An Abundance of K3 Fibrations from Polyhedra with Interchangeable Parts (by Philip Candelas, Andrei Constantin, Harald Skarke),which is purely mathematical – it is a paper about higher-dimensional shapes – but its relevance for high-energy physics is obvious. Philip Candelas in particular is a pioneer of the applications of Calabi-Yau manifolds in string theory. He's also highly active these days. Aside from the newest paper, you may be interested in a paper from December 2011 in which they construct a heterotic string model that produces the MSSM with three generations and absolutely nothing else at low energies!

The today's paper is dedicated to the memory of Maximilian Kreuzer († November 2010) of Vienna who has done lots of work and let me say that I haven't seen a more original work dedicated to a recently deceased person for quite some time.

Let me begin at a natural starting point.

Quantum field theories (and more primitive theories) require that you choose the spacetime dimension at the beginning. This number is a "metaphysical assumption", something you shouldn't think about. At most, you may invent anthropic stories unrelated to QFT that explain how difficult life could be in a higher number of dimensions than those 3+1 dimensions (space+time) that we observe.

But string theory is a theory of everything so it dictates what the number of spacetime dimensions may be or must be. In perturbative superstring theory, the spacetime has to have 9+1 dimensions in total. It implies that there are 6 hidden dimensions of spacetime we don't know. Similarly, in M-theory, there have to be 7 hidden dimensions and in F-theory, there are 8 of them even though 2 of them are predestined to remain infinitely small.

(See Where are we in extra dimensions and other articles for a basic sketch of shapes of extra dimensions in string theory.)

The shape of the extra dimensions may be studied separately from the large 3+1 dimensions we know. This statement is pretty much equivalent to saying that the total 10-, 11-, or 12-dimensional spacetime has the shape given by the Cartesian product\[

{\mathcal M}_{\rm total} = {\mathcal M}^{3+1} \times CY.

\] The letters \(CY\) indicate the shape of hidden or extra dimensions. I said that the spacetime dimension isn't arbitrary in string theory. Even when we choose one of the few possibilities for the spacetime dimension, the shape isn't arbitrary, either. The shape has to obey the equations of string theory, too. When the shape is large enough and/or when one may argue that corrections may be dropped or neglected, the equations of string theory for the shape reduce to Einstein's equations. In the vacuum, they may be written as\[

R_{\mu\nu} = 0.

\] The Ricci curvature tensor vanishes. This condition is a priori valid for the total spacetime. However, under various sensible conditions, it is equivalent to the Ricci-flatness of the compact manifold \(CY\) itself. The hidden dimensions have to obey Einstein's equations of the general theory of relativity. And they're very constraining.

One class of solutions are flat manifolds. They have no curvature; the Riemann curvature tensor vanishes so the Ricci tensor vanishes, too. The infinite flat space is an example but it is phenomenologically excluded; we just don't observe many new flat infinitely large dimensions and we don't observe spin-1 partners of all known fermions etc. However, you may also construct flat compact (i.e. finite-volume) manifolds by making the coordinates periodic. We call these compactifications (specified by a lattice of periodicities, not necessarily "simple periodicity" for each individual coordinate) toroidal compactifications.

Toroidal compactifications are therefore an important and simple enough class of solutions to the equations determining the shape of extra dimensions. For a given dimension of the torus, we may still choose its 6-8 radii and the "angles" in between them. These toroidal compactifications usually preserve all the supersymmetries of the parent, higher-dimensional theory which implies that the spectrum is too organized and simple – more so than the particle spectrum in the real world.

Are there other solutions?

Yes, you may also take "orbifolds" of the tori. But these shapes strictly speaking have a nonzero delta-function-like Riemann tensor at the fixed points of the orbifold action (which is used to identify points to build the orbifold: an orbifold is a coset or quotient or a set of equivalence classes). However, it may be shown that the orbifolds are rather generally just special points in broader moduli spaces of shapes that have nonzero curvature everywhere (although there may be many different ways to describe the same points and it may be debatable which of them is more fundamental if any).

So what are the non-flat but Ricci-flat shapes?

Let's look at them according to their dimension. One-dimensional manifolds are locally a line which can't have a curvature so as long as they remain non-singular, there is nothing else to discuss. An infinite line and a circle are the only two closed one-dimensional manifolds.

