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Feynman sum over Young diagrams (discrete histories)

A covariant definition of M(atrix) theory?

One of the most important difficult outstanding problems in science – OK, at least in theoretical physics if you can't see that it's the ultimate heart of science – is a universal definition of string/M-theory that is valid in all superselection sectors or compactifications and for all values of \(g_s\), the string coupling, and all other moduli.

It is in no way guaranteed that such a single definition exists. There may exist a profound reason why string/M-theory will always be known as a manifold – an atlas of patches that may be glued to each other but that depend on individual definitions with equations and concepts that differ from one patch to another.

The knowledge of a universal definition would turn all questions about the "size of the landscape" and the "existence of a vacuum that matches the observable Universe" into fully rigorous mathematical tasks – something that can't quite be said to be the case today. Overlooked inconsistencies, instabilities, or selection effects could pinpoint the correct vacuum, too.

It seems very likely that our Universe may be usefully described as a compactification of one of the supersymmetric master vacua in a maximum number of dimensions. There seem to be one in 11 spacetime dimensions, M-theory, and five string "theories" in 10 dimensions: type I, IIA, IIB, HE, and HO (the latter two are heterotic).

The existence of a definition of the master vacuum sometimes allows us to define all of its compactifications, too. Well, this is the case for all the string theories at the string coupling \(g_s\ll 1\). All those pieces of string/M landscape seem to be describable by the two-dimensional world sheet conformal field theories (CFT), a formalism that arguably keeps its status of the most important calculational pillar of all string/M-theory. The Duality Revolution of the 1990s has identified the perturbative stringy calculus to be just a "limit" of a broader theory and returned physicists' focus back to the spacetime, while elevating branes and other objects to a new status on par with the strings, except that the fundamental strings were still needed to determine how those things work and why the revolutionary, string-downgrading insights themselves were correct.

This formalism – which, I repeat, seems to be relevant only for perturbative calculations whenever a string coupling \(g_s\ll 1\) may be isolated – boasts a number of wonderful features that unify concepts, preserve consistency, and enable calculations, especially

  1. the state-operator correspondence, a one-to-one map between local operators in the plane and states on the cylinder
  2. operator product expansions (OPEs)
  3. new stringy geometric symmetries such as T-duality
  4. current algebras
  5. derivation of the critical dimension, \(D=10\), and spacetime field theories (and equations such as the Ricci flatness or Einstein's equations) from world sheet conformal symmetry
  6. minimal models and bootstrap methods to determine properties of special world sheet CFTs
  7. finite-dimensional moduli spaces of Riemann surfaces (basic classes of the histories we sum over); the modular invariance is a way to explain why the dangerous corners of this moduli space are absent
  8. classification of D-brane boundary states identified with consistent boundary conditions for boundaries of the world sheet – a whole "transperturbative" framework allows one to study the properties of D-branes
and a few others. A wisdom hiding between a tight skeleton constructed out of these gems is the reason why the Feynman diagrams with propagators replaced by "stringy tubes" are so much more well-behaved and allow one to treat the gravitational force consistently, too.

All these perturbative theories may also be redefined in the light-cone gauge. Some generators of the Lorentz symmetry cease to be manifest symmetries. However, the absence of unphysical excitations (scalar and longitudinal gauge particles and good and bad ghosts) becomes manifest.

This beautiful apparatus of world sheet CFTs is only good for perturbative calculations (to all orders) in the \(g_s\ll 1\) regime even though the D-branes, in some sense, allow us to go beyond the simple expansions and include certain whole classes of nonperturbative corrections – even though all of their description uses the same "perturbative" methods.

I like to say that this great discovery by Polchinski extends the perturbative expansion in a similar way in which the cardinal and ordinal numbers (certain families of "infinite numbers") extend the integers. What do I mean?

Well, the perturbative calculations allow us to calculate contributions of the form \(g_s^N\) where \(n\in\ZZ\) is an integer. But a D-brane may contribute something like \(\exp(-1/g_s)\) to a process – the numerator in the exponent shouldn't really be "one" but another number but it is true that the denominator is just the first power of \(g_s\) and not the second one which you expect for "normal" instantons and solitons.

The function \(\exp(-1/g_s)\) is much smaller than any \(g_s^N\) when \(g_s\ll 1\). You may imagine that it's a term that you get "after" all the integers. So formally,\[

\exp(-1/g_s) = g_s^\omega

\] where \(\omega\) is the smallest ordinal greater than any integer (it's the number of integers). However, the D-brane calculus allows you to calculate terms of the form \(g_s^N \exp(-1/g_s)\) etc. as well – this particular expression may be rewritten as \(g_s^{\omega+N}\). More complicated ordinals may appear as the exponents as well.

