Mathematical aspects of mirror
                    symmetry

Morrison, David R.

doi:10.1090/pcms/003/05

Cited by 36 publications

(22 citation statements)

References 112 publications

(151 reference statements)

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“…The coincidence of their result with mathematically rigorous results 1,7,8 gave a great surprising! On the other hand, also in the recent studies of low energy effective dynamics of N = 2 supersymmetric Yang-Mills theory, N = 2 supersymmetry and duality play a crucial role.…”

Section: Introductionmentioning

confidence: 82%

One-instanton prepotentials from WDVV equations in N=2 supersymmetric SU(4) Yang–Mills theory

Ohta

1999

Journal of Mathematical Physics

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Prepotentials in N = 2 supersymmetric Yang-Mills theories are known to obey non-linear partial differential equations called Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations. In this paper, the prepotentials at one-instanton level in N = 2 supersymmetric SU(4) Yang-Mills theory are studied from the standpoint of WDVV equations. Especially, it is shown that the one-instanton prepotentials are obtained from WDVV equations by assuming the perturbative prepotential and by using the scaling relation as a subsidiary condition but are determined without introducing Seiberg-Witten curve. In this way, various one-instanton prepotentials which satisfy both WDVV equations and scaling relation can be derived, but it turns out that among them there exist one-instanton prepotentials which coincide with the instanton calculus.

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Section: Introductionmentioning

confidence: 82%

One-instanton prepotentials from WDVV equations in N=2 supersymmetric SU(4) Yang–Mills theory

Ohta

1999

Journal of Mathematical Physics

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“…When n = 3, we are dealing with holomorphic curves in Calabi-Yau 3-folds. This has been thoroughly studied (for a recent review see [48]), and our brief presentation of the results will certainly not do justice to the complexity and diversity of the subject.…”

Section: Topological Gauge Theories On Holomorphic Curvesmentioning

confidence: 99%

Aspects of N ⩾ 2 topological gauge theories and D-branes

Blau¹,

1997

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We comment on various aspects of topological gauge theories possessing N T ≥ 2 topological symmetry:1. We show that the construction of Vafa-Witten and Dijkgraaf-Moore of 'balanced' topological field theories is equivalent to an earlier construction in terms of N T = 2 superfields inspired by supersymmetric quantum mechanics.2. We explain the relation between topological field theories calculating signed and unsigned sums of Euler numbers of moduli spaces.3. We show that the topological twist of N = 4 d = 4 Yang-Mills theory recently constructed by Marcus is formally a deformation of four-dimensional super-BF theory.4. We construct a novel N T = 2 topological twist of N = 4 d = 3 Yang-Mills theory, a 'mirror' of the Casson invariant model, with certain unusual features (e.g. no bosonic scalar field and hence no underlying equivariant cohomology) 5. We give a complete classification of the topological twists of N = 8 d = 3 Yang-Mills theory and show that they are realised as world-volume theories of Dirichlet two-brane instantons wrapping supersymmetric three-cycles of Calabi-Yau three-folds and G 2 -holonomy Joyce manifolds.6. We describe the topological gauge theories associated to D-string instantons on holomorphic curves in K3s and Calabi-Yau 3-folds. 1

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“…We then apply our methods to a codimension four determinantal Calabi-Yau threefold in P 7 , recently given a nonabelian gauge theory description by the present authors, for which no mirror Calabi-Yau is currently known. We derive predictions for its Gromov-Witten invariants and verify that our predictions satisfy nontrivial geometric checks.1 The work of Böhm [24, 25] provides a promising proposal, but we have been unable to implement it well enough to produce a mirror for this example.2 Local special Kähler manifolds are also often called projective special Kähler manifolds and are distinct from special Kähler manifolds -see, e.g., [32].3 This geometric structure gives rise to a Hodge filtration F 3 ⊂ F 2 ⊂ F 1 ⊂ F 0 of weight 3, with F 3 ≃ L, F 2 ≃ V, F 1 ≃ L ⊥ , and F 0 ≃ V C , where L ⊥ is the subspace of V C perpendicular to L with respect to the symplectic pairing ·, · -see, e.g., [33,34].The last expression involves the periods of Ω,B J Ω(ξ) , I, J = 0, . .…”

mentioning

confidence: 99%

Two-Sphere Partition Functions and Gromov–Witten Invariants

Jockers¹,

Kumar

Lapan

et al. 2014

Commun. Math. Phys.

136

315

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Many N = (2, 2) two-dimensional nonlinear sigma models with Calabi-Yau target spaces admit ultraviolet descriptions as N = (2, 2) gauge theories (gauged linear sigma models). We conjecture that the two-sphere partition function of such ultraviolet gauge theories -recently computed via localization by Benini et al. and Doroud et al. -yields the exact Kähler potential on the quantum Kähler moduli space for Calabi-Yau threefold target spaces. In particular, this allows one to compute the genus zero Gromov-Witten invariants for any such Calabi-Yau threefold without the use of mirror symmetry. More generally, when the infrared superconformal fixed point is used to compactify string theory, this provides a direct method to compute the spacetime Kähler potential of certain moduli (e.g., vector multiplet moduli in type IIA), exactly in α ′ . We compute these quantities for the quintic and for Rødland's Pfaffian Calabi-Yau threefold and find agreement with existing results in the literature. We then apply our methods to a codimension four determinantal Calabi-Yau threefold in P 7 , recently given a nonabelian gauge theory description by the present authors, for which no mirror Calabi-Yau is currently known. We derive predictions for its Gromov-Witten invariants and verify that our predictions satisfy nontrivial geometric checks.1 The work of Böhm [24, 25] provides a promising proposal, but we have been unable to implement it well enough to produce a mirror for this example.2 Local special Kähler manifolds are also often called projective special Kähler manifolds and are distinct from special Kähler manifolds -see, e.g., [32].3 This geometric structure gives rise to a Hodge filtration F 3 ⊂ F 2 ⊂ F 1 ⊂ F 0 of weight 3, with F 3 ≃ L, F 2 ≃ V, F 1 ≃ L ⊥ , and F 0 ≃ V C , where L ⊥ is the subspace of V C perpendicular to L with respect to the symplectic pairing ·, · -see, e.g., [33,34].The last expression involves the periods of Ω,B J Ω(ξ) , I, J = 0, . . . , h 2,1 , (2.6) with respect to a canonical symplectic basis (A I , B J ) of H 3 (Y ξ , Z) satisfying A I , B J = δ I J , A I , A J = B I , B J = 0 . (2.7) 2.3 The quantum Kähler moduli space M KählerThe main player of this note is the quantum-corrected Kähler moduli space M Kähler of a Calabi-Yau threefold Y , which is defined as the corresponding space of chiral-antichiral and antichiralchiral moduli of the underlying SCFT. It is also a local special Kähler manifold, parametrizing 15 We thank C. Vafa for pointing this out to us. 16 Note that in spacetime, some of these fields become quaternionic and are subject to further gs corrections.

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Mathematical aspects of mirror symmetry

Cited by 36 publications

References 112 publications

One-instanton prepotentials from WDVV equations in N=2 supersymmetric SU(4) Yang–Mills theory

One-instanton prepotentials from WDVV equations in N=2 supersymmetric SU(4) Yang–Mills theory

Aspects of N ⩾ 2 topological gauge theories and D-branes

Two-Sphere Partition Functions and Gromov–Witten Invariants

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