Introduction to $A_{\infty}$-infinity algebras and modules

Keller, Bernhard

doi:10.4310/hha.2001.v3.n1.a1

Cited by 324 publications

(356 citation statements)

References 31 publications

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“…It is known [38,48] that in general topological open string theory can correspond to an A ∞ category [37], as appeared in Kontsevich's original proposal. An explicit construction of an A ∞ structure on the category of coherent sheaves appears in [40,44]; the higher products are essentially correlation functions in holomorphic Chern-Simons theory (the third order product was already discussed in [58]).…”

Section: Topologically Twisted Open String Theorymentioning

confidence: 99%

D-branes, categories and N=1 supersymmetry

Douglas¹

2001

Journal of Mathematical Physics

298

738

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We show that boundary conditions in topological open string theory on Calabi-Yau manifolds are objects in the derived category of coherent sheaves, as foreseen in the homological mirror symmetry proposal of Kontsevich. Together with conformal field theory considerations, this leads to a precise criterion determining the BPS branes at any point in CY moduli space, completing the proposal of Π-stability.

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Section: Topologically Twisted Open String Theorymentioning

confidence: 99%

D-branes, categories and N=1 supersymmetry

Douglas¹

2001

Journal of Mathematical Physics

298

738

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“…The following treatment is based on Fukaya et al [9], and uses their conventions. Two survey papers by Keller [15,16] are useful introductions; note that [16] uses the conventions of [9], as we do, but [15] has different conventions on signs and grading. We restrict to A ∞ algebras over Q, but one can also work over any commutative ring R.…”

Section: Remark 216 (A) Perturbation Data Does Not Involve a Series (Smentioning

confidence: 99%

Immersed Lagrangian floer theory

Akaho¹,

Joyce²

2010

J. Differential Geom.

188

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Abstract. Let (M, ω) be a compact symplectic 2n-manifold, and L a compact embedded Lagrangian submanifold in M . Fukaya, Oh, Ohta and Ono [9] construct Lagrangian Floer cohomology for such M, L, yielding groups HF * (L, b; Λnov) for one Lagrangian or HF * `( L 1 , b 1 ), (L 2 , b 2 ); Λnov´for two, where b, b 1 , b 2 are choices of bounding cochains, and exist if and only if L, L 1 , L 2 have unobstructed Floer cohomology. These are independent of choices up to canonical isomorphism, and have important invariance properties under Hamiltonian equivalence. Floer cohomology groups are the morphism groups in the derived Fukaya category of (M, ω), and so are an essential part of the Homological Mirror Symmetry Conjecture of Kontsevich.The goal of this paper is to extend [9] to immersed Lagrangians L in M with immersion ι : L → M , with transverse self-intersections. In the embedded case, Floer cohomology HF * (L, b; Λnov) is a modified, 'quantized' version of singular homology H n− * (L; Λnov) over the Novikov ring Λnov. In our immersed case, HF * (L, b; Λnov) turns out to be a quantized version ofThe theory becomes simpler and more powerful for graded Lagrangians in Calabi-Yau manifolds, when we can work over a smaller Novikov ring ΛCY. The proofs involve associating a gapped filtered A∞ algebra over Λ 0 nov or Λ 0 CY to ι : L → M , which is independent of nearly all choices up to canonical homotopy equivalence, and is built using a series of finite approximations called A N,0 algebras for N = 0, 1, 2, . . ..

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“…which induce isomorphisms between the respective m 1 -cohomologies. We will need the result that A ∞ -quasi-isomorphisms have homotopy inverses, see [25] and the references therein. We also need the result that an A ∞ -morphism f between minimal A ∞ -algebras is an isomorphism if and only if f 1 is an isomorphism.…”

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confidence: 99%

Computation of Superpotentials for D-Branes

Aspinwall

Katz

2006

Commun. Math. Phys.

160

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We present a general method for the computation of tree-level superpotentials for the world-volume theory of B-type D-branes. This includes quiver gauge theories in the case that the D-brane is marginally stable. The technique involves analyzing the A ∞ -structure inherent in the derived category of coherent sheaves. This effectively gives a practical method of computing correlation functions in holomorphic Chern-Simons theory. As an example, we give a more rigorous proof of previous results concerning 3-branes on certain singularities including conifolds. We also provide a new example.

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Introduction to $A_{\infty}$-infinity algebras and modules

Abstract: These are expanded notes of four introductory talks on A ∞ -algebras, their modules and their derived categories.

Cited by 324 publications

References 31 publications

D-branes, categories and N=1 supersymmetry

D-branes, categories and N=1 supersymmetry

Immersed Lagrangian floer theory

Computation of Superpotentials for D-Branes

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