Issues in Topological Gauge Theory

Andrei Losev; Nikita Nekrasov; Samson Shatashvili

REVIEW

Issues in Topological Gauge Theory

Andrei Losev, Nikita Nekrasov, Samson Shatashvili

High Energy Physics - Theory · Submitted 16 Nov 1997 · 57 references

Pith's verdict: A program paper that names the right problem — contact terms between p-observables — and gives a scheme with two or three real checks, not a theorem; the matter-as-index-bundle section is the part that aged best.

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

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv IDhep-th/9711108 (version 2)

Pith IDV7FX2DYE, Pith's permanent record of this paper and its review

CitationAndrei Losev, Nikita Nekrasov, Samson Shatashvili. "Issues in Topological Gauge Theory." arXiv:hep-th/9711108 (1997). https://pith.science/paper/V7FX2DYE

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 put the contact-term problem in twisted N=2 gauge theory on the table and gave a workable scheme for it. Read it as a program with real checks, not as a theorem, and read section 5.2 first — the matter-as-equivariant-Euler-class-of-a-Dirac-index-bundle idea is the piece that has held up and that people actually build on. The Gromov-Witten framing in the introduction is more aspiration than construction, and the authors say so; the honest content is the abelian effective theory and the contact terms, not the four-dimensional Deligne-Mumford analogue, which they explicitly leave open (section 4.2, the discussion around the blowup of the diagonal). The twisted superfield formalism in 3.4 is genuinely useful: writing the duality as a Gaussian integral over the long superfield and tracking the Det(∂u/∂a)^{χ/2} anomaly makes the modular properties transparent in a way the older component treatments do not. The blowup derivation of the two-observable contact term (4.2) is the best piece of physics in the paper, and it agrees with the Hamilton-Jacobi route, which is a real check rather than a consistency ritual. On the stress-test concern: it lands, but it undercounts. The uniqueness claim in the abstract does outrun the evidence for k+l>2 and r>1, and the authors concede the higher-rank extension is unknown. But there are two independent validations of the degree-2 homogeneous Hamiltonian, not one — the C_{2,2} match against Göttsche-Zagier and Moore-Witten, and the ghost-number anomaly computation in 4.2 (Eqs. 4.9–4.13), which fixes the same normalization from the instanton-charge scaling. The H(a,a_D) = -u^2 versus H|_L = u clash is real but it is a notation collision, not a contradiction: the first is the square of the second up to sign, chosen so that H^r reproduces Tr φ^{2r}. It should be cleaned up, not treated as an error. The other soft spot worth naming is the Künneth formula (5.5), assumed rather than proved and checked only on IP1×IP1; the authors flag that it fails in type A sigma models, so it is a hypothesis, not a result. Same for the τ2 = const contour prescription, which they admit is a choice. Who should read it: anyone working on Donaldson invariants, u-plane integrals, or the four-dimensional/type-B correspondence. The Landau-Ginzburg target with components labelled by the weight lattice (7.2) is a nice structural observation that later work took seriously. Worth a serious referee, and worth citing.

Referee report

Significance

If the scheme is correct, it gives a uniform evaluation of the full tower of Donaldson-type observables and a concrete proposal for the compactified configuration space that plays the role of the Deligne-Mumford space in four dimensions. The paper is ambitious and ties together several threads (duality, contact terms, matter, higher-dimensional compactification, two-dimensional analogies). It deserves credit for the checks it does supply: the two-observable contact term is compared against two independent published computations (Göttsche-Zagier and Moore-Witten, Eqs. 4.16-4.20) and re-derived by a separate blowup argument (§4.2), and the Künneth identity is verified explicitly on IP1 x IP1 (§5.1, Eqs. 5.6-5.9). These are nontrivial consistency tests, not restatements.

Major objections (4)

[Abstract s4; §2.3] The abstract's claim that the scheme 'uniquely fixes all the contact terms' outruns what is established. The authors themselves state in §2.3 that 'in general there is no canonical way of extending the Γ-invariant function on L to the Γ-invariant function on C2r' and that 'in the case of higher rank we do not know how to extend the functions u1,...,ur.' Uniqueness therefore holds only conditional on the two proposed principles (background independence and degree-2 homogeneity), and the only independent validation of the resulting contact-term formula (Eq. 2.33) is the k=l=1 case checked against Göttsche-Zagier and Moore-Witten. For k+l>2 and for r>1 no independent check is presented. The abstract and introduction should state this scope explicitly rather than asserting uniqueness without qualification.
[Eq. 2.6 vs Eq. 2.26] There is an unresolved notational conflict between the normalization in Eq. 2.6, H(a(u),aD(u)) = -u^2, and the homogeneity principle in Eq. 2.26, H(a,aD)|_L = u. As written these assign different restrictions of H to L, and the homogeneity degree is unclear: if u scales with weight 2 (as the ghost-number discussion in §2.3 suggests), a degree-2 homogeneous H restricted to L should be proportional to u, not u^2. The intermediate derivation of Eq. 2.33 uses ∂H/∂aD, so the reader cannot tell which H is meant. Either the two equations refer to two distinct Hamiltonians (in which case the shared symbol H must be disambiguated) or one of them is in error. This should be reconciled before the derivation of Eq. 2.33 can be followed.
[Claims 1-3 (Eqs. 2.14, 2.15); §4.5] The derivation of the contour-integral formulas is sketched rather than carried through. The passage from the u-plane measure (Eq. 4.44) to the residue formula (Eq. 4.55) relies on the universality conjecture and on the identification of the factor a^{(ℓ,ℓ)-σ}/8 with the contribution of the massless hypermultiplet; the argument that this 'is what (4.55) does' is asserted, not demonstrated term by term. For the b2+=1 case, Claim 2 (Eq. 2.15) is stated without derivation beyond the blowup-factor discussion. The authors also explicitly leave the Nf=2 case open ('We hope to elaborate further on this subject elsewhere'), where Eq. 4.20 differs from Eq. 4.28 by a one-instanton contribution. These are acknowledged gaps, but they bound the scope of the headline result and should be reflected in the abstract.
[§4.1-4.2; §2.3] The contact terms are fixed by principles motivated by modular invariance and then checked against modular-invariant known results, which is close to circular. The blowup argument in §4.2 does provide a genuinely independent route for the 2-2 contact term and is the strongest evidence in the paper; however, no comparably independent derivation is given for the higher contact terms or for higher rank. The paper would be strengthened by either an independent check of at least one k+l>2 contact term or an explicit statement that the higher contact terms rest on the universality conjecture alone.

