Five Dimensional Gauge Theories and Relativistic Integrable Systems

Nikita Nekrasov

REVIEW

Five Dimensional Gauge Theories and Relativistic Integrable Systems

Nikita Nekrasov

High Energy Physics - Theory · Submitted 29 Sep 1996 · 37 references

Pith's verdict: A proposal paper, but a specific one: the pure-SU(2) curve is exactly the 4d Seiberg-Witten curve in disguise, and the adjoint-matter identification is the part that still needs work.

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arXiv IDhep-th/9609219 (version 3)

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CitationNikita Nekrasov. "Five Dimensional Gauge Theories and Relativistic Integrable Systems." arXiv:hep-th/9609219 (1996). https://pith.science/paper/VIVGLC3X

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Editor's note

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The thing to know up front: the pure-gauge half of this is stronger than the abstract's 'we propose' suggests. The SU(2) spectral curve is isomorphic to the four-dimensional Seiberg-Witten curve at finite U, with matching differentials, and the one-loop prepotential is an actual computation rather than an ansatz. The adjoint-matter section is where it turns into a guess, and the paper is fairly open about that being a guess. The classifier's record says no limitations are stated; that's wrong — the text says plainly it could not identify the integrable system for fundamental matter, and defers the flow between systems to later work. Read those as acknowledged, not as findings of mine. What is genuinely new: the Kaluza-Klein sum giving F = (i/8π³R²)Σ Li₃(e^{2πR⟨α,a⟩}), with both the 4d logarithm and the 5d cubic intersection form coming out of the limits; the sinh-deformed curve of (2.20)-(2.23) coinciding with the relativistic Toda spectral curve; and the period computation (2.34), which shows the hypermultiplet mass sitting in the symplectic form. That last one is a nice concrete check and I'd point students at it. I'd push back on the reader's weakest-assumption framing. Eqs. (2.10)-(2.12) give an explicit finite-U isomorphism u = R²U² − 1/R² with matching differentials, so for pure SU(2) the periods are pinned at finite U, not just asymptotically. What is not shown is that those periods are the 5d central charges including magnetic string tensions — asserted, never computed independently. That is a physical assumption, not a mathematical gap, and it should be named as such. The genuinely soft section is 2.4. The elliptic Ruijsenaars-Schneider model is adopted as 'the natural candidate' because it degenerates correctly at R→0 and m→∞; two limits do not pin a curve, and another deformation with the same limits would pass the same tests. UV finiteness is hedged ('seems to be'), and the N>2 period identification is handed to [23], a mathematics paper whose bearing on the physical central charges is not demonstrated here. [25] is in preparation and [12] is the author's own thesis, so parts of the support are not independently checkable. None of that sinks the paper; it is a 1996 conference talk and reads like one. Who should read it: anyone working on 5d and 6d compactifications, M-theory on Calabi-Yau, or the gauge/integrable correspondence. It is a seed paper and it has held up as one. Send it to referees — the claims are specific enough to check, and the pure-gauge part already checks out.

Referee report

Significance

If correct, this is an important paper: it puts forward a concrete and checkable correspondence between five-dimensional N=2 gauge theories on a circle and known relativistic integrable systems, and it supplies explicit curves, differentials, and period integrals rather than a qualitative analogy. The perturbative prepotential in Eq. (2.8) is a genuine computation with a stated method, and its two limits are a nontrivial consistency check. The identification of the adjoint hypermultiplet mass with a period of the symplectic form in Eq. (2.34) is a sharp, falsifiable statement, and the reduction to supersymmetric quantum mechanics on the instanton moduli space in the appendix is a useful structural result. The paper is also candid about what is assumed and what is delegated, which makes the proposal easy to test.

Major objections (3)

