Edward Witten

How to use this course

Edward Witten is a theoretical physicist whose work has had a larger influence on pure mathematics than that of any other physicist of his era. He rarely proves theorems in the strict sense. He finds the physical system in which a mathematical object lives, computes with the tools of quantum field theory, and predicts results that mathematicians then prove, often over many years. This course follows that pattern: each technical lesson states the physical idea, then the mathematics it produced.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations or proof sketches, and exercises with folded solutions. Where Witten’s argument is physical and the rigorous proof came from others, the lesson says so. The order is roughly chronological: supersymmetry and geometry (1981–1985), topological field theory (1988–1991), duality (1994–1998), and later work on Langlands duality and scattering amplitudes (2003–2007).

Timeline

Baltimore and Brandeis, 1951–1973

Edward Witten was born in Baltimore, Maryland, on 26 August 1951. His father, Louis Witten, was a theoretical physicist who worked on general relativity and gravitation, so physics was present at home from the start. Witten graduated from the Park School of Baltimore in 1968.

He did not go straight into science. At Brandeis University he studied history, with a minor in linguistics, and graduated in 1971. He was interested in politics and journalism. In 1972 he spent about six months working on George McGovern’s presidential campaign. He then spent one semester as a graduate student in economics at the University of Michigan before leaving.

This detour is a large part of the Witten legend, and it is easy to exaggerate. He had absorbed a good deal of mathematics and physics on his own, and once he chose science he moved very quickly. In 1973 he was admitted to Princeton in applied mathematics, an unusual entry for someone with a history degree, and he soon transferred to the physics department.

Princeton and Harvard, 1973–1980

Witten arrived in Princeton at a remarkable moment. In 1973 David Gross and his student Frank Wilczek, and independently David Politzer at Harvard, had discovered asymptotic freedom: the strong force weakens at short distances, which made quantum chromodynamics (QCD) a credible theory of quarks and gluons. Gross became Witten’s advisor. Witten received his M.A. in 1974 and his Ph.D. in 1976 with the thesis Some problems in the short distance analysis of gauge theories.

He spent 1976–77 as a postdoctoral fellow at Harvard and 1977–1980 as a Junior Fellow of the Harvard Society of Fellows, with a visit to Oxford in 1977–78, where Michael Atiyah and his colleagues were studying the mathematics of instantons. His papers from these years are on QCD and gauge theory. They include the description of baryons in the large-\(N\) limit of QCD (1979) and an explanation of the large mass of the \(\eta'\) meson through topological effects in the large-\(N\) limit, now known as the Witten–Veneziano formula (1979).

These papers already show the habits of his later career. He looks for the simplest limit in which a hard problem becomes tractable, whether large \(N\), weak coupling or supersymmetry. He also gives topology a central role in physics, which most physicists of the time did not.

Princeton professor, 1980–1987

In September 1980 Witten became a Professor of Physics at Princeton University. The next seven years produced a series of papers that connected supersymmetric physics with geometry: the positive energy theorem (1981), supersymmetry and Morse theory (1982), the Witten index (1982), the SU(2) anomaly (1982), and work on global anomalies and index theory. Lessons 04–06 cover them. In 1982 he received a MacArthur Fellowship.

In 1984 Michael Green and John Schwarz showed that type I superstring theory with gauge group \(\mathrm{SO}(32)\) is free of anomalies. Witten had studied anomalies in detail, with Luis Alvarez-Gaumé, and he immediately saw the significance. He threw himself into string theory, and many physicists followed. This was the first superstring revolution. In 1985 Witten, Philip Candelas, Gary Horowitz and Andrew Strominger showed how compactifying the heterotic string on a six-dimensional Calabi–Yau manifold could produce physics resembling the Standard Model. Calabi–Yau manifolds have been central to the interaction of physics and algebraic geometry ever since. In 1987 Green, Schwarz and Witten published the two-volume text Superstring Theory, and Witten moved to the Institute for Advanced Study.

Supersymmetry and Morse theory

For the general reader. The shape of a landscape constrains its hills, valleys and passes: a closed surface with no handles has at least one peak and one pit. Morse theory makes this precise. In 1982 Witten showed that the theorem is a statement about a quantum particle moving on the landscape, whose lowest energy states sit at the critical points. The paper gave mathematicians a new way to prove topological results by analysis.

