Non-trivial Fundamental Group Research Articles

Spaces of positive scalar curvature metrics on totally nonspin manifolds with spin boundary

In this article we study the space of positive scalar curvature metrics on totally nonspin manifolds with spin boundary. We prove that for such manifolds of certain dimensions, those spaces are not connected and have nontrivial fundamental group. Furthermore we show that a well-known propagation technique for detection results on spaces of positive scalar curvature metrics on spin manifolds ceases to work in the totally nonspin case.

The u-plane integral, mock modularity and enumerative geometry

We revisit the low-energy effective U(1) action of topologically twisted {mathcal {N}}=2 SYM theory with gauge group of rank one on a generic oriented smooth four-manifold X with nontrivial fundamental group. After including a specific new set of mathcal Q-exact operators to the known action, we express the integrand of the path integral of the low-energy U(1) theory as an anti-holomorphic derivative. This allows us to use the theory of mock modular forms and indefinite theta functions for the explicit evaluation of correlation functions of the theory, thus facilitating the computations compared to previously used methods. As an explicit check of our results, we compute the path integral for the product ruled surfaces X=Sigma _g times mathbb {CP}^1 for the reduction on either factor and compare the results with existing literature. In the case of reduction on the Riemann surface Sigma _g, via an equivalent topological A-model on mathbb {CP}^1, we will be able to express the generating function of genus zero Gromov–Witten invariants of the moduli space of flat rank one connections over Sigma _g in terms of an indefinite theta function, whence we would be able to make concrete numerical predictions of these enumerative invariants in terms of modular data, thereby allowing us to derive results in enumerative geometry from number theory.

Examples of Calabi-Yau threefolds with small Hodge numbers

We observe some suitable examples of Calabi-Yau threefolds for heterotic superstring compactifications. It is reasonable to seek CY threefolds with Euler characteristic equals ±6 because of generation’s number. Hosotani mechanism for violations of the gauge group by the Wilson loops requires such CY space has a non-trivial fundamental group. These spaces can be obtained by factoring the complete intersection Calabi-Yau spaces by the free action of some discrete group. Also we shortly discuss cases when discrete groups act with fixed point sets.

Direct Measurement of Topological Properties of an Exceptional Parabola

Non-Hermitian systems can produce branch singularities known as exceptional points (EPs). Different from singularities in Hermitian systems, the topological properties of an EP can involve either the winding of eigenvalues that produces a discriminant number (DN) or the eigenvector holonomy that generates a Berry phase. The multiplicity of topological invariants also makes non-Hermitian topology richer than its Hermitian counterpart. Here, we study a parabola-shaped trajectory formed by EPs with both theory and acoustic experiments. By obtaining both the DNs and Berry phases through the measurement of eigenvalues and eigenfunctions, we show that the EP trajectory endows the parameter space with a nontrivial fundamental group. Our findings not only shed light on exotic non-Hermitian topology but also provide a route for the experimental characterization of non-Hermitian topological invariants.

The diffeomorphism type of symplectic fillings

We show that simply connected contact manifolds that are subcritically Stein fillable have a unique symplectically aspherical filling up to diffeomorphism. Various extensions to manifolds with non-trivial fundamental group are discussed. The proof rests on homological restrictions on symplectic fillings derived from a degree-theoretic analysis of the evaluation map on a suitable moduli space of holomorphic spheres. Applications of this homological result include a proof that compositions of right-handed Dehn twists on Liouville domains are of infinite order in the symplectomorphism group. We also derive uniqueness results for subcritical Stein fillings up to homotopy equivalence and, under some topological assumptions on the contact manifold, up to diffeomorphism or symplectomorphism.

A complete description of the antipodal set of most symmetric spaces of compact type

It is known that the antipodal set of a Riemannian symmetric space of compact type $G/K$ consists of a union of $K$-orbits. We determine the dimensions of these $K$-orbits of most irreducible symmetric spaces of compact type. The symmetric spaces we are not going to deal with are those with restricted root system $\mathfrak{a}_r$ and a non-trivial fundamental group, which is not isomorphic to $\mathbb{Z}_2$ or $\mathbb{Z}_{r+1}$. For example, we show that the antipodal sets of the Lie groups $Spin(2r+1)\:\: r\geq 5$, $E_8$ and $G_2$ consist only of one orbit which is of dimension $2r$, 128 and 6, respectively; $SO(2r+1)$ has also an antipodal set of dimension $2r$; and the Grassmannian $Gr_{r,r+q}(\mathbb{R})$ has a $rq$-dimensional orbit as antipodal set if $r\geq 5$ and $r\neq q>0$.