A homework exercise for you: Can one calculate just these effects or all ordinals relevant for the D-brane-like calculations? Which of them are? What is \(\omega^\omega\), if anything? Or should we use \(\aleph\)- i.e. aleph-based cardinals instead of ordinals? ;-)

OK, this world sheet CFT calculus is wonderful but it doesn't allow us to calculate generic things at \(g_s \geq O(1)\). Without some independent information, we should even say that the world sheet CFT doesn't allow us to present a precise definition of the theory. However, the evidence is unambiguous that the theory is fully consistent for \(g_s\geq O(1)\), too. How can it be defined outside the perturbative realm so that we don't rely on the "incomplete" perturbative or transperturbative expressions?

Some vacua of string/M-theory, the \(AdS\) vacua, are known to be equivalent to the boundary CFT which in principle may admit a complete definition, perhaps in terms of a "lattice gauge theory" or something similar. But these vacua are rather special. AdS/CFT doesn't seem to be helpful for finding a rigorous definition for Calabi-Yau-like compactifications of string theory, for example. This unhelpfulness partly boils down to the large dimensions' not being curved as an anti de Sitter space (we would love to understand lots of Minkowski and perhaps de Sitter compactifications, too); it partly boils down to our ignorance about the "right" boundary CFT even if the large dimensions actually are \(AdS\) (vacua with a negative cosmological constant).

However, another set of complete nonperturbative definitions exist for string/M-theory: the BFSS Matrix theory.

These "new matrix models" by Banks, Fischler, Shenker, Susskind give us a perfect definition of M-theory in the 11-dimensional flat spacetime – plus a limited class of the compactifications. All these descriptions arise in the discrete light-cone gauge. A light-like coordinate \(x^+\) is used as the "time coordinate". A complementary light-like coordinate, \(x^-\), is compactified on a very long circle of radius \(R\). That's borderline consistent (when it comes to the absence of grandfather paradoxes i.e. closed time-like curves) and allows us to separate the Hilbert space into sectors with different values of the quantized complementary momentum \(p^-=N/R\).

It may be shown (Seiberg, Sen etc.) that the sector of M-theory with the particular longitudinal momentum \(N/R\) is nothing else than the Hilbert space of a \(U(N)\) maximally supersymmetric quantum mechanical matrix model, i.e. maximally supersymmetric Yang-Mills theory compactified to 0+1 dimensions. Because we want to study states with a finite \(p^-=N/R\) but send \(R\to\infty\) to avoid the unfamiliar light-like compactification, we are primarily interested in the \(N\to\infty\) limit of the matrix models, too.

Certain compactifications may be defined, too. A Hořava-Witten domain wall or two may be added. (I've written several papers on the heterotic matrix models.) The "effective membranes" have 16 real fermions living on the membrane boundary – which translate to the fundamental representation of \(O(N)\) – the group is reduced from \(U(N)\) to \(O(N)\) because the domain wall forces you to consider a \(\ZZ_2\) orientifold – and allow you to prove (if you are a good mathematician) that the \(E_8\) gauge symmetry arises in the spacetime and is localized on the domain wall. The ultimate reason is the same as the reason why \(E_8\) arises in the fermionic description of the heterotic string, of course.

Which compactifications with the \(E_8\) domain walls have been defined by BFSS-like matrix models becomes a bit complicated once you start to add circularly compactified dimensions. There also exists a BFSS-like matrix model for a compactification on a K3 – in terms of the \((2,0)\) theory in \(d=6\) on \(K3\times S^1\times \RR\).

But the most symmetric compactifications that are usually discussed are the toroidal compactifications of M-theory. M-theory on \(T^k\) for \(k=1,2,3,4,5\) may be defined in terms of a BFSS matrix model: it is the same maximally supersymmetric gauge theory compactified to \(k+1\) dimensions and all the \(k\) spatial dimensions are compactified on a dual \(\tilde T^k\).

This claim may be taken literally for \(k=1,2,3\) – the last case is the normal \(d=4\) supersymmetric Yang-Mills, the theory beloved as the most important example of AdS/CFT – compactified on a \(T^3\times \RR\). However, for \(k=4,5\), such a Yang-Mills theory would be non-renormalizable. The precise string/M-derivations automatically produce the right, ultraviolet-complete replacement for the gauge theory. For \(k=4\), the 5-dimensional Yang-Mills is replaced by the 6-dimensional \((2,0)\) supersymmetric CFT on a 5-torus. For \(k=5\), we get the little string theory (a decoupled non-gravitational but non-local theory derived from NS5-branes) on a 5-torus.

