HALF A MAGNETIC MONOPOLE

Christian Ausoni, Bjørn Ian Dundas and John Rognes

September 9th 2008

Abstract. We show that when the gerbe μ representing a magnetic monopole is viewed as a virtual 2-vector bundle, then it decomposes, modulo torsion, as two times a virtual 2-vector bundle ς. We therefore interpret ς as representing half a magnetic monopole, or a semipole.

§1. Introduction

Let A be a connective S-algebra, where S is the sphere spectrum, and let K(A) = K_{0}(π_{0}(A)) × BGL_{∞}(A)^{+} be its algebraic K-theory space. The natural map w: BGL_{1}(A) → K(A) is given by the inclusion of 1 × 1 matrices BGL_{1}(A) → BGL_{∞}(A), followed by the canonical map into the plus construction. Let ku denote the connective complex K-theory spectrum, with π_{∗}ku = ℤ[u], |u| = 2, and let π: ku → Hℤ be the unique 2-connected S-algebra map to the integral Eilenberg– Mac Lane spectrum. Its homotopy fiber is bu, with π_{∗}bu = (u) ⊂ ℤ[u]. We define BSL_{1}(ku) and K(π) as the homotopy fibers of the induced maps π: BGL_{1}(ku) → BGL_{1}(ℤ) and π: K(ku) → K(ℤ), respectively, so that we have the following commutative diagram of horizontal homotopy fiber sequences

(1.1)  \xymatrix{BSL_{1}(ku) \ar[r] \ar[d] & BGL_{1}(ku) \ar[r]^{π} \ar[d]^{w} & BGL_{1}(ℤ) \ar[d]^{w} \\ K(π) \ar[r] & K(ku) \ar[r]^{π} & K(ℤ) .}

We have a map from the Eilenberg–Mac Lane complex K(ℤ, 3) to the upper left hand corner of this diagram, induced by the infinite loop space inclusion BU(1) → BU_{⊗} and the equivalences K(ℤ, 3) ≃ BBU(1) and BBU_{⊗} ≃ BSL_{1}(ku). Recall that the space K(ℤ, 3) represents gerbes with band U(1) [Br93, Ch. V], whereas K(ku) represents virtual 2-vector bundles [BDR04, Thm. 4.10], [BDRR]. A 2-vector bundle of rank 1 is the same as a gerbe, and the composite map

(1.2)  K(ℤ, 3) → K(ku)

represents the construction that views a gerbe as a virtual 2-vector bundle.

Typeset by AMS-TEX

2000 Mathematics Subject Classification. 19D50, 55P43, 81T40, 81S10.

We now consider gerbes and 2-vector bundles over the base space S^{3}. There is a map μ: S^{3} → K(ℤ, 3) representing a generator of H^{3}(S^{3}) = ℤ, or dually, corepresenting a generator of π_{3}K(ℤ, 3) = ℤ. It also represents a U(1)-gerbe over S^{3}, which is interpreted in [Br93, Ch. VII] as a mathematical model for a magnetic monopole, stationary in time and localized at one point.

Parallel transport in this gerbe, around closed loops in S^{3}, defines a holonomy line bundle over the free loop space of S^{3} [Br93, Ch. VI]. Its complex 1-dimensional fibers can be interpreted as the state spaces of these free loops, viewed as strings in S^{3}. Parallel transport over compact surfaces in S^{3}, between tensor products of copies of this line bundle, defines an action functional that makes these state spaces part of a field theory. Here the compact surfaces are viewed as world sheets in S^{3}. In a quantized theory one would consider Hilbert spaces of sections in the holonomy line bundle, rather than its individual fibers, as the state spaces.

Following the point of view explained in [AR, §5.5], we also view 2-vector bundles over a base space as data defining (virtual) state spaces and action functionals for strings in that base. More field theories arise this way, since the state spaces are no longer restricted to being 1-dimensional, hence it is also possible to model more kinds of particles by 2-vector bundles than those arising from gerbes.

In particular we may ask, as the second author did, how the magnetic monopole μ over S^{3} decomposes when viewed as a virtual 2-vector bundle. Does it remain a single particle?

