Ground-state edge-cone metrics and template-topology reflection for anyon tensor networks in geometric models of matter (Atiyah-Marcolli, 2021)

In "Anyon Networks from Geometric Models of Matter" (Q. J. Math. 72 (2021) 717-733), completed posthumously by Matilde Marcolli from joint work with Michael Atiyah, the authors propose realizing anyon tensor networks - the algebraic data structures underlying topological quantum computation - inside four-dimensional Riemannian geometry: orbifold building blocks $(M_v, \Sigma_v)$ carrying edge-cone metrics are glued along discs of their singular surfaces according to a finite directed network template $G$, and the network tensors are read off from a topological quantum field theory with corners based on the Kauffman bracket. The paper explicitly leaves open (i) whether the resulting composite geometry $(M_G, \Sigma_G)$ can occupy a "ground state", that is, carry an Einstein and (anti)self-dual edge-cone metric (a gravitational-instanton-type metric), given the known cone and Seiberg-Witten obstructions, and (ii) whether the topology of the network template is reflected in the topology of the resulting geometric model; it also restricts its braid-theoretic input to standard braid systems, leaving the non-standard Hurwitz-orbit case unworked. The open problem is therefore: determine for which network templates and gluings the composite edge-cone orbifold admits an Einstein and (anti)self-dual metric; establish how the combinatorial topology of the template is reflected in the invariants of $(M_G, \Sigma_G)$; and extend the Temperley-Lieb-Jones representation of the surface-braid data to non-standard Hurwitz-orbit braid systems.

Background

Geometric models of matter form a program, initiated by Atiyah, Manton and Schroers, in which particles are modeled by four-dimensional Riemannian manifolds (or orbifolds, that is, spaces that are locally quotients of Euclidean space by a finite group) rather than by fields on a background: particle quantum numbers arise from topological invariants, with the signature $\tau$ of the 4-manifold (the signature of its intersection form on middle-dimensional homology) playing the role of baryon number, and the self-intersection number of an embedded surface playing the role of electric charge. The distinguished "ground states" of such models are gravitational instantons: complete (in the compact orbifold setting, closed) Einstein 4-manifolds whose Weyl curvature is (anti)self-dual, meaning that the anti-self-dual (respectively self-dual) half of the Weyl 2-form vanishes; classical examples are the Taub-NUT and Atiyah-Hitchin metrics, and Kronheimer proved that the ALE (asymptotically locally Euclidean) gravitational instantons are classified by their hyperkähler data.

The program's orbifold refinement uses edge-cone metrics: on a pair $(M, \Sigma)$ of a smooth 4-manifold $M$ and an embedded surface $\Sigma$ of cone points, the metric is smooth away from $\Sigma$ and modeled on a cone of angle $2\pi/\nu$ transverse to $\Sigma$. Atiyah and LeBrun computed the resulting orbifold characteristic numbers - the orbifold Euler characteristic $\chi_{\mathrm{orb}} = \chi(M) - (1-1/\nu)\chi(\Sigma)$ and orbifold signature $\tau_{\mathrm{orb}} = \tau(M) - (1-1/\nu^2)[\Sigma]^2$, where $[\Sigma]^2$ is the self-intersection of $\Sigma$ - via Kawasaki's orbifold index theorem, showed that an Einstein edge-cone metric forces the cone inequalities

$$2\chi(M) \pm 3\tau(M) \geq (1-1/\nu)\left(2\chi(\Sigma) \pm (1+1/\nu)[\Sigma]^2\right),$$

and, with LeBrun's Seiberg-Witten-theoretic obstructions (invariants built from the monopole equations in the edge-cone setting), gave strong restrictions; LeBrun's explicit self-dual metrics on connected sums $\#_n\mathbb{CP}^2$ and the LeBrun-Singer Kummer-type construction supply the known positive examples, while Donaldson-Friedman twistor deformation theory obstructs self-dual metrics on general connected sums. Abreu's toric extremal metrics provide self-dual edge-cone metrics on toric surfaces relevant to the simplest examples.

In the predecessor paper "Anyons in Geometric Models of Matter", Atiyah and Marcolli showed how anyons - quasiparticles in two spatial dimensions whose exchange statistics are governed by the braid group rather than the symmetric group, and which are the computational resource of topological quantum computation (see the survey of Freedman, Kitaev, Larsen and Wang) - arise in this framework as surface braids: multisections of the orbifold normal bundle of $\Sigma \subset M$ over a branched cover $S \to \Sigma$ determine representations $\rho_S: \pi_1(\Sigma \setminus b(S)) \to B_m(D^2)$, where $B_m(D^2)$ is the braid group on $m$ strands in a disc and $b(S)$ is the branch set. Representations of $B_m$ into Temperley-Lieb algebras (diagram algebras spanned by non-crossing matchings, obtained via the Kauffman bracket of knot theory at a chosen root of unity, as in Zinno's explicit basis) yield anyonic quantum gates; Freedman, Larsen and Wang proved that the modular functor of Chern-Simons theory at the fifth root of unity is universal for quantum computation, and condensed-matter anyonic tensor networks have been simulated numerically by Pfeifer et al., with the Fibonacci anyon model (a simple non-Abelian anyon with two fusion channels; see the survey of Trebst, Troyer, Wang and Ludwig) as the standard computationally universal example. The braid group $B_n$ acts on $n$-tuples of braids (braid systems) by the Hurwitz action, and the orbit of the standard system of generators is the well-understood case; Digne and Gobet treat non-standard systems (Mikado braids) in the Hecke-algebra setting.

