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(29 Apr 2013)
posted to no-tag
by NitinCR
on 2013-04-30 09:09:32
Abstract
We study the questions of how supersymmetry is spontaneously broken in Anti de-Sitter spacetime. We verify that the would-be R-symmetry in $AdS_4$ plays a central role for the existence of meta-stable supersymmetry breaking. To illustrate, some well-known models such as Poloyni models and O'Raifeartaigh models are investigated in detail. Our calculations are reliable in flat spacetime limit and confirm us that meta-stable vacua are generic even though quantum corrections are taken into account. ...
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(29 Apr 2013)
posted to no-tag
by NitinCR
on 2013-04-30 09:09:07
Abstract
We review the construction of braided tensor categories and modular tensor categories from representations of vertex operator algebras, which correspond to chiral algebras in physics. The extensive and general theory underlying this construction also establishes the operator product expansion for intertwining operators, which correspond to chiral vertex operators, and more generally, it establishes the logarithmic operator product expansion for logarithmic intertwining operators. We review the main ideas in the construction of the tensor product bifunctors and the associativity isomorphisms. For rational and logarithmic conformal field theories, we review the ...
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(24 Feb 2013)
posted to no-tag
by NitinCR
on 2013-02-26 09:12:15
Abstract
We study some examples of complex, classical, scalar fields within the new framework that we introduced in a previous work. In these particular examples, we replace the usual functional integral by a complex functional arising from partial Wick rotation of a quantum field. We generalize the Feynman-Kac relation to this setting, and use it to establish the spectral condition on a cylinder. We also consider positive-temperature states. ...
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(25 Feb 2013)
posted to no-tag
by NitinCR ✚
on 2013-02-26 08:56:35
along with 1 person
pvmoniz
Abstract
These lectures from the TASI 2012 summer school outline the basics of supersymmetry (SUSY) in 3+1 dimensions. Starting from a ground-up development of superspace, we develop all of the tools necessary to construct SUSY lagrangians. While aimed at an introductory level, these lectures incorporate a number of "super-tricks" for SUSY aficionados, including SUSY-covariant derivatives, equations of motion in superspace, background field methods, and non-linear realizations of goldstinos. ...
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(18 Feb 2013)
Abstract
A book Chapter consisting of some of the main areas of research in graph theory applied to physics. It includes graphs in condensed matter theory, such as the tight-binding and the Hubbard model. It follows the study of graph theory and statistical physics by means of the analysis of the Potts model. Then, we consider the use of graph polynomials in solving Feynman integrals, graphs and electrical networks, vibrational analysis in networked systems and random graphs. The second part deals with the study of complex networks and includes ...
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(18 Feb 2013)
posted to no-tag
by NitinCR
on 2013-02-19 09:34:26
Abstract
We describe an approach to logarithmic conformal field theories as limits of sequences of ordinary conformal field theories with varying central charge c. Logarithmic behaviour arises from degeneracies in the spectrum of scaling dimensions at certain values of c. The theories we consider are all invariant under some internal symmetry group, and logarithmic behaviour occurs when the decomposition of the physical observables into irreducible operators becomes singular. Examples considered are quenched random magnets using the replica formalism, self-avoiding walks as the n->0 of the O(n) model, and percolation as ...
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(18 Feb 2013)
posted to no-tag
by NitinCR
on 2013-02-19 09:34:16
Abstract
The Hessian's determinant for a version of massive gravity given by an infinite expansion of a square root function of the induced metric, vanishes. We show that it allows us to eliminate one of four scalar fields used to generate the graviton mass. This, however, gives rise to the appearance of extra terms in the action with the squared time derivative of the metric, thus signaling that a nonlinear ghost survives.We demonstrate this phenomenon considering a simple system with constraint, which is supposed to reduces the number of ...
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(4 Feb 2013)
Abstract
In this lecture note, we discuss the distinction and possible equivalence between scale invariance and conformal invariance in relativistic quantum field theories. As of January 2013, our consensus is that there is no known example of scale invariant but non-conformal field theories in d=4 under the assumptions of (1) unitarity, (2) Poincaré invariance (causality), (3) discrete spectrum in scaling dimension, (4) existence of scale current and (5) unbroken scale invariance. We have a perturbative proof based on the higher dimensional analogue of Zamolodchikov's c-theorem, but the non-perturbative proof is ...
