Hochschild and cyclic (co)homology of the quadratic self-duality quotients of the Yang-Mills algebra

The Yang-Mills algebra $A$ introduced by Connes and Dubois-Violette is the cubic algebra generated by the covariant derivatives of a generic Yang-Mills connection; their 2002 paper proved it is Koszul and Gorenstein of global dimension 3, produced its minimal projective bimodule resolution, and stated that this resolution "opens the way for the computation of the Hochschild and of the cyclic (co)homology" of $A$. That computation for $A$ itself was subsequently carried out by Movshev (2008, by geometric methods) and, completely and independently, by Herscovich and Solotar (2012, by algebraic methods, with closed-form Hilbert series for all Yang-Mills algebras $YM(n)$, $n \geq 2$, and corrections to some of Movshev's announced statements). What no later work supplies is the same computation for the quadratic (anti-)self-duality quotients $A^{(\pm)}$ of the 4-dimensional Yang-Mills algebra: the Koszul, non-Gorenstein algebras of global dimension 2 and exponential growth describing covariant derivatives of (anti-)self-dual connections, each the enveloping algebra of a semidirect product of the free Lie algebra on three generators by a derivation. For these quotients the 2002 paper likewise published the minimal projective resolution (its Corollary 4) but extracted only the Hochschild dimension and the Poincaré series. The remaining problem is to compute, explicitly and in closed form, the graded Hochschild (co)homology and cyclic (co)homology of $A^{(\pm)}$, and to identify the resulting cyclic classes (cocycles of Chern-character type) and the noncommutative-geometric information they carry, completing for the quotient algebras what Movshev and Herscovich-Solotar completed for the cubic algebra.

Background

In noncommutative geometry, Connes and Dubois-Violette (2002) attached to the classical Yang-Mills equations on the $(s+1)$-dimensional pseudo-Euclidean space a universal algebra of the equations of motion. A gauge connection is described by covariant derivatives $\nabla_\lambda = \partial_\lambda + A_\lambda$ (the partial derivatives shifted by the components of the gauge field, which do not commute), and the Yang-Mills equations read $g^{\lambda\mu}[\nabla_\lambda,[\nabla_\mu,\nabla_\nu]] = 0$ in terms of commutators. The Yang-Mills algebra $A$ is the algebra generated by symbols $\nabla_\mu$ modulo exactly these cubic relations; every solution of the Yang-Mills equations defines a representation of it, so $A$ encodes the universal underlying noncommutative symmetry of gauge theory (it also appeared independently in string-theoretic work on matrix models).

Two standard invariants of an associative algebra $B$ are relevant here. Hochschild (co)homology $HH_\bullet(B)$ and $HH^\bullet(B)$ is the homology theory of $B$ as a bimodule over itself (Tor and Ext over the enveloping algebra $B \otimes B^{op}$); among other roles, $HH^1$ is the Lie algebra of derivations modulo inner ones (the noncommutative vector fields) and $HH^2$ classifies infinitesimal deformations. Cyclic homology $HC_\bullet(B)$, introduced by Connes, is the noncommutative analogue of de Rham cohomology and is linked to Hochschild homology by Connes' long exact sequence; its dual classes (cyclic cocycles) support Chern characters in noncommutative geometry. For graded algebras these groups are naturally graded, and a compact way to state a complete computation is via Hilbert series (the generating functions of the graded dimensions).

Connes and Dubois-Violette proved that $A$ is a cubic Koszul algebra (its trivial module has a minimal resolution with linear maps, in the generalized sense of homogeneous algebras of degree 3), Gorenstein (a noncommutative analogue of Poincaré duality), of global dimension 3 (equivalently for these algebras, the length of the minimal projective resolution of the trivial module) and of exponential growth, and they wrote down an explicit minimal projective bimodule resolution (their Corollary 1), stating that it "opens the way for the computation of the Hochschild and of the cyclic (co)homology of $A$". In the 4-dimensional Euclidean case the Yang-Mills algebra admits two natural quotients: a connection can be self-dual or anti-self-dual, meaning $[\nabla_0, \nabla_k] = \pm[\nabla_\ell, \nabla_m]$ for $(k,\ell,m)$ a cyclic permutation of $(1,2,3)$, and such connections automatically solve the Yang-Mills equations. The corresponding quadratic (anti-)self-duality algebras $A^{(\pm)}$, exchanged by reversing the orientation of $\mathbb{R}^4$, are Koszul of global dimension 2, not Gorenstein, of exponential growth, with Poincaré series (its Hilbert series, the generating function of the graded dimensions) $1/((1-t)(1-3t))$; $A^{(+)}$ is the universal enveloping algebra of the semidirect product of the free Lie algebra on three generators $\nabla_1, \nabla_2, \nabla_3$ (the Lie algebra on these symbols with no relations beyond antisymmetry and the Jacobi identity) by the derivation $\delta(\nabla_k) = [\nabla_\ell, \nabla_m]$, equivalently a cross-product of the tensor algebra on three generators with that derivation. For $A^{(+)}$ the same paper published the minimal projective bimodule resolution (its Corollary 4) but extracted from it only the Hochschild dimension (2) and the growth.

