Instantons and recursion relations in N=2 Susy gauge theory

Marco Matone

REVIEW

Instantons and recursion relations in N=2 Susy gauge theory

Marco Matone

High Energy Physics - Theory · Submitted 15 Jun 1995 · 12 references

Pith's verdict: This is the Matone relation paper, and the central identity, the modular invariance, and the recursion all hold up; the only real gap is a one-line B=0 check that the author asserts instead of displaying.

T0 · AI review

How this review was made: An AI referee (selfhosted/deepseek-v4.1-flash) read the paper on October 29, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

T0 AI review An AI read the paper and answered a fixed list of questions.
T1 Audit Automatic checks that the references and files the paper relies on exist and match.
T2 Compute The paper's code or calculations re-run with the same results.
T3 Formal statement The claims written out in a formal math language a computer can read.
T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 4 major objections · 7 minor · Ask about this paper · The compiled record · Read the full review →

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv IDhep-th/9506102 (version 1)

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

CitationMarco Matone. "Instantons and recursion relations in N=2 Susy gauge theory." arXiv:hep-th/9506102 (1995). https://pith.science/paper/PNINZOQX

Cite this reviewBibTeX for Pith's review of this paper

Claim paper

Why Pith says this

Highlight any text to comment on it.

Editor's note

read the letter

Short version: this is the paper that gives you u = πi(F − ½a∂_aF), and it holds up. If you work in Seiberg–Witten you already use it; if you don't, it's the cleanest statement of why the instanton series is not an independent input but a consequence of one differential equation seeded by the one-loop term. The abstract's claim at [abstract:s2] is the load-bearing one and it is correct as stated, modulo grammar. What's actually new: the modular invariance of G under Γ(2) (Eq. 13), the nonlinear ODE (35), and the recursion (36) that generates every G_k from G_0 = 1/2. The transformation law in Eqs. (8)–(11) is a short computation and it's right — the a_D², a², and aa_D cross terms cancel using AD − BC = 1. The chain from the uniformizing equation (25) through (30) to (35) is a legitimate change of variables, though the uniformization apparatus is heavier than the result needs; the paper could say plainly that it is a chain-rule substitution dressed in Fuchsian-group language. The soft spot we both flagged — setting B = 0 after Eq. (34) with 'one can check' — is presentational, not substantive. The obstruction sits at order a² with coefficient −4BG_0², so B = 0 follows in one line. The paper's phrasing about 'a^{−2(2k+1)} terms' is imprecise about where the obstruction lives, but the conclusion is right. I'd want that line displayed in a referee report, not a rewrite. Two smaller things. The double integral (7) for F(u) is stated rather than derived; nothing downstream depends on it, so it's minor. And footnote 2's cross-check against [7] is only up to an overall normalization convention, which weakens the independent confirmation more than the text admits — the agreement of G_1, G_2, G_3 is real, but the caveat should be spelled out. The closing line about extending to more general cases is a promissory note with nothing behind it here; fine for a letter, just don't cite it as done. Who should read it: anyone doing Seiberg–Witten or Nekrasov partition functions, and anyone who wants a compact worked example of a prepotential fixed by a single ODE. It's eight pages and the payoff is disproportionate. Recommendation: send it out. The result is correct, the derivation is checkable, and the gaps are the kind a referee closes in one round.

Referee report

Significance

If correct, the result is a genuinely predictive structure: a single differential equation seeded only by the perturbative one-loop coefficient G_0 = ½ generates the entire nonperturbative prepotential of the SU(2) theory, with no fitted parameters. The identification of the modular-invariant combination πi(F − ½ a ∂_a F) with the modulus u is a sharp, falsifiable statement, and the first coefficients produced by Eq. (36) are stated to agree with the independent instanton computation of Ref. [7]. The uniformization-theory route via the Γ(2) uniformizing equation (25) and the reduction from third- to second-order equation (29)–(30) is economical and, as far as I can check, internally consistent: the accessory parameters c_1 = −c_2 = ¼ inserted in Eq. (18) do reproduce the Schwarzian potential (3+u²)/(4(1−u²)²) of Eq. (25), and the Wronskian value 2i/π in Eq. (38) is consistent with G = u and Eq. (12). The paper is terse even by letter standards, and several load-bearing algebraic steps are asserted rather than displayed; these are fixable without changing the conclusions.

Major objections (4)

