| dc.description.abstract | One of the recent fundamental developments in theoretical high energy physics is the AdS/CFT
correspondence [1, 2, 3, 4] which posits a relationship between Quantum Field Theories (QFT)
in a given dimension and String Theory on a higher dimensional anti- de Sitter (AdS) spacetime.
This has revolutionised our understanding of QFTs (more speci fically conformal fi eld
theories (CFTs)) and string theory/gravity, and has far reaching consequences for explorations
into a vast array of physical phenomena. Using the elegant formalism provided by
this powerful duality, often called \holography", one can now use fundamental physical observables
in QFT to better understand the nature of quantum gravity. The theoretical tools
provide a translation of calculable field theoretic observables into the language of gravity
thereby leading to the construction of holographic models for several interesting QFTs.
Entanglement is a fundamental physical property of all quantum systems. From models
of various condensed matter systems to its application as a tool for secure and fast communication
in quantum information theory [5], it serves as an intersection point between different
subfi elds of physics [6]. From the AdS/CFT point of view quantum entanglement connects
geometry with quantum information, providing a window to understand how the bulk gravity
physics emerges from the holographic fi eld theoretic viewpoint. Probing various aspects of
this connection in detail will be the broad theme of this thesis.
For extended, many-body systems, the most well known measure of quantum entanglement
is the \Entanglement Entropy" (EE) which is also the best understood measure within
the holographic framework. In early 2006, Ryu and Takayanagi (RT) gave a simple and
elegant prescription for computing this quantity using AdS/CFT duality within Einstein
gravity [7, 8]. They proposed that EE for a subsystem within an extended system (QFT),
is computed by the (proper) area of a static, codimension- 2, \extremal" surface inside the
dual AdS spacetime. The RT proposal has passed several non-trivial consistency checks, for
example strong sub-additivity, area law to name a few [9]. A remarkable aspect of the proposal
is the ease with which EE can now be calculated, while it is well known that obtaining
EE from fi rst principles in QFT presents several technical challenges which have so far been
surmounted only in some 2d fi eld theories using the \replica method" [10, 11, 12].
The most intriguing aspect of the RT proposal is its striking similarity to Bekenstein-
Hawking (BH) entropy which is proportional to the area of a black hole horizon, further
confi rming an intimate relationship between entropy and geometry [13, 14, 15, 16]. This
leads to the natural question: what is the connection between EE and BH entropy? This
question has been sharpened recently by Lewkowycz and Maldacena (LM) via the concept of
Generalized Gravitational Entropy which extends the QFT replica trick to a replica symmetry
for the dual space-time [17]. This was used to prove the RT conjecture successfully by deriving
the correct extremal surface equation for two derivative gravity theories. In this thesis I have
studied the generalization of LM method for higher derivative gravity theories [18, 19, 20,
21, 22, 23] describing holographic duals (of QFT's with fi nite number of colours) and fi nite
't Hooft coupling which takes the AdS/CFT correspondence beyond the usual supergravity
limit. If one wants to use AdS/CFT to study real life systems then it is absolutely necessary to
incorporate the nite coupling effect into the theory and hence the study of higher derivative
effects becomes very important. In these two papers [21, 22] I have formulated a proof for
the existence of the entropy functionals for certain higher derivative theories extending LM
method. We have shown that the for a certain special class of higher derivative theories
there exist well de fined entropy functionals. To extend this proof for more general theories
of gravity opens up a possibility of breaking replica symmetry in the bulk space-time [24].
For higher derivative gravity, black hole entropy for a large class of stationary black
holes with bifurcate killing horizon is given by the well known Wald prescription [25, 26, 27]
which relates the concept of the Noether charge with the black hole entropy. Iyer and Wald
proposed a generalization for dynamical horizons. This throws up the question whether
there is a relation between these EE functionals and the Noether charge, and whether we can
derive them using the approach of Iyer and Wald. For a certain class of theories I have shown
that there exists a relation between these two [28] but a more rigorous proof is needed. This
somewhat fi rms up the area-entropy relation for arbitrary surfaces and proves the existence of
holographic EE functionals for higher curvature theories thereby extending the applicability
of Iyer-Wald formalism beyond the bifurcation surface.
Apart from this, it is well known that there exist several measures of quantum entanglement,
each satisfying a variety of mathematical inequalities and conditions [5]. Translating
these into the language of holography constrains the dual gravity theory and will lead to
general statements about the consistency of the theory. In this thesis I have discussed one
such measure namely Relative entropy [29], the positivity of which has led to constraints on
the underlying gravity theory [30]. Also entanglement entropy is a very useful tools for probing
renormalization group (RG) flow from the holographic point of view [34, 31, 32, 35, 33].
We end with exploring the concept of renormalized entanglement entropy [36, 37] and its
application in probing RG flow in the context of N = 2 gauged supergravity [38].
References
[1] J. M. Maldacena, \The large N limit of superconformal eld theories and supergravity,"
Adv. Theor. Math. Phys. 2, 231 (1998), [arXiv:hep-th/9711200] | en_US |