Minor objections (4)

[passim] There are numerous typographical errors that impede reading: 'presicely' (footnote 4), 'invaraint' (p. 3), 'statetement' (p. 15), 'multuplet' (p. 15), 'correspodning' (p. 32), 'interpeted' (p. 4), 'Couomb' (p. 60), 'Klewer' (ref. 55), 'Hogh' (ref. 50), and 'th' for 'the' in ref. 26. These should be corrected in a copy-edited version.
[§2.3, 'A note on definitions'] The authors acknowledge that the letter F is overloaded between the prepotential and the generating function of correlators. The note is helpful, but the overload recurs throughout (e.g. F(a;tr) in §2.2 versus F as prepotential in §2.3), and a distinct symbol for the generating function would remove residual ambiguity.
[Eq. 2.17] The table of ∆ and f for Nf = 1,2,3 is difficult to parse in its current layout; a properly typeset table with clear column headings would help the reader verify the Picard-Fuchs equations (Eq. 2.16).
[Introduction; ref. [9]] The overlap with Moore-Witten [9] is acknowledged in the introduction, which is appropriate. It would be useful to state more precisely which of the b2+=1 formulas in §4 are new relative to [9] and which are reproduced for comparison.

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

The paper

The paper claims

The paper proposes that a homogeneity principle plus a conjectured universality of contact terms uniquely fixes the contact terms in twisted $N=2$ abelian gauge theory, yielding contour-integral formulas for all $p$-observable correlators.

The reading

The paper asks how to compute the full tower of topological observables in twisted $N=2$ gauge theories, the setting that produces Donaldson-type invariants of four-manifolds. Its proposal is that the difficult contact terms between observables are not independent data: once one fixes how the Coulomb-branch coordinate $u$ extends to a homogeneous function on the full symplectic space, all contact terms follow from Hamilton-Jacobi evolution, and the correlators collapse to contour integrals over the $u$-plane weighted by Seiberg-Witten invariants. If this is right, the same machinery covers $p$-observables for $0leq pleq4$, some manifolds with $b_2^+=1$, theories with matter, and higher-dimensional compactifications. The authors present the scheme as a first step toward a four-dimensional analogue of the Gromov-Witten paradigm, where contact terms encode how the compactified configuration space of points on the four-manifold is glued together.

What carries the argument

The load-bearing machinery is the twisted superfield formalism combined with the Hamilton-Jacobi deformation of a $Gamma$-invariant Lagrangian submanifold $Lsubsetmathbb{C}^{2r}$. The contact terms are not computed by Feynman diagrams directly; they are read off from the failure of a naive holomorphic observable to be modular invariant, and then fixed by requiring that the deformation be generated by a homogeneous Hamiltonian whose restriction to $L$ is the desired function of $u$. The universality principle says the contact term between $O^{(p)}_{varphi_1}$ and $O^{(q)}_{varphi_2}$ is $O^{(p+q-4)}_{C(varphi_1,varphi_2)}$ with a $C$ independent of $p$ and $q$. This is what turns an infinite tower of contact terms into a single calculable object and produces the contour-integral formulas of Claims 1--3.

Extended reading notes

Core claim

The central claim is that the contact terms between $p$-observables in the low-energy abelian twisted theory are fixed by two principles: background independence, meaning the deformation Hamiltonians are time-independent, and homogeneity, meaning the Coulomb-branch coordinate $u$ extends to a function $H(a,a_D)$ on the symplectic vector space with $H(mu a,mu a_D)=mu^2 H(a,a_D)$. With this extension, the generating function $F(a;T)$ solves the Hamilton-Jacobi system $partial F/partial T_{k,alpha}=-H_k(a,partial F/partial a)e_alpha$, and the pair contact term takes the universal form $F_{(k,alpha),(l,beta)}=kl,u^{k+l-2}bigl(du-a,du/a_D,da-a,da_D,dubigr)e_alpha cdot e_beta$. The paper then claims that correlation functions of all $p$-observables are given by contour integrals over the $u$-plane, with Seiberg-Witten invariants supplying the non-perturbative weights, and that the same scheme extends to some manifolds with $b_2^+=1$ through blowup $\theta$ factors.

Load-bearing premise

The load-bearing premise is that the $Gamma$-invariant function $u$ on the Coulomb branch $L$ has a unique homogeneous extension $H$ to the full symplectic space $mathbb{C}^{2r}$ and that the deformation Hamiltonians can be chosen time-independent; the authors explicitly note that no canonical extension exists in general and that for higher rank they do not know how to extend $u_1,ldots,u_r$.

Editorial extensions

If this is right

Correlation functions of all $p$-observables on manifolds of generalized simple type reduce to contour integrals over the $u$-plane weighted by Seiberg-Witten invariants, as in Eq. (2.14).
On some manifolds with $b_2^+=1$, the same correlators acquire explicit blowup factors built from theta functions, as in Eq. (2.15).
Theories with matter are expressed as integrals over instanton moduli space of the equivariant Euler class of the Dirac index bundle, giving a uniform matter-coupled extension.
Higher-dimensional compactifications are claimed to correspond to insertions of $hat A$-genus or elliptic genus classes into the instanton-moduli integral.
The contact-term scheme supplies a concrete candidate for the four-dimensional analogue of the Deligne-Mumford compactification: intersections of cycles are replaced by blowups of diagonals.

Reading between the lines

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

If the universality claim is right, the compactified configuration space of points on a four-manifold must have a boundary structure whose strata are blowups of diagonals; that gives a concrete target for constructing the four-dimensional analogue of Deligne-Mumford.
The rank-one homogeneity trick suggests a practical algorithm for higher-rank prepotentials, but the paper's own restriction to rank one means this extension remains to be demonstrated.
The proposed type $B$ Landau-Ginzburg dictionary predicts that contact terms can be recomputed from residues in a two-dimensional model; a discrepancy between the two computations would be a direct test of the dictionary.
The matter formula suggests that Donaldson invariants with matter are intersection numbers of universal bundles, which could be checked on rational surfaces where the moduli space is well understood.

Original abstract

We discuss topological theories, arising from the general $\mathcal{N}=2$ twisted gauge theories. We initiate a program of their study in the Gromov-Witten paradigm. We re-examine the low-energy effective abelian theory in the presence of sources and study the mixing between the various $p$-observables. We present the twisted superfield formalism which makes duality transformations transparent. We propose a scheme which uniquely fixes all the contact terms. We derive a formula for the correlation functions of $p$-observables on the manifolds of generalized simple type for $0 \leq p \leq 4$ and on some manifolds with $b_{2}^{+} =1$. We study the theories with matter and explore the properties of universal instanton. We also discuss the compactifications of higher dimensional theories. Some relations to sigma models of type $A$ and $B$ are pointed out and exploited.

