The joy of IIB Matrix Models

Posted by Urs Schreiber

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As readers of sci.physics.research might remember, a while ago I had learned about the IIB Matrix Model and had fallen in love with it. Unfortunately at that time I was busy with other things and didn’t find the time to absorb the technical details. Two events now made me have a second look at the literature on this model. One is, maybe surprisingly, my encounter with Pohlmeyer invariants. The other is the review

T. Azuma, Matrix models and the gravitational interaction

which appeared recently.

What do Pohlmeyer inavariants have to do with proposals for nonperturbative string theory? There is at least one intriguing technical similarity:

Pohlmeyer invariants are classical gauge invariant observables of the bosonic string which map any configuration of the string (at constant worldsheet time) to the number obtained by picking any constant $U(N \to \infty)$ gauge connection on target space and evaluating its Wilson line around the loop formed by the string at the given worldsheet time.

In the paper

K. Pohlmeyer & K.-H. Rehren, The invariant charges of the Nambu-Goto Theory: Their Geometric Origin and Their Completeness

it is shown that from the knowledge of the values of all these Wilson lines one can reconstruct the form of the surface swept out by the string.

What does this have to do with the IIB Matrix Model, though?

For completeness let me recall that the IIB Matrix model is obtained either by a complete dimensional reduction of 10d SYM or of a matrix regularization of the Green-Schwarz IIB superstring. Either way one is left with the simple action

(1)

S = -\frac{1}{g^2} \mathrm{Tr} \left( \frac{1}{4} \left[ A_\mu,A_\nu \right] \left[ A^\mu,A^\nu \right] + \frac{1}{2} \bar \Psi \Gamma^\mu \left[ A_\mu,\Psi \right] \right) \,.

Here $A$ and $\Psi$ are $N\times N$ Hermitian matrices and $\Psi$ is furthermore a Majorana-Weyl spinor in ten dimensions. All these objects are constant, i.e. do not depend on any coordinate parameters - there are none in this model.

There are many possible routes to rederive known string theory from this action. Let me just list a few important papers.

The best starting point to read about the IIB Matrix Model is probably

H. Aoki, S. Iso, H. Kawai, Y. Kitazawa, A. Tsuchiya, T. Tada, IIB Matrix Model.

This is based in part on

M. Fukuma, H. Kawai, Y. Kitazawa, A. Tsuchiya, String Field Theoy from IIB Matrix Model,

where intriguing hints are given, that Wilson loops in this model of constant gauge connections satisfy the equations of motion of closed string field theory. I’ll have to say more about this below. Here I just note that this way of reobtaining strings from the IIB model is complementary to realizing that the action $S$ above is the matrix regularization of the $GS$ action. In fact the authors argue that one way one arrives at F-strings, while the other way one arrives at D-strings.

This is incidentally the point where the idea behind the Pohlmeyer invariants reappears in the IIB Matrix Model: In both cases Wilson loops of large- $N$ constant connections around a loop describe physical configurations of a string which is identified with this loop! Of course this is nothing but an aspect of string/gauge duality, somehow, but it is a particularly nice one, I think.

In order to see how the IIB model fits into the very big picture the paper

A. Connes, M. Douglas, A. Schwarz, Noncommutative Geometry and Matrix Theory: Compactification on Tori

is probably indispensable. Therein it is discussed how the compactified IIB Matrix Model is the same as the BFSS Matrix Model at finite temperature!

T. Azumo has more interesting references in his thesis paper. One of them looks like a valuable review text, apparently private notes by S. Shinohara, but unfortunately (for me!) this postscript is written in Japanese! :-)

I want to say more about the IIB Model soon. Today my aim is to get started by trying to work out the central steps involved in the proof that Wilson loops of the IIB Matrix Model satisfy equations of motion of string field theory. I am motivated by the fact that the respective derivation in the above mentioned papers involves some rather messy looking formulas which unfortunately may obscure the absolutely beautiful mechanisms that are involved. These are what I want to work out.