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(31 Jan 2013)
posted to no-tag
by NitinCR
on 2013-02-01 13:22:47
Abstract
Starting from the Virasoro algebra and its relatives the generalization to higher genus compact Riemann surfaces was initiated by Krichever and Novikov. The elements of these algebras are meromorphic objects which are holomorphic outside a finite set of points. A crucial and non-trivial point is to establish an almost-grading replacing the honest grading in the Virasoro case. Such an almost-grading is given by splitting the set of points of possible poles into two non-empty disjoint subsets. Krichever and Novikov considered the two-point case. Schlichenmaier studied the most general ...
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(19 Feb 2013)
posted to no-tag
by NitinCR ✚
on 2013-02-01 12:49:45
along with 1 person
pvmoniz
Abstract
We show that in many cases nontrivial and complicated supersymmetric quantum mechanical (SQM) models can be obtained from the simple model describing free dynamics in flat complex space by two operations: (i) Hamiltonian reduction and (ii) similarity transformation of the complex supercharges. We conjecture that it is true for any SQM model. ...
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(30 Jan 2013)
posted to no-tag
by NitinCR
on 2013-01-31 11:35:59
Abstract
In a spacetime with no global timelike Killing vector, we do not have a natural choice for the vacuum state of matter fields, leading to the ambiguity in defining the Feynman propagators. In this paper, choosing the vacuum state as the instantaneous ground state of the Hamiltonian at each moment, we develop a method for calculating wave functions associated with the vacuum and the corresponding in-in and in-out propagators. We apply this method to free scalar field theory in de Sitter space and obtain de Sitter invariant ...
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(25 Jan 2013)
Abstract
We study the thermal partition function of level $k$ U(N) Chern-Simons theories on $S^2$ interacting with matter in the fundamental representation. We work in the 't Hooft limit, $N,k\to∞$, with $λ = N/k$ and $\fracT^2 V_2N$ held fixed where $T$ is the temperature and $V_2$ the volume of the sphere. An effective action proposed in <a href="/abs/1211.4843">arXiv:1211.4843</a> relates the partition function to the expectation value of a `potential' function of the $S^1$ holonomy in pure Chern-Simons theory; in several examples we compute the holonomy potential as a function ...
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(28 Jan 2013)
posted to no-tag
by NitinCR
on 2013-01-29 09:49:16
Abstract
Topological defects are interfaces joining two conformal field theories, for which the energy momentum tensor is continuous across the interface. A class of the topological defects is provided by the interfaces separating two bulk systems each described by its own Lagrangian, where the two descriptions are related by a discrete symmetry. In this paper we elaborate on the cases in which the discrete symmetry is a bosonic or a fermionic T- duality. We review how the equations of motion imposed by the defect encode the general bosonic T- ...
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(27 Jan 2013)
posted to no-tag
by NitinCR
on 2013-01-29 09:48:06
Abstract
In this article we review the relation between string theory and non-equilibrium physics based on our previously published work. First we explain why a theory of quantum gravity and non-equilibrium statistical physics should be related in the first place. Then we present the necessary background from the recent research in non-equilibrium physics. The review discusses the relationship of string theory and aging phenomena, as well as the connection between AdS/CFT correspondence and the Jarzynski identity. We also discuss the emergent symmetries in fully developed turbulence and the corresponding non-equilibrium ...
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(21 Jan 2013)
Abstract
If a black hole starts in a pure quantum state and evaporates completely by a unitary process, the von Neumann entropy of the Hawking radiation initially increases and then decreases back to zero when the black hole has disappeared. Here numerical results are given for an approximation to the time dependence of the radiation entropy under an assumption of fast scrambling, for large nonrotating black holes that emit essentially only photons and gravitons. The maximum of the von Neumann entropy then occurs after about 53.81% of the evaporation ...
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(23 Jan 2013)
posted to no-tag
by NitinCR ✚
on 2013-01-24 05:46:25
along with 1 person
pvmoniz
Abstract
This lecture reviews aspects of and prospects for progress towards a theory of quantum gravity from a particle physics perspective, also paying attention to recent findings of the LHC experiments at CERN. ...
