arXiv subjects

Meng-Chwan Tan

Publications and source records attributed to Meng-Chwan Tan.

At least 37 records · Page 2Linked to original sources

arXiv2017

Four-Dimensional N=2 Supersymmetric Theory with Boundary as a Two-Dimensional Complex Toda Theory

We perform a series of dimensional reductions of the 6d, $\mathcal{N}=(2,0)$ SCFT on $S^2\timesΣ\times I\times S^1$ down to 2d on $Σ$. The reductions are performed in three steps: (i) a reduction on $S^1$ (accompanied by a topological twist along $Σ$) leading to a supersymmetric Yang-Mills theory on $S^2\timesΣ\times I$, (ii) a further reduction on $S^2$ resulting in a complex Chern--Simons theory defined on $Σ\times I$, with the real part of the complex Chern-Simons level being zero, and the imaginary part being proportional to the ratio of the radii of $S^2$ and $S^1$, and (iii) a final reduction to the boundary modes of complex Chern--Simons theory with the Nahm pole boundary condition at both ends of the interval $I$, which gives rise to a complex Toda CFT on the Riemann surface $Σ$. As the reduction of the 6d theory on $Σ$ would give rise to an $\mathcal{N}=2$ supersymmetric theory on $S^2\times I\times S^1$, our results imply a 4d-2d duality between four-dimensional $\mathcal{N}=2$ supersymmetric theory with boundary and two-dimensional complex Toda theory.

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arXiv2016

Gauge and Supersymmetry Invariance of N=2 Boundary Chern-Simons Theory

In this paper, we study the restoration of gauge symmetry and up to half the supersymmetry (N=(2,0) or N=(1,1) in two dimensions) for N=2 non-Abelian Chern-Simons theories in the presence of a boundary. We describe the boundary action which is a supersymmetric WZW model coupled to the bulk Chern-Simons theory. Unlike the N=1 case, higher supersymmetry (N=(2,0)) will endow the group manifold of the WZW model with a complex structure. Therefore, the N=(2,0) WZW model in our paper is constructed via a coset space $G_c/G$, where $G$ is the same as the gauge group in the Chern-Simons action.

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arXiv2016

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

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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arXiv2016

An M-Theoretic Derivation of a 5d and 6d AGT Correspondence, and Relativistic and Elliptized Integrable Systems

We generalize our analysis in [arXiv:1301.1977], and show that a 5d and 6d AGT correspondence for SU(N) -- which essentially relates the relevant 5d and 6d Nekrasov instanton partition functions to the integrable representations of a q-deformed and elliptic affine W_N-algebra -- can be derived, purely physically, from the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent. Via an appropriate defect, we also derive a "fully-ramified" version of the 5d and 6d AGT correspondence where integrable representations of a quantum and elliptic affine SU(N)-algebra at the critical level appear on the 2d side, and argue that the relevant "fully-ramified" 5d and 6d Nekrasov instanton partition functions are simultaneous eigenfunctions of commuting operators which define relativistic and elliptized integrable systems. As an offshoot, we also obtain various mathematically novel and interesting relations involving the double loop algebra of SU(N), elliptic Macdonald operators, equivariant elliptic genus of instanton moduli space, and more.

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arXiv2015

Ω-deformation of B-twisted gauge theories and the 3d-3d correspondence

We study Ω-deformation of B-twisted gauge theories in two dimensions. As an application, we construct an Ω-deformed, topologically twisted five-dimensional maximally supersymmetric Yang-Mills theory on the product of a Riemann surface $Σ$ and a three-manifold $M$, and show that when $Σ$ is a disk, this theory is equivalent to analytically continued Chern-Simons theory on $M$. Based on these results, we establish a correspondence between three-dimensional $\mathcal{N} = 2$ superconformal theories and analytically continued Chern-Simons theory. Furthermore, we argue that there is a mirror symmetry between Ω-deformed two-dimensional theories.

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arXiv2014

N=2 supersymmetric gauge theories and quantum integrable systems

We study N=2 supersymmetric gauge theories on the product of a two-sphere and a cylinder. We show that the low-energy dynamics of a BPS sector of such a theory is described by a quantum integrable system, with the Planck constant set by the inverse of the radius of the sphere. If the sphere is replaced with a hemisphere, then our system reduces to an integrable system of the type studied by Nekrasov and Shatashvili. In this case we establish a correspondence between the effective prepotential of the gauge theory and the Yang-Yang function of the integrable system.

