Lattice N=4 super Yang-Mills at Strong Coupling
PPrepared for submission to JHEP preprint
Lattice N = 4 super Yang-Mills at Strong Coupling Simon Catterall, a Joel Giedt, b Goksu Can Toga a a Department of Physics, Syracuse University, Syracuse, NY 13244, USA b Department of Physics and Astronomy, RPI, Troy, NY 12180, USA
Abstract:
In this paper we present results from numerical simulations of N = 4 superYang-Mills for two color gauge theory over a wide range of ’t Hooft coupling 0 < λ ≤ SU (2) rather than U (2) gaugesymmetry. By explicit computations of the fermion Pfaffian we present evidence that thetheory possesses no sign problem and exists in a single phase out to arbitrarily strongcoupling. Furthermore, preliminary work shows that the logarithm of the supersymmetricWilson loop varies as the square root of the ’t Hooft coupling λ for large λ in agreementwith holographic predictions. a r X i v : . [ h e p - l a t ] N ov ontents In this paper we use numerical simulation to explore the phase structure and Wilson loopsof a lattice formulation of N = 4 super Yang-Mills. The lattice action is a generalizationof the formulation described in [1]. The theory preserves both SU ( N ) gauge invariance, a S point group symmetry associated with the underlying A ∗ lattice and most importantlya single exact supersymmetry.The original supersymmetric lattice formulation of N = 4 SYM has been the subjectof a great deal of both numerical and analytical work [2–5]. General arguments have beenput forward that the theory should approach the continuum N = 4 theory after tuning asingle marginal operator. However, after some initial successes the numerical work has beenhandicapped by two problems: the existence of a chirally broken phase for ’t Hooft couplings λ > U (1) sector of the theory [16].In this paper we show that the situation is markedly improved if one adds a newoperator to the lattice action which preserves the S symmetry and exact supersymmetrybut explicitly breaks the U ( N ) gauge symmetry down to SU ( N ).– 1 – Review of the old supersymmetric construction
We start from the supersymmetric lattice action appearing in [1]. S = N λ Q (cid:88) x Tr (cid:18) χ ab F ab + η D a U a + 12 ηd (cid:19) + S closed (2.1)where the lattice field strength F ab ( x ) = U a ( x ) U b ( x + ˆ a ) − U b ( x ) U a ( x + ˆ b ) (2.2)where U a ( x ) denotes the complexified gauge field living on the lattice link running from x → x + ˆ a where ˆ a denotes one of the five basis vectors of the underlying A ∗ lattice.Similarly D a U a = U a ( x ) U a ( x ) − U a ( x − ˆ a ) U a ( x − ˆ a ) . (2.3)The five fermion fields ψ a , being superpartners of the (complex) gauge fields, live on thecorresponding links, while the ten fermion fields χ ab ( x ) are associated with new face linksrunning from x + ˆ a + ˆ b → x . The scalar fermion η ( x ) lives on the lattice site x and isassociated with the conserved supercharge Q which acts on the fields in the following way Q U a → ψ a Q ψ a → Q η → d Q d → Q χ ab → F ab Q U a → Q = 0 which guarantees the supersymmetric invariance of the first part ofthe lattice action. The auxiliary site field d ( x ) is needed for nilpotency of Q offshell. Thesecond term S closed is given by S closed = − N λ (cid:88) x Tr (cid:15) abcde χ ab D c χ de (2.5)where the covariant difference operator acting on the fermion field χ de takes the form D c χ de ( x ) = U c ( x − ˆ c ) χ de ( x + ˆ a + ˆ b ) − χ de ( x − ˆ d − ˆ e ) U c ( x + ˆ a + ˆ b ) (2.6)The latter term can be shown to be supersymmetric via an exact lattice Bianchi identity (cid:15) abcde D c χ de = 0. Carrying out the Q variation and integrating out the auxiliary field d weobtain the supersymmetric lattice action S = S b + S f where S b = N λ (cid:88) x Tr (cid:18) F ab F ab + 12 Tr ( D a U a ) (cid:19) (2.7)and S f = − N λ (cid:88) x (cid:0) Tr χ ab D [ a ψ b ] + Tr η D a ψ a (cid:1) (2.8)– 2 –n the continuum this action can be obtained by discretization of the Marcus or GL twistof N = 4 Yang-Mills but in flat space is completely equivalent to it. In the continuum thetwist is done as a prelude to the construction of a topological quantum field theory but inthe context of lattice supersymmetry it is merely used as a change of variables that allowsfor discretization while preserving a single exact supersymmetry. The twisting removes thespinors from the theory replacing them by the antisymmetric tensor fields η, ψ a , χ ab whichappears as components of a K¨ahler-Dirac field. The latter is equivalent at zero couplingto a (reduced) staggered field and hence describes four physical Majorana fermions in thecontinuum limit - as required for N = 4 Yang-Mills. The twisting procedure also packsthe six scalar fields of the continuum theory together with the four gauge fields into fivecomplex gauge fields corresponding to the lattice fields U a .As described above, the discrete theory is defined on a somewhat exotic lattice - A ∗ .This admits a larger set of rotational symmetries than a hypercubic lattice and this factplays a role in controlling the renormalization of the theory. Finally, to retain exact su-persymmetry all fields reside in the algebra of the gauge group – taking their values in theadjoint representation of U ( N ): f ( x ) = (cid:80) N A =1 T A f A ( x ) with Tr ( T A T B ) = − δ AB .Ordinarily this would be incompatible with lattice gauge invariance because the mea-sure would not be gauge invariant for link based fields. However, in this N = 4 constructionthe problem is evaded since the fields are complexified which ensures that the Jacobiansthat arise after gauge transformation of U and U cancel. However this restriction to the algebra does pose a further problem. Ordinarily thenaive continuum limit is obtained by expanding the group elements about the identity U a ( x ) = I + aA a ( x ) + . . . . The presence of the unit matrix in this expansion is whatgives rise to hopping terms in the lattice theory and derivative operators in the continuumlimit. If the gauge fields live in the group the unit matrix arises naturally on expandingthe exponential but with fields valued in the algebra it is less clear how such an expansionarises. The saving grace is to notice that the gauge fields take their values in GL ( N, C ) sothat this term can arise by giving a vacuum expectation value to the imaginary part of thetrace mode of the field. Typically this is accomplished by adding to the supersymmetricaction a new term of the form S mass = µ (cid:88) x Tr (cid:0) U a ( x ) U a ( x ) − I (cid:1) (2.9)While this breaks the exact supersymmetry softly all counter terms induced by this breakingwill have couplings that are multiplicative in µ and hence vanishing as µ →
0. Noticealso that this term also generates masses for the scalar fields in the theory and hence alsoregulates the usual flat directions of SYM theory. Actually one should qualify this statement. While the complexified bosonic measure is invariant underlattice U ( N ) gauge transformations it is more subtle to show that the fermion measure is invariant whenthe fermions reside on links. We shall show that this issue is completely evaded in the theory with SU ( N )gauge invariance – 3 – The new action
It has been observed that for couplings λ > U (1) monopoles [5]. These features are inconsistent with the expected supercon-formal phase of N = 4 Yang-Mills. Actually, in pure compact QED in four dimensions,this monopole transition is a well known lattice artifact. Various efforts have been madeover the intervening years to remove this monopole phase - typically this has been done byadding supersymmetric or non-supersymmetric terms to the action that force the deter-minant of the plaquette operator to unity. Such a procedure retains the full U ( N ) gaugesymmetry but restricts the fluctuations of the field strength in the U (1) directions. Thesupersymmetric plaquette term introduced in [17] represents the best of these approachesbut can only allow simulation up to λ ∼ .
