Multi-Instanton Calculus and Equivariant Cohomology

Alessandro Tanzini; Francesco Fucito; Jose F. Morales; Ugo Bruzzo

REVIEW

Multi-Instanton Calculus and Equivariant Cohomology

Ugo Bruzzo, Francesco Fucito, Jose F. Morales, Alessandro Tanzini

High Energy Physics - Theory · Submitted 12 Nov 2002 · 36 references

Pith's verdict: The sum-over-Young-diagram formulas are the real product and they check out at k=1,2; the derivation leans on an unproved localization theorem deferred to an in-preparation paper, which is a genuine but not fatal dependency.

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arXiv IDhep-th/0211108 (version 4)

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CitationAlessandro Tanzini, Francesco Fucito, Jose F. Morales, Ugo Bruzzo. "Multi-Instanton Calculus and Equivariant Cohomology." arXiv:hep-th/0211108 (2002). https://pith.science/paper/SBIY5YEX

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Editor's note

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Worth your time, with one caveat you should know before you invest in the derivation. The durable content is the explicit product formulas: (3.16) for pure N=2, (3.22) with fundamentals, (3.26) for N=2*. Those are what people actually use, and they are benchmarked — the k=1,2 evaluations (3.29)-(3.33) reproduce Nekrasov and Chan-D'Hoker, and the m→0 limit (3.35) gives the known Euler characteristic of the moduli space. That is real external validation, not self-agreement. The general SU(N) character computation (3.14), which turns the tangent-space trace into a sum over boxes of the N Young tableaux, is the technical heart and it is done properly; the Nakajima citation there is to a published book and the algebra is standard, so I would not count that as a gap. The N=2* and fundamental-matter results are new relative to the cited prior work, and the observation that the massive spectrum is just the N=2 spectrum shifted by −m is clean and useful. The soft spot is the one the stress-test names, and it lands. Formula (2.17), the supersymmetric localization formula, is stated without proof and deferred to [23], which is listed as in preparation. That matters more than a normal citation gap because the ordinary theorem it extends, (2.14), is explicitly conditioned on compact M and G, and the ADHM moduli space is neither compact nor smooth before the ζ-resolution. The paper resolves the singularities but never compactifies or gives a decay argument at infinity, so the boundary term in the Stokes step is assumed to vanish rather than shown to. The primed superdeterminant is also introduced by fiat, and that is exactly where the boson/fermion cancellation producing the 2kN character sits. The isolation of critical points in §3.1 is a counting argument, not a proof. None of this is fatal — the outputs are independently checked — but the derivation as written is conditional. The authors do flag the deferral themselves, so treat it as a known dependency rather than a hidden one. For a referee: send it out. The formulas are the contribution and they deserve scrutiny on their own terms; ask specifically for the compactness/decay argument and the normalization of the primed superdeterminant.

Referee report

Significance

If the localization step is valid, the paper gives a clean, explicit and general algorithm for multi-instanton calculus: the instanton measure and the correlators reduce to finite combinatorial products over Young diagrams without solving the ADHM constraints explicitly. The results are benchmarked against independent published computations at k=1,2 (Chan–D'Hoker [33], Flume–Poghossian [20]) and the N=4 limit reproduces the known Euler characteristic and the modular-looking generating function (3.35), which is genuine external validation rather than self-consistency. The explicit product formulas (3.16), (3.22), (3.26) are the kind of result that is directly usable by the community, and the paper is a useful bridge between the physics instanton literature and the mathematical equivariant-cohomology literature. The significance is high provided the derivation's load-bearing tool is put on a firm footing.

Major objections (3)

[§2.2, Eq. (2.17)] The supersymmetric localization formula (2.17) is the load-bearing tool of the entire paper, and the authors acknowledge that its proof is deferred: 'A complete proof of this result will be presented in [23]', with [23] listed as 'U. Bruzzo, F. Fucito, in preparation'. The acknowledged deferral is not itself the problem; the unresolved issue is that the ordinary theorem being extended, Eq. (2.14), is explicitly stated under the hypotheses that 'both M, G are compact' and that ξ* has only isolated zeroes. The ADHM moduli space is non-compact, and the ζ parameter of (2.1)/(3.3) resolves the orbifold singularities but does not compactify the space. No decay or boundary argument is supplied to show that the Stokes/Atiyah–Bott boundary term vanishes, so the applicability of (2.17) to this non-compact space is assumed rather than established. This is a correctness risk rather than a refutation: the k=1,2 outputs (3.29)–(3.33) agree with [33] and [20], and (3.35) reproduces the known Euler characteristic, which is meaningful evidence that the formula does apply here. I would ask the authors to state explicitly the growth conditions under which the boundary term is dropped, or to cite a published statement covering the non-compact case.
[§2.2 and §3.1, Eqs. (3.12)–(3.14)] The 'primed' superdeterminant is introduced by fiat: 'the operator Sdet′ is defined as Sdet∘Π ... from now on each time we write the superdeterminant we intend it to be primed'. This definition is precisely where the boson/fermion cancellation that reduces the character to the 2kN-dimensional result is encoded, so a sign or normalization slip there would propagate into every Z_k. The reduction from the character (3.12) to the final eigenvalue sum (3.14) is then quoted after 'a long but straight algebra (see [12] for details)'. Since (3.14) is the step that produces the product formulas (3.16), (3.22), (3.26), I would ask for either the explicit intermediate eigenvalue list or a worked check of the cancellation at k=2, so that a reader can verify the primed-determinant bookkeeping without reconstructing it from [12].
[§3.1, Eqs. (3.6)–(3.9)] The claim that 'critical points of the U(1)^k × U(1)^{N-1} × U(1)^2 action are isolated' is argued by a counting argument (k components fixed by the diagonal part of E_R) rather than proved, and isolation is exactly the hypothesis needed to apply (2.17). The argument is plausible and is consistent with the explicit k=1,2 solutions, but it should be stated as a proposition with its genericity assumptions on a_λ, ǫ_1, ǫ_2 and ζ made explicit, or at least flagged as an assumption. A short paragraph spelling out why the residual components left undetermined by E_C are exhausted by E_R would close this gap.