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So the goal is to derive from the action

(1)

S(A,\Psi) = -\frac{1}{g^2} \mathrm{Tr} \left( \frac{1}{4} \left[ A_\mu,A_\nu \right] \left[ A^\mu,A^\nu \right] + \frac{1}{2} \bar \Psi \Gamma^\mu \left[ A_\mu,\Psi \right] \right)

equations of motion for the expectation values

(2)

\left\langle F\left(A,\Psi\right) \right\rangle = \int \mathcal{D}A\, \mathcal{D}\Psi\, F(A,\Psi) \exp\left( - S\left(A,\Psi\right) \right)

of observables $F(A,\Psi)$ and to show that for $F$ a Wilson line of $A$ around an abstract loop (in the beginning there is no spacetime in which this loop is embedded in this model, this spacetime arises as a derived concept!) these equations of motion describe propagation of relativistic strings as well as their splitting and joining interactions.

I’ll closely follow the papers by Kawai, Tsuchiya et al. but my goal shall be to focus on the cruciual steps that illuminate how strings and their propagation and interaction arises from calculation with matrices only.

To do so, my first stel shall be to ignore the fermionic contribution and concentrate on the bosonic part of the model, i.e. to consider the action

(3)

S(A) = -\frac{1}{g^2} \mathrm{Tr} \frac{1}{4} \left[ A_\mu,A_\nu \right] \left[ A^\mu,A^\nu \right] \,.

In order to get anything like a Wilson loop from this action consider the abstract circle $\mathrm{S}^1$ and any function

(4)

k : \mathrm{S}^1 \to R^{(9,1)}

(5)

\sigma \mapsto k^\mu(\sigma)

on this circle. To any such function $k$ we may associate an observable $w(k)$ on the space of large constant matrices $A_\mu$ by writing

(6)

v(k) := \mathcal{P} \exp\left( \int_{\mathrm{S}^1} d\sigma\, k^\mu(\sigma) A_\mu \right)

(7)

w_k(A) := w(k) := \mathrm{Tr} (v(k)) \,.

As for now the $k^\mu(\sigma)$ have no physical interpretation and are just auxiliary functions that label all kinds of observables in the model. But we will see that $k$ acquires the meaning of the momentum density on a string’s worldsheet - momentum, that is, in a space which is nothing but the Fourier dual of the space of $k$ themselves.

In order to make things a little simpler lets consider a regularized version of these Wilson loop obervables. Introduce a large integer

(8)

M = [2\pi/\epsilon] \,,

set

(9)

k^\mu(n\epsilon) =: k_n^\mu

and approximate

(10)

v(k) \approx \prod_{n=1}^M \exp\left( i \epsilon\, k^\mu_n A_\mu \right) \,.

The point is that this will allow us to conveniently identify matrix multiplication with differentiation with respect to $k$ by means of the formula

(11)

-i \frac{\partial}{\epsilon \partial k^\mu_n} U_n = A_\mu U_n + \mathcal{O}({\epsilon}) \,.

This way the matrices $A_\mu$ become related to the ‘spacetime’ associated with the $k^\mu$ .

Next we need to choose some sort of Schwinger-Dyson equation for the Matrix Model, re-express it but replacing matrix multiplication by differentiation and read off the sought-after equation of motion for the Wilson loops.

It turns out that a useful Schwinger-Dyson equation to consider is

(12)

0 = k_{M\mu} \int \mathcal{D}A\, \frac{\partial} {\partial A_\mu^\alpha} \left\{ \mathrm{Tr}(t^\alpha v(k^{1})) w(k^2)\cdots w(k^l) e^{-S(A)} \right\} \,.

Here we write

(13)

A_\mu = A_\mu^\alpha t^\alpha

where $t^\alpha$ are the generators of U(N) normalized so as to satisfy

(14)

t^\alpha_{ij}t^{\beta}_{ji} = \delta^{\alpha\beta} \,.

Evaluating the above expression using the product rule gives us three different kinds of terms, depending on whether thr derivative acts on the action, the first Wilson loop or the remaining Wilson loops:

(15)

0 = \underbrace{ k_{M\mu} \int \mathcal{D}A\, \left( \frac{\partial}{\partial A_\mu^\alpha} e^{-S(A)} \right) \mathrm{Tr}\left(t^\alpha v\left(k^{1}\right) \right) w\left(k^2\right)\cdots w\left(k^l\right) }_{=(f)} + \underbrace{ k_{M\mu} \int \mathcal{D}A\, \left( \frac{\partial}{\partial A_\mu^\alpha} \mathrm{Tr}\left(t^\alpha v\left(k^{1}\right)\right) \right) w\left(k^2\right)\cdots w\left(k^l\right) e^{-S(A)} }_{=(s)} + \underbrace{ k_{M\mu} \sum_{b=2}^l \int \mathcal{D}A\, \left( \frac{\partial}{\partial A_\mu^\alpha} w\left(k^b\right) \right) \mathrm{Tr}\left(t^\alpha v\left(k^{1}\right)\right) w\left(k^2\right)\cdots \widehat{w\left(k^b\right)} \cdots w\left(k^l\right) e^{-S(A)} }_{=(j)} \,.

