What are the strings in string theory made of?

The question "what is xxx made of" is really asking "what can xxx be decomposed into"?

For example we know matter is made of atoms because it can be decomposed into atoms. We know atoms are made of electrons, protons and neutrons because atoms can be broken down into electrons, protons and neutrons. But electrons can't be decomposed into anything, so it's meaningless to ask what an electron is made of. We can ask what an electron's mass is, or its energy, or its spin, etc etc, but to ask what it's made from is a question that has no answer.

Exactly the same applies to a string. It is an object that has properties, but it's meaningless to ask what it's made from because it can't be broken down into anything.

In string theory the strings are fundamental objects and cannot be made of anything.

However, the strings of string theory, like more general extended objects --e.g., the membranes in brane theory--, can be considered to be made of more fundamental point-like objects.

An interesting picture is given in Point-Like Structure in Strings

I would finally emphasize that the D0-branes used in Banks' matrix theory or the Thorn's bits are not the more fundamental pointlike objects. For instance, believing that universe is really made of D0-branes would be so misguided as when string theorists believed that universe was really made of strings.

(Please note that I am attempting to answer my own question.)

In string theory it has been implicitly hypothesized, but not rigorously proven, that the mathematical constructs used to describe the vibrations of certain of isotropic materials in simple geometric simple shapes (e.g. strings or rings) are examples of mathematical constructs so fundamental that they show up at many different levels and circumstances in physics. If this string hypothesis is true, then the vibrations of ordinary material strings are a distant echo of these far more fundamental mathematical rules.

There is a precedent for this in modern physics. When James Clerk Maxwell first integrated the knowledge of electricity and magnetism of his time into a single unified theory, he did not at first use differential equations. Instead, he intentionally built a purely mechanical rotating-cell model based on generalizations of specific properties encountered in everyday material objects.

It was not until after Maxwell had used his mechanical model to prove that light was electromagnetic radiation that he realized that all the behaviors of his model could instead be expressed using 20 quaternion-based differential equations with 20 variables, which Oliver Heaviside later compressed down to a mere four vector-based equations that are now called (not entirely accurately) "Maxwell's equations".

The situation hypothesized for string theory is roughly comparable. It also starts with a materials-based dynamic analogy, and similarly proposes that the mathematics that describe that dynamic model are expressions of some deeper and more profound mathematical model. However, it is also true that what constitutes a "fundamental" set of mathematical constructs is much less clear in string theory than it was for the far simpler domain of electromagnetism. That is in part because for string theory the analogies from materials science are extended to much higher numbers of dimensions, and because string theory postulates modes of vibration that are not accessible to direct experimentation and refinement.

Lenny Susskind explains that the answer to this question depends on the parameters of the theory at 1:10:50 to the end of this video.

He makes use of the fact that the question if strings are fundamental or if they are composed of something else is analogous to the question if in electrodynamics, electrons or magnetic monopols have to be considered fundamental to be able to develop a perturbation theory with Feynman diagrams. It can be shown that magnetic charges $q$ and electric charges $e$ are related by

$$ e, q = 2 pi$$

This means, that if the charge of the electron (and therefore their mass) is small, the charge of the magnetic monopoles (and their mass) is huge and vice versa. If the charge and the mass of the electron are small, the electron is considered fundamental and a converging theory (QED) can be developed because the coupling constant $e$ is small. At the same time the magnetic monopoles are heavy complicated things composed of whole bunches of photons and magnetic charges because the coupling constant $q$ is large. This regime corresponds to what we observe with QED being a weakly coupled theory and the magnetic monopoles (if they exist) being to large to be observed. Increasing the electric charge of the electron would lead to a transition to a situation, where the electons become heavy and complicated and in this case it would be more useful to consider a quantized electromagnetic theory with the ligth magnetic monopoles described as fundamental particles.

A similar relationship as described to hold for the pair of electric and magnetic charges exists in string theory between fundamental (f-) strings and D-branes. Depending on the parameters of the theory, either it is more appropriate to consider the D-branes as complicated heavy things composed of fundamental strings, or the D-branes are light and fundamental whereas strings are heavy and complicated things composed of D-branes. The technical term describing this ambiguity is S-duality.

In summary, a unique and universally valid answer to the question what strings are made of can not be given; it depends on the parameters of the theory and the context if it is more useful to consider strings or D-branes as fundamental.

This is to complete @Dilaton 's answer.

