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Yang-Mills theory depends on a quantum mechanical prop- erty, the mass gap: the quantum particles have positive masses, even though the classical waves travel at the speed of light. Establishing the existence of the Yang-Mills theory and a mass gap will require the in- troduction of fundamental new ideas both in physics and in . Thirty two years ago, Hitchin [7] reduced Yang-Mills self-duality equations from R4 to R2 imposing invariance translation in two directions, and considering an auxiliar vector field, so-called Higgs field. Hitchin’s reduction of Yang-Mills self-duality equations in R2 have the important property of conformal invariance, which allows them to be defined on Riemann surfaces. Let Σ=Σg be a compact (closed and connected) Riemann surface of genus g > 2. Let K = KΣ = T ∗Σ be the canonical line bundle over Σ (its cotangent bundle).

Remark 0.1. Algebraically, Σg is also a complete irreducible nonsingular over C:

dimC(Σg)=1. The paper is organized as follows: in section 1 we recall some basic facts about differential geometry, specifically about smooth vector bundles and holomorphic vector bundles; in section 2, we present the gauge group action over the set of connections and the induced action over the holomorphic structures; in section 3, we define the of Higgs bundles, and we present the two principal constructions: Hitchin construction in 3.2 and Simpson construction in 3.3; finally, in 4, we present the most recent result of the author, published in [17] in terms of homotopy.

1 Preliminary definitions

We shall recall some basic definitions we will need, like bundles and some of their invariants such as rank, degree and slope. We also present the definition of stability. Definition 1.1. For any smooth E → Σ, we denotethe rankof E by rk(E)= r and the degree of E by deg(E)= d. Then, for any smooth complex bundle E → Σ the slope is defined to be deg(E) d µ(E) := = . (1) rk(E) r Definition 1.2. A smooth vector bundle E → Σ is called semistable if µ(F) 6 µ(E) for any F such that 0 ( F ⊆E. Similarly, a vector bundle E → Σ is called stable if µ(F) < µ(E) for any nonzero proper subbundle 0 ( F ( E. Finally, E is called polystable if it is the direct sum of stable subbundles, all of the same slope. Let E → Σ be a complex smooth vector bundle with a hermitian metric on it.

Definition 1.3. A connection dA on E is a 0 1 dA :Ω (Σ, E) → Ω (Σ, E) such that dA(fs)= df ⊗ s + fdAs for any function f ∈ C∞(Σ) and any section s ∈ Ω0(Σ, E) where Ωn(Σ, E) is the set of smooth n-forms of Σ with values in E. Locally: dA = d + A = d + Cdz + Bdz¯ 1 where A is a matrix of 1-forms: Aij ∈ Ω (Σ, E), and B, C are matrix valued functions depending on the hermitian metric on E.

Definition 1.4. When a connection dA is compatible with the hermitian metric on E, i.e.

dhs,ti = hdAs,ti + hs, dAti for the hermitian inner product h·, ·i and for s,t any couple of sections of E, dA is a unitary connection. Denote A(E) as the space of unitary connections on E, for a smooth bundle E → Σ.

2 Remark 1.5. Some authors call the matrix A as a connection and call dA = d + A as its correspoding covariant derivative. We abuse notation and will not distinguish between them. Definition 1.6. The fundamental invariant of a connection is its curvature: 2 0 2 FA := dA = dA ◦ dA :Ω (Σ, E) → Ω (Σ, E) n where we are extending dA to n-forms in Ω (Σ, E) in the obvious way. Locally: 2 FA = dA + A . ∞ 2 FA is C (Σ, E)-linear andcan be consideredas a 2-formon Σ with values in End(E): FA ∈ Ω (Σ, End(E)), or locally as a matrix-valued 2-form.

