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We prove that the Hilbert scheme of k points on C 2 ( Hilb k [ C 2 ] ) is self-dual under three-dimensional mirror symmetry using methods of geometry and integrability. Namely, we demonstrate that the corresponding quantum equivariant K-theory is invariant upon interchanging its Kähler and equivariant parameters as well as inverting the weight of the C ħ × -action. First, we find a two-parameter family X k , l of self-mirror quiver varieties of type A and study their quantum K-theory algebras. The desired quantum K-theory of Hilb k [ C 2 ] is obtained via direct limit l ⟶ ∞ and by imposing certain periodic boundary conditions on the quiver data. Throughout the proof, we employ the quantum/classical (q-Langlands) correspondence between XXZ Bethe Ansatz equations and spaces of twisted ħ -opers. In the end, we propose the 3d mirror dual for the moduli spaces of torsion-free rank-N sheaves on P 2 with the help of a different (three-parametric) family of type A quiver varieties with known mirror dual. [ABSTRACT FROM AUTHOR]

Using the Baum–Connes conjecture with coefficients, we develop a K-theory formula for reduced C*-algebras of strongly 0-E-unitary inverse semigroups, or equivalently, for a class of reduced partial crossed products. This generalizes and gives a new proof of previous K-theory results of Cuntz, Echterhoff and the author. Our K-theory formula applies to a rich class of C*-algebras which are generated by partial isometries. For instance, as new applications which could not be treated using previous results, we discuss semigroup C*-algebras of Artin monoids, Baumslag-Solitar monoids and one-relator monoids, as well as C*-algebras generated by right regular representations of semigroups of number-theoretic origin, and C*-algebras attached to tilings. [ABSTRACT FROM AUTHOR]

For any n , r ∈ N , we construct an algebra P n r via generators and quadratic relations, and show that it deforms the W–algebra of gl nr with respect to a nilpotent with Jordan block decomposition r + ... + r . We introduce a surjective morphism from half of the quantum toroidal algebra of gl n to P n r , and show that the action of the quantum toroidal algebra on the K–theory groups of the moduli spaces of parabolic sheaves factors through P n r . [ABSTRACT FROM AUTHOR]

A special case of the geometric Langlands correspondence is given by the relationship between solutions of the Bethe ansatz equations for the Gaudin model and opers-connections on the projective line with extra structure. In this paper, we describe a deformation of this correspondence for SL (N) . We introduce a difference equation version of opers called q-opers and prove a q-Langlands correspondence between nondegenerate solutions of the Bethe ansatz equations for the XXZ model and nondegenerate twisted q-opers with regular singularities on the projective line. We show that the quantum/classical duality between the XXZ spin chain and the trigonometric Ruijsenaars–Schneider model may be viewed as a special case of the q-Langlands correspondence. We also describe an application of q-opers to the equivariant quantum K-theory of the cotangent bundles to partial flag varieties. [ABSTRACT FROM AUTHOR]

We study quantum geometry of Nakajima quiver varieties of two different types—framed A-type quivers and ADHM quivers. While these spaces look completely different we find a surprising connection between equivariant K-theories thereof with a nontrivial match between their equivariant parameters. In particular, we demonstrate that quantum equivariant K-theory of A n quiver varieties in a certain n → ∞ limit reproduces equivariant K-theory of the Hilbert scheme of points on C 2 . We analyze the correspondence from the point of view of enumerative geometry, representation theory and integrable systems. We also propose a conjecture which relates spectra of quantum multiplication operators in K-theory of the ADHM moduli spaces with the solution of the elliptic Ruijsenaars–Schneider model. [ABSTRACT FROM AUTHOR]

We analyse the problem of assigning sign choices to O-planes in orientifolds of type II string theory. We show that there exists a sequence of invariant p-gerbes with p ≥ - 1 , which give rise to sign choices and are related by coboundary maps. We prove that the sign choice homomorphisms stabilise with the dimension of the orientifold and we derive topological constraints on the possible sign configurations. Concrete calculations for spherical and toroidal orientifolds are carried out, and in particular we exhibit a four-dimensional orientifold where not every sign choice is geometrically attainable. We elucidate how the K-theory groups associated with invariant p-gerbes for p = - 1 , 0 , 1 interact with the coboundary maps. This allows us to interpret a notion of K-theory due to Gao and Hori as a special case of twisted KR -theory, which consequently implies the homotopy invariance and Fredholm module description of their construction. [ABSTRACT FROM AUTHOR]