Supersymmetric quantum mechanics

A supersymmetric quantum mechanics has a \(\mathbb Z/2\)-graded Hilbert space \(\mathcal H=\mathcal H_B\oplus\mathcal H_F\), an odd operator \(Q\) with \(Q^2=0\), and Hamiltonian \(H=QQ^\dagger+Q^\dagger Q\). Then \(H\ge0\), and a state has \(E=0\) if and only if \(Q\psi=Q^\dagger\psi=0\). Every eigenspace with \(E\gt0\) pairs bosons with fermions, because \(Q\) maps one to the other isomorphically.

The geometric model

For a compact Riemannian manifold \(M\), take \(\mathcal H=\Omega^\bullet(M)\) with \((-1)^F\) given by form degree, \(Q=d\), \(Q^\dagger=d^*\). Then \(H\) is the Hodge Laplacian \(\Delta=dd^*+d^*d\), the zero-energy states are the harmonic forms, and by Hodge theory their number in degree \(k\) is the Betti number \(b_k\). The Witten index is the Euler characteristic \(\chi(M)\).

Now let \(h\) be a Morse function and \(t\) a real parameter. Witten conjugated the differential:

Since \(d_t\) is conjugate to \(d\), its cohomology is still \(H^\bullet(M)\), so \(\Delta_t\) has \(b_k\) zero modes in degree \(k\) for every \(t\). For large \(t\) the potential \(t^2|dh|^2\) confines low-energy states near the critical points of \(h\). Near a critical point of index \(k\), in Morse coordinates \(h=\sum_i\pm\tfrac12x_i^2\), \(\Delta_t\) becomes a supersymmetric harmonic oscillator with exactly one ground state, and that state is a \(k\)-form.

Proof sketch. For large \(t\), the eigenvalues of \(\Delta_t\) that stay bounded as \(t\to\infty\) are, to leading order, the ground states of the local oscillators: \(m_k\) of them in degree \(k\). The span \(S^\bullet\) of the corresponding eigenforms is preserved by \(d_t\), so it is a finite complex with \(\dim S^k=m_k\) whose cohomology is \(H^\bullet(M)\). The strong Morse inequalities hold for any finite complex. Witten went further. Tunnelling between critical points of adjacent index, along gradient flow lines of \(h\), gives the matrix of \(d_t\) on \(S^\bullet\), which recovers the Morse–Smale–Witten complex. Andreas Floer built Floer homology on this idea.

The positive energy theorem

For the general reader. In general relativity the total mass of an isolated system, measured from far away, should never be negative if matter has positive energy locally. If it could be negative, empty space would be unstable. Richard Schoen and Shing-Tung Yau proved this in 1979 with a hard geometric argument. In 1981 Witten found a much shorter proof using spinors, inspired by supergravity.

A complete Riemannian 3-manifold \((M,g)\) is asymptotically flat if outside a compact set it is diffeomorphic to the complement of a ball in \(\mathbb R^3\), with \(g_{ij}=\delta_{ij}+O(r^{-1})\), \(\partial g=O(r^{-2})\), and integrable scalar curvature. Its ADM mass, in units with \(G=c=1\), is

Witten’s argument

The key identity is the Lichnerowicz formula for the Dirac operator \(D\) on spinors:

Choose a constant spinor \(\psi_0\) at infinity and solve \(D\psi=0\) with \(\psi\to\psi_0\). Integrating \(|D\psi|^2=0\) by parts over the region inside \(S_r\), the boundary term at infinity turns out to be a positive multiple of \(m|\psi_0|^2\):

If \(R\ge0\) the left side is non-negative, so \(m\ge0\). If \(m=0\), then \(\nabla\psi=0\) for every choice of \(\psi_0\), giving a full set of parallel spinors, so \(g\) is flat. The hard analytic step, solving \(D\psi=0\) with the right asymptotics, was made rigorous by Thomas Parker and Clifford Taubes in 1982. Every orientable 3-manifold is spin, so no extra hypothesis is needed in the physical case. In higher dimensions the spinor method requires \(M\) to be spin.

Anomalies and the index theorem

For the general reader. A symmetry of a classical theory can fail after quantization. Such a failure is an anomaly. If it affects a gauge symmetry, the theory is inconsistent. Witten showed that some anomalies are invisible to perturbation theory and can only be detected through the global topology of the space of fields. The tools he used, index theorems and spectral flow, became standard in both physics and geometry.

