In June 2012, I emphasized that the Higgs field was the first experimentally observed elementary scalar field – the Higgs boson is the first elementary spinless particle we know – and because string theory loves to predict scalar fields and gives them many roles while many alternative thinkers love to invent legends claiming that there's something wrong about scalar fields, the discovery of the Higgs boson may be viewed as a minor, modest victory for string theory.
I would like to spend some time with some scalar fields in string theory and with their interesting properties. Quite generally, string theory predicts the existence of many dynamical scalar fields that would be replaced by non-dynamical parameters in quantum field theories. Because they're dynamical, string theory either implies the existence of new long-range forces (these vacua are more or less experimentally ruled out); or it implies that there's a potential that prefers some value or values of these scalar fields so these "parameters" suddenly become calculable.
A two-dimensional torus may be obtained from a rectangle (or a square) if we identify the upper edge with the lower one; and the left edge with the right edge. So much like in PacMan-like games, you reappear on the opposite side of the screen if you try to escape from it ("periodic boundary conditions"). The first identification turns the rectangle into a cylinder; the second one bends it into a doughnut.
The simplest scalar fields – such as the type I string theory coupling constant \(g_s\) – take values in the set \(\RR^+\) or, if we take the natural logarithm (the dilaton), in \(\RR\). The one-dimensional lines cannot have any curvature and they are too simple. So we will focus on something that may be the second simplest configuration space for scalar fields – the set of shapes of a two-dimensional torus.
We want to talk about the two-dimensional surfaces of toroidal topology and ask how many shapes they can have. Of course, it depends what we mean by a "shape".
If you ask how many metrics may be put on a two-torus, or a \(T^2=S^1\times S^1\), the answer is obviously "infinitely many". And it's a very large type of an infinity. Why? Well, you may describe the internal geometry of a two-torus by three components of the metric tensor\[
g_{11},\, g_{12},\, g_{22}
\] which are functions of the coordinates \(\sigma=x^1\) and \(\tau=x^2\) that we take to be periodic. But even if you acknowledge that two functions \(x^{\prime 1},x^{\prime 2}\) may be chosen arbitrarily to redefine the coordinates – and to redefine the functions \(g_{ij}\), it is still true that at least one function of \(\sigma,\tau\) remains totally undetermined. For example, for every torus you have, you may create a \(7\) times larger one. But the overall size isn't the only free parameter; at each point of the torus, there is a free parameter.
But let us change the rules of the game a little bit. Let's define the "shape" in such a way that only the local angles on the torus matter. The overall size of the torus doesn't matter and in fact, we may redefine the metric tensor by an independent overall scalar scaling of the whole metric tensor (the Weyl transformation) at each point and we still consider the shape to be the same thing. So we identify the geometries not only by the "diffeomorphisms" (it doesn't matter which coordinates you choose to describe the torus) but also by the Weyl symmetry,\[
g_{ij}(\sigma,\tau) \equiv e^{2\phi(\sigma,\tau)} g_{ij}(\sigma,\tau)
\] Note that the scaling parameter – which I wrote as an exponential for it to be positive and added the coefficient \(2\) to the exponent because the metric really defines \(ds^2\) – may depend on the location on the torus. With these identifications, there are locally three geometric parameters \(g_{ij}(\sigma,\tau)\) but there are also three functions of \(\sigma,\tau\) – namely \(x^{\prime 1},x^{\prime 2},\phi\) – that may be chosen arbitrarily and that define an equivalence. So roughly speaking, the freedom to redefine cancels the parameters describing the shape. Almost nothing is left.
By having noticed that \(3-3=0\), we actually only prove that there isn't an infinite number of shape parameters left – parameters organized into whole arbitrary functions of \(\sigma,\tau\) – that may define the shape of the torus. However, there may still be some finite remaining parameters which aren't allowed to depend on the location arbitrarily. They're "global parameters" and it's impossible to change their values by coordinate or Weyl transformations.
How many parameters like that a torus has?
Consider a manifold that is topologically a torus. You may always choose the coordinates \(\sigma,\tau\) such that their periodicity is \(2\pi\), for example, which means that the metric tensor components obey the following periodicity conditions:\[
g_{ij}(\sigma,\tau) = g_{ij}(\sigma+2\pi,\tau) = g_{ij}(\sigma,\tau+2\pi).
\] Now, in the July 2012 article about the Euler characteristic \(\chi\), I showed that \(\chi\) is a topological invariant so every two-torus must have \(\chi=0\) because the two-tori represented by a flat rectangle have \(\chi=0\) and the Euler characteristic may be computed from integrals of the curvature.
It means that the integral\[
\int d^2x\,\sqrt{|g|} R = 0
\] of the Ricci scalar vanishes. Because a Weyl transformation generally changes the Ricci scalar as a function of the position, we may choose a Weyl transformation that achieves \(R(\sigma,\tau)=0\) for all values of \(\sigma,\tau\). Some extra arguments would be needed to show that such a Weyl transformation doesn't have to contradict the periodic conditions above.
Believe me that all the minor gaps may be filled and it's true that by a Weyl transformation, every two-torus may be brought to a form that has \(R=0\) locally. Because the Riemann tensor in \(d=2\) has just one independent component \(R_{1212}\), the single condition \(R=0\) actually implies \(R_{ijkl}=0\) and the two-torus has therefore been made Ricci-flat. So one can choose coordinates in which the metric tensor is constant and because we didn't have to damage the periodic conditions, the torus has been brought to the form defined by constant values of \(g_{11},g_{12},g_{22}\) which are interpreted as functions of \(\sigma,\tau\) identified with the \(2\pi\) periodicities.
Note that there is one overall Weyl scaling left:\[
(g_{11},g_{12},g_{22}) \approx e^{2\phi}(g_{11},g_{12},g_{22})
\] As a result, the conformal shapes of the two-torus are determined by \(3-1=2\) i.e. two independent real parameters. Well, we may organize them into a single complex parameter we will confusingly call \(\tau\). What is it, except that it's clearly a different \(\tau\) than \(\tau=x^2\)? (Recipe: \(\tau\) only means \(x^2\) if it appears close to \(\sigma\) in a similar role. Otherwise, the meaning of \(\tau\) will be the new one I will describe momentarily.)