0. It also suffers from a sign problem for λ > N λ κ Q (cid:88) x,a Tr ( η ) (Re det ( U a ( x ) − Q variation and integration over d this modifies the second term in the bosonic action S b to: N λ (cid:88) x,a
12 Tr (cid:0) D a U a ( x ) + κ Re det( U a ( x )) I n (cid:1) (3.2)where I N denotes the N × N unit matrix. A corresponding new fermion term is generated δS f = − N λ κ (cid:88) x,a Tr ( η )det ( U a ( x ))Tr ( U − a ( x ) ψ a ( x )) (3.3)The new term has the effect of suppressing the U (1) phase fluctuations of the complexgauge links that were the origin of the monopole problem. Of course this term explicitlybreaks the U (1) gauge symmetry. However since the U (1) is simply a decoupled free theoryin the continuum limit this should cause no real harm since SU ( N ) gauge invariance ispreserved. Indeed, close to the continuum limit, it should be apparent that the new termsmerely generate mass terms for the trace components of the fields.In the original theory the gauge links were valued in GL ( N, C ). After this term is addedthe moduli space of the theory is reduced to SL ( N, C ). Notice that since any matrix in SL ( N, C ) can be written as the exponential of a traceless matrix the presence of this term guarantees that gauge links can be expanded about the unit matrix for vanishing values ofthe lattice spacing. In this light the remaining rationale for keeping S mass is simply to liftthe usual SU ( N ) flat directions. Indeed, as the reader will see, for most of our results µ is taken very small. – 4 – D E T λ L=8 , κ =1µ=0.1µ=0.05µ=0.01 Figure 1 . Expectation value of the determinant vs λ for 8 lattices at µ = 0 . , . , . The breaking of U (1) gauge invariance also clarifies a delicate issue concerning theinvariance of the fermion measure in the original formulation. Consider the integrationmeasure for the five link fermions (cid:81) x,a dψ a ( x ) in the U ( N ) theory. Under a gauge trans-formation ψ a ( x ) → G ( x ) ψ a ( x ) G † ( x + ˆ a ) this measure transforms by a non-trivial Jacobiancorresponding to the product of the determinants of the gauge factors G ( x ) and G † ( x + ˆ a ).On the torus one can arrange an ordering of the fermion fields in the path integral measuresuch that these factors will cancel out along closed loops but this will not be possible for alllattice topologies. Thus the question of the invariance of the measure under the full U ( N )group is a delicate one. However these problems are completely avoided if G is restrictedto lie in just SU ( N ) as in the new action and the fermion measure is then unambiguouslydefined for an arbitrary lattice.Of course the main question is whether such a term is effective at eliminating themonopole phase seen at strong coupling. In the next section we shall show evidence thatthis is true and at least in the case of 2 colors we see no sign of phase transitions out toarbitrarily large ’t Hooft coupling. Our simulations utilize the rational hybrid Monte Carlo (HMC) algorithm where the Pfaf-fian resulting from the fermion integration is replaced byPf ( M ) = (cid:16) det M † M (cid:17) (4.1)where M is the fermion operator. Notice that this representation neglects any Pfaffianphase which is a key issue which we will return to later. Typical ensembles used in ouranalysis consist of 5000 HMC trajectories with 1000 − −
40 bins.– 5 – B o s o n i c A c t i o n λ L=8 , κ =1 µ=0.1µ=0.05µ=0.01 Figure 2 . Expectation value of the bosonic action vs λ for 8 lattices at µ = 0 . , . , . As a test of the new action we first plot the expectation value of the link determinant asa function of ’t Hooft coupling. We show results in fig 1 for 8 lattices at µ = 0 . , . , . λ provided µ is smallenough confirming that we have effectively reduced the gauge fields to SU (2). We notethat we scan out to λ = 30 in order to go beyond the self-dual point λ SD = 4 πN = 8 π .In fig. 2 we plot the expectation value of the bosonic action as a function of λ for 8 lattices at µ = 0 . , . , .
