Chern Simons Quiver gauge theory

August 10, 2008

Here us a look at the paper that appeared in the arxiv yesterday:

Moduli spaces of Chern-Simons quiver gauge theories (arXiv:0808.0912v1)

First of all let me describe what is quiver gauge theories. There are singularities associated with Dbranes and this is described by quiver gauge theories. Infact quiver is basically a diagram with nodes. The nodes are the unitary U(N) gauge  group and the vector multiplets. This connect the nodes representing the U(N) gauge groups and the corresponding vector multiplets. Each link represents a field in bifundamental representation.

The above paper in arxiv looks interesting. Supposedly,  the author claim to take the first step toward identifying candidate N=2 conformal Chern Simons quiver gauge theories with ADS4 X Y7 gravity duals.The three dimension Chern Simons gauge theories for N=2 SUSY or higher couple with matter may be ASD4 gauge theory dual. Apparently the simplest vacua of this kind is ADS4 X Y7 where Y7 is the Sasaki- Einstein manifold. The simplest solution are the type of Freund and Rubin soulution that was found for d=11 supergravity (this compactifies S7 (seven sphere) to four dimensional anti de Sitter space.

Interestingly, this is a vacuum of M theory which arises as the dual to the three dimensional conformal field theory of M2 branes in flat space.  This also has a Calabi Yau fourfold singularity. In order to find the field theory dual of these kind of solutions one needs to understand the degrees of freedom of these M2 branes. It may be relevant to note that this line of thinking had originally failed because of the no go theorem that says that chiral fermions cannot be given by a smooth manifold. Maldecena et al, showed that the gauge theory duals of a class of ADS4 X S7 background is the N=6 or 8 Chern Simons quivers with certian properties.

In the case of type II string theory, we can construct  N = 1 AdS5/CFT4 duals by considering N D3-branes placed at a conical Calabi-Yau 3-fold singularity X. One can also construct gauge theories from string degrees of freedom which sits on a brane. The dual theory is given by N=1 d=4 quiver gauge theory. So if one looks at the vacua of the N=1 d=4 quiver gauge theory, the author tell us that its a symmetric product of Calabi Yau singularity X one started with. Now the gravity dual is ADS5 X Y5 where Y5 is the Einstein Sasaki base of a Calabi Yau cone.

In this paper they look at the classical vacuum moduli space of the Chern Simons gauge theories with arbitrary Chern Simons levels. There could include Coulom branch, Higgs branch or a combination of that. They look at certain branches and look at Chern Simons quiver theories in terms of M2-branes at a CY 4-fold singularity, they believe that this branch should reproduce CY 4-fold as the 1moduli space of the transverse M2-branes.  The result is that the vacuum modulis space has a branch which is related to the moduli space of four dimensional N=1 quiver theory.


String Gas Cosmology

August 7, 2008

Here is something that may be interesting to look into.

String Gas Cosmology (arXiv:0808.0746)

An interesting review in String Gas cosmology by Brandenberger. Any attempt to incorporate stringy idea into cosmological model is interesting. The starting point is always crucial.

First of all, as the review points out there are several problems that is associated with string cosmology. One needs to consider some form of scalar field and in general in particle physics we consider Higgs field which is responsible for spontaneous symmetry breaking. However, Higgs is not appropriate in the case of cosmology (inflationary models) as it is too massive (possible somewhere between (115-150 Gev). In order to really understand string cosmology completely, we need a complete perturbative description of string theory. In the absence of these, the ideas that Brandenberger and company uses is the idea of T-duality and stringy winding modes in order to understand the string cosmology.  So what Brandenberger and company do is to couple a classical background such as graviton and the other dilaton field to gas of strings. This may be extended to other basic degrees of freedom in string theory such as branes. In this way Brandenberger et al., claims that they are using the key feature of string theory into the cosmology. As we know string has three kinds of states, momentum modes (center of mass mode), oscillatory mode (fluctuation of the strings) and winding mode (counting the number of times that the string).  The interesting aspect is that the background of string cosmology is non singular as it never exceeds the Hagedron temperature. I dont have much illuminating things to say about this rather than quoting Brandenberger.

Claim(not verbatim)

If we start evolution as a dense gas of strings in a space in which all dimensions are string scale tori the gas of strings in which all dimensions are string scale tori then there are dynamical arguments according to which only three of the spatial dimensions can become large.So string gas cosmology can shed light to why we just see three extra dimensions and not the compactified dimension.

