Semicontinuity of the 4d N=2 spectrum under renormalization group flow beyond hypersurface-singularity SCFTs

Xie and Yau (JHEP 03 (2016) 094) proved, as a corollary of Steenbrink's semicontinuity theorem for singularity spectra, that the ordered list of scaling dimensions of the Seiberg-Witten parameters — the N=2 spectrum — of four-dimensional N=2 superconformal field theories defined by isolated quasi-homogeneous hypersurface three-fold singularities is componentwise non-increasing under renormalization group flows on the Coulomb branch, and that every half-open interval $(t, t+1]$ is a semicontinuity domain in singularity-spectrum variables. They conjectured that similar semicontinuity behavior is always true for general N=2 theories, and asked for a verification of the stronger domain-counting statement and a determination of the semicontinuity domains for class S theories. The problem: prove or refute the componentwise and domain semicontinuity of the N=2 spectrum for 4d N=2 SCFTs outside the quasi-homogeneous hypersurface class — class S theories, complete-intersection-singularity theories, and general interacting N=2 SCFTs — and identify the semicontinuity domains where it holds.

Background

A four-dimensional $\mathcal{N}=2$ superconformal field theory (SCFT) is a quantum field theory in four spacetime dimensions with eight supercharges and an $SU(2)_R \times U(1)_r$ R-symmetry (the symmetry acting on the supercharges). Its vacua form a moduli space of degenerate ground states; the component parameterized by expectation values of the half-BPS chiral primary operators $E_{r,(0,0)}$ — scalar chiral operators, labeled by their $U(1)_r$ charge, that are annihilated by half of the supercharges and thereby protected from quantum corrections — is called the Coulomb branch. At a generic Coulomb-branch point the theory reduces to a free abelian $U(1)^r$ gauge theory, $r$ being the rank, and the exact low-energy physics — including the singular loci at which extra massless states and new interacting SCFTs appear — is encoded in the Seiberg-Witten solution, i.e. a family of Riemann surfaces fibered over the Coulomb branch together with a holomorphic one-form (the Seiberg-Witten curve and differential).

Following Xie and Yau, the spectrum of an $\mathcal{N}=2$ theory is the ordered list of rational numbers $\Delta_1 \ge \Delta_2 \ge \cdots \ge \Delta_\mu$ formed by the scaling dimensions of all parameters of the Seiberg-Witten description: Coulomb-branch moduli $u$ (dimension $\Delta > 1$), mass parameters $m$ (dimension 1), and coupling constants $\lambda$; for each operator of dimension $\Delta > 2$ the conjugate coupling entry $2-\Delta$ is included, so the list is symmetric about 1, its entries sum to $\mu$, and $\mu = 2r + f$ with $f$ the number of mass parameters. For example, the $A_3$ Argyres-Douglas theory — Argyres-Douglas theories being the prototypical strongly-coupled SCFTs discovered at special singular loci of $\mathcal{N}=2$ gauge theories — has spectrum $(4/3, 1, 2/3)$.

A large class of $\mathcal{N}=2$ SCFTs is defined by isolated quasi-homogeneous hypersurface three-fold singularities $f: (\mathbb{C}^4, 0) \to (\mathbb{C}, 0)$, quasi-homogeneous meaning $f(\lambda^{q_i} z_i) = \lambda f(z_i)$ for suitable weights $q_i$ (Shapere-Vafa; Xie-Yau, Part I of their singularity-theory program). For these theories the Seiberg-Witten geometry is the mini-versal deformation of $f$ — the smallest deformation containing every inequivalent deformation direction — namely $F(z,\lambda) = f + \sum_\alpha \lambda_\alpha \phi_\alpha$, where the $\phi_\alpha$ form a monomial basis of the Jacobi algebra $J_f = \mathbb{C}[z_0,\ldots,z_3]/(\partial f/\partial z_0, \ldots, \partial f/\partial z_3)$, the ring of polynomials modulo the ideal generated by the partial derivatives of $f$; its dimension equals $\mu$. Isolated hypersurface singularities also carry an intrinsic spectrum $(l_1 \le l_2 \le \cdots \le l_\mu) \subset (0,2)$ of rational numbers, defined through the mixed Hodge structure (a canonical linear-algebraic refinement of cohomology from Hodge theory) on the cohomology of the Milnor fiber — the smooth fiber of a small perturbation of $f$, on which going once around the origin of the perturbation parameter acts by the monodromy transformation, whose eigenvalues are $\exp(2\pi i\, l_\alpha)$ — computable from the weights in the quasi-homogeneous case. The physical and singularity spectra are related by $\Delta_\alpha = 1 + (1-l_\alpha)/l_1$.