In Maulik and Thomas (in preparation) the Vafa–Witten theory of complex projective surfaces is lifted to oriented C ∗ -equivariant cohomology theories. Here we study the K-theoretic refinement. It gives rational functions in t 1 / 2 invariant under t 1 / 2 ↔ t - 1 / 2 which specialise to numerical Vafa–Witten invariants at t = 1 . On the "instanton branch" the invariants give the virtual χ - t -genus refinement of Göttsche–Kool, extended to allow for strictly semistable sheaves. Applying modularity to their calculations gives predictions for the contribution of the "monopole branch". We calculate some cases and find perfect agreement. We also do calculations on K3 surfaces, finding Jacobi forms refining the usual modular forms, proving a conjecture of Göttsche–Kool. We determine the K-theoretic virtual classes of degeneracy loci using Eagon–Northcott complexes, and show they calculate refined Vafa–Witten invariants. Using this Laarakker (Monopole contributions to refined Vafa–Witten invariants. arXiv:1810.00385) proves universality results for the invariants. [ABSTRACT FROM AUTHOR]

We consider the hypothesis that the C-field 4-flux and 7-flux forms in M-theory are in the image of the non-abelian Chern character map from the non-abelian generalized cohomology theory called J-twisted Cohomotopy theory. We prove for M2-brane backgrounds in M-theory on 8-manifolds that such charge quantization of the C-field in Cohomotopy theory implies a list of expected anomaly cancellation conditions, including: shifted C-field flux quantization and C-field tadpole cancellation, but also the DMW anomaly cancellation and the C-field's integral equation of motion. [ABSTRACT FROM AUTHOR]

A key open problem in M-theory is the identification of the degrees of freedom that are expected to be hidden at ADE-singularities in spacetime. Comparison with the classification of D-branes by K-theory suggests that the answer must come from the right choice of generalized cohomology theory for M-branes. Here we show that real equivariant Cohomotopy on superspaces is a consistent such choice, at least rationally. After explaining this new approach, we demonstrate how to use Elmendorf's Theorem in equivariant homotopy theory to reveal ADE-singularities as part of the data of equivariant S 4 -valued super-cocycles on 11d super-spacetime. We classify these super-cocycles and find a detailed black brane scan that enhances the entries of the old brane scan to cascades of fundamental brane super-cocycles on strata of intersecting black M-brane species. We find that on each singular stratum the black brane's instanton contribution, namely its super Nambu–Goto/Green–Schwarz action, appears as the homotopy datum associated to the morphisms in the orbit category. [ABSTRACT FROM AUTHOR]

A key open problem in M-theory is to explain the mechanism of "gauge enhancement" through which M-branes exhibit the nonabelian gauge degrees of freedom seen perturbatively in the limit of 10d string theory. In fact, since only the twisted K-theory classes represented by nonabelian Chan–Paton gauge fields on D-branes have an invariant meaning, the problem is really the understanding the M-theory lift of the classification of D-brane charges by twisted K-theory. Here we show that this problem has a solution by universal constructions in rational super homotopy theory. We recall how double dimensional reduction of super M-brane charges is described by the cyclification adjunction applied to the 4-sphere, and how M-theory degrees of freedom hidden at ADE singularities are induced by the suspended Hopf action on the 4-sphere. Combining these, we demonstrate that, in the approximation of rational homotopy theory, gauge enhancement in M-theory is exhibited by lifting against the fiberwise stabilization of the unit of this cyclification adjunction on the A-type orbispace of the 4-sphere. This explains how the fundamental D6 and D8 brane cocycles can be lifted from twisted K-theory to a cohomology theory for M-brane charge, at least rationally. [ABSTRACT FROM AUTHOR]

