## Yang–Mills theory over surfaces and the Atiyah-Segal theorem (2008)

Citations: | 7 - 5 self |

### BibTeX

@MISC{Ramras08yang–millstheory,

author = {Daniel A. Ramras},

title = {Yang–Mills theory over surfaces and the Atiyah-Segal theorem},

year = {2008}

}

### OpenURL

### Abstract

Abstract. In this paper we explain how Morse theory for the Yang-Mills functional can be used to prove an analogue, for surface groups, of the Atiyah-Segal theorem. Classically, the Atiyah-Segal theorem relates the representation ring R(Γ) of a compact group Γ to the complex K-theory of the classifying space BΓ. For infinite discrete groups, it is necessary to take into account deformations of representations, and with this in mind we replace the representation ring by Carlsson’s deformation K-theory spectrum Kdef(Γ) (the homotopytheoretical analogue of R(Γ)). Our main theorem provides an isomorphism in homotopy K ∗ def (π1Σ) ∼ = K ∗ (Σ) for all compact, aspherical surfaces Σ and all ∗> 0. Combining this result with work of Lawson, we obtain homotopy theoretical information about the stable moduli space of flat connections over surfaces. 1.

### Citations

1470 | Principles of Algebraic Geometry - Griffiths, Harris - 1978 |

812 | Comprehensive Introduction to Differential Geometry - Spivak - 1990 |

417 |
The Yang-Mills equations over Riemann surfaces
- Atiyah, Bott
- 1983
(Show Context)
Citation Context ...oduli space of flat unitary connections over Σ is isomorphic to K 1 (Σ) ∼ = K 1 def (π1Σ). The proof of Theorem 1.1 relies on Morse theory for the Yang-Mills functional, as devoped by Atiyah and Bott =-=[5]-=-, Daskalopoulos [9], and R˚ade [33]; the key analytical input comes from Uhlenbeck’s compactness theorem [39, 40]. In the non-orientable case, we rely on recent work of Ho and Liu [19, 22] regarding r... |

205 | Symmetric spectra
- Hovey, Shipley, et al.
(Show Context)
Citation Context ...f the Yang-Mills flow) but also the modern stable homotopy theory underlying Lawson’s cofiber sequence. His results require, for example, the model categories of module and algebra spectra studied in =-=[11, 12, 23]-=-. Assuming Conjecture 5.3, we know that for Riemann surfaces, Kdef(π1M g ) is free as a ku-module. Hence the Bott map is easily calculated, and one may compute the homotopy groups of Hom(π1(M g ), U)/... |

125 |
modules, and algebras in stable homotopy theory
- Rings
- 1997
(Show Context)
Citation Context ...f the Yang-Mills flow) but also the modern stable homotopy theory underlying Lawson’s cofiber sequence. His results require, for example, the model categories of module and algebra spectra studied in =-=[11, 12, 23]-=-. Assuming Conjecture 5.3, we know that for Riemann surfaces, Kdef(π1M g ) is free as a ku-module. Hence the Bott map is easily calculated, and one may compute the homotopy groups of Hom(π1(M g ), U)/... |

105 |
On the cohomology groups of moduli spaces of vector bundles over curves
- Harder, Narasimhan
- 1975
(Show Context)
Citation Context ...e � by < in this definition, one has the definition of a stable bundle. Given a (smooth) holomorphic structure E on the bundle M × C n , there is a unique filtration (the Harder-Narasimhan filtration =-=[16]-=-) 0 = E0 ⊂ E1 ⊂ · · · Er = E of E by holomorphic sub-bundles with the property that each quotient Di = Ei/Ei−1 is semi-stable (i = 1, . . .,r) and µ(D1) > µ(D2) > · · · > µ(Dr), where µ(Di) = deg(Di) ... |

101 | Lie group valued moment maps
- Alekseev, Malkin, et al.
- 1998
(Show Context)
Citation Context ...ivity for the moduli space of flat connections [19, Theorem 20]. For most surfaces, other work of Ho and Liu [22] gives an alternative method, depending on the theory of quasi-Hamiltonian moment maps =-=[3]-=-. A version of that argument, adapted to the present situation, appears in [35]. 3. Representations and flat connections Let M denote an n-dimensional, compact, connected manifold, with a fixed basepo... |

