arXiv research

Let X be a compact complex manifold, $D_c^b(X)$ be the bounded derived category of constructible sheaves on $X$, and $Fuk(T^*X)$ be the Fukaya category of $T^*X$. A Lagrangian brane in $Fuk(T^*X)$ is holomorphic if the underlying Lagrangian submanifold is complex analytic in $T^*X_{\mathbb{C}}$, the holomorphic cotange…

Visible Lagrangians in Hitchin systems are studied for pillowcase covers.

The structure of a diffeomorphism invariant Lagrangians for an extended object W embedded in a bulk space M is discussed by following a close analogy with the relativistic particle in electromagnetic field as a system that is reparametrization-invariant. The current construction naturally contains, relativistic point p…

In this paper we study the geometry of manifolds with vector cross product and its complexification. First we develop the theory of instantons and branes and study their deformations. For example they are (i) holomorphic curves and Lagrangian submanifolds in symplectic manifolds and (ii) associative submanifolds and co…

The paper proposes a noncommutative deformation of toric varieties.

Generalized complex (GC) geometry interpolates between ordinary symplectic and complex geometry. Stable generalized complex manifolds (first introduced by Cavalcanti, Gualtieri in 2015) carry a Poisson structure which is generically symplectic, but degenerates on a (real) codimension-2 submanifold. Up to gauge equivale…

In the quest for the mathematical formulation of M-theory, we consider three major open problems: a first-principles construction of the single (abelian) M5-brane Lagrangian density, the origin of the gauge field in heterotic M-theory, and the supersymmetric enhancement of exceptional M-geometry. By combining technique…

In this thesis, we study a class of special Lagrangian submanifolds of toric Calabi-Yau manifolds and construct their mirrors using some techniques developed in the SYZ programme. We present a justification on the conjecture on the mirror construction of D- branes in Aganagic-Vafa [2]. We apply the techniques employed …

New construction reveals SU(2)-flavor fields in heterotic M5-brane model.

While the study of bordered (pseudo-)holomorphic curves with boundary on Lagrangian submanifolds has a long history, a similar problem that involves (special) Lagrangian submanifolds with boundary on complex surfaces appears to be largely overlooked in both physics and math literature. We relate this problem to geometr…

We consider D-branes in string theory and address the issue of how to describe them mathematically as a fundamental object (as opposed to a solitonic object) of string theory in the realm in differential and symplectic geometry. The notion of continuous maps, $k$-times differentiable maps, and smooth maps from an Azuma…

We propose a class of toric Lagrangian A-branes on the resolved conifold that is suitable to describe torus knots on S^3. The key role is played by the SL(2,Z) transformation, which generates a general torus knot from the unknot. Applying the topological vertex to the proposed A-branes, we rederive the colored HOMFLY p…

The paper explores the Thomas-Yau conjecture using holomorphic curves and Floer theory.

Solves a recursion for Gromov-Witten invariants of the unknot.

Mathematical framework for brane quantization using SYZ mirror symmetry.

Using a hyperKähler rotation on complex structures of a Calabi-Yau 2-fold and rolling of an isotropic 2-submanifold in a symplectic 6-manifold, we construct, by gluing, a natural family of immersed Lagrangian deformations of a branched covering of a special Lagrangian 3-sphere in a Calabi-Yau 3-fold and study how they …

Quantization and reduction for coisotropic A-branes on Hamiltonian manifolds.

We reinterpret the generalised Lie derivative of M-theory $E_6$ generalised geometry as hamiltonian flow on a graded symplectic supermanifold. The hamiltonian acts as the nilpotent derivative of the tensor hierarchy of exceptional field theory. This construction is an M-theory analogue of the Courant algebroid and reve…

Let $X$ be a compact real analytic manifold, and let $T^*X$ be its cotangent bundle. Let $Sh(X)$ be the triangulated dg category of bounded, constructible complexes of sheaves on $X$. In this paper, we develop a Fukaya $A_\infty$-category $Fuk(T^*X)$ whose objects are exact, not necessarily compact Lagrangian branes in…

The abstract discusses constructing 3d N=2 gauge theories using surgeries and M5-branes.

We study the most general supersymmetric warped M-theory backgrounds with non-trivial G-flux of the type R^{1,2} x M_8 and AdS_3 x M_8. We give a set of necessary and sufficient conditions for preservation of supersymmetry which are phrased in terms of G-structures and their intrinsic torsion. These equations may be in…

Three-dimensional N=2 superconformal field theories are constructed by compactifying M5-branes on three-manifolds. In the infrared the branes recombine, and the physics is captured by a single M5-brane on a branched cover of the original ultraviolet geometry. The branch locus is a tangle, a one-dimensional knotted subm…

Study gauged supergravity, M5-branes, and class R theories, constraining supergravity coefficients and calculating partition functions.

