SYK-like Tensor Models on the Lattice

We study large $N$ tensor models on the lattice without disorder. We introduce techniques which can be applied to a wide class of models, and illustrate it by studying some specific rank-3 tensor models. In particular, we study Klebanov-Tarnopolsky model on lattice, Gurau-Witten model (by treating it as a tensor model on four sites) and also a new model which interpolates between these two models. In each model, we evaluate various four point functions at large $N$ and strong coupling, and discuss their spectrum and long time behaviors. We find similarities as well as differences from SYK model. We also generalize our analysis to rank-$D$ tensor models where we obtain analogous results as $D=3$ case for the four point functions which we computed. For $D>5$, we are able to compute the next-to-subleading ${1 \over N}$ corrections for a specific four point function.Comment: 46 pages, 29 figures; v2:typos corrected, reference added; v3:minor revisions, to be published in JHE

Chiral 2D "Strange Metals" from N = 4 SYM

Familiar field theories may contain closed subsectors made out of only fermions, which can be used to explore new and unusual phases of matter in lower dimensions. We focus on the fermionic su(1,1) sector in N=4 SYM and on its ground states, which are Fermi surface states/operators. By computing their spectrum to order $(g_{YM}^2 N)^2$, we argue that fluctuations around this fermi surface, within the sector and in the limit $k_F\rightarrow\infty$, are governed by a chiral 1+1 dimensional sector of the "strange metal" coset $SU(N)_N \otimes SU(N)_N/SU(N)_{2N}$. On the gravity side, the conjectured dual configuration is an $S=0$ degeneration of a rotating black hole. On general grounds we expect that the near horizon excitations of $(S=0,\Omega=1,J\rightarrow\infty)$ degenerations of black holes will be governed by a chiral sector of a 1+1 CFT.Comment: 44 page

On monopole operators in supersymmetric Chern-Simons-matter theories

We discuss monopole operators in $U(N_c)$ Chern-Simons-matter theories in three space-time dimensions. We mention an apparent problem in the matching of such operators in dualities between non-supersymmetric theories, and suggest a possible resolution. A similar apparent problem exists in the mapping of chiral monopole operators in theories with ${\cal N}=2$ supersymmetry. We show that in many theories the lowest naive chiral monopole operator is actually not chiral, and we find the lowest monopole operator that is actually chiral in these theories. It turns out that there are several different forms of this operator, depending on the number of colors, the number of flavours, and the Chern-Simons level. Since we use the supersymmetric index to find the lowest chiral monopoles, our results for these monopoles are guaranteed to be invariant under the dualities in supersymmetric theories. The theories we discuss are believed to be dual in the 't~Hooft large $N_c$ limit to classical high-spin gravity theories. We argue that these theories (supersymmetric or not) should not have classical solutions charged under the $U(1)$ gauge field in the high-spin multiplet.Comment: 39 pages. v2: fixed typo

Dynamic Assessment of Academic Writing for Business Studies

This study explores the application of a formative assessment approach known as Dynamic Assessment (DA), as developed within the Vygotskian sociocultural theory of learning. DA blends instruction with assessment by targeting and further developing students’ Zone of Proximal Development (ZPD). The study investigates whether, and if so, how DA enhances students’ academic writing and conceptual development in business studies over time. DA and Hallidayan Systemic Functional Linguistics (SFL) informed the methodological design of this study, which employed a mixed methods approach in order to track learners’ ZPDs regarding academic writing development. The use of SFL to provide linguistic evidence for student writing development (ZPD) is new in DA and thus an innovative feature of this study. The data consists of six undergraduate business studies students’ three to four drafts of three assessments, which were analysed for textual and ideational meanings, as well as associated text-based interaction (mediation), complemented by student interviews and subject tutors’ written comments. This study extends previous DA studies such as Poehner and Lantolf (2005) in two key ways: i) its explicit focus on the construction of a macrogenre (whole text) as opposed to investigations of decontextualized language fragments, and ii) the range of mediational strategies identified and the consequent expansion of Poehner’s (2005) framework of mediation typologies. The findings suggest that DA, combined with SFL, provides insights into the learners’ maturing writing abilities, which the tutor can nurture further to help the learners internalise them. This study also shows that DA students made more gains than their non-DA counterparts regarding their ability to write a case study analysis genre. Additionally, the findings suggest that students can transfer their academic writing and conceptual knowledge from one assessment task to another, albeit at a varying level. The study, though small in scale, thus supports the view that targeted tutor support enhances students’ academic writing development. Implications are drawn concerning formative writing assessment research and practice in higher education

