Holomorphic field theories and Calabi--Yau algebras

Abstract

We consider the holomorphic twist of the worldvolume theory of flat D$(2k-1)$-branes transversely probing a Calabi--Yau manifold. A chain complex, constructed using the BV formalism, computes the local observables in the holomorphically twisted theory. Generalizing earlier work in the case $k=2$, we find that this complex can be identified with the Ginzburg dg algebra associated to the Calabi--Yau. However, the identification is subtle; the complex is the space of fields contributing to the holomorphic twist of the free theory, and its differential arises from interactions. For $k=1$, this holomorphically twisted theory is related to the elliptic genus. We give a general description for D1-branes probing a Calabi--Yau fourfold singularity, and for $\mathcal{N}=(0,2)$ quiver gauge theories. In addition, we propose a relation between the equivariant Hirzebruch ${\chi }_y$ genus of large-$N$ symmetric products and cyclic homology.

arXiv comment55 pages, 6 figures, 8 tables

References in corpus (50)

• Phases of $N=2$ Theories In Two Dimensions
• An Index for 4 dimensional Super Conformal Theories
• Fivebranes, Membranes and Non-Perturbative String Theory
• D-branes, Quivers, and ALE Instantons
• D-Branes and Topological Field Theories
• Counting chiral primaries in N=1 d=4 superconformal field theories
• D-Branes on Calabi-Yau Spaces and Their Mirrors
• Elliptic Genera of Symmetric Products and Second Quantized Strings
• Elliptic genera of 2d N=2 gauge theories
• Geometry of Batalin-Vilkovisky quantization
• A New Supersymmetric Index
• Elliptic genera of two-dimensional N=2 gauge theories with rank-one gauge groups
• D-branes, Categories and N=1 Supersymmetry
• On the Landau-Ginzburg Description of $N=2$ Minimal Models
• 2d Index and Surface operators
• Supersymmetric Yang-Mills Theory On A Four-Manifold
• D-Branes, Derived Categories, and Grothendieck Groups
• Seiberg Duality for Quiver Gauge Theories
• Computation of Superpotentials for D-Branes
• Supersymmetric gauge theory and the Yangian
• On the Superconformal Index of N=1 IR Fixed Points: A Holographic Check
• Comments on 1/16 BPS Quantum States and Classical Configurations
• On maximally supersymmetric Yang-Mills theories
• 1/16 BPS States in N=4 SYM
• On associated variety for Lie superalgebras
• Brane Box Realization of Chiral Gauge Theories in Two Dimensions
• Twisting of N=1 SUSY Gauge Theories and Heterotic Topological Theories
• Pure spinor superfields -- an overview
• F-theory and 2d (0,2) Theories
• Twisted supergravity and its quantization
• Sequencing BPS Spectra
• Quantization of the Superstring with Manifest U(5) Super-Poincare Invariance
• Exponential Networks and Representations of Quivers
• Global Symmetries in the Antifield-Formalism
• Topological First-Order Systems with Landau-Ginzburg Interactions
• Index for Supergravity on AdS_5 \times T^{1,1} and Conifold Gauge Theory
• An introduction to the Batalin-Vilkovisky formalism
• SU(5)-invariant decomposition of ten-dimensional Yang-Mills supersymmetry
• Topological Chern-Simons/Matter Theories
• Holomorphic Yang-Mills Theory on Compact Kähler Manifolds
• D-Branes and Quotient Singularities of Calabi-Yau Fourfolds
• Cellular resolutions of noncommutative toric algebras from superpotentials
• B-branes and supersymmetric quivers in 2d
• Topological Symmetry of forms, N=1 Supersymmetry and S-duality on Special Manifolds
• Lectures on mathematical aspects of (twisted) supersymmetric gauge theories
• Chiral differential operators via Batalin-Vilkovisky quantization
• The geometry of pure spinor space
• On the non-commutative geometry of topological D-branes
• Brane Tilings and Non-Commutative Geometry
• Quantum Vacua of 2d Maximally Supersymmetric Yang-Mills Theory

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• B-branes and supersymmetric quivers in 2d
• Twisted characters and holomorphic symmetries
• Maximally twisted eleven-dimensional supergravity
• Semi-Chiral Operators in 4d $\mathcal{N}=1$ Gauge Theories
• Graded Quivers, Generalized Dimer Models and Toric Geometry
• On the Classification of Duality Webs for Graded Quivers
• Some Open Questions in Quiver Gauge Theory
• Local Operators from the Space of Vacua of Four Dimensional SUSY Gauge Theories
• Holomorphic field theories and higher algebra
• Twisting BFSS & IKKT