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, 6,694 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.17 · states limitation 0.11 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.19 · compares with prior results 0.13 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We discuss topological theories, arising from the general $\mathcal{N}=2$ twisted gauge theories. We initiate a program of their study in the Gromov-Witten paradigm. We re-examine the low-energy effective abelian theory in the presence of sources and study the mixing between the various $p$-observables. We present the twisted superfield formalism which makes duality transformations transparent. We propose a scheme which uniquely fixes all the contact terms. We derive a formula for the correlation functions of $p$-observables on the manifolds of generalized simple type for $0 \leq p \leq 4$ and on some manifolds with $b_{2}^{+} =1$. We study the theories with matter and explore the properties of universal instanton. We also discuss the compactifications of higher dimensional theories. Some relations to sigma models of type $A$ and $B$ are pointed out and exploited.

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 discuss topological theories, arising from the general $\mathcal{N}=2$ twisted gauge theories.	background 0.72	0.65	0.05	asserted 0.94
s1	We initiate a program of their study in the Gromov-Witten paradigm.	contribution 0.72	0.84	0.05	asserted 0.93
s2	We re-examine the low-energy effective abelian theory in the presence of sources and study the mixing between the various $p$-observables.	contribution 1.00	0.88	0.07	asserted 0.97
s3	We present the twisted superfield formalism which makes duality transformations transparent.	method 0.64	0.90	0.07	asserted 1.00
s4	We propose a scheme which uniquely fixes all the contact terms.	contribution 0.80	0.90	0.16	asserted 0.90
s5	We derive a formula for the correlation functions of $p$-observables on the manifolds of generalized simple type for $0 \leq p \leq 4$ and on some manifolds with $b_{2}^{+} =1$.	result 0.94	0.95	0.39	asserted 1.00
s6	We study the theories with matter and explore the properties of universal instanton.	contribution 1.00	0.89	0.05	asserted 0.98
s7	We also discuss the compactifications of higher dimensional theories.	contribution 0.99	0.75	0.03	asserted 0.96
s8	Some relations to sigma models of type $A$ and $B$ are pointed out and exploited.	contribution 0.65	0.77	0.05	asserted 0.97

What kind of paper the classifier reports this is: theoretical result 0.90 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v2

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 (57)

[1]

N. Seiberg, E. Witten, “Electric-Magnetic Duality, Mon opole Condensation, And Con- ﬁnement in N = 2 Supersymmetric Yang-Mills Theory ” Nucl. Phys. B426 (199 4) 19-52 (and erratum - ibid. B430 (1994) 485-486 ); “Monopoles, Duality and Chiral Symmetry Breaking in N=2 Sup ersymmetric QCD”, hep-th/9408099, Nucl. Phys. B431 (1994) 484-550

View on Pith Pith review arXiv 1994
[2]

A. Losev, N. Nekrasov, S. Shatashvili, “Seiberg-Witten Solution and ADHM integra- tion”, in progress

View on Pith
[3]

N. Dorey, V. Khoze, M. Mattis, “Supersymmetry and Multi- Instanton Measure”, hep- th/9708036

View on Pith
[4]

E. Witten., Comm. Math. Phys. 118 (1988) 411; E. Witten, Nucl. Phys. B340 (1990) 281

View on Pith 1988
[5]

M. Kontsevich, Yu. Manin, “Gromov-Witten classes and en umerative geometry”, CMP 164 (1994) 525

View on Pith 1994
[6]

M. Kontsevich, “Enumeration of rational curves via toru s action”, hep-th/9405035

View on Pith Pith review arXiv
[7]

W. Fulton, R. MacPherson, “A Compactiﬁcation of Conﬁgur ation Spaces”, Annals of Math. (1993)

View on Pith 1993
[8]

L. G¨ ottsche, D. Zagier, “Jacobi forms and the structure of Donaldson invariants for 4-manifolds with b+2 = 1”, alg-geom/9612020

View on Pith Pith review arXiv
[9]

G. Moore, E. Witten, “Integration over u-plane in Donaldson theory”, hep-th/9709193

View on Pith Pith review arXiv
[10]

R. Borcherds, “Automorphic forms with singularities o n Grassmanians”, alg-geom/9609022

View on Pith Pith review arXiv
[11]

E. Witten, “Monopoles and Four-Manifolds”, hep-th/94 11102

View on Pith
[12]

P. Fre, “Lectures on Special Kahler Geometry and Electr ic–Magnetic Duality Rota- tions” , Nucl.Phys.Proc.Suppl. 45BC (1996) 59-114, hep-th /9512043

View on Pith 1996
[13]

M. Matone, “Instantons and recursion relations in in N = 2 susy gauge theory”, Phys. Lett. B357(1995) 342, hep-th/9506102

View on Pith Pith review arXiv 1995
[14]

J. Sonnenschein, S. Theisen, S. Yankielowicz, Phys. Le tt. 367B (1996) 145; T. Eguchi, S.-K. Yang, Mod. Phys. Lett. A11 (1996) 131

View on Pith 1996
[15]

G. Bonelli, M. Matone, Phys. Rev. Lett. 77 (1996) 4712-4715; hep-th/9605090

View on Pith Pith review arXiv 1996
[16]

V. Arnol’d, “Mathematical Methods of Classical Mechan ics”, Springer-Verlag, 1980

View on Pith 1980
[17]

S. Cecotti, C. Vafa, “Topological Anti-Topological Fu sion”, Nucl. Phys. B367 (1991) 359

View on Pith 1991
[18]

S.K. Donaldson and P.B. Kronheimer, “The Geometry of Fo ur-Manifolds”, Clarendon Press, Oxford, 1990

View on Pith 1990
[19]

E. Witten, “On S-Duality in Abelian Gauge Theory,” hep-th/9505186; Select a Math- ematica 1 (1995) 383 68

View on Pith Pith review arXiv 1995
[20]

E. Witten, “Introduction to Cohomological Field Theor ies”, Lectures at Workshop on Topological Methods in Physics, Trieste, Italy, Jun 11-2 5, 1990, Int. J. Mod. Phys. A6 (1991) 2775

View on Pith 1991
[21]

N. Nekrasov, “Holomorphic bundles and many-body syste ms”, hep-th/9503157, Comm. Math. Phys. (1997)

View on Pith Pith review arXiv 1997
[22]

R. Friedman, J. Morgan, “Smooth Four-Manifolds and Com plex Surfaces”, Ergebnisse der. Math. und Grenz. 3, Springer-Verlag, New York, 1994

