Fourier-Mukai transforms

Hille, Lutz; Bergh, Michel Van den

doi:10.1017/cbo9780511735134.007

Cited by 31 publications

(15 citation statements)

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“…The χ = 0 weighted projective lines will be written 3,4,6) or simply X. We have [12,22,32] X p = E/Z p , (2.27) and passing from elliptic orbifolds E/Z p to weighted projective lines X p is just a convenient shift in language.…”

Section: Coherent Sheaves On Weighted Projective Linesmentioning

confidence: 99%

Higher S-dualities and Shephard-Todd groups

Cecotti

Zotto

2015

J. High Energ. Phys.

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Seiberg and Witten have shown that in N = 2 SQCD with N f = 2N c = 4 the S-duality group PSL(2, Z) acts on the flavor charges, which are weights of Spin (8), by triality. There are other N = 2 SCFTs in which SU(2) SYM is coupled to stronglyinteracting non-Lagrangian matter: their matter charges are weights of E 6 , E 7 and E 8 instead of Spin(8). The S-duality group PSL(2, Z) acts on these weights: what replaces Spin(8) triality for the E 6 , E 7 , E 8 root lattices?In this paper we answer the question. The action on the matter charges of (a finite central extension of) PSL(2, Z) factorizes trough the action of the exceptional ShephardTodd groups G 4 and G 8 which should be seen as complex analogs of the usual triality group S 3 ≃ Weyl(A 2 ). Our analysis is based on the identification of S-duality for SU(2) gauge SCFTs with the group of automorphisms of the cluster category of weighted projective lines of tubular type.

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Section: Coherent Sheaves On Weighted Projective Linesmentioning

confidence: 99%

Higher S-dualities and Shephard-Todd groups

Cecotti

Zotto

2015

J. High Energ. Phys.

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“…In this paper we abuse notations, and use the same symbol G to denote the Dynkin graph, the corresponding Lie algebra, and its (simply-connected) Lie group 3. It is convenient to extend the definition to p = 1 by declaring D 1 (G) to be the empty SCFT for all G 4. The elliptic complete N = 2 SCFTs (1.3) are in one-to-one correspondence with the affine Dynkin graphs which are also stars[23].…”

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confidence: 99%

Homological $S$-duality in 4d $\mathcal{N}=2$ QFTs

Caorsi

Cecotti

2018

Advances in Theoretical and Mathematical Physics

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The S-duality group S(F ) of a 4d N = 2 supersymmetric theory F is identified with the group of triangle equivalences of its cluster category C (F ) modulo the subgroup acting trivially on the physical quantities. S(F ) is a discrete group commensurable to a subgroup of the Siegel modular group Sp(2g, Z) (g being the dimension of the Coulomb branch). This identification reduces the determination of the S-duality group of a given N = 2 theory to a problem in homological algebra. In this paper we describe the techniques which make the computation straightforward for a large class of N = 2 QFTs. The group S(F ) is naturally presented as a generalized braid group.The S-duality groups are often larger than expected. In some models the enhancement of S-duality is quite spectacular. For instance, a QFT with a huge S-duality group is the Lagrangian SCFT with gauge group SO(8) × SO(5) 3 × SO(3) 6 and half-hypermultiplets in the bi-and tri-spinor representations.

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“…If C is a smooth elliptic curve, a proof can be found in [22]. For a rational curve with a node or a cusp this was proved in [12].…”

Section: Derived Equivalences For Integral Curves Of Genus Onementioning

confidence: 98%

“…The determination of this group is widely recognized as an important problem and has been considered by several authors (see [8], [22], [35], [34], [24] and [11]). …”

Section: Introductionmentioning

confidence: 99%

Relative Fourier–Mukai transforms for Weierstraß fibrations, abelian schemes and Fano fibrations

Martín¹,

Gómez²,

Prieto³

2013

Math. Proc. Camb. Phil. Soc.

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Abstract. We study the group of relative Fourier-Mukai transforms for Weierstraß fibrations, abelian schemes and Fano or anti-Fano fibrations. For Weierstraß and Fano or anti-Fano fibrations we describe this group completely. For abelian schemes over an arbitrary base we prove that if two of them are relative Fourier-Mukai partners then there is an isometric isomorphism between the fibre products of each of them and its dual abelian scheme. If the base is normal and the slope map is surjective we show that these two conditions are equivalent. Moreover in this situation we completely determine the group of relative Fourier-Mukai transforms and we prove that the number of relative Fourier-Mukai partners of a given abelian scheme over a normal base is finite.

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Fourier-Mukai transforms

Abstract: In this paper we discuss some of the recent developments on derived equivalences in algebraic geometry.

Cited by 31 publications

References 50 publications

Higher S-dualities and Shephard-Todd groups

Higher S-dualities and Shephard-Todd groups

Homological $S$-duality in 4d $\mathcal{N}=2$ QFTs

Relative Fourier–Mukai transforms for Weierstraß fibrations, abelian schemes and Fano fibrations

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