In mathematical terms, the question is: “What is the structure of π_{3}K(ku) = K_{3}(ku), and what is the image of μ ∈ π_{3}K(ℤ, 3) in that group?” The surprising answer, which explains the title of this paper, is that modulo torsion, μ becomes divisible by two as a virtual 2-vector bundle. In other words, there are virtual 2-vector bundles ς and ν over S^{3}, with 24ν = 0, such that

(1.3)  μ + ν = 2ς

in K_{3}(ku). Modulo torsion, ς is therefore half a magnetic monopole. Ignoring torsion is justified in the physical interpretation, since the numerical invariants of a field theory take torsion-free values, and will send ν to zero. On the other hand, both μ and ς have infinite order in K_{3}(ku).

§2. Statement of results

Let i: S → K(ku) be the unit map, and recall that π_{3}(S) = ℤ/24{ν} and K_{3}(ℤ) ≅ ℤ/48{λ} [LS76]. The composite map πi: S → K(ℤ) induces the injection π_{3}(S) → K_{3}(ℤ) that takes ν to 2λ.

By [Wa78, Prop. 1.2], as generalized in [BM94, Prop. 10.9], the homotopy fiber K(π) is 2-connected. Hence K_{i}(ku) → K_{i}(ℤ) is an isomorphism for i ≤ 2. Here is what happens in dimension three:

Theorem 2.1. (a) The maps K(ℤ, 3) → BSL_{1}(ku) → K(π) induce isomorphisms

ℤ{μ} = π_{3}K(ℤ, 3) ⟶^{≅} π_{3}BSL_{1}(ku) ⟶^{≅} K_{3}(π) .

(b) The unit map i: S → K(ku) induces a homomorphism

ℤ/24{ν} = π_{3}(S) ⟶^{i_{∗}} K_{3}(ku)

that identifies the source with the torsion subgroup in the target. We abbreviate i_{∗}(ν) to ν ∈ K_{3}(ku).

(c) The homotopy fiber sequence K(π) → K(ku) → K(ℤ) induces a short exact sequence

0 → K_{3}(π) → K_{3}(ku) ⟶^{π_{∗}} K_{3}(ℤ) → 0

which is isomorphic to the nontrivial extension

0 → ℤ{μ} → ℤ{ς} ⊕ ℤ/24{ν} ⟶^{π_{∗}} ℤ/48{λ} → 0 .

Here the first map takes μ to 2ς − ν, and the second map takes ς to λ and ν to 2λ.

Corollary 2.2. The map K(ℤ, 3) → K(ku) that represents viewing U(1)-gerbes as virtual 2-vector bundles induces the homomorphism

ℤ{μ} = π_{3}K(ℤ, 3) → K_{3}(ku) = ℤ{ς} ⊕ ℤ/24{ν}

that takes μ to 2ς − ν, where 24ν = 0. The image of ς ∈ K_{3}(ku) in K_{3}(ℤ) is the generating element λ of order forty-eight.

Corollary 2.3. There is no “determinant map”

det: K(ku) → BGL_{1}(ku)

such that the composite det ∘ w is an equivalence.

Corollary 2.2 is readily extracted from Theorem 2.1(a) and (c). Corollary 2.3 follows, since det: K_{3}(ku) → π_{3}BGL_{1}(ku) ≅ ℤ{μ} cannot map 2ς − ν to μ.

Remark 2.4. For commutative rings R there is a determinant map det: K(R) → BGL_{1}(R), which is left inverse to w. On the other hand, it follows from [Wa82, Cor. 3.7] that w: BGL_{1}(S) → K(S) admits no such retraction up to homotopy. In [AR, §5.2], the first and third authors used the existence of a rational determinant map det_{ℚ}: K(ku) → BSL_{1}(ku)_{ℚ} ≃ (BBU_{⊗})_{ℚ} to define the rational anomaly bundle of a 2-vector bundle, generalizing the definition of the anomaly line bundle of a gerbe. Corollary 2.3 shows no such generalization can be integrally defined on all of K(ku). This suggests that an integral anomaly bundle will only be defined on a space covering K(ku), classifying 2-vector bundles with some form of higher orientation.

§3. Proofs

Proof of Thm. 2.1(a). In view of the infinite loop space splitting BU_{⊗} ≃ BU(1) × BSU_{⊗} it is clear that K(ℤ, 3) ≃ BBU(1) → BBU_{⊗} ≃ BSL_{1}(ku) is 4-connected. For the second part, we refer to the proof of [BM94, Prop. 10.9] to see that there is an isomorphism

(3.1)  colim_{n} M_{n}(π_{2}(bu))/[GL_{n}(π_{0}(ku)), M_{n}(π_{2}(bu))] ≅ K_{3}(π) .

