3d. Metric aspects

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Following now Goswami and subsequent papers, let us comment on the “metric” aspects of our quantum isometry group construction. There are many things that can be said here, and the present section will be a modest introduction to all this. To start with, we have the following definition, which is something very standard in geometry:

Definition

Given a compact Riemannian manifold [math]X[/math], we denote by [math]\Omega^1(X)[/math] the space of smooth [math]1[/math]-forms on [math]X[/math], with scalar product given by

[[math]] \lt \omega,\eta \gt =\int_X \lt \omega(x),\eta(x) \gt dx [[/math]]

and we construct the Hodge Laplacian [math]\Delta:L^2(X)\to L^2(X)[/math] by setting

[[math]] \Delta=d^*d [[/math]]

where [math]d:C^\infty(X)\to\Omega^1(X)[/math] is the usual differential map, and [math]d^*[/math] is its adjoint.

Observe the notational clash with the comultiplication for Woronowicz algebras, and with solving this clash being actually an open problem. Physicists like to use [math]\nabla^2[/math] for the Laplacian, but this is not very beautiful, as mathematicians we are just so used to [math]\Delta[/math]. And with my hope here that, regardless of your main interests in mathematics, you teach from time to time PDE classes, as any serious mathematician should do.

Talking about notations, the problem comes from quantum groups, which were the last to come into play. Drinfeld, Jimbo and others used [math](\Delta,\varepsilon,S)[/math], obviously algebra-inspired. Then Woronowicz came with [math](\Phi,e,\kappa)[/math], for avoiding confusion with the Laplacian [math]\Delta[/math], with [math]\varepsilon[/math] from analysis, and with [math]S[/math] from Tomita-Takesaki theory. But then a bit later a younger, reckless generation came, including a former myself, thinking among others that Drinfeld-Jimbo is more interesting than Tomita-Takesaki, and reverting back to [math](\Delta,\varepsilon,S)[/math]. And here we are now, in more recent years, thinking about [math]\Delta[/math] and what to do with it.

But hey, to any problem there should be a solution. As already mentioned on several occasions, and more on this in a moment too, there are some deep problems in relation with the Laplacian in the noncommutative setting, namely axiomatization and general theory, including things like free harmonic functions, then heavy PDE theory to be developed, for the free manifolds, and then, as a culmination of all this, applications to physics, and more specifically, conjecturally, to QCD. And the one who will do all this will certainly have the knowledge and authority to decide what [math]\Delta[/math] should stand for.

Back to work now, and to Definition 3.17, we have the following standard result:

Theorem

The isometry group [math]\mathcal G(X)[/math] of a compact Riemannian manifold [math]X[/math] is the group of diffeomorphisms

[[math]] \varphi:X\to X [[/math]]

whose induced action on [math]C^\infty(X)[/math] commutes with the Hodge Laplacian [math]\Delta[/math].

Show Proof

This is something well-known and standard, and for more on all this, basic Riemannian geometry and related topics, we refer to the book of do Carmo , as well as to the book of Connes and the paper of Goswami .

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Based on the above, and following Goswami , we can formulate:

Definition

The quantum isometry group [math]\mathcal G^+(X)[/math] of a compact Riemannian manifold [math]X[/math] is the biggest compact quantum group acting on [math]X[/math], via

[[math]] \Phi:C(X)\to C(X)\otimes C(G) [[/math]]

with this coaction map commuting with the action of [math]\Delta[/math].

This is something quite tricky. First, the coaction map [math]\Phi[/math] is by definition subject to the usual axioms for the algebraic coactions, namely:

[[math]] (\Phi\otimes id)\Phi=(id\otimes\Delta)\Phi [[/math]]

[[math]] (id\otimes\varepsilon)\Phi=id [[/math]]

In addition, [math]\Phi[/math] must be subject as well to the following smoothness assumption:

[[math]] \Phi(C^\infty(X))\subset C^\infty(X)\otimes C(G) [[/math]]

As for the commutation condition with [math]\Delta[/math], this regards the canonical extension of the action to the space [math]L^2(X)[/math]. And finally, and importantly, the above definition is something non-trivial, coming from a theorem, established in , which states that a universal object [math]\mathcal G^+(X)[/math] as above exists indeed. For details here, we refer to .

Regarding now the examples, we first have something that we know, as follows:

Proposition

The quantum isometry group of the [math]N[/math]-simplex [math]X_N[/math] is

[[math]] \mathcal G^+(X_N)=S_N^+ [[/math]]

which is bigger than the usual isometry group [math]\mathcal G(X_N)=S_N[/math], at [math]N\geq4[/math].

Show Proof

Consider indeed the simplex [math]X_N\subset\mathbb R^N[/math], formed by definition by the standard basis [math]\{e_1,\ldots,e_N\}[/math] of [math]\mathbb R^N[/math]. We know from chapter 2 that the symmetry and quantum symmetry groups of [math]X_N[/math], regarded as a set, are [math]S_N\subset S_N^+[/math]. But this shows too that the classical and quantum isometry groups of [math]X_N[/math], regarded either as an algebraic manifold, as in Proposition 3.2 and Theorem 3.3, or as a Riemannian manifold, as in Theorem 3.18 and Definition 3.19, are [math]S_N\subset S_N^+[/math] as well. Finally, the fact that the inclusion [math]S_N\subset S_N^+[/math] is not an isomorphism at [math]N\geq4[/math] is something that we know too from chapter 2.