Minor objections (7)

[§3.4, Eq. (3.36)] There is a sign error in the displayed identity: from Z = ∏_{n≥1}(1−q^n)^{−N} one has ln Z = −N ∑_{n>0} ln(1−q^n), not +N ∑_{n>0} ln(1−q^n) as written. The final result F_k = N ∑_{d|k} 1/d is correct, so this is a typographical slip, but it should be fixed.
[§3.4] The phrase 'the N-colored number of partitions of an integer k' is awkward; 'the number of N-colored partitions of k' would read better.
[§3.1] The sentence 'Plugging (3.11) in (3.10) one recovers (3.10) in [14]' is confusing because it reuses the same equation number for the present paper and for the cited reference; please write 'recovers Eq. (3.10) of [14]' or similar.
[§2.1 and §3.1] The role of the non-commutativity parameter ζ is introduced in (2.1)/(3.3) as a minimal resolution, but the paper never states that the final partition functions Z_k are independent of ζ. A sentence making this explicit would clarify that the resolution is a calculational device.
[Appendix B, Eq. (B.8)] The BRST transformation 'sC = (φ−a) − [C,C]' as printed looks dimensionally odd; please check whether the intended expression is sC = (φ−a) − [C,C] with the second term a commutator of the connection with itself, and clarify the notation.
[Footnote 7 and §3.3.1] The cross-references to '(3.25) in [14]' and '(3.10) in [14]' should be checked against the numbering in the cited arXiv version, since the reader cannot easily verify them.
[Throughout] There are numerous typographical errors that should be corrected: 'estabilished' and 'meausure' in §1, 'his most notable applications' for 'its most notable applications' in §1, 'operatore' in §3.3, 'correponds' in §3.4, 'arbitrarity' in Appendix B, and 'we shall denoted' in §2.2.

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The paper

The paper claims

Multi-instanton amplitudes in supersymmetric gauge theories are claimed to reduce to a finite sum over Young diagrams, each weighted by an explicit product of eigenvalues.

The reading

The paper asks how to compute the multi-instanton contribution to a supersymmetric gauge theory's partition function without ever solving the ADHM constraints, the matrix equations that define the instanton moduli space and that can be solved in closed form only for one or two instantons. The authors' answer is to write the entire ADHM action as a single BRST-exact expression, $S=Q_epsilon alpha$, and then apply a localization formula: an integral of an equivariantly closed form equals a sum over the isolated zeros of the associated vector field, each weighted by the inverse superdeterminant of the linearized action there. Because the deformation includes a rotation of spacetime, the zeros become isolated and can be labelled by $N$-tuples of Young diagrams whose total number of boxes is the instanton number $k$. If the derivation holds, the $k$-instanton measure, previously an intractable integral over a singular non-compact space, becomes a finite product over the boxes of those diagrams, and the same bookkeeping covers pure $mathcal{N}=2$, $mathcal{N}=2$ with fundamental matter, $mathcal{N}=2^*$ and $mathcal{N}=4$.

What carries the argument

The load-bearing device is a supersymmetric version of the equivariant localization formula, Eq. (2.17), in which the manifold is replaced by a supermanifold whose functions are differential forms and the exterior derivative by an odd vector field $Q^*$ with $tfrac12{Q^*,Q^*}=hatxi^*$, the even generator of the group action. The integral of an equivariantly closed form then equals $(-2pi)^{n/2}$ times a sum over the zeros of $Q^*$, each weighted by the inverse square root of the superdeterminant of $L_{x_0}=[Q^*,cdot]$. The second device is the rewriting of the ADHM action as $Q_epsilon$ of a single functional, so that the action vanishes at the critical points and the weight is one. The third is the character $chi=sum_i(-)^F e^{ilambda_i}$ of the linearized operator, reduced by algebra to a sum over the $2kN$ eigenvalues of the tangent space, which is then converted into a product over boxes; this is where the cancellations that would otherwise come from solving the ADHM constraints are automatically accounted for.

