M-Theoretic Derivations of 4d-2d Dualities: From a Geometric Langlands Duality for Surfaces, to the AGT Correspondence, to Integrable Systems

Meng-Chwan Tan

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M-Theoretic Derivations of 4d-2d Dualities: From a Geometric Langlands Duality for Surfaces, to the AGT Correspondence, to Integrable Systems

Meng-Chwan Tan

High Energy Physics - Theory · Submitted 9 Jan 2013

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim In Part I, we extend our analysis in [arXiv:0807.1107], and show that a mathematically conjectured geometric Langlands duality for complex surfaces in [1], and its generalizations -- which relate some cohomology of the moduli space of certain ("ramified") G-instantons to the integrable representations of the Langlands dual of certain affine (sub) G-algebras, where G is any compact Lie group -- can be derived, purely physically, from the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent.

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CitationMeng-Chwan Tan. "M-Theoretic Derivations of 4d-2d Dualities: From a Geometric Langlands Duality for Surfaces, to the AGT Correspondence, to Integrable Systems." arXiv:1301.1977 (2013). https://pith.science/paper/7LMRNMUM

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In Part I, we extend our analysis in [arXiv:0807.1107], and show that a mathematically conjectured geometric Langlands duality for complex surfaces in [1], and its generalizations -- which relate some cohomology of the moduli space of certain ("ramified") G-instantons to the integrable representations of the Langlands dual of certain affine (sub) G-algebras, where G is any compact Lie group -- can be derived, purely physically, from the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent. In Part II, to the setup in Part I, we introduce Omega-deformation via fluxbranes and add half-BPS boundary defects via M9-branes, and show that the celebrated AGT correspondence in [2, 3], and its generalizations -- which essentially relate, among other things, some equivariant cohomology of the moduli space of certain ("ramified") G-instantons to the integrable representations of the Langlands dual of certain affine W-algebras -- can likewise be derived from the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent. In Part III, we consider various limits of our setup in Part II, and connect our story to chiral fermions and integrable systems. Among other things, we derive the Nekrasov-Okounkov conjecture in [4] -- which relates the topological string limit of the dual Nekrasov partition function for pure G to the integrable representations of the Langlands dual of an affine G-algebra -- and also demonstrate that the Nekrasov-Shatashvili limit of the "fully-ramified" Nekrasov instanton partition function for pure G is a simultaneous eigenfunction of the quantum Toda Hamiltonians associated with the Langlands dual of an affine G-algebra. Via the case with matter, we also make contact with Hitchin systems and the "ramified" geometric Langlands correspondence for curves.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In Part I, we extend our analysis in [arXiv:0807.1107], and show that a mathematically conjectured geometric Langlands duality for complex surfaces in [1], and its generalizations -- which relate some cohomology of the moduli space of certain ("ramified") G-instantons to the integrable representations of the Langlands dual of certain affine (sub) G-algebras, where G is any compact Lie group -- can be derived, purely physically, from the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent.	contribution 0.72	0.90	0.37	asserted 1.00
s1	In Part II, to the setup in Part I, we introduce Omega-deformation via fluxbranes and add half-BPS boundary defects via M9-branes, and show that the celebrated AGT correspondence in [2, 3], and its generalizations -- which essentially relate, among other things, some equivariant cohomology of the moduli space of certain ("ramified") G-instantons to the integrable representations of the Langlands dual of certain affine W-algebras -- can likewise be derived from the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent.	contribution 0.78	0.92	0.27	asserted 1.00
s2	In Part III, we consider various limits of our setup in Part II, and connect our story to chiral fermions and integrable systems.	contribution 0.96	0.90	0.06	asserted 0.80
s3	Among other things, we derive the Nekrasov-Okounkov conjecture in [4] -- which relates the topological string limit of the dual Nekrasov partition function for pure G to the integrable representations of the Langlands dual of an affine G-algebra -- and also demonstrate that the Nekrasov-Shatashvili limit of the "fully-ramified" Nekrasov instanton partition function for pure G is a simultaneous eigenfunction of the quantum Toda Hamiltonians associated with the Langlands dual of an affine G-algebra.	result 0.98	0.91	0.16	asserted 1.00
s4	Via the case with matter, we also make contact with Hitchin systems and the "ramified" geometric Langlands correspondence for curves.	result 0.51	0.86	0.06	asserted 0.92

