Popular Science Blog / Blog de Divulgação Científica

Collection of short blog-like popular science articles, written by members of the research group and focusing on themes related to their own research. Either in english or in portuguese. Colecção de artigos de divulgação científica, curtos e em forma de blog, escritos pelos membros do grupo de investigação e em temas relacionados com a sua própria investigação. Textos em português ou em inglês.

BPS Spectra in Quantum Field Theory and Representation Theory

Quantum field theory has a deep connection with representation theory and the underlying algebraic structures. For example, the very definition of a quantum field theory which admits a weakly coupled lagrangian description is based upon Lie algebras and Lie groups. The lagrangian functional, which describes the interactions between light degrees of freedom at a certain energy scale, is mostly determined by a choice of Lie algebras (the "symmetries" of the theory) and their representations (the "fields" of the theory). Even when the theory does not admit a weakly coupled lagrangian functional, such a choice greatly constrain the spectrum of allowed states. Indeed the pattern of hadronic resonances experimentally observed during the '60s, provided strong motivations to consider Yang-Mills theories based on special unitary structure groups. Another example is given by two dimensional conformal field theories, whose states are often organized according to the representation theory of infinite dimensional algebras.

In certain cases representation theory is a tool powerful enough to obtain exact informations about the exact spectrum of a theory, beyond weak coupling. This can happen when a quantum field theory has enough symmetries to cancel most of the quantum corrections to the parameters which describe the particles in the spectrum (that is, their masses and their charges). An example are four-dimensional supersymmetric quantum field theories with eight supercharges. In this case, and under certain conditions, the spectrum of so-called stable BPS states is captured by the representation theory of an algebraic object called quiver.

In supersymmetric theories, physical states are organized according to the representations of a certain Clifford algebra. Those states on which a certain number of generators act trivially, i.e. on which the Clifford algebra is degenerate, are called BPS. BPS representations are generically "rigid", that is invariant under deformations of the physical parameters. In this sense representation theory protects these states from quantum corrections.

To solve for the spectrum of BPS states is equivalent to provide an answer to the question: which BPS particles with fixed electromagnetic charge exist as a physical state for a certain value of the physical parameters? The question is very similar to the study of hadronic resonances in the theory of strong interactions, but thanks to supersymmetry can be solved exactly in terms of auxiliary algebraic structures. The question if a particle corresponds to a stable state or not, can be addressed by studying its effective dynamics. This is relatively standard in quantum field theory, and it involves an auxiliary quantum mechanics model which is defined on the particle world-line, as the particle propagates in time. In general this problem is very hard. However when the quantum field theory has extended supersymmetry, it often happens that this quantum mechanics can be more easily described in terms of a quiver. In this case we say that the quantum field theory has a BPS quiver.

A quiver is a directed graph, given in terms of nodes and arrows between the nodes. More precisely a quiver Q is given in terms of a quadruple (Q₀ , Q₁ , s , t), where Q₀ is the set of vertices, Q₁ the set of arrows and s , t : Q₁ → Q₀ linear maps which identify the starting and terminating node for each arrow. The path algebra of the quiver CQ is simply the algebra generated by the set of all paths, where the product of two paths is defined as their concatenation, or zero if concatenation is impossible. If the quiver has cycles, that is paths which start and end at the same node, one can introduce a superpotential W defined as a formal linear combination of cycles. Then we can define a path algebra with relations CQ / ∂ W by taking the quotient by the ideal ∂ W obtained by taking cyclic derivatives of the superpotential W by all the arrows. In physics the conditions ∂ W = 0 are also known as F-term relations.

A representation of a quiver (or a module over the path algebra) is obtained by the assignment of a vector space V_i at each node i ∈ Q₀, and of linear maps X_a : V_s(a) → V_t(a) to each arrow a ∈ Q₁. If the quiver has a superpotential W, then the maps X_a of the representation are required to obey the same equations as the arrows, obtained from ∂ W = 0. From the point of view of the quantum mechanics model which describe a BPS particle, the nodes of the quiver represent elementary constituents (whose multiplicities are given by the dimensions dim V_i) while the maps between the nodes represent the interactions which can bound them together. Therefore physical BPS states are described by representations of quivers with potentials (plus an additional mathematical condition, called stability, which coincide with the physical notion of stability and makes sure that the energy of the constituents is not too big).

In this way a difficult physical problem is reduced, thanks to supersymmetry, to a purely algebraic question in representation theory.