For \(k\geq 6\) and higher – which includes the compactifications to 4 large spacetime dimensions that we need in phenomenology; and that are exactly the cases in which exceptional Lie-groups \(E_{6,7,8}\) appear as the U-duality symmetries – no BFSS-like matrix model is known. In fact, the derivation that works for \(k\leq 5\) suggests that the problems at \(k\gt 5\) should be more universal. Perhaps no matrix model should exist. But there could exist a "slightly different", BFSS-related definition of the light-cone gauge physics of these compactifications. Such a model is not known and whether such a description exists at all is an open question, too.

For \(k\geq 1\), the compactification of M-theory may also be interpreted as type IIA (and for \(k\geq 2\) also as type IIB) string theory on \(T^{k-1}\). One may see that the usual perturbative string calculus emerges from the matrix models – it's the collection of knowledge referred to as "matrix string theory" that your humble correspondent kickstarted. One may derive the spectrum, constraints, and statistics of the multi-string states and their perturbative interactions in the weakly coupled limit of matrix string theory; but one derives them from equations that work equally well for all values of \(g_s\) and may be treated nonperturbatively, too.

There are tons of other questions related to the new matrix models that remain open.

Lots of the things that are almost certainly true – namely the existence of a \(N\to\infty\) limit of the \(U(N)\) matrix model, its Poincaré symmetry, the existence of ground states, the black-hole-like behavior of all the other quasinormal states of these models etc. – have not been demonstrated quite rigorously. (Well, the existence of the bound state representing the graviton multiplet has been supported by the index theorem proofs and they may be fully rigorous at this point.)

Another extremely important, surprising omission is the apparent non-existence of a Lorentz-covariant description of the M-theory vacuum in \(d=11\). It's bizarre! If you consider a \(d=10\) string theory, e.g. perturbative type IIA string theory, there exists both a Lorentz-covariant definition with \(bc\)-ghosts and similar stuff; and a light-cone gauge definition of it.

But for the \(d=11\) M-theory, there only exists the light-cone gauge definition so far. Shouldn't there also be a "covariant matrix model of M-theory"? And what it is?

There are good reasons to think that this question isn't just another technicality, a description of another vacuum in another set of variables. In fact, there is a big chance that if you solve this problem, you may become able to define all of string/M-theory, the whole landscape of any shape and any value of any moduli – the dream I started with.

Why? Because in the Lorentz-covariant, non-light-cone-gauge definitions, one seems to be allowed to "change the background" at all times. We seem to know it from perturbative string theory. You may add a marginal operator to the world sheet action and deform the vacuum in any allowed way! This seems to be prohibited in the light-cone gauge because the \(p^-=0\) momentum-free excitations needed to change the superselection sector are "frozen".

OK, let us assume that such a covariant matrix-like M-theory definition exists.

What does it look like? It will be some "counterpart" of the usual perturbative string theory rules based on world sheet CFTs. In fact, it's probably not just a counterpart, it may be a generalization. The conformally symmetric world sheets may be a limit of something similar that obeys some generalization of the conformal symmetry, a condition that just "reduces" to the conformal symmetry in the \(g_s\to 0\) limit(s).

This covariant M(atrix) theory should have some counterpart of the state-operator correspondence, some counterpart of the operator product expansions; something instead of the current algebras; some replacement of the integral over the moduli spaces of Riemann surfaces (world sheets); and something to match all the other things I started with. What are all these generalizations if they exist at all?

The scattering amplitudes in perturbative string theory are computed as correlators of the "vertex operators" corresponding to the external states; and integrated over all possible conformal shapes of the world sheets. The amplitude is schematically\[

{\mathcal A}_{fi} = \int_\Sigma \exp(-S_{WS}) \prod_{i} V_i

\] where the action is some world sheet action, the integral involves the sum over all the topologies and the integral over all the shapes of the world sheets, and \(V_i\) is\[

V_i = \int d^2\sigma \, V^{1,1}(z,\bar z)

\] is an integral of a marginal tensor operator over the whole world sheet. This operator \(V^{1,1}\) is typically normal-ordered and contains factors like \(\exp(ik\cdot X(z))\) to remember the momentum of the external particles, and perhaps other factors such as \(\partial_z^k X^\mu(z)\) that remember the internal excitations of the string by the creation operators \(\alpha_{-k}^\mu\) etc.

