from lattice N = 1 SYM calculations using gauge group SU(3) and Wilson-cloverfermions.
Left: ++ ‘glueball’ and fermionic ‘gluino–glue’ particle masses vs. the ‘adjoint pion’ masssquared, from Ref. [108]. The m π → extrapolations of these masses agree within uncertainties even ata fixed lattice spacing, supporting the formation of supermultiplets expected in the chiral continuum limit. Right:
A measure of supersymmetry-breaking discretization artifacts (defined in the text) is consistent withvanishing ∝ a in the a → continuum limit, from Ref. [109]. rogress and prospects of lattice supersymmetry David Schaich thermal and periodic boundary conditions (BCs) for the gauginos, they find evidence that periodicBCs allow the confined, chirally broken phase to persist for weak couplings where analytic semi-classical methods [118] may be reliable. In addition, there is ongoing work to construct a SYMgradient flow that is consistent with supersymmetry in Wess–Zumino gauge [119], which could beused to define a renormalized supercurrent and help guide fine-tuning [120, 121]. The ordinarynon-supersymmetric gradient flow is already used by many lattice N = 1 SYM projects, to setthe scale (as in the right plot of Fig. 3) and improve signals for observables such as the gauginocondensate [117]. Finally, given the progress in algorithms and computing hardware over the pastdecade, it seems worthwhile to revisit calculations with Ginsparg–Wilson fermions, which couldcomplement and check the ongoing Wilson-fermion work.
3. Maximally supersymmetric Yang–Mills ( N = 4 SYM) in four dimensions
In Sec. 1 we discussed why the twisted (and orbifolded) constructions of SYM with exactsupersymmetry at non-zero lattice spacing require Q ≥ d supercharges. In d = 4 dimensions, thisconstraint picks out another special case, N = 4 SYM with Q = 16 , for which a single ‘twisted-scalar’ supercharge Q is preserved. This theory consists of a SU( N ) gauge field, four Majoranafermions and six real scalars, all massless and transforming in the adjoint representation of SU( N )as usual. Thanks to its many supersymmetries, large SU(4) R symmetry and conformal symmetry, N = 4 SYM is widely studied throughout theoretical physics (especially in its large- N planarlimit). Among many other important roles, it is the conformal field theory of the original AdS/CFTholographic duality [122], and provided early insight into S-duality [123]. Lattice field theory inprinciple enables non-perturbative investigations of this theory even away from the planar regime.On the lattice, the bosonic fields are combined into five-component complexified gauge links (cid:8) U , U (cid:9) , implying the A ∗ lattice structure of five basis vectors symmetrically spanning four di-mensions [2, 58 – 61]. A single fine-tuning of a marginal operator may be required to recover thecontinuum twisted rotation symmetry from the S point-group symmetry of the A ∗ lattice, whichin turn restores the 15 supersymmetries broken by the lattice discretization [15, 124, 125]. Mostnumerical calculations so far fix the corresponding coefficient to its classical value. These calcu-lations also have to regulate flat directions in both the SU( N ) and U(1) sectors. A simple (soft Q -breaking) scalar potential suffices to lift the SU( N ) flat directions, and is removed in contin-uum extrapolations. The U(1) sector is more challenging, and ongoing work is searching for goodways to handle it [64 – 66]. The results shown in Fig. 4 lift the U(1) flat directions by modifyingthe moduli equations in a way that preserves the Q supersymmetry. At least for ’t Hooft cou-plings λ lat ≤ this results in effective O ( a ) improvement indicated by Q Ward identity violationsvanishing ∝ a in the continuum limit [64]. The resulting lattice action is rather complicated, moti-vating the public development of high-performance parallel code [63] for lattice N = 4 SYM andlower-dimensional SYM theories at github.com/daschaich/susy .Figure 4 presents some preliminary results from ongoing lattice N = 4 SYM calculations.The left plot considers the static potential V ( r ) , which is found to be coulombic at all accessible’t Hooft couplings [62, 126, 127], as expected. Fitting (tree-level-improved [127]) lattice data to theCoulomb potential V ( r ) = A − C/r predicts the Coulomb coefficient C ( λ ) shown in the figure.There is a famous holographic prediction [128, 129] that in the regime N → ∞ and λ → ∞ rogress and prospects of lattice supersymmetry David Schaich with λ (cid:28) N this quantity should behave as C ( λ ) ∝ √ λ up to O (cid:16) √ λ (cid:17) corrections, and moregeneral analytic results have been obtained in the N = ∞ planar limit [130]. The lattice resultsfor N ≤ and λ lat ≤ do not show such behavior and instead look consistent with leading-orderperturbation theory. The dashed black line is a fit of the U(4) data to the leading perturbativeexpression C ( λ ) = bλ lat / (4 π ) , where the fit parameter b = 0 . converts the input lattice’t Hooft coupling to the expected continuum normalization. Higher-order perturbative correctionsfor C ( λ ) are suppressed by powers of λ π [131 – 133], suggesting that this apparent leading-orderbehavior for λ lat ≤ should not be surprising.The right plot of Fig. 4 considers the scaling dimension ∆ K ( λ ) = 2 + γ K ( λ ) of the sim-plest conformal primary operator of N = 4 SYM, the Konishi operator O K = (cid:80) I Tr (cid:2) X I X I (cid:3) ,where X I are the scalar fields (obtained from a polar decomposition of the complexified latticegauge links). There are again both perturbative [134 – 136] and holographic [137, 138] predictionsfor ∆ K . The former are also relevant for the strong-coupling regime λ (cid:29) N [139], due to theconjectured S-duality of the theory, which relates its spectrum of anomalous dimensions underthe interchange πNλ ←→ λ πN . In addition, the superconformal bootstrap program has obtainedbounds on the maximum value γ K can reach across all λ [140, 141]. The lattice results in thisfigure for λ lat (cid:46) again appear consistent with perturbation theory. They are obtained from MonteCarlo renormalization group (MCRG) stability matrix analyses [142], with systematic uncertaintiesestimated by varying the number of interpolating operators in the stability matrix (with different op-erators obtained by using different amounts of smearing). Additional systematic uncertainties stillto be quantified include sensitivity to the lattice volume and the number of RG blocking steps. Thestability matrix also includes the related ‘SUGRA’ or (cid:48) operator O IJS = Tr (cid:2) X { I X J } (cid:3) , whosescaling dimension is fixed to its protected value ∆ S = 2 .Existing numerical calculations only scratch the surface of the investigations that could in prin-ciple be pursued by lattice N = 4 SYM. One important task is to push existing studies like those inFig. 4 to stronger ’t Hooft couplings, in order to make contact with holographic predictions and ide-ally investigate the behavior of the system around the S-dual point λ sd = 4 πN . The discussion of Figure 4:
Preliminary results from ongoing four-dimensional lattice N = 4 SYM calculations with gaugegroups U(2), U(3) and U(4).
Left:
The static potential Coulomb coefficient, from L × N t lattices with L ≤ and N t ≤ , appears consistent with leading-order perturbation theory (black dashed line) for λ lat ≤ . Right:
The Konishi scaling dimension, from MCRG stability matrix analyses of L lattices with L ≤ , also appears consistent with perturbation theory (and well below bootstrap bounds) for λ lat (cid:46) . rogress and prospects of lattice supersymmetry David Schaich sign problems in the next section suggests that this is likely to be challenging. An alternative possi-bility is to study S-duality at currently accessible couplings by adjusting the scalar potential so thatthe system moves onto the Coulomb branch of the moduli space where its U( N ) gauge invarianceis higgsed to U(1) N . In this context S-duality relates the masses of the U(1)-charged elementary‘ W bosons’ and the magnetically charged topological ’t Hooft–Polyakov monopoles [123], eachof which may be accessible from lattice calculations with either C-periodic or twisted BCs [143].The finite-temperature behavior of lattice N = 4 SYM will also be interesting to explore. In par-ticular, there is motivation [7] to study the free energy, for which the weak-coupling perturbativeprediction [144] and the holographic strong-coupling calculation [137] differ by a factor of .
4. Challenges for the future
Although the recent progress of lattice supersymmetry is substantial, it is largely concentratedin the three areas discussed above where significant simplifications are possible. Within thosethree areas we have already considered several compelling directions for future work, ranging fromimproved control over large- N continuum extrapolations in lower dimensions, to revisiting N = 1 SYM with Ginsparg–Wilson fermions, and reaching stronger ’t Hooft couplings in N = 4 SYMcalculations. In addition, it will be important for efforts to expand beyond these domains and tacklemore challenging subjects where such simplifications do not appear to be available. We concludethis brief review by touching on some of these subjects, highlighting SQCD and the possibility ofsign problems in supersymmetric lattice systems.