Closed two-dimensional orientable manifolds may be visualized as genus \(h\) Riemann surfaces – a sphere with \(h\)

**h**andles attached to it. The integral of the Ricci curvature scalar over the surface is proportional to \[

\chi = 2 - 2h.

\] The number \(\chi\) – which is pronounced as "chi" – is known as the Euler character. If you approximate a surface by a polyhedron, you may calculate \(\chi\) as the number of vertices minus the number of edges plus the number of two-dimensional faces minus the number of three-dimensional faces (for higher-dimensional objects) plus minus plus minus, you get the point.

At any rate, if the Ricci tensor is zero everywhere, it is obvious that \(2-2h\) must be zero, too. So \(h=1\) and the sphere with one handle is nothing else than the two-dimensional torus. It's the only two-dimensional Ricci-flat manifold. Its shape may be specified by a complex number \(\tau\) and all values of \[

\tau' = \frac{a\tau +b}{c\tau +d},\quad ad-bc=1

\] give the same shape. This equivalence is known as an \(SL(2,\ZZ)\) symmetry. Fine. We have already mentioned that tori are simple solutions to Einstein's equations and we returned to them again. I won't do it again. Instead, I just say that \(n\)-dimensional tori exist and are Ricci-flat for every positive integer \(n\); the value may be odd and for \(n=1\), they are known as circles.

What about the other solutions? It turns out that there are many new Ricci-flat manifolds you may construct with the help of complex numbers – by pairing the real coordinates on the manifold into complex coordinates. That means that we obtain new non-flat but Ricci-flat manifolds in (real) dimensions 2,4,6, and so on. In these interesting classes of shapes (and their vicinity), the Ricci flatness condition becomes equivalent to a reduced "holonomy group" (\(SU(2),SU(3),G_2,SU(4),\dots\)) or to some supersymmetry that may be preserved despite the curvature (because there exist "Killing spinors" on the manifold).

An IQ test: What is the next dimension after 2,4,6 for which you may obtain very nice, new, non-flat but Ricci-flat, and in fact supersymmetric manifolds? What is the next number after 2,4,6? Yes, it is 7. :-) Algebraic geometry has the right to be more subtle than kindergarten mathematics! The new 7-dimensional manifolds are those of a \(G_2\) holonomy – the rotations of the tangent space induced by round trips form a group isomorphic to the exceptional automorphism group of the octonions \(\OO\). We will avoid this exceptional case in the rest of this text.

Fine. So I promised you new nontrivial non-flat but Ricci-flat manifolds whose dimension is 4 and 6. In the case of 4, the only new solution (when it comes to the topology) is the so-called K3 manifold.

A two-dimensional visualization above captures a manifold whose topology is completely fixed. There are \(19\times 3\) real parameters that may be adjusted to change the detailed shape of the manifold. That's it. In the article linked above, I described K3 as the simplest possible compactification of extra dimensions after the tori. And this simplicity – and, aside from the tori, uniqueness among the 4-real-dimensional manifolds – implies that the K3 manifold is important in the broader scheme of things, too.

The new paper I want to discuss uses the K3 manifolds in a very important way, too.

Now, let us jump to 6 extra dimensions. That's the first example directly relevant for phenomenology in particle physics. It's because superstring theories have 10 spacetime dimensions so 6 of them have to be compactified to preserve 4 large dimensions we know. In particular, compactifications of heterotic string theory on six-dimensional Calabi-Yau manifolds \(CY\) represented the first realistic class of the Universe around us within string theory. By Calabi-Yau manifolds, we mean special 6-real-dimensional manifolds that preserve some supersymmetry (and therefore Ricci-flat) which are neither \(T^6\) nor \(T^2\times K3\), i.e. neither a six-torus nor the product of a two-torus and a K3 manifold.

Their unusually realistic structure was discovered in a 1985 paper by Candelas, Strominger, Horowitz, and Witten. Yes, it's the same Candelas. A natural question to ask was: What is the shape of the extra dimensions, \(CY\)? Andy Strominger was assigned his task. He went to math libraries and found exactly one example, the quintic hypersurface in a complex projective space. That was great: they apparently had the unique theory of everything. You can't be shocked that Edward Witten would offer the estimate of a "few weeks" for the time needed to calculate the muon mass and all other parameters and complete the quest for a TOE. It was a reasonable guess; we just know today that it was wrong, too.

