## Monday, April 24, 2017

### How and why strings generalize geometry

Erwin reminded us how excited he was by the fact that string theory provides us with a quantum generalization of the rules of geometry. What does it mean and how does it work?

Well, all previous theories in physics have used the classical manifold geometry (whose definition will be sketched momentarily) as one of the basic prerequisites that the theories had to accept and elaborate upon. This made the classical manifold geometry and its calculations directly relevant for all these theories and the rules of the geometry were therefore rigid dogmas.

In other words, the theories followed the template:
Dear theory, listen, here you have a classical manifold with some shape.

What can you achieve with this pre-existing shape?
And the theories just couldn't do anything else. They were dependent on the geometry of a classical manifold. If there were no manifold, there was no physical theory. And if two manifolds were geometrically different, the physical theories on them had to be distinguishable, too.

Before the discovery of special relativity, physics was also dividing spacetime to the absolute time and the space that exists with it. That meant that the "spacetime" as we understood it today had to be basically factorized to $$\RR \times M^3$$ where $$\RR$$ was the real axis representing time and $$M^3$$ was a purely spatial manifold (OK, some time-dependent fibration with a different $$M^3(t)$$ at each moment time was sometimes allowed, too). At most, you could have picked time-dependent coordinates on that $$M^3$$ in order to celebrate the Galilean relativity.

But Newton's laws of motion said$m\frac{d^2 x^i}{dt^2} = F^i (x^j, v^j).$ The second derivative of the position was equal to the force – which was expressed as a function of the positions and velocities of the point masses. You may imagine how the metric tensor could enter if you generalized the equations to the motion on a curved 3-dimensional manifold.

At any rate, there was no way to avoid the appearance of the coordinates $$x^i$$ that parameterized the classical manifold – the necessary arena for the laws of physics to be formulated at all.

When mechanics was largely superseded by field theory – imagine Maxwell's equations – then the coordinates $$x^i$$ and $$t$$ more or less naturally merged to the spacetime coordinates $$x^\mu$$ and the most interesting resulting theories found it easy to be Lorentz-covariant. And these coordinates still had to appear in the equations encoding the laws of Nature. Why? The fields such as $$\vec B(x,y,z,t)$$ had to be functions of the spacetime coordinates – variables that identify points on a classical manifold.

To make the story short, classical field theory was developed to include some really interesting cases – with Yang-Mills symmetries and even the diffeomorphism symmetries of general relativity. The latter allowed the background spacetime manifold to be curved and almost inevitably postulated that all smooth+nice enough coordinate systems are equally good. Finally, quantum field theory (possibly on curved spaces) added hats on top of the fields. The fields became operators (or operator distributions). But they still depended on the spacetime coordinates.

If you want a simple example, a Klein-Gordon equation on a curved spacetime background says$g^{\mu\nu} \nabla_\mu \nabla_\nu \Phi(x^\lambda) = 0$ or something else on the right hand side. Equations like that use a spacetime manifold with the coordinates $$x^\lambda$$ and the metric tensor $$g_{\mu\nu}(x^\lambda)$$ on them as building blocks that you simply need.

I should finally tell you what I mean by the geometry of classical manifolds that is going to be generalized by string theory. Well, it's a collection of patches such that each of them is diffeomorphic to an open set in $$\RR^n$$. The patches may overlap and conditions guarantee that within the intersections, you get the equivalent results if you use one patch or another. From the viewpoint of topology as a branch of abstract mathematics, all manifolds may also be understood as "sets of points" with some topology that tells you which subsets of the manifolds are "open sets". Because of the phrase "sets of points", all the previous theories implicitly assumed that "points are allowed to live on that manifold", too.

Does string theory require the classical background geometry as well?

Yes and no. A particular simple subset of descriptions of string theory, or some situations in it, depend on the pre-existing spacetime arena in the same way as mechanics or field theory. In other words, this way of doing string theory still obeys the template
Dear strings, here you have a background spacetime geometry.