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Original abstract

In Part I, we extend our analysis in [arXiv:0807.1107], and show that a mathematically conjectured geometric Langlands duality for complex surfaces in [1], and its generalizations -- which relate some cohomology of the moduli space of certain ("ramified") G-instantons to the integrable representations of the Langlands dual of certain affine (sub) G-algebras, where G is any compact Lie group -- can be derived, purely physically, from the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent. In Part II, to the setup in Part I, we introduce Omega-deformation via fluxbranes and add half-BPS boundary defects via M9-branes, and show that the celebrated AGT correspondence in [2, 3], and its generalizations -- which essentially relate, among other things, some equivariant cohomology of the moduli space of certain ("ramified") G-instantons to the integrable representations of the Langlands dual of certain affine W-algebras -- can likewise be derived from the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent. In Part III, we consider various limits of our setup in Part II, and connect our story to chiral fermions and integrable systems. Among other things, we derive the Nekrasov-Okounkov conjecture in [4] -- which relates the topological string limit of the dual Nekrasov partition function for pure G to the integrable representations of the Langlands dual of an affine G-algebra -- and also demonstrate that the Nekrasov-Shatashvili limit of the "fully-ramified" Nekrasov instanton partition function for pure G is a simultaneous eigenfunction of the quantum Toda Hamiltonians associated with the Langlands dual of an affine G-algebra. Via the case with matter, we also make contact with Hitchin systems and the "ramified" geometric Langlands correspondence for curves.

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References

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Cited by 4 Pith papers

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

Seiberg-Witten theory as a Fermi gas
hep-th 2016-03 accept novelty 7.0 of 10

The dual Nekrasov-Okounkov partition function of pure SU(2) N=2 Yang-Mills in a self-dual Omega-background is shown to equal the spectral determinant of an ideal Fermi gas, proving a topological-string/spectral-theory...
M2-brane surface operators and gauge theory dualities in Toda
hep-th 2014-07 conditional novelty 7.0 of 10

M2-brane surface operators in class S theories are matched to degenerate vertex operators of A_{Nf-1} Toda CFT, and two-dimensional Seiberg and Kutasov-Schwimmer dualities are realized as momentum conjugation and cros...
Supersymmetric gauge theories, quantization of moduli spaces of flat connections, and conformal field theory
hep-th 2013-02 conditional novelty 7.0 of 10

The paper argues that instanton partition functions of class-S gauge theories equal Liouville conformal blocks because both are wave-functions in the same quantized algebra of functions on the moduli space of flat con...
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...

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25 arXiv papers cite this in 103 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 · 73 sentences without a confident label

Builds on it 2 papers

Our main aim is to furnish in a pedagogical manner, a fundamental M-theoretic derivation of a 5d and 6d analog of the above 4d-2d relations, by generalizing our analysis in [ref] .

An M-Theoretic Derivation of a 5d and 6d AGT Correspondence, and Relativistic and Elliptized Integrable Systems · Introduction · 1.00
The main aim of this paper is to further explore the implications of our framework in [ref] , so as to physically derive, via M-theory, generalizations of the aforementioned 5d and 6d AGT correspondence, and more.

Higher AGT Correspondences, W-algebras, and Higher Quantum Geometric Langlands Duality from M-Theory · Introduction · 0.93

Cites it as background 17 papers

An important example of these relations is the AGT correspondence for gauge theories on $^4$: in [ref] Alday, Gaiotto and Tachikawa conjectured a relation between the instanton partition functions of $=2$ supersymmetric quiver gauge theories on $^4$ and the conformal blocks of two-dimensional $A_r-1$ Toda conformal field theories (see also [ref] ); this conjecture has been explicitly confirmed in some special cases, see e.g. [ref] .

AGT relations for abelian quiver gauge theories on ALE spaces · AGT relations and ALE spaces · 0.97
The original so-called AGT relation proposed in [ref] states the equality between the correlation functions in the Liouville field theory and the partition function of the $𝒩=2$ supersymmetric Yang-Mills theory with the $SU(2)$ gauge group (for the generalizations of the AGT correspondence for the other gauge groups and conformal field theories see [ref] ).

Coset conformal field theory and instanton counting on C^2/Z_p · Introduction · 0.97
The results obtained so far concern few examples of four-manifolds as $ℂ^2$ [ref] , $ℂ^2/Γ$ [ref] , $S^4$ [ref] and $S^2×S^2$ [ref] .

