Heterosis or the hybrid vigor or outbreeding enhancement is the lucky event (and an important component of Darwin's evolution) in which the offspring has qualitites that surpass both parents, usually because it inherits the good characteristics from both.
The parents are on both sides.
If you search for "heterosis" or "hybrid vigor" via Google Images, you get lots of pictures of corn, puppies, cows, fictitious animal species, and Barack Obama, among other things.
In 1985, four Princeton physicists ignited the second part of the first superstring revolution (that began in 1984) when they discovered the cleverly named heterotic string in their two papers. These men, Gross+Harvey+Martinec+Rohm, are sometimes referred to as the Princeton String Quartet. You won't find any concert of theirs on YouTube but there are lots of pieces by the Brentano String Quartet playing at Princeton.
The heterotic strings represent two maximally decompactified (trying to have as few compactified dimensions as possible, in some sense zero) limits of superstring/M-theory among the six. The six limits are:
So the heterotic strings are important, perhaps covering 1/3 of the approaches to the conventional configuration space of string/M-theory. Moreover, some people including myself believe that the \(E_8\times E_8\) heterotic strings remain the most convincing and well-motivated incarnation of the real world and all the qualitative features we know about it within string theory.
What does the heterotic string have to do with heterosis?
What they have in common is that they are hybrids of two very different parents. Their father is the bosonic string theory that requires \(D=26\) spacetime dimensions; their mother is the \(D=10\) superstring. I am not sure whether I attributed the sex to the parents correctly. On one hand, SUSY is a female name and the supersymmetric side is prettier while the bosonic side is more unconstrained, a little bit like males; on the other hand, it's the superstring that contains both bosons and fermions and if you interpret bosons and fermions as the X and Y chromosomes, the side that has both of them (XY) should be male! ;-)
More seriously, how can you hybridize these two very different theories that don't even agree about the spacetime dimension? Very well, thank you for asking.
The fields on the heterotic string
Calculations in perturbative string theory are naturally not performed in the spacetime. They may be directly performed on the world sheet – the 2-dimensional surface or history that the 1-dimensional strings paint in the spacetime as they evolve in the 1-dimensional time. All the calculable quantities may be expressed from correlation functions in the world sheet theory – which is naturally a two-dimensional conformal field theory (conformal means that by rescaling all distances by a factor, even a factor that may depend on the location on the world sheet, doesn't have any physical impact; only the angles matter).
What does the world sheet theory look like? It remembers how the world sheet is embedded in the spacetime. Start with the bosonic string theory which is easier.
The coordinates of the world sheet may be denoted \((\sigma,\tau)\); the metric on this locally Minkowski space has signature \(({+}{-})\). We may also Wick rotate \(\tau\to i\tilde \tau\) which makes the world sheet Euclidean. Even more than in the spacetime, this trick makes many calculations much more well-defined. Such a Euclidean world sheet is very naturally described in terms of a complex coordinate \(z\) and its complex conjugate \(\overline z\). In fact, very many things are either holomorphic (or antiholomorphic) or they almost hermetically (not heterotically) segregate the dependence on \(z\) and \(\overline z\).
Bosonic theory has fields \(X^\mu(z,\overline z)\) where the index \(\mu=0,1,\dots ,25\) labels the directions in the 26-dimensional spacetime. At the beginning, the world sheet has a 2-dimensional coordinate reparameterization symmetry so two spacetime coordinates may be pretty much set to some standardized functions of \(z\) and \(\overline z\); something like that is done in the light-cone gauge, for example.
Alternatively, we may keep all the \(26\) fields \(X^\mu\) but we must also add Faddeev-Popov \(bc\) ghosts to deal with the diffeomorphism and Weyl symmetry on the world sheet – much like we deal with the analogous gauge symmetry in Yang-Mills theories. These \(bc\) ghosts add \(c=-26\) to the "central charge", a quantum one-loop violation of the scaling symmetry of the theory we demand, and that's why we need to add \(26\) bosons with \(c=+1\) each to cancel the central charge and keep the theory scale-invariant and conformal at the quantum level. (There are many other, seemingly very inequivalent ways to derive the critical dimension. One of them, based on the light-cone gauge, produces \(D=2-2/(1+2+3+\dots)\)) which also gives the right value if you recall that the only meaningful finite number that the sum of integers may be equal to is \(-1/12\).
So the bosonic string theory observables are calculated from some theory in 2 dimensions which contains \(26\) Klein-Gordon fields \(X^\mu\) and some extra fermionic fields \(b,c\) (which are Dirac-like but we assign them a different spin than \(1/2\); the spin in two dimensions is a bit more flexible and convention-dependent than in higher dimensions because the minimal multiplets are one-dimensional for any spin). Pretty simple. However, to calculate the scattering amplitudes of string states, you have to learn about all (even "composite") local operators in this 2-dimensional quantum field theory and be able to make the shape of the world sheet arbitrary and integrate over all shapes.
The case of the \(D=10\) superstring – which gives us type I, type IIA, as well as type IIB string theory (those only differ by allowed boundary conditions and relative chiralities etc.) – is analogous. However, the ghosts are not just the fermions \(b,c\) for the diffeomorphism symmetry but also \(\beta,\gamma\) for the local world sheet supersymmetry; and, which is related, there are not just bosonic fields \(X^\mu\) but also their fermionic superpartners \(\psi^\mu\). The central charge from the \(bc\)-system is still \(c=-26\). However, the bosonic \(\beta\gamma\)-system adds \(c=+11\); note that all these central charges have the form \(\mp(1-3k^2)\) where \(\mp\) is the upper or lower sign for the bosonic or fermionic ghosts, respectively, and \(k=1-2J\) where \(J\) is the weight (dimension; the generalized number of lower indices) of a ghost (or the antighost). In total, the ghosts have \(c=-15\) which may be cancelled by \(c=+10\) from the fields \(X^\mu\) and \(c=10/2\) from their fermionic partners \(\psi^\mu\).
Segregating left-movers and right-movers
We could study the bosonic string theory as a single theory and we could also study the superstring. In the latter case, we would be allowed to make several choices for the signs of the needed GSO projections; and allow or forbid unorientable and open strings (closed strings must always be included in a string theory and by default, they're orientable). In this way, we would get type I, type IIA, and type IIB string theories which could be compactified to get realistic theories in less than \(D=10\) – this collection of steps and choices already exhausts all the basic possibilities in string theory.
However, we may also – perhaps shockingly – do something seemingly perverse but ultimately equally consistent. To construct the heterotic string.
To fully appreciate that this is actually an extremely natural and allowed procedure on the two-dimensional world sheet, you have to see how separated the left-moving and right-moving excitations on the world sheet are. For example, the mass Klein-Gordon fields in two dimensions obey the massless Klein-Gordon equation,\[
0 = \square X^\mu = (\partial_\tau^2 - \partial_\sigma^2) X^\mu = (\partial_\tau+\partial_\sigma)(\partial_\tau-\partial_\sigma)X^\mu
\] The box operator may be factorized to a product of two operators \(\partial_\pm\)! Consequently, the solutions to this equation are the configurations annihilated either by \(\partial_+\) or \(\partial_-\). In other words, they're functions of \(\tau-\sigma\) or \(\tau+\sigma\), respectively (or some linear superpositions of both). These two terms contributing to the general solution are called the right-moving and left-moving modes, respectively. When we switch to the Euclidean world sheet, "right-moving" and "left-moving" are translated to "holomorphic" functions of \(z\) and the "antiholomorphic" functions of \(\overline z\).
The general solution for \(X^\mu(z,\overline z)\) may be written as a sum of terms that only depend on the former; and terms that only depend on the latter. No mixed dependence. Similarly, fermions may be completely separated so that one component may be fully required to be holomorphic or right-moving; the other component of the 2-component spinor may have a Dirac equation that says that it is a left-moving mode i.e. one that only depends on \(\tau+\sigma\) or \(\overline z\). (Sorry if my convention differs from someone else's; you have to be careful when you fully verify or study someone's papers and books.)
You may find it hard to write the world sheet action \(S\) for the left-movers only (or the right movers only) and it may be hard, indeed. But the action isn't the final product we're after. We need the correlation functions of the operators and they can be calculated in the segregated way.
The hybrid, heterotic theory effectively uses \(26\) bosonic fields \(X^\mu(z)\), along with \(b,c(z)\), and the \(10+10\) bosonic and fermionic fields \(X^\mu(\overline z),\psi^\mu(\overline z)\), along with \(b,c,\beta,\gamma(\overline z)\). These fields with pretty much the same dynamics that can be determined from the parent theories control all the calculable quantities of the heterotic string.
What about the mismatch?
The first observation is that there's really no problem whatsoever with the separation of fields \(\psi^\mu,b,c,\beta,\gamma\) into the left-movers and right-movers. They came as loose packages of the left-moving and right-moving part even in the non-hybrid superstring (or the bosonic string theory, in the case of \(b,c\)) simply because their field equations are first-order equations. Such field equations effectively say that a component of the field is holomorphic; or another component is antiholomorphic. And we may separate the components.
