Study electric-magnetic duality in M-theory compactifications.
problem Restoring supersymmetry and understanding dualities in compactified M-theory.
method Dualizing M-theory on G2 manifolds with F-theory and studying D3-branes. result Demonstrates correspondence between D3-branes and shrinking surfaces/curves, revealing light particles with electric and magnetic charges.
New Langlands duality conjectures for 3-manifold skein modules.
problem Understanding Langlands duality in 3-manifold skein modules.
method Combining representation theory of double affine Hecke algebras and 1-form symmetry structure.
result Recent special cases confirmed, with detailed proofs.
Study non-orientable surfaces in gauge theories, matching fluxes with homotopy classes.
problem Understanding non-orientable surfaces in gauge theory and their electric-magnetic duality.
method Reduction of twisted N=4 gauge theory on non-orientable manifolds, matching fluxes with homotopy classes.
result Verification of mirror symmetry of branes as predicted by S-duality.
We study generalized electric/magnetic duality in Abelian gauge theory by combining techniques from locally covariant quantum field theory and Cheeger-Simons differential cohomology on the category of globally hyperbolic Lorentzian manifolds. Our approach generalizes previous treatments using the Hamiltonian formalism …
We present an explicit expression for the topological invariants associated to SU(2) monopoles in the fundamental representation on spin four-manifolds. The computation of these invariants is based on the analysis of their corresponding topological quantum field theory, and it turns out that they can be expressed in …
In the first of these two lectures, I describe a gauge theory approach to understanding quantum knot invariants as Laurent polynomials in a complex variable q. The two main steps are to reinterpret three-dimensional Chern-Simons gauge theory in four dimensional terms and then to apply electric-magnetic duality. The var…
It has been argued based on electric-magnetic duality and other ingredients that the Jones polynomial of a knot in three dimensions can be computed by counting the solutions of certain gauge theory equations in four dimensions. Here, we attempt to verify this directly by analyzing the equations and counting their solut…
This paper studies the interplay between the N=2 gauge theories in three and four dimensions that have a geometric description in terms of twisted compactification of the six-dimensional (2,0) SCFT. Our main goal is to construct the three-dimensional domain walls associated to any three-dimensional cobordism. We find t…
We study the cohomological physics of fivebranes in type II and heterotic string theory. We give an interpretation of the one-loop term in type IIA, which involves the first and second Pontrjagin classes of spacetime, in terms of obstructions to having bundles with certain structure groups. Using a generalization of th…
Revisits Vafa-Witten theory, deriving new invariants and homologies.
problem Understanding Vafa-Witten equations and their invariants.
method Physical derivation and mathematical analysis of Vafa-Witten equations.
result Novel invariants and Floer homologies derived from Vafa-Witten equations.
We generalize the analysis by Moore and Witten in [arXiv:hep-th/9709193], and consider integration over the u-plane in Donaldson theory with surface operators on a smooth four-manifold X. Several novel aspects will be developed in the process; like a physical interpretation of the "ramified" Donaldson and Seiberg-Witte…
Let A be the space of irreducible connections (vector potentials) over a SU(n)-principal bundle on a three-dimensional manifold M. Let T be the fiber product of the tangent and cotangent bundles of A. We endow T with a symplectic structure Ωwhich is represented by a vortex formula. The corresponding Poisson bracket wil…
The proper action functional of (4k+3)-dimensional U(1)-Chern-Simons theory including the instanton sectors has a well known description: it is given on the moduli space of fields by the fiber integration of the cup product square of classes in degree-(2k+2) differential cohomology. We first refine this statement from …
Unified view of spectral networks linking geometry and gauge theory.
problem Understanding BPS states in gauge theories.
method Unified geometric and physical approaches, focusing on spectral networks.
result Spectral networks provide a framework for determining BPS spectra.