Topology of electroweak vacua

In the Standard Model, the electroweak symmetry is broken by a complex, SU(2)-doublet Higgs field and the vacuum manifold SU(2)×U(1)/U(1) has the topology of a 3-sphere. We remark that there exist theoretical alternatives that are locally isomorphic, but in which the vacuum manifold is homeomorphic to an arbitrary non-trivial principal U(1)-bundle over a 2-sphere. These alternatives have non-trivial fundamental group and thus feature topologically-stable electroweak strings. An alternative based on the manifold RP3 (with fundamental group Z/2) allows custodial protection of gauge boson masses and their couplings to fermions and has an explicit realisation in the Minimal Composite Higgs Model, in the case of maximal electroweak symmetry breaking. We show that, in common with all alternatives to S3, such models have a problem with fermion masses.

Geometric U-folds in four dimensions

We describe a general construction of geometric U-folds compatible with a non-trivial extension of the global formulation of four-dimensional extended supergravity on a differentiable spin manifold. The topology of geometric U-folds depends on certain flat fiber bundles which encode how supergravity fields are globally glued together. We show that smooth non-trivial U-folds of this type can exist only in theories where both the scalar and space-time manifolds have non-trivial fundamental group and in addition the scalar map of the solution is homotopically non-trivial. Consistency with string theory requires smooth geometric U-folds to be glued using subgroups of the effective discrete U-duality group, implying that the fundamental group of the scalar manifold of such solutions must be a subgroup of the latter. We construct simple examples of geometric U-folds in a generalization of the axion–dilaton model of supergravity coupled to a single vector multiplet, whose scalar manifold is a generally non-compact Riemann surface of genus at least two endowed with its uniformizing metric. We also discuss the relation between geometric U-folds and a moduli space of flat connections defined on the scalar manifold, which involves certain character varieties not studied in the literature.

The Weinstein Conjecture for Connected Sums

We prove the Weinstein conjecture for non-trivial contact connected sums under either of two topological conditions: non-trivial fundamental group or torsion-free homology.

Measure continuous derivations on von Neumann algebras and applications to ${L^2}$-cohomology

We prove that norm continuous derivations from a von Neumann algebra into the algebra of operators affiliated with its tensor square are automatically continuous for both the strong operator topology and the measure topology. Furthermore, we prove that the first continuous L^2-Betti number scales quadratically when passing to corner algebras and derive an upper bound given by Shen's generator invariant. This, in turn, yields vanishing of the first continuous L^2-Betti number for II_1 factors with property (T), for finitely generated factors with non-trivial fundamental group and for factors with property Gamma.

Characterization algorithms for shift radix systems with finiteness property

For d ∈ ℕ and r ∈ ℝd, let τr : ℤd → ℤd, where τr(a) = (a2, …, ad, -⌊ra⌋) for a = (a1, …, ad), denote the (d-dimensional) shift radix system associated with r. τr is said to have the finiteness property if and only if all orbits of τr end up in (0, …, 0); the set of all corresponding r ∈ ℝd is denoted by [Formula: see text], whereas 𝒟d consists of those r ∈ ℝd for which all orbits are eventually periodic. [Formula: see text] has a very complicated structure even for d = 2. In the present paper, two algorithms are presented which allow the characterization of the intersection of [Formula: see text] and any closed convex hull of finitely many interior points of 𝒟d which is completely contained in the interior of 𝒟d. One of the algorithms is used to determine the structure of [Formula: see text] in a region considerably larger than previously possible, and to settle two questions on its topology: It is shown that [Formula: see text] is disconnected and that the largest connected component has non-trivial fundamental group. The other is the first algorithm characterizing [Formula: see text] in a given convex polyhedron which terminates for all inputs. Furthermore, several infinite families of "cutout polygons" are deduced settling the finiteness property for a chain of regions touching the boundary of 𝒟2.

