# INSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD

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1 INSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD () Instanton (definition) (2) ADHM construction (3) Compactification. Instantons.. Notation. Throughout this talk, we will use the following notation: G: a (semi-simple) Lie group, typically SU (n) g: its Lie algebra : a (simply connected) 4-fold (typically, S 4 ) P : a principal G-bundle π : P A: a connection on P F A : its curvature Ω p (g): differential p-forms on with values in g, i.e., Ω (g) := Γ (T g)..2. Primer on Connections. Recall from Peng s talk that a connection on P can be thought of in three ways: () As a field of horizontal subspaces: T p P = H p P V p P, where V P = ker (π ). (Note that V p = g, and Hp = Tπ(p) ). (2) As a g-valued -form A Ω (g) which is invariant w.r.t. the induced G-action on Ω (g). (3) Given a representation of G on W, as a vector bundle connection on the associated bundle E := P G W A : Ω 0 (E) Ω (E), where A is a linear map satisfying the Leibniz rule: (f s) = f s+df s for f C (), s Γ (, E). The first two of these definitions are seen to be equivalent by setting H p P = ker A p for (2) (), or A p = T p P V p P (projection) for () (2). For (3), we think of P as being the frame bundle for E, and then describe a horizontal frame of sections. For concreteness, we will mostly use (3) in this talk, i.e., fix a vector bundle E, and then think of P as the frame bundle of E. However, everything can still be done for principal bundles, too. From A, we can build a new operator d A : Ω k (E) Ω k+ (E) by requiring that d A = A for sections of E and d A (ω θ) = (d A ω) θ+( ) ω ω d A θ. In general, d 2 A 0, and we give this a special name: the curvature F A := d 2 A : E Ω 2 (E)

2 INSTANTON MODULI AND COMPACTIFICATION 2 Locally, d A = d + A, where A Ω (E) is an E-valued -form, and F A = da + A A Ω 2 (E) The covariant derivative along v T of a section s is given by ι v d A s. If has the additional structure of a Riemannian metric, the formal adjoint d A to d A can be defined: d A φ, ψ dµ = φ, d Aψ dµ. If in addition dim = 4, then from Hodge theory, 2-forms on decompose into self-dual and anti-self-dual parts. This extends to E-valued forms so Ω 2 (E) = Ω + (E) Ω (E), F A = F + A + F A. A connection A is called anti-self-dual (ASD) if F + A = 0 (i.e., F A = F A )..3. Yang-Mills Theory. Yang-Mills theory is a field theory defined for principal G bundles π : P. The fields of the theory are connections, and the action is (up to some constants) (.) S (A) := F A 2 dµ. S is conformally invariant in dimension 4: if g cg is a conformal transformation, then dµ c d dµ and F A c 2 F A, so for dim = d = 4, c 4 4 F A 2 dµ = F A 2 dµ. For a G-invariant metric (which can be readily constructed if G is compact), this action is gauge-invariant. F A 2 is sometimes called the Yang-Mills density. Proposition.. The Euler-Lagrange equations of this action are (.2) d AF A = 0. Proof. This is an exercise in variational calculus. Observe that Then, so F A+tτ = d (A + tτ) + (A + tτ) (A + tτ) = F A + td A τ + t 2 τ τ. F A+tτ 2 = F A 2 + 2t d A τ, F A + t 2 ( ), 0 = d dt S (A + tτ) = 2 Hence the equations of motion are I ll skip most of the algebra. d A τ, F A dµ = 2 τ, d AF A dµ. d AF A = 0.

