Completing the cSYK line of CFTs and its rigid AdS$_2$ dual: subleading $1/N$ finiteness, bulk quartic couplings, and rigorous integral prescriptions

Gross and Rosenhaus introduced cSYK, an exactly conformal variant of the Sachdev-Ye-Kitaev (SYK) model obtained by replacing the ultraviolet part of the SYK action with a quadratic bilocal term. The result is a line of fixed points, from a generalized free field at zero coupling to the infrared of SYK at infinite coupling, with a proposed non-gravitational bulk dual: a tower of scalar fields on a rigid AdS$_2$ background whose masses and cubic couplings were extracted at every coupling. Four issues were left open and remain open: (i) whether subleading $1/N$ corrections are UV finite or induce renormalization or anomalies; (ii) the explicit dependence of the bulk quartic and higher couplings on the boundary coupling $J$; (iii) the global construction and interpretation of the bulk theory, including the divergent $\phi_0$-couplings as $J\to\infty$; and (iv) a rigorous prescription for the non-absolutely-convergent convolution integrals on which all computations rest. The problem is to resolve these linked items and thereby complete the boundary line of fixed points and its bulk dual.

Background

The Sachdev-Ye-Kitaev (SYK) model is a quantum mechanics of $N$ Majorana fermions (real, self-conjugate fermionic modes) interacting through random all-to-all $q$-fermion couplings $J_{i_1\cdots i_q}$ whose variance scales as $1/N^{q-1}$. At large $N$ it is solvable: the disorder-averaged two-point function obeys a closed Schwinger-Dyson equation, an integral equation determining the correlator self-consistently, and the dominant large-$N$ diagrams are 'melonic' ladders. In the infrared (IR) the model becomes approximately conformal, i.e. covariant under $SL(2,\mathbb{R})$, the group of conformal reparametrizations of the time line. Its holographic relevance (holography: the equivalence between a quantum gravity theory in anti-de Sitter space and a lower-dimensional conformal field theory, CFT) is that the IR dual involves Jackiw-Teitelboim (JT) gravity, a two-dimensional dilaton gravity theory, on nearly AdS$_2$ spacetime; see Maldacena and Stanford (2016).

Gross and Rosenhaus studied how to read the AdS$_2$ bulk dual of SYK-like models directly from boundary correlators. In a precursor work they extracted the bulk cubic couplings by matching bilinear three-point functions to Witten diagrams (tree diagrams in AdS, the holographic analogue of Feynman diagrams). The paper at issue, 'A line of CFTs: from generalized free fields to SYK' (Gross and Rosenhaus, 2017), then introduced the variant cSYK. A generalized free field is a field whose correlation functions factorize into products of two-point functions, as for a free field, but without underlying fundamental free modes; the SYK bilinears behave this way at large $N$. In cSYK the ultraviolet (UV, short-time) part of the SYK action is replaced by a quadratic bilocal term, written in terms of the two-time function $G(\tau_1,\tau_2)$ viewed as a field. With this modification the model is exactly $SL(2,\mathbb{R})$-invariant for every value of the coupling $J$: a line of fixed points, reducing to a generalized free theory as $J\to 0$ and approaching the IR of SYK as $J\to\infty$. The singlet sector (the $O(N)$-invariant bilinear operators $O_n$ built from fermion pairs, of conformal dimensions $h_n$) was solved at leading order in $1/N$: two-point function, spectrum, operator-product-expansion (OPE) coefficients (the constants governing the short-distance expansion of operator products), and the three-point functions of bilinears are known at all $J$, unifying the generalized-free and SYK-IR answers.