Vanishing of the First Continuous L2-Cohomology for II1 Factors

We prove that the continuous version of the Connes-Shlyakhtenko first L-cohomology for II1 factors, as proposed by A. Thom in [Th06], always vanishes. In [CS03], A. Connes and D. Shlyakhtenko developed an L2-cohomology theory for finite von Neumann algebras M , and more generally for weakly dense ∗-subalgebras A ⊂ M of such von Neumann algebras. Then in [Th06], A. Thom provided an alternative, Hochschild-type characterization of the first such L2-cohomology of M as the quotient of the space of derivations δ : M → Aff(M ⊗Mop) by the space of inner derivations, where Aff(M ⊗Mop) denotes the ∗-algebra of operators affiliated with M ⊗Mop. Thom also proposed in [Th06] a continuous version of the first L2-cohomology, by considering the (smaller) space of derivations δ that are continuous from M with the operator norm to Aff(M ⊗Mop) with the topology of convergence in measure. He noted that in many cases (e.g., when M has a Cartan subalgebra, or when M is not prime), this cohomology vanishes, i.e. any continuous derivation of M into Aff(M ⊗Mop) is inner. Following up on this work, V. Alekseev and D. Kyed have shown in [AK11] that the first continuous L2-cohomology also vanishes whenM has property (T), whenM is finitely generated with nontrivial fundamental group, or when M has property Gamma. Recently, V. Alekseev proved in [Al13] that this is also the case for the free group factors L(Fn). In this article, we prove that in fact the first continuous L2-cohomology vanishes for all finite von Neumann algebras. The starting point of our proof is a key calculation in the proof of [Al13, Proposition 3.1], which provides a concrete sequence of elements yn in the II1 factor M = L(F3) of the free group F3 with generators a, b, c, that tends to 0 in operator norm, but has the property that if a derivation δ : M → Aff(M ⊗Mop) satisfies δ(ua) = ua ⊗ 1 and δ(ub) = δ(uc) = 0, then δ(yn) does not tend to 0 in measure. More precisely, the yn’s in [Al13] are scalar multiples of words wn in a, b, c with the property that δ(wn) is a larger and larger sum of free independent Haar unitaries. In the case of an arbitrary II1 factor M , we fix a hyperfinite II1 factor R ⊂ M with trivial relative commutant, and then use [Po92] to “simulate” (in distribution) L(F3) inside M , with a any fixed unitary in M and bm, cm Haar unitaries in R such that a, bm, cm are asymptotically free. If now δ is a continuous derivation on M , then by subtracting an inner derivation, we may assume δ vanishes on R, thus on bm, cm. If δ(a) 6= 0, and if we formally define yn’s via the same formula as Alekseev’s, with a, bm, cm in lieu of a, b, c, then a careful estimation of norms of yn and δ(yn), which uses results in [HL99], shows that one still has ‖yn‖ → 0, while δ(yn) 6→ 0 in measure. Mathematics Department, UCLA, CA 90095-1555 (United States), popa@math.ucla.edu Supported in part by NSF Grant DMS-1101718 KU Leuven, Department of Mathematics, Leuven (Belgium), stefaan.vaes@wis.kuleuven.be Supported by Research Programme G.0639.11 of the Research Foundation – Flanders (FWO) and KU Leuven BOF research grant OT/13/079.

Obstructions to classically simulating the quantum adiabatic algorithm

We consider the adiabatic quantum algorithm for systems with no sign problem, such as the transverse field Ising model, and analyze the equilibration time for quantum Monte Carlo (QMC) on these systems. We ask: if the spectral gap is only inverse polynomially small, will equilibration methods based on slowly changing the Hamiltonian parameters in the QMC simulation succeed in a polynomial time? We show that this is not true, by constructing counter-examples. In some examples, the space of configurations where the wavefunction has non-negligible amplitude has a nontrivial fundamental group, causing the space of trajectories in imaginary time to break into disconnected components with only negligible probability outside these components. For the simplest example we give with an abelian fundamental group, QMC does not equilibrate but still solves the optimization problem. More severe effects leading to failure to solve the optimization can occur when the fundamental group is a free group on two generators. Other examples where QMC fails have a trivial fundamental group, but still use ideas from topology relating group presentations to simplicial complexes. We define gadgets to realize these Hamiltonians as the effective low-energy dynamics of a transverse field Ising model. We present some analytic results on equilibration times which may be of some independent interest in the theory of equilibration of Markov chains. Conversely, we show that a small spectral gap implies slow equilibration at low temperature for some initial conditions and for a natural choice of local QMC updates.

