An automorphic form is, in the simplest sense, like a trigonometric function. Trigonometric functions are inescapable in both mathematics and physics, so it makes sense that we would see generalizations of them in physics applications...
Strung Out on Automorphic Forms
Holley Friedlander
Dickinson College
Automorphic forms are highly symmetric functions that one typically does not encounter until graduate coursework in mathematics. Though understanding automorphic forms requires concepts from multiple areas of mathematics, this intricacy is part of their appeal. In a single object, automorphic forms link number theory, group theory, analysis, and as we will see, physics!
Valuable tools
Would you like to win a million dollars? Then you may want to learn about automorphic forms. Mathematicians use automorphic forms as analytic tools to, for example, study questions about the distribution of prime numbers. For instance, many attempts to solve the Riemann hypothesis, one of the Clay Mathematics Institute's [million dollar] Millennium Prize Problems, exploit automorphic forms. Though they have wide-ranging applications in number theory, I would like to tell you how automorphic forms arise in the context of string theory from physics.
A theory of everything
Let's start with physics. Physics is the study of how objects in the universe move and interact, and how forces influence these interactions. Physicists have identified four fundamental forces: gravity, electromagnetism, strong interaction, and weak interaction. There are two modern theories that describe how these forces govern interactions of matter: Einstein's theory of general relativity and quantum field theory. General relativity explains gravity as the curvature of space and time. It helps us understand how large objects, like planets and galaxies, interact in the universe.
Earth’s gravitational field bending space and time
NASA, Public domain, via Wikimedia Commons
Quantum field theory deals with electromagnetism, the strong force, and the weak force. It says these forces have associated particles that we can treat as substances spread throughout space called fields. This is the standard model of particle physics that helps us understand interactions of matter on a subatomic level.
A simulation of a proton-proton collision
CMS: Simulated Higgs to two jets and two electrons by Lucas Taylor (CC-BY-SA-4.0)
Experiments to verify general relativity and quantum field theory have thus far been in stunning agreement with the two theories. But there's a problem—quantum field theory and general relativity are incompatible. If you try to treat gravity like other forces in quantum field theory, things go completely awry.
Physicists are still searching for a theory of "quantum gravity."
String theory is a theoretical framework that attempts to create a "theory of everything." In string theory, vibrational modes of a string are used to represent elementary particles. Different vibrating states correspond to different elementary particles. For example, we can describe extremely small hypothetical particles of gravity called gravitons as one of the many possible vibrating states of a string.
Scientists want to know what happens when gravitons (strings) interact. As strings propagate through time and space, and transmit energy from one point to another, they join and split, and sweep out a two-dimensional surface called a world sheet. One view is that string theory is the study of how world-sheets embed in surrounding spacetime.
A world sheet embedding into a target spacetime
Drawing by K. Klinger-Logan
Now things start to get really weird. Our spacetime has four dimensions: three space dimensions and one time dimension. In string theory, spacetime has 10 dimensions! In order to make sense of the six extra dimensions or, in order to have an effective theory that can explain our observable world, we compactify (think of this as squeezing the extra dimensions together). We call this compactified space that takes into account assumptions of the theory the moduli space of string theory.
A symmetric space
To understand the structure of the moduli space, we need some group theory. Group theory is the mathematics of symmetry. A group is a set together with an operation (or rule) that assigns, to each pair of elements in the set, another element in the set. This operation must satisfy certain properties that impose (or encode) symmetries. A typical example of a group is the set of symmetries of a square, that is, motions you can perform on a square so that when you are finished the square takes up the same space it originally occupied. For example, we can rotate a square 90, 180, 270, or 360 degrees and the square will look the same as when we started. We can also flip the corners of the square across the vertical or horizontal midline or either of the two diagonals. This gives us eight motions. Another option is to combine these motions: we can rotate and then flip. It turns out that any combination of two of the eight motions is equivalent to one of the other eight. There are only eight possibilities! Together, these symmetries form a group.