Show us, dear strings, how can you dance and split and join on that geometry.
You saw that in the quote above, Mother Nature was directly talking to strings with the pretense of superiority (indicating that She didn't consider strings to be more than just the little green men or point masses) and as if they were completely well-defined and the only ones that mattered. That means that we're only talking about perturbative string theory where all quantities are computed as power series in the string coupling constant, $$g_s$$. In perturbative string theory defined like that and only in it, strings are well-defined and represent existing and the only degrees of freedom. That way of thinking about string theory is only safe for $$g_s\ll 1$$.

OK, how do strings obey the order of Mother Nature? They may obey it if the theory is defined as the so-called non-linear sigma model, first written by Gell-Mann and Lévy in 1960, full eight years before the birth of string theory. The model is named "sigma" because the field was labeled $$\Sigma$$ by the authors. And it's nonlinear because the equations of motion aren't just linear in $$\Sigma$$ – equivalently, because the Lagrangian$\mathcal{L}={1\over 2}g_{ab}(\Sigma) (\partial^\mu \Sigma^a) (\partial_\mu \Sigma^b) - V(\Sigma).$ which has to be integrated over the 2D world sheet, $$S=\int d^2\sigma {\mathcal L}$$ to get the action, isn't purely bilinear or quadratic in these fields $$\Sigma$$. Note that the bilinear Klein-Gordon Lagrangian is multiplied by additional functions of $$\Sigma^a$$ – imagine that the latter are Taylor-expanded in $$\Sigma^a$$. What does this Lagrangian describe? It describes a field theory living on an auxiliary space parameterized by the coordinates $$\sigma^\mu$$ – note that I had to pick some letter for the coordinates and $$x^\mu$$ is "morally taken" because it looks like a good synonym for $$\Sigma^a$$ (you should also persistently observe the evolving conventions for the indices: at this point, I am using $$\mu,\nu$$ indices for the world sheet coordinates and $$a,b$$ for the spacetime ones) – and at each point of the so-called world sheet or world volume indicated by these coordinates $$\sigma^\mu$$, there are several fields $$\Sigma^a(\sigma^\mu)$$ that behave like some "generalized Klein-Gordon fields" from the world volume point of view.

I hope that you won't get confused by the world volume coordinates $$\sigma^\mu$$ and the fields $$\Sigma^a$$. The model is named after the fields $$\Sigma^a$$. Fine. But if we have a greater number of fields $$\Sigma^a$$ on the point given by $$\sigma^\mu$$, we may understand a particular choice of the fields $$\Sigma^a(\sigma^\mu)$$, it describes an embedding $\text{world sheet} \to \text{spacetime}$ where the world sheet has coordinates $$\sigma^\mu$$ and the spacetime has coordinates $$\Sigma^a$$. When you have such embeddings, i.e. a shape of the string (or membrane or brane) embedded in the spacetime, you may study how it vibrates if it has some tension that tries to shrink it. And the Lagrangian above is morally the same as the Lagrangian for a true piano string or a rubber band that oscillates in the pre-determined classical manifold whose geometry is encoded in the functions (the metric tensor)$g_{ab}(\Sigma^c(\sigma^\mu)).$ So far nothing too original has taken place. The motion of strings – the case of strings is equivalent to a two-dimensional space ("world sheet") parameterized by $$\sigma^\mu$$, i.e. by the situation in which the index $$\mu$$ has two possible values – depends on the pre-existing spacetime manifold just like the mechanics of point masses did, or just like the classical field theory or quantum field theory did. It's just different objects that are "ordered to live and vibrate" on the given manifold.

However, strings are more creative as well as more self-confident when they get the freedom to vibrate on a pre-existing background. All other objects that are given the Lebensraum to live will simply live – barely survive – and keep the geometry of the spacetime manifold. However, vibrating strings come at various energy levels and some of the massless or low-lying ones may be identified with the gravitons. If you create a coherent state of strings in the graviton state of vibration, the effect of these strings (through the splitting and joining interactions of strings) on all other strings will be exactly equivalent to a modification of the geometry of the background spacetime manifold.

While the field $$g_{ab}(\Sigma(\sigma^\mu))$$ seems like an unchangeable classical parameter describing the spacetime geometry for some seemingly "totally generic" objects – strings – moving within it, you will find out that the strings in a particular internal vibration state (in the graviton state) produce fields living on the spacetime $$\Sigma^a$$ and these fields $$h_{ab}(\Sigma^c)$$ may be naturally added to the background $$g_{ab}$$ to get the full, dynamical metric on the spacetime manifold.