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(14 Jan 2013)
posted to no-tag
by NitinCR
on 2013-01-16 13:41:28
Abstract
Four decades after its first postulation by Bekenstein, black hole entropy remains mysterious. It has long been suggested that the entanglement entropy of quantum fields on the black hole gravitational background should represent at least an important contribution to the total Bekenstein-Hawking entropy, and that the divergences in the entanglement entropy should be absorbed in the renormalization of the gravitational couplings. In this talk, we describe how an improved understanding of black hole entropy is obtained by combining these notions with the renormalization group. By introducing an RG ...
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(15 Jan 2013)
posted to no-tag
by NitinCR
on 2013-01-16 13:41:23
Abstract
We introduce gauge networks as generalizations of spin networks and lattice gauge fields to almost-commutative manifolds. The configuration space of quiver representations (modulo equivalence) in the category of finite spectral triples is studied; gauge networks appear as an orthonormal basis in a corresponding Hilbert space. We give many examples of gauge networks, also beyond the well-known spin network examples. We find a Hamiltonian operator on this Hilbert space, inducing a time evolution on the C*-algebra of gauge network correspondences. Given a representation in the category of spectral triples of ...
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(9 Jan 2013)
Abstract
In Part I, we extend our analysis in [<a href="/abs/0807.1107">arXiv:0807.1107</a>], and show that a mathematically conjectured geometric Langlands duality for complex surfaces in [1], and its generalizations -- which relate some cohomology of the moduli space of certain ("ramified") G-instantons to the integrable representations of the Langlands dual of certain affine (sub) G-algebras, where G is any compact Lie group -- can be derived, purely physically, from the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent. In Part II, to the setup in ...
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(11 Jan 2013)
posted to no-tag
by NitinCR
on 2013-01-14 18:11:20
Abstract
We report the results obtained in the study of Alain Connes noncommutative spectral geometry construction focusing on its essential ingredient of the algebra doubling. We show that such a two-sheeted structure is related with the gauge structure of the theory, its dissipative character and carries in itself the seeds of quantization. From the algebraic point of view, the algebra doubling process has the same structure of the deformed Hops algebra structure which characterizes quantum field theory. ...
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(6 Jan 2013)
Abstract
Foundational investigations in quantum mechanics, both experimental and theoretical, gave birth to the field of quantum information science. Nevertheless, the foundations of quantum mechanics themselves remain hotly debated in the scientific community, and no consensus on essential questions has been reached. Here, we present the results of a poll carried out among 33 participants of a conference on the foundations of quantum mechanics. The participants completed a questionnaire containing 16 multiple-choice questions probing opinions on quantum-foundational issues. Participants included physicists, philosophers, and mathematicians. We describe our findings, identify commonly held ...
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(11 Jan 2013)
Abstract
The first part of this text is a gentle exposition of some basic constructions and results in the extended prequantum theory of Chern-Simons-type gauge field theories. We explain in some detail how the action functional of ordinary 3d Chern-Simons theory is naturally localized ("extended", "multi-tiered") to a map on the universal moduli stack of principal connections, a map that itself modulates a circle-principal 3-connection on that moduli stack, and how the iterated transgressions of this extended Lagrangian unify the action functional with its prequantum bundle and with the WZW-functional. ...
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Abstract
Oscillons are massive, long-lived, localized excitations of a scalar field. We show that in a large class of well-motivated single-field models, inflation is followed by self-resonance, leading to copious oscillon generation and a lengthy period of oscillon domination. These models are characterized by an inflaton potential which has a quadratic minimum and is shallower than quadratic away from the minimum. This set includes both string monodromy models and a class of supergravity inspired scenarios, and is in good agreement with the current central values of the concordance cosmology ...
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(1 Jan 2013)
posted to no-tag
by NitinCR
on 2013-01-09 06:35:16
Abstract
The word "complexity" is most often used as a meta--linguistic expression referring to certain intuitive characteristics of a natural system and/or its scientific description. These characteristics may include: sheer amount of data that must be taken into account; visible "chaotic" character of these data and/or space distribution/time evolution of a system etc. This talk is centered around the precise mathematical notion of "Kolmogorov complexity", originated in the early theoretical computer science and measuring the degree to which an available information can be compressed. In the first part, I will ...
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(2 Jan 2013)
posted to no-tag
by NitinCR
on 2013-01-09 06:35:15
Abstract
In applications of AdS/CFT to condensed matter systems in 2+1 dimensions, the ABJM model is often used, however the condensed matter models are usually abelian and contain charged fields. We show that a naive reduction of the ABJM model to N=1 does not have the desired features, but we can find an abelian reduction that has most features, and we can also add fundamental fields to the ABJM model to obtain other models with similar properties. ...