[§3, Eqs. (34)–(35)] The step that sets B = 0 is the weakest link in the chain and is asserted in a single sentence: 'Inserting the expansion (14) one can check that the only way to compensate the a^{−2(2k+1)} terms is to set B = 0.' No check is displayed, yet Eq. (35) and hence the entire recursion (36) depend on it. The check is short and should be shown: writing G = G_0 a² + G_1 a^{−2} + … with G_0 = ½, the product (1−(G+B)²)G″ contains a term −4B G_0² a², while ¼ a G′³ has no a² contribution, so the a² coefficient is −4B G_0² and vanishes only for B = 0. Equivalently, u = G + B together with a ∼ √(2u) gives G = a²/2 − B + O(a^{−2}), which is incompatible with the constant-free expansion (14). Note also that the phrase 'a^{−2(2k+1)} terms' is imprecise: the obstruction sits at order a², not at negative powers. Please display the argument.
[§3, Eq. (36)] The recursion relations are quoted as a finished formula with the combinatorial coefficients c(k,n) = 2k(n−k−1)+n−1 and d(j,k,n) = [2(n−j)−1][2n−3j−1+2k(j−k+1)] and no derivation. Since Eq. (36) is the paper's central deliverable, the reader should be able to verify it without redoing the order-by-order expansion of Eq. (35) from scratch. At minimum, show the intermediate step (the general order-by-order equation obtained by inserting G = Σ G_k a^{2−4k} into (35)) and work out the n = 0 and n = 1 cases explicitly, which give G_1 = ¼ and G_2 = 5/64 and thus exhibit how the coefficients c and d arise.
[§1, Eq. (7); abstract, sentence [abstract:s3]] The abstract advertises that the paper gives 'the explicit expression of F as function of u', but Eq. (7) is an unevaluated double integral over x and y with a logarithm whose branch is not specified, and it is never checked against the asymptotic expansion (2) that defines F_k. Either evaluate the integral using the constancy of η established in §3, or verify that expanding (7) at large u reproduces the leading i/(2π) a² log a² term and the first few F_k. As written, the reader has no way to confirm that (7) is the prepotential of Eq. (4) rather than a formal antiderivative.
[Footnote 2 and §3, final paragraph] The agreement with Ref. [7] is the only independent check of the recursion, and footnote 2 states it only up to an unspecified normalization, referring the reader to 'the invariance of the quantity F_k F^{KLT}_k / (F^{KLT}_{k+1} F_{k+1})'. This is too vague to serve as a cross-check. Please give the explicit dictionary between the two normalizations (the relative factor of Λ, of i, and of 2πi between G_k and F_k) so that the quoted values G_1 = 1/2², G_2 = 5/2⁶, G_3 = 9/2⁷ can be compared term by term with the coefficients of [7]. Note also that Ref. [7] is cited as a preprint; its publication status should be updated if available.

Minor objections (7)

[Abstract] The phrase 'the instanton contribution are determined by recursion relations' has a grammatical error; it should read 'the instanton contributions are determined by recursion relations'.
[§3, Eqs. (31) and (34)] The prime symbol is overloaded: in Eq. (31) it denotes ∂_u, in Eq. (34) it denotes ∂_a, and in Eq. (10) the notation F^{(k)} is used for ∂_a^k F. Please state the convention explicitly at each switch, since the two derivatives are related by a nontrivial Jacobian.
[§3, Eq. (32)] The constant A is fixed to 1 by the asymptotic relation a ∼ √(2u). Please state which asymptotic form of a(u) is being used (the leading term of Eq. (4) at large u) and to what order the matching is performed.
[Abstract, final sentence] The claim that 'These results can be extended to more general cases' is not supported anywhere in the body of the paper; no SU(n) or SO(2n+1) case is discussed. Either sketch the extension or soften the sentence.
[§1, Eqs. (4)–(7)] The branches of the square roots in Eqs. (4)–(7) and the integration contour in the double integrals (5)–(7) are not specified. Since the identification G = u depends on the constancy of η defined by (5), the domain of validity of that statement should be stated.
[§2, Eq. (26)] The Legendre functions P_{−1/2} and Q_{−1/2} in Eq. (26) are written without their argument; presumably it is u, but this should be made explicit, together with the branch used for √(1−u²).
[§3, Eq. (38)] The value 2i/π for the Wronskian is stated without derivation. It follows from combining Eq. (12), Eq. (32) with A = 1, and the constancy of η; a one-line derivation would help the reader and would also fix the normalization convention used in Eq. (36).

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

A single non-linear differential equation, built from the prepotential of N=2 supersymmetric SU(2) gauge theory, is claimed to generate every instanton correction from one seed term.

The reading

The paper asks what actually fixes the nonperturbative part of the low-energy effective action of $N=2$ supersymmetric $SU(2)$ gauge theory. The answer is packaged in the prepotential $F(a)$, whose expansion in powers of $a$ contains a one-loop logarithm plus an infinite tower of instanton corrections. The author isolates the combination $G(a) = i(F - (1/2)a F'(a))$, shows it is invariant under the modular transformations that mix the electric and magnetic periods, and shows it obeys the non-linear equation $(1-G^2)G'' + (1/4)a (G')^3 = 0$. Because the coefficients of $G$ are proportional to the instanton coefficients, that equation is a recursion: once the leading term $G_0 = 1/2$ is inserted, every higher coefficient is determined in turn. The paper also identifies $G$ with the inverse of the modulus map, $u = G(a)$, and writes $F$ explicitly as a function of $u$ through a double integral.

What carries the argument

The load-bearing device is the uniformization of the thrice-punctured sphere: the moduli coordinate $u$ is written as a ratio of two solutions of a second-order equation whose potential is a Schwarzian derivative, so that the periods $a$ and $a_D$ become a flat section with monodromy in $Gamma(2)$. The reduction $(d^2/du^2 + (3+u^2)/(4(1-u^2)^2)) sqrt{1-u^2} d/du = (1/sqrt{1-u^2}) d/du [(1-u^2)d^2/du^2 - 1/4]$ is what makes the periods solve a second-order equation with constant Wronskian, and that constant Wronskian is what lets the author trade the modulus $u$ for the special coordinate $a$ and obtain a closed equation for $G(a)$.