We consider cycles for graded C ∗ , r -algebras (Real C ∗ -algebras) which are compatible with the ∗ -structure and the real structure. Their characters are cyclic cocycles. We define a Connes type pairing between such characters and elements of the van Daele K-groups of the C ∗ , r -algebra and its real subalgebra. This pairing vanishes on elements of finite order. We define a second type of pairing between characters and K-group elements which is derived from a unital inclusion of C ∗ -algebras. It is potentially non-trivial on elements of order two and torsion valued. Such torsion valued pairings yield topological invariants for insulators. The two-dimensional Kane–Mele and the three-dimensional Fu–Kane–Mele strong invariant are special cases of torsion valued pairings. We compute the pairings for a simple class of periodic models and establish structural results for two dimensional aperiodic models with odd time reversal invariance. [ABSTRACT FROM AUTHOR]

We propose the Roe C∗-algebra from coarse geometry as a model for topological phases of disordered materials. We explain the robustness of this C∗-algebra and formulate the bulk-edge correspondence in this framework. We describe the map from the K-theory of the group C∗-algebra of Zd to the K-theory of the Roe C∗-algebra, both for real and complex K-theory. [ABSTRACT FROM AUTHOR]

We study the K0-group of the commutant modulo a normed ideal of an n-tuple of commuting Hermitian operators in some of the simplest cases. In case n = 1, the results, under some technical conditions are rather complete and show the key role of the absolutely continuous part when the ideal is the trace-class. For a commuting n-tuple, n≥3 and the Lorentz (n,1) ideal, we show under an absolute continuity assumption that the commutant determines a canonical direct summand in K0. Also, certain properties involving the compact ideal, established assuming quasicentral approximate units mod the normed ideal, have weaker versions which hold assuming only finiteness of the obstruction to quasicentral approximate units. [ABSTRACT FROM AUTHOR]

We consider periodic Hamiltonians on a three-dimensional (3-D) lattice with a spectral gap not only on the bulk but also on two edges at the common Fermi level. By using K-theory applied for the quarter-plane Toeplitz extension, two topological invariants are defined. One is defined for the gapped bulk and edge Hamiltonians, and the non-triviality of the other means that the corner Hamiltonian is gapless. We prove a correspondence between these two invariants. Such gapped Hamiltonians can be constructed from Hamiltonians of 2-D type A and 1-D type AIII topological insulators, and its corner topological invariant is the product of topological invariants of these two phases. [ABSTRACT FROM AUTHOR]

We generalize to quantum weighted projective spaces in any dimension previous results of us on K-theory and K-homology of quantum projective spaces 'tout court'. For a class of such spaces, we explicitly construct families of Fredholm modules, both bounded and unbounded (that is, spectral triples), and prove that they are linearly independent in the K-homology of the corresponding C-algebra. We also show that the quantum weighted projective spaces are base spaces of quantum principal circle bundles whose total spaces are quantum lens spaces. We construct finitely generated projective modules associated with the principal bundles and pair them with the Fredholm modules, thus proving their non-triviality. [ABSTRACT FROM AUTHOR]

Let Ω be a tiling space and let G be the maximal group of rotations which fixes Ω. Then the cohomology of Ω and Ω/ G are both invariants which give useful geometric information about the tilings in Ω. The noncommutative analog of the cohomology of Ω is the K-theory of a C*-algebra associated to Ω, and for translationally finite tilings of dimension 2 or less, the K-theory is isomorphic to the direct sum of cohomology groups. In this paper we give a prescription for calculating the noncommutative analog of the cohomology of Ω/ G, that is, the K-theory of the crossed product of the tiling C*-algebra by G. We also provide a table with some calculated K-groups for many common examples, including the Penrose and pinwheel tilings. [ABSTRACT FROM AUTHOR]

The behavior of fermionic systems depends on the geometry of the system and the symmetry class of the Hamiltonian and observables. Almost commuting matrices arise from band-projected position observables in such systems. One expects the mathematical behavior of almost commuting Hermitian matrices to depend on two factors. One factor will be the approximate polynomial relations satisfied by the matrices. The other factor is what algebra the matrices are in, either $${{\bf M}_n(\mathbb{A})}$$ for $${\mathbb{A} = \mathbb{R}}$$ , $${\mathbb{A} = \mathbb{C}}$$ or $${\mathbb{A} = \mathbb{H}}$$ , the algebra of quaternions. There are potential obstructions keeping k-tuples of almost commuting operators from being close to a commuting k-tuple.We consider two-dimensional geometries and so this obstruction lives in $${KO_{-2}(\mathbb{A})}$$ . This obstruction corresponds to either the Chern number or spin Chern number in physics. We show that if this obstruction is the trivial element in K-theory then the approximation by commuting matrices is possible. [ABSTRACT FROM AUTHOR]

