Geometric Construction ofN = 2 Gauge Theories

Mayr, Peter

doi:10.1002/(sici)1521-3978(199901)47:1/3<39::aid-prop39>3.0.co;2-e

Cited by 19 publications

(24 citation statements)

References 28 publications

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“…We will show how this works explicitly for charge two, and relate the construction of length two directly to the well-known Morrison-Park geometry [18]. 8 We will then explain how to obtain higher charges from higher-length flopping algebras. The recipe to obtain charge q is summarized in figure 2, whereas the necessary terminology will be explained in later sections.…”

Section: Summary and Outlinementioning

confidence: 99%

“…(For rigid two-spheres, we get light matter multiplets instead, typically charged under the non-Abelian group.) This picture has been exploited for two decades in the field of geometric engineering in string theory and M-theory [3][4][5][6][7][8], as well as in some closely related Ftheory constructions [9][10][11]. In the latter, a direct interpretation of the singular space is not available in all but the simplest cases.…”

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confidence: 99%

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High electric charges in M-theory from quiver varieties

Collinucci

Fazzi

Morrison

et al. 2019

J. High Energ. Phys.

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M-theory on a Calabi-Yau threefold admitting a small resolution gives rise to an Abelian vector multiplet and a charged hypermultiplet. We introduce into this picture a procedure to construct threefolds that naturally host matter with electric charges up to six. These are built as families of Du Val ADE surfaces (or ALE spaces), and the possible charges correspond to the Dynkin labels of the adjoint of the ADE algebra. In the case of charge two, we give a new derivation of the answer originally obtained by Curto and Morrison, and explicitly relate this construction to the Morrison-Park geometry. We also give a procedure for constructing higher-charge cases, which can often be applied to F-theory models. arXiv:1906.02202v1 [hep-th] 5 Jun 20191 Introduction M-theory on certain singular spaces gives rise to non-Abelian gauge symmetries. More specifically, in Calabi-Yau (CY) geometries with non-isolated singularities that admit crepant resolutions (i.e. those that do not alter the CY condition), Dynkin diagrams show up naturally as a pattern traced out by families of small intersecting two-spheres in the resolved geometry (for non-isolated holomorphic two-spheres that come in a complex one-dimensional family). When this occurs, postulating the presence of light M2-branes wrapping such so-called vanishing spheres gives rise to light degrees of freedom that fill out the root system corresponding to the Dynkin diagram. This leads one to expect the effective field theory to contain a non-Abelian gauge multiplet in the limit where areas of the two-spheres approach zero and the geometry becomes singular [1,2]. (For rigid two-spheres, we get light matter multiplets instead, typically charged under the non-Abelian group.) This picture has been exploited for two decades in the field of geometric engineering in string theory and M-theory [3][4][5][6][7][8], as well as in some closely related Ftheory constructions [9][10][11]. In the latter, a direct interpretation of the singular space is not available in all but the simplest cases. However, two strategies connect F-theory to this phenomenon of non-Abelian degrees of freedom: Either one can take a particular limit of M-theory, such that the duality to type IIB compactified on a circle is manifest, and verify the correspondence. Alternatively, one can analyze the F-theory phenomena directly by considering which (p, q)-strings in the type IIB seven-brane background become light. (An important feature of F-theory is in fact the existence of seven-branes of "exotic" types, i.e. beyond the D7-branes and O7-planes of perturbative type IIB.)Much less clear is what to make of possible Abelian gauge symmetries. 1 These can also be related to singularities, but not as directly as in the case of simple Lie algebras. Since the inception of F-theory, it has been known that one way to understand U(1) gauge groups is by seeking sections (or in some cases, multi-sections) of the elliptic fibration of F-theory [11][12][13]. In M-theory, it suffices to think of divisors in the compactification ...

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Section: Summary and Outlinementioning

confidence: 99%

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confidence: 99%

High electric charges in M-theory from quiver varieties

Collinucci

Fazzi

Morrison

et al. 2019

J. High Energ. Phys.

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“…(1) The toric data of the polytope ∆ n have similar features to the ADE Dynkin diagrams leading to non-abelian gauge symmetries in type II superstring compactifications on Calabi-Yau manifolds [7,8,9,10]. (2) The toric fixed loci, which correspond to the vanishing cycles, have been known to be associated with D-brane charges [32].…”

Section: Toric Geometrymentioning

confidence: 99%

“…Since the discovery of superstring dualities, four-dimensional supersymmetric quantum field theories (QF T 4 ) have been a subject of great interest in connection with superstring compactification on Calabi-Yau manifolds and D-brane physics [1,2,3,4]. For example, embedding N = 2 QF T 4 in type IIA superstring compactified on Calabi-Yau threefolds, with K3 fibration, has found a very nice geometric description using the so-called geometric engineering method [5,6,7,8,9,10]. In this program, these models, which give exact results for the moduli space of the type IIA Coulomb branch, are represented by Dynkin quiver diagrams of Lie algebras [7,8,9,10].…”

Section: Introductionmentioning

confidence: 99%

M-theory onG₂manifolds and the method of (p,q) brane webs

Belhaj¹

2004

J. Phys. A: Math. Gen.

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Using a reformulation of the method of (p, q) webs, we study the four-dimensional N = 1 quiver theories from M-theory on seven-dimensional manifolds with G 2 holonomy. We first construct such manifolds as U (1) quotients of eight-dimensional toric hyper-Kähler manifolds, using N = 4 supersymmetric sigma models. We show that these geometries, in general, are given by real cones on S 2 bundles over complex twodimensional toric varieties, V 2 = C r+2 /C * r . Then we discuss the connection between the physics content of M-theory on such G 2 manifolds and the method of (p, q) webs. Motivated by a result of Acharya and Witten [hep-th/0109152], we reformulate the method of (p, q) webs and reconsider the derivation of the gauge theories using toric geometry Mori vectors of V 2 and brane charge constraints. For WP 2 w 1 ,w 2 ,w 3 , we find that the gauge group is given by G = U (w 1 n) × U (w 2 n) × U (w 3 n). This is required by the anomaly cancellation condition.

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“…Various 4d gauge theories can be geometrically engineered [15,16] and mirror symmetry can be used to compute their exact effective actions. (See also [17] and references therein. )…”

Section: Introductionmentioning

confidence: 99%

Lectures on Mirror Symmetry and Topological String Theory

Alim¹

2012

Preprint

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These are notes of a series of lectures on mirror symmetry and topological string theory given at the Mathematical Sciences Center at Tsinghua University. The N = 2 superconformal algebra, its deformations and its chiral ring are reviewed. A topological field theory can be constructed whose observables are only the elements of the chiral ring. When coupled to gravity, this leads to topological string theory. The identification of the topological string A-and Bmodels by mirror symmetry leads to surprising connections in mathematics and provides tools for exact computations as well as new insights in physics. A recursive construction of the higher genus amplitudes of topological string theory expressed as polynomials is reviewed.

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Geometric Construction ofN = 2 Gauge Theories

Cited by 19 publications

References 28 publications

High electric charges in M-theory from quiver varieties

High electric charges in M-theory from quiver varieties

M-theory onG₂manifolds and the method of (p,q) brane webs

Lectures on Mirror Symmetry and Topological String Theory

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Geometric Construction ofN = 2 Gauge Theories

Cited by 19 publications

References 28 publications

High electric charges in M-theory from quiver varieties

High electric charges in M-theory from quiver varieties

M-theory onG2manifolds and the method of (p,q) brane webs

Lectures on Mirror Symmetry and Topological String Theory

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M-theory onG₂manifolds and the method of (p,q) brane webs