The paper extends this to networks. A network template is a finite directed graph $G$; each vertex $v$ carries an orbifold geometry $(M_v, \Sigma_v)$ whose branch-point set $b(S_v)$, of cardinality $\ell \cdot \mathrm{val}(v)$ for a fixed $\ell \geq 2$, is partitioned into blocks $B_{v,f}$ of $\ell$ points, one attached to each flag (half-edge) $f$ of $v$; each edge identifies normal-bundle discs of its endpoint vertices by a fiber-preserving diffeomorphism $\varphi_e$. Gluing all blocks produces a composite pair $(M_G, \Sigma_G)$, and the braid/TQFT data are assembled by a topological quantum field theory with corners - Gelca's Kauffman-bracket TQFT on DAP-decompositions (discs with attached punctures) - whose partition functions, after taking a bulk trace, produce the entries of a tensor network with tensors indexed by the vertices and contracted along the edges. The paper illustrates the construction with tree templates on $\#_T\mathbb{CP}^2$-type connected sums carrying self-dual edge-cone metrics, and with loop templates on class VII$_0^+$ surfaces (minimal compact complex surfaces with first Betti number 1 and positive second Betti number, diffeomorphic to $(S^1 \times S^3)\#_m\overline{\mathbb{CP}}^2$), where Fujiki and Pontecorvo constructed anti-self-dual bihermitian structures on Inoue surfaces and one may ask for edge-cone metrics with the same property. The construction is explicitly a sketch: the paper is a posthumous, unfinished collaboration in which several points appear, in the authors' words, "in the form of questions or sketches of ideas, rather than being fully elaborated".

Problem Statement

Work within the framework of Atiyah-Marcolli, 'Anyon Networks from Geometric Models of Matter' (Q. J. Math. 72 (2021) 717-733): a finite directed network template $G$ with vertex set $V$, each vertex $v$ labeled by an orbifold geometry $(M_v, \Sigma_v)$ with an edge-cone metric of cone angle $2\pi/\nu$ along the surface $\Sigma_v$, branch-point blocks $B_{v,f}$ of $\ell$ points attached to the flags of $v$, so that the branch set at $v$ has $\ell \cdot \mathrm{val}(v)$ points, where $\mathrm{val}(v)$ is the number of half-edges at $v$, and fiber-preserving identifications $\varphi_e$ of normal-bundle discs along the edges, producing a glued pair $(M_G, \Sigma_G)$ and, via a Kauffman-bracket TQFT with corners and a bulk trace, an anyon tensor network on $G$. The open problem has three linked parts. (A) Ground-state existence: for a family of templates that includes at least one tree template and at least one loop template and varies the valences and connectivity of the templates, determine for which templates $G$, building blocks $(M_v, \Sigma_v)$, cone angles, and gluings the composite $(M_G, \Sigma_G)$ carries a 'ground-state' metric, in the source's sense an Einstein and (anti)self-dual edge-cone metric (a gravitational-instanton-type metric) - that is, settle existence and non-existence systematically beyond the paper's tree example on $\#_T\mathbb{CP}^2$-type connected sums and its sketch for loop templates on class VII$_0^+$ surfaces (minimal compact complex surfaces with first Betti number 1 and positive second Betti number), and determine how the answer depends on the template (trees versus loops, valences, connectivity), in particular whether the known obstructions (the Atiyah-LeBrun cone inequalities on $2\chi \pm 3\tau$ ($\chi$ and $\tau$ the Euler characteristic and signature), Seiberg-Witten-type edge-cone obstructions, and Donaldson-Friedman twistor deformation obstructions) or new ones close off entire template classes. (B) Template-topology reflection: determine to what extent the combinatorial topology of the template $G$ is reflected in, or constrains, the topology and invariants of the resulting geometric model - the orbifold characteristic numbers, the topology of $\Sigma_G$ and of the branched covers used in the braid construction, and the twistor geometry - as opposed to features depending only on the building blocks, and in particular whether some non-degenerate invariant or constraint of $(M_G, \Sigma_G)$ - one that is not shared by all templates with the same building blocks, cone angles, and gluings, so that it separates at least two such templates - is determined by the combinatorial topology of the template, or whether no such non-degenerate template-determined invariant exists. (C) Non-standard braid systems: the paper's construction of network tensors from surface braids is restricted to standard braid systems; work out the case of braid systems in the Hurwitz orbit of non-standard systems (the braid group acts on tuples of braids by Hurwitz moves, and a system is standard when each entry is conjugate to a standard generator of the braid group), which is needed when the gluings $\varphi_e$ transport a standard system to a non-standard one, including the resulting Temperley-Lieb-Jones representation data at roots of unity and the well-definedness of the network tensors. Parts (A), (B) and (C) are jointly the source's question of whether, and in what sense, anyon tensor networks can be realized in geometric models of matter.

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