Representation Of PSL Research Articles

A proposal for 3d quantum gravity and its bulk factorization

Recent progress in AdS/CFT has provided a good understanding of how the bulk spacetime is encoded in the entanglement structure of the boundary CFT. However, little is known about how spacetime emerges directly from the bulk quantum theory. We address this question in an effective 3d quantum theory of pure gravity, which describes the high temperature regime of a holographic CFT. This theory can be viewed as a q-deformation and dimensional uplift of JT gravity. Using this model, we show that the Bekenstein-Hawking entropy of a two-sided black hole equals the bulk entanglement entropy of gravitational edge modes. In the conventional Chern-Simons description, these black holes correspond to Wilson lines in representations of PSL(2, ℝ) ⨂ PSL(2, ℝ). We show that the correct calculation of gravitational entropy suggests we should interpret the bulk theory as an extended topological quantum field theory associated to the quantum semi-group {textrm{SL}}_q^{+}left(2,mathbb{R}right)bigotimes {textrm{SL}}_q^{+}left(2,mathbb{R}right) . Our calculation suggests an effective description of bulk microstates in terms of collective, anyonic degrees of freedom whose entanglement leads to the emergence of the bulk spacetime.

Primitive Permutation Representations of PSL(3, p) and Its Applications

The main result of this paper is a determination of the suborbits of primitive permutation representations of PSL(3, p) relative to a maximal subgroup PGL(2, p), where p ≡ 13(mod 24). As an application, a new family of -transitive graphs and semisymmetric graphs are obtained, respectively.

On gl(2)k Current Superalgebra andTwisted Conformal Field Theory

Motivated by the recently discovered hidden symmetry of the type IIBGreen–Schwarz superstring on certain background, the non-semisimpleKac–Moody twisted superalgebra gl(2)k isinvestigated by means of the vector coherent state method andboson-fermion realization. The free field realization of the twistedcurrent superalgebra at general level k is constructed. Thecorresponding Conformal Field Theory (CFT) has zero central charge.According to the classification theory, this CFT is a nonunitaryfield theory. After projecting out a U(1) factor and an outerautomorphism operator, we get the free field representation ofpsl(2)k, which is the algebra ofgl(2)k modulo the ℤ4-outerautomorphism, the CFT has central charge -2.

Constructive recognition of 𝑃𝑆𝐿(2,𝑞)

Existing black box and other algorithms for explicitly recognising groups of Lie type over G F ( q ) \mathrm {GF}(q) have asymptotic running times which are polynomial in q q , whereas the input size involves only log ⁡ q \log q . This has represented a serious obstruction to the efficient recognition of such groups. Recently, Brooksbank and Kantor devised new explicit recognition algorithms for classical groups; these run in time that is polynomial in the size of the input, given an oracle that recognises P S L ( 2 , q ) \mathrm {PSL}(2,q) explicitly. The present paper, in conjunction with an earlier paper by the first two authors, provides such an oracle. The earlier paper produced an algorithm for explicitly recognising S L ( 2 , q ) \mathrm {SL}(2,q) in its natural representation in polynomial time, given a discrete logarithm oracle for G F ( q ) \mathrm {GF}(q) . The algorithm presented here takes as input a generating set for a subgroup G G of G L ( d , F ) \mathrm {GL}(d,F) that is isomorphic modulo scalars to P S L ( 2 , q ) \mathrm {PSL}(2,q) , where F F is a finite field of the same characteristic as G F ( q ) \mathrm {GF}(q) ; it returns the natural representation of G G modulo scalars. Since a faithful projective representation of P S L ( 2 , q ) \mathrm {PSL}(2,q) in cross characteristic, or a faithful permutation representation of this group, is necessarily of size that is polynomial in q q rather than in log ⁡ q \log q , elementary algorithms will recognise P S L ( 2 , q ) \mathrm {PSL} (2,q) explicitly in polynomial time in these cases. Given a discrete logarithm oracle for G F ( q ) \mathrm {GF}(q) , our algorithm thus provides the required polynomial time oracle for recognising P S L ( 2 , q ) \mathrm {PSL}(2,q) explicitly in the remaining case, namely for representations in the natural characteristic. This leads to a partial solution of a question posed by Babai and Shalev: if G G is a matrix group in characteristic p p , determine in polynomial time whether or not O p ( G ) O_p(G) is trivial.

Extensions of Conformal Nets¶and Superselection Structures

Starting with a conformal Quantum Field Theory on the real line, we show that the dual net is still conformal with respect to a new representation of the Moebius group. We infer from this that every conformal net is normal and conormal, namely the local von Neumann algebra associated with an interval coincides with its double relative commutant inside the local von Neumann algebra associated with any larger interval. The net and the dual net give together rise to an infinite dimensional symmetry group, of which we study a class of positive energy irreducible representations. We mention how superselsection sectors extend to the dual net and we illustrate by examples how, in general, this process generates solitonic sectors. We describe the free theories associated with the lowest weight n representations of PSL(2,R), showing that they violate 3-regularity for n>2. When n>1, we obtain examples of non Moebius-covariant sectors of a 3-regular (non 4-regular) net.

Integral geometry and separation of continuous and discrete spectra in regular representation of PSL (2, R)

The separation of continuous and discrete spectra in regular representation of PSL(2, R is constructed with methods of integral geometry.