How do these things get modified if we switch from covariant perturbative string theory to covariant M-theory? There may be infinitely large matrices, like in the BFSS matrix model, but perhaps we're allowed to compute with \(N=\infty\) from the very beginning.

The integral in \(V_i = \int d^2\sigma\dots\) is probably replaced by \(V_i = {\rm Tr} (\dots) \) in M-theory. Instead of integrating over the world sheet, we're simply summing over all the directions in the representation of \(U(N)\). Whether it has to be the adjoint representation or we may talk about others will be discussed below.

The vertex operators replacing \(V^{1,1}\) may be matrix-valued operators such as\[

[X^\mu,X^\nu] \exp(ik\cdot X)

\] where \(X^\mu\) is an \(\infty\times \infty\) matrix instead of a local field that depends on \(z,\bar z\) i.e. the world sheet. Note that the argument \(X\) inside the exponent is a matrix. The commutator was added as an example of the "excitation-dependent" part of the vertex operator. This portion may be much more complicated.

There should probably be some action in the path integral calculating the M-theoretical amplitudes covariantly. And how do we replace the integral over the moduli spaces of Riemann surfaces? Well, we must integrate over all possible "waiting times". But we must also integrate over the infinite-dimensional \(U(\infty)\) group "manifold" a few times. Well, why? Because \(U(\infty)\) is probably a gauge-symmetry and all the periodic world sheets must be allowed to be twisted by any element in the gauge group. This is analogous to the freedom to twist the toroidal partition function by \(\sigma\)-translations of the world sheet. That's the reason why the one-loop amplitudes in string theory are integrals over both real and imaginary part of \(\tau\) over the fundamental domain.

A funny thing is that the product of traces\[

{\rm Tr}(A) {\rm Tr}(B) \dots

\] may be consolidated into a single trace as long as you insert a \(U(\infty)\) transformation in between \(A,B\) etc. If there's just \(A,B\), we may use\[

\int d^{\infty^2} U\,\,{\rm Tr} (UAU^{-1}B),

\] I guess. Now, the integrand only depends on the conjugacy class of \(U\). The conjugacy classes of \(U(\infty)\) may probably be described by the density \(\rho(\phi)\) that tells you how many eigenvalues of the form \(\exp(i\phi)\) the large matrix has. It may be the case that this periodic \(\phi)\) is related to \(\sigma\) labeling a closed string. The integral over conjugacy classes may be reorganized to a sum over irreducible representations.

If you remember your group theory, the number of conjugacy classes is the same as the number of the inequivalent irreducible representations\[

K = R

\] So there is a \(K=R\)-dimensional space and you may pick its basis either to be associated with the \(K\) conjugacy classes of the group; or with the \(R\) irreducible representations. Both are infinite for \(U(\infty)\), of course.

It means that there is a reason to think that it may be possible to write M-theoretical – and perhaps general string/M-theoretical – scattering amplitudes as sums over Young diagrams \(R\) which label the irreducible representations\[

{\mathcal A} = \sum_R {\rm Tr}_R [ \exp(-S) A B \dots ]

\] There could be some additional finite-dimensional integration left. But otherwise, the path integral could boil down to a discrete sum. The Feynman integrals over histories are sometimes referred to as the "sums over histories". It may have sounded weird to you because the "sum" not only isn't a proper "sum" because it is an integral; in fact, it is an infinite-dimensional, functional integral.

But it could be that some theories – perhaps including M-theory in \(d=11\) itself – could be reorganized as true discrete sums over the histories. For each term in the sum, a Young diagram or two, there is some expression that mimics the calculation on a given world sheet. The discreteness of the sum will guarantee the UV-finiteness much like the nice behavior of the moduli space of Riemann surfaces in string theory.

The structure of matrix operators \(A,B\) representing the external states mimics the template from perturbative string theory. They are adapted to the representation \(R\). It's sort of funny that whenever we write the "trace" in the BFSS-like matrix models, we either mean the trace in the adjoint or the fundamental representation. But in principle, the same adjoint degrees of freedom act on any representation of \(U(N)\). Expressions in the other representations seem completely well-defined but haven't been exploited in the literature yet. Note that traces of products of operators in other representations mix with the polynomials or multi-trace expressions in certain ways.

