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Supersymmetry on a Spatial Lattice

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David B. Kaplan, Emanuel Katz, Mithat Unsal

High Energy Physics - Lattice · Submitted 18 Jun 2002 · 39 references

Pith's verdict: A founding paper of orbifold lattice supersymmetry: the 1+1 case is solid, the 2+1 and 3+1 no-fine-tuning arguments thin out, and the authors say so themselves.

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arXiv IDhep-lat/0206019 (version 3)
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CitationDavid B. Kaplan, Emanuel Katz, Mithat Unsal. "Supersymmetry on a Spatial Lattice." arXiv:hep-lat/0206019 (2002). https://pith.science/paper/TCYRLTGG
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The paper

The paper claims

A lattice can keep some of its supersymmetry exactly, so the continuum limit may need little or no fine tuning.

The reading

The paper asks whether supersymmetric gauge theories — including $N=4$ super Yang-Mills in 3+1 dimensions — can be put on a *spatial* lattice (continuous time, discrete space) while retaining some of the supersymmetry as an exact symmetry of the lattice itself. The construction starts from a supersymmetric quantum mechanics with a large $U(kN^d)$ gauge group and a global R-symmetry, then projects out everything that does not respect a chosen $(Z_N)^d$ subgroup; the surviving blocks of the original matrices become site fields and link fields on a $d$-dimensional periodic lattice with an independent $U(k)$ gauge group at each site. Because some supercharges are neutral under the projection, they survive exactly, and they interchange bosons and fermions at the same site or the same link. That exact symmetry, together with the lattice's point group, is what the authors argue forbids the low-dimension operators — scalar masses, tadpoles, and the like — that would otherwise have to be tuned by hand. The lattice spacing is not a parameter of the action: it is set dynamically by the vacuum expectation value of the link scalars, $a = 1/f$, which is what makes the whole scheme a deconstruction of a higher-dimensional theory rather than a conventional lattice.

What carries the argument

The mechanism is the orbifold projection itself, stated as a matrix constraint. An adjoint field $Φ$ of the mother theory carrying R-charges $r_a$ is kept only if $Φ = e^{-2πi r_a/N} γ_a^{-1} Φ γ_a$ for $a=1,‌dots,d$, where $γ_a$ is a clock matrix acting on one of the $d$ $N$-dimensional factors of the $U(kN^d)$ gauge group. In block form this forces the $k×k$ block $φ_{mn}$ to vanish unless $n = m + r$, so a field with $r=(0,0,‌dots)$ becomes a site field and a field with $r$ a unit vector becomes a link field. The R-charges are chosen as $r_a = q_1 - q_{a+1}$ so that they take only the values $0, ±1$ (keeping interactions local) and so that as many fermions as possible have $r=0$ (maximizing the exact supercharges). The lattice spacing then enters only through expanding about the flat direction $φ_n = (f/√2) 1_k$, with $a = 1/f$, $L = Na$, and $g_{d+1}^2 = a^{d-1} g^2$ held fixed as $a → 0$.

Extended reading notes

Core claim

The central claim is that orbifold projection of a 0+1 dimensional supersymmetric matrix model produces a lattice gauge theory in which a subset of the mother theory's supersymmetry is exact, and that this exactness is sufficient to reach the desired continuum theory without fine tuning in the cases examined. Concretely: a mother theory with four, eight, or sixteen real supercharges, obtained by dimensional reduction of $N=1$ super Yang-Mills from 3+1, 5+1, or 9+1 dimensions, is quotiented by $(Z_N)^d$ built from a combination of R-symmetry and gauge rotations. The daughter theory has a $U(k)$ gauge group per site, bifundamental link variables for both bosons and fermions, and a $Z_N^d$ translation symmetry; there are no fermion doublers, because every fermion mode that survives the continuum limit is needed by the target theory. The authors work through the $(2,2)$ theory in 1+1 dimensions in detail, showing that the only potentially dangerous relevant operator allowed by the exact lattice symmetries is a Fayet-Iliopoulos term, which contributes nothing but a cosmological constant. They then argue the same holds for eight and sixteen supercharges in 2+1 dimensions, and give two proposals for $N=4$ super Yang-Mills in 3+1 dimensions: a body-centered cubic lattice, where the question of marginal operator tuning is left open, and an asymmetric lattice where the continuum limit is taken in planes first and then in the third direction, so that the intermediate 2+1 dimensional theory's super-renormalizability and six extra supercharges protect the second step.

Load-bearing premise

The whole scheme assumes the classical flat directions of the link scalars — above all the singlet 'radion' mode $ζ(t)$ that shifts the lattice spacing — can be held fixed as backgrounds, which the authors address by localizing the initial and final wavefunctions and restricting to times of order the lattice size, so that the path integral runs in a chosen excited state rather than the ground state.

Editorial extensions

If this is right

  • The 1+1 dimensional $(2,2)$ lattice model is claimed to reproduce $(2,2)$ super Yang-Mills in the continuum with no fine tuning beyond an unobservable cosmological constant, because the exact lattice supersymmetry and the absence of Fermi multiplets forbid boson tadpoles and boson masses.
  • The eight-supercharge 2+1 dimensional lattice (square cells with horizontal, vertical, and diagonal links) and the sixteen-supercharge hexagonal lattice are argued to have no allowed counterterms of dimension $p ≤ 3$ other than a cosmological constant, again removing the need for tuning.
  • For $N=4$ super Yang-Mills in 3+1 dimensions, the body-centered-cubic construction gives the target theory at tree level with the correct gauge, scalar, and fermion content and no doublers, but the authors explicitly do not settle whether marginal operators require logarithmic tuning.
  • The asymmetric-lattice variant proposes a two-stage continuum limit whose intermediate theory is a 2+1 dimensional linear moose with eight exact supercharges, reducing the tuning question to a single possible hypermultiplet mass parameter.
  • Because the constructions are spatial lattices with Minkowski time, the authors expect the same orbifold techniques to extend to Euclidean spacetime lattices, where the rank of the R-symmetry is one larger and the exact supercharges correspondingly fewer.