Zero modes and the index

For a Dirac operator \(D_A\) on a compact 4-manifold, coupled to a connection on a bundle in a representation \(R\) of the gauge group, the Atiyah–Singer index theorem gives \(\operatorname{ind}D_A=\int\hat A(M)\operatorname{ch}(E_R)\). On \(S^4\) with an \(\mathrm{SU}(2)\) instanton of charge \(k\), this equals \(2T(R)\,k\), where \(T(\text{fund})=\tfrac12\) and \(T(\text{adj})=2\). Each zero mode changes the fermion path integral, which is how instantons violate classically conserved charges.

Argument. A single Weyl doublet contributes \(\sqrt{\det D_A}\), well defined only up to sign. Follow a path of connections \(A_s\), \(0\le s\le1\), from \(A\) to its gauge transform \(A^g\), with \(g\) non-trivial in \(\pi_4\). Gluing the ends gives a connection on a bundle over \(S^4\times S^1\), and on the mapping torus, a five-manifold, the mod 2 index of the relevant real Dirac operator equals the number of pairs of eigenvalues of \(D_{A_s}\) that cross zero along the path. Witten computed this to be 1 for a single doublet. The square root therefore returns with the opposite sign. With an even number of doublets the signs cancel. In the Standard Model each generation contains four \(\mathrm{SU}(2)\) doublets, three coloured quark doublets and one lepton doublet, so the theory is consistent.

Gravitational anomalies and strings

In 1984 Alvarez-Gaumé and Witten computed the perturbative gravitational anomalies of chiral fields in \(4k+2\) dimensions. They found that type IIB supergravity in ten dimensions is anomaly-free through a cancellation between its chiral fields. Green and Schwarz then showed that in type I theory the gauge and gravitational anomalies can cancel only for a gauge group of dimension 496, through a counterterm now called the Green–Schwarz mechanism. In 1985 Witten’s paper Global gravitational anomalies extended the analysis to large diffeomorphisms, using the eta invariant of Atiyah, Patodi and Singer.

Topological field theory and the Jones polynomial

For the general reader. In 1984 Vaughan Jones discovered a new invariant of knots, a polynomial computed from any diagram of the knot. Nobody understood why it existed. Witten explained it in 1989: the Jones polynomial is the average value of a natural physical quantity in a three-dimensional quantum field theory whose answers do not depend on any distances or angles. Such topological field theories now organise large parts of low-dimensional topology.

Topological quantum field theory

In 1988 Witten wrote down a twisted version of \(N=2\) supersymmetric Yang–Mills theory on a four-manifold whose correlation functions are Simon Donaldson’s invariants of smooth four-manifolds. Atiyah abstracted the structure. A \((d+1)\)-dimensional TQFT assigns a finite-dimensional vector space \(Z(\Sigma)\) to each closed \(d\)-manifold \(\Sigma\) and a vector \(Z(M)\in Z(\partial M)\) to each \((d+1)\)-manifold, so that disjoint unions go to tensor products, reversing orientation gives the dual space, and gluing along a common boundary gives composition. In particular \(Z(\Sigma\times S^1)=\dim Z(\Sigma)\).

Chern–Simons theory

For a compact simple gauge group \(G\) and a connection \(A\) on a closed oriented 3-manifold \(M\), the Chern–Simons action is

It uses no metric. Under a large gauge transformation it changes by \(2\pi k\) times an integer winding number, so \(e^{iS}\) is gauge invariant only when the level \(k\) is an integer. For a knot \(K\subset M\) and a representation \(R\), the Wilson loop is \(W_R(K)=\operatorname{tr}_R\,P\exp\oint_KA\).

