Topology is the branch of geometry that deals with large-scale features of shapes. One cliché is that a topologist cannot distinguish a doughnut from a coffee cup: if a coffee cup were made of rubber, one could continuously deform it to a doughnut without tearing. A geometer, equipped with precision tools, can measure local quantities (distances, curvature) to distinguish the coffee cup from the doughnut. A topologist, seemingly handicapped by defective eyes, can only discern that each has one hole, so at least can distinguish both from a two-holed pretzel. But, after all, a topologist is a geometer too, and the lack of close vision can reveal a forest otherwise obscured by trees. For many problems, that global vision provides crucial insights. This has long been true internally in mathematics: topological ideas play an important role in analysis, algebra, and many other areas. In recent years, topological ideas have found applications to many problems outside of mathematics: data analysis, biology, and robotics to name just a few. My concern in this article is a particular application of topology to quantum physics.
The problem described here—the classification of phases of matter—has great current relevance. Beyond that, the story I tell is one small illustration of the many wonders of mathematics: the abstract and artistic impulses, which guide the internal development of mathematical ideas, yield theorems with unanticipated powerful applications to scientific and technological problems far removed from the original source of and inspiration for those ideas.
A simple example from geometry
Before tackling quantum physics, let’s warm up with a toy example. Imagine an infinite line L on which we position n numbered balls, where n is a fixed counting number n = 1, 2, . . . . (The “balls” are actually idealized points with no extent, but of course in the drawings below the points have a finite size.) What we want to study are all possible configurations of these balls. In the simplest case, we have just a single ball, so we know everything by knowing its position on the line L. What if we have n = 2 balls? In that case, a configuration of two balls is a pair of points on the line L. What does the collection of all such pairs of points look like? Answer: a two-dimensional plane. This becomes more familiar if, following Descartes, we label each point of L with a real number x. Therefore, a pair of points on L corresponds to an ordered pair (x,y) of real numbers. The collection of all pairs is the usual x-y plane. It is clear that configurations of n = 3 points on L correspond to triples (x,y,z) and you can go on up from there. The space S of configurations of n balls on L is an example of a moduli space. A point of S has a geometric meaning: it labels a configuration of points on the line L. Significantly, S is itself a space, not just a discrete collection of points, so for example, we can move continuously along paths in S. Such paths correspond to motions of configurations of points, as illustrated in Figure 1. As the red point in the moduli space S moves northwest, the corresponding two balls on the line L approach each other, eventually collide, and then separate again, but now with the 1-ball to the left of the 2-ball.
Figure 1. Motion in a moduli space
What is the shape of all possible configurations? We have already answered that for n = 1, in which case S is a one-dimensional line, and for n = 2, in which case S is a two-dimensional plane. Putting on a topologist’s hat, we can ask if every pair of configurations can be continuously deformed one to the other, or better ask what the set of deformation classes of configurations is. As we have seen, a deformation is a path in S—two configurations are in the same deformation class if and only if the corresponding points in S can be joined by a path. We use the symbol ‘π0(S)’ for the set of path components of S: two points are in the same path component if and only if they can be joined by a path. In our case there is a single deformation class: any two points in S can be joined by a (straight line) path. Try to envision this in the two-dimensional case depicted in Figure 1. Continuously contract the entire plane to a single point—simultaneously move all possible configurations to a single fixed configuration, say one where the two red balls both lie at the same point of the line L.
Figure 2. Moduli space of gapped configurations
So far we have no interesting topology. Now let’s impose a “gap condition.” Namely, let’s forbid the balls (which remember are points) from coinciding. To construct the moduli space S' of gapped configurations, we begin with the moduli space S of all configurations and remove the part that corresponds to configurations in which at least two balls coincide. For n = 2 balls the excluded set is a line in the plane S: the diagonal blue line in Figure 2 that represents two coincident points of L. The complementary space S' has two path components. There is a simple invariant that distinguishes the two path components: the sequence of numbers of the balls on the line L read from left to right. In the case pictured in Figure 2, that sequence is (2 1); the other path component corresponds to the sequence (1 2). This sequence, called a permutation, is a deformation invariant of the gapped configuration, and it is a complete invariant in that it completely determines the deformation class of the configuration.
In summary, a metric geometer tracks the precise positions of distinct points on the line, whereas a deformation topologist only sees the permutation defined by the ordering of the points.
