A Lattice Formulation of Super Yang-Mills Theories with Exact Supersymmetry

Fumihiko Sugino (Seoul National U.)

REVIEW

A Lattice Formulation of Super Yang-Mills Theories with Exact Supersymmetry

High Energy Physics - Lattice · Submitted 14 Nov 2003 · 21 references

Pith's verdict: A clean, honest construction of lattice SYM with exact supercharges; the d=2 no-fine-tuning claim is a perturbative argument, and the paper says so itself.

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arXiv IDhep-lat/0311021 (version 3)

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CitationFumihiko Sugino (Seoul National U.). "A Lattice Formulation of Super Yang-Mills Theories with Exact Supersymmetry." arXiv:hep-lat/0311021 (2003). https://pith.science/paper/CU6WCCMP

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Editor's note

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Worth your time if you work on lattice supersymmetry, and worth citing regardless. The construction is the real contribution: Q-exact (and Q+Q−-exact) lattice actions built from ordinary unitary link variables, with the supercharges nilpotent up to gauge transformations on the lattice. Getting Q² to close to a gauge transformation with compact links is the fiddly part and Sugino does it correctly — the transformation rules in (3.3) and (5.1) are consistent, and the check Q²ψ = [φ,ψ] is there. The doubler removal is also neat: the P_μ term in (3.16) acts like a Wilson term but the action keeps U(1)_R, evading Nielsen–Ninomiya by breaking the fermion-number U(1)_J instead, which the paper states plainly in (3.20). That is a genuinely nice piece of model building, and the TFT/BTFT framing gives it a clean organizing principle rather than a bag of tricks. The Q±-cohomology appendix is a real result, not decoration — it is what makes the p=3 operator analysis in §5.2 and §5.3 work, and it is derived properly from Dijkgraaf–Moore rather than asserted. On the stress-test note: I agree with it and disagree with the reader's framing. The ΔS power counting is internally consistent — each insertion costs ρ = a^s/L^s, so the induced tr φφ̄ goes as a^{ns} and decouples. That is not where the soft spot is. The soft spot is the one the paper names itself in §3.1: without ΔS the vacuum degeneracy is exponential in the number of plaquettes, and the paper says outright that it 'can not say anything on the continuum limit' of that model without nonperturbative study. ΔS is introduced to fix this, but its own nonperturbative effects — phase structure, possible first-order behaviour, whether the lifted vacua really decouple — are estimated, not shown. So the d=2 claim in abstract:s2 is a perturbative statement dressed as a continuum statement. That is a real gap, but it is an acknowledged one, and the paper is scoped honestly: d=4 leaves p=4 operator mixing open, d=3 explicitly needs counterterms, and §6 asks for the simulation that would settle it. Send it out. The construction will be used; the no-fine-tuning claim is a good target for a numerical follow-up.

Referee report

Major objections (2)

[Section 3.1, Section 3.3, Eq. (3.15), abstract:s2] The no-fine-tuning claim in abstract:s2 rests on the perturbative power-counting argument of Section 3.3, performed around the U_mu_nu = 1 vacuum after the degeneracy has been lifted by the Q-breaking term Delta S of Eq. (3.15). The paper itself states in Section 3.1 that without nonperturbative investigation 'we can not say anything on the continuum limit' of the degenerate model, and the entropy argument for rho = a^s/L^s (0<s<2) is a counting estimate rather than a demonstration that the Delta S deformation selects the correct vacuum nonperturbatively (e.g. that no first-order transition or competing phase intervenes at the chosen rho). The abstract's 'we show' therefore overstates what is established. I would ask the author either to qualify the abstract and the Section 3.3 conclusion explicitly as a perturbative renormalization argument, or to add a short paragraph in Section 3.3 stating what nonperturbative check (for instance a numerical study of the plaquette distribution at the chosen rho) would falsify the picture. This is load-bearing for the central claim, but the fix is local and the physics is sound.
[Section 3.1, Eq. (3.15)] The choice rho = a^s/L^s with 0<s<2 is justified by the statement that 'the entropy effect does not obstruct the decoupling', supported by counting the lifted vacua with Delta S = 4 rho/g_0^2 = 4/(g_2^2 L^s) a^{s-2}. It would strengthen the paper to spell out the two scales separately: Delta S itself scales as a^{s+2} tr F^2 and so vanishes in the classical continuum limit, while its coefficient rho/g_0^2 diverges as a^{s-2}. The manuscript should state explicitly that the divergence of the coefficient, not the vanishing of Delta S, is what lifts the vacua, and should note that this is a relevant deformation whose nonperturbative fate is not settled by the counting argument alone.