88 |
Equivariant K-Theory and Completion
- Atiyah, Segal
- 1969
(Show Context)
Citation Context ...ber and Miscenko [7], the right-hand side is isomorphic to the limit of the K-theories of the skeleta BΓ (n), and hence has the structure of a complete ring. The classical theorem of Atiyah and Segal =-=[4, 6]-=- states that the map α becomes an isomorphism after completing R(Γ) appropriately. In this article, we will explore an analogue of this result for discrete groups Γ admitting finite dimensional classi... |

88 |
with L p -bounds on curvature
- Uhlenbeck
- 1982
(Show Context)
Citation Context ...rem 1.1 relies on Morse theory for the Yang-Mills functional, as devoped by Atiyah and Bott [5], Daskalopoulos [9], and R˚ade [33]; the key analytical input comes from Uhlenbeck’s compactness theorem =-=[39, 40]-=-. In the non-orientable case, we rely on recent work of Ho and Liu [19, 22] regarding representation spaces of non-orientable surface groups and Yang-Mills theory over non-orientable surfaces. The lin... |

73 |
Construction of universal bundles
- Milnor
- 1956
(Show Context)
Citation Context ...ed; in this paper we will need to use universal bundles for Sobolev gauge groups, where the simplicial model may not give an actual universal bundle. Hence it is more convenient to use Milnor’s model =-=[29]-=-, which is functorial and applies to all topological groups. There is a natural zig-zag of weak equivalences connecting these two versions of the classifying space, and this gives a zig-zag connecting... |

68 |
Elementary structure of real algebraic varieties, Ann. of Math 66
- Whitney
- 1957
(Show Context)
Citation Context ...le [17]. Since compact CW complexes have finitely many path components, the proof is complete. ✷ Remark 3.8. We note that the the previous lemma holds even if G is not compact, by a result of Whitney =-=[41]-=- regarding components of varieties. We can now prove the result which connects representation theory with YangMills theory. Proposition 3.9. Assume p > n/2 (and if n = 2, assume p > 4/3), k � 1, and k... |

48 |
Categories and cohomology theories
- Segal
- 1974
(Show Context)
Citation Context ...of Lawson’s deformation K-theory spectrum as well. Given any topological abelian monoid A (for which the inclusion of the identity is a cofibration), one may apply Segal’s infinite loop space machine =-=[36]-=- to produce a connective Ω-spectrum; equivalently the bar construction BA is again an abelian topological monoid and one may iterate. In particular, the zeroth space of this spectrum is exactly ΩBA. T... |

38 |
Characters and cohomology of finite groups
- Atiyah
- 1961
(Show Context)
Citation Context ...ber and Miscenko [7], the right-hand side is isomorphic to the limit of the K-theories of the skeleta BΓ (n), and hence has the structure of a complete ring. The classical theorem of Atiyah and Segal =-=[4, 6]-=- states that the map α becomes an isomorphism after completing R(Γ) appropriately. In this article, we will explore an analogue of this result for discrete groups Γ admitting finite dimensional classi... |

29 |
Homology fibrations and the “group-completion” theorem
- McDuff, Segal
(Show Context)
Citation Context ...ory to the Milnor version. 2.1. Group completion in deformation K-theory. The starting point for our work on surface groups is an analysis of the consequences of McDuff-Segal Group Completion theorem =-=[28]-=- for deformation K-theory, as carried out in [34]. Here we recall that result and explain its consequences for surface groups. Given a topological monoid M and an element m ∈ M, we say that M is stabl... |

26 |
The topology of the space of stable bundles on a compact Riemann surface
- Daskalopoulos
- 1992
(Show Context)
Citation Context ... unitary connections over Σ is isomorphic to K 1 (Σ) ∼ = K 1 def (π1Σ). The proof of Theorem 1.1 relies on Morse theory for the Yang-Mills functional, as devoped by Atiyah and Bott [5], Daskalopoulos =-=[9]-=-, and R˚ade [33]; the key analytical input comes from Uhlenbeck’s compactness theorem [39, 40]. In the non-orientable case, we rely on recent work of Ho and Liu [19, 22] regarding representation space... |