Just to be sure, the total geometry is also dynamical – governed by some partial differential equations – in the general theory of relativity so the "amount of flexibility" of the spacetime is comparable in general relativity and in string theory. However, a difference is that in general relativity, you ended up with a dynamical metric tensor because you postulated it. You haven't derived Einstein's gravity from anything else: you just assumed it all along. On the other hand, in string theory, you only assumed some seemingly different objects – strings – and Einstein's gravity was implied by that assumption.

A cool thing is that Yang-Mills and Proca fields, Dirac and Weyl fields, Kaluza-Klein fields, and other fields in particle physics arise as well, from the same assumption that strings exist – they arise from other vibration states of the strings – which means that string theory reduces the number of independent assumptions in physics. Also, all the gauge and Yukawa and Higgs self-coupling and other interactions all arise from the same elementary splitting-and-joining interactions of strings. In some counting, perturbative string theory reduces all assumptions of physics into one: the world is made of vibrating strings with some tension. And all the interactions are allowed as soon as you allow the world sheets to have any topology it wants – which is a very natural freedom or "right" in a theory of gravity (and the theory on the world sheet is a theory of gravity, too). (More technically, the only axiom you need is described by two-dimensional conformal and modular-invariant theories on the world sheet.)

This reduction makes string theory much more constrained, unique, and predictive than all previous theories in physics. I've discussed it in older blog posts.

But I haven't spent much time with the elementary yet profound observation that the classical geometry (with manifolds' patches and points in them) is no longer fundamental and necessary within string theory.

The perturbative stringy generalization of the geometry has many aspects. First, a "not so new" novelty. Einstein's equations say that the Einstein tensor$R_{ab} - \frac 12 R g_{ab} = \dots$ is equal to the stress-energy tensor multiplied by a constant, and so on. In principle, there could be additional terms on the left hand side that look like $$R_{ac} R_b{}^c$$ or something like that. They could be polynomial in the Riemann curvature tensor – and its covariant derivatives of various sorts. Such extra terms don't spoil any symmetries of general relativity. The resulting theory is still invariant under all coordinate transformations.

Well, perturbative string theory allows you to derive Einstein's equations and it indeed produces infinitely many higher-derivative corrections like that. A particular string theory produces particular corrections. Relatively to the existing terms, the new terms are multiplied by various powers of the dimensionless tensor$\ell_{\rm string}^2 R_{abcd}$ with various contractions of the indices. Some $$\ell_{\rm string}^2 \nabla_a\dots \nabla_b \dots$$ may be present, too. The constant $$\ell_{\rm string} = \alpha'$$ is the so-called (squared) string length (the inverse string tension $$\alpha' = 1/2\pi T$$). And this "small" parameter guarantees that whenever the curvature radii are much longer than this length $$\ell_{\rm string}$$, the new terms are negligible.

Because derivatives of arbitrarily high orders do appear, the resulting theory looks "slightly nonlocal" as a field theory. However, because you may prove that all these terms result from the local propagation and splitting/joining of strings at particular spacetime points (processes which don't depend on the behavior of strings at any other points), the resulting theory actually is precisely local or causal in some truly physical sense, even though the complicated structure with the higher derivatives naively indicates something else.

Also, string theory contains terms that are "even smaller" than any $$(R_{abcd})^n$$ for $$n\in \ZZ$$ in the $$\ell_{\rm string}\to 0$$ limit, the world sheet instantons. The two-dimensional world sheets may get wrapped on some locally minimum 2-cycles in the spacetime manifold and produce corrections that scale like $$\exp(- A / \ell_{\rm string}^2)$$. Note that when $$\ell_{string}$$ is a very small number in the SI units, the exponential is really, really tiny. It may be $$0.000\dots$$ followed by $$10^{70}$$ zeroes and then a nonzero digit, if the proper area $$A$$ is a squared meter and the string length is comparable to the Planck length.

Fine. String theory generalizes the geometry of general relativity because the simplest Einstein's equations are no longer accurate – it produces corrections at every order and also corrections beyond all orders. But this is not the most profound "generalization of geometry" that string theory brings us, I think. Another, perhaps deeper point is that

Perturbative string theories may be defined even without any spacetime manifold and its metric tensor.