We study the representation theory of a conformal net $${\mathcal{A}}$$ on S from a K-theoretical point of view using its universal C*-algebra $${C^*(\mathcal{A})}$$. We prove that if $${\mathcal{A}}$$ satisfies the split property then, for every representation π of $${\mathcal{A}}$$ with finite statistical dimension, $${\pi(C^*(\mathcal{A}))}$$ is weakly closed and hence a finite direct sum of type I factors. We define the more manageable locally normal universal C*-algebra $${C_{\rm ln}^*(\mathcal{A})}$$ as the quotient of $${C^*(\mathcal{A})}$$ by its largest ideal vanishing in all locally normal representations and we investigate its structure. In particular, if $${\mathcal{A}}$$ is completely rational with n sectors, then $${C_{\rm ln}^*(\mathcal{A})}$$ is a direct sum of n type I factors. Its ideal $${\mathfrak{K}_\mathcal{A}}$$ of compact operators has nontrivial K-theory, and we prove that the DHR endomorphisms of $${C^*(\mathcal{A})}$$ with finite statistical dimension act on $${\mathfrak{K}_\mathcal{A}}$$, giving rise to an action of the fusion semiring of DHR sectors on $${K_0(\mathfrak{K}_\mathcal{A})}$$. Moreover, we show that this action corresponds to the regular representation of the associated fusion algebra. [ABSTRACT FROM AUTHOR]

We study the structure of abelian extensions of the group L G of q-differentiable loops (in the Sobolev sense), generalizing from the case of the central extension of the smooth loop group. This is motivated by the aim of understanding the problems with current algebras in higher dimensions. Highest weight modules are constructed for the Lie algebra. The construction is extended to the current algebra of the supersymmetric Wess-Zumino-Witten model. An application to the twisted K-theory on G is discussed. [ABSTRACT FROM AUTHOR]

We give a finite-dimensional and geometric construction of a Chern character for twisted K-theory, introducing a notion of connection on a twisted vectorial bundle which can be considered as a finite-dimensional approximation of a twisted family of Fredholm operators. Our construction is applicable to the case of any classes giving the twisting, and agrees with the Chern character of bundle gerbe modules in the case of torsion classes. [ABSTRACT FROM AUTHOR]

We present several results on the geometry of the quantum projective plane. They include: explicit generators for the K-theory and the K-homology; a real calculus with a Hodge star operator; anti-selfdual connections on line bundles with explicit computation of the corresponding ‘classical’ characteristic classes (via Fredholm modules); complete diagonalization of gauged Laplacians on these line bundles; ‘quantum’ characteristic classes via equivariant K-theory and q-indices. [ABSTRACT FROM AUTHOR]

We study D-branes and Ramond-Ramond fields on global orbifolds of Type II string theory with vanishing H-flux using methods of equivariant K-theory and K-homology. We illustrate how Bredon equivariant cohomology naturally realizes stringy orbifold cohomology. We emphasize its role as the correct cohomological tool which captures known features of the low-energy effective field theory, and which provides new consistency conditions for fractional D-branes and Ramond-Ramond fields on orbifolds. We use an equivariant Chern character from equivariant K-theory to Bredon cohomology to define new Ramond-Ramond couplings of D-branes which generalize previous examples. We propose a definition for groups of differential characters associated to equivariant K-theory. We derive a Dirac quantization rule for Ramond-Ramond fluxes, and study flat Ramond-Ramond potentials on orbifolds. [ABSTRACT FROM AUTHOR]

We provide a finite-dimensional model of the twisted K-group twisted by any degree three integral cohomology class of a CW complex. One key to the model is Furuta’s generalized vector bundle, and the other is a finite-dimensional approximation of Fredholm operators. [ABSTRACT FROM AUTHOR]