Extended reading notes

Core claim

The central claim is that the whole instanton expansion of the $SU(2)$ prepotential is governed by one ordinary differential equation. The author first derives the modular transformation law of $F$ under the monodromy group of the quantum moduli space, finding that $F$ shifts by a quadratic polynomial in $a$ and $a_D$ while the combination $G(a) = i(F - (1/2)a F'(a))$ is left untouched. Using uniformization theory, he shows that $a$ and $a_D$ both solve the second-order equation $[(1-u^2)d^2/du^2 - 1/4] = 0$, whose Wronskian is a constant, and that this flatness is what reduces the third-order uniformizing equation to second order. Rewriting that equation in the variable $a$ rather than $u$ turns it into $(1-G^2)G'' + (1/4)a (G')^3 = 0$ for $G$, and expanding $G = sum_k G_k a^{2-4k}$ with $G_k = 2 i k F_k$ converts the differential equation into a recursion for the $G_k$. The author also reports $u = G(a)$ and $a a_D' - a_D a' = 2i/pi$, and checks the first few coefficients against previously computed instanton contributions.

Load-bearing premise

The derivation leaves an additive constant undetermined in the differential equation for $G$, and the paper removes it by asserting that only one choice cancels the negative powers of $a$ in the expansion, without displaying the check.

Editorial extensions

If this is right

Every instanton coefficient of the $SU(2)$ prepotential is fixed by the recursion generated from the single seed $G_0 = 1/2$, with no further input from the Seiberg-Witten curve at each order.
The modular-invariant function $G$ doubles as the inverse modulus map, $u = G(a)$, so the relation between the microscopic modulus and the special coordinate is obtained directly rather than by inverting a series.
The constant Wronskian $a a_D' - a_D a' = 2i/pi$ gives an explicit handle for locating the curve where $Im(a_D/a) = 0$, the locus of marginal stability.
The author states the construction extends to more general gauge groups and matter content, so the same differential-equation route to instanton coefficients is proposed as a template rather than a one-off computation.
The first computed coefficients reproduce those obtained by independent series inversion, which is offered as evidence that the recursion is correctly normalised.

Reading between the lines

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

The identification $u = G(a)$ says the modulus is essentially the Legendre-type transform of the prepotential with respect to $a$; read that way, the paper's equation is a statement about a generating function whose conjugate variable is the coordinate itself, which suggests the pattern could be tested in rank-one theories with matter hypermultiplets.
If the recursion is exact, the radius of convergence of the instanton series should be set by the singular points of $G$, so the singularities of the differential equation ought to line up with the strong-coupling points of the moduli space; that is a structural prediction a reader could check directly.
The step that discards an additive constant in $G$ is asserted rather than shown, and a nonzero constant would change the coefficients of the recursion; computing a few higher orders independently would settle this cheaply.
Because the equation is first order in $G'$ after a substitution, one could look for a closed-form first integral, which would turn the recursion into an explicit generating function rather than an order-by-order procedure.

Original abstract

We find the transformation properties of the prepotential ${\cal F}$ of $N=2$ SUSY gauge theory with gauge group $SU(2)$. In particular we show that ${\cal G}(a)=\pi i\left({\cal F}(a)-{1\over 2}a\partial_a{\cal F}(a)\right)$ is modular invariant. This function satisfies the non-linear differential equation $\left(1-{\cal G}^2\right){\cal G}''+{1\over 4}a {{\cal G}'}^3=0$, implying that the instanton contribution are determined by recursion relations. Finally, we find $u=u(a)$ and give the explicit expression of ${\cal F}$ as function of $u$. These results can be extended to more general cases.

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, 4,340 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.17 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.26 · compares with prior results 0.05 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

We find the transformation properties of the prepotential ${\cal F}$ of $N=2$ SUSY gauge theory with gauge group $SU(2)$. In particular we show that ${\cal G}(a)=\pi i\left({\cal F}(a)-{1\over 2}a\partial_a{\cal F}(a)\right)$ is modular invariant. This function satisfies the non-linear differential equation $\left(1-{\cal G}^2\right){\cal G}''+{1\over 4}a {{\cal G}'}^3=0$, implying that the instanton contribution are determined by recursion relations. Finally, we find $u=u(a)$ and give the explicit expression of ${\cal F}$ as function of $u$. These results can be extended to more general cases.