[Section 2.3, Eq. (2.28); abstract:s0] The abstract advertises matter in the fundamental representation, but Section 2.3 supplies only a curve obtained by the replacement (2.27) with no derivation and no check against any independently computed quantity; the author states plainly that 'we were not able to identify them'. Since Section 2.1.1 already provides the machinery for the one-loop prepotential with matter (the same localization argument, with the hypermultiplet contributing to the determinant), a check of the N_f curve against the perturbative coupling would be feasible and would substantially strengthen the claim. Absent such a check, the abstract should be softened to say that the fundamental-matter case is conjectural.
[Section 2.4, Eqs. (2.29)-(2.34); abstract:s3] The finiteness of the five-dimensional adjoint theory is stated in the abstract as a result, but Section 2.4 only says it 'seems to be ultra-violet finite' and then proceeds 'Assuming this finiteness we introduce the microscopic coupling'. The elliptic Ruijsenaars-Schneider model is adopted as 'the natural candidate' because it reduces to elliptic Calogero-Moser as R goes to 0 and to relativistic Toda as m goes to infinity; these two limits do not fix the finite-(m,R) curve, and a different deformation with the same limits would pass the same checks. A comparison against the finite-m one-loop coupling in Eq. (2.29) would be the natural discriminator, and the paper should state whether the proposed model reproduces it.
[Section 2.2.1, Eqs. (2.10)-(2.12) and (2.3)-(2.9)] The isomorphism E_U isomorphic to E_u is asserted for u = R^2 U^2 - 1/R^2 without intermediate steps, and the degeneration points quoted (U = plus or minus (1/R^2) sqrt(1 plus or minus zeta^2)) do not follow from the stated definitions nu_4 = Lambda^2/u and nu_5 = 2 zeta^2 sinh^2 alpha without further rescalings. More importantly, the argument of the sinh in the perturbative coupling is 2 pi R <alpha,a> in Eq. (2.3), pi R a in Eq. (2.4), and aR/2 in Eq. (2.9); since Eq. (2.9) is the 'perturbatively exact coupling' against which the curve is checked at the end of Section 2.2.1, the claimed asymptotic matching cannot be verified as written. Please fix the conventions for a and alpha once and for all and show the algebra connecting the curve asymptotics to Eq. (2.9).

Minor objections (9)

[Throughout] There are numerous typographical and OCR artifacts that should be cleaned up before publication, including 'Cambridg e', 'compac tiﬁcation', 'abs ence', 'algbera', 'reﬀered', 'Vander monde' (Vandermonde), 'hyper mulitplets', 'ultra-vio let', 'Dimensionbal' in reference [4], 'Polytnomial' in reference [36], the missing hyphen in 'hepth/9401021', the stray dot in 'They count', and 'F uture directions'.
[Eq. (2.5)] The sum over i,j of |a_i - a_j|^3 presumably should run over i<j with an adjusted coefficient; as written it double counts pairs and includes the vanishing i=j terms.
[Section 2.1.1] The sentence 'The supersymmetry guarantees the exactness of the one-loop approximation' should be qualified as exact within perturbation theory, since the instanton corrections are discussed two paragraphs later and are not one-loop.
[Section 2.2.1] The statement that there are 'five singular points as opposed to the notorious three points of the four-dimensional theory' would be clearer if the five points were listed explicitly.
[Section 2.4, 'A bizarre symmetry'] The six-dimensional tensionless-string interpretation of the symmetry (2.35) is explicitly speculative ('We suspect'); it should be flagged as a conjecture rather than presented alongside the derived statements.
[Reference [12]] Reference [12] is a PhD thesis, cited for the details of the trace in the adjoint representation and for the elliptic genus computation; since a thesis is not readily accessible, the essential points should be summarized in the text.
[Section 2.2.2] For N>2 the identification of the relevant periods is delegated to reference [23]; a sentence stating which cycles yield which central charges would make the claim checkable from the text alone.
[Section 2.4, footnote 8] The remark that the period statement is local in m and that monodromy shifts the period is important for interpreting Eq. (2.34) and deserves to be promoted from a footnote into the main text.
[Eqs. (2.20)-(2.22)] The zeta-regularized determinant should be stated with its normalization; the factor (1/R) sinh(2 pi R a_i - x) differs from the standard sinh(R(a_i - x)/2) by normalization and this affects the comparison with Eq. (2.23).

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The paper

The paper claims

The exact quantum answer for a five-dimensional gauge theory is written in the spectral curve of a relativistic integrable chain, with the compactification radius setting the speed of light.