As you may see, the only potential subtleties of the segregation arise in the case of \(X^\mu(z,\overline z)\) whose field equations are second-order equations (Klein-Gordon). How does the separation work here? The hybrid, heterotic string seems to think that it's embedded in the \(10\)-dimensional spacetime according to the (superstring-like) right-moving excitations propagating on the string; but in the \(26\)-dimensional spacetime according to the (bosonic-string-theory-like) left-moving excitations. (The question which is which depends on a convention, one independent from most of the similar binary conventions. Flipping the conventions leads us to equivalent constructions.)
And yes, there are subtle constraints. There seem to be \(16\) bosonic fields \(X^\mu\) on the bosonic, left-moving side that are completely erased on the superstring, right-moving side. How do such \(16\) spacetime coordinates that half-exist, half-not-exist behave?
An interesting feature of these coordinates is that you may still compute the total momentum \[
P^\mu = \int_0^\pi\dd\sigma\,\partial_\tau X^\mu
\] (note that the \(\tau\)-derivative is the velocity which is proportional to the momentum density with a fixed coefficient) and the total winding, \[
W^\mu = \Delta X^\mu = \int_0^\pi\dd\sigma\,\partial_\sigma X^\mu= X^\mu|^\pi_0.
\] Well, that's true even in theories with both-sided \(X^\mu\) – or in the heterotic string theory for those shared \(10\) coordinates \(X^\mu\) that exist on both sides. However, a special feature of the heterotic string is that\[
(\partial_\tau+\partial_\sigma)X^\mu = 0
\] which is the condition that only the left-moving parts of the sixteen coordinates are allowed. When this equation is integrated from \(0\) to \(\pi\) over \(\sigma\), i.e. over the closed string, we realize that – with some normalization factors you must be careful about but I will simplify them a bit\[
P^\mu = W^\mu.
\] The momentum of the heterotic string in the direction of each of the sixteen "asymmetric" coordinates must be equal to the winding number – how many times the string winds around the given direction. That may seem bizarre but it makes a perfect sense.
You can't really prevent the string from having at least some nonzero values of \(P^\mu\); but in combination with \(P^\mu=W^\mu\), that implies that the windings must be allowed to be nonzero, too. So in some sense, these sixteen "lopsided" coordinates parameterize a \(16\)-dimensional torus.
Even self-dual lattices
Are all tori allowed? The answer is a resounding No. In fact, out of the infinitely many choices, only two completely rigid solutions solve the constraints and produce a consistent string theory (a string theory vacuum, to use the modern terminology in which string theory is already recognized as a unified theory with many solutions).
The torus may be represented as \(\RR^{16}/\Gamma^{(16)}\) where \(\Gamma\) is a symbol for lattices which are something like discrete groups \(\ZZ^{16}\) in this case. However, the sixteen independent generators of \(\ZZ^{16}\) don't have to shift the sixteen "lopsided" directions by the same distance; and as 16-dimensional vectors defining the translations, these generators don't have to be orthogonal each other. A sixteen-dimensional lattice is defined as this kind of \(\ZZ^{16}\) group that may be tilted or stretched or shrunk in various ways.
The quotient means that the coordinates in \(\RR^{16}\) become effectively periodic in some sense – but it's still some general sixteen linear combination of these coordinates that are periodic. The division by the lattice means that we're only interested in the coordinates "modulo integers" so we're interested in their fractional parts, kind of.
Fine, what are the allowed shapes of the lattice \(\Gamma^{(16)}\)? The lucky guys may be derived from the modular invariance of the one-loop toroidal stringy diagrams but this is too technical. We have this cool \(W^\mu=P^\mu\) condition which may do pretty much the same job.
If you study quantum mechanics of particles propagating on a circle of radius \(R\) and circumference \(2\pi R\), you will be able to derive that the momentum \(P\) has to be quantized – a number of the form \(N\hbar/R\). Let's set \(\hbar=1\); I just wanted to remind everyone that all those things may be written in everyday life units. This quantization emerges because the wave function \(\psi(x)\) has to be single-valued on the circle. For example, when you study the orbital angular momentum, it's effectively a particle on a circle of circumference \(2\pi\) (the coordinate \(\phi\)) and the dual "momentum" \(L_z\) has to be an integer because of the single-valuedness of the wave function.
Now, if the particle were replaced by a closed (circular) string, it could also wind around the circle. The total winding would be a multiple of the circumference \(2\pi R\) i.e. \(2\pi R w\). Note that the momentum has units of \(1/R\) which is inverse to the unit of the winding, \(2\pi R\). If you make the circle shorter, the spacing of the winding will shrink but the spacing of the momentum will increase by the same factor.
How is this rule generalized for a general lattice? It's generalized by the statement that the lattice in which the momentum lives is dual to the lattice in which the winding lives. For example, the dual lattice to \(k\ZZ\) is \((1/k)\ZZ\): both are effectively additive groups of integers but the physical sizes of the generators of these groups are inverse to one another. What does it mean to have a dual lattice in the general case?
It's not hard. If you have a lattice \(\Gamma\), the dual lattice \(\Gamma^*\) is composed of all the vectors \(W\in\RR^{16}\) in a "dual vector space" (the space of linear forms) that obey \(W\cdot V\in \ZZ\) for each \(V\in \Gamma\); I wrote \(W\cdot V\) as an inner product, assuming the usual \({\rm diag}({+}{+}\cdots {+})\) signature but I could have been more abstract and write it as \(W(V)\), the action of a linear form on a vector. The factors of \(2\pi\) must be dealt with somewhere but they don't change the qualitative message of all these constructions.
Because the momentum and the winding must belong to lattices that are dual to each other but, at the same moment, the momentum must be equal to the winding, the lattice of the allowed momenta must be equal to the lattice of the allowed windings i.e. to the dual lattice to the lattice of the allowed momenta. If a lattice is equal to its dual, \(\Gamma=\Gamma^*\), we say it is self-dual. And that's a hugely constraining condition, it turns out.
For example, the simple \(\ZZ^{16}\) lattice with the unit and orthogonal generators is self-dual because \(W\cdot V\in\ZZ\) for every two vectors \(V,W\) with sixteen integer-valued coordinates. The self-duality would surely disappear if you tried to deform and stretch the lattice in a generic way.
However, we may actually derive one more (significantly weaker) "even" condition from string theory: \(V^2\) must be not just integer but it must be even: \(V^2\in 2\ZZ\). This condition, arising from the need for \(L_0-\tilde L_0=\dots + V^2/2\) to remain integer-valued for the whole spectrum or, equivalently, from the \(\tau\to\tau+1\) part of the modular invariance, bans the simple \(\ZZ^{16}\) lattice. Are there any even self-dual lattices?
Yes, there are.
Use the symbols \(e_i\) where \(i=1,2,\dots,16\) for the usual orthonormal basis of \(\RR^{16}\). And take the lattice to be composed of all the linear combinations of \(e_i+e_j\) for \(i\neq j\), all the \(e_i-e_j\) for \(i\neq j\), and of \[
W_{\rm halfy} = (\frac 12, \frac 12, \frac 12, \dots , \frac 12)
\] where the same coordinate is repeated sixteen times. It's not hard to see that the inner product of any pair of the basis vectors (over integers) is integer. And because the inner product of every vector in the set above with itself is even – most nontrivially, the squared length of the last vector is \(16/2^2=4\) – the lattice of all the integer combinations of the vectors I just described will be even.
It is also self-dual. Try to find the most general vector \(W\) whose inner product with all the vectors in the "integer-based basis" above is integer-valued. Because it must hold for every \(e_i-e_j\), you may see that \(W_i\) and \(W_j\) must differ by an integer. Because it must hold for \(e_i+e_j\) as well, \(W_i\) and \(-W_j\) must also differ by an integer. It follows that the coordinates \(W_j\) and \(-W_j\) differ by an integer i.e. \(W_j\) itself is an integer multiple of \(1/2\) – it is either integer or integer plus \(1/2\). And I have already justified that if one coordinate is an integer, all the others have to be integers; if one of them differs from an integer by \(1/2\), all of them have to.
In the first (integer) case, you may show that the sum of the sixteen coordinates \(W_j\) is even because the inner product with the vector with sixteen \(1/2\) coordinates still has to be integer. But if the sum of the coordinates is even, it follows that you may express the vector as a combination of the \(e_i\pm e_j\) vectors. Similarly, this holds for \(W-W_{\rm halfy}\) if all the coordinates of \(W\) are half-integral because the inner product of \(W_{\rm halfy}\) with itself is an even integer.
The \(SO(32)\) heterotic string
You have to go through this proof yourself to really understand it – you have to rediscover it – but the lattice I defined as the set of integer combinations of all the vectors is even self-dual. It's what we need for the heterotic string. If you "half-compactify" the sixteen purely left-moving excessive bosonic coordinates on this lattice, you will get a string theory producing an exact \(SO(32)\) gauge symmetry in the spacetime.