By a simple linear transformation of \(\sigma,\tau\), we may bring the flat torus to the form where\[
g_{ij} (\sigma,\tau) = \delta_{ij}
\] but this requires us to change the periodic identifications:\[
(x^1,x^2)\sim (x^1+2\pi,x^2)\sim (x^1+2\pi\,{\rm Re}(\tau),x^2+2\pi\,{\rm Im}(\tau))
\] The first identification is still the same thing as \(\sigma\sim \sigma+2\pi\) we started with (and you may always bring the first identification to this simple form by a rotation of the \(x^1\)-\(x^2\) plane combined with a scaling); the second one is tilted. The vector given by the complex number \(2\pi\tau\) in the complex \(z=x^1+ix^2\) plane defines another identification of points on the plane. It's natural to organize the two real parameters into one complex parameter because the metric on the two-plane is given by an expression that nicely "decouples" \(z\) from its complex conjugate:\[
ds^2 = (dx^1)^2 + (dx^2)^2 = dz^* dz.
\] Note that the torus obtained from a "square" is given by \(\tau=i\): the real part of \(\tau\) vanishes and the imaginary part is equal to one which produces the "usual" identification for the second coordinate. However, there are infinitely many shapes of the two-torus given by \(\tau\in\CC\). The nonzero real part of \(\tau\) "tilts" the torus while the imaginary part of \(\tau\) different from one makes the torus "thinner" or "thicker".
We face a critical question: are all complex values of \(\tau\) producing different shapes?
The answer is No, of course. The "modular group" will be the main class of identifications of different values of \(\tau\).
Using the complex variable \(z=x^1+i x^2\) describing the two-plane, we define the torus by the following identifications:\[
z\sim z+2\pi, \quad z\sim z+2\pi \tau.
\] Feel free to erase \(2\pi\) everywhere if it is annoying you; this factor is a convention related to the circumference of the circle (you may change the convention by redefining the normalization of \(z\)). You see that if you change\[
\tau\to-\tau,
\] you will get the same identification of points on the \(z\)-plane because points \(z\) are identified with \(z+2\pi m +2\pi \tau n\) for \(m,n\in\ZZ\) and it's the same set of points as if you change \(n\to-n\) or, equivalently, \(\tau\to-\tau\). So some identifications – some values of \(\tau\) – are identified with other identifications. I hope it's not too confusing! ;-)
We may use the freedom to change the sign of \(\tau\) to impose the usual convention\[
{\rm Im}(\tau) \gt 0.
\] The imaginary part is positive. Note that we may demand it to be strictly positive. If the imaginary part were zero, \(\tau\) would be real and the identifications would only identify points in the horizontal direction. We couldn't get a non-singular finite two-dimensional manifold in this way.
But there exists a wider class of equivalences between different values of \(\tau\). Note that the identifications of the \(z\)-plane are given by vectors in the "lattice"\[
2\pi m + 2\pi n \tau,\quad m,n\in \ZZ.
\] Its integer-based basis vectors are \(\{1,\tau\}\), if I omit the annoying \(2\pi\) myself. However, you may change this basis to any other integer-based basis \(\{A+B\tau,C+D\tau\}\) where \(A,B,C,D\in\ZZ\). Note that every integer combination of these two basis vectors\[
M(A+B\tau) + N(C+D\tau)
\] is an integer combination of the vectors \(1,\tau\). If the converse holds as well, we may say that the two bases are equally valid and they define the same identification. Well, the converse holds if the vectors \(\{1,\tau\}\) may be expressed as integer combinations of the vectors \(\{A+B\tau,C+D\tau\}\). In other words, it holds if the inverse matrix\[
\pmatrix{+A&+B\\+C&+D}^{-1} = \frac{1}{AD-BC} \pmatrix{+D&-B\\-C&+A}
\] has integer entries. And that's true if \(AD-BC=\pm 1\), i.e. if the determinant is equal to one, up to the sign. Well, if we only allow values of \(\tau\) whose imaginary part is positive, as we have said, they may only be related by transformations with \(AD-BC\gt 0\) which means that we require \(AD-BC=1\). Such \(2\times 2\) matrices form the group called \(SL(2,\ZZ)\), the modular group. Well, I am actually identifying matrices \(M\) and \(-M\) (they have the same determinant!) and this quotient is known as \(PSL(2,\ZZ)\) but I will still call it \(SL(2,\ZZ)\).
What do these identifications do with \(\tau\)?
We said that the integer-based basis vectors \(\{1,\tau\}\) are equally good as the integer-based basis vectors \(\{A+B\tau,C+D\tau\}\). Note that in the first case, we used the convention in which the first identification was determined by the complex number \(1\). This may be achieved, without changing the shape of the torus, by a rotation and overall scaling of the torus. These two operations may be unified into the multiplication by a complex number. And indeed, we may multiply the complex numbers \(\{A+B\tau,C+D\tau\}\) by a complex number, namely the inverse of the first one, so that the first one becomes \(1\):\[
\{A+B\tau,C+D\tau\} \sim \{1,\frac{C+D\tau}{A+B\tau}\}
\] And the final two integer-based vectors obey our convention: the first one is equal to one. Let's now change the conventions and relabel \(A,B,C,D\) as \(d,c,b,a\) (I could have chosen a better convention that avoids this doubling of conventions) so that we may say that there is an equivalence\[
\tau\sim\tau' = \frac{a\tau+b}{c\tau+d},\quad a,b,c,d\in\ZZ,\quad ad-bc = 1.
\] The "shape of the torus" parameters \(\tau,\tau'\) related in this way for some \(SL(2,\ZZ)\) matrices produce the same shapes. It may be unexpected for \(\tau\) to transform this "nonlinearly" under the group; we're more used to linear representations. But at the end, \(\tau\) is just some ratio of the two parameters (defining the identifications) that did transform linearly; you should be able to see where the nonlinearity came from. So how many inequivalent values of \(\tau\) are there? Can we choose a region of the \(\tau\)-plane so that this region counts each allowed shape exactly once?