The reading

The paper proposes an exact, non-perturbative solution of $N=2$ supersymmetric gauge theory in five dimensions compactified on a circle of radius $R$, for pure gauge theory, fundamental matter, and adjoint matter. The strategy is inherited from four dimensions, where the Seiberg–Witten solution was recast as the spectral curve of a periodic Toda chain: here the chain is replaced by its relativistic analogue, with $1/R$ playing the role of the speed of light, so that the four-dimensional answer is recovered as $R^{-1}$ grows large. The author first computes the one-loop prepotential in closed form — a sum of trilogarithms, $F_{5d}^{pert}(a) = (i/8π^3R^2)∑_{α} Li_3(e^{2πR⟨α,a⟩})$ — and shows it reproduces both the Kaluza–Klein tower at small $R$ and the known cubic five-dimensional prepotential as $R→∞$. The non-perturbative content then enters through period integrals of a meromorphic one-form on the curve, whose asymptotics reproduce the perturbative coupling and whose degeneration loci encode where the effective description breaks down. For adjoint matter the proposal is the elliptic Ruijsenaars–Schneider model, and the hypermultiplet mass is recovered as a period of the symplectic form, which the author reads as evidence that the identification of the deformation parameter with the mass is correct.

What carries the argument

The load-bearing object is the Lax operator of the relativistic periodic Toda chain and its spectral curve, together with the meromorphic differential $λ = p dq$ whose periods are identified with the supersymmetric central charges $a_i$ and their duals. The relativistic deformation does all the work: because $1/R$ enters as the speed of light, sending $R→∞$ degenerates the relativistic chain into the ordinary Toda chain and the curve into the four-dimensional Seiberg–Witten curve, while keeping $R$ finite reproduces the entire Kaluza–Klein tower through the sinh factors in the deformed curve. The one-loop prepotential, a sum of trilogarithms, is the perturbative anchor that the periods must match in the weak-coupling limit, and the degeneration points of the curve are what the author uses to locate the singular loci of the moduli space — five of them rather than the familiar three of the four-dimensional theory.

Extended reading notes

Core claim

The central claim is that the low-energy effective action of five-dimensional $N=2$ gauge theory on $X×S^1_R$ is captured by a family of spectral curves belonging to a relativistic integrable system, in exactly the way the four-dimensional Seiberg–Witten solution is captured by the periodic Toda chain. For $G=SU(2)$ the author states the Hamiltonian $H(p,q) = (1/R^2)cosh(Rp)√(1 + 2(ΛR)^2 cosh q)$ and asserts that the family of curves $E_U$ defined by $U = H(p,q)$ yields the correct asymptotics of the coupling $τ(U)$, reducing to the four-dimensional curve under the substitution $u = R^2U^2 - 1/R^2$. For $G=SU(N)$ the corresponding Lax operator is written down explicitly and the spectral curve is brought to the form $ζ^N(χ + χ^{-1}) = y^{-N/2}P_N(y)$, which the author shows coincides with the $\sinh$-deformed Seiberg–Witten curve $z + Λ^{2N}/z = ∏_i (1/R)sinh(2πRa_i - x)$ obtained by replacing the Higgs field by the first-order operator $∂_t + A_t + iφ$ on the circle. The same construction produces curves with fundamental matter, and for the adjoint hypermultiplet the elliptic Ruijsenaars–Schneider model is proposed, with the mass identified through the period $(1/2πi)∫_Γ ω = m/2 + (1/2R)(n_1 + n_2τ)$. A recurring structural point is that the five-dimensional moduli space is a $Z_N$ quotient of the four-dimensional one, reflecting the freedom to multiply the Wilson loop by a center element when all matter is in the adjoint.

Load-bearing premise

The periods of the proposed curve's differential are assumed to equal the exact supersymmetric central charges at finite $U$, whereas the paper only checks the weak-coupling asymptotics and the four-dimensional limit, with no five-dimensional instanton computation performed or cited to test the finite-$U$ content.

Editorial extensions

If this is right

The exact prepotential of the compactified five-dimensional theory, including all instanton corrections weighted by $e^{-8π^2R/g_5^2}$, is computable by period integrals on a single curve rather than by direct instanton counting.
The decompactification limit $R→∞$ reproduces the cubic prepotential $F_{5d} = (1/6)∑_{i,j,k}C_{ijk}A^iA^jA^k$ of the five-dimensional theory, tying the curve to the intersection form of a Calabi–Yau threefold.
The theory with a massive adjoint hypermultiplet is ultraviolet finite, with an effective coupling $τ_{eff} ~ τ_{bare} + (i/2π)log[sinh^2(Ra)/(sinh(Ra+2πRm)sinh(Ra-2πRm))]$ that stops running at large $a$.
The adjoint theory possesses a shift symmetry $m → m + i(n_1 + n_2τ)/(2R)$, which the author suggests points to a six-dimensional tensionless string underlying the four-dimensional $N=4$ theory.
The $Z_N$ quotient structure of the moduli space, and its breaking by fundamental matter, gives a group-theoretic handle on the Coulomb branch of the compactified theory.