Definition 1.7. If the curvature vanishes FA = 0, we say that the connection dA is flat. A flat connection gives a family of constant transition functions for E, which in turn defines a representation of the fundamen- tal group π1(Σ) into GLr(C), and the image is in U(r) if dA is unitary. Besides, from Chern-Weil theory, if FA =0, then deg(E)=0.

2 Gauge Group Action

The gauge theory arises first in the context of physics, specifically in general relativity and classical elec- tromagnetism. Here, we present the mathematical formalism. Definition 2.1. A gauge transformation is an automorphism of E. Locally, a gauge transformation g ∈ ∞ Aut(E) is a C -function with values in GLr(C). A gauge transformation g is unitary if g preserves the hermitian inner product. We denote by G the group of unitary gauge transformations. This gauge group G acts on A(E) by conjugation: −1 g · dA = g dAg ∀g ∈ G and for dA ∈ A(E). Remark 2.2. Note that conjugation by a unitary gauge transformation takes a unitary connectionto a unitary connection. We denote by G¯ the quotient of G by the central subgroup U(1), and by GC the complex gauge group. Besides, we denote by BG and by BG¯ the classifying spaces of G and G¯ respectively. There is a homotopical equivalence: G ≃ GC. Definition 2.3. A holomorphic structure on E is a differential operator: 0 0,1 ∂¯A :Ω (Σ, E) → Ω (Σ, E) such that ∂¯A satisfies Liebniz rule: ∂¯A(fs)= ∂f¯ ⊗ s + f∂¯As and the integrability condition, which is precisely vanishing of the curvature

FA =0. ¯ ∂f p,q Here, ∂f = ∂z¯ dz¯, and Ω (Σ, E) is the space of smooth (p, q)-forms with values in E. Locally: 0,1 ∂¯A = ∂¯ + A dz¯ where A0,1 is a matrix valued function. The reader may consult the details in [6],[8]or[9]. Remark 2.4. Denote the space of holomorphic structures on E by A0,1(E), and consider the collection A0,1(r, d)= {A0,1(E): rk(E)= r and deg(E)= d}. Here, the subcollections 0,1 0,1 0,1 Ass (r, d)= {A (E): E is semistable } ⊆ A (r, d),

0,1 0,1 0,1 As (r, d)= {A (E): E is stable } ⊆ A (r, d),

0,1 0,1 0,1 Aps (r, d)= {A (E): E is polystable } ⊆ A (r, d), will be of particular interest for us.

3 3 Moduli Space of Higgs Bundles

A central feature of Higgs bundles, is that they come in collections parametrized by the points of a quasi- : the Moduli Space of Higgs Bundles.

3.1 Moduli space of vector bundles The starting point to classify these collections, is the Moduli Space of Vector bundles. Using Mumford’sGe- ometric InvariantTheory (GIT) from the work of MumfordFogartyand Kirwan [10], first Narasimhan & Se- shadri [11], and then Atiyah & Bott [1], build and charaterize this family of bundles by Morse theory and by their stability. Both classifications are equivalent. We use the one from stability here: Definition 3.1. Define the moduli space of (polystable) vector bundles E → Σ as the quotient 0,1 C N (r, d)= Aps (r, d)/G . Respectively, define the moduli space of stable vector bundles as the quotient 0,1 C Ns(r, d)= As (r, d)/G ⊆ N (r, d). Remark 3.2. The quotient A0,1(r, d)/GC is not Hausdorff so, we may lose a lot of interesting properties. Narasimhan & Seshadri [11] charaterize this variety, and then Atiyah & Bott [1] conclude in their work that: 0,1 0,1 Theorem 3.3 (9. [1]). If GCD(r, d)=1, then Aps = As and N (r, d)= Ns(r, d) becomes a (compact) differential projective algebraic variety with dimension 2 dimC N (r, d) = r (g − 1)+1. ⊟  3.2 Hitchin Construction Hitchin [7] works with the Yang-Mills self-duality equations (SDE) ∗ FA + [φ, φ ] = 0   (2) ∂¯Aφ = 0  1,0 where φ ∈ Ω Σ, End(E) is a complex auxiliary field and FA is the curvature of a connection dA which  is compatible with the holomorphic structure of the bundle E = (E, ∂¯A). Hitchin calls such a φ as a Higgs field, because it shares a lot of the physical and gauge properties to those of the Higgs boson. The set of solutions 0,1 1,0 β(E) := {(∂¯A, φ)| solution of (2)} ⊆ A (E) × Ω (Σ, End(E)) and the collection

