Strings, strings and strings

 Today, on Twitter I came across a post by Thomas Van Riet [see here], that seeing Strings 2023, the name of the conference should be ``the conference previously known as Strings". This is true -- in fact, so much so that Aron Wall at the beginning of his talk said something along the lines of ``... I am going to do something unusual, and actually talk about string theory ...". Of course -- string talks were present quite nicely; for instance, Remmen gave a talk on his work with Cheung on bespoke dual resonance [2308.03833], Wall on off shell strings and black hole entropy, Sen on logarithmic corrections to supersymmetric black holes and Eberhardt for instance. That said, there were a number of talks that weren't exactly strings -- for instance, Suvrat's talk on the Hilbert space of de Sitter quantum gravity, Engelhardt on an algebraic ER=EPR with Hong Liu (something I am excited to read on, since I had some speculation on similar lines), and Strominger on cosmic ER=EPR in dS/CFT with Cotler. For that matter, a stringy context of de Sitter is something I (and presumably some other people) are wary of. The most ``trustworthy" paper I have read on de Sitter compactifications in string theory is the one by Van Riet, Bena and Grana [2303.17680]. However, I think the star of the show (pardon me for this statement; I found many other talks great as well, but... Well..) was Witten's talk on a background independent formalism for quantum gravity, based on the paper [2308.03663]. 

But, unsurprisingly (since it is Twitter), some people pointed out some opposition to string theory. Well.. I suppose this is where I say that politics arises with trying to compete with string theory. Between Republicans and Democrats, I may be able to choose at some point given some understanding. But with string theory, I doubt anyone will ever say ``oh! Loops look better now!". Or indeed any other theory. Anyone saying ``that theory is wrong, this is better" are essentially trying to gamble with this false politics.

P.S. Twitter is messed up. Millions of years of evolution and this is what it boils down to.

On 2 comments:
Posted in ,

Susskind's de Sitter papers and IAS talk

 Only a few days ago, Susskind gave a talk called Observers and Observations in de Sitter space at IAS hep meetings. The following are some papers to read and some other talks to read in close reference to this. 

While I am not aware of DSSYK$_{\infty }$, I think Susskind will give a talk soon at IAS (as mentioned to Witten at the end of the talk). 

On 1 comment:
Posted in ,

Bit Threads, AdS/CFT and de Sitter

 Recently, I came across a very striking formulation of Ryu-Takayanagi's result in AdS/CFT due to Freedman and Headrick [1604.00354]. While I came across this from reading Frederick and Hubeny's paper on Covariant Bit Threads [2208.10507] (and also from Susskind and Shaghoulian's paper on de Sitter entanglement [2201.03603]), I felt that this proposal has some things to be understood even in the framework of AdS/CFT. The idea is summarised as follows: due to a max-flow-min-cut theorem for Riemannian manifolds, one can state the Ryu-Takayanagi area-minimizing term as a max-flow for bit threads. 

Start from Ryu-Takayanagi (where $\sim $ denotes that the surface is homologous to the boundary subregion):

\[S(\partial R)=\max _{\Sigma }\min _{\mathcal{X}\sim \partial R} \text{Area of } \mathcal{X}\;.\]

Usually, the term $\max _{\Sigma }$ is redundant and is instead used in the HRT prescription for covariant holographic entanglement entropy. From the max-flow-min-cut principle, the term $\min _{\mathcal{X}\sim \partial R}$ can be replaced by maximizers of the flow $v$, which has field lines defined as bit threads. These have two properties: (1) they have a fixed ``width" of $1/4G_{N}$, and (2) they cannot cross a horizon. Ryu-Takayanagi then becomes

\[S(\partial R)=\max _{\# \text{ threads}} \int v\;,\]

where by maximizing flows $v$ we mean the number of threads that leave the boundary subregion. These bit threads have an additional property: they always start and end on the boundary. Ryu-Takayanagi then takes the above form, and is fully expressed in terms of the maximum $\#$ of bit threads leaving the subregion into the bulk, forming a ``bottleneck". This is preserved from the max-flow-min-cut theorem, and so the definition of Ryu-Takayanagi is still the same. 