I am not entirely convinced by this argument although I concede that I havent paid attention to the dynamics. Brandenberger also hopes that string gas can explain the orgin of the scale invariant spectrum of cosmological fluctuations. A signature of the SG scenario is the slight blue tilt in the spectrum of gravitational waves which is predicted by the models. More string gas cosmology later…


Black Hole Information Paradox

August 6, 2008

This is the post for general readership. In the mid 70s Hawking merge QFT and GR in a semiclassically approximately to derive Hawking radiation and its spectrum.The entropy of a black hole is given by the equation. “S = \frac{c^{3}kA}{4 \hbar G}” .

One of the interesting discovery of string theory and especially the work of Strominger and Vafa has validated the value of Hawking Bekenstein entropy. There is one disturbing aspect  for physicists. We will discuss it in the text to follow.

First according to no hair theorem.

Black hole solutions of the Einstein-Maxwell equations in GR tells us that black holes are characterized by only three externally observable parameters: mass, electric charge, and angular momentum and all the other paramaters are inaccessible beyond the even horizon in a black hole.

 How could information be lost in a  black hole ?

The bothersome point is that a pure quantum state may evolve into a mixed thermal state. This is what physicist mean when they talk about information loss. This violates the Liouville theorem. In the case of a entangled pure state one part of the entangles system is thrown into the black hole while the other part is kept outside. This results in a mixed state after the partial trace is taken into a black hole. But since within the interior of the black hole will hit the singularity within a fixed time, the party which is traced over partially might disappear and reappear again. It is not clear what goes on at singularities once quantum effects are taken into account.

This violates unitarity and this is one of the most sacrosanct feature of quantum mechanics. Unitarity implies that the operator which describes the progress of a physical system must be a unitary operator. More clearly, its the e^{iHT} operator than one sees in basic quantum mechanics where H is the Hamiltonian. One knows from quantum mechanics and QFT that the S-matrix that describes how the physical system evolves in a scattering process must be a unitary operator. In QFT this is called the optical theorem. The same thing also happens in the case of Schrodinger equation. i.e the probability of finding the states sums to 1.

One of the guess to resolve this apparent paradox was that the information remains in the remnant of a black hole. However, this remnant would carry too much entropy for an extremely small volume and this is not a very favorable conjecture.

Does this means that the fundamental feature of quantum mechanics (unitarity )has to be modified?

The short answer is no. One should not immediately jump into such conclusions as Hawking’s result is just a semiclassical approximation. This resulted in the famous bet between Preskill and Hawking. With Strominger and Vafa the black hole entropy it seems that Hawking’s entropy calculation is coreect.  So one needs to  think more to understand the apparent paradox. For instance one needs to learn if the information is preserved if one considers more than the semiclassical approximation. 

This best way to resolve this paradox is with the use of ADS CFT. Also, if Hawking radiation receives some quantum correction. The other possibility is bizarre and involves allowing non unitary time evolution. This to my mind is a sketchy proposal. Stephen Hawking published a paper and announced a theory that quantum perturbations of the event horizon could allow information to escape from a black hole, which would resolve the information paradox. This assumes unitarity of ADS/CFT correspondence which is dual to a thermal conformal field theory.


Arxiv today

August 6, 2008

Here is the summary of the four interesting paper at arxiv today..

A Simple Proof of the Chiral Gravity Conjecture (0801.4566)

Chiral gravity is the ADS3 gravity with a cosmological constant, Chern Simons term at a critical value of the coupling. Excitations of ADS3 gravity transform non-trivially under an asymptotic symmetry group consisting of a left moving and right moving conformal group. In the paper “Chiral gravity in three dimension” (arXiv:0801.4566), Strominger et al, conjecture that chiral  gravity exist and is dual to a right-moving boundary CFT. In this preprint, they Strominger et al show that at the chiral point, the group of trivial diffeomorphisms is enhanced to include the left-moving conformal transformations, and the asymptotic symmetry group contains only one (right-moving) copy of the conformal group. Diffeomorphism invariance then requires that all physical excitations are annihilated by left-moving conformal transformations. This establishes nonperturbatively the chiral nature of chiral gravity.

Matrix model for the black hole information paradox (arXiv:0808.0530)

Interesting paper by Polchinski and company. They study the matrix model with a charge-charge interaction (this is the gauge dual of ADS black hole). At late time and large N, there is a power law decay of the correlator and this also shows the continuous spectrum. Interestingly they show that when N is finite the spectrum is not continuos and correlators have no recurrence and the information is preserved. This is studied using various techniques like Feynman graph expansion, sum over young tableaux, loop expansion.They claim to have obtained the leading 1/N^2 for the spectrum and correlators. These techniques are suggestive of possible dual bulk description. At fixed order in 1/n^2 the spectrum is continuos and no recurrence occurs. Hence information loss is manifest. The interchange between long time and large N.