3 INSTANTON MODULI AND COMPACTIFICATION 3 An instanton is a topologically nontrivial solution to the classical equations of motion with finite action. Proposition.2. Anti-self-dual connections are instantons, i.e., topologically nontrivial solutions to (.2). Proof. First we show that an ASD connection solves the equations of motion. The main fact is d AF A = d A F A, so if A is ASD, d A F A = d A F A = 0 by the Bianchi identity. When dim = 4 and G = SU (n), ASD connections are topologically nontrivial: for skew-adjoint matrices (A = A, where is conjugate transpose), i.e., u (n), tr (ξ ξ) = ξ 2, so tr ( FA 2 ) ( F = + 2 A F 2) A dµ. F A 2 = F A F A = F + 2 A + F 2 A, so a connection is ASD if and only if tr ( FA) 2 = F A 2 dµ at every x. Recall that for SU (n) bundles, c (E) vanishes because tr (F A ) = 0, so c 2 (E) = 8π 2 tr ( FA) 2 dµ. Hence, S (A) = F A 2 dµ = 2 dµ + 2 dµ 8π 2 c 2 (E), F A with the bound achieved precisely when A is ASD. For this reason, physicists often refer to c 2 (E) as the instanton number. Let F ij := [ i, j ] = F + A 2. ADHM Construction Then, instanton equation F + A = 0 becomes (2.) A j A i + [A i, A j ]. x i x j F 2 + F 34 = 0, F 4 + F 23 = 0, F 3 + F 42 = 0. The ADHM (Atiyah, Drinfeld, Hitchin, and Manin) construction gives a way of producing ASD connections from linear algebraic data. The idea is to take a Fourier transform of the ASD equations to produce a set of matrix equations which can be more readily solved. Substituting D := + i 2, D 2 := 3 + i 4, the equations (2.) become [D, D 2 ] = (F 3 + F 42 ) + i (F 23 + F 4 ) = 0, [D, D ] + [D 2, D 2] = 2i (F 2 + F 34 ) = 0, so we can reduce the ASD equations to these complex covariant derivatives. Because the ASD equation and F A 2 are conformally invariant, an ASD connection on R 4 with S (A) < can be regarded as an ASD connection on S 4.

4 INSTANTON MODULI AND COMPACTIFICATION ADHM Data. Let U = C 2 as a complex manifold, with coordinates (z, z 2 ). The inputs for the ADHM construction consist of: () A k-dimensional complex vector space H with a Hermitian metric. (2) An n-dimensional complex vector space E, with Hermitian metric and symmetry group SU (n). (3) A linear map T V hom (H, H) defining maps τ, τ 2 : H H. (4) Linear maps π : H E and σ : E H. The ADHM equations are (2.2) [τ, τ 2 ] + σπ = 0, [τ, τ ] + [τ 2, τ 2 ] + σσ π π = 0. If τ, τ 2, σ, π satisfy these equations, then the maps α := τ τ 2, β := [ τ 2 τ σ ] π define a complex because H α H U E β βα = [τ, τ 2 ] + σπ = 0. In fact, it defines a whole C 2 -family of complexes because we can replace (τ, τ 2 ) by (τ z, τ 2 z 2 ) for any point (z, z 2 ) =: x U. We can then define a family of maps R x : H U E H H H R x := α x β x and, if α x is injective and β x is surjective, then R x is surjective and ker R x = (im α x ) ker β x. Definition. A collection (τ, τ 2, σ, π, E, H) of ADHM data is an ADHM system if () it satisfies the ADHM equations (2.2), and (2) the map R x is surjective for each x U ADHM Construction. How can we use this information to construct a connection? Suppose that we have an ADHM system. Now, construct the vector bundle E U with fibers E x = ker R x = ker β x / im α x. (E x is the cohomology bundle of α,β). Proposition 2.. There is a holomorphic structure E on E. (Proof omitted in the interest of time). Let i : E x H U E be inclusion, P α x : H U E (im α x ) and P β x : H U E ker β x denote orthogonal projections, and P x := P α P β = P β P α