The isometry of the torus is just \(U(1)^{16}\) which is the "Cartan subgroup" of \(SO(32)\) and it will be marketed as a part of the gauge symmetry because of the standard Kaluza-Klein mechanism. But there will be new gluon-like massless gauge bosons with a nonzero winding – corresponding to \(4\times 16\times 15/2 = 480\) points in the lattice that obey \(V^2=2\). And they will extend the symmetry group to a nice \(SO(32)\).
More precisely, the symmetry group is \(Spin(32)/\ZZ_2\) because the presence of the vectors such as \(W_{\rm halfy}\) in the lattice means that some states transforming as a spinor under \(Spin(32)\) will appear in the heterotic spectrum, too. Because the signs in \(W_{\rm halfy}\) are sort of prescribed (and an even number of them will flip if we add some \(-e_i-e_j\)), we will only obtain one Weyl (chiral) spinor, not the other. That's the origin of the \(\ZZ_2\) in the quotient. All the spinor states will give you massive states because \(W_{\rm halfy}^2=4\) which is greater than \(2\), the level where new massless states may still occur. The "spintensor" states with more general half-integral coordinates will be even heavier.
The spinorial states of the \(SO(32)\) heterotic string have a nice interpretation in terms of D-branes in the dual type I string theory. The duality will be mentioned later.
The \(E_8\times E_8\) heterotic string
Are there some other even self-dual sixteen-dimensional lattices? There is exactly one. In our construction of the lattice above – which is the "weight lattice of \(Spin(32)/\ZZ_2\)" – we used the vector \(W_{\rm halfy}\) with the coordinates \(1/2\) whose squared length was equal to \(4\). This is not a minimum squared length for an even lattice; the squared length equal to \(2\) would be just fine, too.
So you may also construct a totally analogous lattice \(E_8\) – the "root lattice of \(E_8\)" – to the sixteen-dimensional lattice but in eight, not sixteen dimensions. For the heterotic string, you need to do something with sixteen coordinates. But it's easy to divide them to two groups of eight coordinates and compactify each group on an \(E_8\) lattice.
The proofs that the \(E_8\) lattice is even and self-dual are completely analogous to the proof for the \(Spin(32)/\ZZ_2\) root lattice. But there's one really cool surprise. Because \(W_{\rm halfy}^2=2\) for the \(E_8\) lattice, it's the minimum allowed positive result, we may actually get new massless states (e.g. vector bosons and gauginos) from strings whose momentum and winding have half-integer coordinates, i.e. from the spinorial weights.
So some of the gauge bosons may transform as spinors of \(Spin(16)\). The gauge group in the "smaller" construction could be expected to be \(Spin(16)\), just like it was something like \(Spin(32)\) in the first heterotic string. However, the spinors are actually massless so they must correspond to generators of the gauge group, too. And if you combine the \(120\) generators of \(Spin(16)\) with the \(2^{8}/2=128\) generators that transform as a Weyl (chiral) spinor under \(Spin(16)\), you obtain the \(248\) generators of the group \(E_8\). We don't get just \(Spin(16)\times Spin(16)\) here; the gauge group is larger, \(E_8\times E_8\).
A cute detail – which is seen to be no coincidence if you study anomalies in the heterotic string's spacetime – is that both groups have the same dimension\[
\frac{32\times 31}{2} = 248+248 = 496.
\] In fact, some gravitational anomaly in \(D=10\) which must cancel is proportional to \((n-496)\). It's the \(E_8\times E_8\) string that is much more promising as a starting point to realistic phenomenology. One of the groups \(E_8\) may be broken to a subgroup such as \(E_6,SO(16),SU(5)\) which are viable grand unified groups and, when some extra six dimensions are compactified on a Calabi-Yau-like manifold, you get theories with realistic spectra and interactions (grand unificiation and SUSY is automatically built upon the Standard Model).
I should mention that within the purely Euclidean signature spaces, even self-dual lattices only exist in \(8k\) dimensions. I have discussed the unique eight-dimensional even self-dual lattice, \(E_8\), the lattice producing the equally named Lie group, and the two possible sixteen-dimensional even self-dual lattices. The next dimension where even self-dual lattices exist is \(D=24\). Aside from \(E_8\oplus E_8\oplus E_8\) and \(E_8\oplus \Gamma(Spin(32)/\ZZ_2)\) and an analogous \(Spin(48)/\ZZ_2\), one finds new examples, in particular the cool and less trivial Leech lattice which is crucial for the string-theoretical explanation of the monstrous moonshine (more).
Fermionization
A cool feature of the two-dimensional conformal field theories is that the same theory (physically) may often be expressed in many different ways. Instead of using sixteen left-moving bosons (on the bosonic side), we may "fermionize them". A free real boson in 2D CFTs is equivalent to two free real fermions whose operators may be expressed \[
\psi = \exp(+i\phi/2),\quad \bar\psi = \exp(-i\phi/2)
\] in terms of the boson \(\phi\) or, equivalently, the boson may be written as \(\partial_+\phi = \bar\psi\psi\). It may sound crazy: How could a tensor product of two fermionic Fock spaces look like a single bosonic Fock space? For the states of a single point-like particle species occupying one one-particle state, they're different. But if you include all the factors of the Fock spaces corresponding to all the harmonics along the string and you pick the right boundary conditions, it just works.
So a funny thing is that the same conclusion – there are exactly two possible heterotic string theories in ten dimensions – may be derived from the \(32\) real fermions that may be used instead of the \(16\) bosons above. You must only be careful about their allowed boundary/periodicity conditions around the closed heterotic string; and, which is related, about the GSO-like projections that tame the spectrum a little bit.
I won't discuss the details but the \(Spin(32)/\ZZ_2\) heterotic string may be constructed out of \(32\) real left-moving fermions \(\lambda^a\) that are either simultaneously periodic; or simultaneously antiperiodic (there are two sectors). Because all the fermions are treated in the same way (their friendship and common fate isn't perturbed, not even by the boundary conditions), you get the \(SO(32)\) symmetry: the symmetry currents are simply \(\lambda^a \lambda^b\) which is \(ab\)-antisymmetric. The spinorial states arise from "spin fields" or from the "highly degenerate" sector (because it has zero modes) with periodic, and not the simpler antiperiodic, boundary conditions for \(\lambda^a\). There's one GSO-like projection you have to impose.
Similarly, the \(E_8\times E_8\) heterotic string may be obtained if you split the set of \(32\) fermions \(\lambda^a\) to two groups of sixteen fermions and allow the periodic or antiperiodic boundary conditions for each group separately (there are four sectors, AA, AP, PA, PP). The multiplicity of the sectors is inseparable from two independent GSO-like conditions. Again, you could think that this breaks the group to \(Spin(16)\times Spin(16)\) but you will find the extra spinor states and the gauge group gets enhanced to \(E_8\times E_8\). The resulting theory may be shown to be equivalent – even at the level of string interactions, not just degeneracies in the free spectrum – to the heterotic string theories we obtained via the bosonic construction.
T-duality as a unification of both heterotic string theories
If you pick a direction in the "large" ten-dimensional spacetime and compactify it on a circle as well, the two heterotic string theories become smoothly connected into "one heterotic theory". Why?
The bosonic construction makes it a bit easier to explain. In the construction of the heterotic string theories above, we discussed lattices in a \(16+0\)-dimensional Euclidean space. I added the zero to emphasize that the signature was purely positive, Euclidean, and there were no time-like dimensions.
If you compactify the "non-chiral" boson \(X^{9}\) on a circle, it will have both left-moving and right-moving parts. In the physically most natural inner product, these two parts will behave as coordinates with the opposite signature. In effect, we added \(1+1\) dimensions to the \(16+0\)-dimensional lattice. The result is \(17+1\)-dimensional.
A funny fact is that a \(17+1\)-dimensional even self-dual lattice that we may still require for a consistent compactified heterotic string theory of this sort exists and it is... unique. In fact, the only Minkowskian even self-dual lattices exist in \(p+q\) dimensions where \(p-q\) is a multiple of eight and they're unique whenever \(pq\neq 0\). How can it be unique if we had two solutions to start with? Well, it's unique because\[
\Gamma(Spin(32)/\ZZ_2)\oplus \Gamma^{1,1} = \Gamma(E_8)\oplus \Gamma(E_8)\oplus \Gamma^{1,1}.
\] The lattices that you obtain from the two ten-dimensional heterotic string theories' lattices by adding a simple \(\Gamma^{1,1}\) from the compactified \(X^9\) are the same lattices, just rotated by a "Lorentz transformation" in \(17+1\) dimensions.