How is it possible? I have already mentioned that just like point masses or little green men, strings are capable of hearing Mother Nature's order "this spacetime manifold is your Lebensraum, live as you can". But strings are more clever so they not only identify the spacetime manifold as something that is pretty hospitable to life but they naturally start to bend and improve this environment. They're capable of changing its geometry and independently determine the conditions that the curvature of the environment should obey.

But strings can do something else, too. They may replace the axioms for a manifold with completely different ones – new axioms that are effectively equivalent for the non-linear sigma models but that also admit other, non-geometric solutions. What are the old axioms and the new axioms?

By the old axioms of the geometry, I meant the definition of a manifold as something that includes the atlas i.e. collection of overlapping patches and each patch is equivalent to an open set in $$\RR^n$$, a set whose elements may be called "points in the continuum". So even though strings are one-dimensional, they still work on something (the manifold) that equally allows zero-dimensional point masses to live there as well.

However, one may study the properties of the nonlinear sigma models and he finds out that
the non-linear sigma models on a Ricci-flat (plus stringy correction) target spacetime are two-dimensional, modular-invariant, conformal field theories.
Within the class of non-linear sigma models, the Ricci flatness of the target spacetime manifold is basically equivalent to the conformal invariance (basically just the scale invariance under the scalings of the world sheet coordinates). So the Einsteinian "Ricci flatness" (or equivalent "Einstein flatness" if you wish) is equivalent to the "conformal symmetry of the world sheet theory". The beta-functions have to vanish in a scale invariant theory – and the beta-functions of the non-linear sigma-model for the "metric tensor components" coupling constants are all the corresponding Ricci tensors.

However, when you replaced the old-fashioned, field-theory-style "Ricci flatness" by a new axiom, the "conformal symmetry of the world sheet", you may completely forget about the old Einsteinian axiom – the Ricci flatness – and only work with the new one – the conformal symmetry on the world sheet. And you will find out new solutions – theories on the world sheet that don't look like non-linear sigma models at all. In other words, they don't seem to be derived from any old-fashioned manifold with its atlas and its metric tensor.

When you encounter such a deep statement, you should always know some simple enough yet nontrivial examples. OK, what are the theories on the world sheet and their fields that may be used instead of the fields $$\Sigma^a$$ labeling a simple embedding of the world sheet to a target spacetime?

Fermionization, bosonization

One truly simple example are world sheet theories with fermionic fields only. Imagine that instead of (or on top of) the bosonic fields $$\Sigma^a(\sigma^\mu)$$, you have many free fermionic fields$\psi^\alpha(\sigma^\mu)$ on the world sheet. We are formally adding new, fermionic coordinates onto the world sheet. If point-like particles were propagating on this spacetime, the addition of the fermionic coordinates would be a rather trivial thing. Note that if you have $$N$$ fermionic i.e. Grassmann variables, every function of these coordinates $$\psi^\alpha$$ may be Taylor-expanded and because $$(\theta^a)^2=0$$ already vanishes, the exponent of each $$\psi^\alpha$$ may only be zero or one. So this only allows you $$2^N$$ nonzero terms in the Taylor expansion: each possible fermionic coordinate is only allowed to appear zero times or once in the product. For point-like particles, any finite number of fermionic spacetime coordinates is just a bookkeeping device to merge $$2^N$$ fields that depend on the bosonic coordinates into one field of all the "supercoordinates", i.e. into one superfield.

Again, for point-like particles, superspaces are not really needed or they don't produce anything fundamentally new. When you study theories in which the superspaces are helpful, especially supersymmetric theories, you may always work in "components" instead, too.

The situation is different in string theory. If one-dimensional strings probe the target spacetime, one fermionic spacetime coordinate multiplies the total number of degrees of freedom not by two but by infinitely many. And in fact, when it comes to the counting, two free fermionic fields on the world sheet are equivalent to one bosonic one! ;-)

How is it possible that for strings, fermionic spacetime coordinates multiply the number of degrees of freedom by an infinite factor, just like the bosonic ones? It's not hard to see why. The fermions $$\psi^\alpha$$ on the world sheet may be Fourier-expanded in $$\sigma^1$$, the coordinate along the string, and we get the modes $$\psi^\alpha_n$$ where $$n\in \ZZ$$. The $$n=0$$ mode is equivalent to the fermionic spacetime coordinate seen by a point-like particle theory but on top of it, you have infinitely many other new coordinates for nonzero values of $$n$$. It means that the addition of $$\psi^\alpha(\sigma^1)$$ multiplies the number of terms in the "Taylor expansion" not just by $$2$$ but by $$2^\infty$$: the exponent is infinite because the Fourier index $$n$$ can have infinitely many values.