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 find the transformation properties of the prepotential ${\cal F}$ of $N=2$ SUSY gauge theory with gauge group $SU(2)$.	contribution 0.93	0.95	0.14	asserted 1.00
s1	In particular we show that ${\cal G}(a)=\pi i\left({\cal F}(a)-{1\over 2}a\partial_a{\cal F}(a)\right)$ is modular invariant.	result 0.99	0.93	0.27	asserted 1.00
s2	This function satisfies the non-linear differential equation $\left(1-{\cal G}^2\right){\cal G}''+{1\over 4}a {{\cal G}'}^3=0$, implying that the instanton contribution are determined by recursion relations.	result 0.79	0.90	0.45	asserted 1.00
s3	Finally, we find $u=u(a)$ and give the explicit expression of ${\cal F}$ as function of $u$.	result 0.98	0.96	0.15	asserted 1.00
s4	These results can be extended to more general cases.	future work 0.62	0.44	0.06	hedged possibility 0.69

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

What was examined, and what was not

sources read

arXiv listing abstract v1

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

Works cited (12)

[7]

A. Klemm, W. Lerche and S. Theisen, Nonperturbative Eﬀective Actions of N=2 Super- symmetric Gauge Theories , CERN-TH/95-104, LMU-TPW 95-7, hep-th/9505150

View on Pith Pith review arXiv
[1]

N. Seiberg and E. Witten, Nucl. Phys. B426 (1994) 19

View on Pith 1994
[2]

P. Argyres and A. Faraggi, Phys. Rev. Lett. 73 (1995) 3931, hep-th/9411057; A. Klemm, W. Lerche, S. Theisen and S. Yankielowicz, Phys. Lett. B344 (1995) 169, hep-th/9411048

View on Pith Pith review arXiv 1995
[3]

M. Douglas and S. Shenker, Dynamics of SU (N) Supersymmetric Gauge Theory , RU- 95-12, hep-th/9503163

View on Pith Pith review arXiv
[4]

U. Danielsson and B. Sundborg, The Moduli Space and Monodromies of N=2 Super- symmetric SO(2r+1) Yang-Mills Theory , USIP-95-06, UUITP-4/95, hep-th/9504102. 2Notice that we are using diﬀerent normalizations, thus to compare w ith F KLTk in [7] one should check the invariance of the quantity FkF KLTk F KLTk+1Fk+1 . 7

View on Pith Pith review arXiv
[5]

P. Argyres and M. Douglas, New Phenomena in SU(3) Supersymmetric Gauge Theory , IASSNS-HEP-95/31, RU-95-28, hep-th/9505062

View on Pith Pith review arXiv
[6]

N. Seiberg, Phys. Lett. 206B (1988) 75

View on Pith 1988
[8]

L. Bonora and M. Matone, Nucl. Phys. B327 (1990) 415

View on Pith 1990
[9]

M. Matone, Int. J. Mod. Phys. A10 (1995) 289

View on Pith 1995
[10]

A. Ceresole, R. D’Auria and S. Ferrara, Phys. Lett. 339B (1994) 71

View on Pith 1994
[11]

A. Fayyazuddin, Some Comments on N=2 Supersymmetric Yang-Mills , Nordita 95/22, hep-th/9504120

View on Pith Pith review arXiv
[12]

P. Argyres, A. Faraggi and A. Shapere, Curves of Marginal Stability in N=2 super-QCD , IASSNS-HEP-94/103, UK-HEP/95-07, hep-th/9505190. 8

View on Pith Pith review arXiv

Forward citations

Cited by 41 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

The Proof of the Dijkgraaf-Vafa Conjecture and application to the mass gap and confinement problems
hep-th 2006-02 conditional novelty 8.0 of 10

The Dijkgraaf-Vafa glueball superpotential equations are derived from gauge invariance and a U(1)_R Ward identity, proving the strong conjecture modulo the weak conjecture.
Aspects of $\mathcal{N}=2$ SUSY gauge theories and applications to Quantum Field Theory and Gravity
hep-th 2026-10 conditional novelty 7.0 of 10

The thesis derives a new hypergeometric representation of the confluent Heun function via the quantum Seiberg-Witten curve, yielding resummed tail contributions to gravitational waveforms that agree excellently with n...
Projecting Gravitational Fluctuations onto Near-Horizon Throats
hep-th 2026-08 conditional novelty 7.0 of 10

A pair of coordinate changes projects Heun-type gravitational fluctuation equations onto two matched hypergeometric problems, reproducing instanton-counting results and yielding Schwarzian zero modes.
Analytic approaches to perturbations of strongly coupled Yang-Mills plasma
hep-th 2026-06 unverdicted novelty 7.0 of 10

Seiberg-Witten instanton expansions combined with exact WKB period integrals allow analytic computation and continuation of quasinormal modes from large q to q=0.
Quasinormal Modes of Extremal Reissner-Nordstrom Black Holes via Seiberg-Witten Quantization
hep-th 2026-03 unverdicted novelty 7.0 of 10

Quasinormal-mode frequencies of extremal Reissner-Nordström black holes for charged and massive scalar fields are computed through Seiberg-Witten/Nekrasov-Shatashvili quantization.