Michele Cirafici (Email Michele any questions; possibly to be answered in future posts)

Descobrindo Dimensões Diversas

Why Do Strings Like 10 Dimensions?

The very basic ideas of string theory are not so difficult to convey in non technical terms to a general audience. Even the fact that a higher dimensional space-time is compatible with our everyday experience, through the compactification of extra dimensions, can be explained in fairly simple terms.

In fact, much of the recent extremely rich interactions of string theory with geometry arise precisely from the exploration of the possible geometries of these extra dimensions and of the behaviour of strings propagating in those geometries.

However, the precise value D of the dimensionality of space-time that is necessary for an internally consistent string theory is impossible to explain without detailed complex calculations. As is well known, D=10 for superstrings and D=26 for the bosonic string. It is fascinating how this value, which is at the origin of the already mentioned interesting links to geometry in the target space, is deeply rooted in the fact that one-dimensional strings, as they evolve in time, sweep out two-dimensional worldsheets. Note that an important part of the earlier work on the interaction between strings and geometry is related to the worldsheet geometry. In this context, strings appear related to the algebraic and complex geometry of Riemann surfaces in a very non-trivial way.

Let us concentrate on bosonic strings (in flat space) for simplicity.

We would like to give an idea of how the number D=26 is deeply related to the geometry of the moduli space of Riemann surfaces. This is work that appeared in the mid 1980's by Polyakov, Belavin, Knizhnik, Beilinson, Manin and others. Even today, it is worth blogging about it because it is a beautiful connection between strings and geometry that is, perhaps, not so well known.

The fact that D=26 has several well known physical interpretations, which vary according to the quantization method one uses. In light-cone gauge quantization, the classical Lorentz algebra, describing the Lorentz structure in the target space-time, develops anomalies from operator ordering ambiguities. These anomalies vanish only if D=26. On the other hand, in covariant path integral quantization, using the familiar Fadeev-Popov gauge fixing procedure, similar operator ordering ambiguities give rise to an anomaly in the equation for the desired nilpotency of the BRST operator. Again, this anomaly vanishes only if D=26.

There is, however, a more geometric way of understanding the number 26 and this has to do with deep facts about the moduli space of Riemann surfaces of genus g, M_g. (This is a singular space and a rigorous discussion of many aspects of its geometry requires very delicate mathematical care. In the following we will of course ignore such technical points.)

Take g≥2, for simplicity. Morally, over M_g one can consider a tautological family of Riemann surfaces. That is, over each point p in M_g we consider a genus g Riemann surface X_p that it represents. This gives origin to particular line bundles over M_g, roughly as follows. Consider the space of holomorphic differentials of degree n on X_p, H⁰(X_p,Ωⁿ), and denote its dimension by k(n). One can define a line bundle λ_n over M_g by taking as fiber over p the k(n)th antisymmetric tensor product of H⁰(X_p,Ωⁿ).

For example, recall that the space of holomorphic 1-forms on X_p, H⁰(X_p,Ω), has dimension g. Let ω₁,...,ω_g be a basis of holomorphic 1-forms on X_p. Then, the fiber of λ₁ over p is generated by ω₁∧ω₂∧...∧ω_g. On the other hand, for n=2, the space of quadratic differentials on X_p can be identified with the cotangent space of M_g at p, so that λ₂ is isomorphic to the canonical bundle of M_g.

An important result of Mumford gives canonical isomorphisms λ_n≅λ₁^6n2-6n+1. In particular, the line bundle λ₂⊗λ₁ ^-13 has a trivializing holomorphic section, which is unique up to constant since M_g can be compactified in an appropriate way.

The reader has certainly noticed that 13 is half of 26!

The line bundle λ₁ has a canonical Hermitean structure, defined by integration of forms on X_p for each p, and it can be transported to λ_n by the Mumford isomorphism. Note that the resulting (Mumford) Hermitean structures on the λ_n's do not depend on the choice of a metric on X_p.

On the other hand, the bundles λ_n can be equipped with other natural Hermitean structures, from a construction due to Quillen, which arise by viewing them as determinant line bundles. These Hermitean structures depend on a choice of compatible metric on X_p and are computed in terms of zeta function regularized determinants of Laplacians on X_p. These regularized determinants are exactly the determinants that appear in the string path integral after the Gaussian integration of the scalar and ghost fields! The determinant from integrating the scalar fields is related to the Quillen metric on λ₁^-D/2 while the integration of the ghosts relates to the Quillen metric on λ₂.