Minor objections (7)

[Section 2.2, Eq. (2.13)] The identification of the spinor components with (psi_mu, chi, eta/2) is correct, but the index mu on psi_mu is a label and not a Lorentz index; footnote 4 says this, and it would help to repeat the caveat at Eq. (2.13) so the reader does not misread the transformation rule (2.12).
[Section 3.1, Eq. (3.14)] The condition that '−1' appears an even number of times in the diagonal entries is the SU(N) determinant constraint; it should be stated explicitly as such, since for U(N) the constraint would be absent and the counting of vacua would differ.
[Section 3.2, Eq. (3.18)] The Dirac-spinor combination zeta, zeta-bar and the U(1)_J rotation (3.19) should be checked against the component assignment (2.13); a one-line verification of the relative signs and the 1/sqrt(2) normalization would make the doubler-removal argument easier to follow.
[Section 5.2, Eq. (5.9)] The paper notes that the five marginal operators in Eq. (5.9) can receive corrections to their relative weights, which is why d=4 is excluded from the no-fine-tuning claim. It would help to say this explicitly in Section 5.2, so that the d=4 discussion is not read as a competing claim to abstract:s2.
[Appendix B, Eq. (B.5)] The relation between the N=4 matrices omega_mu of Eq. (B.5) and the N=2 matrices nu_mu of Eq. (B.1) is not explained; a sentence connecting the two conventions would help the reader check the doubler-removal mechanism across the two cases.
[Appendix C] The Q±-cohomology argument invokes Theorem 2.1 of Dijkgraaf-Moore (ref. [14]) and then generalizes it to the lattice. Since the cited theorem is stated for continuum topological field theory, the manuscript should state explicitly which hypotheses carry over unchanged to the lattice setting and which are verified by the algebra (C.7).
[Section 6] The suggestion to impose the admissibility condition (6.1) is interesting, but the claim that 'the breaking effect would vanish in the weak coupling limit' should be quantified, for instance by estimating the measure of the boundary region ||1 - U_mu_nu|| = epsilon in the weak-coupling regime.

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The paper

The paper claims

Sugino shows that lattice super Yang-Mills theories can keep one or two supercharges exact, and that in two dimensions the continuum limit needs no fine tuning for $N=2,4,8$.

The reading

Putting supersymmetry on a lattice usually fails for two reasons: the lattice breaks continuous translations, and the Leibniz rule that makes products of fields transform cleanly breaks down. Sugino's route around both is to keep only the supercharges that behave like internal symmetries, the ones that survive a twist into a topological field theory form, and to make those exact on the lattice. The lattice actions are written as $Q$ of something (or $Q_+Q_-$ of something), so the exact supercharge is preserved by construction, with gauge fields as ordinary unitary link variables and the supercharge nilpotent up to a gauge transformation. The payoff claimed is a two-dimensional set of models ($N=2,4,8$) where gauge invariance, $R$ symmetry and the exact supercharge together forbid radiative corrections from generating any relevant or marginal operator except the identity, so the full supersymmetry and rotational invariance come back in the continuum limit without tuning. In three and four dimensions the same logic leaves marginal operators whose relative weights can shift, so fine tuning is not excluded there.

What carries the argument

The load-bearing device is the lattice supercharge that is nilpotent up to a gauge transformation, $Q^2 = $ (infinitesimal gauge transformation with parameter $varphi$), realized on link variables by $QU_mu(x) = i psi_mu(x) U_mu(x)$ and a correspondingly modified $Q psi_mu(x)$. Because the lattice action is written as $Q$ of a gauge-invariant expression (or $Q_+Q_-$ of one), its invariance is automatic, and the same charge constrains what loop corrections can generate. The second device is the power-counting argument: in two dimensions the coupling $g_2^2$ has mass dimension two, so corrections to an operator of dimension $p$ take the form $(a^{p-4}/g_2^2 + c_1 a^{p-2} + c_2 a^p g_2^2 + cdots) int d^2x, phi^a partial^b psi^{2c}$, which terminates at two loops and leaves only $p leq 2$ operators to check. The third device is the supersymmetry-breaking term $Delta S$ proportional to the standard Wilson action with coefficient $rho = a^s/L^s$, added to lift an exponentially large degeneracy of classical vacua in which each plaquette can sit at $pm 1$; its symmetry-breaking contributions carry extra powers of $a^s$ and are argued to be irrelevant in two dimensions.