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arXiv2014

A Topological Chern-Simons Sigma Model and New Invariants of Three-Manifolds

We construct a topological Chern-Simons sigma model on a Riemannian three-manifold M with gauge group G whose hyperkahler target space X is equipped with a G-action. Via a perturbative computation of its partition function, we obtain new topological invariants of M that define new weight systems which are characterized by both Lie algebra structure and hyperkahler geometry. In canonically quantizing the sigma model, we find that the partition function on certain M can be expressed in terms of Chern-Simons knot invariants of M and the intersection number of certain G-equivariant cycles in the moduli space of G-covariant maps from M to X. We also construct supersymmetric Wilson loop operators, and via a perturbative computation of their expectation value, we obtain new knot invariants of M that define new knot weight systems which are also characterized by both Lie algebra structure and hyperkahler geometry.

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arXiv2013

Quasi-Topological Gauged Sigma Models, The Geometric Langlands Program, And Knots

We construct and study a closed, two-dimensional, quasi-topological (0,2) gauged sigma model with target space a smooth G-manifold, where G is any compact and connected Lie group. When the target space is a flag manifold of simple G, and the gauge group is a Cartan subgroup thereof, the perturbative model describes, purely physically, the recently formulated mathematical theory of "Twisted Chiral Differential Operators". This paves the way, via a generalized T-duality, for a natural physical interpretation of the geometric Langlands correspondence for simply-connected, simple, complex Lie groups. In particular, the Hecke eigensheaves and Hecke operators can be described in terms of the correlation functions of certain operators that underlie the infinite-dimensional chiral algebra of the flag manifold model. Nevertheless, nonperturbative worldsheet twisted-instantons can, in some situations, trivialize the chiral algebra completely. This leads to a spontaneous breaking of supersymmetry whilst implying certain delicate conditions for the existence of Beilinson-Drinfeld D-modules. Via supersymmetric gauged quantum mechanics on loop space, these conditions can be understood to be intimately related to a conjecture by Hohn-Stolz [1] regarding the vanishing of the Witten genus on string manifolds with positive Ricci curvature. An interesting connection to Chern-Simons theory is also uncovered, whence we would be able to (i) relate general knot invariants of three-manifolds and Khovanov homology to "quantum" ramified D-modules and Lagrangian intersection Floer homology; (ii) furnish physical proofs of mathematical conjectures by Seidel-Smith [2] and Gaitsgory [3, 4] which relate knots to symplectic geometry and Langlands duality, respectively.

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arXiv2013

Five-Branes in M-Theory and a Two-Dimensional Geometric Langlands Duality

A recent attempt to extend the geometric Langlands duality to affine Kac-Moody groups, has led Braverman and Finkelberg [arXiv:0711.2083] to conjecture a mathematical relation between the intersection cohomology of the moduli space of G-bundles on certain singular complex surfaces, and the integrable representations of the Langlands dual of an associated affine G-algebra, where G is any simply-connected semisimple group. For the A-type groups, where the conjecture has been mathematically verified to a large extent, we show that the relation has a natural physical interpretation in terms of six-dimensional compactifications of M-theory with coincident five-branes wrapping certain hyperkahler four-manifolds; in particular, it can be understood as an expected invariance in the resulting spacetime BPS spectrum under string dualities. By replacing the singular complex surface with a smooth multi-Taub-NUT manifold, we find agreement with a closely related result demonstrated earlier via purely field-theoretic considerations by Witten. By adding OM five-planes to the original analysis, we argue that an analogous relation involving the non-simply-connected D-type groups, ought to hold as well. This is the first example of a string-theoretic interpretation of such a two-dimensional extension to complex surfaces of the geometric Langlands duality for the A-D groups.

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arXiv2012

Supersymmetric Surface Operators, Four-Manifold Theory and Invariants in Various Dimensions

We continue our program initiated in [arXiv:0912.4261] to consider supersymmetric surface operators in a topologically-twisted N=2 pure SU(2) gauge theory, and apply them to the study of four-manifolds and related invariants. Elegant physical proofs of various seminal theorems in four-manifold theory obtained by Ozsvath-Szabo [2,3] and Taubes [4], will be furnished. In particular, we will show that Taubes' groundbreaking and difficult result -- that the ordinary Seiberg-Witten invariants are in fact the Gromov invariants which count pseudo-holomorphic curves embedded in a symplectic four-manifold X -- nonetheless lends itself to a simple and concrete physical derivation in the presence of "ordinary" surface operators. As an offshoot, we will be led to several interesting and mathematically novel identities among the Gromov and "ramified" Seiberg-Witten invariants of X, which in certain cases, also involve the instanton and monopole Floer homologies of its three-submanifold. Via these identities, and a physical formulation of the "ramified" Donaldson invariants of four-manifolds with boundaries, we will uncover completely new and economical ways of deriving and understanding various important mathematical results concerning (i) knot homology groups from "ramified" instantons by Kronheimer-Mrowka [5]; and (ii) monopole Floer homology and Seiberg-Witten theory on symplectic four-manifolds by Kutluhan-Taubes [4,6]. Supersymmetry, as well as other physical concepts such as R-invariance, electric-magnetic duality, spontaneous gauge symmetry-breaking and localization onto supersymmetric configurations in topologically-twisted quantum field theories, play a pivotal role in our story.

