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arXiv:2106.14886 · Physical review. D/Physical review. D.

Wormhole calculus without averaging from O(N)q−1 tensor model

Sayantan Choudhury, K. Shirish

Published Feb 2, 2022• 10 citations
arXivHTMLPDF

Abstract

The Sachdev--Ye--Kitaev (SYK) model has a wormholelike solution after averaging over the fermionic couplings in the nearly ${\mathrm{AdS}}_{2}$ space. Even when the couplings are fixed the contribution of these wormholes continues to exist and new saddle points appear which are interpreted as ``half-wormholes.'' In this paper, we will study the fate of these wormholes in a model without quenched disorder, namely, a tensor model with $O(N{)}^{q\ensuremath{-}1}$ gauge symmetry whose correlation function and thermodynamics in the large $N$ limit are the same as that of the SYK model. We will restate the factorization problem linked with the wormhole threaded Wilson operator, in terms of global charges or nontrivial cobordism classes associated with disconnected wormholes. Therefore for the partition function to factorize especially at short distances, there must exist certain topological defects which break the global symmetry associated with wormholes and make the theory devoid of global symmetries. We will interpret these wormholes with added topological defects as our ``half-wormholes.'' We will also comment on the late time behavior of the spectral form factor, particularly its leading and subleading order contributions coming from higher genus wormholes in the gravitational sector. Finally we will show how the other nontrivial saddles from half-wormhole dominate and give rise to unusual thermodynamics in the bulk sector due to nonperturbative effects.