Moduli spaces in quantum mechanics
The moduli space of interest in our story comes not from geometry but from quantum physics. Quantum systems have parameters—one can continuously adjust temperatures, pressures, magnetic fields, etc.—and so we can envision a moduli space Q whose points represent quantum mechanical systems. As developed in the 1920s and 1930s, in large part by former IAS Professor John von Neumann, a quantum system is described by: a Hilbert space of quantum states; a Hamiltonian operator, which measures the energy; and a group G of symmetries. The Hamiltonian has a spectrum of possible energies, and again we impose a gap condition, this time the condition that there be a gap in the energy spectrum directly above the minimal energy. The situation is pictured schematically in Figure 3: a point of Q represents the entirety of data that defines a quantum system. There is a subspace Q' of Q whose points represent gapped quantum systems. In Figure 3, Q' is the subspace enclosed by the yellow curves; the blue curve depicts gapless quantum systems that lie between the gapped ones. Here is the central problem:
Compute the set π0(Q') of deformation classes of gapped quantum systems.
To obtain a meaningful problem, we fix some discrete parameters that describe the system: the dimension d of space, the symmetry group G, etc.
Figure 3. Moduli space of gapped quantum systems
The classic example of deformation classes of quantum mechanical systems is pictured in Figure 4, which describes the state of water at different temperatures and pressures. In this picture, the liquid region is connected by a path to the vapor (gas) region, so the liquid and gas states are in the same deformation class. In this context, a deformation class is called a phase of matter. There is a distinct phase: the solid phase (ice). Much more recently, physicists have discovered new phases of matter (unrelated to water): quantum Hall states, topological insulators, and topological superconductors, among others. There is much excitement in both theoretical and experimental circles. Possible robust new materials that exhibit these exotic states of matter would find immediate applications, for example, to quantum computation. Whereas traditionally phases of matter are distinguished by their symmetry structure—the Landau paradigm of phase transitions—these newer phases are differentiated via topological invariants. There are many approaches that have been developed over the past decade. The importance of this work led to its being chosen as the topic for the Prospects in Theoretical Physics summer school held at the IAS and Princeton University in July.
Figure 4. Phase diagram of H2O
Transition to topological field theory
Intuition from physics suggests that the deformation class of a quantum system is encoded in its low-energy behavior. Fluctuations at high energy are not meant to change the topology. This resonates well with intuition from geometry. On a smooth curved space, called a manifold, which is equipped with a notion of distance and angle, one can carry out a generalized Fourier analysis and decompose functions (and differential forms) according to their generalized frequency. Only the functions of lowest frequency—those that are locally constant on the manifold—bear on the largescale topology; high-frequency functions detect small-scale geometric features. In quantum mechanics, one passes from frequency to energy, which suggests constructing a topological invariant from the low-energy part of a quantum theory.
Figure 5. A simple lattice system
The next step is a big leap. We assume that the low-energy part of the system is well approximated by a quantum field theory. This is not such a big leap if the original quantum system is itself a quantum field theory. But the systems of interest are discrete: the degrees of freedom live on a lattice and the Hamiltonian is a discrete sum. (Figure 5 illustrates the discrete nature of lattice systems.) Nonetheless, they often have continuum limits in which the degrees of freedom are fields: functions and other familiar objects from smooth geometry. The particular continuum limit we focus on is at low energy, where—without proof—we assume that there is a scale-independent quantum field theory. (Links between discrete and continuous quantum systems have been extensively studied by IAS Professor Tom Spencer.) The limit scales out the energy, and what is left has no scale. But now we impose the energy gap condition depicted in Figure 3. It implies that the quantum field theory has much more than scale invariance: it has topological invariance. In other words, it is a topological field theory, a particularly tractable species of quantum field theory introduced by IAS Professor Edward Witten in the late 1980s. If we jiggle the original quantum system just a bit, it is reasonable to suppose that this topological field theory also jiggles just a bit. So the topological field theory is a topological invariant of our original system, and can be used to detect if two systems are connected by a path. The intuition from physics suggests this low-energy description should precisely detect the deformation class.
Recall in our toy problem the ordering of the balls (Figure 2) is a complete deformation invariant of gapped configurations. Now for gapped quantum systems we have a concrete proposal for a complete deformation invariant: the low-energy topological field theory.