All these things may be fully proven but the implications are sort of remarkable. You may start with the \(Spin(32)/\ZZ_2\) heterotic string in ten dimensions. You compactify one more dimension so that only \(D=9\) coordinates remain noncompact. You break the group to some \(U(1)^{18}\) by changing the Wilson lines and other moduli and when you adjust these scalar fields to some right values, the gauge symmetry will suddenly start to get enhanced again. But you will get \(E_8\times E_8\) instead of \(Spin(32)/\ZZ_2\). An unexpected feature of the construction is that you can't view either of the two groups as a "broken phase" of the other. In fact, they're two equally large groups (when it comes to their dimension, \(496\)). They should be treated democratically; string theory allows you to break groups to smaller ones but also enhance groups to larger ones (at special points of the moduli space where some stringy states happen to go massless) and these two processes seem to be equally fundamental.
If string theorists weren't forced to see that such transitions exist, much like limited experimenters are hit by Mother Nature to their faces when She forces them to see something they should have seen for quite some time, they (or philosophers) would probably never "invent them" themselves. Nature and mathematics are smarter than us, even the smartest among us.
Heterotic-K3 duality
If you compactify the heterotic strings at least on a two-torus, you will get the vacua that, at strong coupling (but with the volume of the tori kept at a certain value in certain units), may be equivalently described by a totally non-heterotic string theory compactified on a seemingly highly nontrivial manifold, the four-dimensional K3 surface.
For example, a heterotic string on \(T^3\) deals with \(19+3\)-dimensional lattices, and exactly this \(22\)-dimensional lattice, with the right signature (if extracted from the intersection numbers of pairs of two-cycles), may be identified in the cohomology of the K3 surface. The heterotic string arose from a particular "freedom to hybridize", a loophole in the regulations who can have offspring with whom. But this "freedom to hybridize" is actually the same loophole as the possibility to find one more hyper-Kähler, real-four-dimensional manifold aside from the torus, the seemingly nontrivial and curved K3 surface. They're really "the same thing" visualized with the help of different geometric pictures and different degrees of freedom. But in the heterotic and K3 case, we're just thinking about the physics in two different ways – it's a difference in our impressions or visualization or conventions for symbols, sort of – but the underlying mathematics and physics is completely isomorphic.
String theory is full of such unifications of things that naively look completely different.
Different fates of heterotic string theories at strong coupling
String theory and its vacua are living organisms that always prepare lots of surprises for us. Already in \(D=10\), we may ask what the physical phenomena look like if we send the string coupling constant (or the string dilaton) to infinity?
Before the second superstring revolution in the mid 1990s, people would guess it's some uninteresting mess they don't want to talk about. For both heterotic theories, we get the same kind of mess.
However, it was shown that string theory – perhaps because it's such a perfectionist consistent theory – never leads you to a mess. Any simply describable legitimate limit of the moduli space must be a theory with so many special properties that it looks as natural as the starting point.
Even if I told you before the mid 1990s that the strong coupling limit of a string theory must be equivalent to some other string theory, you would probably make many wrong guesses about the theories you actually get from the two \(D=10\) heterotic string theories above. The \(Spin(32)/\ZZ_2\) and \(E_8\times E_8\) heterotic string theories are qualitatively the same structures, up to a difference in "technical details", so they should give you similar strong coupling limits, up to a difference in some other "technical details".
But this argument or expectation would be wrong, too. The strong coupling limits of both theories look very different from one another.
If you try to turn the string coupling constant \(g_s\) in the \(SO(32)\) heterotic string theory to a value much larger than one, you must still get a ten-dimensional supersymmetric theory whose spacetime gauge group includes \(SO(32)\). The ten large dimensions couldn't disappear. The supersymmetry couldn't disappear. The gauge group couldn't disappear. What the limit could be?
Well, there's one more supersymmetric string theory with the \(SO(32)\) gauge group, namely type I superstring theory; the \(SO(32)\) group arises from 32 possible half-colors of "quarks" at the end points of open strings or, equivalently, from the 16 spacetime-filling D9-branes and their mirror images (behind the orientifold mirror). This theory is not heterotic. It's purely "fermionic"; no hybrids. But its strings are unorientable and may be open or closed. (Heterotic strings must be orientable because the left-moving and right-moving parts of it have "inequivalent guts" so they can't be confused. For a similar reason, heterotic strings can't be open because an end point would have to "reflect" left-moving waves to some right-moving waves but the allowed waves in the two directions are inequivalent and can't be mapped to one another.)
Still, the theories are completely equivalent. Type I theory with the coupling \(g_s\) is equivalent to the heterotic \(SO(32)\) theory with the coupling \(1/g_s\). Incidentally, the spinorial states of the \(SO(32)\) heterotic strings appear as states of a particular non-supersymmetric (non-BPS) D0-brane in type I theory. The number of such D0-branes is conserved just modulo two, i.e. as an element of \(\ZZ_2\). A single D0-brane of this kind is stable because it's the lightest object/state that gets mapped to minus itself under the 360° rotation in \(SO(32)\) gauge group (the lightest spinor-like object in the theory).
The fate of the \(E_8\times E_8\) heterotic string is completely different and it was only understood by Petr Hořava and Edward Witten at the end of 1995, months after lots of similar discoveries. Again, the supersymmetry can't disappear. The ten large dimensions can't disappear. The \(E_8\times E_8\) gauge bosons can't disappear. So what the other description with these properties may be if I can assure you it's not the same description (the theory can't be S-self-dual)?
The answer is that while the ten dimensions can't disappear, a new dimension can appear. The strong coupling limit of the \(E_8\times E_8\) heterotic string is an 11-dimensional theory, M-theory, whose new dimension has the shape of the line interval of length \(L\). The value of \(L\) is an increasing function of the string coupling \(g_s\). But M-theory seems to have no non-Abelian gauge bosons.
The heterotic string itself becomes a cylindrical membrane, M2-brane of M-theory, stretched between the two ends of the world.
Well, it has gauge bosons if the 11-dimensional spacetime has boundaries. In fact, at the boundaries of such a spacetime, the gravitinos must be constrained to chiral fields. That creates 10D anomalies near the boundaries and they have to be cancelled. The gravitational anomalies may be cancelled by some other chiral fermions you may add, the gauginos, but that may create new gauge anomalies and mixed anomalies etc. At the end, it turns out that you may cancel all of these anomalies but only if \(E_8\) gauge bosons (and their superpartners, gauginos) live on each boundary of the 11-dimensional spacetime! The fact that it works at all boils down to some nontrivial properties of \(E_8\), some difficult and seemingly "very lucky" identities relating traces of products of generators of the group in the adjoint representation.
Because a line interval has two endpoints – a thick "desk" has two surfaces on both sides – you will get two \(E_8\) factors of the gauge group for each point in the remaining 10-dimensional "large" spacetime (yes, the gauge bosons and gauginos are confined to the end-of-the-world boundaries, much like some gauge fields are confined to D-branes or singularities): the gauge group will be \(E_8\times E_8\). The 11-dimensional M-theory with two boundaries – the heterotic M-theory – is also a good starting point to build realistic compactifications of string theory.
(Physicists describe the line interval as \(S^1/\ZZ_2\), a quotient of circle by the group mirroring it from the left to the right. This quotient is a line interval because e.g. the left half of the circle is a "fundamental domain" while the right part of the circle is just a \(\ZZ_2\) copy of it. And yes, one-half of a circle is the same thing topologically as a line interval. In my conventions, the end points of the line interval are the fixed point under \(\ZZ_2\), i.e. the uppermost and lowermost points of the circle.)
You may see that the heterotic string shows the remarkable uniqueness and interconnectedness of string/M-theory. There are just two possible heterotic string theories in \(D=10\) linked to two fully non-Abelian groups that may satisfy difficult anomaly cancellation conditions in the \(D=10\) spacetime. These two precious solutions may be derived in several languages that use different mathematical toolkits – bosons and lattices; grouping of fermions etc. – and they're connected after a compactification of another dimension. Further compactifications may be equivalently described in terms of M-theory or type II strings on K3 surfaces and the strong coupling limits are also equivalent to type I theory or M-theory with boundaries.
String/M-theory always knows what it's doing and it won't leave you in trouble. The maximally supersymmetric vacua of string/M-theory are arguably "prettier" but they're too symmetric and too sterile. The heterotic strings have 1/2 of the maximum supersymmetry (inherited from the mother, the right-moving supersymmetric side of the hybrid) but that's more than enough for SUSY to play its maternal protective role (and to add fermions into the spectrum and eliminate tachyons from the spectrum, the crippling vices that the offspring didn't want to inherit from the bosonic dad). The realistic vacua require even less supersymmetry.
As the amount of supersymmetry decreases, we are losing the ability to compute everything interesting easily but we're gaining lots of structures and new twists. The heterotic string vacua have so much supersymmetry that they still allow us to compute the answers to almost all simple (BPS) questions essentially by classical calculations but the freedom is already high enough to allow us to connect both versions of the heterotic string theory with one another and with type I, type IIA/IIB and M-theory on K3, with M-theory spacetimes that possess boundaries, and with other vacua I haven't discussed: already at this level, we may see the (more than minimal) interconnectedness of the diverse web of solutions to string theory.
The parents are on both sides.