Why was it wrong? Well, tens of thousands of six-dimensional Calabi-Yau spaces – manifolds that are not flat but that manage to be Ricci-flat by preserving 1/4 of the supersymmetries as well – were later found.

The most versatile class of Calabi-Yau manifolds are obtained from the so-called reflexive polytopes (higher-dimensional polyhedra; "reflexive" refers to the usual duality for polytopes) by the Batyrev construction. (This particular machinery is much more well-known to trained mathematicians than trained physicists, of course. Some of its insights overlap with discoveries that were made by physicists, using the physics jargon and the physical intuition.) Max Kreuzer had intensely worked on these things. He listed all the 473,800,776 such polytopes. Again, if physics with a larger number of solutions looks hopelessly hard to you, it is probably just your problem. Here we are talking about less than half a billion of shapes. It's easy to list them – the number is thousands of times smaller than the Greek government's debt.

One may obtain Calabi-Yau manifolds from these polytopes. The most characteristic integer label describing the topology of a manifold is the Euler character \(\chi\) I have already mentioned (think about the vertices, edges, faces, and so on as the most general definition). For Calabi-Yau manifolds, the Euler character may also be calculated as\[

\chi = 2(h^{1,1}-h^{1,2})

\] where \(h^{1,1}\) and \(h^{1,2}\) are the so-called Hodge numbers, the numbers of two-dimensional and three-dimensional cycles (topologically nontrivial, topologically inequivalent, and independent submanifolds of these dimensions: the two indices label holomorphic and antiholomorphic dimensions separately but I don't want to go into that here). The Euler character is always drawn on the \(x\)-axis of the plots known as the Hodge plots. The other coordinate \(y\) is defined to be the "height" \(h^{1,1}+h^{1,2}\) in almost all cases.

Here is the basic Hodge plot of the known Calabi-Yau manifolds: 30,108 distinct points i.e. pairs \( (h^{1,1},h^{1,2}) \) are included on the picture. That means that there are at least 30,108 known Calabi-Yau topologies (but if the two integers don't specify the topology of a manifold uniquely, there may be many more and the new paper talks about 700,000+ Calabi-Yaus, too).

*Click to zoom in.*

You may see that the number of topologies is large but at this moment, it seems reliably finite. The Euler character only goes from \(\chi=-960\) to \(\chi=+960\) and if you think about the patterns on the picture above, you may quickly decide that it is not an accident.

What patterns do you see on the Hodge plot above? First of all, it is left-right-symmetric. This picture makes it obvious that the corresponding symmetry is called "mirror symmetry" – one of the remarkable mathematical relationships that probably wouldn't have been discovered at all if people didn't know string theory.

While the left-right symmetry looks trivial on such a plot, you must realize that it is an incredibly complicated map. For example, take the manifolds in the right upper and left upper corners, the mirror pair \({\mathcal M}_{11,491}\) and \({\mathcal M}_{491,11}\). They're "extreme" in some sense, coming from the most complex polytopes. You may imagine that these manifolds are some sort of ancestors, Adam and Eve. Adam has 11 hair and 491 legs while Eve has 491 hair and 11 legs only. Their shapes are completely different but the existence of one of them guarantees the existence of the other. And in fact, string theory compactified on one of them produces the same physics as (the same or different) string theory compactified on the mirror partner.

However, if you look carefully, you will realize that the map\[

\chi\to -\chi

\] isn't the only apparent symmetry in the Hodge plot. And they did notice it wasn't the only one. The image also seems to have two valleys, a left valley and a right valley, and they're left-right symmetric by themselves. Alternatively, you may combine this left-right reflection with the original \(\chi\to-\chi\) reflection to get a translation\[

(\chi, y) \to (\chi+960,y).

\] The Hodge numbers jump by \(\pm 240\) if you care. That's quite a clear symmetry of many (but not all) points in the Hodge plot and you may ask what's the reason behind this symmetry.