View on Pith 1994
[23]

R. Fintushel and R.J. Stern, “The Blowup Formula For Don aldson Invariants”, alg-geom/9405002; Annals of Math. 143(1996) 529

View on Pith Pith review arXiv 1996
[24]

E. Witten, “On Quantum gauge theories in two dimensions ,” Commun. Math. Phys. 141 (1991) 153; “Two dimensional gauge theories revisited”, J. Geom. Phys. 9 (1992) 303-368

View on Pith 1991
[25]

M. Blau, G. Thomson, “Derivation of the Verlinde Formul a from Chern-Simons Theory and the G/G model”,Nucl. Phys. B408 (1993) 345-390

View on Pith 1993
[26]

A. Gerasimov, “Localization in GWZW and Verlinde formu la”, UUITP 16/1993, hepth/9305090

View on Pith 1993
[27]

J. Polchinski, “Evaluation of the one-loop string path integral”, Comm. Math. Phys. 104 (1986) 37; B. McClain and B.D.B. Roth, “Modular invariance for interac ting bosonic strings at ﬁnite temperature”, Comm. Math. Phys. 111 (1987) 539

View on Pith 1986
[28]

J.A. Harvey, G. Moore, “Algebras, BPS States, and Strin gs,” hep-th/9510182, Nucl. Phys. B463 (1996) 315

View on Pith Pith review arXiv 1996
[29]

I. Krichever, “Universal τ -function of Whitham hierarchy”, LPTENS-92/18, hep- th/9205110, in “30 years of the Landau Institute”, 477 , I.M. Khalatnikov, V.P. Mi- neev, V.P. (eds.)

View on Pith
[30]

S. Cordes, G. Moore, and S. Ramgoolam, “ Lectures on 2D Ya ng Mills theory, Equiv- ariant Cohomology, and Topological String Theory,” hep-th /9411210

View on Pith
[31]

K. Uhlenbeck, “Removable singularities in Yang-Mills ﬁelds”, Comm. Math. Phys. 83 (1982) 11

View on Pith 1982
[32]

A. Losev, G. Moore, N. Nekrasov, S. Shatashvili, “Four- Dimensional Avatars of 2D RCFT,” hep-th/9509151, Talks at Strings’95 and Trieste 1 995

View on Pith Pith review arXiv
[33]

J.M.F. Labastida, P.M. Llatas, “Potentials for topolo gical sigma models”, Phys. Lett. B271 (1991), 101; J.M.F. Labastida, M. Mari˜ no, Nucl. Phys. B448(1995) 373; J.M.F. Labastida, M. Mari˜ no, hep-th/9603169

View on Pith Pith review arXiv 1991
[34]

S. Hyun, J. Park, J.-S. Park, “Topological QCD”, hep-th /9503201

View on Pith
[35]

N. Nekrasov, PhD. Thesis, Princeton 1996

View on Pith 1996
[36]

N. Nekrasov, “Five dimensional gauge theories and rela tivistic integrable systems”, hep-th/9609219 69

View on Pith Pith review arXiv
[37]

L. Baulieu, A. Losev, N. Nekrasov, “Chern-Simons and Tw isted Supersymmetry in Higher Dimensions”, hep-th/9707174

View on Pith Pith review arXiv
[38]

M. Atiyah, R. Bott, “ The Moment Map And Equivariant Coho mology”, Topology 23 (1984) 1-28

View on Pith 1984
[39]

O. Ganor, D. Morrison, N. Seiberg, “Branes, Calabi-Yau Spaces and Compactiﬁca- tion of Six Dimensional E8 Theory”, hep-th/9610251

View on Pith Pith review arXiv
[40]

O. Ganor, “Toroidal Compactiﬁcation of Heterotic 6D No n-Critical Strings Down to Four Dimensions”, hep-th/9608109, Nucl.Phys. B488 (1997) 223-235

View on Pith Pith review arXiv 1997
[41]

A. Givental, “Equivariant Gromov-Witten Invariants” , alg-geom/9603021; “A mirror theorem for toric complete intersections”, alg-g eom/9701016

View on Pith Pith review arXiv
[42]

E. Witten, “Phases of N=2 Theories in Two Dimensions”, N ucl. Phys. B403 (1993) 159, hep-th/9301042

View on Pith Pith review arXiv 1993
[43]

D. Morrison, M. Plesser, hep-th/9412236

View on Pith Pith review arXiv
[44]

L. Alvarez-Gaume, Commun.Math.Phys. 90 (1983)161; D. Friedan, P. Windey, Nucl.Phys. B235 (1984) 395

View on Pith 1983
[45]

A. Hietamaki, A. Niemi, A. Morozov, K. Palo “Geometry Of N = 1/2 Supersymmetry And The Atiyah-Singer Theorem”, Phys.Lett. B263 (1991) 417

View on Pith 1991
[46]

S. Katz, A. Klemm, C. Vafa, “Geometric Engineering of Qu antum Field Theories”, hep-th/9609239, Nucl.Phys. B497 (1997) 173-195

View on Pith Pith review arXiv 1997
[47]

A. Lawrence, N. Nekrasov, “Instanton Sums and Five Dime nsional Gauge Theory”, hep-th/9706025

View on Pith Pith review arXiv
[48]

M. Kontsevich, “Deformation Quantization of Poisson M anifolds”, dg-ga/ and talk at Strings’97

View on Pith
[49]

K. Saito, Publ. RIMS, Kyoto Univ. 19 (1983) 1231; B. Blok, A. Varchenko, Int.J.Mod.Phys. A7 (1992) 1467-1490 ; R. Dijkgraaf, talk on RIMS conference ”Inﬁnite dimensional analysis”, July 1993 ; A. Losev, Theor.Math.Phys. 95 (1993) 595; A. Losev, in ”Integrable models and Strings ”, 172, Proceedi ngs of 1993 Helsinki Conference, Springer-Verlag, 1995 ; A. L...