The homology of the cubic algebra was subsequently computed. Movshev, in a 2005 preprint on deformations of Yang-Mills algebras and in the published article "Cohomology of Yang-Mills algebras" (Journal of Noncommutative Geometry, 2008), computed the (cyclic) (co)homology of the Yang-Mills algebra by geometric methods. Herscovich and Solotar (Journal für die reine und angewandte Mathematik, 2012), developing the representation theory of these algebras from their 2011 Annals of Mathematics paper, gave a complete, self-contained algebraic computation of the Hochschild and cyclic homology of all Yang-Mills algebras $YM(n)$, $n \geq 2$, over an algebraically closed field of characteristic zero: closed-form Hilbert series for every Hochschild homology and cohomology group and every cyclic homology group, the center, and the Lie structure on $HH^1$ (interpreted as the noncommutative vector fields: dilations, translations, rotations), recovering the 2-dimensional Heisenberg case and the results announced by Movshev while correcting some of his statements. The same authors and related groups also studied inhomogeneous Yang-Mills algebras (Berger-Dubois-Violette) and further representation theory (Herscovich). None of these works treats the Hochschild or cyclic (co)homology of the quadratic self-duality quotients $A^{(\pm)}$, whose minimal resolution has been available since 2002: that computation and the identification of the corresponding cyclic classes are the surviving open part of the 2002 program.

Problem Statement

Let $k$ be a field of characteristic zero and let $A^{(\pm)}$ denote the quadratic (anti-)self-duality algebras of Connes-Dubois-Violette: the quotients of the 4-dimensional Yang-Mills algebra generated by $\nabla_0, \nabla_1, \nabla_2, \nabla_3$ with quadratic relations $[\nabla_0, \nabla_k] = \pm[\nabla_\ell, \nabla_m]$ for $(k,\ell,m)$ a cyclic permutation of $(1,2,3)$ (the two algebras are exchanged by reversing the orientation of $\mathbb{R}^4$). Compute, explicitly and in closed form: (1) the graded Hochschild homology $HH_\bullet(A^{(\pm)})$ and Hochschild cohomology $HH^\bullet(A^{(\pm)})$, for instance as Hilbert series or tables of graded dimensions together with a structural description of the groups (generators, grading, and the natural $SO(3)$-module structure coming from the residual symmetry of the presentation); (2) the cyclic homology $HC_\bullet(A^{(\pm)})$ and cyclic cohomology $HC^\bullet(A^{(\pm)})$, including the effect of Connes' periodicity operator; (3) explicit cyclic cocycles of Chern-character type among the computed classes, with a statement of the noncommutative-geometric information they encode: which classes govern deformations of the algebra via the cup product on $HH^\bullet$, and which are realised as noncommutative vector fields via $HH^1$, both identifications being proved and not merely asserted. The computation may proceed by any rigorous method. Both signs must be treated, or an explicit isomorphism or anti-isomorphism between $A^{(+)}$ and $A^{(-)}$ must be given so that a single computation suffices. This is the surviving part of the 2002 program of Connes-Dubois-Violette (compute the Hochschild and cyclic (co)homology of the Yang-Mills algebra and of its quadratic quotients and analyse the resulting cyclic classes in noncommutative geometry) after Movshev (2008) and Herscovich-Solotar (2012) settled the case of the cubic Yang-Mills algebra.

References

1. Alain Connes, Michel Dubois-Violette, "Yang-Mills Algebra", Letters in Mathematical Physics 61 (2002), no. 2, 149-158. DOI: 10.1023/A:1020733628744, https://doi.org/10.1023/A:1020733628744 (preprint arXiv:math/0206205, https://arxiv.org/abs/math/0206205). Source paper.
2. Michael Movshev, "On deformations of Yang-Mills algebras", preprint (2005). arXiv:hep-th/0509119, https://arxiv.org/abs/hep-th/0509119.
3. Michael Movshev, "Cohomology of Yang-Mills algebras", Journal of Noncommutative Geometry 2 (2008), no. 3, 353-404. DOI: 10.4171/jncg/24, https://doi.org/10.4171/jncg/24.
4. Estanislao Herscovich, Andrea Solotar, "Hochschild and cyclic homology of Yang-Mills algebras", Journal für die reine und angewandte Mathematik 665 (2012). DOI: 10.1515/crelle.2011.107, https://doi.org/10.1515/crelle.2011.107 (preprint arXiv:0906.2576, https://arxiv.org/abs/0906.2576).
5. Estanislao Herscovich, Andrea Solotar, "Representations of Yang-Mills algebras", Annals of Mathematics (2) 173 (2011), no. 2, 1043-1080. DOI: 10.4007/annals.2011.173.2.12, https://doi.org/10.4007/annals.2011.173.2.12.
6. Alain Connes, Michel Dubois-Violette, "Yang-Mills and Some Related Algebras", in: Rigorous Quantum Field Theory, Progress in Mathematics 251, Birkhäuser (2007), 65-78. DOI: 10.1007/978-3-7643-7434-1_6, https://doi.org/10.1007/978-3-7643-7434-1_6 (preprint arXiv:math-ph/0411062, https://arxiv.org/abs/math-ph/0411062).
7. Roland Berger, Michel Dubois-Violette, "Inhomogeneous Yang-Mills Algebras", Letters in Mathematical Physics 76 (2006), no. 1, 65-75. DOI: 10.1007/s11005-006-0075-5, https://doi.org/10.1007/s11005-006-0075-5.
8. Estanislao Herscovich, "Some remarks on representations of Yang-Mills algebras", Journal of Mathematical Physics 56 (2015), no. 1, 011702. DOI: 10.1063/1.4905857, https://doi.org/10.1063/1.4905857.