And the authors have understood it. The Calabi-Yau manifolds represented by these points can actually be constructed from polytopes that may be cut into two pieces, top and bottom, or X and Y chromosomes if you prefer my terminology. As long as the boundary between the two chromosomes goes along a K3 fibration – a manifold that may be visualized as a K3 surface (see above) attached to every point of a base manifold, i.e. a Cartesian product in which the shape of the K3 may change and rearrange as we move along the base manifoold – the two chromosomes are independent and in fact, genetic engineering is allowed. You may combine different X chromosomes with different Y chromosomes. One may say that the change of the Euler character corresponds to the replacement of the "simplest" top by the "most complicated one".

The paper contains several such constructions that explain many patterns of this sort. I recommend you to open the paper and look at least at the pictures – various colored versions of the simple Hodge plot above – because they're pretty. The Calabi-Yau manifolds are genetically decomposed to pairs of chromosomes and slices and projections involving K3 fibers play an important role. Moreover, the K3 fibrations may be classified by Lie group labels. So there exist things such as \(E_8\times \{1\}\) fibrations and \(E_7\times SU(2)\).

*Click to magnify the Hodge fountain.*

I couldn't resist to include at least one more picture from their paper. The \(x\)-axis shows the \(\chi\) once again while \(y=\ln (h^{1,1}h^{1,2})\). The blue, red, and yellow points show K3 fibrations of the \(E_8\times SU(1)\), both, and \(E_7\times SU(2)\) type, respectively. You may say that the \(E_8\times SU(1)\) are the "more rare" points near the top – such as Adam and Eve – while the more "generic" Calabi-Yau manifolds divide the maximum group \(E_8\) more evenly.

There are many more relationships and genetic engineering operations mentioned in the paper. They explain lots of visual observations about the picture you could make even without the knowledge of any algebraic geometry – or any advanced maths, for that matter. The number of different relationships between the Calabi-Yau shapes – e.g. various kinds of ancestry – is large. In a sense, it shouldn't be surprising. There are many ways to cut, glue, mill, drill, project [and many more verbs] three-dimensional objects. The six-dimensional engineers have many more verbs that describe what they can do with "shapes" in their factories. ;-) But don't misinterpret this "factory" jargon: the operations discussed here are much more natural and much less "man-made".

At the end, the number of Calabi-Yaus that are left unmatched to others by these operations and relationships is very small. And as I have said many times over the years, these operations and relationships between different shapes (and perhaps between different decorated shapes that also contain branes, fluxes, and everything else that string theory allows to be nontrivial) may be more than just mathematical curiosities or tools to construct various structures. Some of them may be crucial in cosmological mechanisms that one must understand to master the vacuum selection problem.

The membrane nucleation and tunneling on the landscape was often compared to some kind of "mutation" in a reproductive tree of universes – a tree implied by eternal inflation. But it's plausible that the Calabi-Yaus may choose their X,Y chromosomes (top and bottom) kind of separately, that there is a "sexual reproduction" of universes with newer Calabi-Yaus.

Effectively, some of the "reducible shapes" or "shapes that are distant descendants of others" may be favored or disfavored (or unstable). Also, when you try to calculate the probability that different shapes are realized in Nature, it's plausible that there is some mechanism that favors the individual chromosomes separately. For example, it seems "almost obvious" to me that Nature either heavily favors the shapes near the top of the fountain – the \(E_8\times SU(1)\) types – or those near the bottom. It's plausible that there exists a simple enough – and perhaps even justifiable (by arguments we would already agree with) – rule that will allow physicists to pick the right (or much more likely to be right) shapes sometimes in the future.

And that's the memo.

I hope that gif isn't capable of inducing epileptic seizures

ReplyDeleteAah, this looks great and exciting !

ReplyDeleteI`m just not sure if I should dare to read much more than the title ... :-P

Maybe I can try it after reconsidering the nice FM linked in about "where we are in the extra dimensions" ... :-) ?

Cheers

I had added the variable background colors to a pre-existing animation, a code made in Mathematica. ;-) If you're worried that sneezing could kill you as well, I think it's unlikely:

ReplyDeletehttp://physics.stackexchange.com/questions/32579/what-is-energy-sneezing-coughing-during-a-cold/32589

Do they make any comments about what transitions between one point of the web and another might be?

ReplyDeleteThe only thing clear to me is that movement in the moduli space of an F-theory / dual heterotic compactification can take one from one of these polytopes to another via singularity enhancement or Higgsing, but I'd bet that there are more exotic things going on as well, and were curious if you had any ideas.