Extended reading notes

Core claim

The central claim is that the $k$-instanton partition function of an $SU(N)$ supersymmetric gauge theory is an explicit finite sum over $N$-tuples of Young diagrams with $k$ boxes in total. For pure $mathcal{N}=2$ theory the weight is $prod_{lambda,tildelambda}prod_{sin Y_lambda} 1/[E(s)(E(s)-epsilon)]$ with $E(s)=a_{lambda tildelambda}-epsilon_1 h(s)+epsilon_2(v(s)+1)$, where $h(s)$ and $v(s)$ count boxes to the right of and below the box $s$; adding $N_F$ fundamental hypermultiplets multiplies each box by $prod_f(phi(s)+m_f)$; and the $mathcal{N}=2^*$ case multiplies by $(E(s)-m)(E(s)-epsilon+m)$. In the $mathcal{N}=4$ limit $mto 0$ every weight becomes one, so $Z_k$ counts $N$-coloured partitions of $k$, giving $sum_k Z_k q^k=prod_{nge 1}(1-q^n)^{-N}$ and $F_k=Nsum_{dmid k}1/d$. The authors present this as a derivation rather than a fit: measure, constraints and correlators all follow from one BRST-exact action through localization, and the results for $k=1,2$ reproduce earlier computations.

Load-bearing premise

The whole derivation rests on the supersymmetric localization formula, Eq. (2.17), which the paper states without proof and defers to a companion paper listed as in preparation, and which extends a theorem proved for compact manifolds to a non-compact supermanifold whose fixed-point set is singular until the resolution parameter is switched on.

Editorial extensions

If this is right

Instanton corrections to the Seiberg-Witten prepotential become a finite combinatorial sum for every winding number $k$ and every rank $N$, so coefficients can be extracted without solving the ADHM constraints explicitly.
Adding fundamental hypermultiplets only multiplies each box weight by a flavour factor, so the matter sector is brought under the same computational control as the pure gauge sector.
The $mathcal{N}=4$ case yields the Euler characteristic of the instanton moduli space and a generating function $prod_n(1-q^n)^{-N}$, matching the modular structure $ln|tau_2 eta(tau)^4|$ seen in the D3-brane effective action.
Because the argument uses only the fixed-point character of a $Q$-exact action, the same machinery should apply to any moduli problem presented in ADHM-like matrix form.

Reading between the lines

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

The paper uses a resolution parameter $zeta$ to smooth the small-instanton singularities, but the final sums over diagrams contain no $zeta$; making that independence explicit would be a clean internal consistency check the authors do not perform.
The stated localization formula is a mathematical claim in its own right: a proof would be a theorem about equivariant cohomology of supermanifolds with non-compact or singular fixed-point sets, and would stand independently of any gauge theory application.
Since only the character of the linearized operator enters, the same bookkeeping plausibly extends to quiver and orbifold gauge theories and to instantons on curved backgrounds, where the fixed-point combinatorics changes but the logic does not.
The product-over-boxes form makes the large-$k$ asymptotics accessible by standard combinatorial means, which could turn the instanton expansion into a tool for probing the strong-coupling regime of the prepotential.

Original abstract

We present a systematic derivation of multi-instanton amplitudes in terms of ADHM equivariant cohomology. The results rely on a supersymmetric formulation of the localization formula for equivariant forms. We examine the cases of N=4 and N=2 gauge theories with adjoint and fundamental matter.

The compiled record

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,864 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.06 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.06 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v4. Results and the main claim are underlined; show every role.

We present a systematic derivation of multi-instanton amplitudes in terms of ADHM equivariant cohomology. The results rely on a supersymmetric formulation of the localization formula for equivariant forms. We examine the cases of N=4 and N=2 gauge theories with adjoint and fundamental matter.

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 present a systematic derivation of multi-instanton amplitudes in terms of ADHM equivariant cohomology.	contribution 1.00	0.95	0.71	asserted 1.00
s1	The results rely on a supersymmetric formulation of the localization formula for equivariant forms.	method 1.00	0.61	0.09	asserted 1.00
s2	We examine the cases of N=4 and N=2 gauge theories with adjoint and fundamental matter.	contribution 0.98	0.88	0.07	asserted 0.94

What kind of paper the classifier reports this is: theoretical result 0.93 · evidence from proof or derivation 1.00.

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sources read

arXiv listing abstract v4

not examined

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references and citing sentences
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levels run

P abstract

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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

Figures

**Figure 1.** Two generic Young diagrams denoted by the indices

Figures reproduced from arXiv:hep-th/0211108 by the authors.