Reading between the lines

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

If the identification of the curve with the sinh-deformed Seiberg–Witten curve is exact at finite $U$, then the same geometry should compute the equivariant index of the instanton moduli space on $\mathbb{R}^4×S^1$, giving an independent route to the elliptic genus of that moduli space.
The five singular points of the $SU(2)$ curve, versus three in four dimensions, suggest a definite wall-crossing pattern for BPS states as $U$ is continued around the moduli space; this is a concrete prediction that could be checked against the curve's degeneration loci.
The chamber structure inherited from the trilogarithm's monodromy suggests that the effective prepotential is only piecewise smooth in the five-dimensional limit, which would show up as non-analytic behavior at the boundaries of the Weyl chamber where non-abelian gauge symmetry is restored.
The mass-shift symmetry of the adjoint theory implies a tower of states whose spacing is set by $g_5^2$; if such a tower were found in a string compactification, it would be evidence for the six-dimensional origin the author conjectures.

Original abstract

We propose a non-perturbative solution of N=2 supersymmetric gauge theory in five dimensions compactified on circle of a radius $R$. We consider the cases of the pure gauge theory as well as the theories with matter in the fundamental and in the adjoint representations. The pure theory as well as the one with adjoint hypermultiplet give rise to the known relativistic integrable systems with ${1\over{R}}$ playing the r\^ole of the speed of light. The theory with adjoint hypermultiplet exhibits some interesting finiteness properties. Talk given at the III International Conference ``Conformal Field Theories and Integrable Models'', Chernogolovka, June 23-30 1996

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, 4,302 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.11 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.38 · 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 propose a non-perturbative solution of N=2 supersymmetric gauge theory in five dimensions compactified on circle of a radius $R$. We consider the cases of the pure gauge theory as well as the theories with matter in the fundamental and in the adjoint representations. The pure theory as well as the one with adjoint hypermultiplet give rise to the known relativistic integrable systems with ${1\over{R}}$ playing the r\^ole of the speed of light. The theory with adjoint hypermultiplet exhibits some interesting finiteness properties. Talk given at the III International Conference ``Conformal Field Theories and Integrable Models'', Chernogolovka, June 23-30 1996

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We propose a non-perturbative solution of N=2 supersymmetric gauge theory in five dimensions compactified on circle of a radius $R$.	contribution 0.99	0.92	0.44	asserted 0.94
s1	We consider the cases of the pure gauge theory as well as the theories with matter in the fundamental and in the adjoint representations.	contribution 0.92	0.67	0.04	asserted 0.84
s2	The pure theory as well as the one with adjoint hypermultiplet give rise to the known relativistic integrable systems with ${1\over{R}}$ playing the r\^ole of the speed of light.	result 0.92	0.58	0.50	asserted 1.00
s3	The theory with adjoint hypermultiplet exhibits some interesting finiteness properties.	result 0.97	0.77	0.07	asserted 0.96
s4	Talk given at the III International Conference ``Conformal Field Theories and Integrable Models'', Chernogolovka, June 23-30 1996	background 0.99	0.07	0.01	unclear 0.79

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.95.

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

References

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Branes, Calabi-Yau Spaces, and Toroidal Compactification of the N=1 Six-Dimensional E_8 Theory
hep-th 1996-10 accept novelty 8.0 of 10

The SL(2,Z) duality of N=2 SU(2) with four flavors is the geometric SL(2,Z) of the torus used to compactify the six-dimensional E8 theory, and Calabi-Yau moduli singularities are field-theory singularities.
Elliptic spin Ruijsenaars-Schneider integrable models from 5d $\mathcal{N}=1$ gauge theories
hep-th 2026-08 conditional novelty 7.0 of 10

The Coulomb branch algebra of 5d necklace quiver gauge theories on R^3 x T^2 is shown to be the quantum algebra of observables of the elliptic spin Ruijsenaars-Schneider model, giving it a Poisson structure and quantization.
Dimers for Relativistic Toda Models with Reflective Boundaries
hep-th 2025-10 unverdicted novelty 7.0 of 10