Reading between the lines

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

  • If the exact-supercharge argument holds, the practical bottleneck shifts from tuning to simulation: the lattice spacing is a fluctuating dynamical variable, so numerical work would need to control the radion mode explicitly rather than treat $a$ as an input.
  • The claim that the lattice point group plus exact supersymmetry forbids all relevant operators is a statement about the symmetries of the *lattice* action; it would be worth checking case by case whether any allowed operator is merely suppressed by the smallness of $a g^2$ rather than absolutely forbidden.
  • The two-stage continuum limit for $N=4$ suggests a general strategy: build a lattice whose first continuum limit lands on a super-renormalizable theory with more exact supercharges, and let that intermediate theory protect the second limit. The same trick might apply to other theories where a direct four-dimensional lattice has too many marginal operators.
  • The construction's reliance on a $U(kN^d)$ mother theory rather than $SU(kN^d)$ is a technical choice to avoid a nonlocal tracelessness constraint on site variables; whether an $SU$-based version exists with comparably good lattice properties is left open.
Original abstract

We construct a variety of supersymmetric gauge theories on a spatial lattice, including N=4 supersymmetric Yang-Mills theory in 3+1 dimensions. Exact lattice supersymmetry greatly reduces or eliminates the need for fine tuning to arrive at the desired continuum limit in these examples.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,273 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.08 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.15 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We construct a variety of supersymmetric gauge theories on a spatial lattice, including N=4 supersymmetric Yang-Mills theory in 3+1 dimensions. Exact lattice supersymmetry greatly reduces or eliminates the need for fine tuning to arrive at the desired continuum limit in these examples.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We construct a variety of supersymmetric gauge theories on a spatial lattice, including N=4 supersymmetric Yang-Mills theory in 3+1 dimensions. contribution 1.00 0.94 0.12 asserted 1.00
s1 Exact lattice supersymmetry greatly reduces or eliminates the need for fine tuning to arrive at the desired continuum limit in these examples. interpretation 0.61 0.53 0.67 asserted 0.87

What kind of paper the classifier reports this is: method or tool 0.54 · evidence from proof or derivation 0.91.

What was examined, and what was not

sources read
  • arXiv listing abstract v3
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. An example of the orbifold procedure in Eq. (2.3) for a U(9k) gauge group with global symmetry GR = U(1)×U(1), modded out by Z3 ×Z3. The dark squares are the k×k blocks which survive the orbifold projection within a U(9k) adjoint field with GR charges r. The examples shown (from left to right in the figure) correspond to r = (0, 0), (1, 0), (0, 1) and (1, 1). The projected fields may be interpreted respectively as fields living on sites, horizontal links, vertical links, and diagonal links on a 3 × 3 periodic lattice. these qi charges we form d < p linear combinations {r1, . . . ,…
Figure 2. The lattice with N sites obtained by orbifolding a SUSY QM theory with a U(kN) gauge symmetry by ZN . Sites cor￾respond to U(k) gauge groups with adjoint matter, while links correspond to matter su￾perfields transforming as bifundamentals
Figure 4. When the sixteen supercharge 0 + 1 SYM theory is orbifolded by (ZN ) 2 , one obtains a similar lattice to that in
Figure 6. The sixteen supercharge 0 + 1 SYM theory orbifolded by (ZN ) 3 results in a lattice respecting two exact supercharges with the above structure. The target theory is N = 4 SYM theory in 3 + 1 dimensions. Dark blue links correspond to bosonic chiral superfields, light blue links are Fermi super￾fields, and a vector supermultiplet resides at each site
Figure 8. One cell from the lattice of
Figure 10. Each Fermi supermultiplet (light blue link) in the body centered cubic crystal of
Figure 11. The two dimensional sublattice of the bcc lattice is essentially the same as the one encountered in
Figure 13. This hypermultiplet (we use the 3 + 1 dimensional,

Showing 8 of the paper's 13 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:hep-lat/0206019 by the authors.

References
How later papers used it

48 later papers cite it

Evidence: substantial. 47 of the citing papers share no author with it, and 6 of those carry a confident label from Pith's citation classifier.

Pith read 21 of the 97 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Uses its method or tool 1 of 4 papers with this label
How the literature uses it

46 arXiv papers cite this in 102 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers whose main result relates to this one's

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States the same result 3 papers
  • The presence of an exact supersymmetry reduces and in some cases eliminates the need for fine tuning to achieve a continuum limit invariant under the full supersymmetry of the target theory.

    Exact lattice supersymmetry · 0.95

  • Part of the target theory's supersymmetry is realized exactly on the lattice, reducing or eliminating the need for fine tuning.

    Lattice supersymmetry · 0.95

  • The lattice theories themselves respect certain supersymmetries, which in many cases allows the target theory to be obtained in the continuum limit without fine-tuning.

    Recent Developments in Lattice Supersymmetry · 0.87

Papers citing it · 62 sentences without a confident label

Builds on it 1 paper
  • In this paper, we have demonstrated how to apply the orbifold methods used by KKU [ref] for constructing supersymmetric lattice gauge theories to the quantum simulation of ordinary Yang-Mills theory with U($k$) gauge group.

    Quantum simulation of gauge theory via orbifold lattice · Conclusion and Outlook · 0.83

Uses its method or tool 4 papers
Cites it as support 1 paper
Cites it as background 28 papers

Discussion

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