arXiv research

Researchers study rank two theories with eight supercharges using Lefschetz pencils.

Classifies theories with eight supercharges using pseudo-periodic maps and Riemann surfaces.

We obtain all possible solutions of a 1/4 Bogomol'nyi-Prasad-Sommerfield equation exactly, containing configurations made of walls, vortices and monopoles in the Higgs phase. We use supersymmetric U(N_C) gauge theories with eight supercharges with N_F fundamental hypermultiplets in the strong coupling limit. The moduli…

We systematically analyze Riemannian manifolds M that admit rigid supersymmetry, focusing on four-dimensional N=1 theories with a U(1)_R symmetry. We find that M admits a single supercharge, if and only if it is a Hermitian manifold. The supercharge transforms as a scalar on M. We then consider the restrictions imposed…

We consider type II string theory in space-time backgrounds which admit eight supercharges and can be characterized by the existence of an SU(3) x SU(3) structure. We show that the couplings of such backgrounds strongly resemble the couplings of four-dimensional N=2 supergravities and precisely coincide with the N=2 co…

We consider supersymmetric gauge theories with impurities in various dimensions. These systems arise in the study of intersecting branes. Unlike conventional gauge theories, the Higgs branch of an impurity theory can have compact directions. For models with eight supercharges, the Higgs branch is a hyperKahler manifold…

We extend our previous analysis of Riemannian four-manifolds M admitting rigid supersymmetry to N=1 theories that do not possess a U(1)_R symmetry. With one exception, we find that M must be a Hermitian manifold. However, the presence of supersymmetry imposes additional restrictions. For instance, a supercharge that sq…

We construct supersymmetric field theories on Riemannian three-manifolds M, focusing on N=2 theories with a U(1)_R symmetry. Our approach is based on the rigid limit of new minimal supergravity in three dimensions, which couples to the flat-space supermultiplet containing the R-current and the energy-momentum tensor. T…

Study finds modular classes help in proving Berezin volumes for supersymmetric theories.

In this paper we define the analogue of Calabi--Yau geometry for generic $D=4$, $\mathcal{N}=2$ flux backgrounds in type II supergravity and M-theory. We show that solutions of the Killing spinor equations are in one-to-one correspondence with integrable, globally defined structures in $E_{7(7)}\times\mathbb{R}^+$ gene…

Unified view of spectral networks linking geometry and gauge theory.

This work takes place over a conformally flat spin manifold (M,g). We prove existence and uniqueness of the conformally equivariant quantization valued in spinor differential operators, and provide an explicit formula for it when restricted to first order operators. The Poisson algebra of symbols is realized as a space…

We describe the construction of a Lie superalgebra associated to an arbitrary supersymmetric M-theory background, and discuss some examples. We prove that for backgrounds with more than 24 supercharges, the bosonic subalgebra acts locally transitively. In particular, we prove that backgrounds with more than 24 supersym…

This is the written version of a talk given in Bonn on September 11th, 2001 during a workshop on "Special structures in string theory". We report on joint work in progress with George Papadopoulos aimed at classifying the maximally supersymmetric solutions of the ten- and eleven-dimensional supergravity theories with 3…

We review our recent work on solitons in the Higgs phase. We use U(N_C) gauge theory with N_F Higgs scalar fields in the fundamental representation, which can be extended to possess eight supercharges. We propose the moduli matrix as a fundamental tool to exhaust all BPS solutions, and to characterize all possible modu…

Detects figure-eight knot using Khovanov homology.

Study shows $(2,1)$-cable of figure-eight knot can't be smoothly sliced.

The paper contains the computation of the noncommutative A-ideal of the figure-eight knot, a noncommutative generalization of the A-polynomial. We show that if a knot has the same noncommutative A-ideal as the figure-eight knot, then all colored Kauffman brackets are the same as those of the figure-eight knot.

We study the extent to which the gauge symmetry of abelian Yang-Mills can be deformed under two conditions: first, that the deformation depend on a two-form scale. Second, that the deformation preserve supersymmetry. We show that (up to a single parameter) the only allowed deformation is the one determined by the star …

The study classifies all compact hyperbolic polytopes with eight facets.