25 |
On the Yang–Mills heat equation in two and three dimensions
- Råde
- 1992
(Show Context)
Citation Context ...tions over Σ is isomorphic to K 1 (Σ) ∼ = K 1 def (π1Σ). The proof of Theorem 1.1 relies on Morse theory for the Yang-Mills functional, as devoped by Atiyah and Bott [5], Daskalopoulos [9], and R˚ade =-=[33]-=-; the key analytical input comes from Uhlenbeck’s compactness theorem [39, 40]. In the non-orientable case, we rely on recent work of Ho and Liu [19, 22] regarding representation spaces of non-orienta... |

22 |
Triangulations of algebraic sets. In Algebraic geometry (Proc
- Hironaka
- 1974
(Show Context)
Citation Context ...is algebraic. Since π1M is finitely generated (by k elements, say), Hom(π1M, G) is the subvariety of G k cut out by the relations in π1M. So this space is a real algebraic variety, hence triangulable =-=[17]-=-. Since compact CW complexes have finitely many path components, the proof is complete. ✷ Remark 3.8. We note that the the previous lemma holds even if G is not compact, by a result of Whitney [41] re... |

22 |
Vial let, On the bundle of connections and the gauge orbit manifold in Yang–Mills theory
- Mitter, M
- 1981
(Show Context)
Citation Context ... sum with the trivial connection τ on the trivial line bundle. Since the based gauge groups G k+1 0 (n) act freely on Ak (n), and the projection maps are locally trivial principal G k+1 0 (n)-bundles =-=[30]-=-, a basic result about homotopy orbit spaces [5, 13.1] shows that we have a weak equivalence ( k Aflat(n)/G k 0(n) ) (4) EG k+1 (n) × G k+1 (n) A k flat(n) ≃ −→ EU(n) × U(n) It now follows from (4) th... |

14 |
modules, and algebras in infinite loop space theory
- Rings
(Show Context)
Citation Context ...f the Yang-Mills flow) but also the modern stable homotopy theory underlying Lawson’s cofiber sequence. His results require, for example, the model categories of module and algebra spectra studied in =-=[11, 12, 23]-=-. Assuming Conjecture 5.3, we know that for Riemann surfaces, Kdef(π1M g ) is free as a ku-module. Hence the Bott map is easily calculated, and one may compute the homotopy groups of Hom(π1(M g ), U)/... |

12 |
Uhlenbeck compactness
- Wehrheim
- 2004
(Show Context)
Citation Context ...rem 1.1 relies on Morse theory for the Yang-Mills functional, as devoped by Atiyah and Bott [5], Daskalopoulos [9], and R˚ade [33]; the key analytical input comes from Uhlenbeck’s compactness theorem =-=[39, 40]-=-. In the non-orientable case, we rely on recent work of Ho and Liu [19, 22] regarding representation spaces of non-orientable surface groups and Yang-Mills theory over non-orientable surfaces. The lin... |

11 | An application of transversality to the topology of the moduli space of stable bundles - Daskalopoulos, Uhlenbeck - 1995 |

10 |
A local analytic splitting of the holonomy map on flat connections
- Fine, Kirk, et al.
- 1994
(Show Context)
Citation Context ..., but there does not appear to be a complete reference. Some of the results to follow may be found in Morita’s books [32, 31], and a close relative of the main result is stated in the introduction to =-=[13]-=-.YANG-MILLS THEORY OVER SURFACES AND THE ATIYAH-SEGAL THEOREM 23 Most proofs will be left to the reader; these are generally tedious but straightforward unwindings of the definitions. Usually a good ... |

10 |
The real locus of an involution map on the moduli space of flat connections on a Riemann surface
- Ho
(Show Context)
Citation Context ... deck transformation τ : Mg → Mg induces an involution ˜τ : ˜ P → ˜ P, and ˜τ acts on the space Ak ( ˜ P) by pullback. Connections on P pull back to connections on ˜ P, and in fact, as observed by Ho =-=[18]-=-, the image of the pullback map is precisely the set of fixed points of τ. Hence we have a homeomorphism Ak (P) ∼ = Ak ( ˜ P) eτ , which we treat as an identification. The Yang-Mills functional L is i... |

