Higgs branch localization in three dimensions

Francesco Benini; Wolfger Peelaers

REVIEW

Higgs branch localization in three dimensions

Francesco Benini, Wolfger Peelaers

High Energy Physics - Theory · Submitted 20 Dec 2013 · 67 references

Pith's verdict: A careful, honest 3d extension of Higgs branch localization; the one real overreach is the abstract's unconditional phrasing of a result that only holds for |k| ≤ |Nf−Na|/2.

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arXiv ID1312.6078 (version 3)

Pith IDHRCNFMUZ, Pith's permanent record of this paper and its review

CitationFrancesco Benini, Wolfger Peelaers. "Higgs branch localization in three dimensions." arXiv:1312.6078 (2013). https://pith.science/paper/HRCNFMUZ

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Editor's note

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Worth your time if you care about 3d localization or vortex partition functions. The core is a clean 3d lift of the 2d Higgs branch localization of Benini–Cremonesi and Doroud–Gomis–Le Floch–Lee: add a Q-exact deformation H(φ) that pushes the path integral onto vortex strings at the two circles of S^3_b (or the two poles of S^2×S^1), and the partition function falls out as a sum over Higgs vacua of Z_cl Z'_1-loop Z_v Z_av. The genuinely new material is the explicit BPS analysis on both geometries, the one-loop determinants via the equivariant index theorem, and the observation that the very same vortex partition function (2.78) controls both S^3_b and S^2×S^1 with different parameter maps. That last point is the useful takeaway and it is done properly, not hand-waved. The matching against the Coulomb branch integral in §2.5 and §3.5 is a real independent check — Cauchy determinant, residue summation, parity-anomaly cancellation — not a circular restatement. Credit where due: the algebra is explicit and the note added honestly discloses overlap with Fujitsuka–Honda–Yoshida (1312.3627). The stress-test note holds up on reading. The load-bearing condition is suppression of the deformed Coulomb branch as ζ→±∞, which §2.4.3 and appendix D show requires |k| ≤ |Nf−Na|/2 for U(N) with (anti)fundamentals. The paper does disclose this in §4 and derives it, so it is a scope condition, not a hidden flaw. The one place I would push back is the abstract [abstract:s0], which states the result without the bound; that is a small overreach and the authors should tighten it. The residual soft spot the stress-test flags is real but minor: the necessity of the bound is asserted ('some more clever procedure (if possible at all)') rather than demonstrated by a counterexample outside it. That is a scope question, not an internal inconsistency, and it does not break the central claim within its stated regime. Two smaller things: the residue prescription for the Higgsed chiral multiplets is imported from [5] rather than re-derived, and the double-sine/q-Pochhammer manipulations are dense enough that I could not verify them line by line. Neither is a reason to hold the paper. Who should read it: anyone working on 3d N=2 localization, holomorphic blocks, or vortex partition functions, and anyone who wants a second, vortex-based handle on the S^3_b and S^2×S^1 partition functions. It deserves a serious referee; the technical claims are checkable and the matching is the kind of thing a careful referee can audit.

Referee report

Significance

If correct, the result is a useful structural statement: it gives a second, independent localization scheme for 3d N=2 theories and exhibits directly the holomorphic-block factorization that had previously been observed only by manipulating Coulomb-branch integrals [23,29,31]. The paper is strong where it is most concrete: the BPS equations are written out in components (2.29)-(2.30) and (3.25)-(3.26), the deformation term is constructed explicitly and its bosonic part shown to be a sum of squares (2.22) and (3.24), the one-loop determinants are derived from an index theorem with the unregulated expressions displayed (2.50), (C.6), (3.45), (C.12) rather than quoted, and the matching in Sections 2.5 and 3.5 is carried out step by step (Cauchy determinant, residue summation, parity-anomaly sign). The identification of a single vortex partition function (2.78) governing both S^3_b and S^2 x S^1, with the parameter maps (2.68) and (3.63), is a genuinely checkable prediction, and the 2d limit (2.79) reproduces the known Omega-background vortex partition function. The paper also isolates a scope condition, |k| <= |N_f - N_a|/2, that constrains when the factorization holds; flagging this,