Show 36 more citing papers

Bouncing singularities and thermal correlators on line defects
hep-th 2026-03 unverdicted novelty 7.0 of 10

Retarded correlators of displacement operators on line defects in holographic thermal CFTs exhibit bouncing singularities that match between interior-sensitive WKB and boundary-only OPE analyses.
Tame Complexity of Effective Field Theories in the Quantum Gravity Landscape
hep-th 2026-01 conditional novelty 7.0 of 10

Effective field theories consistent with quantum gravity are conjectured to have uniformly bounded 'tame complexity', a quantitative measure of the information needed to specify them.
Eigenfunctions of deformed Schr\"odinger equations
hep-th 2025-11 conditional novelty 7.0 of 10

Explicit entire eigenfunctions are constructed for the difference operators 2Λ^N cosh(p)+V_N(x) with arbitrary polynomial potential; they become square-integrable only at a discrete set of energies.
$\mathcal{N}=2$ Super Yang-Mills in AdS$_4$ and $F_{\text{AdS}}$-maximization
hep-th 2025-06 conditional novelty 7.0 of 10

A new maximization principle, F_AdS-maximization, selects boundary conditions for N=2 SYM in AdS4 and predicts a transition to a U(1)-Higgsed phase at strong coupling.
One-loop corrections to near extremal Kerr thermodynamics from semiclassical Virasoro blocks
hep-th 2024-12 conditional novelty 7.0 of 10

Near-extremal Kerr black holes get a universal (s-1/2) log T_H entropy correction from long-lived modes, derived exactly from Virasoro blocks.
Black Hole Perturbation Theory Meets CFT$_2$: Kerr Compton Amplitudes from Nekrasov-Shatashvili Functions
hep-th 2023-12 conditional novelty 7.0 of 10

Kerr Compton phase shifts and the tree-level point-particle gravitational Compton amplitude through O(a^8) follow from the Nekrasov-Shatashvili function of the confluent Heun equation, with no super-extremal spin cont...
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...
"Waveforms" at the Horizon
gr-qc 2026-02 conditional novelty 6.0 of 10

A probe scattering off a Schwarzschild black hole transfers a definite leading-order post-Minkowskian angular momentum to the horizon, given by new closed formulas (3.24b), (3.29), (3.36).
Phase Transitions at Unusual Values of $\theta$
hep-th 2025-06 conditional novelty 6.0 of 10

Using Seiberg-Witten theory plus anomaly mediation, the authors find first-order phase transitions in the theta-dependent vacuum energy at theta = pi (N_F=0), 0 and pi (N_F=1), pi/2 and 3pi/2 (N_F=2), and pi/4, 3pi/4,...
One loop corrections to the thermodynamics of near-extremal Kerr-(A)dS black holes from Heun equation
hep-th 2025-06 conditional novelty 6.0 of 10

Cold near-extremal Kerr-de Sitter black holes carry universal log(T) entropy corrections while rotating Nariai geometries do not, as derived from Teukolsky/Heun connection coefficients.
A note on rank $\frac{3}{2}$ Liouville irregular block
hep-th 2025-02 conditional novelty 6.0 of 10

The paper derives the rank 3/2 irregular conformal block for H1 Argyres-Douglas theory via holomorphic anomaly recursion and a deformed Seiberg-Witten curve, exact in the coupling.
Black hole perturbation theory and multiple polylogarithms
hep-th 2023-07 conditional novelty 6.0 of 10

A recursive expansion in multiple polylogarithms yields analytic series for black hole quasinormal frequencies in Schwarzschild (anti) de Sitter space, reproducing published numerical values and extending the hydrodyn...
Holographic thermal correlators from supersymmetric instantons
hep-th 2022-06 conditional novelty 6.0 of 10

An exact, instanton-expandable formula for the thermal scalar two-point function in 4D holographic CFTs, checked against direct numerical solution and against known light-cone bootstrap results.
QNMs of branes, BHs and fuzzballs from Quantum SW geometries
hep-th 2021-05 conditional novelty 6.0 of 10

Quasinormal modes of charged, rotating, and brane-like gravitational backgrounds are shown to be computable from the quantum geometry of supersymmetric gauge theories, with three independent methods agreeing for the D...
Black Hole Quasinormal Modes and Seiberg-Witten Theory
hep-th 2020-06 conditional novelty 6.0 of 10

Quasinormal mode frequencies of four-dimensional Schwarzschild and Kerr black holes are shown to follow exact Bohr-Sommerfeld quantization conditions written in terms of the Nekrasov-Shatashvili free energy of N=2 SU(...
Integrability and cycles of deformed ${\cal N}=2$ gauge theory
hep-th 2019-08 conditional novelty 6.0 of 10

Baxter's T and Q functions of self-dual Liouville theory are identified with the two deformed Seiberg-Witten cycles, a and a_D, of pure N=2 SU(2) gauge theory in the Nekrasov-Shatashvili limit.
TBA equations and resurgent Quantum Mechanics
hep-th 2018-11 accept novelty 6.0 of 10

A system of thermodynamic Bethe ansatz equations is derived that computes exact WKB periods for arbitrary polynomial potentials in one-dimensional quantum mechanics, and is used to solve spectral problems for PT-symme...
Candidate Phases for SU(2) Adjoint QCD$_4$ with Two Flavors from $\mathcal{N}=2$ Supersymmetric Yang-Mills Theory
hep-th 2018-06 accept novelty 6.0 of 10

Deforming N=2 SU(2) Yang-Mills by a symmetry-preserving scalar mass generates anomaly-consistent candidate phases for two-flavor adjoint QCD, including a confining CP1 phase with a topological-insulator confining string.
Quantum Geometry of Resurgent Perturbative/Nonperturbative Relations
hep-th 2017-01 conditional novelty 6.0 of 10