9 | Transversal mappings and flows. An appendix by Al - Abraham, Robbin - 1967 |

9 |
An introduction to the topology of the moduli space of stable bundles on a Riemann surface
- Thaddeus
- 1995
(Show Context)
Citation Context .... Connectivity of Hom(π1(M), U(n)) follows from Proposition 3.9, because any U(n) bundle over a Riemann surface which admits a flat connection is trivial (for a beautiful and elementary argument, see =-=[38]-=-). ✷ Corollary 4.12. Let M be a compact, non-orientable, aspherical surface. Then for any n � 1, the representation space Hom(π1M, U(n)) has two connected components, and if ρ ∈ Hom(π1M, U(n)) and ψ ∈... |

8 | Derived representation theory and the algebraic K-theory of fields
- Carlsson
(Show Context)
Citation Context ...oomed to failure, though. Let Rep(Γ) denote the topological monoid of unitary representation spaces, and let Gr denote the Grothendieck group functor. Carlsson’s deformation K-theory spectrum Kdef(Γ) =-=[8]-=- is a lifting of the functor Gr(π0Rep(Γ)) to the category of spectra, or in fact, ku-modules, in the sense that π0Kdef(Γ) ∼ = Gr(π0Rep(Γ)) . As we will describe in Section 2, the deformation K-theory ... |

7 |
Differential and Riemannian manifolds, volume 160 of Graduate Texts in Mathematics
- Lang
- 1995
(Show Context)
Citation Context ...), and at time 0 is V (A). ThisYANG-MILLS THEORY OVER SURFACES AND THE ATIYAH-SEGAL THEOREM 9 is clearly a Lipschitz vector field and hence its integral curves vary continuously in the initial point =-=[24]-=-, completing the proof. ✷ Remark 3.5. With a bit more care, one can prove Lemma 3.4 under the weaker assumptions k � 1 and kp > n. The basic point is that these assumptions give an embedding L p k (M)... |

6 |
Lectures of Smale on Differential Topology, mimeographed lecture notes
- Abraham
- 1961
(Show Context)
Citation Context ...֒→ L1 (M) (and similarly after restricting to a smooth curve in M). Working in local coordinates, one can deduce continuity of the holonomy map from the fact that limits commute with integrals in L1 (=-=[0, 1]-=-). Lemma 3.6. Assume p > n/2 (and if n = 2, assume p � 4/3). If G is connected, then each G k+1,p 0 (P)-orbit in A k,p flat (P) contains a unique G∞ 0 (P)-orbit of smooth connections. Proof. The assum... |

6 |
and Chiu-Chu Melissa Liu. Connected components of spaces of surface group representations
- Ho
- 2005
(Show Context)
Citation Context ...π1(M)) ∼ = K ∗ (M). This result will be proven in Section 5. For non-orientable surfaces, there is actually an isomorphism on π0 as well; this is just a re-interpretation of the results of Ho and Liu =-=[20, 22]-=-. We note that when M = S 1 ×S 1 , Theorem 1.1 follows from Lawson’s product formula Kdef(Γ1 × Γ2) ≃ Kdef(Γ1) ∧ku Kdef(Γ2) [27] together with his calculation of Kdef(Z) as a ku-module [26].YANG-MILLS... |

6 |
Geometry of differential forms, volume 201 of Translations of Mathematical Monographs
- Morita
- 2001
(Show Context)
Citation Context ...he based gauge group into account (Proposition 7.4). This is essentially well-known, but there does not appear to be a complete reference. Some of the results to follow may be found in Morita’s books =-=[32, 31]-=-, and a close relative of the main result is stated in the introduction to [13].YANG-MILLS THEORY OVER SURFACES AND THE ATIYAH-SEGAL THEOREM 23 Most proofs will be left to the reader; these are gener... |

5 |
and Chiu-Chu Melissa Liu. Yang–Mills connections on nonorientable surfaces
- Ho
(Show Context)
Citation Context ... discrete groups. As an application of Theorem 1.1, we obtain homotopy-theoretical information about the stable moduli space of flat unitary connections over a compact, aspherical surface. Ho and Liu =-=[19, 21]-=- have shown that for each n, the moduli space of flat U(n)-connections is connected (in fact, their results apply to flat G-connections for any compact, connected Lie group G). In Section 6, we combin... |