We investigate the C*-algebras associated to aperiodic structures called model sets obtained by the cut-and-project method. These C*-algebras are Morita equivalent to crossed product C*-algebras obtained from dynamics on a disconnected version of the internal space. This construction may be made from more general data, which we call a hyperplane system. From a hyperplane system, others may be constructed by a process of reduction and we show how the C*-algebras involved are related to each other. In particular, there are natural elements in the Kasparov KK-groups for the C*-algebra of a hyperplane system and that of its reduction. The induced map on K-theory fits in a six-term exact sequence. This provides a new method of the computation of the K-theory of such C*-algebras which is done completely in the setting of non-commutative geometry. [ABSTRACT FROM AUTHOR]

We prove that the C*-algebra of a minimal diffeomorphism satisfies Blackadar’s Fundamental Comparability Property for positive elements. This leads to the classification, in terms of K-theory and traces, of the isomorphism classes of countably generated Hilbert modules over such algebras, and to a similar classification for the closures of unitary orbits of self-adjoint elements. We also obtain a structure theorem for the Cuntz semigroup in this setting, and prove a conjecture of Blackadar and Handelman: the lower semicontinuous dimension functions are weakly dense in the space of all dimension functions. These results continue to hold in the broader setting of unital simple ASH algebras with slow dimension growth and stable rank one. Our main tool is a sharp bound on the radius of comparison of a recursive subhomogeneous C*-algebra. This is also used to construct uncountably many non-Morita-equivalent simple separable amenable C*-algebras with the same K-theory and tracial state space, providing a C*-algebraic analogue of McDuff’s uncountable family of II1 factors. We prove in passing that the range of the radius of comparison is exhausted by simple C*-algebras. [ABSTRACT FROM AUTHOR]

We prove that all unital separable continuous fields of C*-algebras over [0,1] with fibers isomorphic to the Cuntz algebra $${\mathcal{O}}_n \, (2 \leq n \leq \infty)$$ are trivial. More generally, we show that if A is a separable, unital or stable, continuous field over [0,1] of Kirchberg C*-algebras satisfying the UCT and having finitely generated K-theory groups, then A is isomorphic to a trivial field if and only if the associated K-theory presheaf is trivial. For fixed $$d\in \{0,1\}$$ we also show that, under the additional assumption that the fibers have torsion free K d -group and trivial K d+1-group, the K d -sheaf is a complete invariant for separable stable continuous fields of Kirchberg algebras. [ABSTRACT FROM AUTHOR]

The infinite matrix ‘Schwartz’ group G −∞ is a classifying group for odd K-theory and carries Chern classes in each odd dimension, generating the cohomology. These classes are closely related to the Fredholm determinant on G −∞. We show that while the higher (even, Schwartz) loop groups of G −∞, again classifying for odd K-theory, do not carry multiplicative determinants generating the first Chern class, ‘dressed’ extensions, corresponding to a star product, do carry such functions. We use these to discuss Bott periodicity for the determinant bundle and the eta invariant. In so doing we relate two distinct extensions of the eta invariant to self-adjoint elliptic operators and to elliptic invertible suspended families and show that the corresponding τ invariant is a determinant in this sense. [ABSTRACT FROM AUTHOR]

The construction of twisted K-theory classes on a compact Lie group is reviewed using the supersymmetric Wess-Zumino-Witten model on a cylinder. The Quillen superconnection is introduced for a family of supercharges parametrized by a compact Lie group and the Chern character is explicitly computed in the case of SU(2). For large euclidean time, the character form is localized on a D-brane. [ABSTRACT FROM AUTHOR]

In the ordinary quantum Maxwell theory of a free electromagnetic field, formulated on a curved 3-manifold, we observe that magnetic and electric fluxes cannot be simultaneously measured. This uncertainty principle reflects torsion: fluxes modulo torsion can be simultaneously measured. We also develop the Hamilton theory of self-dual fields, noting that they are quantized by Pontrjagin self-dual cohomology theories and that the quantum Hilbert space is $${\mathbb{Z}/2\mathbb{Z}}$$ -graded, so typically contains both bosonic and fermionic states. Significantly, these ideas apply to the Ramond-Ramond field in string theory, showing that its K-theory class cannot be measured. [ABSTRACT FROM AUTHOR]

