The degree–rank conjecture for Sasaki–Einstein volumes

Martelli, Sparks and Yau (2008) proved that the volume of any compact Sasaki–Einstein manifold $L$ of dimension $2n-1$, normalized to the round sphere, is an algebraic number, and observed that in all families they examined its degree over $\mathbb{Q}$ obeys $\operatorname{deg}(L)=(n-1)^{\operatorname{rank}(L)-1}$, where $\operatorname{rank}(L)$ is the dimension of the closure of the Reeb orbits; they proposed this degree–rank relation as a conjecture (Eq. (1.19) of their paper) and asked why extremising the explicit volume function yields algebraic numbers of such unexpectedly low degree. This degree–rank relation is a conjecture, and resolving it for all compact Sasaki–Einstein manifolds would explain why the extremal volume is always an algebraic number of such unexpectedly low degree.

Background

A Sasakian manifold is the odd-dimensional analogue of a Kähler manifold (a complex manifold whose metric is compatible with its complex structure): a Riemannian manifold $(L,g_L)$ of dimension $2n-1$ is Sasakian precisely when its metric cone $(X,\bar g)$, built as $\mathbb{R}_{>0}\times L$ with metric $dr^2+r^2 g_L$, is Kähler of complex dimension $n$. If the cone metric is moreover Ricci-flat — equivalently, $g_L$ is Einstein with $\operatorname{Ric}=2(n-1)g_L$ — then $L$ is called Sasaki–Einstein (SE). The Kähler cone of an SE manifold is a Calabi–Yau cone, and it carries a canonical vector field $\xi=J(r\partial_r)$, the Reeb vector field, obtained by rotating the radial direction with the complex structure. Two invariants of the Sasakian structure are central here. The rank, written $\operatorname{rank}(L)$, is the dimension of the closure of the orbits of the Reeb flow: the Reeb field generates a one-parameter subgroup of the compact isometry group whose closure is a torus, of dimension between $1$ and $n$; rank $1$ means the structure is quasi-regular (a circle orbibundle over a Kähler–Einstein orbifold), while rank $>1$ structures are called irregular.

Martelli, Sparks and Yau (2008) showed that the Einstein–Hilbert action (the total scalar curvature) of a Sasakian metric on a link $L$ — the hypersurface of the cone at $r=1$, i.e. the intersection of the cone with the unit sphere — inside a Calabi–Yau cone $X$, restricted to a space of Sasakian metrics, depends only on the Reeb vector field: it is the volume functional $V(\xi)$, and a Sasaki–Einstein metric can exist only at a critical point of $V$ (equivalently, at a zero of the transverse Futaki invariant, the classical obstruction to Kähler–Einstein metrics). Using Duistermaat–Heckman localization — a finite fixed-point formula from equivariant symplectic geometry — and, alternatively, a limit of an equivariant index that counts holomorphic functions on the cone, they expressed $V$ as an explicit function of topological fixed-point data; for a toric cone — one carrying an effective holomorphic action of the complex torus $(\mathbb{C}^{*})^n$ with a dense open orbit, encoded combinatorially by a fan of rational polyhedral cones called the toric diagram — $V$ is a rational function of the components $b_i$ of $\xi$. A key consequence is that the volume of a Sasaki–Einstein manifold relative to the round sphere, $\operatorname{vol}(L)/\operatorname{vol}(S^{2n-1})$, is always an algebraic number, since the critical point of $V$ solves polynomial equations with rational coefficients. Let $\operatorname{deg}(L)$ denote the degree over $\mathbb{Q}$ of this algebraic number, i.e. the degree of its minimal polynomial; thus $\operatorname{deg}(L)=2$ means the normalized volume is a quadratic irrational.

In the known examples the observed degrees are strikingly small. All quasi-regular SE manifolds have rational normalized volume, hence $\operatorname{deg}=1$, and rank $1$. The first irregular examples discovered, the $Y^{p,q}$ metrics on $S^2\times S^3$ in complex dimension $n=3$ — an explicit cohomogeneity-one family constructed by Gauntlett–Martelli–Sparks–Waldram — are irregular, and hence of rank $2$, precisely when $4p^2-3q^2$ is not a square; their normalized volumes lie in the quadratic field $\mathbb{Q}(\sqrt{4p^2-3q^2})$, which has degree $2$ exactly under the same condition. Martelli, Sparks and Yau noticed that in every family they examined the degree and rank obey $\operatorname{deg}(L)=(n-1)^{\operatorname{rank}(L)-1}$, and proposed this as a conjecture, remarking that a priori the degree of the critical value of such a rational function should be much larger. Via the AdS/CFT correspondence — the conjectured duality relating type IIB string theory on $AdS_5\times L$ to a four-dimensional superconformal field theory — the same question concerns the arithmetic of the exact $a$-central charges produced by a-maximization, the field-theoretic extremal principle that fixes the exact superconformal $R$-symmetry by maximizing the $a$ anomaly coefficient (a measure of the theory's number of degrees of freedom); these charges are algebraic by the same finite-dimensional extremal argument.

Problem Statement

Prove or disprove the Martelli–Sparks–Yau degree–rank conjecture: for every compact Sasaki–Einstein manifold $L$ of dimension $2n-1$,

$$\operatorname{deg}(L)=(n-1)^{\operatorname{rank}(L)-1}$$

where $\operatorname{deg}(L)$ is the degree over $\mathbb{Q}$ of the normalized volume $\operatorname{vol}(L)/\operatorname{vol}(S^{2n-1})$ — an algebraic number by the volume-minimisation theorem of Martelli–Sparks–Yau (2008) — and $\operatorname{rank}(L)$ is the dimension of the closure of the orbits of the Reeb vector field. The associated explanatory goal, stated together with the conjecture in the source passage, is to explain why extremising the explicit volume function $V(b)$ (the volume functional written as a rational function of the components of the Reeb vector) — whose critical point is in principle only an isolated root of a large system of polynomial equations with rational coefficients — always yields an algebraic number of such low degree, when a priori the degree would seem to be much larger. The conjecture is stated for quasi-regular and irregular Sasaki–Einstein manifolds alike.

References

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