References

Works cited (36)

[23]

U. Bruzzo, F. Fucito, in preparation

View on Pith
[33]

G. Chan and E. D’Hoker, Nucl. Phys. B 564 (2000) 503, arXiv:hep-th/9906193

View on Pith Pith review arXiv 2000
[20]

R. Flume, R. Poghossian and H. Storch, Mod. Phys. Lett. A 17 (2002) 327, arXiv:hep-th/0112211

View on Pith Pith review arXiv 2002
[12]

H. Nakajima, “Lectures on Hilbert Schemes of Points on Surfac es”, American Math- ematical Society,University Lectures Series v.18 (1999)

View on Pith 1999
[1]

E. Witten, Commun. Math. Phys. 117 (1988) 353

View on Pith 1988
[2]

D. Amati, K. Konishi, Y. Meurice, G. C. Rossi and G. Veneziano, Ph ys. Rept. 162 (1988) 169; M. A. Shifman, “Instantons In Gauge Theories,” Singapore, Singapore: World Sci- entiﬁc (1994)

View on Pith 1988
[3]

N. Seiberg and E. Witten, Nucl. Phys. B426 (1994) 19; ibid. B431 (1994) 484

View on Pith 1994
[4]

N. Dorey, V. V. Khoze and M. P. Mattis, Phys. Rev. D54 (1996) 2921

View on Pith 1996
[5]

F. Fucito and G. Travaglini, Phys. Rev. D55 (1997) 1099

View on Pith 1997
[6]

M. Matone, Phys. Lett. B 357 (1995) 342, arXiv:hep-th/9506102

View on Pith Pith review arXiv 1995
[7]

N. Dorey, T. J. Hollowood, V. V. Khoze and M. P. Mattis, “The calc ulus of many instantons”, arXiv:hep-th/0206063

View on Pith Pith review arXiv
[8]

D. Bellisai, F. Fucito, A. Tanzini and G. Travaglini, Phys. Lett. B 480 (2000) 365, arXiv:hep-th/0002110; JHEP 0007 (2000) 017, arXiv:hep-th/0003272

View on Pith Pith review arXiv 2000
[9]

U. Bruzzo, F. Fucito, A. Tanzini and G. Travaglini, Nucl. Phys. B 611 (2001) 205, arXiv:hep-th/0008225

View on Pith Pith review arXiv 2001
[10]

N. Dorey, T. J. Hollowood and V. V. Khoze, JHEP 0103 (2001) 040, arXiv:hep-th/0011247

View on Pith Pith review arXiv 2001
[11]

F. Fucito, J. F. Morales and A. Tanzini, JHEP 0107 (2001) 012, arXiv:hep-th/0106061

View on Pith Pith review arXiv 2001
[13]

T. J. Hollowood, JHEP 0203 (2002) 038, arXiv:hep-th/0201075; Nucl. Phys. B 639, 66 (2002), arXiv:hep-th/0202197. 28

View on Pith Pith review arXiv 2002
[14]

N. A. Nekrasov, arXiv:hep-th/0206161

View on Pith Pith review arXiv
[15]

G. W. Moore, N. Nekrasov and S. Shatashvili, Commun. Math. Ph ys. 209 (2000) 77, arXiv:hep-th/9803265

View on Pith Pith review arXiv 2000
[16]

G. W. Moore, N. Nekrasov and S. Shatashvili, Commun. Math. Ph ys. 209 (2000) 97, arXiv:hep-th/9712241

View on Pith Pith review arXiv 2000
[17]

A. Losev, N. Nekrasov and S. L. Shatashvili, Nucl. Phys. B 534 (1998) 549, arXiv:hep-th/9711108

View on Pith Pith review arXiv 1998
[18]

G. W. Moore and E. Witten, Adv. Theor. Math. Phys. 1 (1998) 298, arXiv:hep-th/9709193

View on Pith Pith review arXiv 1998
[19]

E. Witten, J. Geom. Phys. 9 (1992) 303, arXiv:hep-th/9204083

View on Pith Pith review arXiv 1992
[21]

R. Flume and R. Poghossian, “An algorithm for the microscopic ev aluation of the coeﬃcients of the Seiberg-Witten prepotential”, arXiv:hep-th/02 08176

View on Pith
[22]

N. Berline, E. Getzler and M. Vergne, “Heat Kernels and Dirac Op erators”, Berlin, Germany: Springer (1996)

View on Pith 1996
[24]

L. Baulieu and I. M. Singer, Nucl. Phys. Proc. Suppl. B5 (1988) 12

View on Pith 1988
[25]

N. Dorey, T.J. Hollowood, V.V. Khoze, M.P. Mattis and S. Vandore n, Nucl. Phys. B552 (1999) 88, arXiv:hep-th/9901128

View on Pith Pith review arXiv 1999
[26]

M. B. Green and M. Gutperle, JHEP 0002 (2000) 014, arXiv:hep-th/0002011

View on Pith Pith review arXiv 2000
[27]

M. Bill´ o, M. Frau, I. Pesando, F. Fucito, A. Lerda, A. Liccardo , “Classical Gauge Instantons From Mixed Disks”, in preparation

View on Pith
[28]

M. R. Douglas, “Branes within branes”, arXiv:hep-th/9512077 . 29

View on Pith Pith review arXiv
[29]

M. R. Douglas, J. Geom. Phys. 28 (1998) 255, arXiv:hep-th/9604198

View on Pith Pith review arXiv 1998
[30]

E. Witten, Nucl. Phys. B 460 (1996) 541, arXiv:hep-th/9511030

View on Pith Pith review arXiv 1996
[31]

E. T. Akhmedov, Phys. Rev. D 59 (1999) 101901, arXiv:hep-th/9812038

View on Pith Pith review arXiv 1999
[32]