Derivation sketch. Cut \(S^3\) along a sphere meeting the knot in four points. The Hilbert space of the sphere with four marked points is two-dimensional, so any three vectors are linearly dependent. Three links that differ only inside a ball, \(L_+\), \(L_-\) and \(L_0\), give three such vectors, and the coefficients of the linear relation come from the braiding matrices of the WZW model. The result is the skein relation

which determines \(V\). Nicolai Reshetikhin and Vladimir Turaev gave a rigorous construction of these 3-manifold invariants in 1991 using quantum groups. The dimension of \(Z(\Sigma_g)\) for a closed surface of genus \(g\) is given by the Verlinde formula

Two-dimensional gravity

In 1990–91 Witten used topological gravity to conjecture that the generating function of intersection numbers on the moduli spaces \(\overline{\mathcal M}_{g,n}\) of curves satisfies the KdV hierarchy. Maxim Kontsevich proved this in 1992. The statement and proof are in the Kontsevich module.

Seiberg–Witten theory

For the general reader. In 1994 Nathan Seiberg and Witten solved exactly the low-energy behaviour of a supersymmetric version of the strong force. Their solution used a duality between electric and magnetic descriptions. Within weeks Witten turned the magnetic side into a new set of equations on four-dimensional spaces. They reproduced, with far less work, what Donaldson’s theory had needed years of hard analysis to achieve, and they proved new theorems almost at once.

The physics

In \(N=2\) supersymmetric \(\mathrm{SU}(2)\) Yang–Mills theory, the vacua form a complex line, the \(u\)-plane, with \(u=\langle\operatorname{tr}\phi^2\rangle\). The low-energy theory at each point is a \(\mathrm{U}(1)\) gauge theory determined by a holomorphic prepotential. Seiberg and Witten argued that it is encoded in the family of elliptic curves

in units where the dynamical scale is 1. The curve degenerates at \(u=\pm1\) and \(u=\infty\). At \(u=\infty\) the theory is weakly coupled. At \(u=\pm1\) a magnetic monopole or a dyon becomes massless, and the natural description is a \(\mathrm{U}(1)\) gauge theory coupled to a light monopole field, weakly coupled in the dual magnetic variables.

The monopole equations

Let \(X\) be a closed oriented Riemannian 4-manifold with a spin\(^c\) structure, spinor bundles \(S^\pm\) and determinant line bundle \(L\). The unknowns are a \(\mathrm{U}(1)\) connection \(A\) on \(L\) and a spinor \(\psi\in\Gamma(S^+)\):

Key estimate. The Weitzenböck formula for the spin\(^c\) Dirac operator is \(D_A^2\psi=\nabla_A^*\nabla_A\psi+\frac s4\psi+\frac12F_A^+\cdot\psi\), with \(s\) the scalar curvature. For a solution, \(F_A^+\cdot\psi=\tfrac12|\psi|^2\psi\). At a maximum of \(|\psi|^2\), \(\Delta|\psi|^2\ge0\), and pairing the formula with \(\psi\) gives \(\frac s4|\psi|^2+\frac14|\psi|^4\le0\). Hence

This uniform bound gives compactness. If \(s\gt0\) everywhere, it forces \(\psi=0\), so all solutions are reducible and, for \(b_2^+\ge2\), are avoided by a generic perturbation. Peter Kronheimer and Tomasz Mrowka used the equations in 1994 to prove the Thom conjecture: a smooth surface in \(\mathbb{CP}^2\) representing \(d\) times the class of a line, \(d\ge1\), has genus at least \((d-1)(d-2)/2\), the genus of a smooth algebraic curve. Taubes proved that on symplectic four-manifolds the Seiberg–Witten invariants equal certain Gromov invariants.

String dualities, M-theory, holography

For the general reader. By the early 1990s there were five consistent superstring theories in ten dimensions, an embarrassment for a supposed unique theory of everything. In March 1995 Witten argued that they are different limits of a single theory, with an eleventh dimension that appears only when the strings interact strongly. Three years later he gave the precise dictionary for Juan Maldacena’s proposal that a theory of gravity can be equivalent to an ordinary quantum field theory living on its boundary.

T-duality

A closed bosonic string on a circle of radius \(R\) has momentum number \(n\) and winding number \(w\). Its mass spectrum is

It is invariant under \(R\mapsto\alpha'/R\) with \(n\leftrightarrow w\): two different circles give the same physics. For superstrings, T-duality exchanges type IIA and type IIB. S-duality relates the type I and \(\mathrm{SO}(32)\) heterotic strings, and type IIB to itself under \(g_s\mapsto1/g_s\).