If you study string theory or conformal field theory seriously, you will have to see at one point why two fermions are equivalent to one boson. Locally, if you have one bosonic field $$x(\sigma,\tau)$$ and/or a theory with two fermionic fields $$\psi,\bar\psi$$, the maps between these two equivalent descriptions of the same theory are basically$\partial_z x = \psi\bar\psi, \quad \psi = :\exp(ix):, \quad \bar\psi = :\exp(-ix):$ It seems totally counterintuitive – why a fermionic bilinear would be indistinguishable from the derivative of a bosonic field, and why the fermion itself would be indistinguishable from the exponential of a boson – but it's true. You may construct the Hilbert spaces (they're free theories in the simplest case) and count the degeneracy at each energy level. Two fermions simply end up being equivalent to one boson.

This equivalence is just the "simplest, moral template" or a "local sketch" of a full-blown equivalence. For well-defined string theories, you have to be careful about all allowed boundary conditions for the fermions and bosons and the corresponding projections on states. And there exist various possible projections for a group of (an even number) of fermions that are equivalent to some theories with bosons (their number is 1/2 of those of the fermions) and the corresponding projections.

Whole classes of theories, including totally realistic string theories, exist that only contain fermionic world sheet fields. In particular, the "free fermionic heterotic models'" only bosonic fields on the world sheet are those for the 4 spacetime coordinates we know. All the other degrees of freedom – equivalent to the supersymmetric $$D=10$$ heterotic spacetime – are carried by fermionic fields on the world sheet.

Ising, minimal, Gepner models

I don't want to be terribly technical and this is covered in many nice textbooks, e.g. Joe Polchinski's textbook of string theory. But free fermions, while simple, are not the only cool and important models on the world sheet that go beyond a non-linear sigma models. An important extra class are the "minimal models". They are generalizations of the Ising model – and in this stringy discussion, by the Ising model we usually mean the long-distance, scale-invariant limit of it. The simplest Ising model is basically equivalent to a fermion as well. But its generalizations with many states (like the Potts model etc.) – are different theories.

All the operator-product expansions and the spectrum of operators may be determined completely by the bootstrap, by the consistency conditions and axioms of conformal field theories, and they may be proven to exist and be unique (having no continuous deformations etc.). So these minimal models – generalization of the Ising model on the world sheet – contain no fields that could be identified as $$\Sigma^a$$, the coordinates on a curved classical background manifold i.e. with the fields of a non-linear sigma models. But according to the stringy axioms of working, all these theories work great.

Various types of the minimal models on the world sheet may be combined in various ways to obtain modular-invariant theories and they're exactly as consistent as a theory e.g. with the compactified 6 dimensions spanning a Calabi-Yau manifold. Except that you cannot see any manifold in these combinations of the minimal models – which are usually called the Gepner models after the physicist who played with them for the first time.

Funnily enough, you may find out for many such constructions that even though there is no visible, self-evident classical manifold underlying the constructions, some of these simple, geometry-independent constructions are actually equivalent to a non-linear sigma model i.e. a classical geometry – except that it's one that you usually wouldn't guess without lots of experience. So some Gepner models (combinations of several minimal models) are equivalent to non-linear models on Calabi-Yau manifolds of particular topologies. The sizes and curvature radii of these manifolds that are equivalent to the Gepner models are of order $$\ell_{\rm string}$$. That also means that all the stringy corrections to Einstein's equations etc. are of order 100% – i.e. as important as Einstein's original terms.

