Chemistry and Complexity for Solitons in AdS$_5$

Abstract

Minimal D = 5 supergravity admits asymptotically globally AdS 5 gravitational solitons (strictly stationary, geodesically complete spacetimes with positive mass). We show that, like asymptotically flat gravitational solitons, these solutions satisfy mass and mass variation formulas analogous to those satisfied by AdS black holes. A thermodynamic volume associated to the non-trivial topology of the spacetime plays an important role in this construction. We then consider these solitons within the holographic "complexity equals action" and "complexity equals volume" conjectures as simple examples of spacetimes with nontrivial rotation and topology. We find distinct behaviours for the volume and action, with the counterterm for null boundaries playing a significant role in the latter case. For large solitons we find that both proposals yield a complexity of formation proportional to a power of the thermodynamic volume, V 3/4 . In fact, up to numerical prefactors, the result coincides with the analogous one for large black holes.

Holographic Complexity and Thermodynamic Volume

“…However, for rotating black holes the volume and entropy are independent and it becomes clear that it is (24) that captures the correct behaviour, and not an analogous expression involving the entropy. Our result also reproduces the behaviour of the complexity of formation for gravitational solitons [16], which are horizonless geometries that possess thermodynamic volume but no entropy.…”

“…The former relates complexity to the volume of extremal codimension-one surfaces in the bulk, while the latter relates complexity to the value of the gravitational action on a region of spacetime known as the Wheeler-DeWitt (WDW) patch. [12] A number of properties of complexity as defined by the CV and CA proposals are now well-understood for black holes, with both proposals generally yielding qualitatively similar results, but not always [13][14][15][16][17][18]. For example, in both proposals it is known that at late times the complexity grows linearly in time at a rate characterized by the mass, or other thermodynamic potentials, of the black hole [11,[19][20][21][22].…”