Spherical T-Duality

We introduce spherical T-duality, which relates pairs of the form (P, H) consisting of a principal SU(2)-bundle \({P \rightarrow M}\) and a 7-cocycle H on P. Intuitively spherical T-duality exchanges H with the second Chern class c ₂(P). Unless \({dim(M) \leq 4}\), not all pairs admit spherical T-duals and the spherical T-duals are not always unique. Nonetheless, we prove that all spherical T-dualities induce a degree-shifting isomorphism on the 7-twisted cohomologies of the bundles and, when \({dim(M) \leq 7}\), also their integral twisted cohomologies and, when \({dim(M) \leq 4}\), even their 7-twisted K-theories. While spherical T-duality does not appear to relate equivalent string theories, it does provide an identification between conserved charges in certain distinct IIB supergravity and string compactifications.     

Nested T-Duality

We identify the obstructions for Kiritsis–Obers T-duality of boundary WZW models. The open string duality pattern is much richer than in the closed strings case since it depends substantially on the geometry of branes. In particular, the duality obstructions disappear for certain brane configurations associated to non-regular elements of the Cartan torus. It is shown in this case that the boundary WZW model is “nested” in the twisted boundary WZW model as the dynamical subsystem of the latter     

State-Dependent Topological Invariants and Anomalous Bulk-Boundary Correspondence in Non-Hermitian Topological Systems with Generalized Inversion Symmetry

Breakdown of bulk-boundary correspondence in non-Hermitian(NH) topological systems with generalized inversion symmetries is a controversial issue. The non-Bloch topological invariants determine the existence of edge states, but fail to describe the number and distribution of defective edge states in non-Hermitian topological systems. The state-dependent topological invariants, instead of a global topological invariant, are developed to accurately characterize the bulk-boundary correspondence of ...     

T-Duality for Orientifolds and Twisted KR-Theory

D-brane charges in orientifold string theories are classified by the KR-theory of Atiyah. However, this is assuming that all O-planes have the same sign. When there are O-planes of different signs, physics demands a “KR-theory with a sign choice” which up until now has not been studied by mathematicians (with the unique exception of Moutuou, who did not have a specific application in mind). We give a definition of this theory and compute it for orientifold theories compactified on S ¹ and T ². We also explain how and why additional “twisting” is implemented. We show that our results satisfy all possible T-duality relationships for orientifold string theories on elliptic curves, which will be studied further in subsequent work.     

Nonassociative Tori and Applications to T-Duality

In this paper, we initiate the study of C^*-algebras endowed with a twisted action of a locally compact abelian Lie group , and we construct a twisted crossed product , which is in general a nonassociative, noncommutative, algebra. The duality properties of this twisted crossed product algebra are studied in detail, and are applied to T-duality in Type II string theory to obtain the T-dual of a general principal torus bundle with general H-flux, which we will argue to be a bundle of noncommutative, nonassociative tori. Nonassociativity is interpreted in the context of monoidal categories of modules. We also show that this construction of the T-dual includes the other special cases already analysed in a series of papers.     

A Groupoid Approach to Noncommutative T-Duality

Topological T-duality is a transformation taking a gerbe on a principal torus bundle to a gerbe on a principal dual-torus bundle. We give a new geometric construction of T-dualization, which allows the duality to be extended in the following two directions. First, bundles of groups other than tori, even bundles of some nonabelian groups, can be dualized. Second, bundles whose duals are families of noncommutative groups (in the sense of noncommutative geometry) can be treated, though in this case the base space of the bundles is best viewed as a topological stack. Some methods developed for the construction may be of independent interest. These are a Pontryagin type duality that interchanges commutative principal bundles with gerbes, a nonabelian Takai type duality for groupoids, and the computation of certain equivariant Brauer groups. The research reported here was supported in part by National Science Foundation grants DMS-0703718 and DMS-0611653.     