We construct a quantum version of the SU(2) Hopf bundle S 7→ S 4. The quantum sphere S 7 q arises from the symplectic group Sp q (2) and a quantum 4-sphere S 4 q is obtained via a suitable self-adjoint idempotent p whose entries generate the algebra A(S 4 q ) of polynomial functions over it. This projection determines a deformation of an (anti-)instanton bundle over the classical sphere S 4. We compute the fundamental K-homology class of S 4 q and pair it with the class of p in the K-theory getting the value −1 for the topological charge. There is a right coaction of SU q (2) on S 7 q such that the algebra A(S 7 q ) is a non-trivial quantum principal bundle over A(S 4 q ) with structure quantum group A( SU q (2)). [ABSTRACT FROM AUTHOR]

The boundary map in K-theory arising from the Wiener-Hopf extension of a crossed product algebra with is the Connes-Thom isomorphism. In this article the Wiener Hopf extension is combined with the Heisenberg group algebra to provide an elementary construction of a corresponding map on higher traces (and cyclic cohomology). It then follows directly from a non-commutative Stokes theorem that this map is dual w.r.t. Connes’ pairing of cyclic cohomology with K-theory. As an application, we prove equality of quantized bulk and edge conductivities for the integer quantum Hall effect described by continuous magnetic Schrödinger operators. [ABSTRACT FROM AUTHOR]

In this paper we construct an explicit geometric model for the group of gerbes over an orbifold X. We show how from its curvature we can obtain its characteristic class in H3(X) via Chern-Weil theory. For an arbitrary gerbe , a twisting Korb(X) of the orbifold K-theory of X is constructed, and shown to generalize previous twisting by Rosenberg [28], Witten [35], Atiyah-Segal [2] and Bowknegt et. al. [4] in the smooth case and by Adem-Ruan [1] for discrete torsion on an orbifold. [ABSTRACT FROM AUTHOR]

The E-theory defined by Connes and Higson provides a realization of K-homology, the generalized homology theory dual to K-theory, based on the notion of asymptotic homomorphisms. With this realization it becomes possible to associate a K-homology element to a quantization scheme. In this article we associate an asymptotic homomorphism and K-homology element to the Berezin quantization of a bounded symmetric domain. Further, we identify this element with the element of K-homology defined by the Dolbeault operator of the domain. [ABSTRACT FROM AUTHOR]

: We use equivariant methods to define and study the orbifold K-theory of an orbifold X. Adapting techniques from equivariant K-theory, we construct a Chern character and exhibit a multiplicative decomposition for K*orb(X)⊗ℚ, in particular showing that it is additively isomorphic to the orbifold cohomology of X. A number of examples are provided. We then use the theory of projective representations to define the notion of twisted orbifold K–theory in the presence of discrete torsion. An explicit expression for this is obtained in the case of a global quotient. [ABSTRACT FROM AUTHOR]

: It was argued in [25, 5] that in the presence of a nontrivial B-field, D-brane charges in type IIB string theories are classified by twisted K-theory. In [4], it was proved that twisted K-theory is canonically isomorphic to bundle gerbe K-theory, whose elements are ordinary Hilbert bundles on a principal projective unitary bundle, with an action of the bundle gerbe determined by the principal projective unitary bundle. The principal projective unitary bundle is in turn determined by the twist. This paper studies in detail the Chern-Weil representative of the Chern character of bundle gerbe K-theory that was introduced in [4], extending the construction to the equivariant and the holomorphic cases. Included is a discussion of interesting examples. [ABSTRACT FROM AUTHOR]

For a space X acted on by a finite group Γ, the product space Xn affords a natural action of the wreath product Γn = Γn [This symbol cannot be presented in ASCII format] Sn. The direct sum of equivariant K-groups ⊗n≥0 KΓn(Xn)⊗C were shown earlier by the author to carry several interesting algebraic structures. In this paper we study the K-groups K&Htilde;Γn (Xn) of Γn-equivariant Clifford supermodules on Xn. We show that FΓ- (X) = ⊗n≥0 K&Htilde;Γn (Xn)⊗C is a Hopf algebra and it is isomorphic to the Fock space of a twisted Heisenberg algebra. Twisted vertex operators make a natural appearance. The algebraic structures on FΓ- (X) , when Γ is trivial and X is a point, specialize to those on a ring of symmetric functions with the Schur Q-functions as a linear basis. As a by-product, we present a novel construction of K-theory operations using the spin representations of the hyperoctahedral groups. [ABSTRACT FROM AUTHOR]