Supersymmetric QCD:
Adding matter multiplets (‘quarks’ and ‘squarks’ not necessarily inthe fundamental representation) to the four-dimensional lattice N = 1 SYM work discussed inSec. 2 would enable investigations of many important phenomena, including (metastable) dy-namical supersymmetry breaking, conjectured electric–magnetic dualities and RG flows to knownconformal IR fixed points. The downside is that many more supersymmetry-violating operatorsappear, and the fine-tuning challenge becomes enormously harder. Even exploiting the continuum-like flavor symmetries offered by Ginsparg–Wilson fermions, Ref. [3] counts O (10) operators tobe fine-tuned, depending on the gauge group and matter content. In this context working withGinsparg–Wilson fermions appears to be especially strongly motivated, with Refs. [3, 14] arguingthat this may allow most or all of the scalar masses, Yukawas and quartic couplings to be fine-tuned“offline” through multicanonical reweighting, which could vastly reduce computational costs.That said, as in the case of N = 1 SYM, work currently underway uses Wilson fermions andhas to face the full fine-tuning head-on. One tactic for approaching this challenge is to use latticeperturbation theory to guide numerical calculations [145 – 147]. Another is to omit the scalar fieldsat first, and warm up by studying the gauge–fermion theory including both (adjoint) gauginos and(fundamental) quarks [148], which also provides connections to composite Higgs investigationsthat are reviewed by another contribution to these proceedings [149]. These four-dimensionalefforts are just getting underway.Following the logic of Sec. 1, it may prove advantageous to first investigate simpler sys-tems in fewer than four dimensions. In 0+1 dimensions, for example, Refs. [150 – 152] considerthe Berkooz–Douglas matrix model [153], which adds N f fundamental multiplets to Q = 16 rogress and prospects of lattice supersymmetry David Schaich
SYM QM (preserving half of the supercharges in the continuum). As for the case of two- andthree-dimensional SYM, more effort has been devoted to constructing clever lattice formulationsof d = 2 and d = 3 SQCD [154 – 160] compared to carrying out numerical calculations [161].That one numerical calculation [161] uses a generalization of the twisted formulation to realizea quiver construction of two-dimensional Q = 4 SQCD that still preserves one of the superchargesat non-zero lattice spacing [154, 155]. The starting point is three-dimensional 8-supercharge SYMon a lattice with only two slices in the third direction. The twisted formulation can be general-ized to have different gauge groups U( N ) and U( F ) on each slice, with the bosonic and fermionicfields that connect the two slices transforming in the bifundamental representation of U( N ) × U( F ).Decoupling the U( F ) slice then leaves behind a two-dimensional U( N ) theory with half the su-percharges ( Q = 4 ) and F massless fundamental matter multiplets. This same procedure worksfor Q = 8 SQCD in two and three dimensions [158, 159], and may be generalizable to higherrepresentations [160]. Ref. [161] compares U(2) SQCD with F = 3 vs. U(3) SQCD with F = 2 ,observing dynamical supersymmetry breaking for N > F and confirming that the resulting gold-stino is consistent with masslessness in the infinite-volume limit. Sign problems:
Another challenge is that some of the supersymmetric lattice systems discussedabove may suffer from a sign problem, at least in certain regimes. Since the gauginos are Majoranafermions, integrating over them produces the pfaffian of the fermion operator, which can fluctu-ate in sign even when the determinant would be positive. Writing a generic complex pfaffian aspf D = | pf D| e iα , only its magnitude is included in the ‘phase-quenched’ RHMC studies presentedabove. The phase-quenched observables (cid:104)O(cid:105) pq need to be reweighted, (cid:104)O(cid:105) = (cid:104)O(cid:105) pq / (cid:10) e iα (cid:11) pq ,with a sign problem appearing when (cid:10) e iα (cid:11) pq = Z/Z pq vanishes within statistical uncertainties.