In Levin-Wen (LW) models, a wide class of exactly solvable discrete models, for two dimensional topological phases, it is relatively easy to describe only single fluxon excitations, but not the charge and dyonic as well as many-fluxon excitations. To incorporate charged and dyonic excitations in (doubled) topological p…
We continue our program initiated in [arXiv:0912.4261] to consider supersymmetric surface operators in a topologically-twisted N=2 pure SU(2) gauge theory, and apply them to the study of four-manifolds and related invariants. Elegant physical proofs of various seminal theorems in four-manifold theory obtained by Ozsvat…
A necessary and sufficient condition for energy-momentum conservation is proved within a topological, pre-metric approach to classical electrodynamics including magnetic as well as electric charges. The extended Lorentz force, consisting of mutual actions by F=(E, B) on the electric current and G=(H, D) on the magnetic…
Proves Poincaré duality for Hopf algebroids with bijective antipode.
problem Proving Poincaré duality for Hopf algebroids.
method Using twisted Poincaré duality and bijective antipode properties.
result Recovering and extending known Poincaré dualities for Hopf algebroids.
New proof of chain duality for simplicial complexes.
problem Proving the existence of chain duality for chain complexes over simplicial complexes.
method Geometric and conceptual treatment of chain duality.
result Fundamental for Ranicki's surgery exact sequence.
Introduces Kähler duality between domains in complex space.
problem None explicitly stated; focuses on concept introduction.
method None explicitly stated; focuses on concept introduction.
result Introduces Kähler duality between domains in complex space.
Research on dualities in geometric stereotypes.
problem Understanding dualities in geometric stereotypes.
method Continuation of previous research on stereotype spaces and algebras.
result New insights into geometric stereotype dualities.
Duality restored in gauge theory, gravity, and string theory models.
problem Restoring duality invariance in theories coupled to matter.
method Extending phase space to allow for violations of the algebraic Bianchi identity and considering the axion as the duality current.
result Duality current in NS-NS gravity is the divergence of the axion.
Unified proof of four Bavard dualities and new results on quasimorphisms.
problem Comparing stable commutator length and quasimorphisms on groups.
method Expository account and new strengthening of Bavard duality, providing complete proofs.
result Generalized mixed Bavard duality, recovering all previous dualities.
Verma Howe duality connects tensor products of Verma modules to LKB representations.
problem Understanding the relationship between tensor products of Verma modules and LKB representations.
method Established a quantized version of Verma Howe duality and used it to prove the simplicity of LKB representations.
result LKB representations arise from the quantized Verma Howe duality and are shown to be simple modules.
Cohomological and homological spectral sequences are shown to be isomorphic.
problem Cohomological and homological Atiyah-Hirzebruch spectral sequences are not always isomorphic.
method Spanier-Whitehead duality is used to establish an isomorphism between the two spectral sequences.
result Cohomological and homological Atiyah-Hirzebruch spectral sequences are isomorphic for finite spectra.
Differential geometry applied to Mukai duality on K3 surfaces.
problem Understanding Mukai duality on K3 surfaces.
method Differential geometric approach.
result New insights into Mukai duality on K3 surfaces.
The paper proves new results on Poincaré duality pairs and spaces.
problem Establishing Poincaré duality in various contexts and dimensions.
method Analyzes Poincaré spaces and CW pairs, proving relative Poincaré duality and related results.
result Found a finite CW pair (X,Y) where Y fails to satisfy Poincaré duality in any dimension. T-duality rules for 2D (2,1) supersymmetric models clarified.
problem Clarify T-duality in (2,1) supersymmetric sigma-models.
method Gauging of sigma-models in (2,1) superspace, finding complexified duality transformations.
result Complexified duality transformations are equivalent to Buscher transformations and diffeomorphisms.
Establishes higher T-duality for super M-branes.
problem Generalizing T-duality for super p-branes.
method Super L-infinity-algebraic T-duality for super WZW-terms.
result Spherical T-duality of super M5-branes.
The paper proves T-duality and Hori formulae for winding loop spaces.
problem Realizing T-duality and Hori formulae for loop spaces.
method Proving T-duality and Hori formulae for winding q-loop spaces.
result T-duality and Hori formulae for winding q-loop spaces are proven.
Equivariant T-duality connects bundles with twists.
problem Establishing a relationship between bundles with twists.
method Formulating T-duality in equivariant K-theory for compact Lie group actions.
result T-duality is an isomorphism in equivariant K-theory for compact Lie group actions.
We give the definition of a duality that is applicable to arbitrary k-forms. The operator that defines the duality depends on a fixed form Ω. Our definition extends in a very natural way the Hodge duality of n-forms in 2n dimensional spaces and the generalized duality of two-forms. We discuss the properties of …
The paper establishes T-duality for 2D σ-models with H-flux.
problem T-duality for 2D σ-models with H-flux.
method Localization and graded T-duality map (graded Hori morphism).
result Establishes the most general version of T-duality for Type II String Theory.