If you search for "heterosis" or "hybrid vigor" via Google Images, you get lots of pictures of corn, puppies, cows, fictitious animal species, and Barack Obama, among other things.
In 1985, four Princeton physicists ignited the second part of the first superstring revolution (that began in 1984) when they discovered the cleverly named heterotic string in their two papers. These men, Gross+Harvey+Martinec+Rohm, are sometimes referred to as the Princeton String Quartet. You won't find any concert of theirs on YouTube but there are lots of pieces by the Brentano String Quartet playing at Princeton.
The heterotic strings represent two maximally decompactified (trying to have as few compactified dimensions as possible, in some sense zero) limits of superstring/M-theory among the six. The six limits are:
- M-theory in 11 dimensions (added as a full member in 1995); all the vacua below are string-theoretical vacua in 10 dimensions (and were added in the 1980s)
- type IIA string
- type IIB string
- type I string
- heterotic \(E_8\times E_8\) string
- heterotic \(SO(32)\) string
So the heterotic strings are important, perhaps covering 1/3 of the approaches to the conventional configuration space of string/M-theory. Moreover, some people including myself believe that the \(E_8\times E_8\) heterotic strings remain the most convincing and well-motivated incarnation of the real world and all the qualitative features we know about it within string theory.
What does the heterotic string have to do with heterosis?
What they have in common is that they are hybrids of two very different parents. Their father is the bosonic string theory that requires \(D=26\) spacetime dimensions; their mother is the \(D=10\) superstring. I am not sure whether I attributed the sex to the parents correctly. On one hand, SUSY is a female name and the supersymmetric side is prettier while the bosonic side is more unconstrained, a little bit like males; on the other hand, it's the superstring that contains both bosons and fermions and if you interpret bosons and fermions as the X and Y chromosomes, the side that has both of them (XY) should be male! ;-)
More seriously, how can you hybridize these two very different theories that don't even agree about the spacetime dimension? Very well, thank you for asking.
The fields on the heterotic string
Calculations in perturbative string theory are naturally not performed in the spacetime. They may be directly performed on the world sheet – the 2-dimensional surface or history that the 1-dimensional strings paint in the spacetime as they evolve in the 1-dimensional time. All the calculable quantities may be expressed from correlation functions in the world sheet theory – which is naturally a two-dimensional conformal field theory (conformal means that by rescaling all distances by a factor, even a factor that may depend on the location on the world sheet, doesn't have any physical impact; only the angles matter).
What does the world sheet theory look like? It remembers how the world sheet is embedded in the spacetime. Start with the bosonic string theory which is easier.
The coordinates of the world sheet may be denoted \((\sigma,\tau)\); the metric on this locally Minkowski space has signature \(({+}{-})\). We may also Wick rotate \(\tau\to i\tilde \tau\) which makes the world sheet Euclidean. Even more than in the spacetime, this trick makes many calculations much more well-defined. Such a Euclidean world sheet is very naturally described in terms of a complex coordinate \(z\) and its complex conjugate \(\overline z\). In fact, very many things are either holomorphic (or antiholomorphic) or they almost hermetically (not heterotically) segregate the dependence on \(z\) and \(\overline z\).
Bosonic theory has fields \(X^\mu(z,\overline z)\) where the index \(\mu=0,1,\dots ,25\) labels the directions in the 26-dimensional spacetime. At the beginning, the world sheet has a 2-dimensional coordinate reparameterization symmetry so two spacetime coordinates may be pretty much set to some standardized functions of \(z\) and \(\overline z\); something like that is done in the light-cone gauge, for example.
Alternatively, we may keep all the \(26\) fields \(X^\mu\) but we must also add Faddeev-Popov \(bc\) ghosts to deal with the diffeomorphism and Weyl symmetry on the world sheet – much like we deal with the analogous gauge symmetry in Yang-Mills theories. These \(bc\) ghosts add \(c=-26\) to the "central charge", a quantum one-loop violation of the scaling symmetry of the theory we demand, and that's why we need to add \(26\) bosons with \(c=+1\) each to cancel the central charge and keep the theory scale-invariant and conformal at the quantum level. (There are many other, seemingly very inequivalent ways to derive the critical dimension. One of them, based on the light-cone gauge, produces \(D=2-2/(1+2+3+\dots)\)) which also gives the right value if you recall that the only meaningful finite number that the sum of integers may be equal to is \(-1/12\).
So the bosonic string theory observables are calculated from some theory in 2 dimensions which contains \(26\) Klein-Gordon fields \(X^\mu\) and some extra fermionic fields \(b,c\) (which are Dirac-like but we assign them a different spin than \(1/2\); the spin in two dimensions is a bit more flexible and convention-dependent than in higher dimensions because the minimal multiplets are one-dimensional for any spin). Pretty simple. However, to calculate the scattering amplitudes of string states, you have to learn about all (even "composite") local operators in this 2-dimensional quantum field theory and be able to make the shape of the world sheet arbitrary and integrate over all shapes.
The case of the \(D=10\) superstring – which gives us type I, type IIA, as well as type IIB string theory (those only differ by allowed boundary conditions and relative chiralities etc.) – is analogous. However, the ghosts are not just the fermions \(b,c\) for the diffeomorphism symmetry but also \(\beta,\gamma\) for the local world sheet supersymmetry; and, which is related, there are not just bosonic fields \(X^\mu\) but also their fermionic superpartners \(\psi^\mu\). The central charge from the \(bc\)-system is still \(c=-26\). However, the bosonic \(\beta\gamma\)-system adds \(c=+11\); note that all these central charges have the form \(\mp(1-3k^2)\) where \(\mp\) is the upper or lower sign for the bosonic or fermionic ghosts, respectively, and \(k=1-2J\) where \(J\) is the weight (dimension; the generalized number of lower indices) of a ghost (or the antighost). In total, the ghosts have \(c=-15\) which may be cancelled by \(c=+10\) from the fields \(X^\mu\) and \(c=10/2\) from their fermionic partners \(\psi^\mu\).
Segregating left-movers and right-movers
We could study the bosonic string theory as a single theory and we could also study the superstring. In the latter case, we would be allowed to make several choices for the signs of the needed GSO projections; and allow or forbid unorientable and open strings (closed strings must always be included in a string theory and by default, they're orientable). In this way, we would get type I, type IIA, and type IIB string theories which could be compactified to get realistic theories in less than \(D=10\) – this collection of steps and choices already exhausts all the basic possibilities in string theory.
However, we may also – perhaps shockingly – do something seemingly perverse but ultimately equally consistent. To construct the heterotic string.
To fully appreciate that this is actually an extremely natural and allowed procedure on the two-dimensional world sheet, you have to see how separated the left-moving and right-moving excitations on the world sheet are. For example, the mass Klein-Gordon fields in two dimensions obey the massless Klein-Gordon equation,\[
0 = \square X^\mu = (\partial_\tau^2 - \partial_\sigma^2) X^\mu = (\partial_\tau+\partial_\sigma)(\partial_\tau-\partial_\sigma)X^\mu
\] The box operator may be factorized to a product of two operators \(\partial_\pm\)! Consequently, the solutions to this equation are the configurations annihilated either by \(\partial_+\) or \(\partial_-\). In other words, they're functions of \(\tau-\sigma\) or \(\tau+\sigma\), respectively (or some linear superpositions of both). These two terms contributing to the general solution are called the right-moving and left-moving modes, respectively. When we switch to the Euclidean world sheet, "right-moving" and "left-moving" are translated to "holomorphic" functions of \(z\) and the "antiholomorphic" functions of \(\overline z\).
The general solution for \(X^\mu(z,\overline z)\) may be written as a sum of terms that only depend on the former; and terms that only depend on the latter. No mixed dependence. Similarly, fermions may be completely separated so that one component may be fully required to be holomorphic or right-moving; the other component of the 2-component spinor may have a Dirac equation that says that it is a left-moving mode i.e. one that only depends on \(\tau+\sigma\) or \(\overline z\). (Sorry if my convention differs from someone else's; you have to be careful when you fully verify or study someone's papers and books.)
You may find it hard to write the world sheet action \(S\) for the left-movers only (or the right movers only) and it may be hard, indeed. But the action isn't the final product we're after. We need the correlation functions of the operators and they can be calculated in the segregated way.
The hybrid, heterotic theory effectively uses \(26\) bosonic fields \(X^\mu(z)\), along with \(b,c(z)\), and the \(10+10\) bosonic and fermionic fields \(X^\mu(\overline z),\psi^\mu(\overline z)\), along with \(b,c,\beta,\gamma(\overline z)\). These fields with pretty much the same dynamics that can be determined from the parent theories control all the calculable quantities of the heterotic string.
What about the mismatch?
The first observation is that there's really no problem whatsoever with the separation of fields \(\psi^\mu,b,c,\beta,\gamma\) into the left-movers and right-movers. They came as loose packages of the left-moving and right-moving part even in the non-hybrid superstring (or the bosonic string theory, in the case of \(b,c\)) simply because their field equations are first-order equations. Such field equations effectively say that a component of the field is holomorphic; or another component is antiholomorphic. And we may separate the components.