S. Hirano and M. Kato, Prog. Theor. Phys. 98 (1997) 1371, arXiv:hep-th/9708039

View on Pith Pith review arXiv 1997
[34]

J. D. Edelstein, M. Marino and J. Mas, Nucl. Phys. B 541 (1999) 671, arXiv:hep-th/9805172; J. D. Edelstein, M. Gomez-Reino and J. Mas , Nucl. Phys. B 561 (1999) 273, arXiv:hep-th/9904087; J. D. Edelstein, M. Gomez-Re ino, M. Marino and J. Mas, Nucl. Phys. B 574 (2000) 587, arXiv:hep-th/9911115

View on Pith Pith review arXiv 1999
[35]

M. Bianchi, F. Fucito and G. Rossi, Phys. Lett. B 359 (1995) 56, arXiv:hep-th/9507003

View on Pith Pith review arXiv 1995
[36]

N. Dorey, T. J. Hollowood and V. V. Khoze, “Notes on soliton bou nd-state problems in gauge theory and string theory”, arXiv:hep-th/0105090. 30

View on Pith Pith review arXiv

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Instanton counting on blowup. I. 4-dimensional pure gauge theory
math.AG 2003-06 accept novelty 7.0 of 10

The paper proves Nekrasov's conjecture for pure N=2 gauge theory by deriving a blowup equation for the instanton partition function and matching its limit to the Seiberg-Witten contact term equation.
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.
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.
Refined Painlev\'e/gauge theory correspondence and quantum tau functions
hep-th 2025-02 conditional novelty 6.0 of 10

Strong-coupling expansions of N=2 SU(2) partition functions in general Omega-background are derived from quantum Painlevé equations obtained via C2/Z2 blowup relations, and match refined holomorphic anomaly and irregu...
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...
On Painlev\'e/gauge theory correspondence
hep-th 2016-12 conditional novelty 6.0 of 10

The isomonodromic connection of Painlevé equations is shown to match the oper limit of the Hitchin connection, and the resulting long-distance tau-function expansions agree with magnetic and dyonic prepotentials of ra...
General instanton counting and 5d SCFT
hep-th 2014-06 conditional novelty 6.0 of 10

Including the antisymmetric hypermultiplet in the ADHM instanton calculus and dividing out a factorized 8d index restores E7 and E8 symmetry enhancement in the superconformal indices of 5d Sp(1) SCFTs at N_f = 6 and 7.
Higgs branch localization of 3d N=2 theories
hep-th 2013-12 conditional novelty 6.0 of 10

Higgs branch localization directly produces the vortex-times-anti-vortex factorization of 3d N=2 partition functions on the squashed sphere and S^1 x S^2, extending it to more general matter content.
Topological strings and 5d T_N partition functions
hep-th 2013-10 conditional novelty 6.0 of 10

Dividing the refined topological vertex amplitude by a computable factor from decoupled M2-branes reproduces 5d Nekrasov partition functions, yielding contour-integral-free Sp(1) expressions and a partition function f...
How instanton combinatorics solves Painlev\'e VI, V and III's
hep-th 2013-02 conditional novelty 6.0 of 10

General solutions of Painlevé VI, V and III are written as sums over pairs of Young diagrams whose terms are c=1 conformal blocks of 2D CFT, giving a practical numerical series for each transcendent.
Deformed N=2 theories, generalized recursion relations and S-duality
hep-th 2013-02 conditional novelty 6.0 of 10

The epsilon-deformed N=2 SU(2) prepotentials satisfy a modular anomaly recursion relation with an epsilon1*epsilon2 term, and S-duality can be restored to Seiberg-Witten form via a modified prepotential and effective ...
5-dim Superconformal Index with Enhanced En Global Symmetry
hep-th 2012-06 conditional novelty 6.0 of 10

The five-dimensional Sp(1) superconformal index, computed by localization including instantons, organizes into E_{N_f+1} characters for N_f up to 5, confirming the conjectured exceptional global symmetry enhancement.
Hilbert Series for Moduli Spaces of Two Instantons
hep-th 2012-05 conditional novelty 6.0 of 10

The Hilbert series of the two-instanton moduli space is expressed as a character expansion living on a small lattice for every simple gauge group, with explicit formulas for classical groups and conjectures for except...
The ABCDEFG of Instantons and W-algebras
hep-th 2011-11 conditional novelty 6.0 of 10

At one instanton, the Nekrasov partition function of pure N=2 gauge theory matches the norm of a Gaiotto-Whittaker coherent state in the corresponding W-algebra for all simple gauge groups checked, with non-simply-lac...
On instantons as Kaluza-Klein modes of M5-branes
hep-th 2011-10 conditional novelty 6.0 of 10

A supersymmetric index for 5d instantons reproduces the Kaluza-Klein spectrum of M5-branes on a circle and reveals new momentum-carrying worldsheet degrees on SU(N) self-dual strings.
Instanton counting with a surface operator and the chain-saw quiver
hep-th 2011-05 conditional novelty 6.0 of 10

The instanton partition function with a general surface operator is computed from chain-saw quiver fixed points and shown to match the norm of a Whittaker vector in the corresponding W-algebra Verma module for partiti...
Vertices, Vortices & Interacting Surface Operators
hep-th 2011-02 conditional novelty 6.0 of 10