Susskind and Shaghoulian introduced this aspect of bit threads and entanglement entropy in de Sitter space, which is an interesting read. In de Sitter, one does not have this conformal boundary nature to describe a nice picture for entanglement entropy. Instead, as Susskind and Shaghoulian pointed out, one can define a Ryu-Takayanagi formula in static patch by identifying a surface between the stretched horizons to find the entanglement of the pode-antipode system. This reduces to the familiar Gibbons-Hawking entropy,

\[S=\frac{\text{Area of } H_{\Lambda }}{4G_{N}}\;.\]

In the bit threads formulation, one now has the ability to define two proposals: first, a monolayer proposal, where each horizon emits bit threads towards the other component in a single-layered fashion, and second, a bilayer proposal, where the largest component (usually the cosmic horizon) emits bit threads in a double-layered fashion. Due to this, one has bit threads going (1) to the other component, and (2) ``backwards" towards the pode or antipode. Of course, these bit threads find a bottleneck of zero area (a discrepancy taken into account while making the condition of the surface homologous to the horizons), and therefore the monolayer and the bilayer proposals are equivalent. For, say, the Schwarzschild-de Sitter case, the second layer encounters an ER bridge (since the black hole in say the pode would imply a black hole in the antipode in an entangled state), where the second bottleneck is located. However, we are far from understanding whether there are other information theoretic aspects in de Sitter. In AdS/CFT, presumably things like strong subadditivity can be explained in bit threads, but maybe with semiclassical corrections there are other things to take into account. In de Sitter, this becomes even more complicated. However, this is a nice proposal and I believe some more attention should be given to these things.

On 2 comments:
Posted in , ,

Higuchi's Modified norm, Asymptotic WDW states and Refined Algebraic Quantization

 In a review I am writing with a collaborator, I am reviewing a bit of Higuchi's proposal (introduced in this paper) for a modified norm. A couple of papers into inspire, I found a very nice set of papers (see this, this and this) that indicated that there was a little more to the story of the group averaging proposal. It was first initiated by Moncrief in his paper ``Space-Time Symmetries and Linearization Stability of the Einstein Equations. 2", where it was conjectured that the infinite norm of a Hilbert space could be modified into a finite norm by dividing by the volume of the group. In the example of the Hilbert space proposal by Suvrat et al [2303.16315], where the asymptotic solutions to the Wheeler-DeWitt equation are of the form

\[\Psi [g, \chi ]=e^{iS[g, \chi ]}\mathcal{Z}[g, \chi ]\;,\]

one has to take into account of the Gauss law, which requires that these WDW states are invariant under the de Sitter isometry group $G_{dS}$. These are of the form (see the paper for a nice discussion about this and a more proper description of how these states are found to be invariant)

  • Translations: $\tilde{x}^{a}=x^{a}+\epsilon ^{a}$,
  • Rotations: $\tilde{x}^{a}=R^{a}_{b}x^{b}$,
  • Dilatations: $\tilde{x}^{a}=\lambda x^{a}$ and 
  • Special conformal transformations: $\tilde{x}^{a}=\frac{x^{a}-\beta ^{a}|x|^{a}}{1-2(\beta \cdot x)+|\beta |^{2}|x|^{2}}$.
For the $t$ coordinate, the above transformations can be found correspondingly. However, if one defines a ``seed" state built on top of the Euclidean vacuum state $|0\rangle $ (called the Bunch-Davies state) by
\[|\mathrm{seed}\rangle =\int dx_{1}\dots dx_{n}\psi (x_{1}\dots x_{n})\chi (x_{1})\dots \chi (x_{n})|0\rangle \;,\]
where $\psi (x_{n})$ is a ``smearing" function with a compact support and $\chi (x_{n})$ are a collection of $n$ massive scalar fields, we see that the only state invariant under $G_{dS}$ (imposed as a constraint) is that of $|0\rangle $. Due to the smearing function, and the nature of the group, the norm $\langle \Psi , \Psi \rangle $ is infinite. This can be interpreted as forming a one-state subspace $\mathcal{H}^{G_{dS}}$ of $G_{dS}$-invariant states from the full Hilbert space $\mathcal{H}$. To correct this, one uses Higuchi's proposal, which proved Moncrief's conjecture, and uses group averaging to define a modified norm
\[\langle \langle \Psi , \Psi \rangle \rangle =\frac{1}{\mathrm{vol}(SO(1, D+1))}\langle \Psi , \Psi \rangle \;,\]
where $\langle \langle \cdot , \cdot \rangle \rangle $ denotes the modified norm (this is somewhat against the usual convention of $(\cdot , \cdot )$, but I will be using this in a different sense below). What one is doing here is defining the space of states formed by group averaging under $G_{dS}$:
\[|\Psi \rangle =\int _{g\in G_{dS}}dg \; U|\mathrm{seed}\rangle \;,\]
where $dg$ denotes the Haar measure and $U(g)$ is a unitary operator. One then has a finite norm set of states which are invariant under $G_{dS}$ isometry group.

