RealTimeChess: Lessons from a Participatory Design Process for a Collaborative Multi-Touch, Multi-User Game

We report on a long-term participatory design process during which we designed and improved RealTimeChess, a collaborative but competitive game that is played using touch input by multiple people on a tabletop display. During the design process we integrated concurrent input from all players and pace control, allowing us to steer the interaction along a continuum between high-paced simultaneous and low-paced turn-based gameplay. In addition, we integrated tutorials for teaching interaction techniques, mechanisms to control territoriality, remote interaction, and alert feedback. Integrating these mechanism during the participatory design process allowed us to examine their effects in detail, revealing for instance effects of the competitive setting on the perception of awareness as well as territoriality. More generally, the resulting application provided us with a testbed to study interaction on shared tabletop surfaces and yielded insights important for other time-critical or attention-demanding applications.

Integrable Matrix Product States from boundary integrability

We consider integrable Matrix Product States (MPS) in integrable spin chains and show that they correspond to "operator valued" solutions of the so-called twisted Boundary Yang-Baxter (or reflection) equation. We argue that the integrability condition is equivalent to a new linear intertwiner relation, which we call the "square root relation", because it involves half of the steps of the reflection equation. It is then shown that the square root relation leads to the full Boundary Yang-Baxter equations. We provide explicit solutions in a number of cases characterized by special symmetries. These correspond to the "symmetric pairs" $(SU(N),SO(N))$ and $(SO(N),SO(D)\otimes SO(N-D))$, where in each pair the first and second elements are the symmetry groups of the spin chain and the integrable state, respectively. These solutions can be considered as explicit representations of the corresponding twisted Yangians, that are new in a number of cases. Examples include certain concrete MPS relevant for the computation of one-point functions in defect AdS/CFT.Comment: 33 pages, v2: minor corrections, references added, v3: minor modifications, v4: minor modification

The continuum limit of aN−1(2)a_{N-1}^{(2)} spin chains

Building on our previous work for $a_2^{(2)}$ and $a_3^{(2)}$ we explore systematically the continuum limit of gapless $a_{N-1}^{(2)}$ vertex models and spin chains. We find the existence of three possible regimes. Regimes I and II for $a_{2n-1}^{(2)}$ are related with $a_{2n-1}^{(2)}$ Toda, and described by $n$ compact bosons. Regime I for $a_{2n}^{(2)}$ is related with $a_{2n}^{(2)}$ Toda and involves $n$ compact bosons, while regime II is related instead with $B^{(1)}(0,n)$ super Toda, and involves in addition a single Majorana fermion. The most interesting is regime III, where {\sl non-compact} degrees of freedom appear, generalising the emergence of the Euclidean black hole CFT in the $a_{2}^{(2)}$ case. For $a_{2n}^{(2)}$ we find a continuum limit made of $n$ compact and $n$ non-compact bosons, while for $a_{2n-1}^{(2)}$ we find $n$ compact and $n-1$ non-compact bosons. We also find deep relations between $a_{N-1}^{(2)}$ in regime III and the gauged WZW models $SO(N)/SO(N-1)$.Comment: 43 pages, 4 figure

Non compact continuum limit of two coupled Potts models

We study two $Q$-state Potts models coupled by the product of their energy operators, in the regime $2 < Q \le 4$ where the coupling is relevant. A particular choice of weights on the square lattice is shown to be equivalent to the integrable $a_3^{(2)}$ vertex model. It corresponds to a selfdual system of two antiferromagnetic Potts models, coupled ferromagnetically. We derive the Bethe Ansatz equations and study them numerically for two arbitrary twist angles. The continuum limit is shown to involve two compact bosons and one non compact boson, with discrete states emerging from the continuum at appropriate twists. The non compact boson entails strong logarithmic corrections to the finite-size behaviour of the scaling levels, the understanding of which allows us to correct an earlier proposal for some of the critical exponents. In particular, we infer the full set of magnetic scaling dimensions (watermelon operators) of the Potts model.Comment: 33 pages, 10 figures v2: reference added, minor typo corrected v3: revised version for publication in JSTAT: section 3.1 added, some technical content moved to appendi