The answer is Yes, we can. What the region exactly is depends on conventions but there exists a rather natural convention for the "fundamental domain", too. In order to see how large it is, we must study the modular group \(SL(2,\ZZ)\) a bit. A key claim is that every matrix in \(SL(2,\ZZ)\) may be written as a product of the matrices\[
S = \pmatrix{0&-1\\+1&0}, \,\, T=\pmatrix{1&+1\\0&+1},\,\, T^{-1} = \pmatrix{1&-1\\0&+1}
\] You may use as many copies of these matrices as you want, in any order. It's useful to notice that they act on \(\tau\) as follows:\[
S:\tau\to -\frac{1}{\tau},\quad T^{\pm 1}:\tau\to \tau\pm 1
\] They either invert \(\tau\) – with the extra sign needed to keep the imaginary part positive, note that \(i^{-1}=-i\); or they change \(\tau\) by plus minus one. Before we will investigate why a general \(SL(2,\ZZ)\) matrix may be written in terms of \(S\) and \(T^{\pm 1}\), let's see how it restricts the fundamental domain for \(\tau\). To be sure, the answer looks like the grey region here:
The region is given by the inequalities\[
-\frac{1}{2}\lt \tau_1\lt +\frac{1}{2},\quad \tau_2\gt 0,\quad |\tau|\gt 1
\] where the subscripts \(1,2\) refer to the real and imaginary parts, respectively. I sort of deliberately jump in between various conventions in order to prepare the dear reader for the messy world where different people – and, quite often, the same people – keep on using tons of inconsistent conventions, notations, and the most heretical ones even dare to speak different languages (than Czech).
A four-minute introduction to the belly of the Universe. And yes, the belly's most important city is my hometown of Pilsen. A better commercial for my country than this piece on the Franz Kafka airport in Prague. (The officer's name is pretty normal here, Mr Macháček Zlámaljelito, or Mr Little-rinser-off He-broke-the-black-pudding.) Or try Czech porn models try to learn and say one English sentence. Or a trilingual interview with a skier.
The middle inequality repeats the assumption that the imaginary part of \(\tau\) is positive. The first condition puts the real part between \(-1/2\) and \(+1/2\). The last part demands that the absolute value of \(\tau\) is greater than one. How did we derive the new two conditions? We must show that the fundamental domain defined by the conditions above counts all possibilities; but it doesn't double-count or multiply-count any of them.
First, why does the fundamental domain count all possibilities? Start with a \(\tau\) possibly outside the fundamental domain but with \(\tau_2\gt 0\). You may transform it by \(T^k:\tau\to\tau+k\) for an integer \(k\) to get it into the width-one strip \(-1/2\lt \tau_1\lt +1/2\). The transformed \(\tau\) belongs to the strip but it may have \(|\tau|\lt 1\). If it's so, you may invert \(\tau\) by \(S:\tau\to -1/\tau\) to get \(|\tau|\gt 1\). That may bring us outside the strip again in which case we apply \(T^k\), and so on.
Ultimately we're guaranteed to end up in the fundamental domain. There are many reasons why it is so. For example, the boundary of the fundamental domain is composed of segments that are \(SL(2,\ZZ)\) transformations of other segments of the boundary. In particular, the left vertical boundary is related to the right vertical boundary by \(\tau\to\tau+1\) and the arc is mapped onto itself by \(\tau\to-1/\tau\). Because we know how to map some "places near the fundamental domain" to the fundamental domain by an appropriate \(SL(2,\ZZ)\) transformation, we know that there's always some "life behind the boundary". The only region where we could get some "new \(\tau\) inequivalent to the fundamental domain" are those where the mirrored boundaries get infinitely dense but one may see that it may only occur near the real axis, a measure-zero region.
I don't claim the paragraph above to be the full proof but be sure I could give you the full proof and it would be a bit boring. Similarly, one may prove that the fundamental domain doesn't double-count any shapes: if \(\tau\) obeys all the inequalities, it's easy to see that \((a\tau+b)/(c\tau+d)\) inevitably violates at least one of them unless the matrix is (plus minus) the unit matrix.
The reader is encouraged to think about these matters by herself. She should realize that the multiplication of matrices by \(S,T^{\pm 1}\) is related to Euclid's algorithm to find the greatest common divisor of two integers. For example, multiply an \(SL(2,\ZZ)\) matrix by a column vector \(\vec v\) composed of integers. The greatest common divisor of \(v^1,v^2\) doesn't change if we exchange these two components – this pretty much corresponds to the action by \(S\); but it also remains invariant if we replace them by \(v^1-v^2,v^2\) which corresponds to the action by \(T^{-1}\). My previous claim that you ultimately get to the fundamental domain by cleverly combining \(S\) and \(T^{\pm 1}\) is nothing else than the claim that Euclid's algorithm succeeds after a finite number of steps.
And yes, the relationship to Euclid's algorithm is just one of the manifestations of the broader fact that the modular group and all the insights sketched in this blog entry are tightly linked to number theory, the science about prime numbers and other special properties of integers.
Applications
The fundamental domain of \(\tau\) describing the "shape of a torus" is omnipresent in modern theoretical physics and in string theory in particular. Many of these contexts used to be thought to be independent but they were often found to be intrinsically the same thing, by dualities. In some of the situations, we didn't really "see" the torus whose shape is expressed by \(\tau\); however, because of the newly found dualities, the torus suddenly emerged and became as tangible as all other tori. Many of the insights of string theory since the mid 1990s were able to "geometrize" many concepts that previously didn't admit such a visualization.
The oldest application of the fundamental domain in string theory is the shape of one-loop closed string Feynman diagrams. Note that in string theory, the Feynman diagrams – collections of propagators joined at vertices – become "thickened" and "smoothly connected" Riemann surfaces, systems of pants, world sheets of any topology. The one-loop point-like-particle Feynman diagrams get replaced by a torus with extra tubes attached to it; those correspond to the external particles/lines.