(Of course, this is just a maths paper, but there should be implications for many types of string transitions!)

Well,I'm imune but can't guarantee for others :-)

ReplyDeleteJust to say that I appreciate the trouble you take over these physics memos. They are better than listening to a theorist's lecture since one can read them again any time. I only get the gist, but that is what I can get from a theory lecture too :) .

ReplyDeleteAre the symmetries between the CY's related to the dualities of String/M theory or are these separate things.

ReplyDeleteThe mirror symmetry, as reflected by the left-right symmetry of the Hodge chart, is surely considered a (major) duality in string theory. In fact, Strominger, Yau, and Zaslow showed that mirror symmetry *is* T-duality

ReplyDeletehttp://arxiv.org/abs/hep-th/9606040

applied to the three-dimensional toroidal "fibers". Whether the other patterns they mention are exactly dualities (equivalences of physics theories) in any sense is unclear to me.

This was a stackexchange classic!

ReplyDelete"The gauge group, SO(32), is just too awkward a starting point for realistic Yang-Mills model building."

ReplyDeleteI'm glad to see there are some things that are too complex for even you guys to want to try to think about. ha ha.

BTW, for us lit majors, what does the word "singularity" mean when you write: "If the 7-dimensional manifold is singular, the P- and CP-violating objects may happily live at the singularities." You mention that there are "also singularities that are not point-like." So I take it there is such a thing as a "point" in string theory? Two lines intersect at a point, is it something like that?

Apologies for disappointing you, Luke, but SO(32) is awkward not because it's hard to work with. Folks have no trouble to work with SO(32) - and not even with its 32,768-dimensional chiral spinor representations.

ReplyDeleteAfter all, Steven Weinberg, who is not even a full-fledged string theorist, had no trouble to analyze a string theory whose gauge group is SO(8,192).

http://www.sciencedirect.com/science/article/pii/0370269387910963

The group is awkward because one may pretty much demonstrate that it is impossible to break the symmetry in such a way that a realistic spectrum emerges. It's not that we are lazy, we don't know how to work with it, so we choose to believe it can't be done.

Instead, it may be worked with and proved that it can't lead to realistic models.

...

By singularities, I mean special points in the geometry that don't look like a regular piece of flat space with low curvature because something is infinitely curved or identified. An example is the center of a black hole or the Big Bang. These singularities are lines in spacetime - singularities may have any number of dimensions and may be both spacelike and timelike.

In that sentence, I am talking about 0-dimensional points within a 7-dimensional space - which becomes a 3+1-dimensional singularity in the 10+1-dimensional spacetime. One may always try to talk about geometries with points - string theory just adds something to the points and modifies their behavior if you look at the situation with a great, stringy resolution (short distances). But it doesn't mean that the word "point" becomes completely taboo.

Lubos writes: "Type IIA (and also type IIB) has too much supersymmetry. Even if you compactify it on a Calabi-Yau manifold, you would get 8 supercharges which is too much: N=2 supersymmetry in four dimensions. This amount of residual supersymmetry couldn't be broken, wouldn't allow any P/CP-violation, and would lead to huge and experimentally ruled out multiplets, anyway."

ReplyDeleteSo some forms of string theory can be ruled out by experiment. That shows it is wrong to say string theories cannot be tested. The ones that are left are, I presume, consistent with everything we know about the world so far. Ideally I suppose you would like to rule out more of them, leaving you with fewer? How do you envision progress in string theory. Is more experimental data essential? Or do you look for logical and mathematical inconsistencies to rule out some forms? Again, this is just for the amateur non-physicists among your readers.

What does "unoriented" mean as in "Unoriented strings"? No preferred direction?

ReplyDeleteWhen you speak of branes being "stacked" is that like thinking of a line as a stack of points?

ReplyDeleteHi Luke, an unoriented or unorientable string is one that doesn't carry an "arrow" so you may merge two pieces of a string in one way; as well as if you reverse the front and the back of one piece, too.

ReplyDeleteTo get unoriented strings in string theory, you actually have to make the vacuum more complicated by the addition of "orientifolds". Type I string theory is an example. More elementary versions of string theory - type II and heterotic strings - contain only oriented strings. Oriented strings carry an intrinsic arrow on each piece of the string - an arrow that remembers, for every small or large piece of the string, which side or which end is the "front" (direction "to" of the arrow) and which side is the "rear" (direction "from" of the arrow).