A mathematical framework for quantum field theory
We are closer to abstracting a mathematical problem from the physics classification question. But now we need to give a mathematical definition of the objects—topological quantum field theories—that we hope to classify. Almost since the beginning of quantum theory, there have been mathematical formulations capturing particular aspects of quantum field theory. In the late 1980s, a geometric axiom system for scaleinvariant quantum field theories emerged—many important contributions came from the Oxford school—and in particular topological theories became an object of intense mathematical study. (This geometric axiom system derives from Wick-rotated quantum field theory on compact manifolds and applies as well to scale-dependent theories, but that aspect is much less developed.) Many variations have been studied, particularly including strong forms of locality, and powerful theorems such as the Baez-Dolan-Hopkins-Lurie cobordism hypothesis have been proved. This body of mathematics is still under intense internal development, but there are already impressive applications to other parts of mathematics and back to physics.
Figure 6. QFT from a bordism point of view
We only need mention here the basic framework, which is depicted in Figure 6. The dimension of space is d, so that of spacetime is d +1. On the left of the figure are spaces Y0 and Y1 and an evolution X which begins at Y0 and ends at Y1. The field theory F outputs linear information from this diagram: vector spaces H0, H1 from the spaces, and a linear map F(X) between them from the evolution. This captures the quantum information in a geometric form. In a topological theory, the map F is invariant under deformations of the input data.
The input pictures are familiar in topology: they are bordisms as studied by Lev Pontryagin and René Thom in the 1940s and 1950s. This link to bordism goes beyond these pictures in a particular case: if the field theory is not only topological but also invertible. The invertibility refers to the natural composition law, which juxtaposes quantum systems. Invertible field theories, even if not topological, are a very special and particularly simple baby case of general quantum field theories, yet they arise in several important mathematical and physical contexts. My collaborators Michael Hopkins, Constantin Teleman, and I observed that for invertible topological theories the map F can be interpreted in the world of stable homotopy theory, an important and highly developed branch of topology. The key point is that bordisms depicted in Figure 6 have a moduli space that resides in stable homotopy theory—an “infinite loop space.” This particular bordism space is closely related to that introduced by Thom, and in fact is precisely the bordism space studied by Galatius-Madsen-Tillmann-Weiss (with mathematical motivations completely divorced from applications to physics).
Short-range entanglement and stable homotopy theory
We have already seen that gapped quantum systems, in particular gapped lattice systems, may have low-energy topological field theory approximations. The single hypothesis of a gap already makes possible the application of powerful mathematics. A second hypothesis—short-range entanglement as defined by Alexei Kitaev, Xiao-Gang Wen, and other condensed matter physicists—implies that the low-energy topological theory is invertible. One facet is the uniqueness of the ground state of the system on any compact space. Many gapped systems of interest satisfy this additional strong constraint. For these systems, our arguments show that the classification of phases becomes a question in stable homotopy theory! The techniques developed over many years by topologists— generalized cohomology theories, the Adams spectral sequence, and much more—are now brought to bear on specific computations of interest in physics: the computation of π0Q' for particular moduli spaces Q' of gapped quantum systems. Many computations have already been done and others are in progress. In ongoing work with Hopkins, we implement unitarity in this context and obtain strong agreement with a different approach to topological phases due to Kitaev, who arrives at stable homotopy theory directly from lattice systems. Work of Anton Kapustin and collaborators arrives at similar answers. Much work remains to reconcile the different viewpoints.
I have outlined the main ideas, which, in one approach, lead from phases of complicated quantum systems to problems in stable homotopy theory. I want to emphasize one of the key ingredients: the mathematical framework for quantum field theory depicted by the image in Figure 6. It has bordism built in and for that reason leads in the invertible case to bordism spaces in topology. It allows us to tackle the quantum physics problem head-on: with a geometric definition of a quantum field theory, we can directly classify these quantum systems with few inputs—dimension, symmetry group, etc. The success of this mathematical framework in this classification problem gives some hope that it captures essential features of scale-dependent quantum field theories as well, a topic for future investigations.
Recommended Viewing: Is the abstract mathematics of topology applicable to the real world? Topology is the only major branch of modern mathematics that wasn’t anticipated by the ancient mathematicians. Throughout most of its history, topology has been regarded as strictly abstract mathematics, without applications. However, illustrating Eugene Wigner’s principle of “the unreasonable effectiveness of mathematics in the natural sciences,” topology is now beginning to come up in our understanding of many different real-world phenomena. In a mini-symposium organized in May, Robert MacPherson, Hermann Weyl Professor in the School of Mathematics, described the history and pervasiveness of topology, Raúl Rabadán described how topology modifies our understanding of evolution and disease, and Randall Kamien discussed the relationship between topology and liquid crystals, like those in computer displays. Videos of the talks are available at https://video.ias.edu/mini-symposium-topology-2015.