Extended reading notes

Core claim

The paper constructs lattice versions of super Yang-Mills theories with extended supersymmetry in which one supercharge $Q$ (for $N=2$) or two supercharges $Q_+$, $Q_-$ (for $N=4$ and its dimensional reductions) are exact symmetries of the lattice action, not merely symmetries restored in the continuum limit. The actions are $Q$-exact, or $Q_+Q_-$-exact, mirroring the 'topological field theory' and 'balanced topological field theory' rewritings of the continuum theories, and the transformation rules close on the lattice: $Q^2$ equals an infinitesimal gauge transformation with parameter $varphi$. Fermion doublers are removed by a Wilson-like term built from matrices $P_mu$ that does not require breaking the $U(1)_R$ chiral symmetry, evading one assumption of the Nielsen-Ninomiya theorem by giving up a conserved fermion number. The central claim is that for $N=2,4,8$ in two dimensions the symmetries of the lattice action are enough to exclude all relevant and marginal operators other than the identity, so the desired continuum theory is reached with no fine tuning; in three dimensions the operators $Q_+Q_-tr((B_1+B_2+B_3)A_4)$ and the mass-like terms in (5.13) can be generated, and in four dimensions the five marginal structures in (5.9) can be reweighted, so tuning is not excluded there.

Load-bearing premise

The whole two-dimensional no-fine-tuning claim rests on the added supersymmetry-breaking term being negligible: it assumes that the extra powers of the lattice spacing attached to every insertion of that term really do make its symmetry-breaking effects vanish in the continuum limit, and the paper itself notes that without a nonperturbative study of the degenerate vacuum sum it cannot say anything about that limit.

Editorial extensions

If this is right

Lattice formulations of two-dimensional $N=2$, $N=4$ and $N=8$ super Yang-Mills can be taken to the continuum limit by tuning only the coupling, with no counterterms for supersymmetry restoration.
The exact supercharge provides a non-renormalization handle: operators forbidden by $Q$ or by $Q_+$ and $Q_-$ cannot be radiatively generated even though the lattice regulator breaks translation invariance.
Fermion doublers can be removed while keeping a chiral $U(1)_R$ symmetry, at the price of abandoning a conserved fermion number, which shows which assumption of the Nielsen-Ninomiya theorem is actually needed.
The construction extends to $N=2$ and $N=4$ supersymmetric QCD with matter, since those actions also admit topological and balanced topological rewritings.
The three- and four-dimensional versions are explicitly not fine-tuning-free: the paper lists the specific operators that can be generated and would need counterterms.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The two-dimensional results suggest a practical numerical target: measuring whether the $tr,varphi bar{varphi}$ mass term and the doubler spectrum vanish as $a to 0$ would be a direct test of the whole renormalization argument, and the paper's own conclusion invites exactly this simulation.
Because the argument leans on power counting rather than on a nonperturbative treatment of the degenerate vacuum sum, the natural extension is to check whether summing over the $pm 1$ plaquette sectors, or imposing an admissibility bound $||1 - U_{mu nu}(x)|| < epsilon$ as the paper suggests, changes the operator content near the critical point.
If the exact-supercharge strategy works as claimed in two dimensions, the same logic should identify which lower-dimensional reductions of four-dimensional $N=4$ are tuning-free; the paper's three-dimensional analysis already shows the boundary of the method, with mass operators surviving at one loop once $Delta S$ vertices are inserted.
The mechanism is a template for other theories whose actions can be written as an exact supercharge of a gauge-invariant functional, so the natural question is which additional matter content or gauge groups preserve the cohomology statement in Appendix C.