Because the diffeomorphisms and Weyl symmetries are symmetries of the world sheet, the torus may be transformed to the standard form given by the parameter \(\tau\). To calculate the contribution of a one-loop diagram in string theory, we still need to integrate over \(\tau\) – it's a real-two-dimensional integral. But there's a cool surprise.
In quantum field theory, the corresponding integral would go over the whole half-strip given by\[
-\frac{1}{2}\lt \tau_1\lt +\frac{1}{2},\quad \tau_2\gt 0.
\] This half-strip also contains lots of points with arbitrarily small values of \(\tau_2\), the imaginary part of \(\tau\), and the region near the real axis is the source of all the ultraviolet divergences. In string theory, you must be careful not to double-count the shapes of the tori so the integration only goes over the fundamental domain which is defined by the conditions above as well as the inequality\[
|\tau|\gt 1
\] and the region close to the real axis is completely omitted from the integral. That's one way to explain why string theory is free of ultraviolet divergences. The corresponding region of the interval is completely omitted. By the modular \(SL(2,\ZZ)\) transformations, one may show that it's equivalent to some region that has the potential to produce infrared divergences. But there aren't any independent ultraviolet divergences in string theory. Ever. Never. The argument holds for arbitrarily complicated \(n\)-loop diagrams and has been proven rigorously.
But the shape of the one-loop world sheets is just one example of the place in string theory where the fundamental domain of the modular group appears.
The maximally supersymmetric gauge theory in \(d=4\) has a coupling constant \(g_{YM}\) but also a \(\theta\)-angle. They may be combined into\[
\tau = \frac{i}{g_{YM}^2} + \frac{\theta}{2\pi}
\] so that the \(\tau\to -1/\tau\) transformation gets naturally identified with the S-duality ("electromagnetic duality" of a sort) acting as \(g_{YM}\to 1/g_{YM}\) and where the \(\tau\to\tau\pm 1\) gets reinterpreted as the \(2\pi\) angular periodicity of the \(\theta\)-angle. These basic identifications generate the whole \(SL(2,\ZZ)\) which is therefore a symmetry, the S-duality group, of the gauge theory.
A totally analogous situation appears in type IIB string theory: just replace \(1/g_{YM}^2\) by \(1/g_s=\exp(-\phi)\) (the string coupling constant is the exponential of the dilaton) and rename the \(\theta\)-angle as the Ramond-Ramond axion scalar field and the maths is the same.
Although we know that such values of \(\tau\) identified by the \(SL(2,\ZZ)\) modular transformations describe the "possible shapes of some torus", there used to be times when the people didn't know what the torus actually was. Moreover, the two cases – the \(\NNN=4\) gauge theory and type IIB string theory – were thought to be independent of one another.
The revolutionary insights of the last 20 years have completely changed the thinking. The two cases are closely related. Moreover, we know how to visualize "the torus" in both cases.
First, type IIB string theory is related to the \(\NNN=4\) Yang-Mills theory in a simple way: the Yang-Mills theory simply describes the low-energy dynamics of D3-branes, objects that may exist within type IIB string theory. The Yang-Mills coupling constant (and the \(\theta\)-angle) on the D3-brane simply inherits the string coupling constant (and the Ramond-Ramond axion field) from the surrounding type IIB stringy spacetime. This relationship becomes particularly obvious in the AdS/CFT correspondence which claims that the Yang-Mills theory is physically equivalent to type IIB string theory on the \(AdS_5\times S^5\) background.
Moreover, we also know how to properly visualize the "torus" whose possible shapes are parameterized by the complex number \(\tau\). In the case of the gauge theory, we may say that this gauge theory may be thought of as the compactification of a 6-dimensional \((2,0)\) theory on a vanishingly small two-torus. In this way, the proof of the S-duality group of the gauge theory may be reduced to a proof of the existence of this 6-dimensional theory (and its compactifications): the duality is completely manifest because it results from the tori's being the same.
Similarly, type IIB string theory which has a 10-dimensional spacetime may be understood as the compactification of a 12-dimensional theory, F-theory, on a similar 2-torus. The difference is that F-theory can't exist in the decompactified form – in an infinitely large 12-dimensional spacetime, at least people think it can't. This theory forces you to compactify its two dimensions on a two-torus.
The letter "F" in F-theory stands for vaFa, Father, or Fiber, according to your taste.
While F-theory or the \((2,0)\) theory may look too abstract, there exist various additional dualities that translate them to equivalent compactifications of M-theory or gauge theories in which the two-torus still has a well-defined role. The main punch line is that we have tools that allow us to "imagine" the two-torus and give it a geometric, physical interpretation whenever there is a parameter \(\tau\) identified by the modular group. In most cases, string theory provides us with many seemingly inequivalent ways to "geometrize" the way how we think about the physics. The physical phenomena are still exactly equivalent but we may look at them from many directions, directions which provide us with many different expansions or ways to calculate the same thing. These surprising equivalences – dualities – are extremely powerful.
F-theory, the clever way of rephrasing type IIB string theory as a 12-dimensional theory that makes the origin of the torus manifest, also allows us to construct many new solutions to string theory which may be imagined as compactifications of the 12-dimensional theory. If you want 3+1 dimensions to remain large, you may compactify a 12-dimensional theory on an 8-manifold. Essentially because the number of compactified dimensions is so large, eight, F-theory is the class of string vacua that is the "largest" one and produces the richest classes of stringy vacua in the "landscape".
Note that an obvious virtue of F-theory relatively to type IIB string theory is that it encourages us to think about non-constant values of \(\tau\) as functions of the remaining 9+1 spacetime coordinates (not participating in the torus). Not only \(\tau\) may change; if you make a trip around a 7-brane (a co-dimension-two singularity in the spacetime), \(\tau\) doesn't have to return to its original value. It may come to \(\tau'\) which is \(\tau\) transformed by the \(SL(2,\ZZ)\) transformation. We call this change of a variable induced by a round trip "a monodromy". Monodromies are omnipresent in F-theory and the possible monodromies essentially classify the possible types of 7-branes – and be sure that there are many types of such branes in F-theory and they usually produce cool enhanced non-Abelian symmetries and gauge fields that are confined to the singular branes.