Oriented strings are simpler because the winding number - how many times a string winds around a cylinder, for example - is conserved. For unoriented strings, it can't be conserved because +1 and -1 times wound string is the same thing. Also, for unoriented strings, the histories - "world sheets" of merging and splitting strings - may be both orientable and unorientable surfaces. Unorientable surfaces include the Mobius strip, the Klein bottle, and others (you may go around those sheets of paper and reverse an object such as "b" to its mirror image such as "d").

Oriented string theories only add up histories with oriented world sheets, genus h Riemann surfaces (perhaps with boundaries if open strings - those with end points - are allowed).

Hi, I haven't used the term "stacked" in this blog entry, at most "Physics Stack Exchange". But it is being used. A stack of branes is just a collection of branes that are coincident or nearly coincident - parallel with transverse distances between them that are sent to zero so that they occupy the same space. This can be done with branes of any dimensions, including D0-branes which are points.

ReplyDeleteWhen you do so with D-branes, there are new degrees of freedom emerging from open strings that start on the M-th brane of the stack and end on any, N-th brane of the stack. That's why all fields living on the branes, including their position in the transverse direction, are promoted to matrices.

Thanks, Twistor. Is Phil Jones your real name!? ;-) That's quite a name.

ReplyDeleteAlmost all string vacua may be easily ruled out by experiments - and all string vacua except for one may be ruled out by experiments when one works sufficiently hard. For an ever smaller set of ever more realistic vacua, it's increasingly hard to show that they're wrong if they're wrong. In fact, I am confident that are quite a few of known - and precisely described - compactifications about which we're not quite sure that they're wrong because they pass an amazing number of tests. For example, the "pure MSSM heterotic model" by Braun Candelas Davies Donagi

ReplyDeletehttp://arxiv.org/abs/arXiv:1112.1097

that I mentioned looks like MSSM plus some high-energy stuff and quantum gravity. If this model disagrees with a detailed feature of the reality, it's pretty hard to show it. Such models are just damn promising. But there are still many of them - although pure MSSM is rare - so even in this elite set, almost all of them must be ultimately wrong.

The falsification of N=2 vacua, however beautiful they are, is really a matter of seconds because they disagree in some key general properties of the known forces such as fermions with chiral interactions.

Of course it's complete nonsense to say that string theories can't be tested.

Thanks for your kind patience on this and my other questions. Your remarks on "unoriented" were particularly helpful.

ReplyDeleteGreat post! This is some awesome mathematics.

ReplyDeleteWell Lubos, if the theory should describe everything why 6 dimensions have to be compacted by hand? Shouldn´t they emerge from some inevitable physical mechanism spontaneously?

ReplyDeleteDear NumCracker, we don't know whether there is a mechanism that determines the number of large dimensions. It was surely the proposal by Brandenberger and Vafa in 1989

ReplyDeletehttp://www.sciencedirect.com/science/article/pii/0550321389900370

that goes under the name "string gas cosmology" these days - and there are various attempts to update it and generalize such as "brane gas cosmology" - but we don't know for sure whether this mechanism really operates because the arguments in favor of the Yes answer remain pretty vague as of today.

Even if we don't know whether such a thing exists, we may say that 11-dimensional spacetime and spacetimes of the form M4 x CY3 are equally good solutions of the fundamental equations so if you don't talk about their evolution from the initial state or any cosmology at all, you must admit that both possibilities - with the right number of large dimensions as well as wrong ones - are equally plausible.

It seems kind of surprising sometimes how much of this is just classical. It almost seems like the only real "quantum" feature of strings is T-duality. Of course there is S-duality but that seems like a property shared by at least some point theories too.

ReplyDeleteDear Will, I believe that this algorithm for you to get surprised is self-deceptive and the conclusion is completely wrong.

ReplyDeleteEverything in string theory boils down to quantum mechanics. The fact that there are discrete "particle species" in the effective field theories describing string theory couldn't work without string theory: different particle species emerge as discrete quantum states of an excited string and the spectrum of a string couldn't be discrete without quantum mechanics.