Here M_{n} denotes the ring of n × n matrices, and GL_{n} acts on M_{n} by conjugation. Furthermore, under the isomorphism (3.1), π_{3}BSL_{1}(ku) → K_{3}(π) factors as

(3.2)  π_{3}BSL_{1}(ku) ≅ π_{2}(bu) ≅ M_{1}(π_{2}(bu))/[GL_{1}(π_{0}(ku)), M_{1}(π_{2}(bu))] ,

followed by the canonical map from the term n = 1 into the colimit in (3.1). For each n ≥ 1 the matrix trace induces an isomorphism

M_{n}(π_{2}(bu))/[GL_{n}(π_{0}(ku)), M_{n}(π_{2}(bu))] ⟶^{≅} π_{2}(bu)/[π_{0}(ku), π_{2}(bu)] ,

hence each structure map in the colimit is an isomorphism, and therefore the canonical map from (3.2) to K_{3}(π) is also an isomorphism. □

To proceed, we make use of the natural trace map tr: K(A) → T HH(A) to topological Hochschild homology [BHM93]. We define T HH(π) as the homotopy fiber of π: T HH(ku) → T HH(ℤ), so as to obtain the following commutative diagram of horizontal homotopy fiber sequences

(3.3)  \xymatrix{K(π) \ar[r] \ar[d] & K(ku) \ar[r]^{π} \ar[d]^{tr} & K(ℤ) \ar[d]^{tr} \\ T HH(π) \ar[r] & T HH(ku) \ar[r]^{π} & T HH(ℤ) .}

Proof of Thm. 2.1(b) and (c). Passing to homotopy groups, we get the following vertical map of short exact sequences

(3.4)  \xymatrix{0 \ar[r] & K_{3}(π) \ar[r] \ar[d]^{(≅)} & K_{3}(ku) \ar[r]^{π_{∗}} \ar[d]^{tr_{∗}} & K_{3}(ℤ) \ar[r] \ar[d]^{tr_{∗}} & 0 \\ 0 \ar[r] & T HH_{3}(π) \ar[r] & T HH_{3}(ku) \ar[r]^{π_{∗}} & T HH_{3}(ℤ) \ar[r] & 0 .}

Here K_{3}(π) → K_{3}(ku) is injective because K_{4}(ℤ) = 0 [Ro00], and K_{3}(ku) → K_{3}(ℤ) is surjective because K_{2}(π) = 0. Furthermore, T HH_{3}(π) → T HH_{3}(ku) is injective because T HH_{4}(ℤ) = 0, and T HH_{3}(ku) → T HH_{3}(ℤ) is surjective because

ℤ/48{λ} = K_{3}(ℤ) ⟶^{tr_{∗}} T HH_{3}(ℤ) = ℤ/2{e}

takes λ to e [BM94, Thm. 10.14], [Ro98, Thm. 1.1] and the right hand square commutes. The left hand vertical map K_{3}(π) → T HH_{3}(π) is split injective, by [BM94, Thm. 10.12]. We shall soon see that it is in fact an isomorphism.

The 2-primary homotopy of T HH(ku) is fully computed in [AHL], but in low dimensions the following direct argument suffices. The homotopy cofiber ku/S of S → ku is 1-connected, with π_{2}(ku/S) ≅ ℤ. By construction, T HH(ku) is the geometric realization of a simplicial spectrum, and the map from the (n − 1)- skeleton to the n-skeleton has cofiber Σ^{n}ku ∧ (ku/S) ∧ ⋯ ∧ (ku/S), with n copies of ku/S, which is (3n − 1)-connected. By induction, the map from the 1-skeleton to all of T HH(ku) is 5-connected. Furthermore, the 0-simplices ku split off from the 1-skeleton of T HH(ku), so T HH_{3}(ku) ≅ π_{3}(ku) ⊕ π_{3}(Σku ∧ (ku/S)) ≅ ℤ{ϵ}, for some choice of generator ϵ.

Diagram (3.4) is therefore isomorphic to

(3.5)  \xymatrix{0 \ar[r] & ℤ\{μ\} \ar[r] \ar[d]^{≅} & K_{3}(ku) \ar[r]^{π_{∗}} \ar@{->>}[d]^{tr_{∗}} & ℤ/48\{λ\} \ar[r] \ar@{->>}[d]^{tr_{∗}} & 0 \\ 0 \ar[r] & ℤ\{2ϵ\} \ar[r] & ℤ\{ϵ\} \ar[r]^{π_{∗}} & ℤ/2\{e\} \ar[r] & 0 ,}

where the split injection ℤ{μ} → ℤ{2ϵ} must be an isomorphism. (We assume that we have chosen our orientations so that μ maps to 2ϵ, rather than −2ϵ.)