Major objections (2)

[Abstract vs. Sections 2.4.3 and 4] The abstract states without qualification that the partition function 'can be computed with the so-called Higgs branch localization method' and 'directly takes the form of a sum ... of a vortex-string times an antivortex-string partition functions' (abstract:s0, abstract:s2). The body shows that this reduction requires the deformed Coulomb branch to be suppressed in the zeta -> +/- infinity limit, which Section 2.4.3 establishes only under the bound (2.59)-(2.60), and Section 4 restates as 'factorize into VPFs only for |k| <= |N_f - N_a|/2'. The abstract's only qualifier is complete Higgsing by FI terms, which is a different condition. Since the bound is a genuine restriction of the theory space, the abstract should state it, or at least say that the reduction applies to the maximally chiral theories of [49]. This is a scope-of-claim issue rather than an error in the derivation, and it is straightforward to fix.
[Section 4 and Appendix D] Section 4 asserts that the partition functions 'factorize into VPFs only for |k| <= |N_f - N_a|/2', which reads as a statement about factorization itself. What Sections 2.4.3 and 3.4.3 together with Appendix D actually establish is narrower: that this particular contour manipulation (shifting the integration variable by -i zeta kappa and closing the contour) fails outside the bound, because the essential singularity at z = 0 or z = infinity then contributes, and Appendix D explicitly says 'we will not attempt to perform the complete computation' in that regime. The distinction matters, because a reader could take the Section 4 sentence as evidence that factorization is impossible beyond the bound, whereas the paper only shows that its own residue argument does not apply. I would ask the authors to rephrase the Section 4 statement as a limitation of the method as implemented, and to say explicitly that necessity of the bound is not demonstrated. A single worked example outside the bound, even a U(1) case where the Higgs-branch sum is shown to disagree with the Coulomb integral, would settle the question; absent that, the weaker phrasing is the honest one.

Minor objections (6)

[Eq. (2.66) and Eq. (2.68)] Equation (2.66) assigns the same rotational equivariant parameter to both circles, epsilon_N = epsilon_S = 2*pi*b^2, while the vortex and antivortex weights in (2.68) carry b^{-2} and b^2 respectively in their quadratic terms. Given that the two circles are treated asymmetrically everywhere else in Section 2, please confirm that epsilon_S is not 2*pi*b^{-2} (or otherwise explain why the same value appears at both poles).
[Section 2.4.1 and footnote 10] The applicability of the equivariant index theorem is delicate here because the Killing vector xi generates a non-compact isometry for generic b, and this is acknowledged only in footnote 10. Since the whole one-loop computation rests on the reduction along the Hopf fiber, a sentence in the main text stating the caveat and pointing to the footnote would help the reader judge how much of the result is theorem-backed and how much is fixed by the independent Coulomb-branch matching.
[Section 2.3] The argument that there are no smooth solutions with non-trivial phi for positive R-charge relies on taking l and l-tilde incommensurable so that at most one Fourier mode survives the constraint q = 2(m*l + n*l-tilde)/(l + l-tilde). Please state explicitly that for commensurable squashing the conclusion follows by analytic continuation in the real masses, as is done for R-charges in Section 2.4.4.
[Section 2.4.4] The statement that 'each BPS solution is a smooth configuration with no moduli, we simply sum over them with weight 1' deserves one more sentence of justification. On S^3_b the vortex sits on a fixed circle, so the absence of collective coordinates is plausible, but the reader would benefit from being told which moduli are excluded by the geometry and why no Jacobian accompanies the sum.
[Section 2.5 and Section 4] The claim that the bound |k| <= |N_f - N_a|/2 'apparently has been overlooked before' in [23,29,31] is a statement about the prior literature and should either be substantiated with a specific citation of where those works state or fail to state a regime of validity, or softened to a neutral comparison.
[Appendix D] The appendix title contains the typo 'deteminants', and the summarizing paragraph near the end contains 'toward sz = 0 and/or z = infinity'. Both should be corrected.