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It is possible to obtain more examples along the same lines, by looking at more general disconnected manifolds, and with the computation of [math]\mathcal G^+(X)[/math] for disconnected manifolds being actually a very interesting question. See . In what regards the connected case, however, there have been a lot of computations by Bhowmick, Goswami and others, leading to the conjecture that we should have rigidity, [math]\mathcal G^+(X)=\mathcal G(X)[/math], in this case. And with this rigidity conjecture being now a theorem, due to Goswami :

Theorem

For a compact, connected Riemannian manifold [math]X[/math], the inclusion

[[math]] \mathcal G(X)\subset\mathcal G^+(X) [[/math]]

is an isomorphism. That is, [math]X[/math] cannot have genuine quantum isometries.

Show Proof

There is a long story with this result, which solves an old conjecture, and whose proof is non-trivial, and for details, we refer to Goswami's paper .

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In short, tough mathematics here, that we won't get into. This being said, in order to get a feeling for this, here is a particular case of Theorem 3.21, coming with proof:

Proposition

A compact connected Riemannian manifold [math]X[/math] cannot, in particular, have genuine group dual isometries.

Show Proof

Assume indeed that we have a group dual coaction, as follows:

[[math]] \Phi:C(X)\to C(X)\otimes C^*(\Gamma) [[/math]]

Let [math]E=E_1\oplus E_2[/math] be the direct sum of two eigenspaces of the Laplacian [math]\Delta[/math]. Pick a basis [math]\{x_i\}[/math] such that the corresponding corepresentation on [math]E[/math] becomes diagonal, in the sense that we have, for certain group elements [math]g_i\in\Gamma[/math]:

[[math]] \Phi(x_i)=x_i\otimes g_i [[/math]]

The formula [math]\Phi(x_ix_j)=\Phi(x_jx_i)[/math] reads then:

[[math]] x_ix_j\otimes g_ig_j=x_ix_j\otimes g_jg_i [[/math]]

Now since the eigenfunctions of [math]\Delta[/math] are well-known to form a domain, we obtain:

[[math]] g_ig_j=g_jg_i [[/math]]

Similarly, [math]\Phi(x_i\bar{x}_j)=\Phi(\bar{x}_jx_i)[/math] gives [math]g_ig_j^{-1}=g_j^{-1}g_i[/math]. Thus [math]\{g_i,g_i^{-1}\}[/math] pairwise commute, and with the eigenspace [math]E[/math] varying, this shows that [math]\Gamma[/math] must be abelian, as claimed.

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The above is nice and fun, and there are probably some more things to be done here, along the lines of Theorem 3.4. However, as a word of warning, such ideas lead nowhere in the general context of Theorem 3.21. For the proof of that theorem, see .

Getting back now to Goswami's foundational paper , let us discuss the extension of the construction of [math]\mathcal G^+(X)[/math], to the case where [math]X[/math] is a noncommutative compact Riemannian manifold in the sense of Connes . This is again heavy mathematics, with the full understanding of Connes' axiomatization in requiring the reading of his subsequent “reconstruction” paper , and also of his papers with Chamseddine , for examples and motivations, and with the work of Goswami coming on top of that. So, let us be a bit informal here, and formulate things as follows:

Theorem

The theory of compact Riemannian manifolds [math]X[/math] can be extended into a theory of noncommutative compact Riemannian manifolds [math]X[/math], using spectral triples

[[math]] X=(A,H,D) [[/math]]

in the sense of Connes. In this framework, we can talk about the corresponding quantum isometry groups [math]\mathcal G^+(X)[/math], constructed by using commutation with [math]D[/math].

Show Proof

This is something well-beyond the purposes of the present book, with the main references, including , , being those indicated above.

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Now back to our spheres, the free ones do not have spectral triples in the sense of Connes, but there are a few ways of talking about geometry and [math]\mathcal G^+(X)[/math], as follows:

(1) As explained in , it is possible to construct a Laplacian filtration for [math]S^{N-1}_{\mathbb R,+}[/math], meaning eigenspaces but no eigenvalues, and so no operator itself, as being the pullback of the Laplacian filtration for [math]S^{N-1}_\mathbb R[/math], via the embedding [math]S^{N-1}_\mathbb R\subset S^{N-1}_{\mathbb R,+}[/math]. But that is enough in order to talk about [math]\mathcal G^+(S^{N-1}_{\mathbb R,+})[/math], with the result of course that this quantum group coincides with [math]G^+(S^{N-1}_{\mathbb R,+})=O_N^+[/math]. For details here, and extensions, we refer to , .

(2) More recently, the paper of Das-Franz-Wang contains a proposal for the eigenvalues of the Laplacian of [math]S^{N-1}_{\mathbb R,+}[/math], motivated by their questions there, which is non-trivial, beautiful, and most likely correct, from a physical viewpoint. So, as a hot topic now, we have the question of extending the theory in , up to the limits of what can be done. And sky is the limit, when talking about what can be done with [math]\Delta[/math].

So, this is the situation, things doing well, and we refer to and related papers for more on all this. Let us mention, however, as a final comment on the subject, that something not to be ignored is Nash's theorem in . That is one big result in mathematics, and extending it to the noncommutative setting is a key problem. And with this problem being not exactly ours, because are manifolds have coordinates, by definition.

General references

Banica, Teo (2024). "Affine noncommutative geometry". arXiv:2012.10973 [math.QA].