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1. Holonomy theorem for finite semigroups

   https://doi.org/10.1142/S0218196722500217  

   Rhodes, John; Schilling, Anne; Silva, Pedro V. (May 2022, International Journal of Algebra and Computation)

   In this paper, we provide a simple proof of the Holonomy theorem using a new Lyndon–Chiswell length function on the Karnofsky–Rhodes expansion of a semigroup. Unexpectedly, we have both a left and a right action on the Chiswell tree by elliptic maps. 

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2. Which cross fields can be quadrangulated?: global parameterization from prescribed holonomy signatures

   https://doi.org/10.1145/3528223.3530187  

   Shen, Hanxiao; Zhu, Leyi; Capouellez, Ryan; Panozzo, Daniele; Campen, Marcel; Zorin, Denis (July 2022, ACM Transactions on Graphics)

   We describe a method for the generation of seamless surface parametrizations with guaranteed local injectivity and full control over holonomy. Previous methods guarantee only one of the two. Local injectivity is required to enable these parametrizations' use in applications such as surface quadrangulation and spline construction. Holonomy control is crucial to enable guidance or prescription of the parametrization's isocurves based on directional information, in particular from cross-fields or feature curves, and more generally to constrain the parametrization topologically. To this end we investigate the relation between cross-field topology and seamless parametrization topology. Leveraging previous results on locally injective parametrization and combining them with insights on this relation in terms of holonomy, we propose an algorithm that meets these requirements. A key component relies on the insight that arbitrary surface cut graphs, as required for global parametrization, can be homeomorphically modified to assume almost any set of turning numbers with respect to a given target cross-field. 

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3. G2 manifolds with nodal singularities along circles

   https://doi.org/10.1007/s12220-019-002833-3  

   Chen, Gao (September 2019, The Journal of geometric analysis)

   This paper finds matching building blocks for the construction of a compact manifold with G2 holonomy and nodal singularities along circles using twisted connected sum method by solving the Calabi conjecture on certain asymptotically cylindrical manifolds with nodal singularities. However, by comparison to the untwisted connected sum case, it turns out that the obstruction space for the singular twisted connected sum construction is infinite dimensional. By analyzing the obstruction term, there are strong evidences that the obstruction may be resolved if a further gluing is performed in order to get a compact manifold with G2 holonomy and isolated conical singularities with link S3×S3. 

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4. Partition functions of Chern-Simons theory on handlebodies by radial quantization

   https://doi.org/10.1007/JHEP07(2021)194  

   Porrati, Massimo; Yu, Cedric (July 2021, Journal of High Energy Physics)

   null (Ed.)

   A bstract We use radial quantization to compute Chern-Simons partition functions on handlebodies of arbitrary genus. The partition function is given by a particular transition amplitude between two states which are defined on the Riemann surfaces that define the (singular) foliation of the handlebody. The final state is a coherent state while on the initial state the holonomy operator has zero eigenvalue. The latter choice encodes the constraint that the gauge fields must be regular everywhere inside the handlebody. By requiring that the only singularities of the gauge field inside the handlebody must be compatible with Wilson loop insertions, we find that the Wilson loop shifts the holonomy of the initial state. Together with an appropriate choice of normalization, this procedure selects a unique state in the Hilbert space obtained from a Kähler quantization of the theory on the constant-radius Riemann surfaces. Radial quantization allows us to find the partition functions of Abelian Chern-Simons theories for handlebodies of arbitrary genus. For non-Abelian compact gauge groups, we show that our method reproduces the known partition function at genus one. 

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5. An almost flat manifold with a cyclic or quaternionic holonomy group bounds

   Davis, James F; Fang, Fuquan (January 2016, Journal of differential geometry)

   A long-standing conjecture of Farrell and Zdravkovska and independently S.T. Yau states that every almost flat manifold is the boundary of a compact manifold. This paper gives a simple proof of this conjecture when the holonomy group is cyclic or quaternionic. The proof is based on the interaction between flat bundles and involutions. 

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