Dimer graphs are constructed for relativistic Toda chains of listed Lie algebra types, and Seiberg-Witten curves of 5d N=1 pure SYM for group G are identified as spectral curves of the dual Toda chain for G^vee.
$Q$-operators, $q$-opers, and R-matrices in 5d $\mathcal{N}=1$ gauge theory
hep-th 2025-07 conditional novelty 7.0 of 10

In 5d N=1 U(N) gauge theory, codimension-two defects produce Q-operators whose q-difference equations are the Baxter TQ equations of XXZ spin chains built on bi-infinite evaluation modules of quantum affine algebras.
Coulomb and Higgs Branches from Canonical Singularities: Part 0
hep-th 2020-07 conditional novelty 7.0 of 10

The magnetic quiver of a 5d SCFT engineered by a canonical threefold singularity is the electric quiver of the 4d SCFT from the same singularity, with one abelian flavor symmetry gauged.
Exact quantization conditions for cluster integrable systems
hep-th 2015-12 conditional novelty 7.0 of 10

A conjectural exact quantization condition, written in terms of the Nekrasov-Shatashvili free energy of the associated toric Calabi-Yau, reproduces the numerically computed spectra of the Goncharov-Kenyon integrable s...
Membranes and Sheaves
math.AG 2014-04 conditional novelty 7.0 of 10

A conjectural correspondence identifies the M-theory membrane index on a Calabi-Yau fivefold with K-theoretic Donaldson-Thomas theory on a threefold, with the boxcounting parameter realized as a circle-equivariant weight.
Quantum geometry and quiver gauge theories
hep-th 2013-12 conditional novelty 7.0 of 10

In the limit where one Omega-deformation parameter goes to zero, the Y-functions of 4d, 5d, and 6d quiver gauge theories satisfy the q-character equations of the Yangian, quantum affine, or elliptic algebra attached t...
Instanton effects and quantum spectral curves
hep-th 2013-08 conditional novelty 7.0 of 10

The exact WKB quantization condition for the local P1×P1 quantum spectral curve requires non-perturbative instanton corrections that the authors conjecture are the standard (unrefined) topological string free energy, ...
Five-dimensional AGT Conjecture and the Deformed Virasoro Algebra
hep-th 2009-10 conditional novelty 7.0 of 10

The 5D N=1 pure SU(2) instanton partition function appears to equal the inner product of a deformed Gaiotto state in the q-Virasoro algebra, checked through level 9 in the coupling expansion.
Quantization of Integrable Systems and Four Dimensional Gauge Theories
hep-th 2009-08 accept novelty 7.0 of 10

Four-dimensional N=2 gauge theory in the Omega background quantizes the classical integrable system behind its Seiberg-Witten moduli space, with the Omega parameter as Planck's constant and the twisted superpotential ...
Quantum integrability and supersymmetric vacua
hep-th 2009-01 conditional novelty 7.0 of 10

Supersymmetric vacua of a class of two-dimensional gauge theories are identified one-to-one with eigenstates of integrable spin chains, with the XXX chain matched to the equivariant quantum cohomology of the cotangent...
Matrix Models, Geometric Engineering and Elliptic Genera
hep-th 2003-10 accept novelty 7.0 of 10

For N=2* theory compactified from six to four dimensions on a torus, matrix models, geometric engineering via the topological vertex, and elliptic-genus instanton calculus all yield the same prepotential, encoded in a...
Small Instantons, Little Strings and Free Fermions
hep-th 2003-02 conditional novelty 7.0 of 10

For the U(1) N=2 gauge theory with all chiral couplings, the partition function equals a free fermion correlator on a sphere and the all-genus topological string free energy on CP^1.
D-particle bound states and generalized instantons
hep-th 1998-04 conditional novelty 7.0 of 10

The principal contribution to the Witten index of the D0-brane quantum mechanics is 1/N^2 for D=4,6 and the divisor sum over 1/d^2 for D=10, computed by localization and residue evaluation.
Issues in Topological Gauge Theory
hep-th 1997-11 conditional novelty 7.0 of 10

The authors propose a scheme, built on a homogeneity principle and a conjectured universality of contact terms, that fixes all contact terms in twisted N=2 gauge theory and yields contour-integral formulas for correla...
Webs of (p,q) 5-branes, Five Dimensional Field Theories and Grid Diagrams
hep-th 1997-10 accept novelty 7.0 of 10