The right hand square is a pullback, so there is a short exact sequence

(3.6)  0 → ℤ/24{ν} ⟶^{i_{∗}} K_{3}(ku) ⟶^{tr_{∗}} ℤ{ϵ} → 0 ,

where the image of the injective homomorphism i_{∗}: π_{3}(S) → K_{3}(ku) is identified under π_{∗}: K_{3}(ku) → K_{3}(ℤ) with the kernel of tr_{∗}: K_{3}(ℤ) → T HH_{3}(ℤ). Hence the image of i_{∗} equals the kernel of tr_{∗}: K_{3}(ku) → T HH_{3}(ku).

To fix a splitting of (3.6), we let ς ∈ K_{3}(ku) be the class mapping to ϵ in T HH_{3}(ku) and to λ in K_{3}(ℤ). This is admissible, since both classes map to e in T HH_{3}(ℤ). Then

K_{3}(ku) ≅ ℤ{ς} ⊕ ℤ/24{ν} ,

and μ ∈ K_{3}(π) maps to 2ς in K_{3}(ku) modulo the image of i_{∗}. Since μ continues to 0 in K_{3}(ℤ), the exact formula must be 2ς − ν. □

References

[AHL] Vigleik Angeltveit, Michael Hill and Tyler Lawson, Topological Hochschild homology of ℓ and ko, arXiv:0710.4368v1 preprint.

[AR] Christian Ausoni and John Rognes, Rational algebraic K-theory of topological K-theory, arXiv:0708.2160v1 preprint.

[BDRR] Nils A. Baas, Bjørn Ian Dundas, Birgit Richter and John Rognes, Two-vector bundles define a form of elliptic cohomology, arXiv:0706.0531v2 preprint.

[BDR04] Nils A. Baas, Bjørn Ian Dundas and John Rognes, Two-vector bundles and forms of elliptic cohomology, Topology, geometry and quantum field theory, London Math. Soc. Lecture Note Ser., vol. 308, Cambridge Univ. Press, Cambridge, 2004., pp. 18–45.

[BHM93] Marcel Bökstedt, Wu Chung Hsiang and Ib Madsen, The cyclotomic trace and algebraic K-theory of spaces, Invent. Math. 111 (1993), 465–539.

[BM94] Marcel Bökstedt and Ib Madsen, Topological cyclic homology of the integers, K-theory (Strasbourg, 1992), Astérisque, vol. 226, 1994, pp. 57–143.

[Br93] Jean-Luc Brylinski, Loop spaces, characteristic classes and geometric quantization, Progress in Mathematics, vol. 107, Birkhäuser Boston, Inc., Boston, MA, 1993.

[LS76] Ronnie Lee and Robert H. Szczarba, The group K_{3}(ℤ) is cyclic of order forty-eight, Ann. of Math. (2) 104 (1976), 31–60.

[Ro00] John Rognes, K_{4}(ℤ) is the trivial group, Topology 39 (2000), 267–281.

[Ro98] John Rognes, Trace maps from the algebraic K-theory of the integers (after Marcel Bökstedt), J. Pure Appl. Algebra 125 (1998), 277–286.

[Wa78] Friedhelm Waldhausen, Algebraic K-theory of topological spaces. I, Algebraic and geometric topology (Stanford Univ., Stanford, Calif., 1976), Part 1, Proc. Sympos. Pure Math., vol. XXXII, Amer. Math. Soc., Providence, RI, 1978, pp. 35–60.

[Wa82] Friedhelm Waldhausen, Algebraic K-theory of spaces, a manifold approach, Current trends in algebraic topology, Part 1, CMS Conf. Proc., vol. 2, Amer. Math. Soc., Providence, R.I., 1982, pp. 141–184.

Institute of Mathematics, University of Bonn, DE-53115 Bonn, Germany

E-mail address: ausoni@math.uni-bonn.de

Department of Mathematics, University of Bergen, NO-5008 Bergen, Norway

E-mail address: dundas@math.uib.no

Department of Mathematics, University of Oslo, NO-0316 Oslo, Norway

E-mail address: rognes@math.uio.no