On topology of manifolds admitting gradient-like calscades with surface dynamics and on growth of the number of non-compact heteroclinic curves

We consider a class GSD(M3) of gradient-like diffeomorphisms with surface dynamics given on closed oriented manifold M3 of dimension three. Earlier it was proved that manifolds admitting such diffeomorphisms are mapping tori under closed orientable surface of genus g, and the number of non-compact heteroclinic curves of such diffeomorphisms is not less than 12g. In this paper, we determine a class of diffeomorphisms GSDR(M3)⊂GSD(M3) that have the minimum number of heteroclinic curves for a given number of periodic points, and prove that the supporting manifold of such diffeomorphisms is a Seifert manifold. The separatrices of periodic points of diffeomorphisms from the class GSDR(M3) have regular asymptotic behavior, in particular, their closures are locally flat. We provide sufficient conditions (independent on dynamics) for mapping torus to be Seifert. At the same time, the paper establishes that for any fixed g geq1, fixed number of periodic points, and any integer n≥12g, there exists a manifold M3 and a diffeomorphism f∈GSD(M3) having exactly n non-compact heteroclinic curves.

Topological Entropy of Pseudo-Anosov Maps from a Typical Thurston’s Construction

In this paper, we develop a way to extract information about a random walk associated with a typical Thurston’s construction. We first observe that a typical Thurston’s construction entails a free group of rank 2. We also present a proof of the spectral theorem for random walks associated with Thurston’s construction that have finite 2nd moment with respect to the Teichmüller metric. Its general case was remarked by Dahmani and Horbez. Finally, under a hypothesis not involving moment conditions, we prove that random walks eventually become pseudo-Anosov. As an application, we first discuss a random analogy of Kojima and McShane’s estimation of the hyperbolic volume of a mapping torus with pseudo-Anosov monodromy. As another application, we discuss non-probabilistic estimations of stretch factors from Thurston’s construction and the powers for Salem numbers to become the stretch factors of pseudo-Anosovs from Thurston’s construction.

S-folds and holographic RG flows on the D3-brane

Type IIB S-folds of the form AdS4 × S1 × S5 are conjectured to correspond to new strongly coupled three-dimensional CFT’s on a localised interface of SYM4. In this work we construct holographic RG flows on the D3-brane that generically connect anisotropic deformations of SYM4 in the UV to various S-fold CFT’s in the IR with different amounts of supersymmetry and flavour symmetries. Examples of holographic RG flows between S-fold CFT’s are also presented. Lastly a geometric interpretation of axion deformations is provided in terms of monodromies on the internal S5 when moving around the S1. Special attention is paid to the monodromy-induced patterns of symmetry breaking as classified by the mapping torus Th(S5).

Omega vs. pi, and 6d anomaly cancellation

In this note we review the role of homotopy groups in determining non-perturbative (henceforth ‘global’) gauge anomalies, in light of recent progress understanding global anomalies using bordism. We explain why non-vanishing of πd(G) is neither a necessary nor a sufficient condition for there being a possible global anomaly in a d-dimensional chiral gauge theory with gauge group G. To showcase the failure of sufficiency, we revisit ‘global anomalies’ that have been previously studied in 6d gauge theories with G = SU(2), SU(3), or G2. Even though π6(G) ≠ 0, the bordism groups $$ {\Omega}_7^{\mathrm{Spin}}(BG) $$ Ω 7 Spin BG vanish in all three cases, implying there are no global anomalies. In the case of G = SU(2) we carefully scrutinize the role of homotopy, and explain why any 7-dimensional mapping torus must be trivial from the bordism perspective. In all these 6d examples, the conditions previously thought to be necessary for global anomaly cancellation are in fact necessary conditions for the local anomalies to vanish.

Double-Janus linear sigma models and generalized reciprocity for Gauss sums

We study the supersymmetric partition function of a 2d linear σ-model whose target space is a torus with a complex structure that varies along one worldsheet direction and a Kähler modulus that varies along the other. This setup is inspired by the dimensional reduction of a Janus configuration of 4d $$ \mathcal{N} $$ N = 4 U(1) Super-Yang-Mills theory compactified on a mapping torus (T2 fibered over S1) times a circle with an SL(2, ℤ) duality wall inserted on S1, but our setup has minimal supersymmetry. The partition function depends on two independent elements of SL(2, ℤ), one describing the duality twist, and the other describing the geometry of the mapping torus. It is topological and can be written as a multivariate quadratic Gauss sum. By calculating the partition function in two different ways, we obtain identities relating different quadratic Gauss sums, generalizing the Landsberg-Schaar relation. These identities are a subset of a collection of identities discovered by F. Deloup. Each identity contains a phase which is an eighth root of unity, and we show how it arises as a Berry phase in the supersymmetric Janus-like configuration. Supersymmetry requires the complex structure to vary along a semicircle in the upper half-plane, as shown by Gaiotto and Witten in a related context, and that semicircle plays an important role in reproducing the correct Berry phase.

COHERENCE, LOCAL INDICABILITY AND NONPOSITIVE IMMERSIONS

We examine 2-complexes $X$ with the property that for any compact connected $Y$ , and immersion $Y\rightarrow X$ , either $\unicode[STIX]{x1D712}(Y)\leqslant 0$ or $\unicode[STIX]{x1D70B}_{1}Y=1$ . The mapping torus of an endomorphism of a free group has this property. Every irreducible 3-manifold with boundary has a spine with this property. We show that the fundamental group of any 2-complex with this property is locally indicable. We outline evidence supporting the conjecture that this property implies coherence. We connect the property to asphericity. Finally, we prove coherence for 2-complexes with a stricter form of this property. As a corollary, every one-relator group with torsion is coherent.