6. See Dayna Mason’s Flickr page: http://www.flickr.com/photos/daynoir
7. This gauge group SU(3) should not be confused with another group SU(3) discussed in Chapter 2, which was used by Gell-Mann and others to classify elementary particles (it is called the “flavor group”). Gauge group SU(3) has to do with a characteristic of quarks called “color.” It turns out that each quark can have three different colors, and gauge group SU(3) is responsible for changing those colors. Because of that, the gauge theory describing the interaction of quarks is called quantum chromodynamics. David Gross, David Politzer, and Frank Wilczek were awarded a Nobel Prize for their stunning discovery of the so-called asympthotic freedom in quantum chromodynamics (and other non-abelian gauge theories), which helped explain quarks’ mysterious behavior.
8. D.Z. Zhang, C.N. Yang and contemporary mathematics, Mathematical Intelligencer, vol. 15, No. 4, 1993, pp. 13–21.
9. Albert Einstein, Geometry and Experience, Address to the Prussian Academy of Sciences in Berlin, January 27, 1921. Translated in G. Jeffrey and W. Perrett, Geometry and Experience in Sidelights on Relativity, Methuen, 1923.
10. Eugene Wigner, The unreasonable effectiveness of mathematics in the natural sciences, Communications on Pure and Applied Mathematics, Vol. 13, 1960, pp. 1–14.
11. C. Montonen and D. Olive, Magnetic monopoles as gauge particles? Physics Letters B, vol. 72, 1977, pp. 117–120.
12. P. Goddard, J. Nuyts, and D. Olive, Gauge theories and magnetic charge, Nuclear Physics B, vol. 125, 1977, pp. 1–28.
13. Se is the set of complex one-dimensional representations of the maximal torus of G, and Sm is the fundamental group of the maximal torus of G. If G is the circle group, then its maximal torus is the circle group itself, and each of these two sets is in one-to-one correspondence with the set of integers.
Chapter 17. Uncovering Hidden Connections
1. The space M(X,G) may be described in several ways; for example, as the space of solutions of a system of differential equations on X first studied by Hitchin (see the article in endnote 19 below for more details). A description that will be useful to us in this chapter is that M(X,G) is the moduli space of representations of the fundamental group of the Riemann surface S in the complexification of the group G (see endnote 10 to Chapter 15). This means that such a representation is assigned to each point of M(X,G).
2. See the video of Hitchin’s lecture at the Fields Institute: http://www.fields.utoronto.ca/video-archive/2012/10/108-690
3. Here I am referring to the recent work of Ngô Bao Châu on the proof of the “fundamental lemma” of the Langlands Program. See, for example, this survey article: David Nadler, The geometric nature of the fundamental lemma, Bulletin of the American Mathematical Society, vol. 49, 2012, pp. 1–50.
4. Recall that in sigma model, everything is computed by summing over all maps from a fixed Riemann surface ∑ to the target manifold S. In string theory, we make one more step: in addition to summing over all maps from a fixed ∑ to S, as we normally do in the sigma model, we also sum up further over all possible Riemann surfaces ∑ (the target manifold S remains fixed throughout – this is our space-time). In particular, we sum over the Riemann surfaces of arbitrary genus.
5. For more on superstring theory, see Brian Greene, The Elegant Universe, Vintage Books, 2003; The Fabric of the Cosmos: Space, Time, and the Texture of Reality, Vintage Books, 2005.
6. For more on Calabi-Yau manifolds and their role in superstring theory, see Shing-Tung Yau and Steve Nadis, The Shape of Inner Space, Basic Books, 2010, Chapter 6.
7. A torus also has two continuous parameters: essentially, the radii R1 and R2 that we discuss in this chapter, but we will ignore them for the purpose of this discussion.
8. One resolution that has been actively discussed recently is the idea that each of these manifolds gives rise to its own universe with its own physical laws. This is then coupled with a version of the anthropic principle: our universe is selected among them by the fact that physical laws in it allow for intelligent life (so that the question “why is our universe like this?” could be asked). However, this idea, dubbed “string theory landscape” or “multiverse,” has been met with a lot of skepticism on both scientific and philosophical grounds.