βps(r, d) := {β(E)|E polystable, rk(E)= r, deg(E)= d} allow Hitchin to construct the Moduli space of solutions to SDE (2) Y M C M (r, d)= βps(r, d)/G , and Y M C Y M Ms (r, d)= βs(r, d)/G ⊆M (r, d) the moduli space of stable solutions to SDE (2). 0,1 0,1 Y M Y M Remark 3.4. Recall that GCD(r, d)=1 implies Aps = As and so M (r, d)= Ms (r, d). Nitsure [12] computes the dimension of this space in his work: Theorem 3.5 (Nitsure (1991)). The space M(r, d) is a quasi–projective variety of complex dimension 2 dimC(M(r, d)) = (r − 1)(2g − 2).

4 3.3 Simpson Construction An alternative concept arises in the work of Simpson [13]: Definition 3.6. A Higgs bundle over Σ is a pair (E, ϕ) where E → Σ is a holomorphic vector bundle together with ϕ, an endomorphism of E twisted by K = T ∗(Σ) → Σ, the canonical line bundle of the surface Σ: ϕ: E → E ⊗ K. The field ϕ is what Simpson calls Higgs field. There is a condition of stability analogous to the one for vector bundles, but with reference just to subbundles preserved by the endomorphism ϕ: Definition 3.7. A subbundle F ⊂ E is said to be ϕ-invariant if ϕ(F ) ⊂ F ⊗ K. A Higgs bundle is said to be semistable [respectively, stable] if µ(F ) 6 µ(E) [resp., µ(F ) < µ(E)] for any nonzero ϕ-invariant subbundle F ⊆ E [resp., F ( E]. Finally, (E, ϕ) is called polystable if it is the direct sum of stable ϕ-invariant subbundles, all of the same slope. With this notion of stability in mind, Simpson [13] constructs the Moduli space of Higgs bundles as the quotient MH (r, d)= {(E, ϕ)| E polystable }/GC and the subspace H C H Ms (r, d)= {(E, ϕ)| E stable }/G ⊆M (r, d) of stable Higgs bundles. H H Remark 3.8. Again, GCD(r, d)=1 implies M (r, d)= Ms (r, d). See Simpson [13] for details. Finally, Simpson [13] concludes: Proposition 3.9 (Prop. 1.5. [13]). There is a homeomorphism of topological spaces MH (r, d) =∼ MY M (r, d). ⊟

4 Recent Results

1 If we consider a fixed point p ∈ X as a divisor p ∈ Sym (X)= X, and Lp the line bundle that corresponds to that divisor p, we get a complex of the form