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The paper

The paper claims

By adding a Fayet-Iliopoulos-style deformation to the localizing action, the authors show that the path integral of a three-dimensional supersymmetric Chern-Simons-matter theory can be made to collapse onto vortex strings, so that the exact

The reading

The paper asks whether the exact partition function of a three-dimensional supersymmetric gauge theory can be computed by a route that lands on vortices rather than on the usual matrix integral over a Coulomb-branch scalar. The authors add one extra term to the localizing action — a Fayet-Iliopoulos-style deformation built out of the matter scalars — and show that on both the squashed three-sphere $S^3_b$ and on $S^2 times S^1$ the path integral then collapses onto a finite set of Higgs vacua dressed by vortex strings sitting at two circles. The payoff is a formula in which the answer is written directly as a sum over Higgs vacua of a classical piece, a one-loop piece, and a vortex times an antivortex partition function, all objects one already knows how to compute. If correct, this gives a second and more geometrically transparent handle on the same quantities that Coulomb-branch localization produces, and it explains why the three-sphere partition function and the superconformal index factorize into the very same vortex building block.

What carries the argument

The load-bearing device is the extra $Q$-exact operator $S_{text{def}H} = int Q,mathrm{Tr}[i(epsilon^dagger lambda - lambda^dagger tilde{epsilon}) H(varphi)/2]$, whose bosonic piece is not positive definite on its own but becomes a sum of squares once the auxiliary field $D$ is integrated out along a shifted contour. That shift is what moves the localization locus from the Coulomb branch to the Higgs branch, and the parameter $zeta$ in $H(varphi)$ acts as a knob interpolating between the two pictures. The one-loop determinants around the non-constant vortex backgrounds are obtained from an equivariant index theorem for transversally elliptic operators, reduced along a Hopf or thermal circle to a calculation on the base $S^2$, and the determinants for the chiral multiplets that acquire a vacuum expectation value are extracted as residues. The resulting vortex and antivortex factors are the three-dimensional vortex partition function on the twisted $mathbb{R}^2_epsilon times S^1$ — a K-theoretic refinement of the two-dimensional $Omega$-background vortex partition function — and the same function governs both geometries, only the map of parameters changing.

Extended reading notes

Core claim

The central claim is that for three-dimensional $mathcal{N}=2$ R-symmetric Chern-Simons-matter theories that can be completely Higgsed by a Fayet-Iliopoulos term, the supersymmetric partition function on $S^3_b$ and the superconformal index on $S^2 times S^1$ can both be computed by a Higgs branch localization that is an alternative to the standard Coulomb branch one. The mechanism is a new $Q$-exact deformation whose bosonic part replaces the auxiliary-field equation by $D + sigma_R/f = iH(varphi)$ with $H(varphi) = zeta - sum_i varphi_i^dagger T_a varphi_i$, so that the localization locus now contains, besides a deformed Coulomb branch, discrete Higgs vacua and towers of vortex strings at the northern and southern circles. For finite $zeta$ the allowed vortex numbers are bounded, and the vortex contributions exactly account for the jumps the deformed Coulomb-branch integral undergoes as the contour is pushed past the poles of the chiral one-loop determinant. Taking $zeta to pminfty$ so that the deformed Coulomb branch is suppressed leaves a pure sum $Z = sum_{text{Higgs vacua}} Z_{text{cl}} Z'_{text{1-loop}} Z_v Z_{av}$, and the authors verify in sections 2.5 and 3.5 that for $U(N)$ with $N_f$ fundamentals and $N_a$ antifundamentals this reproduces the Coulomb-branch matrix integral, provided $|k| le |N_f - N_a|/2$.

Load-bearing premise

That the deformed Coulomb-branch contribution can be made to vanish by sending the deformation parameter $zeta$ to $pminfty$, which the paper shows requires the Chern-Simons level to satisfy $|k| le |N_f - N_a|/2$; outside that bound the authors state that reduction to a sum of residues would require 'some more clever procedure (if possible at all).'