For quantum potentials with a genus-one classical elliptic curve, the all-orders WKB action determines the all-orders dual (tunneling) action, and for four special arithmetic potentials the relation takes a universal ...
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.
Resurgence and the Nekrasov-Shatashvili Limit: Connecting Weak and Strong Coupling in the Mathieu and Lam'e Systems
hep-th 2015-01 conditional novelty 6.0 of 10

Exact WKB analysis unifies the three spectral regimes of the Mathieu and Lamé equations and proves that non-perturbative effects are encoded in perturbation theory through the Picard-Fuchs equation for the quantized e...
Wild Quiver Gauge Theories
hep-th 2011-12 conditional novelty 6.0 of 10

The prepotential of SU(2) gauge theories coupled to non-Lagrangian Dn SCFTs is computed from irregular Virasoro conformal blocks built from generalized coherent states, matching Seiberg-Witten curve integrals in three...
Quantum Hitchin Systems via beta-deformed Matrix Models
hep-th 2011-04 conditional novelty 6.0 of 10

In the Nekrasov-Shatashvili limit, the Schwinger-Dyson loop equations of the beta-deformed Dotsenko-Fateev matrix model yield the Schroedinger-type equations whose potentials are the quantum Hitchin Hamiltonians on th...
Deformed Prepotential, Quantum Integrable System and Liouville Field Theory
hep-th 2010-06 conditional novelty 6.0 of 10

The null state condition for a degenerate field in Liouville theory yields the Schrödinger equations of the integrable systems tied to N=2 gauge theories, and the monodromies of the conformal block reproduce the defor...
Lectures on Instanton Counting
math.AG 2003-11 conditional novelty 6.0 of 10

The paper computes higher Casimir insertions and the genus-one coefficients of Nekrasov's instanton partition function, matching the latter to the A and B measure factors of the u-plane integral for rank two.
Multi-Instanton Calculus and Equivariant Cohomology
hep-th 2002-11 conditional novelty 6.0 of 10

A supersymmetric localization formula on the ADHM moduli space yields closed sum-over-Young-tableaux expressions for multi-instanton partition functions of N=2, N=2* and N=4 gauge theories.
Tameness and Complexity in Quantum Field Theory and Gravity
hep-th 2026-09 conditional novelty 5.0 of 10

Physical functions and effective quantum gravity theories are tame (definable in o-minimal structures) with finite complexity measurable via sharp o-minimality.
Scalar quasinormal modes of Kerr--AdS$_{\bf 7}$ via accessory parameter expansions
gr-qc 2026-09 accept novelty 5.0 of 10

Scalar quasinormal modes of Kerr-AdS7 with three rotations are computed via accessory parameter expansions of five-singular-point Fuchsian equations.
CFT description of BH's and ECO's: QNMs, superradiance, echoes and tidal responses
hep-th 2022-06 conditional novelty 5.0 of 10

Exact connection matrices for Schrödinger-like wave equations with arbitrary Fuchsian singularities are written in terms of CFT braiding and fusion rules, yielding closed-form black-hole quasinormal modes, amplificati...
More on the SW-QNM correspondence
hep-th 2021-09 conditional novelty 5.0 of 10

Quantum periods of an SU(2) Seiberg-Witten curve reproduce the quasinormal mode frequencies of asymptotically flat Kerr-Newman black holes to within a few percent of direct numerical solutions.
Non-perturbative approaches to the quantum Seiberg-Witten curve
hep-th 2019-08 conditional novelty 5.0 of 10

The Borel-resummed quantum periods and the Fredholm determinant of the modified Mathieu operator are computed from TBA equations and topological string theory, with high-precision numerical tests against WKB and direc...
On Gauge Theory and Topological String in Nekrasov-Shatashvili Limit
hep-th 2012-05 conditional novelty 5.0 of 10

The saddle-point equations of the Nekrasov-Shatashvili limit imply the holomorphic anomaly equation, and the previously derived higher-genus amplitudes satisfy them at low genus and low order.
Gauge theories on Omega-backgrounds from non commutative Seiberg-Witten curves
hep-th 2011-03 conditional novelty 5.0 of 10

The prepotential and chiral correlators of N=2 SU(N) gauge theories in the Omega background follow from a difference equation that acts as a non-commutative Seiberg-Witten curve.
Penner Type Matrix Model and Seiberg-Witten Theory
hep-th 2009-11 accept novelty 5.0 of 10

The Penner-type matrix model's spectral curve and free energy are shown to match the Seiberg-Witten curve and prepotential of N=2 SU(2) gauge theory for Nf=4, 3, and 2, with the UV/IR coupling relation q_UV = theta_2^...
On AGT Relations with Surface Operator Insertion and Stationary Limit of Beta-Ensembles
hep-th 2010-11 conditional novelty 4.0 of 10

Inserting the simplest degenerate operator into a conformal block yields a Schrödinger-like equation whose Bohr-Sommerfeld periods generate the one-parameter deformed Seiberg-Witten prepotential.
The Calculus of Many Instantons
hep-th 2002-06 accept novelty 4.0 of 10

The paper assembles a unified geometric treatment of multi-instanton calculus, from ADHM moduli spaces to supersymmetric partition functions and D-brane relations.