(See Ref. [167] for a brief introduction to sign problems.) In particular, in lattice calculations withperiodic BCs for all fields, the partition function Z is the Witten index and must vanish for any the-ory that can exhibit spontaneous supersymmetry breaking [168], implying a severe sign problem.For Wilson-fermion N = 1 SYM the pfaffian is real and its sign can be computed effi-ciently [169]. Recent clover calculations report (cid:10) e iα (cid:11) pq ≈ , with the situation improving furtheras the lattice spacing decreases [108]. However, (cid:10) e iα (cid:11) pq is expected to decrease exponentially inthe lattice volume, and the situation is likely to be worse for SQCD. Directly evaluating the pfaffianis much more computationally expensive, and has been done mostly for SYM QM and d = 2 SYM,where sign problems also appear to be well under control [36, 38, 80, 81, 86, 95, 97].Figure 5 presents results for the pfaffian phase of lattice N = 4 SYM in four dimensions,adapted from Refs. [125, 170], where only small N and small lattice volumes are computationallyaccessible. (Each pfaffian measurement for a single lattice with N = 2 takes approximately50 hours on 16 cores, and costs scale with the cube of the number of fermion degrees of free-dom [63].) In the left plot, only small per-mille-level phase fluctuations are observed on all ac-cessible volumes with fixed ’t Hooft coupling λ lat = 0 . . In particular, the expected exponentialsuppression of (cid:10) e iα (cid:11) pq with the lattice volume is not visible; instead the largest volumes for gaugegroup U(2) produce results that are constant within uncertainties. In the right plot, however, we seephase fluctuations increasing significantly for stronger ’t Hooft couplings λ lat (cid:38) . This appears to Lower-dimensional Wess–Zumino models could also be a useful setting for future lattice studies of dynamicalsupersymmetry breaking and goldstinos [162 – 166]. rogress and prospects of lattice supersymmetry David Schaich be one of the main obstacles to reaching the stronger couplings of interest in order to directly probeholography and S-duality, with calculations using this lattice action largely limited to λ lat (cid:46) . Final remarks:
Non-perturbative lattice investigations of supersymmetric QFTs are importantand challenging, making this a field in which we can expect to see a great deal more work inthe future. It is encouraging that there has been so much recent progress in lattice studies offour-dimensional N = 1 SYM and N = 4 SYM, along with their dimensional reductions to d < . This brief overview has also omitted coverage of advances in other areas, including theorieswithout gauge invariance such as Wess–Zumino models and sigma models [162 – 166, 171 – 174],the lattice regularization of the Green–Schwarz superstring worldsheet sigma model [175 – 177],and proposals for lattice formulations of a mass-deformed N = 2 ∗ SYM theory with Q = 8 in four dimensions [178] and of Q = 16 SYM in five dimensions [179]. While there are clearchallenges that will be difficult to overcome, in particular concerning supersymmetric QCD andsign problems, overall the prospects of lattice supersymmetry are bright, with many compellingdirections for future investigations.A
CKNOWLEDGMENTS : I thank the organizers of Lattice 2018 for the invitation to present thisoverview, and for all their work to manage the conference. My participation in the conference wassupported by a travel grant from the Faculty of Science at the University of Bern. Enrico Rinaldi,Jun Nishimura, Hiroto So, Marios Costa, Georg Bergner and Björn Wellegehausen provided helpfulinformation about their recent work. I have benefited from collaboration on lattice supersymmetrywith Simon Catterall, Poul Damgaard, Tom DeGrand, Joel Giedt, Raghav Jha, Anosh Joseph andToby Wiseman.
Figure 5:
Results for the phase of the pfaffian (cid:10) Re (cid:0) e iα (cid:1)(cid:11) pq ≈ (cid:10) e iα (cid:11) pq from lattice N = 4 SYM in fourdimensions.
Left:
With fixed ’t Hooft coupling λ lat = 0 . , only per-mille-level fluctuations are observed forU( N ) gauge groups with N = 2 , 3 and 4, up to the largest accessible volumes. Adapted from Ref. [125]. Right:
On a fixed lattice volume, the phase fluctuations increase significantly for stronger couplings λ lat (cid:38) , obstructing studies of λ lat (cid:38) with this lattice action. Adapted from Ref. [170]. rogress and prospects of lattice supersymmetry David Schaich
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