In the sense of Refined Algebraic Quantization (abbreviated to RAQ), the whole consideration above reduces into a mathematical problem: given an auxiliary Hilbert space $\mathcal{H}_{aux}$ and a corresponding set of $*$-algebra of observables $\mathcal{A}_{obs}$, one has to find the ``physical" Hilbert space $\mathcal{H}_{phys}$ such that the inner product satisfies the finiteness constraint, and so that the inner product on $\mathcal{H}_{aux}$ is related to $\mathcal{H}_{phys}$. The constraints in the above construction are that the states are invariant under $G_{dS}$, and essentially what we have done is to find the physical Hilbert space, which is a subspace of the initial (in the RAQ terminology, auxiliary) Hilbert space. In the sense of an $\eta $-mapping as discussed in Marolf's paper, one would find a measure so that the following modified norm converges:
\[(\Psi , \Psi )_{\eta }=\int _{g\in G_{dS}} dg\; \langle \Psi |U|\Psi \rangle \;,\]
where $(\cdot , \cdot )_{\eta }$ refers to the modified inner product defined in terms of $\eta $. Group averaging in the above sense of WDW states becomes the norm $(\cdot , \cdot )_{\eta }\to \langle \langle \cdot , \cdot \rangle \rangle $ of $\mathcal{H}_{phys}\to \mathcal{H}^{G_{dS}}$. 

On No comments:
Posted in , ,

Hamilton vs Smale's Poincare Program

 The Poincare conjecture was one of the seven Millenium problems that was solved in 2003 by Grisha Perelman in his papers [math/0211159], [math/0303109] and [math/0307245], which showed that when performing Ricci flow on a closed $N$-manifold, it reduces to a type II singularity in finite-time, in the sense that under Ricci flow any closed $N$-manifold is homeomorphic (and diffeomorphic) to $\mathbb{S}^{N}$. This is following the Ricci flow program, initiated by Hamilton (who Perelman stated deserved the Millenium prize, declining upon being told that (1) Hamilton could not be awarded for the complete general proof of Poincare, and (2) since the papers were solely arXived in math.DG and not published, the prize required that he publish the papers if Perelman accepted). While Ricci flow now seems to be the generic approach to Poincare conjecture, in 1961 Smale announced his results that under the h-cobordism theorem, in $N\geq 6$ one can show that for certain kinds of manifolds, using the Alexandrov trick one can show that in general, any closed $N$-manifold is homeomorphic to $\mathbb{S}^{N}$. See my (highly naive) notes on this result, linked below. It must be said that, while the h-cobordism theorem holds for topological and smooth manifolds alike (from Rourke and Sanderson), the Poincare conjecture in Smale's proof need not hold for smooth manifolds. Despite several other reasons why this proof does not hold for all kinds of manifolds (and why this proof fails for certain dimensional cases in $N\geq 6$), I believe that Smale's result was a prime motivator for all the other works leading up to the proof of the Poincare conjecture, whether directly or indirectly. As of the superiority of Hamilton over Smale, I choose to believe that Hamilton's works proved to be far more fundamental and intrinsic to Poincare conjecture (and math.DG) than Smale's work. However, this is only my belief. 

Notes on Poincare conjecture and the h-cobordism theorem

On No comments:
Posted in ,

Things in hep-th -- Strings 2023 edition

Some topics I am currently reading on, and things to read. Strings 2023 edition.

Some more works, although I am not aware of these as of yet due to more mathematical complexity. I have included them solely for further interest. 