Editorial extensions

If this is right

The $S^3_b$ partition function and the $S^2 times S^1$ index of the same theory are both expressible through one and the same three-dimensional vortex partition function, with the difference absorbed entirely into how the equivariant and flavour parameters are identified.
For maximally chiral $U(N)$ theories the Coulomb-branch matrix integral can be rewritten as a finite sum over classical Higgs vacua, giving a residue formula that sidesteps the contour-closing bookkeeping of the original integral.
The deformation parameter $zeta$ gives a continuous picture of the two localizations: as $zeta$ grows, deformed Coulomb-branch solutions branch into vortex solutions at thresholds proportional to $b n + b^{-1} m$ on $S^3_b$ and to $m+n$ on $S^2$, so the jumps in the matrix integral are literally the appearance of new vortices.
Taking the equivariant parameter to zero and the Chern-Simons level to zero reduces the three-dimensional vortex partition function to the standard two-dimensional vortex partition function in $Omega$-background, tying the three-dimensional result to the two-dimensional story it was modelled on.

Reading between the lines

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

The restriction $|k| le |N_f - N_a|/2$ reads less like a statement about factorization itself than about which theories admit a clean large-$zeta$ limit; one plausible reading is that for larger $|k|$ the correct statement needs an extra contribution from the essential singularity at $z=0$ or $z=infty$, which the paper explicitly declines to compute.
Because the same vortex partition function appears on two different three-manifolds, the construction supplies a path-integral derivation of the 'holomorphic blocks' structure that was previously established by manipulating Coulomb-branch integrals; the factorization is then a consequence of where the deformation pushes the fields, not of contour gymnastics.
The authors' remark that the Chern-Simons level bound 'apparently has been overlooked before' implies that previously claimed factorizations of three-dimensional partition functions outside that bound deserve a second look.
If the method extends to lens spaces and to four-dimensional $mathcal{N}=1$ theories as suggested, the same deformation should produce vortex-membrane sums there, and an analogous restriction on Chern-Simons levels or their four-dimensional counterparts should reappear as the condition for the deformed Coulomb branch to be suppressible.

Original abstract

We show that the supersymmetric partition function of three-dimensional N=2 R-symmetric Chern-Simons-matter theories on the squashed S^3 and on S^2 x S^1 can be computed with the so-called Higgs branch localization method, alternative to the more standard Coulomb branch localization. For theories that could be completely Higgsed by Fayet-Iliopoulos terms, the path integral is dominated by BPS vortex strings sitting at two circles in the geometry. In this way, the partition function directly takes the form of a sum, over a finite number of points on the classical Coulomb branch, of a vortex-string times an antivortex-string partition functions.

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,097 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.44 · human participants 0.01 · controlled comparison 0.07 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.91 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We show that the supersymmetric partition function of three-dimensional N=2 R-symmetric Chern-Simons-matter theories on the squashed S^3 and on S^2 x S^1 can be computed with the so-called Higgs branch localization method, alternative to the more standard Coulomb branch localization. For theories that could be completely Higgsed by Fayet-Iliopoulos terms, the path integral is dominated by BPS vortex strings sitting at two circles in the geometry. In this way, the partition function directly takes the form of a sum, over a finite number of points on the classical Coulomb branch, of a vortex-string times an antivortex-string partition functions.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We show that the supersymmetric partition function of three-dimensional N=2 R-symmetric Chern-Simons-matter theories on the squashed S^3 and on S^2 x S^1 can be computed with the so-called Higgs branch localization method, alternative to the more standard Coulomb branch localization.	contribution 0.92	0.93	0.84	asserted 1.00
s1	For theories that could be completely Higgsed by Fayet-Iliopoulos terms, the path integral is dominated by BPS vortex strings sitting at two circles in the geometry.	result 0.61	0.73	0.12	asserted 0.91
s2	In this way, the partition function directly takes the form of a sum, over a finite number of points on the classical Coulomb branch, of a vortex-string times an antivortex-string partition functions.	result 0.84	0.82	0.26	asserted 1.00

What kind of paper the classifier reports this is: method or tool 0.70 · evidence from proof or derivation 0.99.