As you may see, the only potential subtleties of the segregation arise in the case of \(X^\mu(z,\overline z)\) whose field equations are second-order equations (Klein-Gordon). How does the separation work here? The hybrid, heterotic string seems to think that it's embedded in the \(10\)-dimensional spacetime according to the (superstring-like) right-moving excitations propagating on the string; but in the \(26\)-dimensional spacetime according to the (bosonic-string-theory-like) left-moving excitations. (The question which is which depends on a convention, one independent from most of the similar binary conventions. Flipping the conventions leads us to equivalent constructions.)
And yes, there are subtle constraints. There seem to be \(16\) bosonic fields \(X^\mu\) on the bosonic, left-moving side that are completely erased on the superstring, right-moving side. How do such \(16\) spacetime coordinates that half-exist, half-not-exist behave?
An interesting feature of these coordinates is that you may still compute the total momentum \[
P^\mu = \int_0^\pi\dd\sigma\,\partial_\tau X^\mu
\] (note that the \(\tau\)-derivative is the velocity which is proportional to the momentum density with a fixed coefficient) and the total winding, \[
W^\mu = \Delta X^\mu = \int_0^\pi\dd\sigma\,\partial_\sigma X^\mu= X^\mu|^\pi_0.
\] Well, that's true even in theories with both-sided \(X^\mu\) – or in the heterotic string theory for those shared \(10\) coordinates \(X^\mu\) that exist on both sides. However, a special feature of the heterotic string is that\[
(\partial_\tau+\partial_\sigma)X^\mu = 0
\] which is the condition that only the left-moving parts of the sixteen coordinates are allowed. When this equation is integrated from \(0\) to \(\pi\) over \(\sigma\), i.e. over the closed string, we realize that – with some normalization factors you must be careful about but I will simplify them a bit\[
P^\mu = W^\mu.
\] The momentum of the heterotic string in the direction of each of the sixteen "asymmetric" coordinates must be equal to the winding number – how many times the string winds around the given direction. That may seem bizarre but it makes a perfect sense.
You can't really prevent the string from having at least some nonzero values of \(P^\mu\); but in combination with \(P^\mu=W^\mu\), that implies that the windings must be allowed to be nonzero, too. So in some sense, these sixteen "lopsided" coordinates parameterize a \(16\)-dimensional torus.
Even self-dual lattices
Are all tori allowed? The answer is a resounding No. In fact, out of the infinitely many choices, only two completely rigid solutions solve the constraints and produce a consistent string theory (a string theory vacuum, to use the modern terminology in which string theory is already recognized as a unified theory with many solutions).
The torus may be represented as \(\RR^{16}/\Gamma^{(16)}\) where \(\Gamma\) is a symbol for lattices which are something like discrete groups \(\ZZ^{16}\) in this case. However, the sixteen independent generators of \(\ZZ^{16}\) don't have to shift the sixteen "lopsided" directions by the same distance; and as 16-dimensional vectors defining the translations, these generators don't have to be orthogonal each other. A sixteen-dimensional lattice is defined as this kind of \(\ZZ^{16}\) group that may be tilted or stretched or shrunk in various ways.
The quotient means that the coordinates in \(\RR^{16}\) become effectively periodic in some sense – but it's still some general sixteen linear combination of these coordinates that are periodic. The division by the lattice means that we're only interested in the coordinates "modulo integers" so we're interested in their fractional parts, kind of.
Fine, what are the allowed shapes of the lattice \(\Gamma^{(16)}\)? The lucky guys may be derived from the modular invariance of the one-loop toroidal stringy diagrams but this is too technical. We have this cool \(W^\mu=P^\mu\) condition which may do pretty much the same job.
If you study quantum mechanics of particles propagating on a circle of radius \(R\) and circumference \(2\pi R\), you will be able to derive that the momentum \(P\) has to be quantized – a number of the form \(N\hbar/R\). Let's set \(\hbar=1\); I just wanted to remind everyone that all those things may be written in everyday life units. This quantization emerges because the wave function \(\psi(x)\) has to be single-valued on the circle. For example, when you study the orbital angular momentum, it's effectively a particle on a circle of circumference \(2\pi\) (the coordinate \(\phi\)) and the dual "momentum" \(L_z\) has to be an integer because of the single-valuedness of the wave function.
Now, if the particle were replaced by a closed (circular) string, it could also wind around the circle. The total winding would be a multiple of the circumference \(2\pi R\) i.e. \(2\pi R w\). Note that the momentum has units of \(1/R\) which is inverse to the unit of the winding, \(2\pi R\). If you make the circle shorter, the spacing of the winding will shrink but the spacing of the momentum will increase by the same factor.
How is this rule generalized for a general lattice? It's generalized by the statement that the lattice in which the momentum lives is dual to the lattice in which the winding lives. For example, the dual lattice to \(k\ZZ\) is \((1/k)\ZZ\): both are effectively additive groups of integers but the physical sizes of the generators of these groups are inverse to one another. What does it mean to have a dual lattice in the general case?
It's not hard. If you have a lattice \(\Gamma\), the dual lattice \(\Gamma^*\) is composed of all the vectors \(W\in\RR^{16}\) in a "dual vector space" (the space of linear forms) that obey \(W\cdot V\in \ZZ\) for each \(V\in \Gamma\); I wrote \(W\cdot V\) as an inner product, assuming the usual \({\rm diag}({+}{+}\cdots {+})\) signature but I could have been more abstract and write it as \(W(V)\), the action of a linear form on a vector. The factors of \(2\pi\) must be dealt with somewhere but they don't change the qualitative message of all these constructions.
Because the momentum and the winding must belong to lattices that are dual to each other but, at the same moment, the momentum must be equal to the winding, the lattice of the allowed momenta must be equal to the lattice of the allowed windings i.e. to the dual lattice to the lattice of the allowed momenta. If a lattice is equal to its dual, \(\Gamma=\Gamma^*\), we say it is self-dual. And that's a hugely constraining condition, it turns out.
For example, the simple \(\ZZ^{16}\) lattice with the unit and orthogonal generators is self-dual because \(W\cdot V\in\ZZ\) for every two vectors \(V,W\) with sixteen integer-valued coordinates. The self-duality would surely disappear if you tried to deform and stretch the lattice in a generic way.
However, we may actually derive one more (significantly weaker) "even" condition from string theory: \(V^2\) must be not just integer but it must be even: \(V^2\in 2\ZZ\). This condition, arising from the need for \(L_0-\tilde L_0=\dots + V^2/2\) to remain integer-valued for the whole spectrum or, equivalently, from the \(\tau\to\tau+1\) part of the modular invariance, bans the simple \(\ZZ^{16}\) lattice. Are there any even self-dual lattices?
Yes, there are.
Use the symbols \(e_i\) where \(i=1,2,\dots,16\) for the usual orthonormal basis of \(\RR^{16}\). And take the lattice to be composed of all the linear combinations of \(e_i+e_j\) for \(i\neq j\), all the \(e_i-e_j\) for \(i\neq j\), and of \[
W_{\rm halfy} = (\frac 12, \frac 12, \frac 12, \dots , \frac 12)
\] where the same coordinate is repeated sixteen times. It's not hard to see that the inner product of any pair of the basis vectors (over integers) is integer. And because the inner product of every vector in the set above with itself is even – most nontrivially, the squared length of the last vector is \(16/2^2=4\) – the lattice of all the integer combinations of the vectors I just described will be even.
It is also self-dual. Try to find the most general vector \(W\) whose inner product with all the vectors in the "integer-based basis" above is integer-valued. Because it must hold for every \(e_i-e_j\), you may see that \(W_i\) and \(W_j\) must differ by an integer. Because it must hold for \(e_i+e_j\) as well, \(W_i\) and \(-W_j\) must also differ by an integer. It follows that the coordinates \(W_j\) and \(-W_j\) differ by an integer i.e. \(W_j\) itself is an integer multiple of \(1/2\) – it is either integer or integer plus \(1/2\). And I have already justified that if one coordinate is an integer, all the others have to be integers; if one of them differs from an integer by \(1/2\), all of them have to.
In the first (integer) case, you may show that the sum of the sixteen coordinates \(W_j\) is even because the inner product with the vector with sixteen \(1/2\) coordinates still has to be integer. But if the sum of the coordinates is even, it follows that you may express the vector as a combination of the \(e_i\pm e_j\) vectors. Similarly, this holds for \(W-W_{\rm halfy}\) if all the coordinates of \(W\) are half-integral because the inner product of \(W_{\rm halfy}\) with itself is an even integer.
The \(SO(32)\) heterotic string
You have to go through this proof yourself to really understand it – you have to rediscover it – but the lattice I defined as the set of integer combinations of all the vectors is even self-dual. It's what we need for the heterotic string. If you "half-compactify" the sixteen purely left-moving excessive bosonic coordinates on this lattice, you will get a string theory producing an exact \(SO(32)\) gauge symmetry in the spacetime.