The bulk dual proposed for cSYK is unusual: because cSYK has a line of fixed points, the bulk is a non-gravitational field theory on a rigid (non-dynamical) AdS$_2$ background, with no dilaton (unlike the SYK IR, whose dual is JT gravity). Its content is a boundary scalar $\phi_0$ dual to $O_0=\chi^i\partial_\tau\chi^i$, plus a tower of scalars $\phi_n$ dual to $O_n$, $n\ge 1$, with masses fixed by the dimensions $h_n$. The cubic couplings $\lambda_{nmk}$ were computed at all $J$ and become $J$-independent at large operator indices; at strong coupling the $\lambda_{nmk}$ with $n,m,k\ne 0$ approach the SYK-IR values, so the terms involving $\phi_0$ are the novel part. Three structural features drive the open problems. First, the full couplings involving $\phi_0$ diverge as $J\to\infty$: 'the divergence of these couplings as $J\to\infty$ is simply a consequence of the divergence of the OPE coefficient for $O_0$', whose dimension $h_0$ approaches 2 while its OPE coefficient $c_2^{(0)}$ blows up; in SYK the analogous divergence forced a departure from the exact IR, breaking $SL(2,\mathbb{R})$ and producing large gravitational backreaction (the deformation of the bulk geometry by matter). Second, in the later analysis of SYK at the quartic level (Gross and Rosenhaus, 'All point correlation functions in SYK', 2017), derivative quartic interactions could not be eliminated: 'there should generically be an infinite number of independent quartic terms, with various combinations of derivatives', and the bulk 'should be regarded as a theory of extended objects, rather than local fields'. Third, the diagrammatic computations rest on frequency-space convolution integrals that are UV finite but not absolutely convergent, so their values depend on the evaluation method; the standard treatment turns the convolution into a product of Fourier transforms, and a more rigorous approach due to Gurau (2018) multiplies the frequency-space two-point function by a decaying regulator $e^{-\epsilon|\omega|}$ before taking $\epsilon\to 0$. The finiteness arguments themselves rely on fermion antisymmetry and the randomness of the couplings; the paper observes that unlike the super-renormalizable SYK action, cSYK is classically scale invariant, so possible quantum anomalies (quantum violations of classical symmetries) must be excluded.

Problem Statement

The problem is to complete the cSYK program of Gross and Rosenhaus (2017): establish the boundary line of fixed points beyond leading order in $1/N$, complete and interpret the non-gravitational rigid-AdS$_2$ bulk dual, and place the underlying integral manipulations on a rigorous footing. The primary source states four open items. They are linked components of one objective and are stated here as in the source, with their relations made explicit.

(i) Subleading $1/N$ corrections. The singlet-sector diagrams of cSYK are UV finite at the leading nontrivial order in $1/N$. Determine whether the subleading $1/N$ corrections are also UV finite for generic coupling $J$, or whether they diverge and induce renormalization of the cSYK action or conformal anomalies; if they diverge, identify the divergent diagrams and the induced counterterms or anomaly. The source states: 'It is conceivable that sub-leading 1/N corrections could also be UV finite, but we have not checked this.'

(ii) Bulk quartic and higher couplings. The bulk masses and cubic couplings $\lambda_{nmk}$ are known at all $J$. Determine the explicit dependence of the bulk quartic couplings - and, structurally, the higher-point couplings - on the boundary coupling $J$, at least at leading order in $1/N$, including their behavior at large operator indices and in the limits $J\to 0$ and $J\to\infty$. The source states: 'It will be interesting to see how the form of the bulk quartic couplings depends on the coupling J', pointing to a follow-up work 'in progress' that never appeared. The quartic couplings are those required to reproduce, in the boundary four-point function of the $O_n$, the coefficients of the double-trace blocks (the conformal-block contributions of products of two bilinear operators). Determine whether the expected infinitude of derivative quartic terms is compatible with a local bulk Lagrangian containing finitely many terms.

(iii) Full construction and interpretation of the non-gravitational bulk dual. Beyond masses and cubic couplings, construct the bulk theory and its global interpretation: the degrees of freedom (local fields, or extended objects as anticipated by the quartic-level analysis), the full interaction structure, the treatment of the divergent $\phi_0$-couplings as $J\to\infty$, and the fate of the operator $O_0$ - whose dimension $h_0$ approaches 2 at strong coupling while its OPE coefficient diverges - as the line approaches the SYK infrared, where the SYK dual becomes JT gravity and the geometry becomes dynamical. The construction must cover the whole cSYK line at general coupling $J$ and reproduce the cSYK boundary data (the masses, the couplings $\lambda_{nmk}$, and the leading-order four-point coefficients); a bulk theory valid only in the $J\to\infty$ (SYK conformal-fixed-point) limit, or one that does not reproduce the cSYK couplings along the line, does not resolve this item.

(iv) Rigorous treatment of the non-absolutely-convergent integrals. The convolution integrals in the cSYK diagrammatic expansion are UV finite but not absolutely convergent. Provide a rigorous, systematic prescription that defines these integrals unambiguously, agrees with the standard Fourier-transform evaluation wherever that evaluation is well defined, and is sufficient for the computations required by items (i)-(iii).

Scope and conventions. The model is the cSYK theory defined in the primary source (its action, disorder average, and normalization conventions), with the coupling $J$ ranging over the line of fixed points from the generalized free limit $J\to 0$ to the SYK-infrared limit $J\to\infty$. Progress on individual items is valuable partial progress. The full objective is resolved when the boundary line is established beyond leading order in $1/N$ (items i and iv, which decide whether the line is more than a leading-$1/N$ artifact) and the bulk dual is complete in both its layers - interactions and global structure (items ii and iii).

References

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