k Φ ⊗k E −−−−−→ E ⊗ K ⊗ Lp k 0 ⊗k where Φ ∈ H (X, End(E) ⊗ K ⊗ Lp ) is a Higgs field with poles of order k. So, we call such a complex as a k-Higgs bundle and Φk as its k-Higgs field. As well as fora Higgs bundle,a k-Higgs bundle (E, Φk) is stable (respectively semistable) if the slope of any Φk-invariant subbundle of E is strictly less (respectively less or equal) than the slope of E : µ(E). Finally, (E, Φk) is called polystable if E is the direct sum of stable Φk-invariant subbundles, all of the same slope. Motivated by the results of Bradlow, García–Prada, Gothen [3] and the work of Hausel [5], in terms of the homotopy groups of the moduli space of Higgs bundles and k-Higgs bundles respectively, the author has tried to improve the result of Hausel to higher rank. So far, here is one of the main results. Recall that here, only announce the result, no proof is given. The reader can find the proof in [17]. Theorem 4.1 (Cor. 4.14. [17]). Suppose the rank is either r =2 or r =3, and GCD(r, d)=1. Then, for all n exists k0, depending on n, such that ∼ k = ∞ πj MΛ(r, d) −−−−−→ πj (MΛ (r, d))  for all k > k0 and for all j 6 n − 1. ⊟ There is a lot of interesting recent results related with the homotopy of the moduli space of Higgs bundles, in terms of its stratifications (see [4]), and in terms of Morse Theory and Variations of Hodge Structures (see [17]). Nevertheless, the introduction of a wide bunch of notation would be needed. The author encourages the reader to explore those results in the references above mentioned.

5 References

[1] M. F. Atiyah and R. Bott, Yang-Mills equations over Riemann surfaces, Phil. Trans. Roy. Soc. Lon- don,A 308 (1982), 523–615. [2] S.B. Bradlow, O. García-Prada and P.B. Gothen, What is a Higgs Bundle?, Notices of the AMS, 54, No. 8 (2007), 980–981. [3] S. B. Bradlow, O. García-Prada and P. B. Gothen, Homotopy groups of moduli spaces of represen- tations, Topology 47 (2008), 203–224. [4] P. B. Gothen and R. A. Zúñiga-Rojas, Stratifications on the moduli space of Higgs bundles, Portu- galiae Mathematica, EMS, 74, (2017), 127–148. [5] T. Hausel, Geometry of Higgs bundles, Ph.D. thesis, University of Cambridge, UK, 1998. [6] N.J. Hitchin, Gauge Theory on Riemann Surfaces, Proceedings of the First College of Riemann Surfaces held in Trieste, Italy (1987) 99-118. [7] N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc., 55 (1987), 59–126. [8] S. Kobayashi, Differential Geometry of Complex Vector Bundles, Publications of the Mathematical Society of Japan, Vol. 15, Iwanami Shoten Publ. and Princeton Univ. Press, 1987. [9] M. Lübke and A. Teleman, The Kobayashi-Hitchin Correspondence, World Scientific Publishing Co., 1995. [10] D. Mumford, J. Fogarty and F. Kirwan, Geometric Invariant Theory, Springer, 1994. [11] M.S. Narasimhan and C.S. Seshadri, Stable and Unitary Vector Bundles on a Compact Riemann Surface, The Annals of Mathematics, Second Series Vol. 82 (1965), pp. 540-567. [12] Nitsure, N., Moduli Space of Semistable Pairs on a Curve, Proc. London Math. Soc., (3), 62, (1991), pp. 275-300. [13] C. T. Simpson, Higgs bundles and local systems, Publ. Math. IHÉS 75 (1992), 5–95. [14] C. N. Yang and R. L. Mills, Conservation of isotopic spin and isotopic gauge invariance, Phys. Rev. 96, (1954), 191–195. [15] R. A. Zúñiga-Rojas, Homotopy groups of the moduli space of Higgs bundles, Ph. D. thesis, Univer- sidade do Porto, Portugal, 2015. [16] R. A. Zúñiga-Rojas, On the of The Moduli Space of sigma-Stable Triples and (1,2)- VHS, to appear. arXiv:1803.01936 [math.AG]. [17] R. A. Zúñiga-Rojas, Stabilization of the Homotopy Groups of The Moduli Space of k-Higgs Bun- dles, Revista Colombiana de Matemáticas, 52 (2018) 1, 9-31.

Ronald A. Zúñiga-Rojas Centro de Investigaciones Matemáticas y Metamatemáticas CIMM Project: 820-B8-224 Escuela de Matemática Universidad de Costa Rica UCR San José 11501, Costa Rica e-mail: [email protected]

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