Perturbation semigroup structure and generalized inner fluctuations for genuinely noncommutative spectral triples without the first-order condition

In Connes' noncommutative geometry, inner fluctuations of the Dirac operator generate gauge fields, but the standard construction assumes the first-order condition, which many spectral triples fail to satisfy, including the canonical triples on quantum groups and on Podleś quantum spheres. Chamseddine, Connes and van Suijlekom (J. Geom. Phys. 73 (2013) 222-234) extended inner fluctuations to triples without the first-order condition by adding a quadratic term to the fluctuated Dirac operator, and showed that the resulting fluctuations are governed by a semigroup Pert(A) of self-adjoint normalized elements of the algebraic tensor product of A with its opposite algebra, a semigroup that extends the unitary group of A. The paper's declared open directions remain open: the structure theory of Pert(A) and of its group of invertible elements, the application of the formalism to genuinely noncommutative spaces such as quantum groups and quantum spheres (genuinely noncommutative here means an infinite-dimensional involutive algebra that is not a finite direct sum of matrix algebras over R, C or H, nor an operator system), and the general question of when the quadratic term generates composite rather than fundamental scalar degrees of freedom. The k = 4 Standard-Model-relevant algebra H+H+M4(C) was treated by the companion paper (JHEP 11 (2013) 132, yielding Pati-Salam unification), and Pert(A) has since been computed for matrix algebras and operator systems; the genuinely noncommutative direction and the general composite-versus-fundamental classification remain open, with neither direction appearing to have been treated in the follow-up literature cited below.

Background

Connes' noncommutative geometry encodes a compact Riemannian spin manifold through a spectral triple $(\mathcal{A},\mathcal{H},D)$: an involutive algebra $\mathcal{A}$ of coordinate functions represented on a Hilbert space $\mathcal{H}$ of spinors, together with a self-adjoint operator $D$ (the Dirac operator) whose spectrum replaces the metric data. A real spectral triple carries in addition an antilinear isometry $J:\mathcal{H}\to\mathcal{H}$ implementing charge conjugation, with $JDJ^{-1}=\epsilon D$ for $\epsilon=\pm 1$. The first-order condition requires $[[D,a],JbJ^{-1}]=0$ for all $a,b\in\mathcal{A}$; it is the algebraic counterpart of the fact that a classical Dirac operator is a first-order differential operator.

Geometry on a spectral triple is probed through inner fluctuations: replacing $D$ by a fluctuated operator $D+A+\epsilon JAJ^{-1}$, where $A=\sum_j a_j[D,b_j]$ represents a self-adjoint universal one-form (an element of the bimodule $\Omega^1(\mathcal{A})$ of finite sums $\sum_j a_j\,\delta b_j$). Under the unitary group $U(\mathcal{A})$ these fluctuations transform like a gauge field with gauge group the unitaries $U=uJuJ^{-1}$, and they are transitive: fluctuations of fluctuations are again inner fluctuations. Combined with the spectral action principle (the bosonic action taken as the trace of a cutoff function of the fluctuated Dirac operator), this produces gauge and gravity actions, and on almost-commutative manifolds (products of an ordinary manifold with a finite geometry, i.e. a spectral triple with finite-dimensional algebra and Hilbert space) it yields the Standard Model coupled to gravity. In the classification of irreducible finite geometries of KO-dimension 6 (dimension counted modulo 8 through the commutation signs of $J$, of the grading operator that splits $\mathcal{H}$ into even and odd subspaces, and of $D$, in the sense of real K-theory) performed by Chamseddine and Connes in 'Why the Standard Model' (J. Geom. Phys. 58 (2008) 38-47), the algebra $\mathbb{H}\oplus\mathbb{H}\oplus M_4(\mathbb{C})$ arises naturally, and imposing the first-order condition breaks it to the Standard Model algebra $\mathbb{C}\oplus\mathbb{H}\oplus M_3(\mathbb{C})$.

Chamseddine, Connes and van Suijlekom (2013) analyzed what happens without the first-order condition. The naive linear fluctuation is then neither gauge-covariant nor transitive; both defects are cured by adding a quadratic term, giving the fluctuated operator

$$D' = D + A_{(1)} + \tilde A_{(1)} + A_{(2)},\quad A_{(1)}=\sum_j a_j[D,b_j],\quad \tilde A_{(1)}=\sum_j \hat a_j [D,\hat b_j],\quad A_{(2)}=\sum_j \hat a_j [A_{(1)}, \hat b_j],$$

where $\hat T=JTJ^{-1}$ gives the action of the conjugate copy of the algebra (so $\tilde A_{(1)}=\epsilon JA_{(1)}J^{-1}$). The term $A_{(2)}$ vanishes precisely when the first-order condition holds. Conceptually, the fluctuations come from the action $T\mapsto\sum_i a_i T b_i$ of a semigroup $\operatorname{Pert}(\mathcal{A})$ of inner perturbations that depends only on the involutive algebra: writing elements of $\mathcal{A}\otimes\mathcal{A}^{\mathrm{op}}$ (the algebraic tensor product with the opposite algebra, which multiplies in reverse order) as $\sum_j a_j\otimes b_j^{\mathrm{op}}$, one defines $\operatorname{Pert}(\mathcal{A})$ as the set of self-adjoint normalized elements, where normalization means $\sum_j a_j b_j=1$ and self-adjointness refers to the involution $(\sum_j a_j\otimes b_j^{\mathrm{op}})^*=\sum_j b_j^*\otimes a_j^{*\mathrm{op}}$. This set is closed under multiplication, extends the unitary group via $u\mapsto u\otimes (u^*)^{\mathrm{op}}$, and acts on real spectral triples through a homomorphism into $\operatorname{Pert}(\mathcal{A}\otimes\hat{\mathcal{A}})$; its invertible elements form a group.