I plan to write an introduction on F-theory, elliptic curves, etc. sometime in the future.
I would like to spend some time with some scalar fields in string theory and with their interesting properties. Quite generally, string theory predicts the existence of many dynamical scalar fields that would be replaced by non-dynamical parameters in quantum field theories. Because they're dynamical, string theory either implies the existence of new long-range forces (these vacua are more or less experimentally ruled out); or it implies that there's a potential that prefers some value or values of these scalar fields so these "parameters" suddenly become calculable.
A two-dimensional torus may be obtained from a rectangle (or a square) if we identify the upper edge with the lower one; and the left edge with the right edge. So much like in PacMan-like games, you reappear on the opposite side of the screen if you try to escape from it ("periodic boundary conditions"). The first identification turns the rectangle into a cylinder; the second one bends it into a doughnut.
The simplest scalar fields – such as the type I string theory coupling constant \(g_s\) – take values in the set \(\RR^+\) or, if we take the natural logarithm (the dilaton), in \(\RR\). The one-dimensional lines cannot have any curvature and they are too simple. So we will focus on something that may be the second simplest configuration space for scalar fields – the set of shapes of a two-dimensional torus.
We want to talk about the two-dimensional surfaces of toroidal topology and ask how many shapes they can have. Of course, it depends what we mean by a "shape".
If you ask how many metrics may be put on a two-torus, or a \(T^2=S^1\times S^1\), the answer is obviously "infinitely many". And it's a very large type of an infinity. Why? Well, you may describe the internal geometry of a two-torus by three components of the metric tensor\[
g_{11},\, g_{12},\, g_{22}
\] which are functions of the coordinates \(\sigma=x^1\) and \(\tau=x^2\) that we take to be periodic. But even if you acknowledge that two functions \(x^{\prime 1},x^{\prime 2}\) may be chosen arbitrarily to redefine the coordinates – and to redefine the functions \(g_{ij}\), it is still true that at least one function of \(\sigma,\tau\) remains totally undetermined. For example, for every torus you have, you may create a \(7\) times larger one. But the overall size isn't the only free parameter; at each point of the torus, there is a free parameter.
But let us change the rules of the game a little bit. Let's define the "shape" in such a way that only the local angles on the torus matter. The overall size of the torus doesn't matter and in fact, we may redefine the metric tensor by an independent overall scalar scaling of the whole metric tensor (the Weyl transformation) at each point and we still consider the shape to be the same thing. So we identify the geometries not only by the "diffeomorphisms" (it doesn't matter which coordinates you choose to describe the torus) but also by the Weyl symmetry,\[
g_{ij}(\sigma,\tau) \equiv e^{2\phi(\sigma,\tau)} g_{ij}(\sigma,\tau)
\] Note that the scaling parameter – which I wrote as an exponential for it to be positive and added the coefficient \(2\) to the exponent because the metric really defines \(ds^2\) – may depend on the location on the torus. With these identifications, there are locally three geometric parameters \(g_{ij}(\sigma,\tau)\) but there are also three functions of \(\sigma,\tau\) – namely \(x^{\prime 1},x^{\prime 2},\phi\) – that may be chosen arbitrarily and that define an equivalence. So roughly speaking, the freedom to redefine cancels the parameters describing the shape. Almost nothing is left.
By having noticed that \(3-3=0\), we actually only prove that there isn't an infinite number of shape parameters left – parameters organized into whole arbitrary functions of \(\sigma,\tau\) – that may define the shape of the torus. However, there may still be some finite remaining parameters which aren't allowed to depend on the location arbitrarily. They're "global parameters" and it's impossible to change their values by coordinate or Weyl transformations.
How many parameters like that a torus has?
Consider a manifold that is topologically a torus. You may always choose the coordinates \(\sigma,\tau\) such that their periodicity is \(2\pi\), for example, which means that the metric tensor components obey the following periodicity conditions:\[
g_{ij}(\sigma,\tau) = g_{ij}(\sigma+2\pi,\tau) = g_{ij}(\sigma,\tau+2\pi).
\] Now, in the July 2012 article about the Euler characteristic \(\chi\), I showed that \(\chi\) is a topological invariant so every two-torus must have \(\chi=0\) because the two-tori represented by a flat rectangle have \(\chi=0\) and the Euler characteristic may be computed from integrals of the curvature.
It means that the integral\[
\int d^2x\,\sqrt{|g|} R = 0
\] of the Ricci scalar vanishes. Because a Weyl transformation generally changes the Ricci scalar as a function of the position, we may choose a Weyl transformation that achieves \(R(\sigma,\tau)=0\) for all values of \(\sigma,\tau\). Some extra arguments would be needed to show that such a Weyl transformation doesn't have to contradict the periodic conditions above.
Believe me that all the minor gaps may be filled and it's true that by a Weyl transformation, every two-torus may be brought to a form that has \(R=0\) locally. Because the Riemann tensor in \(d=2\) has just one independent component \(R_{1212}\), the single condition \(R=0\) actually implies \(R_{ijkl}=0\) and the two-torus has therefore been made Ricci-flat. So one can choose coordinates in which the metric tensor is constant and because we didn't have to damage the periodic conditions, the torus has been brought to the form defined by constant values of \(g_{11},g_{12},g_{22}\) which are interpreted as functions of \(\sigma,\tau\) identified with the \(2\pi\) periodicities.
Note that there is one overall Weyl scaling left:\[
(g_{11},g_{12},g_{22}) \approx e^{2\phi}(g_{11},g_{12},g_{22})
\] As a result, the conformal shapes of the two-torus are determined by \(3-1=2\) i.e. two independent real parameters. Well, we may organize them into a single complex parameter we will confusingly call \(\tau\). What is it, except that it's clearly a different \(\tau\) than \(\tau=x^2\)? (Recipe: \(\tau\) only means \(x^2\) if it appears close to \(\sigma\) in a similar role. Otherwise, the meaning of \(\tau\) will be the new one I will describe momentarily.)