01. This expectation value can be calculated exactly by exploitingthe (almost) Q -exact nature of the lattice action and yields V < S b > = N for an N colortheory on a system with (lattice) volume V independent of coupling λ . For SU (2) thisimplies S B = 18 . λ . The results are clearly consistent with this prediction to afraction of a percent as µ → U (2) theory. Indeed all the observables wehave looked at show smooth dependence on λ providing evidence that the lattice theorypossesses only a single phase out to arbitrarily arbitrarily strong coupling. It is interestingto note that the bosonic action is proportional to N and not N − U (1) modes. That is because they are still present in this formulation; ratherthan being removed, they are being tamed. The new terms added to the action mostlyaffect the vacuum of these fields—which is why they still contribute to the counting ofdegrees of freedom.Further confidence in this finding comes from studying a simple bilinear Ward identitygiven by (cid:10) Q Tr ( η U a U a ) (cid:11) = 0. Fig. 3 shows this quantity as a function of λ for several µ at L = 8. It falls slowly with λ and decreases more quickly with decreasing µ . To clarify itsdependence on lattice size we plot the Ward identity for λ = 10 . L for two values of µ in Fig. 4. This plot makes it clear that the Ward identity decreases with increasing L .Indeed, comparing L = 6 with L = 12 at µ = 0 .
005 the change is consistent with a 1 /L – 6 –ependence on lattice size. W a r d I d e n t i t y λ L=8 , κ =1µ=0.1µ=0.05µ=0.01 Figure 3 . Bilinear Q -susy ward identity vs λ for 8 lattices for µ = 0 . , . , . W I L κ =1 µ=0.025µ=0.005 Figure 4 . Bilinear Q -susy ward identity vs L at λ = 10 . µ = 0 . , . Of course these results are derived from simulations of a model in which the phase of thePfaffian that results from fermion integration is neglected. To check for the presence ofsuch a phase we have computed it using the ensemble of configurations generated in ourphase quenched Monte Carlo. Writing the Pfaffian phase as e iα ( λ,U ) we plot the quantity1 − cos α as a function of µ at λ = 10 . κ = 1 . , 3 × , 3 × × respectively. When measuring– 7 – .00000010.00000100.00001000.00010000.00100000.01000000.10000001.0000000 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 - c o s α µ λ =10, κ =1 L=2 ,L=3 x4L=3 x4L=3x4 Figure 5 . Pfaffian phase vs µ at λ = 10 . the phase of the Pfaffian we set κ = 0 in the fermion operator. Clearly the phase angle isdriven towards very small values for small enough µ . We have observed this for all valuesof λ – the analogous plot fig. 11 for λ = 30 is shown in the appendix. Of course thelattices used in these tests are quite small and one should worry whether the sign problemreturns on larger volumes. Our results suggest that this is not the case – the average phaseappears to saturate as the volume increases. Systems with sign problems typically exhibitphase fluctuations that increase exponentially with volume. This lattice model seems verydifferent in this regard. - c o s α µ λ =10 L=2 ,L=3 x4L=3 x4 Figure 6 . Pfaffian phase vs µ at λ = 10 .