Feynman's "sum over histories" that looks like the sum over Young diagrams wouldn't be quite new in the literature. In 2003, Iqbal, Nekrasov, Okounkov, and Vafa released their work on the quantum foam and topological strings. Curiously enough, this arXiv paper only appeared in a semiclassical journal, JHEP, in 2008.

They wrote a partition sum as a sum over 3-dimensional generalizations of the Young diagrams. Some "typical" shapes of these diagrams could be seen at high "temperature", so to say, when a big portion of the corner of the "infinite octant" is melted away, resemble the perturbative calculations in topological string theory. But they had the full alternative, "non-perturbative" formula for the partition sum.

Moreover, each 3-dimensional Young diagram could have been thought of as defining a 3-complex-dimensional geometry – a configuration of the topological string theory that works well in 3 complex dimensions.

All of this is cool and for quite some time, I have believed that there should be an analogous identity or duality but with the true, 2-dimensional Young diagrams. One nice aspect of this reduced dimension, 2 complex dimensions, is that it seems to be right for the "extended world sheets" or the 2+2-dimensional "membrane" theories that may be the mother of the world sheet according to some logic based on either "double field theory" or the \(\NNN=2\) or \(\NNN=(2,1)\) strings of Kutasov and Martinec. (And there could exist links to the del Pezzo surfaces in the mysterious duality.)

Another nice feature has been mentioned above: the "true" Young diagrams may be linked to representations of \(U(N)\) and therefore admit "alternative" formulae in terms of integrals over the \(U(N)\) group manifold and/or its conjugacy classes.

I got several steps further but I won't reveal these things as long as the "theory of everything" remains incomplete enough for me to have doubts. However, what I can reveal is something I encountered today. I finally ran into a relatively unknown 2008 paper by Iizuka, Okuda, and Polchinski – which I couldn't have read any carefully when it first appeared. They studied some matrix models, e.g. matrix extensions of the harmonic oscillator, that were supposed to be nice toy models of the black hole information mysteries. And they did manage to rewrite the Feynman path integral of one of their models in terms of the sum over Young diagrams.

(Like many other physicists, they call it "Young tableau" in the singular or "Young tableaux" in the plural because people think that they look more intelligent when they use the French words and when they drink the French champagne instead of the 10 times cheaper and physically indistinguishable Bohemia Sekt. However, this jargon is actually mathematically incorrect because the "Young tableaux" have extra numerical labels in the boxes and label specific basis vectors in the representation. The label-free boxed pictures labeling the whole representation are called the Young diagrams LOL!)

There are quite some cool formulae in Section 5 of that paper (pages 19-27). They actually demonstrate that a Green's function of a simple matrix model may be rewritten as a sum over the Young diagrams. And on the page 20, you even encounter the Frobenius relation involving the characters and their orthogonality and other wonderful things.

This paper is a relatively unknown one (relatively to the fame of at least one co-author) and the authors themselves used this mathematics just to move with a seemingly technical aspect of a toy model. But I do think that there exists an extension of this calculus where the sum over the Young diagrams etc. plays the analogous role to the integral over the Riemann surfaces (shapes of world sheets) and – with some better, more structure summands – allows one to covariantly calculate scattering amplitudes in M-theory in \(d=11\), and perhaps, with some general "action" constrained by a counterpart of the world sheet conformal symmetry, the scattering amplitudes in all compactifications of string/M-theory.

You're invited to compete with me and complete this "theory of everything". What's the form of the "action" for the matrix degrees of freedom that is relevant for M-theory? Are there replacements of the \(bc\)-ghosts or are they absent due to the discrete, sum-like character of the Feynman sum over histories? What are the simplest "MCFTs", the "MOPEs" in them, and how does the MSOC work?

How does one prove that the amplitudes (Feynman diagrams) correctly factorize? The decompositions of tensor products of representations should be useful for that, much like the representation-theoretical formula (5.8) in Polchinski et al. (this is the gluing recipe for two histories). Irreducible representations correspond to "connected world sheets"; reducible ones "are" the multi-component world sheets. What is the M-theory counterpart of the modular invariance of the torus (and higher-genus surfaces), if anything?

When you write the covariant definition of M-theory in \(d=11\), can you actually find some of the other poles of the scattering amplitudes? It seems that in \(d=11\), all the other singularities in the S-matrix should correspond to black hole microstates and all of them should be unstable (at least a little bit).

Can you compactify M-theory on a circle and rederive string theory with its conformal symmetry on the world sheet? Can you compactify on \(T^k\) and rederive the exceptional U-duality groups? OK, and after a few steps, can you find the appropriate vacuum for the Universe around us, too? :-)

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