O(D, D) and the string α′ expansion: an obstruction

Double Field Theory (DFT) is an attempt to make the O(d, d) T-duality symmetry of string theory manifest, already before reducing on a d-torus. It is known that supergravity can be formulated in an O(D, D) covariant way, and remarkably this remains true to the first order in α′. We set up a systematic way to analyze O(D, D) invariants, working order by order in fields, which we carry out up to order α′3. At order α′ we recover the known Riemann squared invariant, while at order α′2 we find no independent invariant. This is compatible with the α′ expansion in string theory. However, at order α′3 we show that there is again no O(D, D) invariant, in contradiction to the fact that all string theories have quartic Riemann terms with coefficient proportional to ζ (3). We conclude that DFT and similar frameworks cannot capture the full α′ expansion in string theory.

The Square-Zero Basis of Matrix Lie Algebras

A method is presented that allows one to compute the maximum number of functionally-independent invariant functions under the action of a linear algebraic group as long as its Lie algebra admits a basis of square-zero matrices even on a field of positive characteristic. The class of such Lie algebras is studied in the framework of the classical Lie algebras of arbitrary characteristic. Some examples and applications are also given.

Divisibility of Dynamical Maps with Time Independent Invariant State

A new condition for dynamical maps with time independent invariant state displaying Heisenberg divisibility is provided. This condition is trivially satisfied for a Markovian semigroup. However, beyond Markovian semigroups it provides a nontrivial characteristics of quantum Markovianity. Interestingly, when the invariant state is pure one recovers a condition based on Wigner–Yanase skew information.

Cut points: An invariant of virtual links

Checkerboard framings are an extension of checkerboard colorings for virtual links. From checkerboard framings, we obtain an independent invariant of virtual links: the cut point number. Checkerboard framings and cut points can be used as a tool to extend other classical invariants to virtual links.

The semantics of human facial expressions

This paper points out that a major shift of paradigm is currently going on in the study of the human face and it seeks to articulate and to develop the fundamental assumptions underlying this shift. The main theses of the paper are: 1) Facial expressions can convey meanings comparable to the meanings of verbal utterances. 2) Semantic analysis (whether of verbal utterances or of facial expressions) must distinguish between the context-independent invariant and its contextual interpretations. 3) Certain components of facial behavior (“facial gestures”) do have constant context-independent meanings. 4) The meanings of facial components and configurations of components have an inherent first-person and present tense orientation. 5) The basis for the interpretation of facial gestures is, above all, experiential. 6) The meanings of some facial expressions are universally intelligible and can be interpreted without reference to any local conventions. 7) To be fruitful, the semantic analysis of facial expressions needs a methodology. This can be derived from the methodological experience of linguistic semantics. The author illustrates and supports these theses by analyzing a range of universally interpretable facial expressions such as the following ones: “brow furrowed” (i.e. eyebrows drawn together); eyebrows raised; eyes wide open; corners of the mouth raised; corners of the mouth lowered; mouth open (while not speaking); lips pressed together; upper lip and nose “raised” (and, consequently, nose wrinkled).