By a simple linear transformation of \(\sigma,\tau\), we may bring the flat torus to the form where\[
g_{ij} (\sigma,\tau) = \delta_{ij}
\] but this requires us to change the periodic identifications:\[
(x^1,x^2)\sim (x^1+2\pi,x^2)\sim (x^1+2\pi\,{\rm Re}(\tau),x^2+2\pi\,{\rm Im}(\tau))
\] The first identification is still the same thing as \(\sigma\sim \sigma+2\pi\) we started with (and you may always bring the first identification to this simple form by a rotation of the \(x^1\)-\(x^2\) plane combined with a scaling); the second one is tilted. The vector given by the complex number \(2\pi\tau\) in the complex \(z=x^1+ix^2\) plane defines another identification of points on the plane. It's natural to organize the two real parameters into one complex parameter because the metric on the two-plane is given by an expression that nicely "decouples" \(z\) from its complex conjugate:\[
ds^2 = (dx^1)^2 + (dx^2)^2 = dz^* dz.
\] Note that the torus obtained from a "square" is given by \(\tau=i\): the real part of \(\tau\) vanishes and the imaginary part is equal to one which produces the "usual" identification for the second coordinate. However, there are infinitely many shapes of the two-torus given by \(\tau\in\CC\). The nonzero real part of \(\tau\) "tilts" the torus while the imaginary part of \(\tau\) different from one makes the torus "thinner" or "thicker".
We face a critical question: are all complex values of \(\tau\) producing different shapes?
The answer is No, of course. The "modular group" will be the main class of identifications of different values of \(\tau\).
Using the complex variable \(z=x^1+i x^2\) describing the two-plane, we define the torus by the following identifications:\[
z\sim z+2\pi, \quad z\sim z+2\pi \tau.
\] Feel free to erase \(2\pi\) everywhere if it is annoying you; this factor is a convention related to the circumference of the circle (you may change the convention by redefining the normalization of \(z\)). You see that if you change\[
\tau\to-\tau,
\] you will get the same identification of points on the \(z\)-plane because points \(z\) are identified with \(z+2\pi m +2\pi \tau n\) for \(m,n\in\ZZ\) and it's the same set of points as if you change \(n\to-n\) or, equivalently, \(\tau\to-\tau\). So some identifications – some values of \(\tau\) – are identified with other identifications. I hope it's not too confusing! ;-)
We may use the freedom to change the sign of \(\tau\) to impose the usual convention\[
{\rm Im}(\tau) \gt 0.
\] The imaginary part is positive. Note that we may demand it to be strictly positive. If the imaginary part were zero, \(\tau\) would be real and the identifications would only identify points in the horizontal direction. We couldn't get a non-singular finite two-dimensional manifold in this way.
But there exists a wider class of equivalences between different values of \(\tau\). Note that the identifications of the \(z\)-plane are given by vectors in the "lattice"\[
2\pi m + 2\pi n \tau,\quad m,n\in \ZZ.
\] Its integer-based basis vectors are \(\{1,\tau\}\), if I omit the annoying \(2\pi\) myself. However, you may change this basis to any other integer-based basis \(\{A+B\tau,C+D\tau\}\) where \(A,B,C,D\in\ZZ\). Note that every integer combination of these two basis vectors\[
M(A+B\tau) + N(C+D\tau)
\] is an integer combination of the vectors \(1,\tau\). If the converse holds as well, we may say that the two bases are equally valid and they define the same identification. Well, the converse holds if the vectors \(\{1,\tau\}\) may be expressed as integer combinations of the vectors \(\{A+B\tau,C+D\tau\}\). In other words, it holds if the inverse matrix\[
\pmatrix{+A&+B\\+C&+D}^{-1} = \frac{1}{AD-BC} \pmatrix{+D&-B\\-C&+A}
\] has integer entries. And that's true if \(AD-BC=\pm 1\), i.e. if the determinant is equal to one, up to the sign. Well, if we only allow values of \(\tau\) whose imaginary part is positive, as we have said, they may only be related by transformations with \(AD-BC\gt 0\) which means that we require \(AD-BC=1\). Such \(2\times 2\) matrices form the group called \(SL(2,\ZZ)\), the modular group. Well, I am actually identifying matrices \(M\) and \(-M\) (they have the same determinant!) and this quotient is known as \(PSL(2,\ZZ)\) but I will still call it \(SL(2,\ZZ)\).
What do these identifications do with \(\tau\)?
We said that the integer-based basis vectors \(\{1,\tau\}\) are equally good as the integer-based basis vectors \(\{A+B\tau,C+D\tau\}\). Note that in the first case, we used the convention in which the first identification was determined by the complex number \(1\). This may be achieved, without changing the shape of the torus, by a rotation and overall scaling of the torus. These two operations may be unified into the multiplication by a complex number. And indeed, we may multiply the complex numbers \(\{A+B\tau,C+D\tau\}\) by a complex number, namely the inverse of the first one, so that the first one becomes \(1\):\[
\{A+B\tau,C+D\tau\} \sim \{1,\frac{C+D\tau}{A+B\tau}\}
\] And the final two integer-based vectors obey our convention: the first one is equal to one. Let's now change the conventions and relabel \(A,B,C,D\) as \(d,c,b,a\) (I could have chosen a better convention that avoids this doubling of conventions) so that we may say that there is an equivalence\[
\tau\sim\tau' = \frac{a\tau+b}{c\tau+d},\quad a,b,c,d\in\ZZ,\quad ad-bc = 1.
\] The "shape of the torus" parameters \(\tau,\tau'\) related in this way for some \(SL(2,\ZZ)\) matrices produce the same shapes. It may be unexpected for \(\tau\) to transform this "nonlinearly" under the group; we're more used to linear representations. But at the end, \(\tau\) is just some ratio of the two parameters (defining the identifications) that did transform linearly; you should be able to see where the nonlinearity came from. So how many inequivalent values of \(\tau\) are there? Can we choose a region of the \(\tau\)-plane so that this region counts each allowed shape exactly once?