I have called these terms $(f)$ , $(s)$ and $(j)$ because they will be seen to describe the free propagation, the splitting and the joining of the strings desribed by the various $k^i$ .

This works as follows:

The $(f)$ term yields

(16)

(f) = k_{M\mu} \frac{1}{g^2} \int \mathcal{D}A\, \mathrm{Tr} \left(t^\alpha v(k^{1})\right) \mathrm{Tr}\left( \left[t^\alpha,A_\nu\right] \left[A^\mu,A^\nu\right]\right) w\left(k^2\right)\cdots w\left(k^l\right) e^{-S(A)} = k_{M\mu} \frac{1}{g^2} \int \mathcal{D}A\, \mathrm{Tr} \left( \left[A_\nu, \left[A^\mu,A^\nu\right]\right] v\left(k^{1}\right) \right) w\left(k^2\right)\cdots w\left(k^l\right) e^{-S(A)} \approx \frac{i}{g^2 \epsilon} \left( -i\frac{\partial}{\epsilon \partial k_{M-1}} - -i\frac{\partial}{\epsilon \partial k_{1}} \right)^2 \left\langle w\left(k^1\right) w\left(k^2\right)\cdots w\left(k^l\right) \right\rangle \,.

By replacing matrices by derivatives these can be taken out of the integral and become differential operators on the space of multi-loops. In fact, recalling that in the end $k^\mu$ will be identified with a momentum density on the string define the operator

(17)

x^\mu(\sigma) := -i \frac{\delta}{\delta k^\mu(\sigma)}

on the space of (expectation values of) Wilson loops. The the above can be rewritten as

(18)

(f) = \frac{i \epsilon}{g^2} x^{\prime 2}(0) \left\langle w\left(k^1\right) w\left(k^2\right)\cdots w\left(k^l\right) \right\rangle \,.

It’s nice how the big old matrix term condenses to this concise form, but this is not quite the free equation of motion for the string that we expected to see.

No problem. The reason is that the splitting term $(s)$ contains a contributions which doesn’t describe any splitting at all and has hence to be included in the free piece $(f)$ :

One finds

(19)

(s) = k_{M\mu} i \epsilon \int \mathcal{D}A\, \sum_{j=1}^M \mathrm{Tr}\left(t^\alpha \left( \prod_{n=1}^j U_n\left(k^1\right) \right) k^\mu_j t^\alpha \left( \prod_{n=j+1}^M U_n\left(k^1\right) \right) \right) w\left(k^2\right)\cdots w\left(k^l\right) e^{-S(A)} = k_{M\mu} i \epsilon \int \mathcal{D}A\, \sum_{j=1}^M k^\mu_j \mathrm{Tr}\left( \prod_{n=1}^j U_n(k^1) \right) \mathrm{Tr}\left( \prod_{n=j+1}^M U_n(k^1) \right) w\left(k^2\right)\cdots w\left(k^l\right) e^{-S(A)} = k_{M\mu} i \epsilon \int \mathcal{D}A\, \sum_{j=1}^{M-1} k^\mu_j \mathrm{Tr}\left( \prod_{n=1}^j U_n(k^1) \right) \mathrm{Tr}\left( \prod_{n=j+1}^M U_n(k^1) \right) w\left(k^2\right)\cdots w\left(k^l\right) e^{-S(A)} + { iN\epsilon (k_M)^2 \left\langle w\left(k^1\right) w\left(k^2\right)\cdots w\left(k^l\right) \right\rangle } \,.

It is fun to see how the partial derivative with respect to $A_\mu$ splits the string associated with $k^1$ in half and how the contraction of the $t^\alpha$ glues the remaining open ends so as to form two new closed strings. However in one of these processes a piece of string of vanishing length is split off and produces not another string but a kinematical term proportional to $k\cdot k$ . Taking this term together with previously found $(f)$ gives

(20)

(f) + (s^\prime) = i \left( N\epsilon \left(k\left(0\right)\right)^2 + \frac{ \epsilon}{g^2} x^{\prime 2}\left(0\right) \right) \left\langle w\left(k^1\right) w\left(k^2\right)\cdots w\left(k^l\right) \right\rangle \,.