How later papers used it

100 later papers cite it

Evidence: substantial. 100 of the citing papers share no author with it, and 31 of those carry a confident label from Pith's citation classifier.

Pith read 19 of the 118 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

8 later papers by other authors are listed below: 8 used its method or tool.

Uses its method or tool 8 of 23 papers with this label

Used the Matone relation discussed in its appendix to include instanton corrections when requiring the Gamma-function part to vanish.

Quasinormal Modes of C-metric from SCFTs (2023) · 0.91 · paraphrased: the licence does not allow quoting
Used Matone's relation to express its measure factor variable in terms of modular functions.

Mocking the u-plane integral (2019) · 0.89 · paraphrased: the licence does not allow quoting
Used the Matone relation to invert its parameter as a function of the quantum mirror map.

Massive vector field perturbations in the Schwarzschild spacetime from supersymmetric gauge theory (2025) · 0.93 · paraphrased: the licence does not allow quoting
To connect them, one has to pass through the intermediate region, whose monodromy properties are encoded in the parameter $a$. $a$ is related to the instanton part of the Nekrasov-Shatashvili free-energy, $F(1/t)$, through Matone relation [ref] , [equation] Further details of this procedure and of the computation of $F(1/t)$ are given in the supplemental material.

Exact low-temperature Green's functions in AdS/CFT: From Heun to confluent Heun (2024) · Method and Analysis · 0.92
To make use of this quantisation condition, we substitute [ref] in the Matone relation [ref] , which relates the parameters $a$ and $c$:[We remark that only $a^2$ appears in the right hand side of [ref], so that the choice for the sign in front of $a$ in [ref] does not make any difference.] [equation] This relation provides an expansion of the parameter $c$, which in the gauge theory language is related to the energy of the Seiberg-Witten curve, in powers of the instanton counting parameter $t$.

Analytic approaches to perturbations of strongly coupled Yang-Mills plasma (2026) · QNMs from the quantisation condition · 0.95
Used an identity from this paper to rewrite an equation.

Donaldson-Witten theory and indefinite theta functions (2017) · 0.80 · paraphrased: the licence does not allow quoting
To be more precise, $a$ can be computed by inverting the Matone relation [ref] perturbatively in the instanton parameter $Λ≡t$.

Black hole scattering amplitudes via analytic small-frequency expansion and monodromy (2024) · Monodromy in the near-spatial infinity region · 0.96
Computed the instanton parts of its deformed prepotential by a generalized Matone relation.

On Gauge Theory and Topological String in Nekrasov-Shatashvili Limit (2012) · 0.83 · paraphrased: the licence does not allow quoting

How the literature uses it

96 arXiv papers cite this in 115 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 · 70 sentences without a confident label

Builds on it 5 papers

Let us analyze the $N=2^⋆$ theory on a generic $Ω$-deformation starting from the generalized Matone's relation [ref] 7.11 2 = 2 a^2-2 _1 _2 q ∂_q logZ_inst, where $Z_inst$ is the instanton partition function of the gauge theory.

Chiral trace relations in $\Omega$-deformed $\mathcal N=2$ theories · Hints from Matone's relation · 0.94
These relations generalize the quantum Matone equation [ref] to the AD case.

On irregular states and Argyres-Douglas theories · Irregular conformal blocks · 0.97
([ref]) is Matone's relation [ref] generalized to the case with $Ω$-background [ref] .

A note on rank $\frac{3}{2}$ Liouville irregular block · The deformed SW curve · 0.90

Uses its method or tool 23 papers

Using the connection formulas of Heun equation [ref] , the retarded thermal correlator can be formally written as (see [ref] ) G_R(ω,p)=r_h^4a_1 e^-∂_a_1F ∑_σ'=±ℳ_-σ'(a_t_0,a;a_0)ℳ_(-σ')+(a,a_1;a_∞)t^σ' ae^-σ'/2∂_a F/∑_σ=±ℳ_-σ(a_t_0,a;a_0)ℳ_(-σ)-(a,a_1;a_∞)t^σae^-σ/2∂_a F , where $a$ satisfies the Matone relation [ref] u=-a^2+a_t_0^2-14+a_0^2+t_0 ∂_t_0 F , We also have ℳ_θθ'(α_0,α_1;α_2)=Γ(-2θ'α_1)Γ(1+2θ'α_0)Γ(12+α_2+θα_0-θ' α_1)Γ(12-α_2+θα_0-θ' α_1) .

Bouncing singularities and thermal correlators on line defects · Wave equations as Heun equations · 1.00
This is not trivial, but it can be done systematically by using recursive techniques, akin to the Matone relation [ref] .