T-duality, mirror symmetry, and many-to-one equivalences of the geometries

When point masses or little green men "live" on a predetermined spacetime manifold, you may be pretty sure that they may probe or measure its geometry and they will be able to distinguish each 2 manifolds from each other. Two manifolds are the same if there is a diffeomorphism between them – which maps a point of the first manifold to the corresponding point of the second manifold and vice versa, in such a way that the distances between all the points end up being the same in both languages.

Two field theories or configurations on the same classical manifold may be different – because the physical theory or configuration adds extra data on top of the classical geometry – but two field theories or arrangements on two different geometries are unavoidably different, too, because the spacetime geometry is classically equivalent to the "constitutional rules" that no other rules or phenomena may ever overrule.

However, string theory goes beyond this "predetermined" rule to decide about the equivalence of two manifolds. In my constitutional analogy, strings are capable of sensibly editing their constitution as well to deal with curved and other spaces. In the footnote, I am grateful to Barack Obama for discussions. ;-) Perturbative string theory – even if we talk about the non-linear sigma models – on two manifolds that are clearly different according to the old rules of equivalence of manifolds (sketched in the previous paragraphs) may still be exactly and perfectly physically equivalent.

Even though strings are usually "as sensitive to the scaling of distances" as point-like particles, string theory on a spacetime with a circular dimension of radius $$R$$ is exactly equivalent to the theory on a manifold whose circular dimension has the radius of $$\ell_{\rm string}^2 / R$$. That's known as the T-duality. The equivalence holds because the momentum modes and the winding modes get perfectly interchanged. You may apply this T-duality on 3-dimensional toroidal fibers of a more complex manifold, a Calabi-Yau three-fold, and you will get the mirror symmetry – the equivalence of string theory on two target spacetimes whose geometry is completely (even topologically) completely different according to the old-fashioned rules of the geometry of manifolds!

You should appreciate how deep this result is and what it means. It means that while string theory often agrees with the old-fashioned rules what it means for two spacetime manifolds to be physically equivalent, it stands above the most straightforward, old-fashioned factoids. String theory thinks different and in some cases, it may say that two geometries are completely physically equivalent even though all other objects, point-like particles, and little green men could think that those are completely different.

To get a precise physical agreement in all physically measurable quantities in a grand theory that contains everything you need in a theory of everything is highly nontrivial but string theory often does it. These equivalences – T-duality, mirror symmetry, and perhaps other dualities – also show that the selection of the consistent theories of quantum gravity is extremely limited and constrained. The number of vacua of a consistent theory of quantum gravity is so low that string theory prefers to sell each solution "several times". It may sell it as a Calabi-Yau manifold, or as its mirror. There is only one physical theory, not two inequivalent ones, but this one theory is capable of adopt "at least two jobs". The same theory may be presented as the consistent quantum theory of gravity associated with two (or more) distinct spacetime manifolds etc.

As I said, many of the non-linear sigma models (i.e. old-fashioned geometries, as probed by vibrating strings) are precisely equivalent to seemingly non-geometric theories on the world sheet such as the Gepner models. Within string theory, the old-fashioned rules dictating "which two manifolds are physically the same" no longer hold. You must use string theory's own, more advanced rules – and they only reduce to the old-fashioned rules if all the characteristic distances such as radii of compact dimensions and curvature radii are much longer than $$\ell_{\rm string}$$ at all times. In other words, string theory tells you that the old axioms and rules may only be trusted in the long-distance limit and it explicitly tells you what the rules have to be replaced with when some radii or curvature radii are short enough.

In the text above, I discussed both non-geometric vacua like the Gepner models (collections of minimal models) as well as the geometric, non-linear sigma models. One may combine the phenomena in various ways. For example, unorientable theories may be defined on a Möbius strip – which is a strip with a $$\ZZ_2$$ twist applied before the reconnection. However, aside from the left-right $$\ZZ_2$$ map that defines the Möbius strip, the stringy geometry allows you other gauge transformations, i.e. the $$\ZZ_2$$ T-duality transformation. You go around a Möbius-like strip and the world becomes T-dual to what it was when you get back: the momentum and winding modes get reverted if you walk around your house! ;-) Compactifications of this kind cover a big class of clearly non-geometric vacua of string theory and as far as I know, the true experts in this sub-industry still disagree whether all non-geometric vacua of perturbative string theory may be constructed in this way, as "locally geometric ones" connected by some T-dualities.