The vortex partition function of N=(2,2) gauge theories with adjoint and anti-fundamental matter equals the field-theory limit of the topological vertex on a strip with column-diagram boundary conditions, and resums i...
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...
A & B model approaches to surface operators and Toda theories
hep-th 2010-04 conditional novelty 6.0 of 10

Multiple surface operator insertions in 4d N=2 quiver gauge theories are matched, order by order, to open topological string amplitudes with multiple toric branes and to degenerate-operator insertions in Toda conforma...
D-brane Instantons on the T^6/Z_3 orientifold
hep-th 2007-04 conditional novelty 6.0 of 10

A detailed string-amplitude computation shows that only the U(4) and Sp(6)xU(2) gauge theories at the Z_3 orientifold singularity get ADS superpotentials from gauge instantons, while ED3 stringy instantons generate Ma...
Exact expressions for the 5-Point Liouville conformal block with a level-two degenerate field insertion
hep-th 2025-06 conditional novelty 5.0 of 10

A rigorous inductive proof that the 5-point Liouville conformal block with a level-2 degenerate insertion can be expressed exactly in terms of one hypergeometric function and its derivative.
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...
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...
Line defects and 5d instanton partition functions
hep-th 2016-01 conditional novelty 5.0 of 10

The partition function of a 5d gauge theory with a Tong-Wong line defect is shown to be a degree-N polynomial in the defect fugacity, using the Jeffrey-Kirwan residue prescription.
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.
Gauge Theories on ALE Space and Super Liouville Correlation Functions
hep-th 2011-07 conditional novelty 5.0 of 10

The partition function of N=2 quiver gauge theories on the ALE space O_{P^1}(-2) matches, order by order, the correlation functions of N=1 super Liouville conformal field theory.
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.
Refined Topological Vertex and Instanton Counting
hep-th 2007-10 conditional novelty 4.0 of 10

The refined topological vertex, with a modified framing factor, reproduces the K-theoretic Nekrasov partition function for general SU(N) geometries.

How later papers used it

117 later papers cite it

Evidence: substantial. 80 of the citing papers share no author with it, and 22 of those carry a confident label from Pith's citation classifier.

Pith read 12 of the 173 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: 2 used its method or tool and 1 compared its own results with it. 1 more paper by the same authors is listed: 1 used its method or tool.

Uses its method or tool 3 of 27 papers with this label · 1 by the same authors

Computed the vortex partition function in N=(2,2) super Yang-Mills theories by the equivariant localization formula for supersymmetric systems.

Localization of Vortex Partition Functions in $\mathcal{N}=(2,2) $ Super Yang-Mills theory (2011) · 1.00 · paraphrased: the licence does not allow quoting
Gave the vortex partition function for (N) gauge theory with N fundamental chiral multiplets by applying this paper's localization formula.

Vortex counting from field theory (2012) · 0.99 · paraphrased: the licence does not allow quoting

By the same authors

Used the refined topological vertex together with this paper to write the instanton partition function as a sum with coefficients of the Nekrasov partition function.

Seiberg-Witten theory as a Fermi gas (2016) (same authors) · 0.99 · paraphrased: the licence does not allow quoting

Compares with it 1 of 5 papers with this label

Compared with eqn.(3.26) of this paper, with a recent modification in the hypermultiplet mass contribution, and found that the sinh arguments in the numerator of its own N=2* partition function are shifted by m+ϵ_R.

On instantons as Kaluza-Klein modes of M5-branes (2011) · 0.98 · paraphrased: the licence does not allow quoting

How the literature uses it

115 arXiv papers cite this in 171 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 · 108 sentences without a confident label

Builds on it 2 papers

The instanton partition function is [equation] By Equation [ref] and the corresponding expression for the Nekrasov partition function of $U(r)$ gauge theory on $ℝ^4$ with $N$ fundamental hypermultiplets of masses $μ_1, …, μ_N$ [ref] [equation] we get [box=]multline ^N, inst_X_k(ε_1, ε_2, a⃗,μ⃗; , ξ⃗ ) =∑_v⃗∈_w⃗ ξ⃗^ v⃗ ∑_v⃗ ^1/2 ∑_α=1^r v⃗_α·Cv⃗_α ∏_s=1^N ∏_α=1^r ∏_n=1^k-1 ℓ^(n)_v⃗_α(ε_1^(n), ε_2^(n), a_α+μ_s )/∏_α,β=1^r ∏_n=1^k-1 ℓ^(n)_v⃗_βα(ε_1^(n), ε_2^(n), a_βα ) ∏_i=1^k ^N, inst_^2 (ε_1^(i), ε_2^(i), a⃗^(i),μ⃗; ) .

Framed sheaves on root stacks and supersymmetric gauge theories on ALE spaces · Instanton partition function · 0.89
In the case with adjoint hypermultiplet instead of ([ref]) one starts with the (super) character [ref] χ= (1-T_m)× ∑_α,β=1^N e_βe_α^-1 {∑_s∈Y_α(T_1^-l_β(s)T_2^a_α(s)+1)+∑_s∈Y_β(T_1^l_α(s)+1T_2^-a_β(s))}.