QP perspective on Poisson-Lie T-duality topology changes.
problem Understanding Poisson-Lie T-duality through QP manifolds.
method QP manifolds and canonical transformations for symplectic reductions.
result Canonical transformations mediate Poisson-Lie T-duality.
Unified framework for T-duality in both trivial and non-trivial topologies.
problem Unified description of T-duality for metrics and B-fields in non-trivial topology.
method Developed a new unifying framework for T-duality.
result Unified description of T-duality for metrics and B-fields in non-trivial topology.
Geometric duality connects graph isomorphism and knot equivalence.
problem Understanding the equivalence of graph isomorphism and knot equivalence.
method Observation of geometric duality in planar graphs and links.
result The equivalence relation defined by isomorphisms of checkerboard graphs is the same as 2-isomorphisms of checkerboard graphs.
Koszul duality for manifold modules proven.
problem Proving Koszul self duality of manifold modules.
method Using generalized Thom complexes and operads in Top.
result Koszul self duality of little disk modules proven.
This dissertation explores T-duality between hyperkähler structures and branes on algebraic integrable systems.
problem Investigates T-duality between hyperkähler structures and branes on algebraic integrable systems.
method Uses techniques of generalized geometry and Fourier-Mukai transform to show T-duality between semi-flat hyperkähler structures and generalized branes.
result Shows T-duality between semi-flat hyperkähler structures and generalized branes on algebraic integrable systems.
Maps self-duality in little disks operad to framed manifolds.
problem Self-duality of little disks operad.
method Configuration space level Pontryagin--Thom constructions.
result Existence of compatible self-duality map for framed manifolds.
We study generalized complex structures and T-duality (in the sense of Bouwknegt, Evslin, Hannabuss and Mathai) on Lie algebras and construct the corresponding Cavalcanti and Gualtieri map. Such a construction is called "Infinitesimal T-duality". As an application we deal with the problem of finding symplectic stru…
New spherical T-duality for higher degree forms in fiber bundles.
problem Extending T-duality to higher degree forms in fiber bundles.
method Generalizing T-duality to S2n−1-bundles with closed odd forms of arbitrary degree. result Existence and isomorphic twisted cohomology of T-dual spaces. The paper establishes a duality between non-compact and compact symmetric pairs.
problem Understanding the relationship between non-compact and compact symmetric pairs.
method Developed a duality theorem between non-compact pseudo-Riemannian semisimple symmetric pairs and commutative compact semisimple symmetric triads.
result Explicit description of a one-to-one correspondence between non-compact and compact symmetric pairs.
Paper constructs counterexamples of higher solutions to self-duality equations.
problem Whether all real sections give rise to global solutions of self-duality equations.
method Using Willmore surfaces and generalized Whitham flow.
result Higher solutions exist on an open dense subset of the Riemann surface, not globally.
We study discrete group actions on coarse Poincare duality spaces, e.g. acyclic simplicial complexes which admit free cocompact group actions by Poincare duality groups. When G is an (n-1) dimensional duality group and X is a coarse Poincare duality space of formal dimension n, then a free simplicial action of G on X d…
The paper shows plentiful non-homotopy finite Poincaré duality spaces.
problem The existence of non-homotopy finite Poincaré duality spaces.
method Constructing a finitely dominated Poincaré space with a non-trivial 2-divisible element in the reduced Grothendieck group.
result The existence of finitely dominated Poincaré spaces that are not homotopy finite.
Extends duality preserving singular set images and first fundamental forms to generalized cuspidal edges.
problem Preserving singular set images and first fundamental forms on generalized cuspidal edges.
method Extends previous isometric duality to generalized cuspidal edges including cuspidal cross caps and 5/2-cuspidal edges.
result New geometric insights on the duality.
Extends T-duality to exotic Courant algebroids.
problem Lack of T-duality for exotic Courant algebroids.
method Extends T-duality isomorphism to exotic exact Courant algebroids.
result Exchange of momentum and winding numbers.
Study S-duality and supersymmetry on curved manifolds using localization.
problem Understanding S-duality and supersymmetry on curved manifolds.
method Localization and Fourier transform interpretation of S-duality.
result Evidence for interpreting S-duality as Fourier transform.