What was examined, and what was not

sources read

arXiv listing abstract v3

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full text
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references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
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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

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How later papers used it

75 later papers cite it

Evidence: substantial. 65 of the citing papers share no author with it, and 12 of those carry a confident label from Pith's citation classifier.

Pith read 14 of the 141 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

3 later papers by other authors are listed below: 3 used its method or tool.

Uses its method or tool 3 of 9 papers with this label

Regularized this paper's expression using the q-Pochhammer symbol and Hurwitz zeta function regularization, similar to other works.

Partition Functions of $\mathcal{N}=1$ Gauge Theories on $S^2 \times \mathbb{R}^2_\varepsilon$ and Duality (2018) · 0.93 · paraphrased: the licence does not allow quoting
Closely followed the presentation of appendix B of this paper.

Black hole microstates and supersymmetric localization (2018) · 0.86 · paraphrased: the licence does not allow quoting
Worked in this paper's Higgs branch localization scheme, noting it does not affect the matter BPS equations.

Interfaces and Quantum Algebras, I: Stable Envelopes (2021) · 0.89 · paraphrased: the licence does not allow quoting

How the literature uses it

75 arXiv papers cite this in 141 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 · 102 sentences without a confident label

Builds on it 1 paper

The equations [ref] and [ref] generalize to arbitrary $A$-type backgrounds those of [ref] [ref] .

$\mathcal{N}=2$ supersymmetric field theories on 3-manifolds with A-type boundaries · Real and gauge multiplets · 0.99

Uses its method or tool 9 papers

In order to evaluate its large $N$ limit, though, we find more convenient to recast it in a different form, called Bethe Ansatz formula [ref] (see also [ref] for a 3d analog, and [ref] for similar Higgs branch localization formulas).

Black holes in 4d $\mathcal{N}=4$ Super-Yang-Mills · The dual field theory and its index · 0.96
Similar to Refs. [ref] , the expression [ref] can be regularized as [equation] where we used the $q$-Pochhammer symbol $(a; q)_n$ defined as (a; q)_n ≡∏_k = 0^n-1 (1 - a q^k) , and the Hurwitz zeta function regularization: ∑_q ≥0 (q + x) = ζ_H (-1, x) = -1/2 (x^2 - x + 1/6) .

Partition Functions of $\mathcal{N}=1$ Gauge Theories on $S^2 \times \mathbb{R}^2_\varepsilon$ and Duality · 1-Loop Determinants · 0.93
In this section we will closely follow the presentation of appendix B of [ref] .

Black hole microstates and supersymmetric localization · Supersymmetric Chern-Simons-matter theories in three dimensions · 0.86

Compares with it 4 papers

We thus recover the same picture as was found in [ref] in three dimensions.

Higgs branch localization of $\mathcal{N}=1$ theories on $S^3 \times S^1$ · Deformed Coulomb branch · 0.83
By comparing 3d vortex partition functions obtained by localization [ref] , we notice that on Higgs branch 3d brane webs of nonabelian theories are abelianized and some internal lines on toric diagrams (brane webs) need to be assigned with empty Young tableaux.

3d $\mathcal{N}=2$ Brane Webs and Quiver Matrices · Introduction · 0.86
We emphasize that although similar to [ref] this particular form of $𝒱_H$ with the insertion of the $Γ_5$-matrix as well as the separation of the supercharges $Σ$ and $Σ$ is carefully chosen and has not appeared in the literature before.

Partition Functions of $\mathcal{N}=1$ Gauge Theories on $S^2 \times \mathbb{R}^2_\varepsilon$ and Duality · $δ$-Exact Action and BPS Equations · 0.93

Cites it as support 1 paper

Indeed recent works on SUSY localization [ref] imply that CSS sometimes give important contributions [ref] .