The evidence is the spectrum of BPS states, whose masses are fixed by supersymmetry and can be followed to strong coupling. Type IIA has D0-branes of mass \(1/(g_s\ell_s)\) and bound states of \(n\) of them with mass \(n/(g_s\ell_s)\). These form exactly the Kaluza–Klein tower of a circle of radius \(R_{11}=g_s\ell_s\), which decompactifies as \(g_s\to\infty\).

Holography

Maldacena conjectured in 1997 that type IIB string theory on \(\mathrm{AdS}_5\times S^5\) is equivalent to \(N=4\) super Yang–Mills theory in four dimensions. Witten, and independently Steven Gubser, Igor Klebanov and Alexander Polyakov, gave the dictionary in 1998: the generating function of correlators of a boundary operator \(\mathcal O\) equals the gravitational partition function with boundary value \(\phi_0\) for the dual bulk field,

For a scalar of mass \(m\) in \(\mathrm{AdS}_{d+1}\) of radius \(L\), solving the wave equation near the boundary shows \(\phi\sim z^{d-\Delta}\phi_0\), with

Witten also showed that black holes in AdS correspond to the deconfined high-temperature phase of the gauge theory, giving a geometric picture of confinement.

S-duality, Langlands, twistors

For the general reader. The Langlands program is a vast web of conjectures linking number theory, representation theory and geometry. Its geometric form pairs every gauge group with a dual group. In 2006 Anton Kapustin and Witten showed that this pairing is the same as a duality of four-dimensional gauge theory that exchanges electric and magnetic charges. In a separate direction, Witten’s twistor string theory of 2003 led to new methods for computing particle collisions.

S-duality

\(N=4\) super Yang–Mills theory with gauge group \(G\) has a coupling \(\tau=\frac{\theta}{2\pi}+\frac{4\pi i}{g^2}\). Claus Montonen and David Olive conjectured in 1977 that it is equivalent to the theory with the Langlands dual group \(G^\vee\) and coupling \(-1/(n_g\tau)\), where \(n_g\) is 1 for simply laced \(G\). Combined with \(\tau\mapsto\tau+1\), this generates an action of \(\mathrm{SL}(2,\mathbb Z)\) (or a related group) on the space of theories.

The argument is physical. A proof of the de Rham geometric Langlands conjecture by purely mathematical means was announced in 2024 by Dennis Gaitsgory, Sam Raskin and collaborators. The Kapustin–Witten framework still guides much of the field, including Witten’s later work with Davide Gaiotto on knot homology from gauge theory.

Twistors and amplitudes

For gluons with momenta \(p_i=\lambda_i\tilde\lambda_i\) written in spinor form, with \(\langle ij\rangle=\epsilon_{ab}\lambda_i^a\lambda_j^b\), the tree-level colour-ordered amplitude with two negative-helicity gluons \(r,s\) and the rest positive is, up to coupling and the momentum delta function,

This Parke–Taylor formula (1986) is far simpler than the Feynman diagrams that produce it. Witten observed in 2003 that, transformed to Roger Penrose’s twistor space, these amplitudes are supported on complex lines. He proposed a string theory in twistor space that generates them. In 2005 Ruth Britto, Freddy Cachazo, Bo Feng and Witten found the BCFW recursion, which builds all tree amplitudes from three-point ones and is now a standard tool.

The Institute, 1987–2022

In 1987 Witten became a professor in the School of Natural Sciences at the Institute for Advanced Study in Princeton, the institution once associated with Einstein. He held the Charles Simonyi Professorship from 1997 and spent 1999–2001 as a visiting professor at Caltech. He became Professor Emeritus in 2022 and has continued to publish.

The Fields Medal came in 1990, at the International Congress of Mathematicians in Kyoto. He was the first physicist to receive it. In his laudation, Michael Atiyah described how Witten’s physical insight had repeatedly led to new and deep mathematical theorems. The award was controversial with some mathematicians, because much of the work it honoured did not consist of proofs. Later developments, from Seiberg–Witten invariants to mirror symmetry, made the choice look prescient.

His doctoral students include Jonathan Bagger, Cumrun Vafa, Xiao-Gang Wen, Dror Bar-Natan, Shamit Kachru, Eva Silverstein and Sergei Gukov, who between them work in string theory, condensed matter physics and knot theory. His influence spreads further through his lectures and expository papers, and through programmes such as the 1996–97 IAS year on quantum fields and strings for mathematicians.