It turns out that the Mumford isomorphisms are isometries for the Quillen metrics: in particular, only for D=26, does the determinant coming from the scalar fields appear with the correct power (-13) to be paired up with the determinant coming from the ghosts, so that the string path integrand is independent of the choices of metric on each X_p and the conformal anomaly vanishes. Only in this case, is the integrand a well defined function on M_g.

Note that for a different value of D, the integrand would be described in terms of the Quillen norm of a section of a non-trivial bundle on M_g and it would depend on the choice of metrics on the X_p's producing a non-vanishing conformal anomaly.

After discarding, as usual, the integration over the gauge group, the string path integral for D=26 can then be written solely in terms of holomorphic data and without any metric ingredients, making conformal invariance explicit. The Polyakov string measure is actually a top form on M_g which is the Mumford norm on λ₁^-13 of the trivializing holomorphic section of λ₂⊗λ₁^-13.

It is remarkable that the same number 26 can be obtained both from intricate quantum field theoretic calculations related to operator ordering ambiguities, and from delicate pure algebro-geometric calculations which deal only with the holomorphic geometry of Riemann surfaces! This is yet another magical feature of string theory!

(Some) Mathematical Problems of Quantization

If the histories of Mathematics and Physics have been interwoven from very early on, this is even more so after the advent of Quantum Mechanics. There is hardly any part of Mathematics that has not been enriched by ideas or problems of quantum-mechanical origin, and this takes place in Algebra, Analysis, Geometry and Topology. And yet, the question of what is quantization in mathematical terms remains very much open:

Given a "classical system" what does it mean (mathematically) to "quantize it"?

From a mathematical point of view, such a classical system can be described by a phase space represented by a symplectic manifold, that is a manifold of dimension 2n equipped with a symplectic form ω. (A symplectic form is a differential 2-form obeying some simple properties.) Locally, all such manifolds are modeled on R²ⁿ, with n position coordinates (q's) and n momenta coordinates (p's). Associated to ω one obtains a Poisson bracket on the space of smooth functions on M.

According to the general ideology of quantum mechanics, quantizing such a phase space should correspond to a promotion of (say, smooth) functions on M to operators on some Hilbert space H, such that the Poisson bracket of functions on M corresponds to the commutator of the associated operators, but with a constant of proportionality that is the universal Planck's constant. Regretably, it is well known that such procedure is impossible. Firstly, if one considers operators acting on smooth functions on M, there are well-known ordering problems; these can be solved for functions quadratic in p's and q's; however, the notion of quadratic function is not even invariant under canonical transformations. Also, as is well known from the study of the Heisenberg group (irreducible) representations in quantum mechanics, the quantum Hilbert space H should consist, instead, of functions "only of q's or only of p's", which is again a coordinate dependent characterization.

Geometric quantization is a mathematical framework that attempts to partially answer, or at least formulate, these questions. To give an example, geometric quantization gives the correct quantization of the harmonic oscillator. (There are other mathematically very rich approaches to quantization, for instance deformation quantization.)

In order to select "half of the variables" on which the elements of H can depend, one chooses a polarization P of M. This is a choice of Lagrangian distribution on the complexified tanget bundle of M, obeying some natural properties. In practice, locally, one is selecting an appropriate set of n "q's" and/or "p's" on which the elements of H can depend. Often, M is a complex manifold and local holomorphic coordinates "z's" are chosen; in this case one has a "holomorphic polarization". On the other hand, it is very interesting to consider "real polarizations" that really correspond to Lagrangian foliations of M.

Given some restrictions on ω, one can find an Hermitian line bundle L→M with compatible connection of curvature iω. The elements of H are then taken to be the sections of L which are covariantly constant along P. When working with holomorphic polarizations one looks for holomorphic sections of L, which are familiar objects from algebraic and complex geometry. Quantization in a real polarization, on the other hand, has quite a different flavour: if M is compact, covariantly constant sections are supported on a discrete set of leaves of the foliation defined by P (Bohr-Sommerfeld leaves). Here, the description becomes necessarily more analytical. Note that, when M is compact, geometric quantization produces a finite-dimensional quantum Hilbert space H.