Supersymmetric solutions and Borel singularities for N=2 supersymmetric Chern-Simons theories · Introduction · 1.00

Cites it as background 18 papers

It is known that in the (Coulomb branch) localization formula for $Z_S_b^3$, picking up poles of the one-loop determinant in a half complex plane gives rise to Higgs branch representation of the partition function including a product of vortex and anti-vortex partition functions for some theories [ref] .

Supersymmetric solutions and Borel singularities for N=2 supersymmetric Chern-Simons theories · Conclusion and discussions · 0.85
The study of Higgs branch localisation for 2d $𝒩=(2,2)$ theories on $S^2$ was initiated by [ref] and later extended to $S^3$ partition functions and superconformal indices on $S^2 ×S^1$ in the works [ref] .

Boundaries & Localisation with a Topological Twist · Introduction · 0.84
See [ref] for several theoretical perspectives on the factorization.

Anomalies and Holomorphy in Supersymmetric Chern-Simons-Matter Theories · Introduction · 0.95

What it cites · 106 sentences without a confident label

Builds on 1 paper

In this paper we have extended the Higgs branch localization framework of [ref] to three-dimensional $=2$ R-symmetric theories on $S^3_b$ and $S^2 ×S^1$.

Partition functions of N=(2,2) gauge theories on S^2 and vortices · Discussion · 1.00

Uses the method or tool of 8 papers

As in [ref] , Higgs branch localization can be achieved by adding another $$-exact term to the deformation action.

Partition functions of N=(2,2) gauge theories on S^2 and vortices · The BPS equations · 0.88
Following [ref] we consider the factorized ansatz $ϵ_±= f(τ) ϵ_±^S^2(θ,φ)$, where the 2d spinor satisfies $D_μ̂ ϵ_±^S^2 = ±1/2r γ_μ̂ γ^3 ϵ_±^S^2$ with $μ̂=θ,φ$.

Index for three dimensional superconformal field theories with general R-charge assignments · Killing spinors on $S^2 ×S^1$, supersymmetric index and deformed background · 0.94
Identifying the equivariant parameters of the $U(1)_∂_ϕ_H×U(1)_R ×G$ action as $ε= 1/ℓ- 1/ℓ̃$, $ε̌= 1/ℓ + 1/ℓ̃$ and $a = -i( 1/ℓ A_φ+ 1/ℓ̃ A_χ) -$, following [ref] we obtain (see appendix [ref]) the one-loop determinant for a chiral multiplet of R-charge $q$ in gauge representation $$:

Exact results for vortex loop operators in 3d supersymmetric theories · One-loop determinants from an index theorem · 0.92

Compares with 4 papers

When this work was under completion, we became aware of [ref] which has substantial overlap with our paper.

Higgs branch localization of 3d N=2 theories · Introduction · 0.89
This expression is precisely the standard two-dimensional vortex partition function in $Ω$-background, see [ref] .

Partition functions of N=(2,2) gauge theories on S^2 and vortices · Comparison with the two-dimensional vortex partition function · 0.88
Such an expression for $Z_vortex$ can be compared with [ref] .

Vortices and 3 dimensional dualities · Matching with the Coulomb branch integral · 0.98

Cites as background 29 papers

Lagrangians invariant under the supersymmetry transformations were studied in [ref] .

Notes on SUSY Gauge Theories on Three-Sphere · Supersymmetric theories on three-manifolds · 0.84
Lagrangians invariant under the supersymmetry transformations were studied in [ref] .

SUSY Gauge Theories on Squashed Three-Spheres · Supersymmetric theories on three-manifolds · 0.84
After the seminal work of Pestun on $S^4$ [ref] , the techniques have been developed in many different contexts, essentially from two to five dimensions (see [ref] for a non-exhaustive list).

Exact Results for Wilson Loops in Superconformal Chern-Simons Theories with Matter · Introduction · 0.89

Discussion

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