Two observations from the source paper drive the open directions examined here. Physically, applying the formalism to the $k=2$ finite geometry of the classification (a toy model with algebra $\mathbb{C}\oplus\mathbb{C}\oplus M_2(\mathbb{C})$) showed that $A_{(2)}$ generates no new scalar fields but is parametrized by the fields already present in $A_{(1)}$, the fluctuations having rank 1, so that the scalar sector is composite rather than fundamental. Mathematically, genuinely noncommutative spaces typically violate the first-order condition: the coordinate algebra of the quantum group $SU_q(2)$ ($SU(2)$ with its multiplication deformed by a real parameter $0<q<1$) carries a spectral triple whose Dirac operator was constructed by Dąbrowski, Landi, Sitarz, van Suijlekom and Várilly, and the Podleś quantum spheres (a family of $q$-deformed two-spheres homogeneous under $SU_q(2)$) carry spectral triples whose Dirac operators were constructed by D'Andrea, Dąbrowski, Landi and Wagner; these are the canonical test cases for the generalized formalism. Applying the framework to such spaces requires, first of all, understanding $\operatorname{Pert}(\mathcal{A})$ for their infinite-dimensional noncommutative algebras; the published structure theory covers only finite-dimensional algebras (direct sums of matrix algebras over $\mathbb{C}$, $\mathbb{R}$ and $\mathbb{H}$) and, in a modified setting, operator systems (concrete unital self-adjoint subspaces of the bounded operators on a Hilbert space, the structures appearing in the spectral-truncation approach to noncommutative geometry).

Problem Statement

Within the generalized inner-fluctuation framework of Chamseddine, Connes and van Suijlekom, in which the fluctuated Dirac operator of a real spectral triple not satisfying the first-order condition is $D'=D+A_{(1)}+\tilde A_{(1)}+A_{(2)}$ with $A_{(2)}=\sum_j \hat a_j[A_{(1)},\hat b_j]$ and $\hat T=JTJ^{-1}$, determine what the framework produces away from the almost-commutative (matrix-algebra) world, in three linked parts:

(1) Structure of the perturbation semigroup. For the genuinely noncommutative, infinite-dimensional algebras of the quantum group $SU_q(2)$ and of the Podleś quantum spheres, in the algebraic setting and in the $C^*$-completion where the semigroup is naturally defined, determine the structure of $\operatorname{Pert}(\mathcal{A})$, the semigroup of self-adjoint normalized elements of $\mathcal{A}\otimes\mathcal{A}^{\mathrm{op}}$ (normalization of $\sum_j a_j\otimes b_j^{\mathrm{op}}$ meaning $\sum_j a_j b_j=1$; self-adjointness meaning invariance under the involution $\sum_j a_j\otimes b_j^{\mathrm{op}}\mapsto\sum_j b_j^*\otimes a_j^{*\mathrm{op}}$): classify its elements and its group of invertible elements, and determine the relation of that group to the unitary group $U(\mathcal{A})$ and to the representation theory of the underlying noncommutative space.

(2) Generalized inner fluctuations on genuinely noncommutative spaces. For the canonical spectral triples on such spaces, the Dirac operators on the Podleś quantum spheres and on $SU_q(2)$, which violate the first-order condition, compute the generalized inner fluctuations including the quadratic term $A_{(2)}$; verify that the fluctuated operator is self-adjoint, that the fluctuation is gauge-covariant under $U=uJuJ^{-1}$, and that it is transitive through the $\operatorname{Pert}(\mathcal{A})$-action; and determine the gauge and scalar content produced.

(3) Composite versus fundamental scalars. Establish a general criterion determining which classes of non-first-order spectral triples yield composite scalar degrees of freedom, with $A_{(2)}$ parametrized by the fields of $A_{(1)}$ as in the $k=2$ toy model, and which yield genuinely new fundamental scalar fields, as occurs for generic initial Dirac operators in the $k=4$ model where additional fundamental Higgs fields in the $(3,1,10)$, $(1,1,6)$ and $(2,2,1+15)$ representations appear; short of a general criterion, determine the answer for the classes of triples in part (2).

These parts form one objective, namely making the generalized inner-fluctuation framework effective for genuinely noncommutative spectral triples: part (1) is its mathematical prerequisite, part (2) its target application, and part (3) the physical content question of that same application. The $k=4$ computation that the source paper deferred is excluded from the problem, having been completed by the companion Pati-Salam paper.

References

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