In Type IIB string theory (one of the standard formalizations of string theory), the moduli space starts with a Lie group. A Lie group is a group that at small scales looks like a vector space where we can do calculus. Thankfully, many Lie groups have concrete descriptions. For example, the special linear group $SL_2(\mathbb{R})$ is a Lie group. This is the group of $2 \times 2$ real matrices with determinant one, where we combine elements using matrix multiplication. As a set,
$$SL_2(\mathbb{R})=\left\{\begin{bmatrix} a&b\\c&d\end{bmatrix}: a,b,c,d \in \mathbb{R}, ad-bc=1 \right\}.$$
Another example of a Lie group is a subgroup of the special linear group: the special orthogonal group $SO_2(\mathbb{R})$. This is the group of $2\times 2$ real orthogonal matrices with determinant one. A square matrix is orthogonal if its product with its transpose gives the identity. Thus we can write
\[SO_2(\mathbb{R})=\{A \in SL_2(\mathbb{R}): AA^T=I\}.\]
In general, the moduli space of string theory starts with a Lie group $G$ and is expressed using the notation $\Gamma \backslash G /K$, where $\Gamma$ and $K$ are certain subgroups of $G$. You can think of elements in $\Gamma\backslash G/K$ as "glued together" collections of elements from $G$ that satisfy the same symmetries with respect to the two subgroups $\Gamma$ and $K$. When we choose not to compactify our moduli space, the group $G$ is the special linear group $SL_2(\mathbb{R})$. When we compactify more, we see more complicated groups, such as the exceptional Lie group $E_8$, play the role of $G$. The picture below is an attempt to visualize the root system $E_8$ (the root system of a Lie group is a configuration of vectors in Euclidean space that characterizes the behavior of the group).
Jgmoxness, CC BY-SA 3.0, via Wikimedia Commons
Remember, we want to understand what happens when gravitons interact in the moduli space. A string scattering amplitude computes the probability of a certain outcome based on string interactions. These outcomes depend on various quantities including the string coupling, which measures the strength of a string-to-string interaction and something called the string scale, which depends on the length of a string. For the experts in the room, what I am describing what is known as the four-graviton scattering amplitude.
One common approach to compute a string scattering amplitude is called the low energy expansion. In this case, energies are measured in powers of the string scale. The analytic part of the expansion in $D$ dimensional spacetime looks like this:
\[\mathcal{A}^{(D)}(s,t,u;g)=\mathcal{E}_{(0,-1)}^{(D)}(g)\frac{\mathcal{R}^4}{\sigma_3} +\sum_{p=0}^{\infty}\sum_{q=0}^{\infty} \mathcal{E}_{(p,q)}^{(D)}(g)\,\sigma_2^p\sigma_3^q\mathcal{R}^4.\]
There's a lot going on here! The $g$ is an element of the moduli space; $s$, $t$, and $u$ are what are called Mandelstam variables that encode the energy, momentum, and angles of the particles. You may have noticed that $s$, $t$, and $u$ do not appear on the right side of the equation. They are hidden in $\sigma_2$ and $\sigma_3$, which can be written in terms of these variables. Finally, the $\mathcal{R}^4$ denotes a particular combination of what are called polarization tensors. Here is the takeaway: To compute the scattering amplitude we must understand the coefficient functions $\mathcal{E}^{(D)}_{(p,q)} (g)$. These are functions whose domain is the moduli space, which we now know is highly symmetric. This brings us to automorphic forms.
From amplitudes to automorphic forms
An automorphic form is, in the simplest sense, like a trigonometric function. Trigonometric functions are inescapable in both mathematics and physics, so it makes sense that we would see generalizations of them in physics applications! One thing we learn early on about trigonometric functions, like our familiar friends sine and cosine, is that they are periodic. Periodic functions have translational symmetries: If you shift the graph of a periodic function a horizontal distance $P$, called the period, the overall shape of the graph is unchanged. For a function $f$ with period $P$, we can write this as
\[f(x+P)=f(x).\]
In the case of the sine function, the period $P=2\pi$—the value of the sine function at any real number is equal to a value at a real number between $0$ and $2\pi$. Automorphic forms are like all periodic functions in that they satisfy a collection of symmetries, but the symmetries of an automorphic form are determined by a group.
To get a feel for how an automorphic form behaves, it is easiest to start with an example. A standard example of an automorphic form is the non-holomorphic Eisenstein series on $SL_2(\mathbb{R})$. This is actually a function on the upper half-plane, which we denote by $\mathcal{H}$. A point in $\mathcal{H}$ can be written as $z=x+iy$, where $x$ and $y$ are real numbers and $y$ is positive. Or we could say that $z$ is a complex number with positive imaginary part.