5 | On the connectedness of moduli spaces of flat connections over compact surfaces
- Ho, Liu
(Show Context)
Citation Context ... discrete groups. As an application of Theorem 1.1, we obtain homotopy-theoretical information about the stable moduli space of flat unitary connections over a compact, aspherical surface. Ho and Liu =-=[19, 21]-=- have shown that for each n, the moduli space of flat U(n)-connections is connected (in fact, their results apply to flat G-connections for any compact, connected Lie group G). In Section 6, we combin... |

4 |
String topology of classifying spaces
- Gruher, Salvatore
(Show Context)
Citation Context ...he homotopy orbit space of the adjoint action of G on itself. (This result is well-known for any group G, but the only reference of which I am aware is the elegant proof given by Gruher in her thesis =-=[15]-=-). To begin, note that (G Ad )hG = Hom(Z, G)hG. Connections A over the circle are always flat, and hence give rise to holonomy representations of π1S 1 = Z: A ↦→ (ρA : Z → G). After modding out based ... |

4 |
Derived Representation Theory of Nilpotent Groups
- Lawson
- 2004
(Show Context)
Citation Context ...es of the group in question. This spectrum may be constructed as the K-theory spectrum associated to a topological permutative category of representations. This approach originated in Lawson’s thesis =-=[25]-=-, and is explained in detail in [34]. Here we will take a more naive, but essentially equivalent, approach. The present viewpoint makes clear the precise analogy between deformation K-theory and the c... |

4 | The product formula in unitary deformation K-theory - Lawson - 2006 |

4 |
Geometry of characteristic classes, volume 199 of Translations of Mathematical Monographs
- Morita
- 2001
(Show Context)
Citation Context ...he based gauge group into account (Proposition 7.4). This is essentially well-known, but there does not appear to be a complete reference. Some of the results to follow may be found in Morita’s books =-=[32, 31]-=-, and a close relative of the main result is stated in the introduction to [13].YANG-MILLS THEORY OVER SURFACES AND THE ATIYAH-SEGAL THEOREM 23 Most proofs will be left to the reader; these are gener... |

4 |
Stable representation theory of infinite discrete groups
- Ramras
- 2007
(Show Context)
Citation Context ...aces, other work of Ho and Liu [22] gives an alternative method, depending on the theory of quasi-Hamiltonian moment maps [3]. A version of that argument, adapted to the present situation, appears in =-=[35]-=-. 3. Representations and flat connections Let M denote an n-dimensional, compact, connected manifold, with a fixed basepoint m0 ∈ M. Let G be a compact Lie group, and P π → M be a smooth principal G-b... |

3 |
A K-theory on the category of infinite cell complexes
- Buhˇstaber, Miˇsčenko
- 1968
(Show Context)
Citation Context ...lasses” of representations. Each representation ρ : Γ → U(n) induces a vector bundle Eρ over the classifying space BΓ, and this provides a map R(Γ) α → K0 (BΓ). By a theorem of Buhstaber and Miscenko =-=[7]-=-, the right-hand side is isomorphic to the limit of the K-theories of the skeleta BΓ (n), and hence has the structure of a complete ring. The classical theorem of Atiyah and Segal [4, 6] states that t... |

2 |
The Bott cofiber sequence in deformation K-theory. Available at http://math.mit.edu/∼tlawson
- Lawson
- 2006
(Show Context)
Citation Context ...and Liu [20, 22]. We note that when M = S 1 ×S 1 , Theorem 1.1 follows from Lawson’s product formula Kdef(Γ1 × Γ2) ≃ Kdef(Γ1) ∧ku Kdef(Γ2) [27] together with his calculation of Kdef(Z) as a ku-module =-=[26]-=-.YANG-MILLS THEORY OVER SURFACES AND THE ATIYAH-SEGAL THEOREM 3 Theorem 1.1 suggests that deformation K-theory is the proper setting in which to study Atiyah-Segal type results. In particular, we exp... |

2 | Topology for analysis. Robert E. Krieger Publishing Co - Wilansky - 1983 |

1 | Excision in deformation K-theory
- Ramras
- 2007
(Show Context)
Citation Context ...or any discrete group Γ with a compact classifying space, the deformation K-groups of Γ will agree with K∗ def (BΓ) in high enough degrees. We note that the author’s excision result for free products =-=[34]-=- and Lawson’s product formula indicate that this phenomenon should be stable under both free and direct products of discrete groups. As an application of Theorem 1.1, we obtain homotopy-theoretical in... |