0. Ensembles were generated without including the newfermionic term
Retaining the new U (1) breaking fermion term in the evolution but neglecting it whenmeasuring the phase is clearly a questionable procedure. However, the modification that– 8 –s neglected relates to the trace modes, which decouple from the SU ( N ) theory in thecontinuum in any case. So, in some sense we are discarding an irrelevant piece. Neverthe-less, we have also generated ensembles in which the new fermion term is dropped from thefermion action in both the evolution and the measurement of the phase. A typical plot ofthe resultant phase versus for µ at λ = 10 . Q -symmetry softly (proportional to κ ) it leads to larger deviations in the Ward identitiesand so we have reinstated the new fermion term in our later simulations used for study-ing Wilson loops. The fact that eliminating the new fermion term from both the Pfaffianmeasurement and the simulation still preserves the good behavior can be understood asthe new bosonic term accomplishing the most important task: stabilizing and suppressingthe U (1) modes of the link fields in a Q -symmetric way that is only softly broken.It is interesting to try and understand theoretically why the observed phase fluctuationsare so small. We start by writing the expectation value of the phase measured in the phasequenched ensemble as < e iα ( λ,κ,U ) > phase quenched = (cid:90) D U D U e iα ( κ,λ, U ) | Pf( U ) | e − S B ( λ,κ, U ) = 1 (5.1)where we have chosen the normalization of the measure so that the full partition withsusy preserving periodic boundary conditions (the Witten index) is unity. Furthermore, Q -invariance ensures that this expectation value of the phase factor is independent of κ andcan be computed for κ → ∞ where the partition function is saturated by configurationswith unit determinant - the SU (2) theory. Finally, the topological character of this partitionfunction can be exploited to localize the integral to configurations which are constant overthe lattice – the integral reducing to a Yang-Mills matrix model integral. The resultantPfaffian for the SU (2) matrix model is known to be real, positive definite [18]. Of courseour simulations are performed at finite κ , and use a thermal boundary condition, but thenumerical results strongly suggest that as a practical matter the phase fluctuations aresmall for the relevant range of parameters.The encouraging results for the phase of the Pfaffian may also be related to the factthat out to very large λ the center symmetry is unbroken, so that Eguchi-Kawai reduction[19] may be valid. In that case the theory is equivalent to a single-site lattice, wherethe gauge theory is in fact just the matrix model that has been indicated in the previousparagraph. This may also explain why we are able to obtain results consistent with large N predictions (below), since the fact that we are in volumes larger than a single site mayin fact translate into larger N in the reduced model. The previous results provide strong evidence that the lattice theory exists in a single phasewith unbroken supersymmetry out to very large values of the gauge coupling and that themodel can be simulated with a Monte Carlo algorithm without encountering a sign problem.– 9 – l o g ( W il s o n L oo p s ) Sqrt( λ )L=12 , µ=0.025, κ =16x65x54x43x32x21x1 Figure 7 . Supersymmetric n × n Wilson loops on 12 lattice at µ = 0 . With this in hand we turn to whether the lattice simulations can provide confirmation ofknown results for N = 4 Yang-Mills at strong coupling. Most of these analytic resultswere obtained by exploiting the AdS/CFT correspondence which allows strong couplingresults in the gauge theory to be obtained by solving a classical gravity problem in anti-de Sitter space. Using this duality a variety of results for supersymmetric Wilson loopshave been obtained over the last twenty years. Such Wilson loops generalize the usualWilson loops by including contributions from the scalars and are realized in the twistedconstruction by forming path ordered products of the complexified lattice gauge fields U a .In the continuum the generic feature of such Wilson loops is that for strong coupling theydepend not on λ as one would expect from perturbation theory but instead vary like √ λ .In fig. 7 we show the logarithm of the n × n supersymmetric Wilson loops W ( n, n ) for a -0.0016-0.0014-0.0012-0.001-0.0008-0.0006-0.0004-0.0002 0 0 5 10 15 20 25 30 l o g ( χ ( , )) λ L=12 , µ=0.025, κ =1 Figure 8 . Estimated string tension vs λ for 12 lattice at µ = 0 .