The critical dimension emerges from quantum effects, too. It has to be D=10 for the conformal anomaly to cancel and the conformal anomaly is a one-loop, purely quantum effect.

Gravity - GR - follows from string theory at long distances. In the vacuum, for example, the Ricci tensor has to vanish. How do you derive it's the Ricci tensor, i.e. how do you derive the equations of motion? They are the beta-function for the worldsheet coupling constants. But all beta-functions are loop effects, purely quantum effects caused by virtual particles. So the Ricci tensor appears in Einstein's equations due to purely quantum effects, too.

Dualities are purely quantum, too. T-duality forces us to switch a large and small hbar on the worldsheet; S-duality does the same in the spacetime.

Even if you forget about the "actual string theory" in which all elementary facts and patterns result from quantum mechanics, as sketched by the major examples above, your reasoning is completely irrational. For example, you say that "only T-duality [and S-duality] is a quantum feature" of the theory. But the word "only" only reflects your prejudices, not the actual weight of the ideas. In the typical energy regime of string theory, e.g. at the string scale etc., things like T-duality are omnipresent and they affect everything. The only justification for combining them with the word "only" is that you actually chose to focus on the studying of another energy regime, one that may be approximated by some classical theories.

So no surprise, for this classical regime, you find out that classical notions are enough. But this shows absolutely nothing about string theory or any other theory. It just shows that you are not really interested in physics beyond the classical approximation. Almost every complicated enough quantum theory has some classical limits and you may choose various attitudes to it. You clearly chose the attitude that the classical limit is what you're primarily interested in, so the classical aspect of any theory will have a greater weight in your eyes. But in the actual structure of the theory - in the phenomena where it really matters, at the typical length/energy scale of the theory - it is absolutely impossible to be satisfied with the classical notions or the classical laws.

Your comment about the S-duality misses the point, too. Field theories may exhibit S-duality, too. But field theories of this kind, like the N=4 gauge theory, are actually *equivalent* to string theory. They are exactly the same thing. Because this equivalence exists, it would be totally silly to be surprised that the "pointlike particle field theories" may exhibit the stringy dualities. Of course that they can because they're just ways to describe string theory.

A point I want to make is that your perception of a "classical dominance" is just a self-inflicted wound. Almost everything we can actually talk about in terms of normal observations boils down to classical observables and gadgets that de facto obey the classical limits of the laws of Nature. But that doesn't mean that there's no quantum mechanics beneath the phenomena. However, one must look deeply enough to see the quantum mechanisms that underlie seemingly simple things. And in string theory, all mundane seemingly classical things such as the discreteness of the particle types, specific critical dimension, and low-energy effective equations boil down to quantum effects.

This long comment, worth a blog post by its own ;-), strongly relumes my feeling that it is so beautiful and cool when lower energy or classical physics can be explained by underlying deeper theories. It occured to me for the first time when considering the relationship between thermodynamics and statistical mechanics. Before knowing about statistical mechanics, parts of thermodynamics seemed some kind of a mess I just had to learn by heart to me :-P (overstating a little bit ...)

ReplyDeleteThe examples you mention at the beginning of this comment are cool, and it picks me that at present, I do not understand STuff well enough such that I can see at a slighly technical level how and why they work myself (to be happy and satisfied) ... :-/.

But such things always motivate me to learn more :-D

Exactly, Dilaton, and a great example.

ReplyDeleteThermodynamics is analogous to the classical limit - and the thermodynamic and classical limits are often taken together. And the underlying statistical and/or quantum mechanics removes the "mess" and vague ad hoc explanations and choices that exist in the messy, classical or large-N theory.

The messiness really boils down to some ambiguity or ignorance and the random choices it requires us to make to get the right answers. In thermodynamics, the entropy is only determined up to an additive shift: S goes to S+constant doesn't seem to influence anything and we don't know the constant. In statistical quantum mechanics, there is a fixed constant. The ambiguity goes away, things start to make sense.

But as long as we only deal with big heat engines, of course that thermodynamics and classical physics are still OK. We just know what to do with the ignorance and seeming messiness hiding at short distance scales etc. It's really the desire to remove the messiness that drives theoretical physics.

Wow, this article is indeed very nice and it makes a perfect appendix to my "Shape of Inner Space" book, so thanks for this Lumo :-).