The answer is Yes, we can. What the region exactly is depends on conventions but there exists a rather natural convention for the "fundamental domain", too. In order to see how large it is, we must study the modular group \(SL(2,\ZZ)\) a bit. A key claim is that every matrix in \(SL(2,\ZZ)\) may be written as a product of the matrices\[
S = \pmatrix{0&-1\\+1&0}, \,\, T=\pmatrix{1&+1\\0&+1},\,\, T^{-1} = \pmatrix{1&-1\\0&+1}
\] You may use as many copies of these matrices as you want, in any order. It's useful to notice that they act on \(\tau\) as follows:\[
S:\tau\to -\frac{1}{\tau},\quad T^{\pm 1}:\tau\to \tau\pm 1
\] They either invert \(\tau\) – with the extra sign needed to keep the imaginary part positive, note that \(i^{-1}=-i\); or they change \(\tau\) by plus minus one. Before we will investigate why a general \(SL(2,\ZZ)\) matrix may be written in terms of \(S\) and \(T^{\pm 1}\), let's see how it restricts the fundamental domain for \(\tau\). To be sure, the answer looks like the grey region here:
The region is given by the inequalities\[
-\frac{1}{2}\lt \tau_1\lt +\frac{1}{2},\quad \tau_2\gt 0,\quad |\tau|\gt 1
\] where the subscripts \(1,2\) refer to the real and imaginary parts, respectively. I sort of deliberately jump in between various conventions in order to prepare the dear reader for the messy world where different people – and, quite often, the same people – keep on using tons of inconsistent conventions, notations, and the most heretical ones even dare to speak different languages (than Czech).
A four-minute introduction to the belly of the Universe. And yes, the belly's most important city is my hometown of Pilsen. A better commercial for my country than this piece on the Franz Kafka airport in Prague. (The officer's name is pretty normal here, Mr Macháček Zlámaljelito, or Mr Little-rinser-off He-broke-the-black-pudding.) Or try Czech porn models try to learn and say one English sentence. Or a trilingual interview with a skier.
The middle inequality repeats the assumption that the imaginary part of \(\tau\) is positive. The first condition puts the real part between \(-1/2\) and \(+1/2\). The last part demands that the absolute value of \(\tau\) is greater than one. How did we derive the new two conditions? We must show that the fundamental domain defined by the conditions above counts all possibilities; but it doesn't double-count or multiply-count any of them.
First, why does the fundamental domain count all possibilities? Start with a \(\tau\) possibly outside the fundamental domain but with \(\tau_2\gt 0\). You may transform it by \(T^k:\tau\to\tau+k\) for an integer \(k\) to get it into the width-one strip \(-1/2\lt \tau_1\lt +1/2\). The transformed \(\tau\) belongs to the strip but it may have \(|\tau|\lt 1\). If it's so, you may invert \(\tau\) by \(S:\tau\to -1/\tau\) to get \(|\tau|\gt 1\). That may bring us outside the strip again in which case we apply \(T^k\), and so on.
Ultimately we're guaranteed to end up in the fundamental domain. There are many reasons why it is so. For example, the boundary of the fundamental domain is composed of segments that are \(SL(2,\ZZ)\) transformations of other segments of the boundary. In particular, the left vertical boundary is related to the right vertical boundary by \(\tau\to\tau+1\) and the arc is mapped onto itself by \(\tau\to-1/\tau\). Because we know how to map some "places near the fundamental domain" to the fundamental domain by an appropriate \(SL(2,\ZZ)\) transformation, we know that there's always some "life behind the boundary". The only region where we could get some "new \(\tau\) inequivalent to the fundamental domain" are those where the mirrored boundaries get infinitely dense but one may see that it may only occur near the real axis, a measure-zero region.
I don't claim the paragraph above to be the full proof but be sure I could give you the full proof and it would be a bit boring. Similarly, one may prove that the fundamental domain doesn't double-count any shapes: if \(\tau\) obeys all the inequalities, it's easy to see that \((a\tau+b)/(c\tau+d)\) inevitably violates at least one of them unless the matrix is (plus minus) the unit matrix.
The reader is encouraged to think about these matters by herself. She should realize that the multiplication of matrices by \(S,T^{\pm 1}\) is related to Euclid's algorithm to find the greatest common divisor of two integers. For example, multiply an \(SL(2,\ZZ)\) matrix by a column vector \(\vec v\) composed of integers. The greatest common divisor of \(v^1,v^2\) doesn't change if we exchange these two components – this pretty much corresponds to the action by \(S\); but it also remains invariant if we replace them by \(v^1-v^2,v^2\) which corresponds to the action by \(T^{-1}\). My previous claim that you ultimately get to the fundamental domain by cleverly combining \(S\) and \(T^{\pm 1}\) is nothing else than the claim that Euclid's algorithm succeeds after a finite number of steps.
And yes, the relationship to Euclid's algorithm is just one of the manifestations of the broader fact that the modular group and all the insights sketched in this blog entry are tightly linked to number theory, the science about prime numbers and other special properties of integers.
Applications
The fundamental domain of \(\tau\) describing the "shape of a torus" is omnipresent in modern theoretical physics and in string theory in particular. Many of these contexts used to be thought to be independent but they were often found to be intrinsically the same thing, by dualities. In some of the situations, we didn't really "see" the torus whose shape is expressed by \(\tau\); however, because of the newly found dualities, the torus suddenly emerged and became as tangible as all other tori. Many of the insights of string theory since the mid 1990s were able to "geometrize" many concepts that previously didn't admit such a visualization.
The oldest application of the fundamental domain in string theory is the shape of one-loop closed string Feynman diagrams. Note that in string theory, the Feynman diagrams – collections of propagators joined at vertices – become "thickened" and "smoothly connected" Riemann surfaces, systems of pants, world sheets of any topology. The one-loop point-like-particle Feynman diagrams get replaced by a torus with extra tubes attached to it; those correspond to the external particles/lines.
Because the diffeomorphisms and Weyl symmetries are symmetries of the world sheet, the torus may be transformed to the standard form given by the parameter \(\tau\). To calculate the contribution of a one-loop diagram in string theory, we still need to integrate over \(\tau\) – it's a real-two-dimensional integral. But there's a cool surprise.