The exact 8d chiral ring from 4d recursion relations · Introduction · 0.97
In order to compute it, we have to Fourier transform [ref] bringing it in the form of a difference equation [ref] , whose integrability condition can be written as a continued fraction: P_0(a)=M(a+1)/P_0(a+1)-M(a+2)/P_0(a+2)-...+M(a)/P_0(a-1)-M(a-1)/P_0(a-1)-... with M(x)=ΛP_L(x-1/2)P_R(x-1/2) Equation [ref] with [ref] can be solved for $u(a,Λ)$ order by order in $Λ$, getting [equation] We can invert this series in order to obtain [equation] The Nekrasov-Shatashvili (NS) prepotential $ℱ_NS(a,Λ)$ is obtained by integrating the quantum version of the Matone relation [ref] u=-Λ∂ℱ_NS(a,Λ)/∂Λ Furthermore, we have to add a $Λ$-indipendent constant obtained from the one-loop prepotential, so that ℱ_NS(a,Λ)=ℱ_tree(a,Λ)+ℱ_1-loop(a)+ℱ_inst(a,Λ) where [equation] Finally, the $a_D-$period is given by a_D=-1/2πi ∂ℱ_NS/∂a QNM frequencies are obtained by imposing the exact quantization condition [ref]

Charge (in)stability and superradiance of Topological Stars · Quantum Seiberg-Witten periods · 0.85

Compares with it 3 papers

We also note that using the Matone relation [ref] , the result for $n=2$ matches perfectly with the mass expansion of the prepotential obtained in [ref] .

Chiral observables and S-duality in N = 2* U(N) gauge theories · Chiral observables from localization · 0.93
This nicely matches with the prepotential of the theory with $N_f = 3$ [ref] up to the irrelevant factor $πi$.

Penner Type Matrix Model and Seiberg-Witten Theory · Free energy · 1.00
Overall, we find [equation] The first terms can be recognized as reproducing the Matone relation [ref] , with the critical exponent at the matching region $κ$ playing the role of the Coulomb branch parameter $a$.

Projecting Gravitational Fluctuations onto Near-Horizon Throats · Deriving $ΔC_0$: Comparison with Instanton Computation · 0.90

Cites it as background 12 papers

More recently a remarkable connection with quantum Seiberg-Witten (SW) curves of $N = 2$ super-symmetric Yang-Mills (SYM) theory on a Nekrasov-Shatashvili (NS) $Ω$-background ($ϵ_1 = ħ$, $ϵ_2 = 0$) [ref] was revealed in [ref] .

QNMs of branes, BHs and fuzzballs from Quantum SW geometries · Introduction · 0.91
We recall that the quantum periods satisfy the so-called quantum Matone relation [ref] .

Non-perturbative approaches to the quantum Seiberg-Witten curve · The all-orders WKB method · 0.83
The quantum Matone relation [ref] ,

Painlev\'e Kernels and Surface Defects at Strong Coupling · Known: differential operators and the NS phase of the Omega-background · 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.

[1] [7]

A. Klemm, W. Lerche and S. Theisen, Nonperturbative Eﬀective Actions of N=2 Super- symmetric Gauge Theories , CERN-TH/95-104, LMU-TPW 95-7, hep-th/9505150

View on Pith Pith review arXiv

[2] [1]

N. Seiberg and E. Witten, Nucl. Phys. B426 (1994) 19

View on Pith 1994

[3] [2]

P. Argyres and A. Faraggi, Phys. Rev. Lett. 73 (1995) 3931, hep-th/9411057; A. Klemm, W. Lerche, S. Theisen and S. Yankielowicz, Phys. Lett. B344 (1995) 169, hep-th/9411048

View on Pith Pith review arXiv 1995

[4] [3]

M. Douglas and S. Shenker, Dynamics of SU (N) Supersymmetric Gauge Theory , RU- 95-12, hep-th/9503163

View on Pith Pith review arXiv

[5] [4]

U. Danielsson and B. Sundborg, The Moduli Space and Monodromies of N=2 Super- symmetric SO(2r+1) Yang-Mills Theory , USIP-95-06, UUITP-4/95, hep-th/9504102. 2Notice that we are using diﬀerent normalizations, thus to compare w ith F KLTk in [7] one should check the invariance of the quantity FkF KLTk F KLTk+1Fk+1 . 7

View on Pith Pith review arXiv

[6] [5]

P. Argyres and M. Douglas, New Phenomena in SU(3) Supersymmetric Gauge Theory , IASSNS-HEP-95/31, RU-95-28, hep-th/9505062

View on Pith Pith review arXiv

[7] [6]

N. Seiberg, Phys. Lett. 206B (1988) 75

View on Pith 1988

[8] [8]

L. Bonora and M. Matone, Nucl. Phys. B327 (1990) 415

View on Pith 1990

[9] [9]

M. Matone, Int. J. Mod. Phys. A10 (1995) 289

View on Pith 1995

[10] [10]

A. Ceresole, R. D’Auria and S. Ferrara, Phys. Lett. 339B (1994) 71

View on Pith 1994

[11] [11]

A. Fayyazuddin, Some Comments on N=2 Supersymmetric Yang-Mills , Nordita 95/22, hep-th/9504120

View on Pith Pith review arXiv

[12] [12]

P. Argyres, A. Faraggi and A. Shapere, Curves of Marginal Stability in N=2 super-QCD , IASSNS-HEP-94/103, UK-HEP/95-07, hep-th/9505190. 8

View on Pith Pith review arXiv