ReplyDeleteI particularely like the nice explanation of the Hodge plot and the description of things one can read off from it :-).

I'll probably look at the colored pictures in the paper later today ... :-D

Awesome discussion Dilaton and Lubos :-) I can't help but making an analogy : QM is a "new world" like Columbus' America in 1492. Today some people are still questioning if the natives are human beings (or "God's creatures") like us... we might still be in the 1920s ;-)

ReplyDeleteNo, they may be mathematically consistent, as I believe they are ... however, the theory may still be lacking a mechanism to point the correct compactification scheme. Well, in fact, if that already existed and it was known we would have been derived physics from math, then Plato would be delighted by the power of abstraction! Do you have any hint how it would be implemented?

ReplyDeletePrecisely, Lubos. If anything in the universe "looks" classical to you it is only because you are not looking deeply enough.

ReplyDeleteI am amused by arguments that our minds may operate quantum mechanically. How ridiculous. Why should our minds be an exception? Trying to explain conscienceness as a quantum effect is silly. Trying to explain it as a non-quantum effect is sillier.

My experience of thermodynamics was just the opposite, Dilaton. I first learned chemical thermodynamics and found it to be quite beautiful. When I later learned statistical mechanics I understood why it was so beautiful.

ReplyDeleteGene, true. Thermodynamics is beautiful. Einstein would count it among the "principle-based theories of Nature", as opposed to the constructive ones. It's a great axiomatic framework to understand many things. However, it is also incomplete. The incompleteness reaches the technical details about how all the things work, of course - it's not the purpose of thermodynamics. But sometimes the incompleteness goes even to the questions that thermodynamics may be expected to say something about...

ReplyDeleteYour points are well taken. It's true that quantum effects do underlie and determine everything, even if one does wind up using a (to me) surprising number of classical considerations. However if I can make one more point. When you're tempted to reply to a relatively innocuous and short post with an immense rant employing the word "fucking" (twice), it says more about you than about the original post. In fact I don't have any of the attitudes which you imputed to and which apparently have made you so upset. I share your views in general when it comes the various anti-string and anti-quantum heretics in the world. I'm sorry that you've chosen long ago to make abrasiveness be your preferred style of communication. It's a style that seems to work better in blog-world than in real life, but still I think you could have a bigger audience with a more moderate style, and I think everyone would benefit. For example, I was considering donating, and I have donated to other blogs, but now I doubt I will.

ReplyDeleteWill Nelson,

ReplyDeleteYes, our host could turn it down a little but he would be less interesting. A preferable alternative would be for you to be less sensitive. How about it?

I've been a TRF fan for so,long that I've become inured to Lubos' "abrasive" style and find it refreshing.

It is far preferable to the sugar-coated namby-pamby style that is prevalent elsewhere and, besides, he uses the f-word mainly when he responds to something stupid or to someone who just refuses to learn. Occasionally, he uses the word just for emphasis so try not to be so easily offended. Actually, Lubos is remarkably tolerant of diverse views.

So, stick around; you can learn a lot.

Dear Will, for pedagogical reasons, I used the terms "fucking atoms" and "fucking uninterested" to emphasize that the words that follow play a critical role in the argument. What does it say about me except that in similar situations, I am using the same words as 90% of the English speakers?

ReplyDeleteBe assured that the hostility is mutual. I despite people who think that they're better people by avoiding language or the content of their speech that could be found controversial by anyone. In general, they're shitty fucking hypocritical pretentious opportunist assholes.

Hey Will,

ReplyDeleteI completely agree with Gene.

Lumo can be a little bit impatient sometimes because the worst trolls who really want to destroy cool physics are often those who have no clue about it. And they are not in the slightest interested in understanding it either ...

But if he sees that you just do not jet understand things the correctly but you mean no harm by it and you are willing to learn, he writes you very nice and detailed technical explanations as he has done it for you ... So you should really not feel insulted or something.

There are just way too many evil trolls and sourballs out in the world wild web today and I appreciate it that Lumo calls them by their appropriate names and does not allow them to come in here. Other (otherwise nice) physics bloggers are much too nice and patient with trolls and sourballs, they allow them to pester their sites and discuss politely with them insted of blowing them away !

I mean banning them from their sites for example ...

Cheers

thanks for sharing.

ReplyDelete