In quantum field theory, the corresponding integral would go over the whole half-strip given by\[
-\frac{1}{2}\lt \tau_1\lt +\frac{1}{2},\quad \tau_2\gt 0.
\] This half-strip also contains lots of points with arbitrarily small values of \(\tau_2\), the imaginary part of \(\tau\), and the region near the real axis is the source of all the ultraviolet divergences. In string theory, you must be careful not to double-count the shapes of the tori so the integration only goes over the fundamental domain which is defined by the conditions above as well as the inequality\[
|\tau|\gt 1
\] and the region close to the real axis is completely omitted from the integral. That's one way to explain why string theory is free of ultraviolet divergences. The corresponding region of the interval is completely omitted. By the modular \(SL(2,\ZZ)\) transformations, one may show that it's equivalent to some region that has the potential to produce infrared divergences. But there aren't any independent ultraviolet divergences in string theory. Ever. Never. The argument holds for arbitrarily complicated \(n\)-loop diagrams and has been proven rigorously.
But the shape of the one-loop world sheets is just one example of the place in string theory where the fundamental domain of the modular group appears.
The maximally supersymmetric gauge theory in \(d=4\) has a coupling constant \(g_{YM}\) but also a \(\theta\)-angle. They may be combined into\[
\tau = \frac{i}{g_{YM}^2} + \frac{\theta}{2\pi}
\] so that the \(\tau\to -1/\tau\) transformation gets naturally identified with the S-duality ("electromagnetic duality" of a sort) acting as \(g_{YM}\to 1/g_{YM}\) and where the \(\tau\to\tau\pm 1\) gets reinterpreted as the \(2\pi\) angular periodicity of the \(\theta\)-angle. These basic identifications generate the whole \(SL(2,\ZZ)\) which is therefore a symmetry, the S-duality group, of the gauge theory.
A totally analogous situation appears in type IIB string theory: just replace \(1/g_{YM}^2\) by \(1/g_s=\exp(-\phi)\) (the string coupling constant is the exponential of the dilaton) and rename the \(\theta\)-angle as the Ramond-Ramond axion scalar field and the maths is the same.
Although we know that such values of \(\tau\) identified by the \(SL(2,\ZZ)\) modular transformations describe the "possible shapes of some torus", there used to be times when the people didn't know what the torus actually was. Moreover, the two cases – the \(\NNN=4\) gauge theory and type IIB string theory – were thought to be independent of one another.
The revolutionary insights of the last 20 years have completely changed the thinking. The two cases are closely related. Moreover, we know how to visualize "the torus" in both cases.
First, type IIB string theory is related to the \(\NNN=4\) Yang-Mills theory in a simple way: the Yang-Mills theory simply describes the low-energy dynamics of D3-branes, objects that may exist within type IIB string theory. The Yang-Mills coupling constant (and the \(\theta\)-angle) on the D3-brane simply inherits the string coupling constant (and the Ramond-Ramond axion field) from the surrounding type IIB stringy spacetime. This relationship becomes particularly obvious in the AdS/CFT correspondence which claims that the Yang-Mills theory is physically equivalent to type IIB string theory on the \(AdS_5\times S^5\) background.
Moreover, we also know how to properly visualize the "torus" whose possible shapes are parameterized by the complex number \(\tau\). In the case of the gauge theory, we may say that this gauge theory may be thought of as the compactification of a 6-dimensional \((2,0)\) theory on a vanishingly small two-torus. In this way, the proof of the S-duality group of the gauge theory may be reduced to a proof of the existence of this 6-dimensional theory (and its compactifications): the duality is completely manifest because it results from the tori's being the same.
Similarly, type IIB string theory which has a 10-dimensional spacetime may be understood as the compactification of a 12-dimensional theory, F-theory, on a similar 2-torus. The difference is that F-theory can't exist in the decompactified form – in an infinitely large 12-dimensional spacetime, at least people think it can't. This theory forces you to compactify its two dimensions on a two-torus.
The letter "F" in F-theory stands for vaFa, Father, or Fiber, according to your taste.
While F-theory or the \((2,0)\) theory may look too abstract, there exist various additional dualities that translate them to equivalent compactifications of M-theory or gauge theories in which the two-torus still has a well-defined role. The main punch line is that we have tools that allow us to "imagine" the two-torus and give it a geometric, physical interpretation whenever there is a parameter \(\tau\) identified by the modular group. In most cases, string theory provides us with many seemingly inequivalent ways to "geometrize" the way how we think about the physics. The physical phenomena are still exactly equivalent but we may look at them from many directions, directions which provide us with many different expansions or ways to calculate the same thing. These surprising equivalences – dualities – are extremely powerful.
F-theory, the clever way of rephrasing type IIB string theory as a 12-dimensional theory that makes the origin of the torus manifest, also allows us to construct many new solutions to string theory which may be imagined as compactifications of the 12-dimensional theory. If you want 3+1 dimensions to remain large, you may compactify a 12-dimensional theory on an 8-manifold. Essentially because the number of compactified dimensions is so large, eight, F-theory is the class of string vacua that is the "largest" one and produces the richest classes of stringy vacua in the "landscape".
Note that an obvious virtue of F-theory relatively to type IIB string theory is that it encourages us to think about non-constant values of \(\tau\) as functions of the remaining 9+1 spacetime coordinates (not participating in the torus). Not only \(\tau\) may change; if you make a trip around a 7-brane (a co-dimension-two singularity in the spacetime), \(\tau\) doesn't have to return to its original value. It may come to \(\tau'\) which is \(\tau\) transformed by the \(SL(2,\ZZ)\) transformation. We call this change of a variable induced by a round trip "a monodromy". Monodromies are omnipresent in F-theory and the possible monodromies essentially classify the possible types of 7-branes – and be sure that there are many types of such branes in F-theory and they usually produce cool enhanced non-Abelian symmetries and gauge fields that are confined to the singular branes.
I plan to write an introduction on F-theory, elliptic curves, etc. sometime in the future.
Shapes of torus and modular group
Reviewed by DAL
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April 11, 2013
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Reviewed by DAL
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April 11, 2013
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