New insights into holography from supersymmetric localization

Abstract

Maldacena's conjecture, often known as the holographic duality or the AdS/CFT correspondence, proposes an equivalence between gravitational theories in a hyperbolic space of a certain dimensionality and gauge theories living on its boundary. The manner in which this connection is established makes the duality specially remarkable: both sides are thought to describe the very same string theoretic physics, but the validity regimes of the two descriptions are disjoint, and one expects either framework to be in its regime of validity the appropriate way to effectively reexpress the physics of the other. The nonintersection of these applicability regimes makes the duality very useful, but also very hard to verify and materialize.

Notice the potential implications of this framework for theoretical physics: one direction, strongly coupled quantum field theories become in a certain regime describable as semiclassical gravitational spacetimes, while on the other direction certain string theories without semiclassical background obtain a clean and workable definition as gauge field theories. As a consequence of these facts, holography has played a central role in research since its appearance, almost two decades ago.

However, despite numerous efforts devoted to its characterization, general understanding of the duality has only been majoritarely achieved around the regimes where the gravitational description becomes semiclassical. Consequently, the gauge-to-gravity direction of the duality is far less exploited than the opposite one, despite its conceptual relevance.

Having available results for strongly coupled gauge theories would be of a great help in addressing holography in this comparatively underdevelopped direction, and they would set a fertile ground to test, refine and understand the holographic conjecture. These type of predictions are hard to come by, but not inexistent: non-renormalized magnitudes constitute their most paradygmatical example, and recently different techniques have obtained exact results at arbitrary coupling for specific obserable sectors.

This thesis studies specifically one of these techniques, known as supersymmetric localization, and its role in shoring AdS/CFT. In particular, it restricts its analysis to a specific type of theory: Lagrangian N. = 2 SYM, and for specific results: half-BPS Wilson circular loops. Several interesting insights are put forward by its results.

A first observation is that the exact functional dependence we obtain from localization offer a guide on how to extend holographic predictions from their validity regime to a finite gauge range which produces plausible results, although a great care has to be taken in this process.

Complementarily, the study of this parametrical dependence for gauge N. = 4 theories with gauge Lie algebras presents two suggestive patterns: On the one hand, 't Hooft's topological expansion presents, at least for charges in fundamental representations, an underlying structure that relates sectors with different number of crosscaps among themselves. On the other hand, the matrix model structure obtained in the localization process can be interpreted in terms of a fermionic quantum mechanics, which at the 't Hooft limit matches the "bubbling geometry" structure of Lin, Lunin and Maldacena, but which persists at finite gauge group range.

Finally, the comparison of localization results within a more general type of construction is pre-sented. The specific set of theories considered contains both examples with semiclassical holographic duals and examples where this type of geometry is precluded. Supersymmetric field predictions in this case differentiate both groups with qualitatively different behaviours in the matrix model. This suggests a possible connection between the matrix model structure and the semiclassical spacetime codification in the dual field theory. Similar observations have been made in the literature.

This thesis, therefore, explicits a wide list of suggestions for holography that are motivated by lo-calization results in different regimes, even though the latter have been severely restricted to particular examples.

Aquesta tesi estudia les lliçons que hom pot extreure, des del punt de vista de la dualitat hologràfica, de l'anomenada localització supersimètrica. La conjunció d'aquests dos formalismes ha estat utilitzada anteriorment per verificar els resultats del primer, però aquesta tesi va un pas més enllà i qüestiona fins a quin punt els resultats exactes oferts per la localització poden ser útils per extendre l'aplicabilitat de l'holografia gravitatòria. L'anàlisi d'aquests resultats en diferents règims posa de relleu diversos aspectes d'interès.

Una primera observació és que la dependència funcional exacta que hom pot llegir dels resultats de la localització ofereixen, quan són utilitzats amb cura, una guia per estendre les prediccions holgràfiques des del seu règim de validesa a un rang de gauge finit mitjançant la qual es pot obtenir resultats plausibles.

Complementàriament, l'estudi d'aquesta dependència paramètrica per teories maximalment super-simètriques amb grups de Lie clàssics presenta dos patrons suggerents: d'una banda, el desenvolupa-ment de 't Hooft té, almenys per càrregues en representació fonamental, una estructura subjacent que relaciona els sectors amb diferent nombre d'inversions topològiques entre ells. De l'altra, l'estructura dels models de matrius obtinguts en localitzar pot ésser interpretada en termes d'una mecànica quàntica de fermions, que encaixa amb l'estructura dels "universos de bombolles" de Lin, Lunin i Maldacena al límit de 't Hooft, però que manté la seva forma a qualsevol rang finit.

Per últim, es compara resultats de localització entre ells en un marc de teories més generals, en què algunes admeten dual semiclàssic i d'altres no és d'esperar que ho facin. Les prediccions supersimètriques en aquest cas presenten diferències qualitatives que permeten distingir aquests dos grups de teories, suggerint una possible connexió entre la codificació de l'estructura gravitatòria a la teoria de camps i el model de matrius que s'obté en localitzar que complementa altres indicis en el mateix sentit obtinguts a la literatura.

En definitiva, doncs, es posa de manifest en aquesta tesi que, tot i haver restringit l'anàlisi a sectors i observables específics, els resultats obtinguts per mitjà de la localització supersimètrica ofereixen una rica gamma de suggeriments sobre l'estructura hologràfica.