The very basic reason theoretical physicist are entranced by string models and their extensions is because they promise to be the framework of the Theory Of Everything, the holy grail of theoretical physics.

String theories and their extensions provide for the quantization of gravity, the long standing difficulty in formulating a TOE. If there existed a competing theory with composite preons or what not, which included the quantization of gravity in a unified manner, one would estimate the merits of each as a TOE. At the moment string theory has no competition, and strings are the fundamental entities of reality in these theories.

In short, string is a dislocation defect of the vacuum crystalline structure.

This by the way explains why apparent full angle around a string seems to be smaller than $2 pi$.

This question from 2012, "What are the strings in string theory made of?", now has a 2018 addendum in which the author proposes (I paraphrase) that the only real strings are QCD strings made of gluons, and that the strings of string theory are unreal and a mistake. The addendum is a diatribe which departs from the original question, but I would like nonetheless to respond to it.

It is asserted that string theory took a wrong turn when Scherk and Schwarz proposed to interpret the massless spin-2 object as a graviton, and the characteristic scale of the theory as the Planck scale rather than the QCD scale; and that this was a step away from science because now we have a theory with a googol or more ground states, unconstrained by the need to match hadron behavior. Instead, theorists should have stuck with trying to explain strong force phenomenology, and in particular the Regge trajectories which suggested strings in the first place.

Now, initially, as is well known, strings fell out of favor once the standard model came together. So theorists of the strong interaction didn't get involved with string theory; they developed QCD, a field theory.

Meanwhile, once string theory was reborn as a theory of everything with extra dimensions, only a literal handful of people dared to work on it. The mainstream effort to obtain a unified theory stayed with field theory. The possibilities of field theory were explored - grand unification, supersymmetry, supergravity, extra dimensions.

So investigations into field theory had reached the point where it looked a lot like string theory anyway. And at that point the mainstream was ready to investigate string theory properly. What they found was a flood of new information. Strings resolved the formal problems of quantum gravity; they could in principle predict the couplings and the particle masses; via branes they had a multitude of connections to field theory that are still being uncovered.

There was already a long history of interplay between field theory and string theory. 't Hooft had described the 1/N expansion of a strongly coupled field theory, in which Feynman diagrams with numerous vertices became meshlike approximations to string worldsheets. Polyakov wanted to understand flux tubes in gauge theories in terms of strings in an extra dimension. Maldacena combined these ideas in the context of superstring theory, producing his famous AdS/CFT duality.

Sakai and Sugimoto described a string-theory setting - an intersection of two stacks of branes, in a particular geometry - which has become the standard way to mimic QCD within string theory. It does a good job of imitating the very complex spectrum of bound states that occur within QCD, and has connections to Skyrme's model, a respected non-stringy approach to the strong force.

I mention all this to emphasize that the application of string theory to the real world is not just about Calabi-Yau spaces and supersymmetric grand unification. There is a parallel line of development which pertains much more directly to strong-force physics.

Nonetheless, there is, empirically, more to the world than just the strong force. So it's nice that string theory can give us other things we need, like gravity and chiral fermions, as well as shedding light on the dynamics of strongly coupled field theories.

The expectations of the mainstream unification program have been somewhat confounded by the LHC's failure, so far, to reveal anything beyond the Higgs boson. For string theory as a theory of everything, this may mean that people need to look at different vacua than the ones they are used to considering, like nonsupersymmetric ones. Maybe people should try to extend the Sakai-Sugimoto model to encompass the whole standard model, and see what happens.

Regardless of what twists and turns lie ahead, it's something of a mistake to think of string theory as an entirely separate world of theory, so far unproven, contrasted to the realm of field theory, which is experimentally validated every day. On the contrary, you could literally write a book about the ways in which string theory has directly enriched the study of field theory.

And given field theory's high-energy ("ultraviolet") problems, it is highly plausible that string theory is simply the framework in which field theory becomes truly well-defined. The relationship is most acutely formulated in terms of Vafa's swampland concept. The swampland is the complement to the landscape of string vacua; it consist of all those field theories which do not exist as the low-energy limit of a string vacuum. If string theory is the unique way to UV-complete a field theory, then the field theories in the landscape have a special status mathematically.