The isometry of the torus is just \(U(1)^{16}\) which is the "Cartan subgroup" of \(SO(32)\) and it will be marketed as a part of the gauge symmetry because of the standard Kaluza-Klein mechanism. But there will be new gluon-like massless gauge bosons with a nonzero winding – corresponding to \(4\times 16\times 15/2 = 480\) points in the lattice that obey \(V^2=2\). And they will extend the symmetry group to a nice \(SO(32)\).
More precisely, the symmetry group is \(Spin(32)/\ZZ_2\) because the presence of the vectors such as \(W_{\rm halfy}\) in the lattice means that some states transforming as a spinor under \(Spin(32)\) will appear in the heterotic spectrum, too. Because the signs in \(W_{\rm halfy}\) are sort of prescribed (and an even number of them will flip if we add some \(-e_i-e_j\)), we will only obtain one Weyl (chiral) spinor, not the other. That's the origin of the \(\ZZ_2\) in the quotient. All the spinor states will give you massive states because \(W_{\rm halfy}^2=4\) which is greater than \(2\), the level where new massless states may still occur. The "spintensor" states with more general half-integral coordinates will be even heavier.
The spinorial states of the \(SO(32)\) heterotic string have a nice interpretation in terms of D-branes in the dual type I string theory. The duality will be mentioned later.
The \(E_8\times E_8\) heterotic string
Are there some other even self-dual sixteen-dimensional lattices? There is exactly one. In our construction of the lattice above – which is the "weight lattice of \(Spin(32)/\ZZ_2\)" – we used the vector \(W_{\rm halfy}\) with the coordinates \(1/2\) whose squared length was equal to \(4\). This is not a minimum squared length for an even lattice; the squared length equal to \(2\) would be just fine, too.
So you may also construct a totally analogous lattice \(E_8\) – the "root lattice of \(E_8\)" – to the sixteen-dimensional lattice but in eight, not sixteen dimensions. For the heterotic string, you need to do something with sixteen coordinates. But it's easy to divide them to two groups of eight coordinates and compactify each group on an \(E_8\) lattice.
The proofs that the \(E_8\) lattice is even and self-dual are completely analogous to the proof for the \(Spin(32)/\ZZ_2\) root lattice. But there's one really cool surprise. Because \(W_{\rm halfy}^2=2\) for the \(E_8\) lattice, it's the minimum allowed positive result, we may actually get new massless states (e.g. vector bosons and gauginos) from strings whose momentum and winding have half-integer coordinates, i.e. from the spinorial weights.
So some of the gauge bosons may transform as spinors of \(Spin(16)\). The gauge group in the "smaller" construction could be expected to be \(Spin(16)\), just like it was something like \(Spin(32)\) in the first heterotic string. However, the spinors are actually massless so they must correspond to generators of the gauge group, too. And if you combine the \(120\) generators of \(Spin(16)\) with the \(2^{8}/2=128\) generators that transform as a Weyl (chiral) spinor under \(Spin(16)\), you obtain the \(248\) generators of the group \(E_8\). We don't get just \(Spin(16)\times Spin(16)\) here; the gauge group is larger, \(E_8\times E_8\).
A cute detail – which is seen to be no coincidence if you study anomalies in the heterotic string's spacetime – is that both groups have the same dimension\[
\frac{32\times 31}{2} = 248+248 = 496.
\] In fact, some gravitational anomaly in \(D=10\) which must cancel is proportional to \((n-496)\). It's the \(E_8\times E_8\) string that is much more promising as a starting point to realistic phenomenology. One of the groups \(E_8\) may be broken to a subgroup such as \(E_6,SO(16),SU(5)\) which are viable grand unified groups and, when some extra six dimensions are compactified on a Calabi-Yau-like manifold, you get theories with realistic spectra and interactions (grand unificiation and SUSY is automatically built upon the Standard Model).
I should mention that within the purely Euclidean signature spaces, even self-dual lattices only exist in \(8k\) dimensions. I have discussed the unique eight-dimensional even self-dual lattice, \(E_8\), the lattice producing the equally named Lie group, and the two possible sixteen-dimensional even self-dual lattices. The next dimension where even self-dual lattices exist is \(D=24\). Aside from \(E_8\oplus E_8\oplus E_8\) and \(E_8\oplus \Gamma(Spin(32)/\ZZ_2)\) and an analogous \(Spin(48)/\ZZ_2\), one finds new examples, in particular the cool and less trivial Leech lattice which is crucial for the string-theoretical explanation of the monstrous moonshine (more).
Fermionization
A cool feature of the two-dimensional conformal field theories is that the same theory (physically) may often be expressed in many different ways. Instead of using sixteen left-moving bosons (on the bosonic side), we may "fermionize them". A free real boson in 2D CFTs is equivalent to two free real fermions whose operators may be expressed \[
\psi = \exp(+i\phi/2),\quad \bar\psi = \exp(-i\phi/2)
\] in terms of the boson \(\phi\) or, equivalently, the boson may be written as \(\partial_+\phi = \bar\psi\psi\). It may sound crazy: How could a tensor product of two fermionic Fock spaces look like a single bosonic Fock space? For the states of a single point-like particle species occupying one one-particle state, they're different. But if you include all the factors of the Fock spaces corresponding to all the harmonics along the string and you pick the right boundary conditions, it just works.
So a funny thing is that the same conclusion – there are exactly two possible heterotic string theories in ten dimensions – may be derived from the \(32\) real fermions that may be used instead of the \(16\) bosons above. You must only be careful about their allowed boundary/periodicity conditions around the closed heterotic string; and, which is related, about the GSO-like projections that tame the spectrum a little bit.
I won't discuss the details but the \(Spin(32)/\ZZ_2\) heterotic string may be constructed out of \(32\) real left-moving fermions \(\lambda^a\) that are either simultaneously periodic; or simultaneously antiperiodic (there are two sectors). Because all the fermions are treated in the same way (their friendship and common fate isn't perturbed, not even by the boundary conditions), you get the \(SO(32)\) symmetry: the symmetry currents are simply \(\lambda^a \lambda^b\) which is \(ab\)-antisymmetric. The spinorial states arise from "spin fields" or from the "highly degenerate" sector (because it has zero modes) with periodic, and not the simpler antiperiodic, boundary conditions for \(\lambda^a\). There's one GSO-like projection you have to impose.
Similarly, the \(E_8\times E_8\) heterotic string may be obtained if you split the set of \(32\) fermions \(\lambda^a\) to two groups of sixteen fermions and allow the periodic or antiperiodic boundary conditions for each group separately (there are four sectors, AA, AP, PA, PP). The multiplicity of the sectors is inseparable from two independent GSO-like conditions. Again, you could think that this breaks the group to \(Spin(16)\times Spin(16)\) but you will find the extra spinor states and the gauge group gets enhanced to \(E_8\times E_8\). The resulting theory may be shown to be equivalent – even at the level of string interactions, not just degeneracies in the free spectrum – to the heterotic string theories we obtained via the bosonic construction.
T-duality as a unification of both heterotic string theories
If you pick a direction in the "large" ten-dimensional spacetime and compactify it on a circle as well, the two heterotic string theories become smoothly connected into "one heterotic theory". Why?
The bosonic construction makes it a bit easier to explain. In the construction of the heterotic string theories above, we discussed lattices in a \(16+0\)-dimensional Euclidean space. I added the zero to emphasize that the signature was purely positive, Euclidean, and there were no time-like dimensions.
If you compactify the "non-chiral" boson \(X^{9}\) on a circle, it will have both left-moving and right-moving parts. In the physically most natural inner product, these two parts will behave as coordinates with the opposite signature. In effect, we added \(1+1\) dimensions to the \(16+0\)-dimensional lattice. The result is \(17+1\)-dimensional.
A funny fact is that a \(17+1\)-dimensional even self-dual lattice that we may still require for a consistent compactified heterotic string theory of this sort exists and it is... unique. In fact, the only Minkowskian even self-dual lattices exist in \(p+q\) dimensions where \(p-q\) is a multiple of eight and they're unique whenever \(pq\neq 0\). How can it be unique if we had two solutions to start with? Well, it's unique because\[
\Gamma(Spin(32)/\ZZ_2)\oplus \Gamma^{1,1} = \Gamma(E_8)\oplus \Gamma(E_8)\oplus \Gamma^{1,1}.
\] The lattices that you obtain from the two ten-dimensional heterotic string theories' lattices by adding a simple \(\Gamma^{1,1}\) from the compactified \(X^9\) are the same lattices, just rotated by a "Lorentz transformation" in \(17+1\) dimensions.
