D branes in Landau-Ginzburg models and algebraic geometry

By A. Kapustin, Yi Li. 2002

272 citations recorded by the source corpus.

We study topological D-branes of type B in N = 2 Landau-Ginzburg models, focusing on the case where all vacua have a mass gap. In general, tree-level topological string theory in the presence of topological D-branes is described mathematically in terms of a triangulated category. For example, it has been argued that B-branes for an N = 2 sigma-model with a Calabi-Yau target space are described by the derived category of coherent sheaves on this space. M. Kontsevich previously proposed a candidate category for B-branes in N = 2 Landau-Ginzburg models, and our computations confirm this proposal. We also give a heuristic physical derivation of the proposal. Assuming its validity, we can completely describe the category of B-branes in an arbitrary massive Landau-Ginzburg model in terms of modules over a Clifford algebra. Assuming in addition Homological Mirror Symmetry, our results enable one to compute the Fukaya category for a large class of Fano varieties. We also provide a (somewhat trivial) counter-example to the hypothesis that given a closed string background there is a unique set of D-branes consistent with it.

Subfields: Geometry and Topology, Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/9730dea82f1631a08d84bc8cf2cdb60795f83083

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Citations in the corpus

A category of kernels for equivariant factorizations and its implications for Hodge theory (2011, background): “d Y. Li conﬁrmed Kontsevich’s prediction and gave a mathematically conjectural description of the Chern character map and the pairing on Hochschild homology for the category of matrix factorizations, [KL03a] [KL03b]. In mathematics, several foundational papers by Orlov soon followed: [Orl04, Orl06, ”
Matrix Factorizations and Representations of Quivers I (2005, background): “z mA 2 (x ⊗mA 2 (y ⊗z)). Put u := −mA 0 (1), d(x) := (−1)|x|+1mA 1 (x) and x · y := (−1)|y|mA 2 (x ⊗ y). Then a triple (u,d,·) deﬁnes on ⊕a,bA(a,b) a curved diﬀerential graded (CDG) algebra structure [KL1]. Example-Deﬁnition 2.3. Let f ∈ C[x1,...xn] be a polynomial which satisﬁes the following quasi”
Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves (2005, result): “Namely, viewing now X as a symplectic manifold and M as a complex manifold, the derived category of B-branes of the Landau-Ginzburg model W : M → C, which was defined algebraically in [10, 15] following ideas of Kontsevich, should be equivalent to the derived Fukaya category of X.”
Chern characters and Hirzebruch-Riemann-Roch formula for matrix factorizations (2010, background): “In the context of the present paper the most relevant interpretation is the one suggested Kontsevich: to view matrix factorizations as D-branes in topological Landau-Ginzburg models (see [12]).”
Phases Of N=2 Theories In 1+1 Dimensions With Boundary (2008, background): “65) by adding a suitable boundary term to the action [10, 33].”
Triangulated categories of singularities and equivalences between Landau-Ginzburg models (2005, unknown): “The equivalence of this definition with the physics notion of Bbranes in LG models was verified in the paper [11] in the case of the usual quadratic superpotential and physical arguments were given supporting Kontsevich’s proposal for a general superpotential.”
Matrix factorizations and link homology (2004, background): “Matrix factorizations appeared in commutative algebra in early and mideighties [E1], [B], [Kn], [S], [BEH] in the study of isolated hypersurface singularities, and much more recently in string theory, as boundary conditions for strings in Landau-Ginzburg models [ KL1 -3].”
Matrix factorizations and Cohomological Field Theories (2011, unknown): “izations of complexes obtained by replacing the condition d2 = 0 with d2 = w (see [14], [7]). Matrix factorizations appeared in physics in connection with open-closed topological string theories (see [27], [28], [29]). They are also related to an important invariant of 1Sometimes this theory is refe”
Derived Categories of Coherent Sheaves and Triangulated Categories of Singularities (2005, background): “d are not needed, we do not ﬁx them. With any point λ ∈A1 we can associate a triangulated category DBλ(W). We give a construction of these categories under the condition that X = Spec(B) is aﬃne (see [17, 22]). The category of coherent sheaves on an aﬃne scheme X = Spec(B) is the same as the categor”
Homological algebra of knots and BPS states (2011, background): “y from crossings. For example, for the fundamental representation of g= sl(N) the potential is a function of a single variable, (3.1) Wsl(N), (x) = x N+1 . In physics, matrix factorizations are known [41, 42, 43, 44, 45, 46] to describe D-branes and topological defects in Landau-Ginzburg models whic”
Dirichlet Branes and Mirror Symmetry (2009, methodology): “In this case the category of B-branes is equivalent to the category of representations of the Clifford algebra with n generators [240, 286].”
Opening Mirror Symmetry on the Quintic (2006, methodology): “In [15, 16], the continuation of these boundary states over the moduli space was analyzed using matrix factorizations [17,18] in the mirror B-model Landau-Ginzburg description.”
D-branes and K-theory in 2D topological field theory (2006, background): “The resulting enlarged category is essentially the bounded derived category of the category of coherent sheaves on X .”
Landau-Ginzburg Realization of Open String TFT (2003, background): “Extending the work of [30, 34], we will in particular show that the underlying cohomology problems are isomorphic, and thus lead to the same open string spectrum.”
Khovanov-Rozansky Homology and Topological Strings (2004, background): “In fact, the category hmfW is a simple example of a “D-brane category”, whose objects are D-branes in N = 2 Landau-Ginzburg model with a superpotential W [27,28,29,30].”
Singularity Categories, Schur Functors and Triangular Matrix Rings (2007, unknown): “framework of Homological Mirror Symmetry Conjecture (compare [11]).”