A reflection of the fundamental problems that were mentioned above is then the dependence of quantization on the choice of P. Ideally, one would like any two choices of polarization, P and P', to yield quantum Hilbert spaces, H and H', which should be unitarily isomorphic in a natural way. This question turns out to be related to very nice mathematics lying on the intersection of algebraic and complex geometry, symplectic geometry, analysis and group theory.

An important class of examples is when M has a Kähler structure, that is a complex structure compatible with ω. One can then study how the quantization depends on the choice of complex structure. Typically, once ω is fixed, M can be equipped with many different compatible complex structures. (For example, if one builds a torus by glueing sides of a rectangle on the complex plane — by identifying z→z+1 and z→z+t, where Im t>0 — then such a torus is naturally equipped with local complex coordinates, z and translations of z. The resulting complex structure is characterized by t, with t,t' yielding isomorphic complex structures iff the are related by a fractional linear transformation in PSL(2,Z). These complex structures are all compatible with the usual phase space structure of R².)

There is no general theorem giving a natural unitary relation between quantizations of M associated to different choices of polarization; however, in some cases and for some families of polarizations one can at least show that dim H is independent of the polarization.

There are a few examples of manifolds where these issues can be studied quite explicitly and answered positively: abelian varieties, cotangent bundles of Lie groups and toric varieties. In all of these cases, some particularly relevant set of real polarizations can be interpreted as lying on the boundary of the space of compatible complex structures. (In the example of the torus above, take t to the boundary of the upper half-plane.) One can then follow the complex structure dependence of the states in H. In the two compact families (abelian and toric varieties), as the complex structure approaches the boundary, the holomorphic sections are seen to degenerate into distributional states suported along Bohr-Sommerfeld leaves, with unitarity being preserved. Surprisingly, a special kind of "piecewise linear" geometry called tropical geometry, seems to emerge as the complex structure degenerates.

It would be desirable to extend these results to more general classes of symplectic manifolds. An example that is particularly relevant is the moduli space of flat connections on a Riemann surface. This is the phase space of 3d Chern-Simons theory that is very much related to knot invariants and 3-manifold topological invariants. (See the previous entry in the blog, "What's Happening in Knot Theory", by Roger Picken.)

Surprisingly, geometric quantization also makes an appearence in topological string theory (see the blog entry by Gabriel Lopes Cardoso, "Black Holes and String Theory"). The same partition function that relates to black-hole microstate counting is also a vector in a quantum Hilbert space H. Its dependence on the complex structure of the string target space has received a lot of attention in recent studies.

One hundred years after its discovery, quantum-mechanics will thus continue to be a source of beautiful mathematics and of strong interdisciplinary interaction.

What's Happening in Knot Theory

Take a piece of rope, tie it up any way you like, and fuse the two ends together, to make a closed loop. Can we classify all the possibilities, i.e. given two such "knots", can we tell whether they are the same (up to rearranging the rope without cutting it) or different? This incredibly simple question, at the basis of knot theory, has led to a huge number of mathematical developments and all sorts of fascinating connections with various branches of physics.

One of the great things about knot theory is that there is an easy-to-state fundamental theorem, due to Reidemeister, telling you exactly how 2D drawings of knots, so-called knot diagrams, are related for equivalent knots. Thus if you can find some number that is invariant under three simple diagram changes — the famous Reidemeister moves — you can tell two knots apart straight away, if this invariant takes on a different value for each of them. Lots of results can be obtained by hand this way, just by colouring knot diagrams according to some simple rules and counting the possibilities. Curiously, algebraic relations corresponding to the third, most subtle, Reidemeister move crop up all over the place, for instance in the Yang-Baxter equation, which guarantees the solvability of 2D statistical mechanics systems.

Classifying knots is intimately related to understanding the shape of the complementary space of the knot in 3D space, i.e. the space with a hole shaped like the knot dug out. You can go a long way towards distinguishing different knot complements by working out their fundamental group (a group made up of closed loops sitting inside the complement). However the same group can be presented in a variety of ways, so two presentations may appear to be different but actually describe the same group. Way back in 1923, Alexander found a way to express key information about the fundamental group of the complement as something unmistakeable: a polynomial in one variable with integer coefficients. For years this Alexander polynomial was the main tool for knot theorists. In the 1980's knot theory really got going again with the discovery of a new knot polynomial, the Jones polynomial, that was much more powerful than the Alexander polynomial at distinguishing knots. Also Kauffman found an efficient approach to calculating this polynomial, through relations between the polynomials of three almost identical knots — this method of "skein relations" had been pioneered by Conway as an approach to the Alexander polynomial.