View on Pith 1983
[50]

M. Blau and G. Thompson, “Lectures on 2d Gauge Theories: Topological Aspects and Path Integral Techniques”, Presented at the Summer Scho ol in Hogh Energy Physics and Cosmology, Trieste, Italy, 14 Jun - 30 Jul 1993, h ep-th/9310144

View on Pith 1993
[51]

K. Intriligator, “Fusion residues”, Mod. Phys. Lett. A6 (1991) 3541-3556, hep- th/9108005

View on Pith 1991
[52]

E. Witten, “The Verlinde Algebra And The Cohomology Of T he Grassmannian”, hep-th/9312104

View on Pith Pith review arXiv
[53]

M. Bershadsky, A. Johansen, V. Sadov and C. Vafa, “Topol ogical Reduction of 4D SYM to 2D σ–Models”, hep-th/9501096 70

View on Pith Pith review arXiv
[54]

M. Atiyah, R. Bott, “The Yang-Mills Equations Over Riem ann Surfaces”, Phil. Trans. R.Soc. London A 308, 523-615 (1982)

View on Pith 1982
[55]

A. Polyakov, “Gauge ﬁelds and strings”, Klewer Academi c Press

View on Pith
[56]

M. Gromov, Invent. Math. 82 (1985) 307

View on Pith 1985
[57]

A. Gorsky, I. Krichever, A. Marshakov, A. Morozov, A. Mi ronov, “Integrablity and Seiberg-Witten Exact Solution”, hep-th/9505035, Phys .Lett.B355: 466-474, 1995 71

View on Pith Pith review arXiv 1995

Forward citations

Cited by 42 Pith papers

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

Seiberg-Witten Prepotential From Instanton Counting
hep-th 2002-06 conditional novelty 9.0 of 10

Equivariant localization on the instanton moduli space yields a closed combinatorial formula for the Seiberg-Witten prepotential, matching known instanton expansions up to five instantons.
Topological Twisting of 4d $\mathcal{N}=2$ Supersymmetric Field Theories
hep-th 2024-11 conditional novelty 8.0 of 10

For any 4d N=2 theory, topologically twisted partition functions depend on the diffeomorphism type, 't Hooft fluxes, and a generalized spin-c structure.
The Coulomb Branch of 3d $\mathcal{N}=4$ Theories
hep-th 2015-03 conditional novelty 8.0 of 10

An abelianization map computes the Coulomb branch chiral ring of a general 3d N=4 gauge theory, with nonabelian monopole operators given by equivariant integrals over BPS moduli spaces, verified in many examples.
Seiberg-Witten Geometry of Four-Dimensional $\mathcal N=2$ Quiver Gauge Theories
hep-th 2012-11 accept novelty 8.0 of 10

The vacuum moduli space of mass-deformed N=2 ADE quiver gauge theories equals the moduli space of genus-zero quasimaps to Bun_G(E), with the underlying integrable systems identified as monopole and instanton moduli spaces.
The Omega Deformation, Branes, Integrability, and Liouville Theory
hep-th 2010-02 conditional novelty 8.0 of 10

The Omega-deformation is equivalent, away from fixed points, to a field redefinition plus a rotated supersymmetry, and the resulting theory is captured by a coisotropic A-brane that explains its links to integrability...

Show 37 more citing papers

Supersymmetric vacua and Bethe ansatz
hep-th 2009-01 conditional novelty 8.0 of 10

The vacuum equations of two-dimensional N=2 supersymmetric gauge theories coincide with the Bethe equations of integrable spin chains, yielding a dictionary between the two subjects.
Four-dimensional wall-crossing via three-dimensional field theory
hep-th 2008-07 conditional novelty 8.0 of 10

The Kontsevich-Soibelman wall-crossing formula for BPS states in N=2 gauge theories follows from the continuity of the exact hyperkähler metric on the moduli space of the theory on R^3 x S^1.
Localization of gauge theory on a four-sphere and supersymmetric Wilson loops
hep-th 2007-12 accept novelty 8.0 of 10

A localization computation on the four-sphere proves the Erickson-Semenoff-Zarembo/Drukker-Gross conjecture and gives exact matrix model formulas for N=2 and N=2* supersymmetric gauge theories.
Seiberg-Witten Theory and Random Partitions
hep-th 2003-06 accept novelty 8.0 of 10

The low-energy Seiberg-Witten prepotential of N=2 gauge theories is derived from the exact instanton partition function by a saddle-point analysis of random partitions.
Far Beyond the Planar Limit in Strongly-Coupled $\mathcal{N}=4$ SYM
hep-th 2020-03 conditional novelty 7.0 of 10

A new exact localization constraint on the N=4 SYM sphere free energy fixes the stress-tensor four-point correlator at order 1/N^8, giving the first genus-three holographic correlator coefficient.
Modular Invariance in Superstring Theory From ${\cal N} = 4$ Super-Yang-Mills
hep-th 2019-12 conditional novelty 7.0 of 10

The large-N expansion of a four-point correlator in N=4 super-Yang-Mills reproduces the SL(2,Z)-invariant Eisenstein series coefficients of the R^4 and D^4R^4 interactions in type IIB string theory, including D-instan...
Genus-2 Holographic Correlator on $AdS_5 \times S^5$ from Localization
hep-th 2019-08 conditional novelty 7.0 of 10

The genus-two D4R4 coefficient in the strongly coupled N=4 SYM stress-tensor correlator is fixed to 7/3072 by combining supersymmetric localization with the flat-space IIB S-matrix.
AdS$_4$/CFT$_3$ from Weak to Strong String Coupling
hep-th 2019-06 accept novelty 7.0 of 10

The R^4 correction to the stress-tensor four-point function in ABJM theory is fixed at finite type IIA string coupling and matches the known genus-zero and genus-one string S-matrix.
$\mathcal{N}=4$ Super-Yang-Mills Correlators at Strong Coupling from String Theory and Localization
hep-th 2019-02 conditional novelty 7.0 of 10

Localization and string-theory computations of 1/λ corrections to half-BPS four-point functions in N=4 SYM agree at R^4 order, confirming AdS/CFT beyond supergravity and fixing the D^4R^4 correction.
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...
Two-Sphere Partition Functions and Gromov-Witten Invariants
hep-th 2012-08 accept novelty 7.0 of 10

The two-sphere partition function of a gauged linear sigma model equals the exact Kahler potential of its Calabi-Yau target, giving genus-zero Gromov-Witten invariants without mirror symmetry.
Exact Results for 't Hooft Loops in Gauge Theories on S^4
hep-th 2011-05 conditional novelty 7.0 of 10

The exact path integral of an N=2 gauge theory on S^4 with a supersymmetric 't Hooft loop factorizes into pole instanton contributions and an equator monopole-screening contribution, matching Toda CFT predictions.
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 ...
Central charges of N=2 superconformal field theories in four dimensions
hep-th 2008-04 conditional novelty 7.0 of 10

The central charges a and c of a strongly coupled N=2 SCFT are read off from the R-charge of the topological path-integral measure factor A^chi B^sigma, with no holography required.
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.
Generalised $\Omega$-backgrounds in twisted $N=4$ super Yang-Mills
hep-th 2026-10 conditional novelty 6.0 of 10