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(29 Dec 2012)
posted to no-tag
by NitinCR
on 2013-01-09 06:35:01
Abstract
We study the large N anomalous dimensions of operators in a Leigh-Strassler deformation of N=4 super Yang-Mills theory. The operators that we study have a bare dimension of order N (so that the large N limit is not captured by planar diagrams) and are AdS/CFT dual to giant gravitons. The diagonalization of the dilatation operator factorizes into two problems. One of these problems is solved using a double coset ansatz. The second problem is equivalent to a set of decoupled harmonic oscillators. ...
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(2 Jan 2013)
posted to no-tag
by NitinCR
on 2013-01-03 02:09:55
Abstract
In this work a thorough study of a number of specific supersymmetric sigma-models with extended supersymmetry is performed within the context of generalised complex geometry, more specially the supersymmetric Wess-Zumino-Witten model on a variety of group manifolds. By explicitly calculating the admissible complex structures and the associated pure spinors on the target manifold a full characterisation of the different possible geometries is provided. By using this approach the various aspects of generalised Kaehler geometry can be studied in detail. Also considered are the various isometries present in the model ...
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(2 Jan 2013)
Abstract
This is a concise review of holographic superconductors and superfluids. We highlight some predictions of the holographic models and the emphasis is given to physical aspects rather than to the technical details, although some references to understand the latter are systematically provided. We include gapped systems in the discussion, motivated by the physics of high-temperature superconductivity. In order to do so we consider a compactified extra dimension (with radius R), or, alternatively, a dilatonic field. The first setup can also be used to model cylindrical superconductors; when these ...
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(21 Dec 2012)
Abstract
We establish a direct connection between scattering amplitudes in planar four-dimensional theories and a remarkable mathematical structure known as the positive Grassmannian. The central physical idea is to focus on on-shell diagrams as objects of fundamental importance to scattering amplitudes. We show that the all-loop integrand in N=4 SYM is naturally represented in this way. On-shell diagrams in this theory are intimately tied to a variety of mathematical objects, ranging from a new graphical representation of permutations to a beautiful stratification of the Grassmannian G(k,n) which generalizes the notion ...
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(31 Dec 2012)
Abstract
These notes introduce the fundamentals of black hole geometry, the thermality of the vacuum, and the Hawking effect, in spacetime and its analogues. Stimulated emission of Hawking radiation, the trans-Planckian question, short wavelength dispersion, and white hole radiation in the setting of analogue models are also discussed. No prior knowledge of differential geometry, general relativity, or quantum field theory in curved spacetime is assumed. ...
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posted to no-tag
by NitinCR
on 2013-01-01 09:56:43
Abstract
Minimal representations of a real reductive group G are the "smallest" irreducible unitary representations of G. The author suggests a program of global analysis built on minimal representations from the philosophy: small representation of a group = large symmetries in a representation space. This viewpoint serves as a driving force to interact algebraic representation theory with geometric analysis of minimal representations, yielding a rapid progress on the program. We give a brief guidance to recent works with emphasis on the Schroedinger model. ...
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(31 Dec 2012)
posted to no-tag
by NitinCR
on 2013-01-01 09:56:42
Abstract
We propose a superfield method to construct off-shell actions for N-extended conformal supergravity theories in three space-time dimensions. It makes use of the superform technique to engineer supersymmetric invariants. The method is specifically applied to the case of N=1 conformal supergravity and provides a new realization for the actions of conformal and topological massive supergravities. ...
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(17 Mar 2013)
Abstract
Cluster algebras are commutative rings with a set of distinguished generators having a remarkable combinatorial structure. They were introduced by Fomin and Zelevinsky in 2000 in the context of Lie theory, but have since appeared in many other contexts, from Poisson geometry to triangulations of surfaces and Teichmüller theory. In this expository paper we give an introduction to cluster algebras, and illustrate how this framework naturally arises in Teichmüller theory. We then sketch how the theory of cluster algebras led to a proof of the Zamolodchikov periodicity conjecture ...
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(17 Dec 2012)
posted to no-tag
by NitinCR
on 2012-12-18 15:57:24
Abstract
We prove that the problem of determination of factor gauge groups given the rank of the gauge group at any given vacuum in the Landscape is in the computational complexity class NPC. This extends a result of Denef and Douglas on the computational complexity of determination of the value of the cosmological constant in the Landscape. ...