You may be wondering why a function on the upper half-plane would be called an Eisenstein series on $SL_2(\mathbb{R})$. It turns out that the upper half-plane is isomorphic (equivalent) to the coset space $SL_2(\mathbb{R})/SO_2(\mathbb{R})$. Fittingly, the elements of this coset space are called cosets, which you can think of as glued together collections of elements of $SL_2(\mathbb{R})$. We put matrices $A$ and $B$ in $SL_2(\mathbb{R})$ in the same coset if there is a matrix $O$ in $SO_2(\mathbb{R})$ satisfying $A=BO$.
We can now, finally, define our non-holomorphic Eisenstein series $E_s(z)$. For $z=x+iy$ in $\mathcal{H}$,
\[E_s(z)=\frac{1}{2}\sum_{\substack{m,n \in \mathbb{Z}\\(m,n)=1}} \frac{y^s}{|mz+n|^{2s}}.\]
The infinite sum is over all pairs of relatively prime integers $m$ and $n$, meaning $m$ and $n$ should not have any common factor. The notation $|mz+n|$ denotes the magnitude of the complex number $mz+n$, which tells us, if we plot $mz+n$ in the plane, how far it is from the origin. You probably noticed the subscript $s$—this is a complex parameter. The sum converges absolutely when the real part of $s$ is greater than one, meaning, in those cases there is an actual (complex) value that we can associate to the infinite sum. As you can see, it is helpful to know some complex analysis to fully understand this automorphic form. If this is all new to you, the good news is we can appreciate the symmetries of the Eisenstein series without totally comprehending its definition.
The group that determines the symmetries of $E_s(z)$ is $SL_2(\mathbb{Z})$, the group of $2\times 2$ integer matrices with determinant one:
\[SL_2(\mathbb{Z})=\left\{\begin{bmatrix}a&b\\c&d\end{bmatrix}:a,b,c,d \in \mathbb{Z},ad-bc=1\right\}.\]
Elements of $SL_2(\mathbb{Z})$ give transformations of the upper half-plane. The mathematical term is to say the group $SL_2(\mathbb{Z})$ acts on the upper half-plane. For a matrix $\gamma=\begin{bmatrix}a&b\\c&d\end{bmatrix}$ in $SL_2(\mathbb{Z})$ and a point $z$ in $\mathcal{H}$, we define
\[\gamma z=\frac{az+b}{cz+d}.\]
For example, if $\gamma=\begin{bmatrix} 1&1\\0&1\end{bmatrix}$ then $\gamma z=z+1$.
Just as the sine function is invariant under translations that take a real number $x$ to another real number $x+2\pi$, the Eisenstein series $E_s(z)$ is invariant under these transformations that, given a matrix $\gamma$ in $SL_2(\mathbb{Z})$, send a point $z$ in the upper half-plane to another point $\gamma z$ in the upper half-plane. For any $\gamma \in SL_2(\mathbb{Z})$, we have
\[E_s\left(\gamma z\right)=E_s(z).\]
If $\gamma=\begin{bmatrix} 1&1\\0&1\end{bmatrix}$, this says that $E_s(z+1)=E_s(z).$ In other words, the Eisenstein series is periodic! But this is only part of the story.
For a better understanding of the symmetries of $E_s(z)$, we need to understand the structure of $SL_2(\mathbb{Z}).$ The group $SL_2(\mathbb{Z})$ is generated by two elements:
\[T=\begin{bmatrix} 1&1\\0&1\end{bmatrix}
\mbox{ and } S=\begin{bmatrix} 0&-1\\1&0\end{bmatrix}.\] This means every one of the infinitely many elements in $SL_2(\mathbb{Z})$ can be written as the product of some number of $S$'s and $T$'s. In terms of transformations of the upper half-plane, the element $T$ corresponds to horizontal translation by one unit and $S$ corresponds to inversion in the unit circle followed by reflection about the $y$-axis. The fundamental domain for this action is the region in the upper half-plane containing all points whose real part is between $-1/2$ and $1/2$ and whose distance from the origin is greater than or equal to one (the fundamental domain is gray in the picture below). In other words, every point in the upper half-plane can be reached starting from a point in the fundamental domain and acting with some $\gamma$ in $SL_2(\mathbb{Z})$.
