<p>We investigate pinched geometries in a two-dimensional Lorentzian model of quantum Regge calculus using the tensor renormalization group method. A pinched geometry refers to a configuration with an infinitely long temporal extent, even when the total spacetime area is fixed. We examine several choices of integration measures and triangulations to study whether such geometries can dominate in the limit of infinitely many triangles. Our results indicate that pinched geometries are strongly suppressed, and this suppression is observed across different integral measures and triangulations. These results suggest the possible emergence of smooth geometries as well as a sort of universality for infinitely many triangles.</p>

Yoshiyasu Ito

Daisuke Kadoh

Yuki Sato

Pinched geometries in 2D Lorentzian quantum Regge calculus

We demonstrate a tensor renormalization group (TRG) calculation for a two-dimensional Lorentzian model of quantum Regge calculus (QRC). This model is expressed in terms of a tensor network by discretizing the continuous edge lengths of simplicial manifolds and identifying them as tensor indices. The expectation value of the space-time area, which is obtained through the higher-order TRG method, nicely reproduces the exact value. The Lorentzian model does not have the spike configuration that was an obstacle in the Euclidean QRC, but it still has a length-divergent configuration called a pinched geometry. We find a possibility that the pinched geometry is suppressed by checking the average edge length squared in the limit where the number of simplices is large. This implies that the Lorentzian model may describe smooth geometries, although the investigation of the higher moments is required to make the statement more conclusive. Our results also indicate that TRG is a promising approach to the numerical study of simplicial quantum gravity.

Japan

Tensor network approach to 2D Lorentzian quantum Regge calculus

We quantize the two-dimensional projectable Horǎva-Lifshitz gravity with a bi-local as well as space-like wormhole interaction. The resulting quantum Hamiltonian coincides with the one obtained through summing over all genus in the string field theory for two-dimensional causal dynamical triangulations. This implies that our wormhole interaction can be interpreted as a splitting or joining interaction of one-dimensional strings.

Denmark

Netherlands

Jan Ambjorn

Yuki Hiraga

Wormholes in 2d Horǎva-Lifshitz quantum gravity

The Bannai–Ito algebra can be defined as the centralizer of the coproduct embedding of <math><mrow><mi>osp</mi></mrow><mo>(</mo><mn>1</mn><mo>|</mo><mn>2</mn><mo>)</mo></math> in <math><mrow><mi>osp</mi></mrow><msup><mrow><mo>(</mo><mn>1</mn><mo>|</mo><mn>2</mn><mo>)</mo></mrow><mrow><mo>⊗</mo><mi>n</mi></mrow></msup></math>. It will be shown that it is also the commutant of a maximal Abelian subalgebra of <math><mi>o</mi><mo>(</mo><mn>2</mn><mi>n</mi><mo>)</mo></math> in a spinorial representation and an embedding of the Racah algebra in this commutant will emerge. The connection between the two pictures for the Bannai–Ito algebra will be traced to the Howe duality which is embodied in the <math><mi>P</mi><mi>i</mi><mi>n</mi><mo>(</mo><mn>2</mn><mi>n</mi><mo>)</mo><mo>×</mo><mrow><mi>osp</mi></mrow><mo>(</mo><mn>1</mn><mo>|</mo><mn>2</mn><mo>)</mo></math> symmetry of the massless Dirac equation in <math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msup></math>. Dimensional reduction to <math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math> will provide an alternative to the Dirac–Dunkl equation as a model with Bannai–Ito symmetry.