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(12 Dec 2012)
posted to no-tag
by NitinCR
on 2012-12-14 09:37:01
Abstract
We study the partition function of the N=6 supersymmetric U(N_1)_k x U(N_2)_-k Chern-Simons-matter (CSM) theory, also known as the ABJ theory. For this purpose, we first compute the partition function of the U(N_1) x U(N_2) lens space matrix model exactly. The result can be expressed as a product of q-deformed Barnes G-function and a generalization of multiple q-hypergeometric function. The ABJ partition function is then obtained from the lens space partition function by analytically continuing N_2 to -N_2. The answer is given by min(N_1,N_2)-dimensional integrals and generalizes the ...
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(3 Jul 2012)
Abstract
On January 4, 2012, the centenary of Henri Poincaré's death, a colloquium was held in Nancy, France the subject of which was "Vers une biographie d'Henri Poincaré". Scholars discussed several approaches for writing a biography of Poincaré. In this paper I present a personal and scientific biographical sketch of Poincaré, which does not in any way reflect Poincaré's rich personality and immense activity in science: When Poincaré traveled to parts of Europe, Africa and America, his companions noticed that he knew well everything from statistics to history and ...
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(7 Dec 2012)
posted to no-tag
by NitinCR
on 2012-12-10 10:45:48
Abstract
We highlight the general notion of a relative quantum field theory, which occurs in several contexts. One is in gauge theory based on a compact Lie algebra, rather than a compact Lie group. This is relevant to the maximal superconformal theory in six dimensions. ...
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(7 Dec 2012)
Abstract
We reformulate eleven-dimensional supergravity, including fermions, in terms of generalised geometry, for spacetimes that are warped products of Minkowski space with a $d$-dimensional manifold $M$ with $d≤7$. The reformation has a $E_d(d) × \mathbbR^+$ structure group and is has a local $H_d$ symmetry, where $H_d$ is the double cover of the maximally compact subgroup of $E_d(d)$. The bosonic degrees for freedom unify into a generalised metric, and, defining the generalised analogue $D$ of the Levi-Civita connection, one finds that the corresponding equations of motion are the vanishing of the ...
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(30 Nov 2012)
posted to no-tag
by NitinCR
on 2012-12-05 09:29:09
Abstract
We study the combined effect of world-volume and background fluxes on Euclidean D3-brane instantons in F-theory compactifications. We derive an appropriate form of the fermionic effective action, in which the fermions are topologically twisted and the dynamical effect of fluxes, non-trivial axio-dilaton and warping is taken into account. We study the structure of fermionic zero modes, which determines the form of possible non-perturbative superpotential and F-terms in the four-dimensional effective action. Invariance under SL(2,Z) is discussed in detail, which allows for an interpretation of the results in terms of ...
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(4 Dec 2012)
posted to no-tag
by NitinCR
on 2012-12-05 09:28:59
Abstract
Eugene Wigner's revolutionary vision predicted that the energy levels of large complex quantum systems exhibit a universal behavior: the statistics of energy gaps depend only on the basic symmetry type of the model. Simplified models of Wigner's thesis have recently become mathematically accessible. For mean field models represented by large random matrices with independent entries, the celebrated Wigner-Dyson-Gaudin-Mehta (WDGM) conjecture asserts that the local eigenvalue statistics are universal. For invariant matrix models, the eigenvalue distributions are given by a log-gas with potential $V$ and inverse temperature $β = 1, ...
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(3 Dec 2012)
Abstract
We derive formulas for counting certain classes of vacua in the string/M theory landscape. We do so in the context of the moduli space of M-theory compactifications on singular manifolds with G_2 holonomy. Particularly, we count the numbers of gauge theories with different gauge groups but equal numbers of U(1) factors which are dual to each other. The vacua correspond to various symmetry breaking patterns of grand unified theories. Counting these dual vacua is equivalent to counting the number of conjugacy classes of elements of finite order inside ...