All these things may be fully proven but the implications are sort of remarkable. You may start with the \(Spin(32)/\ZZ_2\) heterotic string in ten dimensions. You compactify one more dimension so that only \(D=9\) coordinates remain noncompact. You break the group to some \(U(1)^{18}\) by changing the Wilson lines and other moduli and when you adjust these scalar fields to some right values, the gauge symmetry will suddenly start to get enhanced again. But you will get \(E_8\times E_8\) instead of \(Spin(32)/\ZZ_2\). An unexpected feature of the construction is that you can't view either of the two groups as a "broken phase" of the other. In fact, they're two equally large groups (when it comes to their dimension, \(496\)). They should be treated democratically; string theory allows you to break groups to smaller ones but also enhance groups to larger ones (at special points of the moduli space where some stringy states happen to go massless) and these two processes seem to be equally fundamental.
If string theorists weren't forced to see that such transitions exist, much like limited experimenters are hit by Mother Nature to their faces when She forces them to see something they should have seen for quite some time, they (or philosophers) would probably never "invent them" themselves. Nature and mathematics are smarter than us, even the smartest among us.
Heterotic-K3 duality
If you compactify the heterotic strings at least on a two-torus, you will get the vacua that, at strong coupling (but with the volume of the tori kept at a certain value in certain units), may be equivalently described by a totally non-heterotic string theory compactified on a seemingly highly nontrivial manifold, the four-dimensional K3 surface.
For example, a heterotic string on \(T^3\) deals with \(19+3\)-dimensional lattices, and exactly this \(22\)-dimensional lattice, with the right signature (if extracted from the intersection numbers of pairs of two-cycles), may be identified in the cohomology of the K3 surface. The heterotic string arose from a particular "freedom to hybridize", a loophole in the regulations who can have offspring with whom. But this "freedom to hybridize" is actually the same loophole as the possibility to find one more hyper-Kähler, real-four-dimensional manifold aside from the torus, the seemingly nontrivial and curved K3 surface. They're really "the same thing" visualized with the help of different geometric pictures and different degrees of freedom. But in the heterotic and K3 case, we're just thinking about the physics in two different ways – it's a difference in our impressions or visualization or conventions for symbols, sort of – but the underlying mathematics and physics is completely isomorphic.
String theory is full of such unifications of things that naively look completely different.
Different fates of heterotic string theories at strong coupling
String theory and its vacua are living organisms that always prepare lots of surprises for us. Already in \(D=10\), we may ask what the physical phenomena look like if we send the string coupling constant (or the string dilaton) to infinity?
Before the second superstring revolution in the mid 1990s, people would guess it's some uninteresting mess they don't want to talk about. For both heterotic theories, we get the same kind of mess.
However, it was shown that string theory – perhaps because it's such a perfectionist consistent theory – never leads you to a mess. Any simply describable legitimate limit of the moduli space must be a theory with so many special properties that it looks as natural as the starting point.
Even if I told you before the mid 1990s that the strong coupling limit of a string theory must be equivalent to some other string theory, you would probably make many wrong guesses about the theories you actually get from the two \(D=10\) heterotic string theories above. The \(Spin(32)/\ZZ_2\) and \(E_8\times E_8\) heterotic string theories are qualitatively the same structures, up to a difference in "technical details", so they should give you similar strong coupling limits, up to a difference in some other "technical details".
But this argument or expectation would be wrong, too. The strong coupling limits of both theories look very different from one another.
If you try to turn the string coupling constant \(g_s\) in the \(SO(32)\) heterotic string theory to a value much larger than one, you must still get a ten-dimensional supersymmetric theory whose spacetime gauge group includes \(SO(32)\). The ten large dimensions couldn't disappear. The supersymmetry couldn't disappear. The gauge group couldn't disappear. What the limit could be?
Well, there's one more supersymmetric string theory with the \(SO(32)\) gauge group, namely type I superstring theory; the \(SO(32)\) group arises from 32 possible half-colors of "quarks" at the end points of open strings or, equivalently, from the 16 spacetime-filling D9-branes and their mirror images (behind the orientifold mirror). This theory is not heterotic. It's purely "fermionic"; no hybrids. But its strings are unorientable and may be open or closed. (Heterotic strings must be orientable because the left-moving and right-moving parts of it have "inequivalent guts" so they can't be confused. For a similar reason, heterotic strings can't be open because an end point would have to "reflect" left-moving waves to some right-moving waves but the allowed waves in the two directions are inequivalent and can't be mapped to one another.)
Still, the theories are completely equivalent. Type I theory with the coupling \(g_s\) is equivalent to the heterotic \(SO(32)\) theory with the coupling \(1/g_s\). Incidentally, the spinorial states of the \(SO(32)\) heterotic strings appear as states of a particular non-supersymmetric (non-BPS) D0-brane in type I theory. The number of such D0-branes is conserved just modulo two, i.e. as an element of \(\ZZ_2\). A single D0-brane of this kind is stable because it's the lightest object/state that gets mapped to minus itself under the 360° rotation in \(SO(32)\) gauge group (the lightest spinor-like object in the theory).
The fate of the \(E_8\times E_8\) heterotic string is completely different and it was only understood by Petr Hořava and Edward Witten at the end of 1995, months after lots of similar discoveries. Again, the supersymmetry can't disappear. The ten large dimensions can't disappear. The \(E_8\times E_8\) gauge bosons can't disappear. So what the other description with these properties may be if I can assure you it's not the same description (the theory can't be S-self-dual)?
The answer is that while the ten dimensions can't disappear, a new dimension can appear. The strong coupling limit of the \(E_8\times E_8\) heterotic string is an 11-dimensional theory, M-theory, whose new dimension has the shape of the line interval of length \(L\). The value of \(L\) is an increasing function of the string coupling \(g_s\). But M-theory seems to have no non-Abelian gauge bosons.
The heterotic string itself becomes a cylindrical membrane, M2-brane of M-theory, stretched between the two ends of the world.
Well, it has gauge bosons if the 11-dimensional spacetime has boundaries. In fact, at the boundaries of such a spacetime, the gravitinos must be constrained to chiral fields. That creates 10D anomalies near the boundaries and they have to be cancelled. The gravitational anomalies may be cancelled by some other chiral fermions you may add, the gauginos, but that may create new gauge anomalies and mixed anomalies etc. At the end, it turns out that you may cancel all of these anomalies but only if \(E_8\) gauge bosons (and their superpartners, gauginos) live on each boundary of the 11-dimensional spacetime! The fact that it works at all boils down to some nontrivial properties of \(E_8\), some difficult and seemingly "very lucky" identities relating traces of products of generators of the group in the adjoint representation.
Because a line interval has two endpoints – a thick "desk" has two surfaces on both sides – you will get two \(E_8\) factors of the gauge group for each point in the remaining 10-dimensional "large" spacetime (yes, the gauge bosons and gauginos are confined to the end-of-the-world boundaries, much like some gauge fields are confined to D-branes or singularities): the gauge group will be \(E_8\times E_8\). The 11-dimensional M-theory with two boundaries – the heterotic M-theory – is also a good starting point to build realistic compactifications of string theory.
(Physicists describe the line interval as \(S^1/\ZZ_2\), a quotient of circle by the group mirroring it from the left to the right. This quotient is a line interval because e.g. the left half of the circle is a "fundamental domain" while the right part of the circle is just a \(\ZZ_2\) copy of it. And yes, one-half of a circle is the same thing topologically as a line interval. In my conventions, the end points of the line interval are the fixed point under \(\ZZ_2\), i.e. the uppermost and lowermost points of the circle.)
You may see that the heterotic string shows the remarkable uniqueness and interconnectedness of string/M-theory. There are just two possible heterotic string theories in \(D=10\) linked to two fully non-Abelian groups that may satisfy difficult anomaly cancellation conditions in the \(D=10\) spacetime. These two precious solutions may be derived in several languages that use different mathematical toolkits – bosons and lattices; grouping of fermions etc. – and they're connected after a compactification of another dimension. Further compactifications may be equivalently described in terms of M-theory or type II strings on K3 surfaces and the strong coupling limits are also equivalent to type I theory or M-theory with boundaries.
String/M-theory always knows what it's doing and it won't leave you in trouble. The maximally supersymmetric vacua of string/M-theory are arguably "prettier" but they're too symmetric and too sterile. The heterotic strings have 1/2 of the maximum supersymmetry (inherited from the mother, the right-moving supersymmetric side of the hybrid) but that's more than enough for SUSY to play its maternal protective role (and to add fermions into the spectrum and eliminate tachyons from the spectrum, the crippling vices that the offspring didn't want to inherit from the bosonic dad). The realistic vacua require even less supersymmetry.
As the amount of supersymmetry decreases, we are losing the ability to compute everything interesting easily but we're gaining lots of structures and new twists. The heterotic string vacua have so much supersymmetry that they still allow us to compute the answers to almost all simple (BPS) questions essentially by classical calculations but the freedom is already high enough to allow us to connect both versions of the heterotic string theory with one another and with type I, type IIA/IIB and M-theory on K3, with M-theory spacetimes that possess boundaries, and with other vacua I haven't discussed: already at this level, we may see the (more than minimal) interconnectedness of the diverse web of solutions to string theory.
In the honor of the heterotic string
![In the honor of the heterotic string]()
Reviewed by DAL
on
May 09, 2013
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