9. Many interesting properties of quantum field theories in various dimensions have been discovered or elucidated by connecting these theories to superstring theory, using dimensional reduction or studying branes. In a sense, superstring theory has been used as a factory for producing and analyzing quantum field theories (mostly, supersymmetric). For example, this way one obtains a beautiful interpretation of the electromagnetic duality of four-dimensional supersymmetric gauge theories. So, even though we don’t know yet whether superstring theory can describe the physics of our universe (and still don’t fully understand what superstring theory is), it has already produced many powerful insights into quantum field theory. It has also led to numerous advances in mathematics.
10. The dimension of the Hitchin moduli space M(X,G) is equal to the product of the dimension of the group G (which is the same as the dimension of LG) and (g−1), where g denotes the genus of the Riemann surface X.
11. For more on branes, see Lisa Randall, Warped Passages: Unraveling the Mysteries of the Universe’s Hidden Dimensions, Harper Perennial, 2006; especially, Chapter IV.
12. More precisely, the A-branes on M(X,G) are objects of a category, the concept we discussed in Chapter 14. The B-branes on M(X,LG) are objects of another category. The statement of homological mirror symmetry is that these two categories are equivalent to each other.
13. Anton Kapustin and Edward Witten, Electric-magnetic duality and the geometric Langlands Program, Communications in Number Theory and Physics, vol. 1, 2007, pp. 1–236.
14. For more on the T–duality, see Chapter 7 of the book by Yau and Nadis referenced in endnote 6.
15. For more on the SYZ conjecture, see Chapter 7 of the book by Yau and Nadis referenced in endnote 6.
16. More precisely, each fiber is the product of n circles, where n is an even natural number, so it is an n-dimensional analogue of a two-dimensional torus. Note also that the dimension of the base of the Hitchin fibration and the dimension of each toric fiber will always be equal to each other.
17. In Chapter 15, we discussed a different construction, in which automorphic sheaves were obtained from representations of Kac–Moody algebras. It is expected that the two constructions are related, but as of the time of writing, this relation was still unknown.
18. Edward Frenkel and Edward Witten, Geometric endoscopy and mirror symmetry, Communications in Number Theory and Physics, vol. 2, 2008, pp. 113–283, available online at http://arxiv.org/pdf/0710.5939.pdf
19. Edward Frenkel, Gauge theory and Langlands duality, Astérisque, vol. 332, 2010, pp. 369–403, available online at http://arxiv.org/pdf/0906.2747.pdf
20. Henry David Thoreau, A Week on the Concord and Merrimack Rivers, Penguin Classics, 1998, p. 291.
Chapter 18. Searching for the Formula of Love
1. C.P. Snow, The Two Cultures, Cambridge University Press, 1998.
2. Thomas Farber and Edward Frenkel, The Two-Body Problem, Andrea Young Arts, 2012. See http://thetwobodyproblem.com/ for more details.
3. Michael Harris, Further investigations of the mind–body problem, a chapter from an upcoiming book, available online at http://www.math.jussieu.fr/~harris/MindBody.pdf
4. Henry David Thoreau, A Week on the Concord and Merrimack Rivers, Penguin Classics, 1998, p. 291.
5. E.T. Bell, Men of Mathematics, Touchstone, 1986, p. 16.
6. Robert Langlands, Is there beauty in mathematical theories?, in The Many Faces of Beauty, ed. Vittorio Hösle, University of Notre Dame Press, 2013, available online at http://publications.ias.edu/sites/default/files/ND.pdf
7. Yuri I. Manin, Mathematics as Metaphor: Selected Essays, American Mathematical Society, 2007, p. 4.
8. Philosophers have debated the ontology
of mathematics for centuries. The point of view that I advocate in this book is usually referred to as mathematical Platonism. Note however that there are different kinds of Platonism, and there are also other philosophical interpretations of mathematics. See, for example, Mark Balaguer, Mathematical Platonism, in Proof and Other Dilemmas: Mathematics and Philosophy, Bonnie Gold and Roger Simons (eds.), Mathematics Association of America, pp. 179–204, and references therein.