Original abstract

We construct super Yang-Mills theories with extended supersymmetry on hypercubic lattices of various dimensions keeping one or two supercharges exactly. Gauge fields are represented by ordinary unitary link variables, and the exact supercharges are nilpotent up to gauge transformations. Among the models, we show that the desired continuum theories are obtained without any fine tuning of parameters for the cases ${\cal N}=2, 4, 8$ in two-dimensions.

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The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; show every role.

We construct super Yang-Mills theories with extended supersymmetry on hypercubic lattices of various dimensions keeping one or two supercharges exactly. Gauge fields are represented by ordinary unitary link variables, and the exact supercharges are nilpotent up to gauge transformations. Among the models, we show that the desired continuum theories are obtained without any fine tuning of parameters for the cases ${\cal N}=2, 4, 8$ in two-dimensions.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We construct super Yang-Mills theories with extended supersymmetry on hypercubic lattices of various dimensions keeping one or two supercharges exactly.	contribution 1.00	0.96	0.17	asserted 1.00
s1	Gauge fields are represented by ordinary unitary link variables, and the exact supercharges are nilpotent up to gauge transformations.	method 0.97	0.79	0.07	asserted 1.00
s2	Among the models, we show that the desired continuum theories are obtained without any fine tuning of parameters for the cases ${\cal N}=2, 4, 8$ in two-dimensions.	result 1.00	0.94	0.67	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.80 · evidence from proof or derivation 0.97.

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Figures

**Figure 1.** The plaquette structure of the term ~χ(x) · QΦ( ~ x) for d = 2 case. The variable Uµ is assigned on each link. Fermions χ and ψµ are distributed in the depicted four patterns. where fermions were combined as (2.13). Pµ are hermitian matrices satisfying {Pµ, Pν} = 2δµν . For their explicit form, see appendix B. ∆µ and ∆∗ µ represent forward and backward difference operators respectively: ∆µf(x) ≡ 1 a (f(x + ˆµ) − f(x)), ∆ ∗ µf(x) ≡ 1 a (f(x) − f(x − µˆ)). (3.17) The last term containing Pµ has a similar structure to the Wilson term, and plays a role of removing fermion doublers. The…

Figures reproduced from arXiv:hep-lat/0311021 by the authors.

References

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Forward citations

Cited by 9 Pith papers

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Twisted Supersymmetric Gauge Theories and Orbifold Lattices
hep-th 2006-03 accept novelty 7.0 of 10

Orbifold lattice regularizations of supersymmetric Yang-Mills theories reproduce, in the continuum limit, the Marcus and Blau-Thompson twisted versions of those theories.
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hep-lat 2005-03 accept novelty 7.0 of 10

An orbifold projection of the sixteen-supercharge matrix model produces explicit Euclidean lattices whose classical continuum limits are the maximally supersymmetric Yang-Mills theories in d=1,2,3,4.
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hep-th 2020-11 conditional novelty 6.0 of 10

Reusing the orbifold lattice construction recasts U(k) Yang-Mills as coupled harmonic oscillators whose Hamiltonian is a sum of Pauli strings, offering a group-theory-free route to quantum simulation.
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hep-th 2011-02 conditional novelty 6.0 of 10

At one loop, the twisted lattice formulation of N=4 super Yang-Mills requires only a single common wavefunction renormalization, with no logarithmically divergent fine tuning of its four couplings.
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hep-th 2008-07 conditional novelty 6.0 of 10

N=4 super Yang-Mills on R x S^3 is regularized by the plane wave matrix model around a concentric-fuzzy-sphere vacuum, and one-loop tadpole and fermion self-energy calculations reproduce the continuum coefficients.

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hep-lat 2007-06 conditional novelty 6.0 of 10

A naive lattice discretization of 4-supercharge Yang-Mills quantum mechanics recovers continuum supersymmetry without fine tuning, and the thermal theory shows a single deconfined phase for N up to 12.
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hep-lat 2015-03 conditional novelty 5.0 of 10

Lattice simulations of the BFSS matrix model with an exact-supersymmetry action reproduce the gravity-side black hole energy and yield a next-to-leading power consistent with the predicted 4.6.
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hep-th 2007-12 conditional novelty 5.0 of 10

The supersymmetric orbifold lattice actions of Cohen et al. and Kaplan et al. are shown to coincide with the geometric discretization of the corresponding twisted supersymmetric Yang-Mills theories.
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hep-lat 2009-03 accept novelty 3.0 of 10

Twisting and orbifold projection produce the same class of lattice gauge theories with one exact nilpotent supercharge, and for two and three dimensions that exact symmetry removes the need for fine tuning.