5 INSTANTON MODULI AND COMPACTIFICATION 5 be projection onto E x. H U E comes equipped with a flat product connection d, and we can define an induced connection A on E by, for a section s : U E, d A s := P di (s). By virtue of this construction, A is unitary and compatible with the holomorphic structure on E. It is also ASD. Theorem 2.2 (ADHM). The assignment (τ, τ 2, σ, π) d A = P di sets up a oneto-one correspondence between () equivalence classes of ADHM data for group SU (n) and index k, and (2) gauge equivalence classes of finite energy ASD SU (n)-connections A over R 4 with c 2 (A) = k. Note that (g, h) U (k) SU (n) acts on ADHM data by (τ, τ 2, σ, π) ( gτ g, gτ 2 g, gσh, hπg ), so we mean classes of ADHM data up to this equivalence Example: BPST Instanton. The simplest example is to take k = and n = 2. This corresponds to solutions on SU (2) bundles with c 2 =. Then, τ, τ 2 are just complex numbers, σ and π are complex vectors, and the ADHM equations become σ π = 0, σ 2 = π 2. Pick π = (, 0) and σ = (0, ), then for (τ, τ 2 ), have τ αx = τ 2 0 = [ τ τ 2 0 ], β x = [ τ 2 τ 0 ], and in general: replace (τ, τ 2 ) by (τ z, τ 2 z 2 ) for any point (z, z 2 ) =: x U, so [ ] τ z R x = τ 2 z 2 0. τ 2 + z 2 τ z 0 In particular, for (τ, τ 2 ) = (0, 0), have [ ] z z R x = 2 0. z 2 z 0 A unitary basis for R x is 0 {σ, σ 2 } = 0 + x 2 z, + x 2 z 2. z 2 z Suppose that in this trivialization we let z = x + ix 2, z 2 = x 3 + ix 4, and the connection matrix is A = A i dx i, so A i is the matrix with (p, q)th entry i σ p, σ q = σp, σ q. x i

6 INSTANTON MODULI AND COMPACTIFICATION 6 Then, written out in full, the connection form is A = + x 2 (θ i + θ 2 j + θ 3 k), where i, j, k are a standard basis for su (2) and θ = x dx 2 x 2 dx x 3 dx 4 + x 4 dx 3 θ 2 = x dx 3 x 3 dx x 4 dx 2 + x 2 dx 4, θ 3 = x dx 4 x 4 dx x 2 dx 3 + x 3 dx 2 is such that dθ, dθ 2, dθ 3 is a basis for the ASD two-forms on R 4. The curvature F A = da + A A is then ( ) 2 F A = + x 2 (dθ i + dθ 2 j + dθ 3 k), and we can recover the other degrees of freedom lost in our choices of π, σ, τ, τ 2 by translations x x y and dilations x x/λ to obtain other connections with F A(y,λ) λ 2 = (λ 2 + x y 2) Moduli Space of ASD Connections Definition 3.. Let E be a bundle over a compact, oriented Riemannian 4-manifold. The moduli space of ASD connections M E is the set of gauge equivalence classes of ASD connections on E. Recall that a gauge transformation is an automorphism u : E E respecting the structure on the fibers and reducing to the identity map on. It acts on a connection by the rule u(a) s = u A ( u s ) = A s ( A u) u s, where the covariant derivative A u is formed by regarding it as a section of the vector bundle End (E). In local coordinates, this looks like u (A) = uau (du) u. The curvature transforms as a tensor under gauge transformations: F u(a) = uf A u. For connections on principal bundles P, this has a somewhat nicer expression: If u : P P satisfies () u (p g) = u (p) g and (2) π (u (p)) = π (p) for all g G, and A Ω P (g) is a connection, u (A) := ( u ) A. Now we turn to some results about the structure of this moduli space.