Fibre Inflation: Observable Gravity Waves from IIB String Compactifications  (arXiv:0808.0706)

This is all in the spirit of  stringy inflation model. In this model inflaton field can take trans-Planckian values while driving a period of slow-roll inflation. This leads naturally to a realisation of large field inflation, in as much as the inflationary epoch is well described by the single-field scalar potential. V = V_0 (3 – 4 exp{-phi/\sqrt{3}}).  This model rises in the context of  type IIB string compactifications with large-volume moduli stabilisation. Kahler moduli is generically existent in such models which appears as a string loop correction to the Kahler potential which appears in scalar potential. In this particular example they consider the Kahler potential of certain K3 fibered Calabi Yau manifold. They think that “there are likely to be a great number of models in this class – `high-fibre models’ – in which the inflaton starts off far enough up the fibre to produce observably large primordial gravity waves.” They approximate the ratio of tensor to scalar perturbations to be r = 0.0264, 0.0189, 0.00797 and 0.00528.

 Gravity Waves and Linear Inflation from Axion Monodromy (arXiv:0808.0691)

A  mechanism for chaotic inflation driven by monodromy using extended closed string axions is proposed. They analyze this and also claim that this is compatible with moduli stabilization and can be realized in many types of compactifications, including warped Calabi-Yau manifolds and more general Ricci-curved spaces. In this broad class of models, the potential is linear in the canonical inflaton field, predicting a tensor to scalar ratio r=0.07 accessible to upcoming cosmic microwave background (CMB) observations. 

Compare the tensor to scalar ration in the above paper by Cicoli, Burgess and Quevedo.


Black holes–Quasinormal modes (Part I)

July 8, 2008

I was asked to discuss quasinormal modes in the context of black holes.

Think about a linearized differential equation like the one in general relativity or a Schrodinger tyoe equation. The solutions of these equations with some boundary conditions are called quasinormal modes. In the case of quasinormal modes that the eigenvalue may be complex. The imaginary part of the eigenvalue (frequency) relates to the exponential decrease of the wavefunction in time. A geometry of a perturbed black hole will be a ring after a damped oscillation. The frequencies and the damping time are independent of the the initual perubation. The field can fall radiate to infinity or fall into the blackhole. The modes decay and the corresponding frequencies are complex. These oscillations are known as quasinormal modes.A quasinormal mode of a blackhole would exponentially decrease the asymmetry of blackhole. This occurs as the black hole evolves in time to be a spherical object. One of the successed of string theory is to (re)derive the Bekenstein-Hawking entropy by counting black hole microstates. This was derived by for a five dimensional extremal black holes in string theory by counting the degeneracy of BPS soliton bound states. More on quasinormal modes/black holes to follow….


Topological Open Strings

July 7, 2008

There is an interesting paper in arxiv by Bouchard et al., titled topological open strings on orbifold. As I believe that the readership of the this blog is varied, I need to describe briefly what is topological string.

Topological Strings

Topological string   has operators which  is the algebra of operators in string theory. The algebra preserves certain SUSY and this can be obtained by certain “topological” twist of the worldsheet in String Theory. These operators are associated with certain spin. This is analogous to topological field theory and hence the topological string theory (TS) do not have any degrees of freedom. There are basically two topological string theory( A-model and B-model). Also, there are only certain kinds of admissible spacetime admits the criteria for topological string theory. Strings on ordinary background are not the topological strings. A procedure known as topological twist is required. These theories have two U(1) symmetries. The R-symmetry (two U(1) symmetries) may lead to two A model and B models. This will make the theory BRST exact. Topological theory are related to Chern Simons theory and Gromov Witten invariants. Also there exist relations to mirror symmetry.

Topological String Theory on Calabi Yaus

The good thing about the topological string sector is that one can exactly calculate the amplitude and one can study the moduli spaces.  Topological string amplitudes are modular object  with special transformation properties under the modular group. At large radius limits topological string amplitudes are generating functions of Gromov Witten invariants. When going to orbifold points of the moduli space the amplitudes become generating functions for orbifold Gromov–Witten invariants. In this paper they try to relate the Gromov Witten invariants to orbifold to Gromov witten invariants. They extend the result by  Aganagic, Bouchard and Klemm of closed strings to open strings.