Five-dimensional supersymmetric gauge theories can be encoded in planar webs of (p,q) five-branes, where monopole tensions are face areas, the effective coupling is a weighted edge mass, and the Coulomb branch dimensi...
Quantum curves of 6d SCFTs on a -3 curve
hep-th 2026-10 conditional novelty 6.0 of 10

Quantum Seiberg-Witten curves for pure su(3), g2, and so(N) with N from 7 to 10, 6d SCFTs on a -3 curve, are proposed as elliptic difference equations verified to three-string order.
Refined Vafa-Witten invariants for toric surfaces from supersymmetric localization in 5D gauge theory
hep-th 2026-07 conditional novelty 6.0 of 10

5D N=1* supersymmetric localization on toric surfaces equals refined Vafa–Witten invariants for odd first Chern class; the even-c1 case fails without a hand-added constant.
On 5d SCFTs and their BPS quivers. Part I: B-branes and brane tilings
hep-th 2019-12 conditional novelty 6.0 of 10

The BPS spectrum of a 5d SCFT on a circle is organized by the derived category of coherent sheaves on its Calabi-Yau singularity, realized concretely as a fractional-brane quiver for toric cases.
Five-dimensional SCFTs and gauge theory phases: an M-theory/type IIA perspective
hep-th 2018-12 accept novelty 6.0 of 10

A type IIA reduction of M-theory on toric Calabi-Yau threefolds yields a checkable map from each resolution's geometry to the masses, couplings and instanton spectrum of the corresponding five-dimensional gauge theory.
5d Partition Functions with A Twist
hep-th 2018-08 conditional novelty 6.0 of 10

The authors derive the exact partition function of 5d N=1 gauge theories on S^3_b x Sigma_g as a sum over Bethe-Ansatz solutions, including instantons, and use it to compute 4d superconformal indices and holographic f...
Topologically twisted indices in five dimensions and holography
hep-th 2018-08 conditional novelty 6.0 of 10

A contour-integral gluing formula for the 5D topologically twisted index on M4 x S1, with large-N limits reproducing AdS7 holography and predicting AdS6 black hole entropy.
BPS relations from spectral problems and blowup equations
hep-th 2016-09 accept novelty 6.0 of 10

The K-theoretic blowup equations of SU(N) gauge theory imply the compatibility condition linking S-dual quantization and spectral-determinant quantization for the Y^{N,m} geometries, proving a prior conjecture.
Quiver elliptic W-algebras
hep-th 2016-08 conditional novelty 6.0 of 10

Elliptic W-algebras W_{q1,q2,p}(Gamma) are defined for arbitrary quivers via the six-dimensional quiver gauge theory partition function, with holomorphic currents shown to commute with screening charges.
Exact quantization conditions for the relativistic Toda lattice
hep-th 2015-11 accept novelty 6.0 of 10

A conjectural exact quantization condition for the relativistic Toda lattice, combining the Nekrasov-Shatashvili free energy with an S-dual non-perturbative term, reproduces numerically computed spectra for three particles.
Defects and Quantum Seiberg-Witten Geometry
hep-th 2014-12 conditional novelty 6.0 of 10

The Nekrasov partition function of 5d U(N) gauge theory with a maximal monodromy defect is an eigenfunction of the elliptic Ruijsenaars-Schneider Hamiltonian, with the Wilson loop expectation value as eigenvalue.
Fiber-Base Duality and Global Symmetry Enhancement
hep-th 2014-11 conditional novelty 6.0 of 10

The 5D Nekrasov partition function for SU(2) with Nf ≤ 7 flavors, expanded in a duality-invariant Coulomb modulus, decomposes into E_{Nf+1} characters, with fiber-base duality generating the enhancement.
Non-Lagrangian Theories from Brane Junctions
hep-th 2013-10 conditional novelty 6.0 of 10

The paper derives Seiberg-Witten curves for the 5D T_N brane junctions at general N and gives a refined-topological-vertex formula for their Nekrasov partition functions, after removing a conjectured non-full-spin factor.
5D super Yang-Mills theory and the correspondence to AdS$_7$/CFT$_6$
hep-th 2013-04 conditional novelty 6.0 of 10

5D super Yang-Mills on S5 reproduces the AdS7 supergravity Wilson loop at the symmetry-enlargement point, but the free energy agreement requires fitting the compactification radius to the free energy at a different hy...
Walls, Lines, and Spectral Dualities in 3d Gauge Theories
hep-th 2013-01 conditional novelty 6.0 of 10