The paper constructs Omega-deformed rigid supergravity backgrounds for the Donaldson-Witten, Vafa-Witten and Geometric Langlands twists of N=4 super Yang-Mills on general Euclidean manifolds.
Anomalous Symmetries End at the Boundary
hep-th 2020-12 conditional novelty 6.0 of 10

A symmetry with an 't Hooft anomaly cannot be preserved by any boundary condition of the theory, proven for perturbative and discrete anomalies and conjectured for general gravitational ones.
Vertex algebras and 4-manifold invariants
hep-th 2017-05 conditional novelty 6.0 of 10

4-manifold invariants, including Seiberg-Witten and a new equivariant multi-monopole family, are argued to equal chiral correlators of impurity vertex operators in a 2d theory from fivebranes, with the multi-monopole ...
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.
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.
Exact results for boundaries and domain walls in 2d supersymmetric theories
hep-th 2013-08 conditional novelty 6.0 of 10

The hemisphere partition function of a 2d N=(2,2) gauge theory with B-type boundary conditions computes the exact D-brane central charge and depends only on the brane's K-theory class.
Counting Exceptional Instantons
hep-th 2012-05 conditional novelty 6.0 of 10

By conjecturally extending the Nakajima-Yoshioka blow-up recursion to arbitrary gauge groups, the authors compute one- and two-instanton partition functions for E6, E7, E8, F4 and G2, matching known one-instanton resu...
Spectral Duality in Integrable Systems from AGT Conjecture
hep-th 2012-04 conditional novelty 6.0 of 10

The N-site Heisenberg XXX spin chain and a reduced gl(N) Gaudin model are shown to be spectrally dual, with matching spectral curves and Seiberg-Witten differentials, and with a claimed matching of their quantized Bax...
Direct integration for general Omega backgrounds
hep-th 2010-09 conditional novelty 6.0 of 10

A generalized holomorphic anomaly equation plus a conifold gap condition is shown sufficient to compute refined topological string amplitudes on several non-compact Calabi-Yau spaces, matching independent refined-vert...
Topological strings in generalized complex space
hep-th 2006-03 conditional novelty 6.0 of 10

A BV/AKSZ construction defines a topological sigma-model on generalized Calabi-Yau targets, recovering the A- and B-models and deriving the Kontsevich star product in the open-string sector.
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.
Gauge Origami and BPS/CFT correspondence
hep-th 2025-02 conditional novelty 5.0 of 10

The gauge origami partition function on C4 is realized as a correlation function of BPS qq-character operators, with D6 and D8 characters labeled by plane and solid partitions.
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.
Ding-Iohara-Miki symmetry of network matrix models
hep-th 2016-03 conditional novelty 5.0 of 10

Ward identities of network matrix models are generated by the Ding-Iohara-Miki algebra, and an elliptic version of the construction yields elliptic qq-characters.
A direct proof of AGT conjecture at beta = 1
hep-th 2010-12 conditional novelty 5.0 of 10

The AGT correspondence between 2d conformal blocks and 4d Nekrasov functions is proved at the special point epsilon_1 + epsilon_2 = 0 by expanding both sides in Schur polynomials and matching term by term.
Explicit examples of DIM constraints for network matrix models
hep-th 2016-04 conditional novelty 4.0 of 10

The authors show, with explicit formulas, that Ward identities for balanced network matrix models all follow from the commutation properties of the Ding-Iohara-Miki algebra.
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.
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.
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.

How later papers used it

87 later papers cite it

Evidence: some. 79 of the citing papers share no author with it, and 15 of those carry a confident label from Pith's citation classifier.

Pith read 8 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

2 later papers by other authors are listed below: 2 built on it.

Builds on it 2 of 5 papers with this label

Realized this paper's paradigm in its context, interpreting homology classes of another work's variety of triples via virtual fundamental cycles from moduli problems of theories with matter.

Coulomb Branches in 3d $\mathcal{N} = 4$ Revisited (2024) · 0.83 · paraphrased: the licence does not allow quoting
Followed ideas from this paper closely and generalized them to the case of equivariant cohomological field theory.

S-duality and supersymmetry on curved manifolds (2020) · 0.99 · paraphrased: the licence does not allow quoting

How the literature uses it

84 arXiv papers cite this in 165 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 · 110 sentences without a confident label

Builds on it 4 papers

We obtain the super-analog of the Losev–Moore-Nekrasov–Shatashvili (LMNS) integral formula [ref] , and also the Nekrasov-type combinatorial formula [ref] for the instanton partition function.

Super instanton counting and localization · Introduction · 0.93
In our treatment we follow closely ideas from [ref] which we generalize to the case of equivariant cohomological field theory.

S-duality and supersymmetry on curved manifolds · S-duality in cohomological variables · 0.99
The result can be readily extended to the case with surface defects [ref] or theories with matter hypermultiplets [ref] and even class $𝒮$ theories [ref] , although we leave this for future work.

The $u$-plane integral, mock modularity and enumerative geometry · Conclusions · 0.91

Uses its method or tool 7 papers

The authors of [ref] made a particular choice that allowed them to exploit a fermionic symmetry to localize the path integral (as in [ref] ) and compute the exact partition function for $(2,2)$ gauge theories with charged matter on $S^2$.

Two-Sphere Partition Functions and Gromov-Witten Invariants · The conjecture $Z_S^2 = e^-K$ · 0.81
The problem is solved through the localization technique, this can be achieved only after embedding the $𝒩=2$ gauge theory into the so called $Ω$-background [ref] .

Magnetic expansion of Nekrasov theory: the SU(2) pure gauge theory · Introduction · 0.85
Subsequently the technique of “lattice reduction” can be applied to evaluate the integral when $b_2(M)>1$ [ref] .

Donaldson-Witten theory and indefinite theta functions · Introduction · 0.95

Compares with it 4 papers

One motivation of our work is to compare with results for topological correlators obtained using the low energy approach by evaluating the $u$-plane integral [ref] .

Equivariant Localization for N=2 Theories with Hypermultiplets · Introduction · 0.88
We already recognize two contact terms, unlike in [ref] where there is only one, with the second one here being precisely due to the presence of the surface defect.

Donaldson-Witten theory, surface operators and mock modular forms · The Grassmann and photon path integral · 0.99
This should be contrasted with the calculation of the Nekrasov partition function on $ℂ^2$ [ref] (see also earlier works [ref] ) where a well-defined moduli space exists and there is good control (both physical and mathematical) over the singular configurations.

