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Motivated by string theory on the orbifold ${cal M}/G$ in presence of a Kalb-Ramond field strength $H$, we define the operators that lift the group action to the twisted sectors. These operators turn out to generate the quasi-quantum group $D_{omega}[G]$, introduced in the context of conformal field theory by R. Dijkgraaf, V. Pasquier and P. Roche, with $omega$ a 3-cocycle determined by a series of cohomological equations in a tricomplex combining de Rham, u{C}ech and group cohomologies. We further illustrate some properties of the quasi-quantum group from a string theoretical point of view.
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We present the general form of the operators that lift the group action on the twisted sectors of a bosonic string on an orbifold ${cal M}/G$, in the presence of a Kalb-Ramond field strength $H$. These operators turn out to generate the quasi-quantum group $D_{omega}[G]$, introduced in the context of orbifold conformal field theory by R. Dijkgraaf, V. Pasquier and P. Roche. The 3-cocycle $omega$ entering in the definition of $D_{omega}[G]$ is related to $H$ by a series of cohomological equations in a tricomplex combining de Rham, Cech and group coboundaries. We construct magnetic amplitudes for the twisted sectors and show that $omega=1$ arises as a consistency condition for the orbifold theory. Finally, we recover discrete torsion as an ambiguity in the lift of the group action to twisted sectors, in accordance with previous results presented by E. Sharpe.
We investigate the underlying quantum group symmetry of 2d Liouville and dilaton gravity models, both consolidating known results and extending them to the cases with $mathcal{N} = 1$ supersymmetry. We first calculate the mixed parabolic representation matrix element (or Whittaker function) of $text{U}_q(mathfrak{sl}(2, mathbb{R}))$ and review its applications to Liouville gravity. We then derive the corresponding matrix element for $text{U}_q(mathfrak{osp}(1|2, mathbb{R}))$ and apply it to explain structural features of $mathcal{N} = 1$ Liouville supergravity. We show that this matrix element has the following properties: (1) its $qto 1$ limit is the classical $text{OSp}^+(1|2, mathbb{R})$ Whittaker function, (2) it yields the Plancherel measure as the density of black hole states in $mathcal{N} = 1$ Liouville supergravity, and (3) it leads to $3j$-symbols that match with the coupling of boundary vertex operators to the gravitational states as appropriate for $mathcal{N} = 1$ Liouville supergravity. This object should likewise be of interest in the context of integrability of supersymmetric relativistic Toda chains. We furthermore relate Liouville (super)gravity to dilaton (super)gravity with a hyperbolic sine (pre)potential. We do so by showing that the quantization of the target space Poisson structure in the (graded) Poisson sigma model description leads directly to the quantum group $text{U}_q(mathfrak{sl}(2, mathbb{R}))$ or the quantum supergroup $text{U}_q(mathfrak{osp}(1|2, mathbb{R}))$.
Previously we have indicated the relationship between quantum groups [Phys. Lett A272, (2000)] and strings via WZWN models in the context of applications to cuprates and related materials.The connection between quantum groups and strings is one way of seeing the validity of our previous conjecture [i.e. that a theory for cuprates may be constructed on the basis of quantum groups]. The cuprates seems to exhibit statistics, dimensionality and phase transitions in novel ways. The nature of excitations [i.e. quasiparticle or collective] must be understood. The Hubbard model captures some of the behaviour of the phase transitions in these materials. On the other hand the phases such as stripes in these materials bear relationship to quantum group or string-like solutions. One thus expects that the relevant solutions of Hubbard model may thus be written in terms of stringy solutions. In short this approach may lead to the non-perturbative formualtion of Hubbard and other condensed matter Hamiltonians. The question arises that how a 1-d based symmetry such as quantum groups can be relevant in describing a 3-d [spatial dimensions] system such as cuprates. The answer lies in the key observation that strings which are 1-d objects can be used to describe physics in $d$ dimensions. For example gravity [which is a 3-d [spatial] plus time] phenomenon can be understood in terms of 1-d strings. Thus we expect that 1-d quantum group object induces physics in 2-d and 3-d which may be relevant to the cuprates. We present support for our contention using [numerical] variational Monte-Carlo [MC] applied to 2d d-p model. We also briefly discuss others ways to formulate a string picture for cuprates, namely by exploiting connection between gauge theories and strings and tHooft picture of quark confinement.
We consider a Hayden & Preskill like setup for both maximally chaotic and sub-maximally chaotic quantum field theories. We act on the vacuum with an operator in a Rindler like wedge $R$ and transfer a small subregion $I$ of $R$ to the other wedge. The chaotic scrambling dynamics of the QFT Rindler time evolution reveals the information in the other wedge. The holographic dual of this process involves a particle excitation falling into the bulk and crossing into the entanglement wedge of the complement to $r=R backslash I$. With the goal of studying the locality of the emergent holographic theory we compute various quantum information measures on the boundary that tell us when the particle has entered this entanglement wedge. In a maximally chaotic theory, these measures indicate a sharp transition where the particle enters the wedge exactly when the insertion is null separated from the quantum extremal surface for $r$. For sub-maximally chaotic theories, we find a smoothed crossover at a delayed time given in terms of the smaller Lyapunov exponent and dependent on the time-smearing scale of the probe excitation. The information quantities that we consider include the full vacuum modular energy $R backslash I$ as well as the fidelity between the state with the particle and the state without. Along the way, we find a new explicit formula for the modular Hamiltonian of two intervals in an arbitrary 1+1 dimensional CFT to leading order in the small cross ratio limit. We also give an explicit calculation of the Regge limit of the modular flowed chaos correlator and find examples which do not saturate the modular chaos bound. Finally, we discuss the extent to which our results reveal properties of the target of the probe excitation as a ``stringy quantum extremal surface or simply quantify the probe itself thus giving a new approach to studying the notion of longitudinal string spreading.
We study the ground states and left-excited states of the A_{k-1} N=(2,0) little string theory. Via a theorem by Atiyah [1], these sectors can be captured by a supersymmetric nonlinear sigma model on CP^1 with target space the based loop group of SU(k). The ground states, described by L^2-cohomology classes, form modules over an affine Lie algebra, while the left-excited states, described by chiral differential operators, form modules over a toroidal Lie algebra. We also apply our results to analyze the 1/2 and 1/4 BPS sectors of the M5-brane worldvolume theory.
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