Obstructions to classically simulating the quantum adiabatic algorithm

We consider the adiabatic quantum algorithm for systems with ``no sign problem", such as the transverse field Ising model, and analyze the equilibration time for quantum Monte Carlo (QMC) on these systems. We ask: if the spectral gap is only inverse polynomially small, will equilibration methods based on slowly changing the Hamiltonian parameters in the QMC simulation succeed in a polynomial time? We show that this is {\it not} true, by constructing counter-examples. In some examples, the space of configurations where the wavefunction has non-negligible amplitude has a nontrivial fundamental group, causing the space of trajectories in imaginary time to break into disconnected components with only negligible probability outside these components. For the simplest example we give with an abelian fundamental group, QMC does not equilibrate but still solves the optimization problem. More severe effects leading to failure to solve the optimization can occur when the fundamental group is a free group on two generators. Other examples where QMC fails have a {\it trivial} fundamental group, but still use ideas from topology relating group presentations to simplicial complexes. We define gadgets to realize these Hamiltonians as the effective low-energy dynamics of a transverse field Ising model. We present some analytic results on equilibration times which may be of some independent interest in the theory of equilibration of Markov chains. Conversely, we show that a small spectral gap implies slow equilibration at low temperature for some initial conditions and for a natural choice of local QMC updates.

Localization on three-manifolds

We consider supersymmetric gauge theories on Riemannian three-manifolds with the topology of a three-sphere. The three-manifold is always equipped with an almost contact structure and an associated Reeb vector field. We show that the partition function depends only on this vector field, giving an explicit expression in terms of the double sine function. In the large N limit our formula agrees with a recently discovered two-parameter family of dual supergravity solutions. We also explain how our results may be applied to prove vortex-antivortex factorization. Finally, we comment on the extension of our results to three-manifolds with non-trivial fundamental group.

GEOMETRY OF SPIN ANDSPINcSTRUCTURES IN THE M-THEORY PARTITION FUNCTION

We study the effects of having multiple Spin structures on the partition function of the spacetime fields in M-theory. This leads to a potential anomaly which appears in the eta invariants upon variation of the Spin structure. The main sources of such spaces are manifolds with nontrivial fundamental group, which are also important in realistic models. We extend the discussion to the Spinccase and find the phase of the partition function, and revisit the quantization condition for the C-field in this case. In type IIA string theory in 10 dimensions, the (mod 2) index of the Dirac operator is the obstruction to having a well-defined partition function. We geometrically characterize manifolds with and without such an anomaly and extend to the case of nontrivial fundamental group. The lift to KO-theory gives the α-invariant, which in general depends on the Spin structure. This reveals many interesting connections to positive scalar curvature manifolds and constructions related to the Gromov–Lawson–Rosenberg conjecture. In the 12-dimensional theory bounding M-theory, we study similar geometric questions, including choices of metrics and obtaining elements of K-theory in 10 dimensions by pushforward in K-theory on the disk fiber. We interpret the latter in terms of the families index theorem for Dirac operators on the M-theory circle and disk. This involves superconnections, eta forms, and infinite-dimensional bundles, and gives elements in Deligne cohomology in lower dimensions. We illustrate our discussion with many examples throughout.

Classification of compact homogeneous spaces with invariant G2-structures

In this note we classify all homogeneous spaces $G/H$ admitting a $G$-invariant $G_2$-structure, assuming that $G$ is a compact Lie group and $G$ acts effectively on $G/H$. They include a subclass of all homogeneous spaces $G/H$ with a $G$-invariant $\tilde G_2$-structure, where $G$ is a compact Lie group. There are many new examples with nontrivial fundamental group. We study a subclass of homogeneous spaces of high rigidity and low rigidity and show that they admit families of invariant coclosed $G_2$-structures (resp. $\tilde G_2$-structures).