7 INSTANTON MODULI AND COMPACTIFICATION Uhlenbeck s Theorems. First, there are a few technical results due to Uhlenbeck that allow us to leverage tools from the study of elliptic differential equations to make statements about ASD connections. Theorem 3.2 (Uhlenbeck). There are constants ɛ, M > 0 such that any connection A on the trivial bundle over B 4 with F A L 2 < ɛ is gauge equivalent to a connection Ã over B4 with () d Ã = 0, (2) lim x Ã r = 0, and (3) Ã L 2 M F Ã L 2. Moreover for suitable constants ɛ, M, Ã is uniquely determined by these properties, up to Ã u0ãu 0 for a constant u 0 in U (n). First, some notes about the theorem: Ã 2 L = Ã 2 + Ã 2 dµ 2 B 4 is the Sobolev norm. d Ã is the Coulomb gauge condition (the importance of which will be explained in the following section). Finally, lim x Ã r = 0 means that, for Ãr (ρ, σ) a function on S 3, this function tends to 0 as r. The main power of Uhlenbeck s Theorem is that it turns a system of nonlinear, nonelliptic differential equations into an elliptic one. This section provides a sketch of why that might be a desirable thing to do. Recall the d + operator, defined by ( ) d + = ( + ) d, 2 which maps The ASD equation F + A d + : Ω Ω +. = 0 then becomes, in local coordinates, (3.) d + A + (A A) + = 0. This is a nonlinear, non-elliptic equation. When d Ã = 0, d + d : i Ω 2i+ i Ω 2i is elliptic, so if H () = 0, then all -forms are othogonal to ker (d + d ). Elliptic differential operator theory implies that ( ) (3.2) A L 2 const. d A k L 2 + da L 2 k k for all k. When d A = 0, this becomes A L 2 k const. F A L 2 k, and the ASD equation can be replaced by the elliptic differential equation δa = 0, where δ = d + + d is an elliptic operator. The main consequence of Uhlenbeck s Theorem relevant to the discussion of ASD connections comes from combining it with the following theorem:

8 INSTANTON MODULI AND COMPACTIFICATION 8 Theorem 3.3 (Uhlenbeck). There exists a constant ɛ 2 > 0 such that if Ã is any ASD connection on the trivial bundle over B 4 which satisfies d Ã = 0 and Ã L 4 ɛ 2, then for all interior domains D B 4 and l, Ã L 2 l (D) M l,d FÃ L 2 (B 4 ) for a constant M l,d depending only on l and D. Combining this with Theorem 3.2 gives Corollary 3.4. For any sequence of ASD connections A α over B 4 with F (A α ) L 2 ɛ, there is a subsequence α and gauge equivalent connections Ãα which converge in C on the open ball Results about the Moduli Space. Putting our previous results together, we get the following statements: Theorem 3.5 (Uhlenbeck s Removable Singularities). Let A be a unitary connection over the punctured ball B 4 \ {0} which is ASD with respect to a smooth metric on B 4. If F A 2 <, B 4 \{0} then there is a smooth ASD connection over B 4 gauge equivalent to A over the punctured ball. Note that this theorem implies that, for example, the ADHM construction gives all of the ASD connections on S 4 (not just R 4 ). Let M k (G) denote the moduli space of ASD connections up to gauge transformation with c 2 = k, and M k (G) denote the closure of M k (G) in the space of ideal connections. An ideal connection is a connection with curvature densities possibly having δ-measure concentrations at up to k points of, i.e., of the form Then, F A 2 + 8π 2 n Theorem 3.6. Any infinite sequence in M k has a weakly convergent subsequence in M k, with limit point in M k. Corollary 3.7. The space M k is compact. What do these spaces look like locally? Let G denote the group of gauge transformations of E, and the isotropy group of A. Then, i= δ xi Γ A = {u G : u (A) = A}, Proposition 3.8. If A is an ASD connection over, a neighborhood of [A] in M is modeled on a quotient f (0) /Γ A, where f : ker δ A coker d + A is a Γ A -equivariant map and δ A = d A + d+ A.

9 INSTANTON MODULI AND COMPACTIFICATION 9 References [ADHM] M. Atiyah, V. Drinfeld, N. Hitchin, and Y. Manin, Construction of Instantons, Physics Letters A 65 (3) [DK] S. K. Donaldson and P. B. Kronheimer, The Geometry of Four-Manifolds, Oxford Univ. Press, 99. [L] A. J. Lindenhovius, Instantons and the ADHM Construction, URL:

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