025 using χ (6 , – 10 –2 lattice at κ = 1 . √ λ . The straight lines correspond to fitswith √ λ ≥
3. It is clear that all the loops show a √ λ dependence at strong coupling inagreement with the holographic prediction. This is encouraging. It is also clear that thefits show a linear dependence on the length of the perimeter of the loop. If we parametrizethe static potential defined by W ( R, T ) = e − V ( R ) T in the form V ( R ) = σ ( λ ) R + α ( λ ) /R + M ( λ ) (6.1)The presence of the constant term M ( λ ) will yield the observed perimeter scaling pro-vided the string tension is small or zero. Such a perimeter term also occurs in continuumtreatments where it corresponds to the energy of a static probe source in the fundamentalrepresentation and has to be explicitly subtracted out to see the non-abelian Coulombbehavior hidden in α ( λ ) [20].One way to remove the perimeter dependence is to consider Creutz ratios defined by χ ( R, T ) = W ( R, T ) W ( R − , T − W ( R, T − W ( R − , T ) (6.2)For a theory with Wilson loops containing both perimeter, area and Coulomb behaviorsone finds ln χ ( R, R ) ∼ − σ ( λ ) + α ( λ ) /R (6.3)Thus we can read of the string tension by examining the large R behavior of ln χ ( R, R ).In fig. 8 we plot ln χ (6 ,
6) = − σ versus λ for a 12 lattice at λ = 10 . µ = 0 . l o g ( W / P R / L ) Sqrt( λ )L=8 ,µ=0.025, κ =14x42x2 Figure 9 . Renormalized supersymmetric 4 × × lattice at µ = 0 . interesting question is whether we can see evidence for a non-abelian Coulomb potentialat small R . Direct fits to the Creutz ratio are consistent with the presence of such a termbut the errors in α ( λ ) are large. – 11 –n alternative way to probe for this is is to divide the original Wilson loops by anappropriate power of the measured Polyakov line P which is given by product of gaugelinks along a thermal cycle. The (logarithm of the) Polyakov line also picks up a termlinear in the length of the lattice due to a massive source and hence can used to subtractthe linear divergence in the rectangular Wilson loop. We thus define a renormalized Wilsonloop on a L lattice of the form W R ( R, R ) = W ( R, R ) P RL (6.4)These are shown in fig. 9 for a 8 lattice. Notice that the 2 × × √ λ . This result can also be seen on the larger 12 lattice shown in fig. 10. Notice thatthe average slope in this case is somewhat larger than the data on 8 . This presumablyreflects the residual breaking of conformal invariance due to finite volume as well as finitelattice spacing. However it may also indicate that our definition of a renormalized Wilsonloop does not do a perfect job of subtracting all the linear divergences needed to reveal anunderlying Coulombic term. Further work is needed on larger lattices to clarify this issue. l o g ( W / P R / L ) Sqrt( λ )L=12 , κ =1, µ=0.025 6x63x3 Figure 10 . Renormalized supersymmetric 6 × × lattice at µ = 0 . Details of the fits for the different Wilson loops and lattice are shown in tables 1,2.The square root behavior at large λ is consistent with the result for circular Wilsonloops in N = 4 SYM derived by Gross and Drukker [21] and Maldacena’s holographicargument [22]. There are also explicit calculations using holography for the rectangularWilson loop in [20]. The strange √ λ dependence cannot be seen in perturbation theoryand this (admittedly) very preliminary result is a very non-trivial test of the correctnessof the lattice approach in a non-perturbative regime.– 12 –oop Size a √ λ +b Reduced- χ × √ λ + 8.0(4) 8.112 × √ λ + 8.8(2) 2.25 Table 1 . Normalized Supersymmetric Wilson loop fits on 8 lattice at µ = 0 .
025 for f ( λ ) = a √ λ + b Loop Size a √ λ +b Reduced- χ × √ λ + 12.4(3) 6.583 × √ λ +12.94(9) 0.90 Table 2 . Normalized Supersymmetric Wilson loop fits on 12 lattice at µ = 0 .
025 for f ( λ ) = a √ λ + b We have found that a supersymmetric modification of the lattice action enables us to extendour simulations to what seem to be arbitrarily large values of the ’t Hooft coupling withoutencountering difficulties that had previously limited our studies to modest λ . This seemsto be attributable to stabilizing the potential for the U (1) modes in a way that preservesthe essential Q supersymmetry of the construction. The current study has been limitedto gauge group SU (2). It is natural to inquire what occurs for this construction for other SU ( N ). We will investigate this in future studies; however, we expect that a sign problemwill reemerge since in the zero-dimensional matrix models for N >
Acknowledgments
This work was supported by the US Department of Energy (DOE), Office of Science,Office of High Energy Physics, under Award Numbers DE-SC0009998 (SC,GT) and DE-SC0013496 (JG). Numerical calculations were carried out on the DOE-funded USQCDfacilities at Fermilab. The authors would like to thank David Schaich for help with theparallel code used in this work. – 13 –
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