9. Roger Penrose, The Road to Reality, Vintage Books, 2004, p. 15.
10. Ibid., pp. 13–14.
11. Kurt Gödel, Collected Works, volume III, Oxford University Press, 1995, p. 320.
12. Ibid., p. 323.
13. Roger Penrose, Shadows of the Mind, Oxford University Press, 1994, Section 8.47.
14. In the landmark Gottschalk v. Benson decision, 409 U.S. 63 (1972), the U.S. Supreme Court stated (quoting earlier cases before the court): “a scientific truth, or the mathematical expression of it, is not a patentable invention.... A principle, in the abstract, is a fundamental truth; an original cause; a motive; these cannot be patented, as no one can claim in either of them an exclusive right.... He who discovers a hitherto unknown phenomenon of nature has no claim to a monopoly of it which the law recognizes.”
15. Edward Frenkel, Andrey Losev, and Nikita Nekrasov, Instantons beyond topological theory I, Journal of the Institute of Mathematics of Jussieu, vol. 10, 2011, 463–565; there is a footnote in the article explaining that formula (5.7) played the role of “formula of love” in Rites of Love and Math.
16. We consider the supersymmetric quantum mechanical model on the sphere (denoted here by ) and the correlation function of two observables, denoted by F and ω. This correlation function is defined in our theory as the integral appearing on the left-hand side of the formula. However, our theory also predicts a different expression for it: a sum over the “intermediate states” appearing on the right-hand side. Consistency of our theory requires that the two sides be equal to each other. And indeed they are; that’s what our formula says.
17. Le Monde Magazine, April 10, 2010, p. 64.
18. Laura Spinney, Erotic equations: Love meets mathematics on film, New Scientist, April 13, 2010, available online at http://ritesofloveandmath.com
19. Hervé Lehning, La dualité entre l’amour et les maths, Tangente Sup, vol. 55, May–June 2010, pp. 6–8, available online at http://ritesofloveandmath.com
20. We used the poem To the Many by Anna Akhmatova, a great Russian poet of the first half of the twentieth century.
21. Norma Farber, A Desperate Thing, The Plowshare Press Incorporated, 1973, p. 21.
22. Einstein’s letter to Phyllis Wright, January 24, 1936, as quoted in Walter Isaacson, Einstein: His Life and Universe, Simon & Schuster, 2007, p. 388.
23. David Brewster, Memoirs of the Life, Writings, and Discoveries of Sir Isaac Newton, vol. 2, Adamant Media Corporation, 2001 (reprint of a 1855 edition by Thomas Constable and Co.), p. 407.
Epilogue
1. Edward Frenkel, Robert Langlands, and Ngô Bao Châu, Formule des Traces et Fonctorialité: le Début d’un Programme, Annales des Sciences Mathématiques du Québec 34 (2010) 199–243, available online at http://arxiv.org/pdf/1003.4578.pdf
Edward Frenkel, Langlands Program, trace formulas, and their geometrization, Bulletin of AMS, vol. 50 (2013) 1–55, available online at http://arxiv.org/pdf/1202.2110.pdf
Glossary of Terms
Abelian group. A group in which multiplication of any two elements does not depend on the order in which they are multiplied. For example, the circle group.
Automorphic function. A particular kind of function that appears in harmonic analysis.
Automorphic sheaf. A sheaf that replaces an automorphic function in the geometric Langlands relation in the right column of Weil’s Rosetta stone.
Category. An algebraic structure comprised of “objects” and “morphisms” between any pair of objects. For instance, vector spaces form a category, and so do sheaves on a manifold.
Circle. A manifold that may be described as the set of all points on the plane equidistant from a given point.
Circle group. The group of rotations of any round object, such as a round table. It is a circle with a special point, the identity element of this group. The circle group is the simplest example of a Lie group.
Complex number. A number of the form , where a and b are two real numbers.
Composition (of two symmetries). The symmetry of a given object obtained by applying two symmetries of that object one after another.
Correspondence. A relation between objects of two different kinds, or a rule that assigns objects of one kind to objects of another kind. For example, a one-to-one correspondence.
Cubic equation. Equation of the form P(y) = Q(x), where P(y) is a polynomial of degree two and Q(x) is a polynomial of degree three. An example, which is studied in detail in this book, is the equation y2 + y = x3 − x2.