Fundamental domain for $SL_2(\mathbb{Z})$ acting on the upper half-plane
Fropuff, CC BY-SA 3.0, via Wikimedia Commons
Because $E_s(z)$ is invariant under the action of $SL_2(\mathbb{Z})$, the domain of $E_s(z)$ is effectively the coset space $SL_2(\mathbb{Z})\backslash \mathcal{H}$. We think of this coset space as one where we have glued together certain points in $\mathcal{H}$. In this case, we put two points in $\mathcal{H}$ in the same coset if there is an element of $SL_2(\mathbb{Z})$ whose associated transformation moves us from one point to the other. Based on our previous discussion, we can rewrite $SL_2(\mathbb{Z})\backslash \mathcal{H}$ as the (double) coset space $SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R})/SO_2(\mathbb{R})$.
Now we are ready to generalize. Given a Lie group $G$, an automorphic form on $G$ is a function on a coset space $G/K$, that
- satisfies a symmetry (invariance) under the action of a subgroup $\Gamma$ on $G/K$;
- satisfies certain differential equations; and
- satisfies some growth conditions.
A few notes. First, the subgroups $\Gamma$ and $K$ have to meet specific conditions in relation to $G$. Second, condition 1) effectively says that the domain of our automorphic form is $\Gamma\backslash G/K$. Third, we make a distinction between an automorphic function, which need only satisfy this symmetry condition 1), and an automorphic form, which also satisfies the differential and growth conditions 2) and 3).
We already saw that $E_s(z)$ satisfies the first condition in the definition of an automorphic form for $G=SL_2(\mathbb{R})$, $K=SO_2(\mathbb{R})$ and $\Gamma=SL_2(\mathbb{Z})$, but what about the other two conditions? The second condition says that $E_s(z)$ is an eigenfunction for the Laplacian $\Delta=-y^2(\partial_x^2+\partial_y^2)$ on $\mathcal{H}$. In general, a function $f$ is an eigenfunction for the differential operator $\Delta$ if $\Delta f=\lambda f$, where $\lambda$ is a scalar. In our case,
\[\Delta E_s(z)=\lambda_sE_s(z),\]
where $\lambda_s=s(1-s)$. I will leave it up to the eager reader to verify this. The third condition says that $E_s(z)$ satisfies some growth conditions as $y$ (the imaginary part of $z$) approaches infinity. To see that this is true, we have to look at the Fourier expansion of the Eisenstein series. That requires a lot more machinery, so I won't discuss it here.
Stringing it all together
What have we learned so far? In string theory, we model hypothetical particles of gravity called gravitons with vibrating strings. To understand how gravitons interact, we look at string scattering amplitudes. To compute these amplitudes, we must determine some coefficient functions, which we wrote as $\mathcal{E}^{(D)}_{(p,q)} (g)$. The domain of these coefficient functions is the highly symmetric moduli space of string theory. In order to be defined on this symmetric (double) coset space, the coefficient functions must be automorphic functions.
Remember, to get an automorphic form we need more than just the symmetry, we need the function to satisfy differential and growth conditions. It turns out that supersymmetry, which treats equations for force and matter as the same, imposes differential conditions on the coefficient functions. When $D=10$, the coefficient $\mathcal{E}^{(D)}_{(0,0)} (g)$ is the non-holomorphic Eisenstein series on $SL_2(\mathbb{Z})$!
Don't get too excited. Not all the coefficient functions are automorphic forms. As $p$ and $q$ increase, the differential conditions we get from supersymmetry get more complicated. But in the cases we understand so far, which are only for low-order coefficients such as when $p$ and $q$ are $0$ or $1$, we need to understand automorphic forms like Eisenstein series in order to find the coefficient functions and in turn, the scattering amplitudes. In particular, we need to understand automorphic forms associated to more complicated groups, like $E_8$.
And, if we understand automorphic forms for groups like $E_8$, then we can solve the Riemann hypothesis and win a million dollars! Just kidding.
Further reading
- “What is a Lie group?” American Institute of Mathematics
- “The Origins of the Universe: Quantum origins,” Centre for Theoretical Cosmology
- M. B. Green, S. D. Miller, J. G. Russo, and P. Vanhove, “Eisenstein series for higher-rank groups and string theory amplitudes,” Commun. Num. Theor. Phys. 4 (2010) 551-596, arXiv:1004.0163
- K. Martin, “A brief overview of modular and automorphic forms”