Exact results for ${\cal N}=2$ supersymmetric gauge theories on compact toric manifolds and equivariant Donaldson invariants · Introduction · 0.89

What it cites · 161 sentences without a confident label

Builds on 2 papers

Our derivation reproduces and generalizes to nonsimply-laced gauge groups the conjecture by Nishioka-Tachikawa in [ref] , that the Nekrasov instanton partition function for pure simply-laced $G$ on an $A_m-1$-type ALE space is given by the norm of a coherent state in a Verma module over the sum of a parafermionic coset affine algebra $RCFT[A_m-1, G]$ and the $m$-th para-$W$-algebra derived from the affine $G$-algebra.

Para-Liouville/Toda central charges from M5-branes · Introduction · 0.97
First, by considering the topological string limit in our derivation of the AGT correspondence for a conformal necklace quiver with $n$ $SU(N)$ gauge groups, we will reproduce and generalize a purely field-theoretic result by Nekrasov-Okounkov in [ref] , that relates the corresponding Nekrasov instanton partition function of the $N = 2^∗$ $SU(N)$ theory to the theory of $N$ chiral fermions on a torus.

Seiberg-Witten Theory and Random Partitions · Introduction · 0.94

Uses the method or tool of 5 papers

In $$5, to the setup in $$3, we will introduce Omega-deformation via the fluxbrane background studied in [ref] , add half-BPS boundary defects realized by M9-branes [ref] , and go on to show that the pure AGT correspondence for the $A$, $B$, $C$, $D$ and $G$ groups, can likewise be derived from the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent.

The Omega Deformation From String and M-Theory · Introduction · 0.92
In $$5, to the setup in $$3, we will introduce Omega-deformation via the fluxbrane background studied in [ref] , add half-BPS boundary defects realized by M9-branes [ref] , and go on to show that the pure AGT correspondence for the $A$, $B$, $C$, $D$ and $G$ groups, can likewise be derived from the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent.

General Omega Deformations from Closed String Backgrounds · Introduction · 0.92
In $$5, to the setup in $$3, we will introduce Omega-deformation via the fluxbrane background studied in [ref] , add half-BPS boundary defects realized by M9-branes [ref] , and go on to show that the pure AGT correspondence for the $A$, $B$, $C$, $D$ and $G$ groups, can likewise be derived from the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent.

Open M5-branes · Introduction · 0.92

Compares with 8 papers

In so doing, we would be able to reproduce exactly the mathematical result by Braverman in [ref] , that relates the “fully-ramified” Nekrasov instanton partition function for pure $G$ to the norm of a coherent state in the Verma module over the Langlands dual of an affine $G$-algebra.

Instanton counting via affine Lie algebras I: Equivariant J-functions of (affine) flag manifolds and Whittaker vectors · Introduction · 0.81
Our derivation physically reproduces the mathematical conjecture by Braverman et al. in [ref] , that the Nekrasov instanton partition function for pure $G$ is given by the norm of a coherent state in the Verma module over the Langlands dual affine $W$-algebra.

A finite analog of the AGT relation I: finite W-algebras and quasimaps' spaces · Introduction · 0.82
Indeed, our results agree with [ref] , $$3.2, first paragraph, where for $SO(3)$, it was shown that the instanton number takes values in $ℤ_≥0/2$; moreover, our results also agree with [ref] , where for all other $SO(N+1)$, it was shown that the instanton numbers take only non-negative integer values.

$S$-duality in Vafa-Witten theory for non-simply laced gauge groups · Introduction · 0.97

Cites as background 13 papers

Right at about the same time, in an attempt to generalize the geometric Langlands duality for Lie groups [ref] to affine Kac-Moody groups, mathematicians Braverman and Finkelberg were also led to formulate a conjecture in [ref] , which asserts that the intersection cohomology of the moduli space of $G$-instantons on $ℝ^4 / ℤ_k$ should be related to the integrable representations of the Langlands dual of an affine $G$-algebra.

math.RT/0401222 · Introduction · 0.84
The basis for the AGT correspondence for $SU(N)$ – as first pointed out by Alday and Tachikawa in [ref] – is a conjectured relation between the equivariant cohomology of the moduli space of $SU(N)$-instantons and the integrable representations of an affine $W_N$-algebra.

Affine SL(2) conformal blocks from 4d gauge theories · Introduction · 0.87
This conjectured relation was first proved mathematically for finite $W_N$-algebras in [ref] , and later proved mathematically for affine $W_N$-algebras in [ref] .

Quantum Groups and Quantum Cohomology · Introduction · 0.80

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