A proposed spectral duality identifies 3d N=2 gauge theories with integrable systems through their shared spectral curve, and period integrals on knot A-polynomial curves yield the spectrum of line operators and domain walls.
Twisted supersymmetric 5D Yang-Mills theory and contact geometry
hep-th 2012-02 accept novelty 6.0 of 10

The full perturbative partition function of twisted 5D supersymmetric Yang-Mills on the five-sphere is computed exactly and expressed as a finite-dimensional matrix model.
ABCD of instantons
hep-th 2004-04 accept novelty 6.0 of 10

The paper derives the low-energy prepotential and Seiberg-Witten curves for N=2 supersymmetric Yang-Mills theory with SU(N), SO(N), and Sp(N) gauge groups by instanton counting in an Omega-background, without resolvin...
On classification of rank two theories with eight supercharges Part III: Seiberg-Witten geometry
hep-th 2025-08 unverdicted novelty 5.0 of 10

The abstract claims rank-two Seiberg-Witten geometries can be systematized via one-parameter curve families y^2 = f(x,t), with f fixed by singular fibers at t = infinity.
The $U$-plane of rank-one 4d $\mathcal{N}=2$ KK theories
hep-th 2021-07 accept novelty 5.0 of 10

The Mordell-Weil group of the Seiberg-Witten elliptic fibration determines the global form of the flavour symmetry group for rank-one 4d N=2 theories and their 5d parents on a circle.
Instantons, Symmetries and Anomalies in Five Dimensions
hep-th 2020-09 accept novelty 5.0 of 10

Five-dimensional non-abelian gauge theories have a finite-order mixed 't Hooft anomaly between the instanton number symmetry and the one-form center symmetry, which constrains their phases and the symmetry enhancement...
Coherent states in quantum $\mathcal{W}_{1+\infty}$ algebra and qq-character for 5d Super Yang-Mills
hep-th 2016-06 conditional novelty 5.0 of 10

The 5d qq-characters built from Gaiotto states, intertwiners and flavor vertex operators of the quantum W_{1+∞} algebra are shown to be polynomials, yielding Ward identities for 5d N=1 super Yang-Mills.
M5-branes, toric diagrams and gauge theory duality
hep-th 2011-12 accept novelty 5.0 of 10

The paper derives an explicit map between the parameters of 5D N=1 SU(N)^{M-1} and SU(M)^{N-1} linear quiver gauge theories, showing they describe the same low-energy physics.
Proving AGT conjecture as HS duality: extension to five dimensions
hep-th 2011-05 conditional novelty 4.0 of 10

A q-deformed five-dimensional version of the AGT-as-Hubbard-Stratonovich-duality argument is constructed and checked term by term against Nekrasov functions at β=1, with β≠1 left unresolved.
Matrix Model Conjecture for Exact BS Periods and Nekrasov Functions
hep-th 2009-11 conditional novelty 4.0 of 10

The exact Seiberg-Witten differential for the doubly deformed prepotential is conjectured to be the one-point resolvent of a quiver beta-ensemble in its Dijkgraaf-Vafa phase, matching Nekrasov functions to Virasoro co...
Nekrasov Functions from Exact BS Periods: the Case of SU(N)
hep-th 2009-11 conditional novelty 4.0 of 10

The deformed Seiberg-Witten period construction, with the differential replaced by its exact WKB quantum version, reproduces Nekrasov's prepotential to first order in the deformation parameter and first instanton orde...
Quantum field theory and four-manifolds: Perspectives from physics and geometry
hep-th 2026-10 accept novelty 1.0 of 10

Four summer-school lecture series, presented as a connected account, surveying the route from twisted supersymmetric Yang-Mills theory to four-manifold invariants and their algebro-geometric formulations.
Spectral Networks: Bridging higher-rank Teichm\"uller theory and BPS states
math-ph 2024-11 unverdicted

A comprehensive introduction to spectral networks that develops higher-rank Teichmüller theory in parallel with class S gauge theory and BPS spectra.

How later papers used it

86 later papers cite it

Evidence: substantial. 82 of the citing papers share no author with it, and 13 of those carry a confident label from Pith's citation classifier.

Pith read 28 of the 148 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

1 later paper by other authors is listed below: 1 used its method or tool.

Uses its method or tool 1 of 3 papers with this label

Imitated the arguments in the appendix to this paper to localize the theory with its action in the presence of monopole singularities to the minimal supersymmetric quantum mechanics on the compactified moduli space of Hecke modifications.