So to sum up, although string theorists are very far from agreeing on which vacuum, or even on what type of vacuum, describes the real world; and although theorists are of course free to pursue other hypotheses (which may nonetheless turn out to be string theory in a new guise); it doesn't really make sense to say field theory good, string theory bad, let's focus on the Regge trajectories. There is a landscape of possible field theories even greater than the landscape of string vacua, and the latter landscape tells us something important about the former landscape. And it is likely that the QCD explanation of Regge trajectories will come via the application of string theory - the ten-dimensional string theory that all those millions of dollars have been spent studying - in the appropriate form.

You can derive string theory from time travel and $mathrm{NP}$-completeness. Time travel forces nature to solve hard problems and from what we know of computational complexity, strings are needed as the primitive objects of an efficient algorithm for $mathrm{NP}$-complete problems. All point-particle interactions in more than one dimension are damped exponentially due to jamming.

Suppose that physics takes place in a region with CTCs. (You can derive something that at least looks like quantum physics from this assumption, but that's another question.) According to a paper by Aaronson and Watrous (https://arxiv.org/abs/0808.2669) CTCs can efficiently solve $mathrm{PSPACE}$ problems both in classical and quantum physics. Assuming the problems aren't solved by magic, there must be some polytime algorithm that solves $mathrm{PSPACE}$ problems for time travel to be realizable.

If we narrow our attention to time travel problems whose initial values make them $mathrm{NP}$-complete (rather than $mathrm{PSPACE}$-complete or manifestly in $mathrm{P}$), we can approximate the initial value problem and the evolution problem as satisfiability problems, presented as CNFs.

If we assume quantum field theory (or for that matter classical field theory in a spacetime with CTCs) has no local mechanism of solving hard problems. This likely requires the additional assumption that $mathrm{BPP}=mathrm{BQP}$ to show that QFT can only efficiently solve the same class of problems as classical field theory and showing that field theory has the same proof power as Resolution (see below). We can then infer that (almost) all interactions must be carried through strings. The $mathrm{BPP}$ in $mathrm{BPP}=mathrm{BQP}$ also lets us use a PCP approach whereby the physically realistically task of approximating a $mathrm{NP}$-complete problems beyond a certain threshold is just as hard.

If we look at any time slice of a time travel spacetime, the time evolution (and any interactions) will be $mathrm{NP}$-hard. This is because a local change will, through time travel, propagate non-locally and non-causally to other regions in the time slice. This will be true even if the SAT problem is arranged on a 2-dimensional grid. (Planar SAT is $mathrm{NP}$-complete: https://en.wikipedia.org/wiki/Planar_SAT) Only one-dimensional objects will be tractable. Actually the one-dimensional objects can have a constantly bounded thickness because bounded tree-width CNFs will still manifestly be in $mathrm{P}$. This can be thought of a way to encode information like the boson and fermion modes of the string. (The original field data is likely encoded in the coordinate data of the strings.) For spin glasses similar intractability issues arise, however for time travel systems, nature must find a solution, so there must be long range interactions.

In proof complexity a major distinction is made between Resolution and equivalently weak proof systems and stronger proof system like Extended Resolution, Frege, and Extended Frege. Resolution is proven to require exponentially long refutations for some satisfiability problems, but stronger proof systems like Extended Resolution are known to have short proofs where Resolution does not.

Resolution uses a very simple rule for reasoning about clauses.

Resolution Step:

Given $A lor l$ and $lnot l lor B$ conclude $A lor B$, where $l$ is any literal (possibly negated variable). For example, $(x_1 lor lnot x_2 lor x_3) $ and $(lnot x_3 lor x_4 lor lnot x_5)$ implies $(x_1 lor lnot x_2 lor x_4 lor lnot x_5)$.

Resolution can easily lead to a blow up in proof size. This is controlled by the maximum number of literals in the clause, width. There are exponential lower bounds on proofs using resolution.

Extended Resolution is the same as resolution, except that new variables can be defined. Define new variables $w$ which is equivalent to $l_1 land l_2$: $$ (lnot w lor l_1) land (lnot w lor l_2) land (w lor lnot l_1 lor lnot l_2) $$

or that $$ w Longleftrightarrow ( l_1 land l_2) $$

where $l_1, l_2$ are literals, variables that can optionally be negated. Note that $lnot w Longleftrightarrow (lnot l_1 lor lnot l_2) Longleftrightarrow lnot l_1 implies l_2$, so that Extended Resolution variables are equivalent to implications. As such it is easy to think of strings as 'implications' which offer a continuous equivalent of Extended Resolution variables. Connected groups of strings must be acting in concert to perform an efficient SAT algorithm which makes $mathrm{P} = mathrm{NP}$.