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arXiv2012

Notes on the "Ramified" Seiberg-Witten Equations and Invariants

In these notes, we carefully analyze the properties of the "ramified" Seiberg-Witten equations associated with supersymmetric configurations of the Seiberg-Witten abelian gauge theory with surface operators on an oriented closed four-manifold X. We find that in order to have sensible solutions to these equations, only surface operators with certain parameters and embeddings in X, are admissible. In addition, the corresponding "ramified" Seiberg-Witten invariants on X with positive scalar curvature and b^+_2 > 1, vanish, while if X has b^+_2 = 1, there can be wall-crossings whence the invariants will jump. In general, for each of the finite number of basic classes that corresponds to a moduli space of solutions with zero virtual dimension, the perturbed "ramified" Seiberg-Witten invariants on Kahler manifolds will depend - among other parameters associated with the surface operator - on the monopole number "l" and the holonomy parameter "alpha". Nonetheless, the (perturbed) "ramified" and ordinary invariants are found to coincide, albeit up to a sign, in some examples.

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arXiv2011

Integration Over The u-Plane In Donaldson Theory With Surface Operators

We generalize the analysis by Moore and Witten in [arXiv:hep-th/9709193], and consider integration over the u-plane in Donaldson theory with surface operators on a smooth four-manifold X. Several novel aspects will be developed in the process; like a physical interpretation of the "ramified" Donaldson and Seiberg-Witten invariants, and the concept of curved surface operators which are necessarily topological at the outset. Elegant physical proofs -- rooted in R-anomaly cancellations and modular invariance over the u-plane -- of various seminal results in four-dimensional geometric topology obtained by Kronheimer and Mrowka [1,2] -- such as a universal formula relating the "ramified" and ordinary Donaldson invariants, and a generalization of the celebrated Thom conjecture -- will be furnished. Wall-crossing and blow-up formulas of these "ramified" invariants which have not been computed in the mathematical literature before, as well as a generalization and a Seiberg-Witten analog of the universal formula as implied by an electric-magnetic duality of trivially-embedded surface operators in X, will also be presented, among other things.

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arXiv2010

Equivariant Cohomology Of The Chiral de Rham Complex And The Half-Twisted Gauged Sigma Model

In this paper, we study the perturbative aspects of the half-twisted variant of Witten's topological A-model coupled to a non-dynamical gauge field with Kahler target space X being a G-manifold. Our main objective is to furnish a purely physical interpretation of the equivariant cohomology of the chiral de Rham complex, recently constructed by Lian and Linshaw in [arXiv:math/0501084], called the "chiral equivariant cohomology". In doing so, one finds that key mathematical results such as the vanishing in the chiral equivariant cohomology of positive weight classes, lend themselves to straightforward physical explanations. In addition, one can also construct topological invariants of X from the correlation functions of the relevant physical operators corresponding to the non-vanishing weight-zero classes. Via the topological invariance of these correlation functions, one can verify, from a purely physical perspective, the mathematical isomorphism between the weight-zero subspace of the chiral equivariant cohomology and the classical equivariant cohomology of X. Last but not least, one can also determine fully, the de Rham cohomology ring of X/G, from the topological chiral ring generated by the local ground operators of the physical model under study.

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arXiv2009

Surface Operators in Abelian Gauge Theory

We consider arbitrary embeddings of surface operators in a pure, non-supersymmetric abelian gauge theory on spin (non-spin) four-manifolds. For any surface operator with a priori simultaneously non-vanishing parameters, we explicitly show that the parameters transform naturally under an SL(2, Z) (or a congruence subgroup thereof) duality of the theory. However, for non-trivially-embedded surface operators, exact S-duality holds only if the quantum parameter effectively vanishes, while the overall SL(2, Z) (or a congruence subgroup thereof) duality holds up to a c-number at most, regardless. Via the formalism of duality walls, we furnish an alternative derivation of the transformation of parameters - found also to be consistent with a switch from Wilson to 't Hooft loop operators under S-duality. With any background embedding of surface operators, the partition function and the correlation functions of non-singular, gauge-invariant local operators on any curved four-manifold, are found to transform like modular forms under the respective duality groups.