We establish a direct map between refined topological vertex and sl(N) homological invariants of the of Hopf link, which include Khovanov-Rozansky homology as a special case. This relation provides an exact answer for homological invariants of the of Hopf link, whose components are colored by arbitrary representations …

We study complex Lagrangian submanifolds of a compact hyper-Kähler manifold and prove two results: (a) that an involution of a hyper-Kähler manifold which is antiholomorphic with respect to one complex structure and which acts non-trivially on the corresponding symplectic form always has a fixed point locus which is co…

In the Lagrangian approach to 2-dimensional sigma models, B-fields and D-branes contribute topological terms to the action of worldsheets of both open and closed strings. We show that these terms naturally fit into a 2-dimensional, smooth open-closed functorial field theory (FFT) in the sense of Atiyah, Segal, and Stol…

Generalized complex geometry, introduced by Hitchin, encompasses complex and symplectic geometry as its extremal special cases. We explore the basic properties of this geometry, including its enhanced symmetry group, elliptic deformation theory, relation to Poisson geometry, and local structure theory. We also define a…

We propose a new partially topological theory in three dimensions which couples Chern-Simons theory to matter. The 3-manifolds needed for this construction admit transverse holomorphic foliation (THF). The theory depends only on the choice of such a structure, but not on a choice of metric and in this sense, it is topo…

We study 1-parameter families of holomorphic curves with Lagrangian boundary in Calabi-Yau 3-folds. We show that the expected codimension one phenomena can be organized to match the HOMFLYPT skein relations from quantum topology. It follows that counting holomorphic curves by the class of their boundaries in the skein …

This paper connects symplectic and Kähler manifolds via brane quantization.

Researchers create holographic super-embeddings for M5 and M2 branes.

Introduces D-branes in para-Hermitian geometries using T-duality.

This is the written version of my talk at SUSY '98. It presents a geometric characterisation of the allowed near-horizon geometries of supersymmetric branes. We focus primarily on the M2-brane, but results for other branes (e.g., the D3-brane) are also presented. Some new examples are discussed.

We describe mirror symmetry on higher dimensional tori, paying special attention to the behaviour of D-branes under mirror symmetry. To find the mirror D-branes the description of mirror symmetry on D-branes due to Ooguri, Oz en Yin is used. This method allows us to deal with the coisotropic D-branes recently introduce…

We formalize higher dimensional and higher gauge WZW-type sigma-model local prequantum field theory, and discuss its rationalized/perturbative description in (super-)Lie n-algebra homotopy theory (the true home of the "FDA"-language used in the supergravity literature). We show generally how the intersection laws for s…

We classify and construct all the smooth Kaluza-Klein reductions to ten dimensions of the M2- and M5-brane configurations which preserve some of the supersymmetry. In this way we obtain a wealth of new supersymmetric IIA backgrounds describing composite configurations of D-branes, NS-branes and flux/nullbranes; bound s…

We analyze global anomalies for elementary Type II strings in the presence of D-branes. Global anomaly cancellation gives a restriction on the D-brane topology. This restriction makes possible the interpretation of D-brane charge as an element of K-theory.

Differential Cohomotopy theory predicts brane interactions via chord diagrams.

We construct a gauge fixed action for topological membranes on $G_2$-manifold such that its bosonic part is the standard membrane theory in a particular gauge. We prove that quantum mechanically the path-integral in this gauge localizes on associative submanifolds. Moreover on $M\times S^1$ the theory naturally reduces…

New K-theory framework reveals exotic brane charges and conformal blocks.

It has been shown recently that the geometry of D-branes in general topologically twisted (2,2) sigma-models can be described in the language of generalized complex structures. On general grounds such D-branes (called generalized complex (GC) branes) must form a category. We compute the BRST cohomology of open strings …

The relation between open topological strings and representation theory of symmetric quivers is explored beyond the original setting of the knot-quiver correspondence. Multiple cover generalizations of the skein relation for boundaries of holomorphic disks on a Lagrangian brane are observed to generate dual quiver desc…

Geometric quantization on hyperKähler manifolds via brane quantization.

Constructs brane current algebras from QP-manifolds, generalizing string currents.

We consider branes $N=I\times\so$, where $\so$ is an $n$\ndash dimensional space form, not necessarily compact, in a Schwarzschild-AdS_{(n+2)} bulk $\mc N$. The branes have a big crunch singularity. If a brane is an ARW space, then, under certain conditions, there exists a smooth natural transition flow through the sin…

In this lecture I review how a matrix/Azumaya-type noncommutative geometry arises for D-branes in string theory and how such a geometry serves as an origin of the master nature of D-branes; and then highlight an abundance conjecture on D0-brane resolutions of singularities that is extracted and purified from a work of …

Cohomotopy charge quantization implies String structure on M5-branes.

This is the next step of uncovering the relation between string theory and exotic smooth R^4. Exotic smoothness of R^4 is correlated with D6 brane charges in IIA string theory. We construct wild embeddings of spheres and relate them to a class of topological quantum Dp-branes as well to KK theory. These branes emerge w…