Coulomb Branches in 3d $\mathcal{N} = 4$ Revisited (2024) · 0.83 · paraphrased: the licence does not allow quoting

How the literature uses it

83 arXiv papers cite this in 144 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 · 101 sentences without a confident label

Builds on it 3 papers

Our work is a generalization and covariantization of previous works by Nekrasov [ref] , by Baulieu, Losev and Nekrasov [ref] and of earlier works by Nair and Schiff [ref] .

Twisted supersymmetric 5D Yang-Mills theory and contact geometry · Summary and outlook · 1.00
In this Section, we consider a higher dimensional analog of the geometric analysis discussed in this Chapter [ref] .

Instanton Counting, Quantum Geometry and Algebra · Eight supercharge theory in higher dimensions · 0.80
We could generalize this setup to 5d $𝒩=1$ and 6d $𝒩=(1,0)$ theories with supergroup gauge symmetry to explore the correspondence to the trigonometric and elliptic integrable systems [ref] .

Quantum Integrable Systems from Supergroup Gauge Theories · Discussions and Future Directions · 0.83

Uses its method or tool 2 papers

The prepotential can be derived in a slightly roundabout way, by compactifying the theory on a circle, thus obtaining a 4d $=2$ KK theory, and then taking the 5d limit on the 4d $=2$ prepotential [ref] .

Five-dimensional SCFTs and gauge theory phases: an M-theory/type IIA perspective · The 5d $=1$ gauge-theory prepotential, revisited · 0.92
We take both coordinates $x,y$ and moduli $Q_b, Q_f$ to depend on $R$ through the parametrization adopted in [ref] x=e^q , y = e^√(2) i R p , Q_b = -Λ^2 R^2 , Q_f = 1/2 R^4 U(R)^2 .

Exponential BPS graphs and D-brane counting on toric Calabi-Yau threefolds: Part II · Four-dimensional limit · 0.91

Compares with it 5 papers

The spectral curve is [equation] Scaling $y →(x-x^-1)^2 y$ gives [equation] This spectral curve coincides with the Seiberg-Witten curve of five dimensional pure $SO(2N)$ supersymmetric gauge theory on $^4 ×S^1$ [ref] .

Dimers for Relativistic Toda Models with Reflective Boundaries · $D^(1)_N$ · 0.86
We can easily see that the curve described by [ref] is a special case of the general 5D $SU(3)$ $N_f=6$ SW curve [ref] [equation] if we constrain the additional parameters as [equation] This is the 5D uplift of the statement in [ref] that the 4D $E_6$ CFT is obtained as the strong coupling limit of the 4D SU(3) theory with six flavors.

Non-Lagrangian Theories from Brane Junctions · Seiberg-Witten curves from M-theory · 0.86
In this limit, the curve ([ref]) reduces to [equation] If we choose $C=1$ and assume that $b=ã_1+ã_2$ with $ã_1 = ã_2^-1$ the curve ([ref]) coincides with the one previously given in [ref] .

M5-branes, toric diagrams and gauge theory duality · $SU(2)$ gauge theory with four flavors · 1.00

Cites it as support 1 paper

A second motivation for studying the half-geometry is that taking $τ→0$ corresponds to a weak-coupling limit for 5d $=1$ $SU(2)$ Yang-Mills theory [ref] .

The threefold way to quantum periods: WKB, TBA equations and q-Painlev\'e · Half-geometry limit · 0.98

Cites it as background 27 papers

Nekrasov [ref] proposed a non-perturbative solution for 5d gauge theories compactified on a circle using relativistic integrable systems.

Dimer Models and Integrable Systems · 5d Gauge Theories and Dimers · 0.81
Our starting point is the five-dimensional Seiberg-Witten description [ref] , where the Seiberg-Witten curve $Σ$ is identified with the mirror curve of the local Calabi-Yau geometry “geometrically engineering" the 5d theory on $^4×S^1$ [ref] .

The threefold way to quantum periods: WKB, TBA equations and q-Painlev\'e · The general picture · 0.89
It is known [ref] that by using the supersymmetry the partition function of the five dimensional topological gauge theory is expressed as a partition function of a quantum mechanics of a non-charged particle moving on a instanton moduli space.

Coadjoint Orbits and Wilson Loops in Five Dimensional Topological Gauge Theories · Introduction · 0.83

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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