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arXiv2009

Surface Operators in N=2 Abelian Gauge Theory

We generalise the analysis in [arXiv:0904.1744] to superspace, and explicitly prove that for any embedding of surface operators in a general, twisted N=2 pure abelian theory on an arbitrary four-manifold, the parameters transform naturally under the SL(2,Z) duality of the theory. However, for nontrivially-embedded surface operators, exact S-duality holds if and only if the "quantum" parameter effectively vanishes, while the overall SL(2,Z) duality holds up to a c-number at most, regardless. Nevertheless, this observation sets the stage for a physical proof of a remarkable mathematical result by Kronheimer and Mrowka--that expresses a "ramified" analog of the Donaldson invariants solely in terms of the ordinary Donaldson invariants--which, will appear, among other things, in forthcoming work. As a prelude to that, the effective interaction on the corresponding u-plane will be computed. In addition, the dependence on second Stiefel-Whitney classes and the appearance of a Spin^c structure in the associated low-energy Seiberg-Witten theory with surface operators, will also be demonstrated. In the process, we will stumble upon an interesting phase factor that is otherwise absent in the "unramified" case.

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arXiv2008

Gauging Spacetime Symmetries On The Worldsheet And The Geometric Langlands Program -- II

We generalise the analysis carried out in [arXiv:0710.5796], and find that our previous results can be extended beyond the case of SL(N,C). In particular, we show that an equivalence--at the level of the holomorphic chiral algebra--between a bosonic string on a smooth coset manifold G/B and a B-gauged version of itself on G, will imply an isomorphism of classical W-algebras and a level relation which underlie a geometric Langlands correspondence for the simply-laced, complex ADE-groups. In addition, as opposed to line operators and branes of an open topological sigma-model, the Hecke operators and Hecke eigensheaves, can, instead, be physically interpreted in terms of the correlation functions of local operators in the holomorphic chiral algebra of a closed, quasi-topological sigma-model. Our present results thus serve as an alternative physical interpretation--to that of an electric-magnetic duality of four-dimensional gauge theory demonstrated earlier by Kapustin and Witten in [arXiv:hep-th/0604151]--of the geometric Langlands correspondence for complex ADE-groups. The cases with tame and mild "ramifications" are also discussed.

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arXiv2008

Chiral Algebras of (0,2) Sigma Models: Beyond Perturbation Theory

We explore the nonperturbative aspects of the chiral algebras of N = (0,2) sigma models, which perturbatively are intimately related to the theory of chiral differential operators (CDOs). The grading by charge and scaling dimension is anomalous if the first Chern class of the target space is nonzero. This has some nontrivial consequences for the chiral algebra. As an example, we study the case where the target space is CP^1, and show that worldsheet instantons trivialize the chiral algebra entirely. Consequently, supersymmetry is spontaneously broken in this model. We then turn to a closer look at the supersymmetry breaking from the viewpoint of Morse theory on loop space. We find that instantons interpolate between pairs of perturbative supersymmetric states with different fermionic numbers, hence lifting them out of the supersymmetric spectrum. Our results reveal that a "quantum" deformation of the geometry of the target space leads to a trivialization of the kernels of certain twisted Dirac operators on CP^1.

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arXiv2008

The Half-Twisted Orbifold Sigma Model and the Chiral de Rham Complex

In this paper, we study the perturbative aspects of the half-twisted variant of Witten's topological A-model on a complex orbifold X/G, where G is an isometry group of X. The objective is to furnish a purely physical interpretation of the mathematical theory of the Chiral de Rham complex on orbifolds recently constructed by Frenkel and Szczesny in [Chiral de Rham complex and orbifolds, arXiv: math.AG/0307181]. In turn, one can obtain a novel understanding of the holomorphic (twisted) N=2 superconformal structure underlying the untwisted and twisted sectors of the quantum sigma model, purely in terms of an obstruction (or a lack thereof) to a global definition of the relevant physical operators which correspond to G-invariant sections of the sheaf of Chiral de Rham complex on X. Explicit examples are provided to help illustrate this connection, and comparisons with their non-orbifold counterparts are also made in an aim to better understand the action of the G-orbifolding on the original half-twisted sigma model on X.

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