Gravitational Edge Mode in Asymptotically AdS2_2: JT Gravity Revisited

We study the gravitational edge mode of the Jackiw-Teitelboim (JT) gravity and the constrained $sl(2,\mathbb{R})$ BF theory for the asymptotically AdS$_2$. We revisit the derivation of the Schwarzian theory from the wiggling boundary as an action for the gravitational edge mode. We present an alternative description for the gravitational edge mode from the metric fluctuation with the fixed boundary, which is also known as the would-be gauge mode in the gravity. We clarify the relation between the wiggling boundary and the would-be gauge mode. We demonstrate a natural top-down derivation of $PSL(2,\mathbb{R})$ gauging and the path integral measure of the Schwarzian theory. In the constrained $sl(2,\mathbb{R})$ BF theory, we develop a method for incorporating the gravitational edge mode in the BF theory. In this BF theory coupled to the edge mode, we derive the Schwarzian theory with $PSL(2,\mathbb{R})$ gauging. We show that the Haar measure for the Iwasawa decomposition of $PSL(2,\mathbb{R})$ leads to the path integral measure.Comment: 32 pages, 6 figure

Thermal out-of-time-order correlators, KMS relations, and spectral functions

We describe general features of thermal correlation functions in quantum systems, with specific focus on the fluctuation-dissipation type relations implied by the KMS condition. These end up relating correlation functions with different time ordering and thus should naturally be viewed in the larger context of out-of-time-ordered (OTO) observables. In particular, eschewing the standard formulation of KMS relations where thermal periodicity is combined with time-reversal to stay within the purview of Schwinger-Keldysh functional integrals, we show that there is a natural way to phrase them directly in terms of OTO correlators. We use these observations to construct a natural causal basis for thermal n-point functions in terms of fully nested commutators. We provide several general results which can be inferred from cyclic orbits of permutations, and exemplify the abstract results using a quantum oscillator as an explicit example.Comment: 36 pages + appendices. v2: minor changes + refs added. v3: minor changes, published versio

Quantum groups, non-commutative AdS2AdS_2, and chords in the double-scaled SYK model

We study the double-scaling limit of SYK (DS-SYK) model and elucidate the underlying quantum group symmetry. The DS-SYK model is characterized by a parameter $q$, and in the $q\rightarrow 1$ and low-energy limit it goes over to the familiar Schwarzian theory. We relate the chord and transfer-matrix picture to the motion of a ``boundary particle" on the Euclidean Poincar{\'e} disk, which underlies the single-sided Schwarzian model. $AdS_2$ carries an action of $\mathfrak{s}\mathfrak{l}(2,{\mathbb R}) \simeq \mathfrak{s}\mathfrak{u}(1,1)$, and we argue that the symmetry of the full DS-SYK model is a certain $q$-deformation of the latter, namely $\mathcal{U}_{\sqrt q}(\mathfrak{s}\mathfrak{u}(1,1))$. We do this by obtaining the effective Hamiltonian of the DS-SYK as a (reduction of) particle moving on a lattice deformation of $AdS_2$, which has this $\mathcal{U}_{\sqrt q}(\mathfrak{s}\mathfrak{u}(1,1))$ algebra as its symmetry. We also exhibit the connection to non-commutative geometry of $q$-homogeneous spaces, by obtaining the effective Hamiltonian of the DS-SYK as a (reduction of) particle moving on a non-commutative deformation of $AdS_3$. There are families of possibly distinct $q$-deformed $AdS_2$ spaces, and we point out which are relevant for the DS-SYK model.Comment: 70 pages, 6 figure

Schwinger-Keldysh superspace in quantum mechanics

We examine, in a quantum mechanical setting, the Hilbert space representation of the BRST symmetry associated with Schwinger-Keldysh path integrals. This structure had been postulated to encode important constraints on influence functionals in coarse-grained systems with dissipation, or in open quantum systems. Operationally, this entails uplifting the standard Schwinger-Keldysh two-copy formalism into superspace by appending BRST ghost degrees of freedom. These statements were previously argued at the level of the correlation functions. We provide herein a complementary perspective by working out the Hilbert space structure explicitly. Our analysis clarifies two crucial issues not evident in earlier works: firstly, certain background ghost insertions necessary to reproduce the correct Schwinger-Keldysh correlators arise naturally. Secondly, the Schwinger-Keldysh difference operators are systematically dressed by the ghost bilinears, which turn out to be necessary to give rise to a consistent operator algebra. We also elaborate on the structure of the final state (which is BRST closed) and the future boundary condition of the ghost fields.Comment: 30 page