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Supersymmetry on a Euclidean Spacetime Lattice I: A Target Theory with Four Supercharges

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Andrew G. Cohen, David B. Kaplan, Emanuel Katz, Mithat Unsal

High Energy Physics - Lattice · Submitted 25 Feb 2003 · 41 references

Pith's verdict: A solid construction paper: the Euclidean orbifold lattice for 2D (2,2) SYM works at tree level, and the no-fine-tuning argument is plausible but the soft-mass sector is asserted rather than shown.

T0 · AI review

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arXiv IDhep-lat/0302017 (version 1)
Pith IDCSZI472Y, Pith's permanent record of this paper and its review
CitationAndrew G. Cohen, David B. Kaplan, Emanuel Katz, Mithat Unsal. "Supersymmetry on a Euclidean Spacetime Lattice I: A Target Theory with Four Supercharges." arXiv:hep-lat/0302017 (2003). https://pith.science/paper/CSZI472Y
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The paper

The paper claims

A Euclidean lattice action for two-dimensional (2,2) supersymmetric Yang-Mills is constructed that preserves one exact supersymmetry and is argued to reproduce the target theory in the continuum limit without fine-tuning.

The reading

Supersymmetric theories are hard to put on a computer. A lattice, which is a grid of points, breaks the continuous symmetries that supersymmetry depends on. Without an exact symmetry on the grid, a numerical simulation drifts away from the theory you wanted unless you fine-tune many parameters by hand. This paper builds a lattice for a specific two-dimensional supersymmetric gauge theory, the (2,2) theory with four supercharges. The construction starts from a larger matrix model in zero dimensions, which has four supersymmetries, and projects it down using a discrete symmetry, an orbifold projection. The projection keeps exactly one of the four supersymmetries intact on the lattice. The resulting lattice action has bosons and fermions on links and sites, and it looks unusual compared to a standard square lattice. The authors then expand the lattice action around a particular classical configuration and take the lattice spacing to zero. They show, mode by mode, that the continuum limit is the target supersymmetric Yang-Mills theory: the right kinetic terms, the right interactions, and no extra copies of particles (no doublers). They then argue that the exact lattice supersymmetry plus the other lattice symmetries forbid any unwanted counterterms, so no fine-tuning is needed. The only allowed counterterm shifts the vacuum energy. Two caveats are stated by the authors. The flat directions of the theory need a small soft supersymmetry-breaking mass to stabilize them, and the argument that this mass does not affect physics is a controlled approximation. Also, the paper is analytic: no numerical simulation is performed, and the authors note that the fermion determinant sign is not guaranteed.
Extended reading notes

Core claim

The lattice action of Eq. (1.2) respects one exact supersymmetry and its continuum limit is the (2,2) supersymmetric Yang-Mills theory of Eq. (1.1) without fine-tuning. This is the abstract's sentence s1 and the paper's load-bearing assertion: the orbifold projection of the four-supercharge matrix model yields a lattice action whose exact Q=1 supersymmetry, together with the U(1)^3, Z2 and Z_N x Z_N symmetries, forbids all relevant and marginal counterterms except a vacuum-energy shift, so the target theory emerges as a -> 0 with no tuned parameters.

Load-bearing premise

The counterterm classification in §5 is complete: the only allowed local counterterm is the Fayet-Iliopoulos term of Eq. (5.3). This rests on the superfield dimension assignments (S and V of dimension 1, Lambda and Xi of dimension 3/2), on the requirement that the operator be Grassmann with U(1)_R charge -1/2, and on the Z2 crystal symmetry forbidding Tr Xi. It is a perturbative power-counting argument, not a nonperturbative proof. If any additional relevant or marginal operator with the right quantum numbers survived the lattice symmetries, the no-fine-tuning claim would fail.

Original abstract

We formulate a Euclidean spacetime lattice whose continuum limit is (2,2) supersymmetric Yang-Mills theory in two dimensions, a theory which possesses four supercharges and an anomalous global chiral symmetry. The lattice action respects one exact supersymmetry, which allows the target theory to emerge in the continuum limit without fine-tuning. Our method exploits an orbifold construction described previously for spatial lattices in Minkowski space, and can be generalized to more complicated theories with additional supersymmetry and more spacetime dimensions.

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, 3,030 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.26 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.43 · compares with prior results 0.47 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We formulate a Euclidean spacetime lattice whose continuum limit is (2,2) supersymmetric Yang-Mills theory in two dimensions, a theory which possesses four supercharges and an anomalous global chiral symmetry. The lattice action respects one exact supersymmetry, which allows the target theory to emerge in the continuum limit without fine-tuning. Our method exploits an orbifold construction described previously for spatial lattices in Minkowski space, and can be generalized to more complicated theories with additional supersymmetry and more spacetime dimensions.

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 formulate a Euclidean spacetime lattice whose continuum limit is (2,2) supersymmetric Yang-Mills theory in two dimensions, a theory which possesses four supercharges and an anomalous global chiral symmetry. contribution 0.99 0.83 0.27 asserted 1.00
s1 The lattice action respects one exact supersymmetry, which allows the target theory to emerge in the continuum limit without fine-tuning. contribution 0.36 0.87 0.38 asserted 1.00
s2 Our method exploits an orbifold construction described previously for spatial lattices in Minkowski space, and can be generalized to more complicated theories with additional supersymmetry and more spacetime dimensions. method 0.84 0.56 0.11 asserted 0.64

What kind of paper the classifier reports this is: method or tool 0.67 · evidence from proof or derivation 0.91.

What was examined, and what was not

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  • figures and tables
  • references and citing sentences
  • relations to other papers
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  • 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. The two dimensional Euclidean lattice for the two dimensional SYM theory with four supercharges. The lattice respects one exact supercharge. Arrows on the links signify the orientation of the r charge vectors of the link fermions, as discussed in §2. direction, y, y and β are variables on the links in theˆ = {0, 1} direction, ξ sits on the diagonal links, and λ is a site variable. The parameter a appearing in the action has dimension of length, and is interpreted as the lattice spacing. The continuum limit is defined as a → 0, N → ∞ while holding the 2-dimensional coupling g2 ≡ ga…

Figures reproduced from arXiv:hep-lat/0302017 by the authors.

References
How the literature uses it

53 arXiv papers cite this in 98 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 citing it · 61 sentences without a confident label

Uses its method or tool 6 papers
Compares with it 3 papers
  • This formulation is identical to what was given in [ref] .

    Classification of Supersymmetric Lattice Gauge Theories by Orbifolding · Conclusions · 1.00

  • Acting with the $$-transformation shown above and again integrating out the auxiliary field $d$ we derive the gauge and $$-invariant lattice action S=∑_(^L†_μν^L_μν+ 1/2(^(-)_μ_μ)^2- χ_μν^(+)_[μ.ψ_.ν]- η^(-)_μψ_μ) But this is precisely the orbifold action arising in [ref] with the modified deconstruction step described in [ref] and [ref] which we will describe in detail in the next section.

    Exact lattice supersymmetry · Lattice theory for $(2,2)$ SYM · 0.92

  • This is slightly different from those in the original papers [ref] reflecting a difference of motivation (the target of the original papers was exact supersymmetry on the lattice).

    Exponential speedup in quantum simulation of Kogut-Susskind Hamiltonian via orbifold lattice · Kogut-Susskind Hamiltonian from the orbifold lattice · 0.95

Cites it as background 21 papers

Discussion

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