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(4 Dec 2012)
posted to no-tag
by NitinCR ✚
on 2012-12-05 08:55:58
along with 1 person
pvmoniz
Abstract
It is a common wisdom that properties of macroscopic bodies are well described by (semi)classical physics. As we have suggested this wisdom is not applicable to black holes. Despite being macroscopic, black holes are quantum objects. They represent Bose-Einstein condensates of N-soft gravitons at the quantum critical point, where N Bogoliubov modes become gapless. As a result, physics governing arbitrarily-large black holes (e.g., of galactic size) is a quantum physics of the collective Bogoiliubov modes. This fact introduces a new intrinsically-quantum corrections in form of 1/N, as opposed ...
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(3 Dec 2012)
posted to no-tag
by NitinCR ✚
on 2012-12-05 08:53:35
along with 1 person
pvmoniz
Abstract
We review particle physics model building in type IIB string theory and F-theory. This is a region in the landscape where in principle many of the key ingredients required for a realistic model of particle physics can be combined successfully. We begin by reviewing moduli stabilisation within this framework and its implications for supersymmetry breaking. We then review model building tools and developments in the weakly coupled type IIB limit, for both local D3-branes at singularities and global models of intersecting D7-branes. Much of recent model building work ...
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(29 Nov 2012)
Abstract
If quantum mechanics governs nature, black holes must evolve unitarily, providing a powerful constraint on the dynamics of quantum gravity. Such evolution apparently must in particular be nonlocal, when described from the usual semiclassical geometric picture, in order to transfer quantum information into the outgoing state. While such transfer from a disintegrating black hole has the dangerous potential to be violent to generic infalling observers, this paper proposes the existence of a more innocuous form of information transfer, to relatively soft modes in the black hole atmosphere. Simplified ...
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(1 Dec 2012)
posted to no-tag
by NitinCR
on 2012-12-04 15:20:57
Abstract
The total mass, the Witten type gauge conditions and the spectral properties of the Sen--Witten and the 3-surface twistor operators in closed universes are investigated. A non-negative expression, built from the norm of the 3-surface twistor operator and the energy-momentum tensor of the matter fields on a spacelike hypersurface, is found which, in the asymptotically flat/hyperboloidal case, provides a lower bound for the ADM/Bondi--Sachs mass. In closed universes the analogous expression coincides with the first eigenvalue of the Sen--Witten operator, and it is vanishing if and only if ...
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(28 Nov 2012)
Abstract
We describe the experience of an observer falling into a black hole using the AdS/CFT correspondence. In order to do this, we reconstruct the local bulk operators measured by the observer along his trajectory outside the black hole. We then extend our construction beyond the black hole horizon. We show that this is possible because of an effective doubling of the observables in the boundary theory, when it is in a pure state that is close to the thermal state. Our construction allows us to rephrase questions ...
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(29 Nov 2012)
posted to no-tag
by NitinCR
on 2012-11-30 07:14:36
Abstract
The purpose of this thesis is to explore the properties of multiple coincident M2- and M5-branes. We begin with a review of the BLG and ABJM models of multiple M2-branes and our focus will be on their formulation in terms of 3-algebras. We then examine the coupling of multiple M2-branes to the background 3-form and 6-form gauge fields of eleven-dimensional supergravity. In particular we show in detail how a natural generalisation of the Myers flux-terms, along with the resulting curvature of the background metric, leads to mass terms ...
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(28 Nov 2012)
Abstract
These lecture notes recall the conceptual developments which led from the discovery of the neutron to our present understanding of strong interaction physics. ...
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(27 Nov 2012)
posted to no-tag
by NitinCR
on 2012-11-30 07:10:55
Abstract
We describe a novel approach to dimensional reduction in classical field theory. Inspired by ideas from noncommutative geometry, we introduce extended algebras of differential forms over space-time, generalized exterior derivatives and generalized connections associated with the "geometry" of space-times with discrete extra dimensions. We apply our formalism to theories of gauge- and gravitational fields and find natural geometrical origins for an axion- and a dilaton field, as well as a Higgs field. ...
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(29 Nov 2012)
posted to no-tag
by NitinCR ✚
on 2012-11-30 07:10:03
along with 1 person
pvmoniz
Abstract
We study the probability distribution P(Λ) of the cosmological constant Λ in a specific set of KKLT type models of supersymmetric IIB vacua. P(Λ) is divergent at Λ =0^- and the likely value of Λ drops exponentially as the number of complex structure moduli h^2,1 increases. Also, owing to the hierarchical and approximate no-scale structure, the probability of having a positive Hessian (mass squared matrix) approaches unity as h^2,1 increases. ...
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