Non-perturbative aspects of gauge and string theories and their holographic relations · ADHM construction · 0.84

Uses its method or tool 28 papers

The proof will rely on multi instanton formulae obtained via localization [ref] .

Multi instanton tests of holography · Introduction and Summary · 1.00
The instanton partition function follows from the localization formula [ref] Z_inst =∑_Y q^|Y| ∏_u,v=1^N Z_∅,Y_v(μ̅_u-a_v ) Z_Y_u,∅(a_u-μ_v )Z_Y_u,Y_v(a_u-a_v)

Exact results in N=2 gauge theories · The gauge theory on $^2$ · 0.91
To linear order in $δT$ one gets [equation] In order to perform the integration over the moduli space and obtain explicit results, we exploit the localization formula and adopt Nekrasov's approach to the multi-instanton calculus Ref.s [ref] (see also Ref.s [ref] ).

Non-perturbative gauge/gravity correspondence in N=2 theories · The non-perturbative $t$ profile · 1.00

Compares with it 4 papers

The corresponding bra is given by [equation] Now, it can be checked that [equation] Therefore, [equation] which is indeed the one-instanton contribution of Nekrasov's partition function [ref] calculated from the geometry of the one-instanton moduli space.

The ABCDEFG of Instantons and W-algebras · $A_n$ · 0.86
Comparing with eqn.(3.26) of [ref] , with a recent modification in the hypermultiplet mass contribution [ref] , we find that the $sinh$ arguments in the numerator of the $𝒩=2^∗$ partition function are shifted by $m+ϵ_R$.

On instantons as Kaluza-Klein modes of M5-branes · The index · 0.98
The action in [ref] contains the quartic terms of $a'_m, χ_a$ and the quadratic term of $D^c̃$ which are absent in our action.

N=4 Instanton Calculus in Omega and R-R Backgrounds · Conclusions and Discussion · 1.00

Cites it as background 20 papers

Recently Alday Gaotto and Tachikawa [ref] have found a very remarkable relation between generalized partition functions [ref] in certain classes of $N=2$ conformal $SU(2)$ quiver gauge theories [ref] (see [ref] for instanton counting in quiver theories) and correlation functions of 2d Liouville theory on Riemann surfaces.

Recursion relations in CFT and N=2 SYM theory · Introduction · 0.81
This line of research is founded on the so called AGT correspondence [ref] which relates the results of localization for the partition functions of a $N=2$ supersymmetric quiver gauge theories [ref] with certain correlators of the Liouville CFT.

Post Newtonian emission of gravitational waves from binary systems: a gauge theory perspective · Introduction · 0.85
The instanton part of the partition function [ref] of this theory can be represented as [ref] [equation] where $Y⃗$ is a pair of Young diagrams $Y⃗=(Y_1,Y_2)$ and $|Y⃗|$ is the total number of boxes.

Recursion relation for instanton counting for $SU(2)$ ${\cal N}=2$ SYM in NS limit of $\Omega$ background · The deformed prepotential in the NS limit · 0.81

Discussion

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[1] [23]

U. Bruzzo, F. Fucito, in preparation

View on Pith

[2] [33]

G. Chan and E. D’Hoker, Nucl. Phys. B 564 (2000) 503, arXiv:hep-th/9906193

View on Pith Pith review arXiv 2000

[3] [20]

R. Flume, R. Poghossian and H. Storch, Mod. Phys. Lett. A 17 (2002) 327, arXiv:hep-th/0112211

View on Pith Pith review arXiv 2002

[4] [12]

H. Nakajima, “Lectures on Hilbert Schemes of Points on Surfac es”, American Math- ematical Society,University Lectures Series v.18 (1999)

View on Pith 1999

[5] [1]

E. Witten, Commun. Math. Phys. 117 (1988) 353

View on Pith 1988

[6] [2]

D. Amati, K. Konishi, Y. Meurice, G. C. Rossi and G. Veneziano, Ph ys. Rept. 162 (1988) 169; M. A. Shifman, “Instantons In Gauge Theories,” Singapore, Singapore: World Sci- entiﬁc (1994)

View on Pith 1988

[7] [3]

N. Seiberg and E. Witten, Nucl. Phys. B426 (1994) 19; ibid. B431 (1994) 484

View on Pith 1994

[8] [4]

N. Dorey, V. V. Khoze and M. P. Mattis, Phys. Rev. D54 (1996) 2921

View on Pith 1996

[9] [5]

F. Fucito and G. Travaglini, Phys. Rev. D55 (1997) 1099

View on Pith 1997

[10] [6]

M. Matone, Phys. Lett. B 357 (1995) 342, arXiv:hep-th/9506102

View on Pith Pith review arXiv 1995

[11] [7]

N. Dorey, T. J. Hollowood, V. V. Khoze and M. P. Mattis, “The calc ulus of many instantons”, arXiv:hep-th/0206063

View on Pith Pith review arXiv

[12] [8]