S-duality and supersymmetry on curved manifolds · Introduction · 0.93

Cites it as background 36 papers

As originally defined in [ref] , building on [ref] , the $Ω$ deformation involves a deformation of the Lagrangian that preserves part of the supersymmetry.

The Omega Deformation, Branes, Integrability, and Liouville Theory · Introduction · 0.96
The $u$-plane and its contribution to the path integral were studied in detail in [ref] .

The $u$-plane integral, mock modularity and enumerative geometry · Introduction, Summary and Plan · 0.97
The full path integral $_DW$ of this theory can be decomposed in to a continuous integral $_u$ over the Coulomb branch of the theory, where the gauge group is spontaneously broken to U(1) by a non-vanishing expectation value of the order parameter $u=1/16π^2<Tr ϕ^2>$, and a contribution[We use the subscript “SW”, since this contribution is fully determined by the so-called Seiberg-Witten invariants of $M$ [ref] .] $_SW$ from the points where the effective theory on the Coulomb branch becomes singular [ref] .

Donaldson-Witten theory and indefinite theta functions · Introduction · 0.80

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] [1]

N. Seiberg, E. Witten, “Electric-Magnetic Duality, Mon opole Condensation, And Con- ﬁnement in N = 2 Supersymmetric Yang-Mills Theory ” Nucl. Phys. B426 (199 4) 19-52 (and erratum - ibid. B430 (1994) 485-486 ); “Monopoles, Duality and Chiral Symmetry Breaking in N=2 Sup ersymmetric QCD”, hep-th/9408099, Nucl. Phys. B431 (1994) 484-550

View on Pith Pith review arXiv 1994

[2] [2]

A. Losev, N. Nekrasov, S. Shatashvili, “Seiberg-Witten Solution and ADHM integra- tion”, in progress

View on Pith

[3] [3]

N. Dorey, V. Khoze, M. Mattis, “Supersymmetry and Multi- Instanton Measure”, hep- th/9708036

View on Pith

[4] [4]

E. Witten., Comm. Math. Phys. 118 (1988) 411; E. Witten, Nucl. Phys. B340 (1990) 281

View on Pith 1988

[5] [5]

M. Kontsevich, Yu. Manin, “Gromov-Witten classes and en umerative geometry”, CMP 164 (1994) 525

View on Pith 1994

[6] [6]

M. Kontsevich, “Enumeration of rational curves via toru s action”, hep-th/9405035

View on Pith Pith review arXiv

[7] [7]

W. Fulton, R. MacPherson, “A Compactiﬁcation of Conﬁgur ation Spaces”, Annals of Math. (1993)

View on Pith 1993

[8] [8]

L. G¨ ottsche, D. Zagier, “Jacobi forms and the structure of Donaldson invariants for 4-manifolds with b+2 = 1”, alg-geom/9612020

View on Pith Pith review arXiv

[9] [9]

G. Moore, E. Witten, “Integration over u-plane in Donaldson theory”, hep-th/9709193

View on Pith Pith review arXiv

[10] [10]

R. Borcherds, “Automorphic forms with singularities o n Grassmanians”, alg-geom/9609022

View on Pith Pith review arXiv

[11] [11]

E. Witten, “Monopoles and Four-Manifolds”, hep-th/94 11102

View on Pith

[12] [12]

P. Fre, “Lectures on Special Kahler Geometry and Electr ic–Magnetic Duality Rota- tions” , Nucl.Phys.Proc.Suppl. 45BC (1996) 59-114, hep-th /9512043

View on Pith 1996

[13] [13]

M. Matone, “Instantons and recursion relations in in N = 2 susy gauge theory”, Phys. Lett. B357(1995) 342, hep-th/9506102

View on Pith Pith review arXiv 1995

[14] [14]

J. Sonnenschein, S. Theisen, S. Yankielowicz, Phys. Le tt. 367B (1996) 145; T. Eguchi, S.-K. Yang, Mod. Phys. Lett. A11 (1996) 131

View on Pith 1996

[15] [15]

G. Bonelli, M. Matone, Phys. Rev. Lett. 77 (1996) 4712-4715; hep-th/9605090

View on Pith Pith review arXiv 1996

[16] [16]

V. Arnol’d, “Mathematical Methods of Classical Mechan ics”, Springer-Verlag, 1980

View on Pith 1980

[17] [17]

S. Cecotti, C. Vafa, “Topological Anti-Topological Fu sion”, Nucl. Phys. B367 (1991) 359

View on Pith 1991

[18] [18]

S.K. Donaldson and P.B. Kronheimer, “The Geometry of Fo ur-Manifolds”, Clarendon Press, Oxford, 1990

View on Pith 1990

[19] [19]

E. Witten, “On S-Duality in Abelian Gauge Theory,” hep-th/9505186; Select a Math- ematica 1 (1995) 383 68

View on Pith Pith review arXiv 1995

[20] [20]

E. Witten, “Introduction to Cohomological Field Theor ies”, Lectures at Workshop on Topological Methods in Physics, Trieste, Italy, Jun 11-2 5, 1990, Int. J. Mod. Phys. A6 (1991) 2775

View on Pith 1991

[21] [21]

N. Nekrasov, “Holomorphic bundles and many-body syste ms”, hep-th/9503157, Comm. Math. Phys. (1997)

View on Pith Pith review arXiv 1997

[22] [22]

R. Friedman, J. Morgan, “Smooth Four-Manifolds and Com plex Surfaces”, Ergebnisse der. Math. und Grenz. 3, Springer-Verlag, New York, 1994

View on Pith 1994

[23] [23]

R. Fintushel and R.J. Stern, “The Blowup Formula For Don aldson Invariants”, alg-geom/9405002; Annals of Math. 143(1996) 529

View on Pith Pith review arXiv 1996

[24] [24]

E. Witten, “On Quantum gauge theories in two dimensions ,” Commun. Math. Phys. 141 (1991) 153; “Two dimensional gauge theories revisited”, J. Geom. Phys. 9 (1992) 303-368

View on Pith 1991

[25] [25]

M. Blau, G. Thomson, “Derivation of the Verlinde Formul a from Chern-Simons Theory and the G/G model”,Nucl. Phys. B408 (1993) 345-390

View on Pith 1993

[26] [26]

A. Gerasimov, “Localization in GWZW and Verlinde formu la”, UUITP 16/1993, hepth/9305090

View on Pith 1993

[27] [27]

J. Polchinski, “Evaluation of the one-loop string path integral”, Comm. Math. Phys. 104 (1986) 37; B. McClain and B.D.B. Roth, “Modular invariance for interac ting bosonic strings at ﬁnite temperature”, Comm. Math. Phys. 111 (1987) 539

View on Pith 1986

[28] [28]

J.A. Harvey, G. Moore, “Algebras, BPS States, and Strin gs,” hep-th/9510182, Nucl. Phys. B463 (1996) 315

View on Pith Pith review arXiv 1996

[29] [29]

I. Krichever, “Universal τ -function of Whitham hierarchy”, LPTENS-92/18, hep- th/9205110, in “30 years of the Landau Institute”, 477 , I.M. Khalatnikov, V.P. Mi- neev, V.P. (eds.)