Then in 1989, in one of the most influential papers of this generation, Witten found an entirely new approach to the Jones polynomial, via quantum field theory. The idea had come about in conversations with Atiyah and Segal about associating knots to Wilson loop observables in the Chern-Simons model in 3 dimensions (Atiyah often uses this story to encourage people to talk to each other at conferences!). It is remarkable that this viewpoint also turned out to encompass so-called Vassiliev invariants of knots (this is a very large class of knot invariants, that can be expressed in terms of a finite but arbitrarily large set of "chord diagrams"). One way of seeing how quantum field theory enters the picture is due to Kontsevich and Bar-Natan, who established relations between Feynman diagrams in the Chern-Simons perturbation theory and the chord diagrams of the Vassiliev theory. Chern-Simons theory also provides a link between knots and (open topological) string theory, as shown e.g. by Mariño. Incidentally, the Witten breakthrough of 1989 is being commemorated at a conference starting next week in Bonn, with an outstanding line-up of speakers.

The latest exciting development in knot theory is the notion of knot homology, due to Khovanov. Here the Jones polynomial is revealed as "merely" the Euler characteristic of a far more subtle homology theory based on the diagrams used by Kauffman to calculate the Jones polynomial of a given knot. The "categorification" philosophy underlying this construction looks likely to be influential in further developments beyond knot theory, both in mathematical areas like representation theory, and in physics. Many other interesting areas of investigation related to knots are emerging, like so-called virtual knots and higher-dimensional knots (the next step up is 2D surfaces in 4D space). Watch this space for more news about knots and their relatives.

Black Holes and String Theory

Recent astrophysical measurements have revealed the existence of a supermassive black hole in the galactic center of our galaxy, the Milky Way. These measurements also show that black holes are ubiquitous in our Universe. Thus, black holes do not just represent exotic solutions to Einstein's theory of General Relativity: they really exist in Nature.

A black hole possesses a surface, called the event horizon, that separates the interior of the black hole from the outside region. The area of this surface determines the thermodynamic Bekenstein-Hawking entropy of the black hole. This entropy should, according to Boltzmann, have a statistical interpretation in terms of microstates of the system, i.e. of the black hole. Thus, any candidate theory for a consistent theory of quantum gravity has to be able to identify these microstates, and their subsequent counting has to reproduce the thermodynamic entropy of the black hole.

String theory is a leading contender for a consistent theory of quantum gravity. With the pioneering work of Strominger and Vafa it has become evident that it is possible to identify and to count black hole microstates in the context of string theory. At the current juncture, the black holes that are best understood are those that arise as solutions to theories of gravity with a certain amount of supersymmetry. Supersymmetry is a symmetry that relates bosons and fermions, and the amount of supersymmetry is counted by the number of generators of this symmetry. These generators are called supercharges, and the black holes arising in string theory that are best understood are those with eight supercharges. Black holes of this type are supported by scalar fields, and these exhibit an interesting flow mechanism termed the attractor mechanism. This mechanism states that as one moves towards the event horizon of the black hole, the scalar fields flow to specific values at the event horizon, thereby loosing all memory of their initial values far away from the horizon.

The attractor mechanism is at the heart of the recent progress in string theory in reproducing the thermodynamic black hole entropy by microstate counting. The latter exhibits fascinating connections with topological string theory and with the theory of automorphic forms. Topological string theory is a simplified topological version of full-fledged string theory that appears to capture the microstates of supersymmetric black holes. Automorphic forms are an extension to several complex variables of the concept of analytic functions on the upper half-plane satisfying a certain functional equation.

In addition, the microstates should also be captured by the enigmatic AdS₂/CFT₁ correspondence. The AdS/CFT correspondence, discovered by Maldacena, describes a deep connection between gravitational theories in d-dimensional spacetimes and field theories in one dimension lower. It states that quantum gravity (closed string theory) in anti-de Sitter spacetimes has an equivalent (dual) description in terms of conformal field theories (open string theory) in one dimension lower, living at the boundary of anti-de Sitter spacetime. The AdS₂/CFT₁ correspondence is a poorly understood example of this duality.

Despite much work in this area, our understanding of black holes with eight supercharges is far from being complete, and it is likely that many more surprises will emerge. Ultimately the goal is to obtain an understanding of realistic black holes, i.e. of black holes with no supersymmetry at all.