Uniqueness Theorem for BMS-Invariant States of Scalar QFT on the Null Boundary of Asymptotically Flat Spacetimes and Bulk-Boundary Observable Algebra Correspondence

This work concerns some features of scalar QFT defined on the causal boundary of an asymptotically flat at null infinity spacetime and based on the BMS-invariant Weyl algebra .(a) (i) It is noticed that the natural BMS invariant pure quasifree state λ on , recently introduced by Dappiaggi, Moretti and Pinamonti, enjoys positivity of the self-adjoint generator of u-translations with respect to every Bondi coordinate frame on , ( being the affine parameter of the complete null geodesics forming and complex coordinates on the transverse 2-sphere). This fact may be interpreted as a remnant of the spectral condition inherited from QFT in Minkowski spacetime (and it is the spectral condition for free fields when the bulk is the very Minkowski space). (ii) It is also proved that the cluster property under u-displacements is valid for every (not necessarily quasifree) pure state on which is invariant under u displacements. (iii) It is established that there is exactly one algebraic pure quasifree state which is invariant under u-displacements (of a fixed Bondi frame) and has positive self-adjoint generator of u-displacements. It coincides with the GNS-invariant state λ. (iv) Finally it is shown that in the folium of a pure u-displacement invariant state ω (like λ but not necessarily quasifree) on is the only state invariant under u-displacement.(b) It is proved that the theory can be formulated for spacetimes asymptotically flat at null infinity which also admit future time completion i ⁺ (and fulfill other requirements related with global hyperbolicity). In this case a -isomorphism ı exists - with a natural geometric meaning - which identifies the (Weyl) algebra of observables of a linear field propagating in the bulk spacetime with a sub algebra of . Using ı a preferred state on the field algebra in the bulk spacetime is induced by the BMS-invariant state λ on .     

T-Duality for Torus Bundles with H-Fluxes via Noncommutative Topology

It is known that the T-dual of a circle bundle with H-flux (given by a Neveu-Schwarz 3-form) is the T-dual circle bundle with dual H-flux. However, it is also known that torus bundles with H-flux do not necessarily have a T-dual which is a torus bundle. A big puzzle has been to explain these mysterious missing T-duals. Here we show that this problem is resolved using noncommutative topology. It turns out that every principal T²-bundle with H-flux does indeed have a T-dual, but in the missing cases (which we characterize), the T-dual is non-classical and is a bundle of noncommutative tori. The duality comes with an isomorphism of twisted K-theories, just as in the classical case. The isomorphism of twisted cohomology which one gets in the classical case is replaced by an isomorphism of twisted cyclic homology.VM was supported by the Australian Research Council.JR was partially supported by NSF Grant DMS-0103647, and thanks the Department of Pure Mathematics of the University of Adelaide for its hospitality in January 2004, which made this collaboration possible.     

The Newtonian Correspondence

It is shown that the field equations of general theory of relativity in the Einstein tensor form and the unimodular theory of gravity do not fulfill the correspondence principle commitment completely. The consistent formalisms are briefly discussed.     

T-Duality: Topology Change from <Emphasis Type="Italic">H</Emphasis>-Flux

T-duality acts on circle bundles by exchanging the first Chern class with the fiberwise integral of the H-flux, as we motivate using E₈ and also using S-duality. We present known and new examples including NS5-branes, nilmanifolds, lens spaces, both circle bundles over Pⁿ, and the AdS⁵×S⁵ to AdS⁵×P²×S¹ with background H-flux of Duff, Lü and Pope. When T-duality leads to M-theory on a non-spin manifold the gravitino partition function continues to exist due to the background flux, however the known quantization condition for G₄ receives a correction. In a more general context, we use correspondence spaces to implement isomorphisms on the twisted K-theories and twisted cohomology theories and to study the corresponding Grothendieck-Riemann-Roch theorem. Interestingly, in the case of decomposable twists, both twisted theories admit fusion products and so are naturally rings.     

Schoenberg Correspondence on Dual Groups

As in the classical case of Lévy processes on a group, Lévy processes on a Voiculescu dual group are constructed from conditionally positive functionals. It is essential for this construction that Schoenberg correspondence holds for dual groups: The exponential of a conditionally positive functional is a convolution semigroup of states.     

"对应原理"的普遍意义——科学与人文漫话之十四

论述了"对应原理"提出的历史背景,讨论了"对应原理"的基本内容、普遍意义及其推广到社会领域的表述.     

Correspondence arguments for large angle scattering

The correspondence principle of Bjorken and Kogut is used to relate inclusive and exclusive processes at large angle. Each inclusive cross section of a certain type is shown to lead to a definite result for an exclusive cross section. A study of models suggests that diagrams exhibiting leading particle structure are important at least near the edge of phase space. pp, πp and γp scattering are discussed using such diagrams, and correspondence enables tests of dimensional counting arguments for large angle hadron-parton exclusive amplitudes to be made.