Archive for September 2021

Misconception between Bosonic String and Susperstring in RNS Formalism

There are many common misconceptions (or carelessness) that amateur string readers have. One of those is that bosonic string theory on worldsheet is for bosons, and superstring theory on superspace is for fermions (well, only fermions). This is technically wrong. Superstring theory has both bosonic sector (with Neveu-Schwarz - NS- boundary condition) and fermionic sector (with Ramond boundary condition). However, the NS bosonic sector (which uses the same $X^\mu$ worldsheet of the free bosonic theory of D=26) of Superstring is different from the bosonic theory in D=26. One of the things that differentiate the two is the presence of an extra oscillator in the former.  NS bosonic sector does not have the critical dimension $26$ but $10$.

In superstring theory, we add an extra wave-function $\psi^\mu$ which is related to $X^\mu$ by world-sheet supersymmetry (space-time SUSY is used in GS formalism). The fermionic sector (of course of Superstring) is also ghosts-free at $D=10$. And the Virasoro algebra of the free bosonic theory is replaced by the Super-Virasoro algebra.

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Random Matrices

I have been lately studying the random matrices and their application that widely defaults for the JT gravity. Though, random matrices need to be started with Wigner's idea of the random matrix in nuclear theory. Right now, random matrix theory can be considered an important subject, at least from my learning view. In the following (short), I present my rough ideas of random matrix theory, extracted from here.


Random matrix theory (RMT) is a classic example of statistical group theory in general physics. The most recent development for RMT is the equivalence of JT gravity with RMT, see here. From the correspondence of AdS/CFT, one learns that a bulk theory with gravity lives on the boundary of a quantum system. However, the equivalence of JT gravity is not given to a boundary theory (or a bulk theory). In fact, RMT shares the correspondence with JT gravity; hence JT gravity is dual to random matrix integral of Hamiltonian $ H $, where $ H $ is a random matrix.

(RMT is mainly concerned about groups' statistics, at least for us, whose applications are wide in physics, as indicated.)

Consider a matrix $ \sf M $, from linear algebra we know that $ \sf M$ holds eigenvalues $ {\sf {m}}_{ij} $ (for $ 2 \times 2 $ matrix). Suppose the elements of matrix $ \sf M $ are random variables, to which we say to obey some specific outlined properties. In that case, the study of those ($\mathsf{m}_{ij} $) eigenvalues are called the "random matrix theory'' problem. Now one can ask what the practical application of RMT is. Actually, there are many practical applications. Consider the well-studied example of the nucleus using these random matrices, which Wigner (and Dyson) developed. And the recent example of the success of RMT is JT gravity. I suggest the reader to read the most interesting book on this subject by  Madan Lal Mehta.


From a mathematical perspective, random matrices serve great as well. However, it tends that it is currently revolutionizing the physics. Nuclear physics had first exploitation of random matrices. And now, it is used as a tool in black hole information problems, JT gravity (see the Saad, Shenker, Stanford), and other statistics problems (include the pedastriation and all that researches).  The deformations of these matrices were done by Witten, and he also did the volume problems for the subject. 

Radom matrices can be classified into classes. See sec. 4 in this and Witten's paper on JT gravity. For a full study, consider the Mehta's book.

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Few Updates

Let us catch with some updates. 

  • Susskind has a good paper on de-Sitter space here. He has been pretty active in de-Sitter lately. One should first start with his and colleagues' paper on the causal patch and then follow up on his recent writings. (I recently saw a video of Susskind highly appraising Juan Maldacena here.)
  • I wrote a fine and short piece of introduction (and basically mixed up) article on field theory and beyond, which can be found here and here.
  • Subir Sachdev, TIFR, and IAS have organized a course on Quantum phases of matter. The lectures can be viewed at http://qpt.physics.harvard.edu/qpm/.
  • Breakthrough Prizes have been awarded https://breakthroughprize.org/News/65.
Ig Nobel Prizes are out, check it to chuckle a little.

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