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Closely followed the discretization prescription given by this paper.

Lattice Formulation of ${\cal N} = 2^*$ Yang-Mills (2017) · 1.00 · paraphrased: the licence does not allow quoting

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Showed that one particular truncation followed by field redefinitions yields a theory very closely related to this paper's formulation of four-dimensional N=2 supersymmetric Yang-Mills theory.

Relations among Supersymmetric Lattice Gauge Theories via Orbifolding (2007) · 0.99 · paraphrased: the licence does not allow quoting

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Said that its idea that the fermion mass must vanish in the continuum limit is in line with this paper.

Mass spectrum of $2$-dimensional $\mathcal{N}=(2,2)$ super Yang-Mills theory on the lattice (2018) · 0.96 · paraphrased: the licence does not allow quoting

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In this paper, we modify the formulation for two-dimensional $=(2,2)$ and $(4,4)$ SYM theories [ref] so that the vacuum degeneracy is resolved without imposing the admissibility condition.

Lattice Formulation for 2d N=(2,2), (4,4) Super Yang-Mills Theories without Admissibility Conditions · Introduction · 0.84
In particular, the application to a similar lattice formulation of 2d $=(4,4), (8,8)$ super Yang-Mills theories that preserves two supercharges [ref] should be straightforward.

O(a) Improvement of 2D N=(2,2) Lattice SYM Theory · Conclusion and discussions · 0.86

Compares with it 9 papers

The improved action ([ref]) coincides with the unimproved lattice action given in [ref] for $_μ=_12=1$, and the $(a)$ terms of $_μ$ and $_12$ provide the improvement.

O(a) Improvement of 2D N=(2,2) Lattice SYM Theory · Summary of the improved theory · 0.89
We note that it does not conflict with the fact that the continuum limit of Sugino's lattice formulation is the physical supersymmetric gauge theory [ref] .

Topologically Twisted $N=(2,2)$ Supersymmetric Yang-Mills Theory on Arbitrary Discretized Riemann Surface · Continuum two-dimensional $N=(2,2)$ supersymmetric Yang-Mills theory · 0.84
By using the backward difference $Δ_μ^*$ ($Δ_μ^* φ(x)≡φ(x) - φ(x-μ̂)$), the fermion kinetic terms are expressed as [equation] where $Ψ(x)=( ψ_1(x), ψ_2(x), χ(x), 1/2η(x) )^T$, and matrices [equation] satisfy the algebra: [equation] This is the same kinetic action obtained in [ref] , and thus no fermion doubler appears.

Lattice Formulation for 2d N=(2,2), (4,4) Super Yang-Mills Theories without Admissibility Conditions · $U(N)$ gauge group · 0.81

Cites it as support 2 papers

In addition, even if we consider the square lattice, it is shown that fermion doubler is absent [ref] .

Topologically Twisted $N=(2,2)$ Supersymmetric Yang-Mills Theory on Arbitrary Discretized Riemann Surface · Conclusion and discussion · 0.98
Note that this idea is in line with [ref] , as the fermion mass must vanish in the continuum limit to recover the chiral limit, as it is not a relevant operator.

Mass spectrum of $2$-dimensional $\mathcal{N}=(2,2)$ super Yang-Mills theory on the lattice · Supersymmetry restoration in the continuum limit · 0.96

Cites it as background 23 papers

In the next section, we give a short review of the lattice formulation for two-dimensional $=(2,2)$ and $(4,4)$ SYM theories in [ref] .

Lattice Formulation for 2d N=(2,2), (4,4) Super Yang-Mills Theories without Admissibility Conditions · Introduction · 0.91
A number of approaches that allow to preserve exactly one supersymmetry on the lattice in theories with an extended supersymmetry have been proposed in recent years [ref] .

Lattice supersymmetry in 1D with two supercharges · Introduction · 0.97
Two approaches have been pursued to produce such supersymmetric actions; one based on ideas drawn from the field of topological field theory [ref] and another pioneered by David B.

Supersymmetric lattices - a brief introduction · Introduction · 0.90

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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