His later work has ranged widely: noncommutative geometry in string theory with Seiberg (1999), twistor strings (2003), the BCFW recursion (2005), geometric Langlands (2006), three-dimensional gravity, knot homology from gauge theory, analytic continuation of Chern–Simons theory, the Sachdev–Ye–Kitaev model and its tensor-model relatives (2016), and expository papers on quantum information and entanglement in field theory for physicists.

Honours and influence

Witten’s honours span physics and mathematics. Besides the Fields Medal (1990), they include a MacArthur Fellowship (1982); the Dirac Medal of the ICTP and the Einstein Medal of the Einstein Society of Berne (1985); the Alan T. Waterman Award (1986); the Dannie Heineman Prize for Mathematical Physics (1998); the Nemmers Prize in Mathematics (2000); the National Medal of Science (2003); the Harvey Prize and the Henri Poincaré Prize (2006); the Crafoord Prize in Mathematics, shared with Maxim Kontsevich (2008); the Lorentz Medal and the Isaac Newton Medal (2010); the Fundamental Physics Prize (2012); the Kyoto Prize in Basic Sciences (2014); and the Hamburg Prize for Theoretical Physics (2023). He is a member of the National Academy of Sciences (1988), a Foreign Member of the Royal Society (1998), an associate of the French Academy of Sciences (2000) and a member of the Pontifical Academy of Sciences (2006). He has served on the board of Americans for Peace Now since 1992.

Legacy

• A new style of mathematics: quantum field theory as a machine for conjectures. Donaldson theory, Seiberg–Witten invariants, Gromov–Witten theory, mirror symmetry and quantum invariants of knots all bear his mark.
• String theory: anomaly cancellation, Calabi–Yau compactification, M-theory and the AdS/CFT dictionary shaped three decades of research.
• Topology in physics: the Witten index, global anomalies and topological field theory are now standard tools in particle and condensed matter physics, including the study of topological phases of matter.
• Communication: his survey papers and lecture notes are a principal route by which mathematicians learn physics and physicists learn modern geometry.

Sources and further reading

Biography

1. Institute for Advanced Study, Edward Witten, curriculum vitae (2024).
2. J. J. O’Connor and E. F. Robertson, Edward Witten, MacTutor History of Mathematics.
3. Edward Witten, Wikipedia.
4. Inamori Foundation, 2014 Kyoto Prize laureate: Edward Witten.
5. M. Atiyah, On the work of Edward Witten, Proceedings of the ICM, Kyoto 1990.
6. MacArthur Foundation, Edward Witten, Class of 1982.

Physics and mathematics

1. E. Witten, A new proof of the positive energy theorem, Comm. Math. Phys. 80 (1981) 381–402.
2. E. Witten, Supersymmetry and Morse theory, J. Differential Geom. 17 (1982) 661–692.
3. E. Witten, An SU(2) anomaly, Phys. Lett. B 117 (1982) 324–328.
4. L. Alvarez-Gaumé and E. Witten, Gravitational anomalies, Nucl. Phys. B 234 (1984) 269–330.
5. E. Witten, Topological quantum field theory, Comm. Math. Phys. 117 (1988) 353–386.
6. E. Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989) 351–399.
7. N. Seiberg and E. Witten, Electric–magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang–Mills theory, Nucl. Phys. B 426 (1994).
8. E. Witten, Monopoles and four-manifolds, Math. Res. Lett. 1 (1994) 769–796.
9. E. Witten, String theory dynamics in various dimensions, Nucl. Phys. B 443 (1995).
10. E. Witten, Anti de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998).
11. E. Witten, Perturbative gauge theory as a string theory in twistor space, Comm. Math. Phys. 252 (2004).
12. A. Kapustin and E. Witten, Electric-magnetic duality and the geometric Langlands program, Commun. Number Theory Phys. 1 (2007).
13. M. B. Green, J. H. Schwarz and E. Witten, Superstring Theory, 2 vols., Cambridge University Press, 1987.
14. P. Deligne et al. (eds.), Quantum Fields and Strings: A Course for Mathematicians, 2 vols., American Mathematical Society and IAS, 1999.
15. J. W. Morgan, The Seiberg–Witten Equations and Applications to the Topology of Smooth Four-Manifolds, Princeton University Press, 1996.