Curve over a finite field. An algebraic object comprised of all solutions of an algebaric equation in two variables (such as a cubic equation) with values in a finite field of p elements and all of its extensions.
Dimension. The number of coordinates needed to describe points of a given object. For example, a line and a circle have dimension one, and a plane and a sphere have dimension two.
Duality. Equivalence between two models (or theories) under a prescribed exchange of parameters and objects.
Fermat’s Last Theorem. The statement that for any natural number n greater than 3, there are no natural numbers x, y, z such that xn + yn = zn.
Fibration. Suppose we have two manifolds M and B, and a map from M to B. For any point in B, we have the set of points in M mapping to this point, called the “fiber” over this point. M is called a fibration (or a fiber bundle) over the “base” B if all of these fibers can be identified with each other (and each point in B has a neighborhood U whose preimage in M can be identified with the product of U and a fiber).
Finite field. The set of natural numbers between 0 and p−1, where p is a prime number, or its extension obtained by adjoining solutions of a polynomial equation in one variable.
Function. A rule that assigns a number to each point of a given set or manifold.
Fundamental group. The group of all continuous closed paths on a given manifold starting and ending at a given point.
Galois group. The group of symmetries of a number field preserving the operations of addition and multiplication.
Gauge group. A Lie group that appears in a given gauge theory and determines, in particular, the particles and the interactions between them within that theory.
Gauge theory. A physical model of a particular kind, describing certain fields and interactions between them. There is such a theory (or model) for any Lie group, called a gauge group. For example, the gauge theory corresponding to the circle group is the theory of electromagnetism.
Group. A set with an operation (variably called composition, addition, or multiplication) that assigns an element of this set to any pair of elements. (For example, the set of all integers with the operation of addition.) This operation must satisfy the following properties: the existence of the identity element, the existence of the inverse of each element, and associativity.
Harmonic analysis. A branch of mathematics studying decomposition of functions in terms of harmonics, such as the sine and cosine functions.
Hitchin moduli space. The space (or manifold) whose points are representations of the fundamental group of a given Riemann surface in a given Lie group.
Integer. A number that is either a natural number, or 0, or the negative of a natural number.
Kac–Moody algebra. The Lie algebra of the loop group of a given Lie group, extended by an extra line.
Langlands dual group. A Lie group assigned to any given Lie group G by a special procedure. It is denoted by LG.
Langlands relation (or Langlands correspondence). A rule assigning an automorphic function (or an
automorphic representation) to a representation of a Galois group.
Lie algebra. The tangent space to a Lie group at the point corresponding to the identity element of this group.
Lie group. A group that is also a manifold, such that the operation in the group gives rise to a smooth map.
Loop. A closed curve, such as a circle.
Manifold. A smooth geometric shape such as a circle, a sphere, or the surface of a donut.
Map from one set (or manifold), M, to another set (or manifold), N. A rule that assigns a point of N to each point of M. (A map is sometimes referred to as a mapping.)
Modular form. A function on the unit disc satisfying special transformation properties under a subgroup of the group of symmetries of the disc (called the modular group).
Natural number. Number 1 or any number obtained by adding 1 to itself several times.
Non-abelian group. A group in which multiplication of two elements depends in general on the order in which they are multiplied. For example, the group SO(3).
Number field. A numerical system obtained by adjoining to rational numbers all solutions of a finite collection of polynomials in one variable whose coefficients are rational numbers.
Polynomial in one variable. An expression of the form anxn + an−1xn−1 + ... + a1x + a0, where x is a variable and an,an−1,...,a1,a0 are numbers. Polynomials in several variables are defined similarly.
Polynomial equation. An equation of the form P = 0, where P is a polynomial in one or more variables.
Prime number. A natural number that is not divisible by any natural number other than itself and 1.
Representation of a group. A rule that assigns a symmetry of a vector space to each element of a given group so that some natural properties are satisfied. More generally, a representation of a group G in another group H is a rule that assigns an element of H to each element of G, so that some natural properties are satisfied.
Quantum field theory. This term may refer to one of two things. First, it may be a branch of physics that studies models of interactions of quantum particles and fields. Second, it may be a particular model of this type.
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