A classical theorem of singularity theory — conjectured by Arnold and proved by Steenbrink in 1985, after partial results by others — states that every half-open interval $(t, t+1]$ is a semicontinuity domain: under any deformation of an isolated hypersurface singularity, the number of singularity-spectrum numbers lying in such an interval cannot increase. Since a renormalization-group (RG) flow on the Coulomb branch of a singularity-defined theory is exactly a deformation of $f$, Xie and Yau deduced that under any flow triggered by relevant deformations (perturbations by operators of scaling dimension smaller than four, which grow in strength toward the infrared) or by Coulomb-branch expectation values, the ordered UV and IR spectra obey the componentwise inequalities $\Delta_i \ge \Delta_i'$ with $\mu' \le \mu$ (for an IR theory that is a product of decoupled sectors, the sector spectra are concatenated and re-ordered; e.g. the flow $A_3 \to A_2$ has $(4/3, 1, 2/3) \to (6/5, 4/5)$). They derived as consequences the absence of dangerous irrelevant deformations (a coupling of negative scaling dimension can never become relevant in the IR), the four-dimensional a-theorem — monotonicity along RG flow of the Weyl-anomaly coefficient $a$, a number measuring the field content of the theory that appears in the trace of the energy-momentum tensor on curved backgrounds, known in general from Komargodski-Schwimmer — for these flows, and a necessary condition that the spectra of a UV theory and of any candidate RG endpoint must satisfy. The componentwise inequality is the weaker consequence of Steenbrink's theorem; the stronger statement is the full domain-counting semicontinuity, and which sets of real numbers are semicontinuity domains beyond the hypersurface class is precisely what the open problem below asks.

What is missing. The proof rests on the hypersurface-singularity dictionary and does not cover: (a) class S theories, the large class of $\mathcal{N}=2$ theories obtained by compactification of the six-dimensional $(2,0)$ SCFT (the maximally supersymmetric superconformal theory in six dimensions) on punctured Riemann surfaces, with the genus, puncture, and twist data labeling the theory — most of these theories admit no hypersurface-singularity definition; (b) theories defined by isolated complete-intersection three-fold singularities (ICIS), i.e. zero loci of several equations in a higher-dimensional affine space, for which the SCFT dictionary was established with strong evidence by Wang-Xie-Yau-Yau while the corresponding spectrum semicontinuity is, in that paper's words, much less understood; and (c) general interacting $4d$ $\mathcal{N}=2$ SCFTs, for which no singularity model exists and the statement must be made purely in terms of the ordered lists of scaling dimensions of the Seiberg-Witten parameters. Xie and Yau state the expectation that similar semicontinuity behavior is always true, note that the weaker componentwise statement can be checked for some class S flows using the class S moduli-space analysis of Xie-Yonekura, and leave the stronger statement, the semicontinuity domains, and the general case open. The question belongs to the broader program of finding monotonicity laws along RG flow beyond the central charges, such as the a-theorem and Gukov's proposed $\mu$-theorem on the number of relevant operators.

Problem Statement

Let $T_{UV}$ be an interacting $4d$ $\mathcal{N}=2$ SCFT and define its spectrum as the ordered list $\Delta_1 \ge \Delta_2 \ge \cdots \ge \Delta_\mu$ of scaling dimensions of its Seiberg-Witten parameters (Coulomb-branch moduli, mass parameters, and coupling constants, including for each operator of dimension $\Delta > 2$ the conjugate coupling entry $2-\Delta$); for an IR theory that is a product of decoupled sectors, the spectra of the sectors are concatenated and re-ordered. Xie and Yau proved, for theories defined by isolated quasi-homogeneous hypersurface three-fold singularities, that under every renormalization-group (RG) flow on the Coulomb branch — triggered by relevant deformations (couplings of operators of scaling dimension smaller than four) or by expectation values of Coulomb-branch operators — the IR spectrum satisfies the componentwise inequalities

$$\Delta_i \ge \Delta_i' \quad (i = 1, \ldots, \mu'), \qquad \mu' \le \mu,$$

and that, in singularity-spectrum variables, every half-open interval $(t, t+1]$ is a semicontinuity domain, i.e. the number of spectrum numbers in such an interval cannot increase along the flow.

Determine whether the corresponding statements hold for $4d$ $\mathcal{N}=2$ SCFTs outside the quasi-homogeneous hypersurface class, and identify the precise semicontinuity domains where they do: (a) class S theories, i.e. theories obtained by compactification of the six-dimensional $(2,0)$ SCFT on punctured Riemann surfaces; (b) theories defined by isolated complete-intersection three-fold singularities (ICIS), i.e. zero loci of several equations in a higher-dimensional affine space; (c) arbitrary interacting $4d$ $\mathcal{N}=2$ SCFTs, where the statement must be formulated directly in terms of the ordered lists of scaling dimensions of the Seiberg-Witten parameters (Coulomb-branch moduli, mass parameters, and coupling constants).

For any one of the classes (a)-(c), the statement is settled either by a proof, for that class, of the componentwise inequalities together with an identification of the semicontinuity domains for that class, or by a counterexample within that class: a UV SCFT, a deformation of the stated type, and an IR endpoint that either violates the componentwise inequalities or increases the number of spectrum numbers in some half-open interval $(t,t+1]$.

References

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