D. Bellisai, F. Fucito, A. Tanzini and G. Travaglini, Phys. Lett. B 480 (2000) 365, arXiv:hep-th/0002110; JHEP 0007 (2000) 017, arXiv:hep-th/0003272

View on Pith Pith review arXiv 2000

[13] [9]

U. Bruzzo, F. Fucito, A. Tanzini and G. Travaglini, Nucl. Phys. B 611 (2001) 205, arXiv:hep-th/0008225

View on Pith Pith review arXiv 2001

[14] [10]

N. Dorey, T. J. Hollowood and V. V. Khoze, JHEP 0103 (2001) 040, arXiv:hep-th/0011247

View on Pith Pith review arXiv 2001

[15] [11]

F. Fucito, J. F. Morales and A. Tanzini, JHEP 0107 (2001) 012, arXiv:hep-th/0106061

View on Pith Pith review arXiv 2001

[16] [13]

T. J. Hollowood, JHEP 0203 (2002) 038, arXiv:hep-th/0201075; Nucl. Phys. B 639, 66 (2002), arXiv:hep-th/0202197. 28

View on Pith Pith review arXiv 2002

[17] [14]

N. A. Nekrasov, arXiv:hep-th/0206161

View on Pith Pith review arXiv

[18] [15]

G. W. Moore, N. Nekrasov and S. Shatashvili, Commun. Math. Ph ys. 209 (2000) 77, arXiv:hep-th/9803265

View on Pith Pith review arXiv 2000

[19] [16]

G. W. Moore, N. Nekrasov and S. Shatashvili, Commun. Math. Ph ys. 209 (2000) 97, arXiv:hep-th/9712241

View on Pith Pith review arXiv 2000

[20] [17]

A. Losev, N. Nekrasov and S. L. Shatashvili, Nucl. Phys. B 534 (1998) 549, arXiv:hep-th/9711108

View on Pith Pith review arXiv 1998

[21] [18]

G. W. Moore and E. Witten, Adv. Theor. Math. Phys. 1 (1998) 298, arXiv:hep-th/9709193

View on Pith Pith review arXiv 1998

[22] [19]

E. Witten, J. Geom. Phys. 9 (1992) 303, arXiv:hep-th/9204083

View on Pith Pith review arXiv 1992

[23] [21]

R. Flume and R. Poghossian, “An algorithm for the microscopic ev aluation of the coeﬃcients of the Seiberg-Witten prepotential”, arXiv:hep-th/02 08176

View on Pith

[24] [22]

N. Berline, E. Getzler and M. Vergne, “Heat Kernels and Dirac Op erators”, Berlin, Germany: Springer (1996)

View on Pith 1996

[25] [24]

L. Baulieu and I. M. Singer, Nucl. Phys. Proc. Suppl. B5 (1988) 12

View on Pith 1988

[26] [25]

N. Dorey, T.J. Hollowood, V.V. Khoze, M.P. Mattis and S. Vandore n, Nucl. Phys. B552 (1999) 88, arXiv:hep-th/9901128

View on Pith Pith review arXiv 1999

[27] [26]

M. B. Green and M. Gutperle, JHEP 0002 (2000) 014, arXiv:hep-th/0002011

View on Pith Pith review arXiv 2000

[28] [27]

M. Bill´ o, M. Frau, I. Pesando, F. Fucito, A. Lerda, A. Liccardo , “Classical Gauge Instantons From Mixed Disks”, in preparation

View on Pith

[29] [28]

M. R. Douglas, “Branes within branes”, arXiv:hep-th/9512077 . 29

View on Pith Pith review arXiv

[30] [29]

M. R. Douglas, J. Geom. Phys. 28 (1998) 255, arXiv:hep-th/9604198

View on Pith Pith review arXiv 1998

[31] [30]

E. Witten, Nucl. Phys. B 460 (1996) 541, arXiv:hep-th/9511030

View on Pith Pith review arXiv 1996

[32] [31]

E. T. Akhmedov, Phys. Rev. D 59 (1999) 101901, arXiv:hep-th/9812038

View on Pith Pith review arXiv 1999

[33] [32]

S. Hirano and M. Kato, Prog. Theor. Phys. 98 (1997) 1371, arXiv:hep-th/9708039

View on Pith Pith review arXiv 1997

[34] [34]

J. D. Edelstein, M. Marino and J. Mas, Nucl. Phys. B 541 (1999) 671, arXiv:hep-th/9805172; J. D. Edelstein, M. Gomez-Reino and J. Mas , Nucl. Phys. B 561 (1999) 273, arXiv:hep-th/9904087; J. D. Edelstein, M. Gomez-Re ino, M. Marino and J. Mas, Nucl. Phys. B 574 (2000) 587, arXiv:hep-th/9911115

View on Pith Pith review arXiv 1999

[35] [35]

M. Bianchi, F. Fucito and G. Rossi, Phys. Lett. B 359 (1995) 56, arXiv:hep-th/9507003

View on Pith Pith review arXiv 1995

[36] [36]

N. Dorey, T. J. Hollowood and V. V. Khoze, “Notes on soliton bou nd-state problems in gauge theory and string theory”, arXiv:hep-th/0105090. 30

View on Pith Pith review arXiv