On No comments:
Posted in , ,

Thoughts on de Sitter

In the famous 1997 paper by Maldacena [hep-th/9711200] introducing AdS/CFT (which also happens to be the most cited paper in hep-th), the notion of duality between bulk quantum gravity and a CFT on the boundary was found. In some sense, this can be viewed as the equivalence between the gravitational partition function $Z_{\text{grav}}$ and the CFT partition function $Z_{\text{CFT}}$. Some interesting things have also come out of this, most prominently (in my opinion) the calculation of holographic entanglement entropy by Ryu-Takayanagi [hep-th/0603001] and Hubeny-Rangamani-Takayanagi [0705.0016]. Since the inception of AdS/CFT itself, however, people have looked for a dS/CFT correspondence -- a holographic duality between some field theory on the conformal boundary $\mathcal{I}^{+}$ and the bulk gravitating region described by the $D$-dimensional de Sitter space. See for instance, the original paper putting forward dS/CFT by Strominger: [hep-th/0106113]. However, this is something that seems too much of a long-shot, namely, for the following reasons:
  • Holography at $\mathcal{I}^{+}$ means that the partition function $Z_{\text{CFT}}$ requires non-unitarity, which in some sense can be imagined by the lack of a minimal spacelike geodesic separating two distinct points on the boundary. Therefore, any holographic entanglement entropy proposal would define ``pseudo-entropy" and not something as straightforward as the Ryu-Takayanagi formula. Which of course, one can put up with, but at least for a superficial reason I am petty enough to say that the ``actual" dS/CFT correspondence (if one exists at all) gives a slightly more satisfying description of entanglement entropy than the timelike entanglement entropy - pseudo-holographic entropy equivalence by Harper et al [2210.09457]. 
  • Far too many proposals. Again, it might just be that I am being petty with multiple descriptions of holography in de Sitter space, but if a proper description of holography did exist at all, it should be somewhat like AdS/CFT in the sense that the duality is concrete. Right now we have multiple descriptions -- such as global dS/CFT, static patch holography (which is a good enough thing due to things like entanglement entropy calculations and so on), half-de Sitter (introduced just recently by KRST, [2306.07575]), and recently, Cauchy slice holography, which I will talk about in a different post. Perhaps the answer is one of two things: either, that such a correspondence simply does not exist, or that a different kind of holography is to be considered, which would be holography of information from asymptotic quantization. 
  • Finally, because it seems out of bound from a quantum gravity description without supposing things. Which holographic proposal would be have to pay attention to from, say, canonical quantum gravity? The answer for me, seems to be -- none. I have talked below. 
What is this aspect of quantum gravity I mentioned that does not work for holography? I cannot make sense of this as of yet, due to some reasons that will be apparent soon. However, if one builds holography in the sense of the GPKW relation,
\[\Psi [g]\sim Z[g]\;,\]
where $\Psi [g]$ satisfies the Hamiltonian and momentum constraints (referred to as a Wheeler-DeWitt state, satisfying the annihilation of the Hamiltonian constraint or the Wheeler-DeWitt equation) and $Z[g]$ is a CFT partition function in dS. Now, here is the question: if one assumes this to be true (which is, for well-known case of the Hartle-Hawking state), what is the Hilbert space of dS quantum gravity? One could suppose to do Cauchy slice holography [2204.00591], which for AdS/CFT works amazingly -- you start from the CFT partition function and use a $T\overline{T}$-deformation to move into a bulk Cauchy slice, and identify holographic duality between $\Psi [g]$ and the deformed partition function $Z^{\Sigma }[g]$. Note a thing above, which is that I have not mentioned the dependence of $\Psi [g]$ or $Z[g]$ on matter contributions -- if we consider the case of gravity coupled to a massive scalar field, one would have in those contributions as well, but for the sake of discussion I have not included it. For de Sitter, this deformation from the boundary $\mathcal{I}^{+}$ is not clear, at least for me. But even if one does identify these things, which holographic proposal would we derive? Static patch? Global dS/CFT? This is not clear for me yet. But even if this is clear, this does not answer the question, ``what is the Hilbert space of de Sitter quantum gravity?" This was answered just recently (discussions on which I was fortunate to hear before the paper was arXived at lunch at ICTS) in a paper by Suvrat and collaborators [2303.16315], which is a nice result, and this uses asymptotic quantization, which is to work with late-time slices (essentially by identifying a conformal factor $\Omega $ in the metric so as to identify an intrinsic ``clock" or York time) instead of $\mathcal{I}^{+}$. This has a nice result, that in some sense one has a proper identification of the WDW states in de Sitter. However, more on this later.
On No comments:
Posted in , ,

John Baez's paper on Hoang Xuan Sinh’s Thesis

 On my reading list is the paper by John Baez that came out just today. While I am not very familiar with Category theory, I find this to be an interesting read. This is on Hoang Xuan Sinh's thesis work that she did with Grothendieck, knowing him from his teaching weeks during the Vietnam war near Hanoi. 

[2308.05119] Hoàng Xuân Sính's Thesis: Categorifying Group Theory (arxiv.org)

On No comments:
Posted in ,
Subscribe to: Posts (Atom)