Now, with a little tweaking of $N,\epsilon$ and $g$ , taking careful limits, etc., this begins to look like the Hamiltonian constraint of the free superstring! (The limits are subtle and I think Aoki et al. dont have them under rigorous control, either. But they give a lot of arguments for why the advertised string field theory should be obtained in the correctly taken limit. Furthermore, my prefactors might contain errors. They are not precisely what I see in the literature, which, on the other hand, does not seem to be fully consistent, either. I need to check that.)

Good, finally one can convince oneself that the last term, $(j)$ , really does describe the joining of two strings. One gets:

(21)

(j) = k_{M\mu} i\epsilon \int \mathcal{D}A\, \sum_{b=2}^l \sum_{j=1}^M \mathrm{Tr}\left(t^\alpha v\left(k^1\right)\right) \mathrm{Tr}\left( \left( \prod_{n=1}^j U_n\left(k^b\right) \right) k^\mu_j t^\alpha \left( \prod_{n=j+1}^M U_n\left(k^b\right) \right) \right) w\left(k^2\right) \cdots \widehat{ w\left(k^{b}\right)} \cdots w\left(k^l\right) e^{-S} = k_{M\mu} i\epsilon \int \mathcal{D}A\, \sum_{b=2}^l \sum_{j=1}^M k^\mu_j \mathrm{Tr}\left( \left( \prod_{n=1}^j U_n\left(k^b\right) \right) v\left(k^1\right) \left( \prod_{n=j+1}^M U_n\left(k^b\right) \right) \right) w\left(k^2\right) \cdots \widehat{ w\left(k^{b}\right)} \cdots w\left(k^l\right) e^{-S(A)} \,.

Here now the partial derivative with respect to $A_\mu$ cuts open one of the strings and the contraction over $t^\alpha$ glues the ends with the open ends of the first string.

The claim is that the three terms together give proper closed string field theory in an appropriate limit $\epsilon \to 0$ ,. $N \to \infty$ . I very much enjoy the train of thoughts that leads to this result.

Posted at February 20, 2004 8:08 PM UTC

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Re: The joy of IIB Matrix Models

BTW, does anyone know of any attempts to study the IIB matrix
model on nontrivial backgrounds like pp-waves? Is there any
literature on that? I know that there is considerable activity
in BFSS models on pp-wave backgrounds, but did anybody look at,
say, the calculation which I sketched above for cases where the
SYM theory is defined on a nontrivial background?

Posted by: Urs Schreiber on March 5, 2004 8:30 PM | Permalink | PGP Sig | Reply to this

The Deep Thought Project

Everybody knows Douglas Adams’ famous story about the people who set up a computer to calculate the answer to simply everything.

Interestingly, there are really people trying to do that, in a sense. The idea is that if the Matrix Models of string theory contain the full non-perturbative information about the theory, and if they can in principle be solved explicitly - well, then just set up a computer to solve them and see what happens!

I was indirectly asked to provide some references regarding these attempts. Here they are:

Yoshihisa Kitazawa, Yastoshi Takayama, Dan Tomino, Correlators of Matrix Models on Homogeneous Spaces (2004)

Jun Nishimura, Toshiyuki Okubo, Fumihiko Sugino Testing the Gaussian expansion method in exactly solvable matrix models (2003)

Jun Nishimura, Lattice Superstring and Noncommutative Geometry (2003)

H. Kawai, S. Kawamoto, T. Kuroki, T. Matsuo, S. Shinohara, Mean Field Approximation of IIB Matrix Model and Emergence of Four Dimensional Space-Time (2002)

H. Kawai, S. Kawamoto, T. Kuroki & S. Shinohara, Improved perturbation theory and four-dimensional space-time in IIB matrix model (2002)

Werner Krauth, Hermann Nicolai, Matthias Staudacher, Monte Carlo Approach to M-Theory (1998)

H. Aoki, S. Iso, H. Kawai, Y. Kitazawa, T. Tada, Space-Time Structures from IIB Matrix Model (1998)

Posted by: Urs Schreiber on March 26, 2004 6:36 PM | Permalink | PGP Sig | Reply to this

The String Coffee Table

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February 20, 2004