Homotopical complexity of 2D billiard orbits

Traditionally, rotation numbers for toroidal billiard flows are defined as the limiting vectors of average displacements per time on trajectory segments. Naturally, these creatures live in the (commutative) vector space ℝn, if the toroidal billiard is given on the flatn-torus. The billiard trajectories, being curves, often getting very close to closed loops, quite naturally define elements of the fundamental group of the billiard table. The simplest non-trivial fundamental group obtained this way belongs to the classical Sinai billiard, i.e. the billiard flow on the 2-torus with a single, strictly convex obstacle (with smooth boundary) removed. This fundamental group is known to be the groupF2freely generated by two elements, which is a heavily noncommutative, hyperbolic group in Gromov’s sense. We define the homotopical rotation number and the homotopical rotation set for this model, and provide lower and upper estimates for the latter one, along with checking the validity of classically expected properties, like the density (in the homotopical rotation set) of the homotopical rotation numbers of periodic orbits.The natural habitat for these objects is the infinite cone erected upon the Cantor set Ends (F2) of all ŋds" of the hyperbolic groupF2. An element of Ends (F2) describes the direction in (the Cayley graph of) the groupF2in which the considered trajectory escapes to infinity, whereas the height functiont(t≧ 0) of the cone gives us the average speed at which this escape takes place.The main results of this paper claim that the orbits can only escape to infinity at a speed not exceeding\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\sqrt 2 $ \end{document}, and any directione∈ Ends (F2) for the escape is feasible with any prescribed speeds, 0 ≦s≦\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\sqrt 2 $ \end{document}/2. This means that the radial upper and lower bounds for the rotation setRare actually pretty close to each other.

The topology of homophase misorientation spaces

Homophase misorientation spaces are investigated with a focus on the effect of symmetry operations on their topology and their minimum embedding dimensions in Euclidean space. Whereas the topology of rotation space is well established and requires a minimum of five variables for a one-to-one and continuous mapping, the spaces of orientations and misorientations are quotient spaces of the rotation space and are obtained by applying various equivalence relations. The equivalence relations for orientation spaces only involve the rotational symmetries of the underlying crystals. These spaces are classified under the three-dimensional manifolds called the spherical 3-manifolds, which have a non-trivial fundamental group, are not simply connected spaces, and do not embed in three-dimensional Euclidean space. In the case of homophase misorientation spaces, however, in addition to rotational symmetry operations there is a further ‘grain exchange symmetry’, which is shown to simplify the topology considerably. In some important cases this symmetry also reduces the number of Euclidean dimensions required to embed these misorientation spaces. The homophase misorientation spaces for the dihedral point groups D 2(222), D 4(422) and D 6(622), the tetrahedral point group T(23), and the octahedral group O(432) are all found to be embeddable in only three dimensions, two dimensions less than required for rotations. Hence, these misorientation systems can be represented using three variables in a one-to-one and continuous manner.

Vogan duality for nonlinear type B

Let G = S p i n [ 4 n + 1 ] \mathbb {G}=\mathrm {Spin}[4n+1] be the connected, simply connected complex Lie group of type B 2 n B_{2n} and let G = S p i n ( p , q ) G=\mathrm {Spin}(p,q) ( p + q = 4 n + 1 ) (p+q=4n+1) denote a (connected) real form. If q ∉ { 0 , 1 } q \notin \left \{0,1\right \} , G G has a nontrivial fundamental group and we denote the corresponding nonalgebraic double cover by G ~ = S p i n ~ ( p , q ) \tilde {G}=\widetilde {\mathrm {Spin}}(p,q) . The main purpose of this paper is to describe a symmetry in the set of genuine parameters for the various G ~ \tilde {G} at certain half-integral infinitesimal characters. This symmetry is used to establish a duality of the corresponding generalized Hecke modules and ultimately results in a character multiplicity duality for the genuine characters of G ~ \tilde {G} .