Smooth figure-eight knot cables have infinite order.

We construct a geometric model of eight-dimensional manifolds and realize them in the context of type II string theory. These eight-manifolds are constructed by non-trivial $T^{4}$ fibrations over Calabi-Yau two-folds. These give rise to eight-dimensional non-Kahler Hermitian manifolds with $SU(4)$ structure. The eight…

New constructions show stable geodesics and figure-eights in convex hypersurfaces.

Proved colored HOMFLY-PT polynomials for specific knots.

In this paper we show that some open set of the representations of the fundamental group of figure-eight knot complement found in \cite{Ballas12a} are the holonomies of a family of finite volume properly convex projective structures on the figure-eight knot complement.

We show that most cabled knots over the figure eight knot in $S^3$ satisfy the AJ-conjecture, in particular, any $(r,s)$-cabled knot over the figure eight knot satisfies the $AJ$-conjecture if $r$ is not a number between $-4s$ and $4s$.

We present supersymmetric, curved space, quantum mechanical models based on deformations of a parabolic subalgebra of osp(2p+2|Q). The dynamics are governed by a spinning particle action whose internal coordinates are Lorentz vectors labeled by the fundamental representation of osp(2p|Q). The states of the theory are t…

Study of lambda lengths in figure eight knot complement using Eisenstein integers.

Verifies a conjecture for the figure eight knot.

We obtain a branched spherical CR structure on the complement of the figure eight knot with a given holonomy representation (called rho_2). There are essentially two boundary unipotent representations from the complement of the figure eight knot into PU(2,1), we call them rho_1 and rho_2. We make explicit some fundamen…

Study on the growth of colored Jones polynomial for figure-eight knot cables.

The paper proves rigidity of surgeries on the figure-eight knot complement.

Authors prove quantum invariant conjecture for figure-eight knot complement.

Study shortest non-simple geodesics on 2-orbifolds, finding unique shortest curve.

Jones slopes detect figure eight knot, and characterize alternating knots.

We describe a simple fundamental domain for the holonomy group of the boundary unipotent spherical CR uniformization of the figure eight knot complement, and deduce that small deformations of that holonomy group (such that the boundary holonomy remains parabolic) also give a uniformization of the figure eight knot comp…

3-manifolds transform into figure-eight knot complements through complex hyperbolic deformations.

In this paper, we compute the symplectic Floer homology of the figure eight knot. This provides first nontrivial knot with trivial symplectic Floer homology.

Proves volume conjectures for figure-eight knot surgeries.

We give explicit equations that describe the character variety of the figure eight knot for the groups SL(3,C), GL(3,C) and PGL(3,C). This has five components of dimension 2, one consisting of totally reducible representations, another one consisting of partially reducible representations, and three components of irred…

In this paper we find infinitely many lattices in $SL(4,\mathbb{R})$ each of which contains thin subgroups commensurable with the figure-eight knot group.

New example shows figure eight knot not smoothly concordant but homology cobordant.

Study on colored Jones polynomial of figure-eight knot for complex parameters.

Study shows quantum modularity in figure-eight knot's colored Jones polynomial.

This paper determines the minimal degree sequence for two compact rational knots, namely the trefoil and figure-eight knots. We find explicit projections with the minimal degree sequence of each knot. This is done by modifying a non-compact rational minimal-degree parameterization of the trefoil and figure-eight knots …

Study shows shortest periodic geodesic on hyperbolic orbisphere complements figure-eight knot.

The colored Jones polynomial of the figure-eight knot connects to an SL(2;R) representation.

In this paper we investigate the asymptotic behavior of the colored Jones polynomials and the Turaev-Viro invariants for the figure eight knot. More precisely, we consider the $M$-th colored Jones polynomials evaluated at $(N+1/2)$-th root of unity with a fixed limiting ratio, $s$, of $M$ and $(N+1/2)$. We find out the…