View on Pith

[30] [30]

S. Cordes, G. Moore, and S. Ramgoolam, “ Lectures on 2D Ya ng Mills theory, Equiv- ariant Cohomology, and Topological String Theory,” hep-th /9411210

View on Pith

[31] [31]

K. Uhlenbeck, “Removable singularities in Yang-Mills ﬁelds”, Comm. Math. Phys. 83 (1982) 11

View on Pith 1982

[32] [32]

A. Losev, G. Moore, N. Nekrasov, S. Shatashvili, “Four- Dimensional Avatars of 2D RCFT,” hep-th/9509151, Talks at Strings’95 and Trieste 1 995

View on Pith Pith review arXiv

[33] [33]

J.M.F. Labastida, P.M. Llatas, “Potentials for topolo gical sigma models”, Phys. Lett. B271 (1991), 101; J.M.F. Labastida, M. Mari˜ no, Nucl. Phys. B448(1995) 373; J.M.F. Labastida, M. Mari˜ no, hep-th/9603169

View on Pith Pith review arXiv 1991

[34] [34]

S. Hyun, J. Park, J.-S. Park, “Topological QCD”, hep-th /9503201

View on Pith

[35] [35]

N. Nekrasov, PhD. Thesis, Princeton 1996

View on Pith 1996

[36] [36]

N. Nekrasov, “Five dimensional gauge theories and rela tivistic integrable systems”, hep-th/9609219 69

View on Pith Pith review arXiv

[37] [37]

L. Baulieu, A. Losev, N. Nekrasov, “Chern-Simons and Tw isted Supersymmetry in Higher Dimensions”, hep-th/9707174

View on Pith Pith review arXiv

[38] [38]

M. Atiyah, R. Bott, “ The Moment Map And Equivariant Coho mology”, Topology 23 (1984) 1-28

View on Pith 1984

[39] [39]

O. Ganor, D. Morrison, N. Seiberg, “Branes, Calabi-Yau Spaces and Compactiﬁca- tion of Six Dimensional E8 Theory”, hep-th/9610251

View on Pith Pith review arXiv

[40] [40]

O. Ganor, “Toroidal Compactiﬁcation of Heterotic 6D No n-Critical Strings Down to Four Dimensions”, hep-th/9608109, Nucl.Phys. B488 (1997) 223-235

View on Pith Pith review arXiv 1997

[41] [41]

A. Givental, “Equivariant Gromov-Witten Invariants” , alg-geom/9603021; “A mirror theorem for toric complete intersections”, alg-g eom/9701016

View on Pith Pith review arXiv

[42] [42]

E. Witten, “Phases of N=2 Theories in Two Dimensions”, N ucl. Phys. B403 (1993) 159, hep-th/9301042

View on Pith Pith review arXiv 1993

[43] [43]

D. Morrison, M. Plesser, hep-th/9412236

View on Pith Pith review arXiv

[44] [44]

L. Alvarez-Gaume, Commun.Math.Phys. 90 (1983)161; D. Friedan, P. Windey, Nucl.Phys. B235 (1984) 395

View on Pith 1983

[45] [45]

A. Hietamaki, A. Niemi, A. Morozov, K. Palo “Geometry Of N = 1/2 Supersymmetry And The Atiyah-Singer Theorem”, Phys.Lett. B263 (1991) 417

View on Pith 1991

[46] [46]

S. Katz, A. Klemm, C. Vafa, “Geometric Engineering of Qu antum Field Theories”, hep-th/9609239, Nucl.Phys. B497 (1997) 173-195

View on Pith Pith review arXiv 1997

[47] [47]

A. Lawrence, N. Nekrasov, “Instanton Sums and Five Dime nsional Gauge Theory”, hep-th/9706025

View on Pith Pith review arXiv

[48] [48]

M. Kontsevich, “Deformation Quantization of Poisson M anifolds”, dg-ga/ and talk at Strings’97

View on Pith

[49] [49]

K. Saito, Publ. RIMS, Kyoto Univ. 19 (1983) 1231; B. Blok, A. Varchenko, Int.J.Mod.Phys. A7 (1992) 1467-1490 ; R. Dijkgraaf, talk on RIMS conference ”Inﬁnite dimensional analysis”, July 1993 ; A. Losev, Theor.Math.Phys. 95 (1993) 595; A. Losev, in ”Integrable models and Strings ”, 172, Proceedi ngs of 1993 Helsinki Conference, Springer-Verlag, 1995 ; A. L...

View on Pith 1983

[50] [50]

M. Blau and G. Thompson, “Lectures on 2d Gauge Theories: Topological Aspects and Path Integral Techniques”, Presented at the Summer Scho ol in Hogh Energy Physics and Cosmology, Trieste, Italy, 14 Jun - 30 Jul 1993, h ep-th/9310144

View on Pith 1993

[51] [51]

K. Intriligator, “Fusion residues”, Mod. Phys. Lett. A6 (1991) 3541-3556, hep- th/9108005

View on Pith 1991

[52] [52]

E. Witten, “The Verlinde Algebra And The Cohomology Of T he Grassmannian”, hep-th/9312104

View on Pith Pith review arXiv

[53] [53]

M. Bershadsky, A. Johansen, V. Sadov and C. Vafa, “Topol ogical Reduction of 4D SYM to 2D σ–Models”, hep-th/9501096 70

View on Pith Pith review arXiv

[54] [54]

M. Atiyah, R. Bott, “The Yang-Mills Equations Over Riem ann Surfaces”, Phil. Trans. R.Soc. London A 308, 523-615 (1982)

View on Pith 1982

[55] [55]

A. Polyakov, “Gauge ﬁelds and strings”, Klewer Academi c Press

View on Pith

[56] [56]

M. Gromov, Invent. Math. 82 (1985) 307

View on Pith 1985

[57] [57]

A. Gorsky, I. Krichever, A. Marshakov, A. Morozov, A. Mi ronov, “Integrablity and Seiberg-Witten Exact Solution”, hep-th/9505035, Phys .Lett.B355: 466-474, 1995 71

View on Pith Pith review arXiv 1995