D-brane generalized geometry

So far in this text, I discussed the "generalizations of geometry" that are encoded in perturbative string theory i.e. in a modular-invariant, two-dimensional, conformal field theory on a world sheet. But when $$g_s\ll 1$$ no longer holds, nonperturbative string phenomena start to be important. Lots of facts are known about non-perturbative physics of string theory – for example, the $$g_{\rm string}\to \infty$$ limits of many string vacua are exactly known (and they are usually other, weakly coupled vacua of string theory or M-theory). And we even have full nonperturbatively exact definitions of string theory for any finite $$g_{\rm string}$$ such as matrix string theory.

But it's still fair to say that the "exact rules" of the generalized geometry – as probed not just by strings but all the objects in the theory we still call string theory – are much less understood than the rules in perturbative string theory where the conformal fields theories on the world sheet "know about everything" in the game. Perturbative string theory – i.e. conformal theories on the world sheet – are so well understood that even rigorous mathematicians basically have an axiomatic framework to study it. Non-perturbative string theory remains so incompletely understood that even physicists – who can survive with much less rigor – don't know of any universally applicable set of axioms (yet).

Inside and even outside perturbative string theory, we know lots of things that may be considered "stringy generalizations of geometries". For example, the interactions of strings on a background with a nonzero B-field may be seen to be equivalent to field theories (with all the excited string fields) on a noncommutative geometry. These are field theories with fields that are multiplied not in the pointwise way – but using the star-product or, equivalently, as operators that depend on $$\hat x,\hat p$$ noncommuting coordinates on a phase space. And these theories with the star-product are also equivalent to the "matrix-like generalizations" of the spacetime coordinates that are seen by $$N$$ D-branes.

Coincident D-branes generalize the geometry in yet another way. If you have $$N$$ ordinary, old-fashioned particles living on a manifold, their coordinates are given by $$x^a_i$$ where $$a=1,2,\dots D$$ [thanks, Bill] and $$i=1,2,\dots N$$. However, if you replace these particles by the stringy Dirichlet particles i.e. D0-branes, the degrees of freedom get organized to $$x^a_{ij}$$, matrix elements labeled by two and not just one indices $$i,j=1,2,\dots N$$. It's similar to the way how quantum mechanics generalizes probability distributions: it replaces them with the density matrix that has the old probabilities on the diagonal but may also have anything (Hermitian) outside the diagonal.

The spacetime geometry as probed by D-branes brings you new equivalences between geometries – and non-geometries – that should be added to the equivalences seen by the perturbative strings.

To summarize, perturbative strings, D-branes, and other objects in string theory can see the geometry of spacetime manifolds that you "force upon them" in a similar way as on any objects propagating on the geometry. But they behave in a much more mature way on this geometry than any other objects – from "non-string theories" – you could think of. They are capable of living "without the crutches of an explicit geometry" and they are also able to see that two or more geometries (or non-geometries) are actually physically equivalent even if all the other probes (point masses and little green men) would think that they are not equivalent.

Much is known about the new rules with which string/M-theory modifies or replaces the old-fashioned rules of geometry. Much remains unknown. By the stringy or quantum generalized geometry, we may mean any class of rules or phenomena that reduce to those of the classical geometries when the radii and curvature radii are kept much longer than the string length at all times but that also say something coherent when this condition is relaxed. With this definition, the seemingly mathematical rules of "stringy or quantum generalized geometry" become pretty much a synonym for the physical laws of string/M-theory itself. Physical phenomena become primary.

You may see that because of its ability to produce the old-fashioned rules of geometry in some limits but also imply something else, string theory and string theorists are no longer reduced to uncritical users of the products of mathematics and mathematicians. They become active builders or discoverers of new rules of mathematics and new structures in mathematics – rules that mathematicians were often ignorant about because they were only probing the geometry by clumsy probes (and clumsy ways of thinking which emulated the clumsy probes in people's hands). String theory and string theorists can do better, be more accurate, map the structure of all consistent theories more accurately, and get rid of axioms that are sometimes not quite true and shouldn't be believed uncritically.

The spacetime and all rules governing it are emergent in string theory. String theory may modify, generalize, and even live without these structures and assumptions – yet remain equally well-defined and even more consistent at the same moment.