DOI
10.1103/physrevd.105.046002

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The Sachdev--Ye--Kitaev (SYK) model has a wormholelike solution after averaging over the fermionic couplings in the nearly AdS2{\mathrm{AdS}}_{2}AdS2​ space. Even when the couplings are fixed the contribution of these wormholes continues to exist and new saddle points appear which are interpreted as half-wormholes.'' In this paper, we will study the fate of these wormholes in a model without quenched disorder, namely, a tensor model with $O(N{)}^{q\ensuremath{-}1}$ gauge symmetry whose correlation function and thermodynamics in the large $N$ limit are the same as that of the SYK model. We will restate the factorization problem linked with the wormhole threaded Wilson operator, in terms of global charges or nontrivial cobordism classes associated with disconnected wormholes. Therefore for the partition function to factorize especially at short distances, there must exist certain topological defects which break the global symmetry associated with wormholes and make the theory devoid of global symmetries. We will interpret these wormholes with added topological defects as our half-wormholes.'' We will also comment on the late time behavior of the spectral form factor, particularly its leading and subleading order contributions coming from higher genus wormholes in the gravitational sector. Finally we will show how the other nontrivial saddles from half-wormhole dominate and give rise to unusual thermodynamics in the bulk sector due to nonperturbative effects.
In this note we study the SYK model with one time point, recently considered by Saad, Shenker, Stanford, and Yao. Working in a collective field description, they derived a remarkable identity: the square of the partition function with fixed couplings is well approximated by a ``wormhole'' saddle plus a ``pair of linked half-wormholes'' saddle. It explains factorization of decoupled systems. Here, we derive an explicit formula for the half-wormhole contribution. It is expressed through a hyperpfaffian of the tensor of SYK couplings. We then develop a perturbative expansion around the half-wormhole saddle. This expansion truncates at a finite order and gives the exact answer. The last term in the perturbative expansion turns out to coincide with the wormhole contribution. In this sense the wormhole saddle in this model does not need to be added separately, but instead can be viewed as a large fluctuation around the linked half-wormholes.
A bstract After averaging over fermion couplings, SYK has a collective field description that sometimes has “wormhole” solutions. We study the fate of these wormholes when the couplings are fixed. Working mainly in a simple model, we find that the wormhole saddles persist, but that new saddles also appear elsewhere in the integration space — “half-wormholes.” The wormhole contributions depend only weakly on the specific choice of couplings, while the half-wormhole contributions are strongly sensitive. The half-wormholes are crucial for factorization of decoupled systems with fixed couplings, but they vanish after averaging, leaving the non-factorizing wormhole behind.
Abstract In this paper, we introduce a “partially disorder-averaged” SYK model. This model has a real parameter that smoothly interpolates between the ordinary totally disorder-averaged SYK model and the totally fixed-coupling model. For the large N effective description, in addition to the usual bi-local collective fields, we also introduce a new additional set of local collective fields. These local fields can be understood as “half” of the bi-local collective fields, and in the totally fixed-coupling limit, they represent the “half-wormholes” which were found in recent studies. We find that the large N saddles of these local fields vanish in the total-disorder-averaged limit, while they develop nontrivial profiles as we gradually fix the coupling constants. We argue that the bulk picture of these local collective fields represents a correlation between a spacetime brane and the asymptotic AdS boundary. This illuminates how the half-wormhole saddles emerge in the SYK model with fixed couplings.
We briefly introduce and review the Sachdev-Ye-Kitaev (SYK) model. This model exhibits quenched disorder, and many quantities, like the Euclidean path integral, are computed as an average over disorder or an average over many different unitary theories corresponding to different realizations of the disorder couplings. This places the interpretation of the Euclidean path integral as a thermal partition function under scrutiny, which we promptly investigate. We reach he seeming conclusion that the Euclidean path integral indeed does not support a Hilbert space interpretation, which is corroborated by other investigations in the literature. We then move one dimension higher, to two-dimensional Jackiw-Teitelboim (JT) gravity, investigating its salient features and most importantly, the problem of Hilbert space factorization. This puzzle implies that, due to constraints coming from gauge symmetry, a bulk gravitational theory cannot be dual to a theory with a tensor-product structure on disconnected boundary components. We then briefly introduce and review spin glasses, and review the investigations into the presence of a spin glass phase in the SYK model. Finally, we discuss some natural outgrowths of ideas presented in this thesis
We derive a large-N wormhole saddle in the Sachdev-Ye-Kitaev (SYK) model for the variance of the thermal one-point function of a heavy operator across realizations of the microscopic couplings. We propose that this saddle describes a double trumpet geometry. In the double-scaling limit, we express the wormhole as a transition amplitude in an emergent bulk Hilbert space of chords connecting the two boundaries. In the Schwarzian limit, the chords proliferate and the wormhole defines a quantum transition amplitude in the open-universe Hilbert space of Jackiw-Teitelboim (JT) gravity. Comparing the SYK wormhole amplitude with its counterpart in JT gravity coupled to matter provides a direct test of the heavy-operator randomness hypothesis in SYK. The microscopic SYK couplings specify the α-parameters of the wormhole. For a fixed realization of the couplings, we compute the observable in the corresponding α-state. We provide the bulk wavefunction of the associated half-wormhole, which exhibits erratic correlations in the length basis that do not self-average. We extend the bulk description to the Hilbert space of a closed universe. We point out that the correlation chords from which the wormhole geometry emerges structurally resemble local observer worldlines.
Recently it has been shown by Almheiri and Lin [1] that the reconstruction of a black-hole interior is sensitive to knowing the exact coupling of the boundary theory even if the coupling is irrelevant. This motivates us to enlarge the set of the one-time and two-time toy models inspired from the Sachdev-Ye-Kitaev by deforming the same with ``irrelevant'' coupling. We find that half-wormholes, as well as the wormholes, persist in the presence of the deformation, leading to a similar mechanism for curing the factorization problem. While for the one-time case, the deformed partition function and its moments change by an overall factor, which can completely be absorbed into a renormalization of coupling; for the two-time (or coupled one-time) Sachdev-Ye-Kitaev we find nontrivial dynamics of the saddles as the couplings are varied. Curiously, the irrelevant deformations that we consider can also be thought of as an ensemble average over an overall scaling of the original undeformed Hamiltonian with an appropriate probability distribution; this allows for the possibility that half-wormholes may also be present in suitably defined ensemble of theories.