Equivariant map

About: Equivariant map is a research topic. Over the lifetime, 9205 publications have been published within this topic receiving 137115 citations.

Equivariant stable homotopy and Sullivan's conjecture

Abstract: Let G be a p-group, and let X be a G-complex Let EG denote a contractible space on which G acts freely By the "homotopy fixed point set" of X, we mean the fixed point set F(EG, X) G, where F(EG, X) denotes the function space of all maps EG ~ X, equipped with a G-action by 9f = 9f9-1 If we let * denote the one point space with trivial G-action, we may also consider the G-space F ( , ,X ) ; it is canonically G-homeomorphic to X The G-map E G ~ , induces a map ~/: X G ~F( , , X) G In [19], D Sullivan proposed the following

On the Coulomb and Higgs branch formulae for multi-centered black holes and quiver invariants

Abstract: In previous work we have shown that the equivariant index of multi-centered N=2 black holes localizes on collinear configurations along a fixed axis. Here we provide a general algorithm for enumerating such collinear configurations and computing their contribution to the index. We apply this machinery to the case of black holes described by quiver quantum mechanics, and give a systematic prescription -- the Coulomb branch formula -- for computing the cohomology of the moduli space of quiver representations. For quivers without oriented loops, the Coulomb branch formula is shown to agree with the Higgs branch formula based on Reineke's result for stack invariants, even when the dimension vector is not primitive. For quivers with oriented loops, the Coulomb branch formula parametrizes the Poincare polynomial of the quiver moduli space in terms of single-centered (or pure-Higgs) BPS invariants, which are conjecturally independent of the stability condition (i.e. the choice of Fayet-Iliopoulos parameters) and angular-momentum free. To facilitate further investigation we provide a Mathematica package "CoulombHiggs.m" implementing the Coulomb and Higgs branch formulae.

LieTransformer: Equivariant Self-Attention for Lie Groups

Abstract: Group equivariant neural networks are used as building blocks of group invariant neural networks, which have been shown to improve generalisation performance and data efficiency through principled parameter sharing. Such works have mostly focused on group equivariant convolutions, building on the result that group equivariant linear maps are necessarily convolutions. In this work, we extend the scope of the literature to self-attention, that is emerging as a prominent building block of deep learning models. We propose the LieTransformer, an architecture composed of LieSelfAttention layers that are equivariant to arbitrary Lie groups and their discrete subgroups. We demonstrate the generality of our approach by showing experimental results that are competitive to baseline methods on a wide range of tasks: shape counting on point clouds, molecular property regression and modelling particle trajectories under Hamiltonian dynamics.

Mass Generation in Continuum SU(2) Gauge Theory in Covariant Abelian Gauges

Abstract: The local action of an SU(2) gauge theory in general covariant Abelian gauges and the associated equivariant BRST symmetry that guarantees the perturbative renormalizability of the model are given. I show that a global SL(2,R) symmetry of the model is spontaneously broken by ghost-antighost condensation at arbitrarily small coupling and leads to propagators that are finite at Euclidean momenta for all elementary fields except the Abelian ``photon''. The Goldstone states form a BRST-quartet. The mechanism eliminates the non-abelian infrared divergences in the perturbative high-temperature expansion of the free energy.

An Equivariant Sphere Theorem

Abstract: Let M be a connected 3-manifold acted on by a group G. Suppose M has a triangulation invariant under G. In this paper it is shown that if there exists an embedded 2-sphere S which does not bound a 3-ball, then there exists such an S for which gS = S or gS 0 S = 0 for every g e G. This result was proved by Meeks, Simon and Yau [3] using analytic techniques. The proof given here is self-contained and elementary. The proof involves looking at embedded 2-spheres which are in general position with respect to the given triangulation. Such a sphere is called minimal if it does not bound a 3-ball and the number of intersections with the 1-skeleton of the triangulation is the smallest possible. The key result proved in this paper is that given a finite set of minimal spheres satisfying a general position condition, there is a finite set of 'standard' disjoint minimal spheres whose union has the same intersection with the 1-skeleton as the original spheres. The set of disjoint spheres is unique up to a homeomorphism of M which fixes the 2-skeleton. In §4 it is shown that if G\\K is finite then there is a G-equivariant decomposition of M with irreducible factors. We are then able to deduce from the ordinary loop theorem an equivariant version of the projective plane theorem. In §5 the arguments of the previous sections are modified to provide a proof of the equivariant loop theorem [2]. I think that many of the topological results obtained using analytic minimal surface theory can also be derived using the techniques of this paper. I am grateful to Andrew Bartholomew for pointing out an error in an earlier version of this paper. I thank both Peter Scott and the referee for their helpful comments.