26

The very short answer is that there is no direct connection between gauge theory (which is living on some perhaps hypothetical smooth 4-manifold) and triangulation of some high-dimensional topological manifold. Here are some remarks to justify that statement. The question addressed in Quinn's expository article is not whether 4-manifolds can be ...


19

Often in the literature by "Chern-Simons theory" is meant by default $G$-Chern-Simons theory whose gauge group is a connected and simply connected semisimple compact group $G$, such as $G = SU$. In this case it so happens that all $G$-principal bundles on a 3-manifold $\Sigma_3$ are trivializable, and hence one can identify the space of G-principal ...


18

Please do not ignore the other author, Peter Kronheimer. Based on all of the material I've read, I do not agree with your belief about the book. I think it is more detailed than you will find elsewhere which covers all of that material. Here are some useful alternatives, though: Take that book and replace the structure group $SU(2)$ by $SO(3)$. This route ...


17

Here is a mid 1970s point of view, courtesy of Atiyah-Patodi-Singer. Suppose you have a complex vector bundle $E$ of rank $r$ over a smooth manifold $M$. A polynomial function $P$ on the space of $r\times r$ matrices is called invariant if $P(T AT^{-1})=P(A)$ for any $r\times r$ complex matrix $A$ and any invertible $r\times r$ matrix $T$. ...


17

Proof of (1): (a). Suppose $X$ and $Y$ are $G$-spaces, the action of $G$ on $X$ is free, and $X\to X/G$ is a principal bundle, then the space of $G$-equivariant maps $$ F(X,Y)^G $$ is the same thing as the space of sections of the fibration $X\times_G Y \to X/G$. (b). If $E\to B$ is a Hurewicz fibration, with $B$ connected and the fiber contractible, ...


17

By itself, a (Yang-Mills) instanton is a classical concept. It is a solution of the classical Yang-Mills equations (considered on a manifold with a Riemannian, rather than a Lorentzian, metric), such that the classical Yang-Mills action functional evaluated on this solution is finite (not divergent). Also, the concept of instanton is not restricted to Yang-...


13

There's something about the notation you should know before you get confused when trying to do non-abelian gauge theory. The second term in the field strength should involve a combination of the wedge product of forms and the Lie bracket: the field strength (in the case of an arbitrary gauge group $G$ with Lie algebra $\mathfrak{g}$) should actually be $$F = ...


13

Yes, such a Kähler form always exists: Embed $G$ as a matrix group in $\mathrm{SU}(n)$ for some $n$ and then let $G_\mathbb{C}\subset \mathrm{SL}(n,\mathbb{C})\subset M_n(\mathbb{C})$ be the complexification. Choose a Kähler form on this latter vector space, pull it back to $G_\mathbb{C}$ and then, using the compactness of $G$, average its pullbacks under ...


13

This was conjectured by Witten in his paper in his paper Monopoles and four-manifolds. The conjecture says that if $X$ has Seiberg-Witten simple type (meaning that $SW_X(\mathfrak{s})$ is nonzero only when the moduli space associated to $X$ is 0-dimensional) and satisfies some mild homological conditions (including $b_1(X)=0$ and $b^+(X)>1$ odd), then $X$...


13

I'm afraid that that article does not do a lot of details in the introductory Section 0, just because more complete explanations were already available in earlier articles of mine. Here are some brief answers. Right now, I don't have the time to write out the explanations in greater detail. This is a consequence of the claim, proved in the reference [Br$_2$...


11

If $G$ is a connected compact Lie group and the base $M$ is homotopy equivalent to a finite cell complex, then a rough count of the set $[M, BG]$ is given by the rational homotopy theory. The point is that the classifying space $BG$ is rationally homotopy equivalent to the product of Eilenberg-MacLane spaces, e.g. for $G=U(n)$ each Eilenberg-MacLane ...


11

It's been a while but I remember that I found John Morgan's "An introduction to gauge theory" quite helpful when I was first trying to read about 4-manifolds and gauge theory. While it doesn't go nearly as far as Donaldson-Kronheimer, it does give a gentle introduction to gauge theory. Here's the complete reference: Morgan, John W., An introduction to gauge ...


10

I may be missing something :-) I'm not seeing where Kronheimer shows $$ \text{hyper-Kähler + simply connected} \Rightarrow \text{ALE} \tag{?} $$ as you claim. But in [2] he shows that "every hyper-Kähler ALE 4-manifold is isometric to a member of one of the families obtained in [1]." As said members are (I believe) simply connected, this means that $$ \text{...


10

Let me clarify a couple of issues from the previous answers/comments: 1) The linearization of $Rc-\tfrac{1}{2}Rg$ has mixed signs, and for this reason the equation $\partial_t g=-(Rc-\tfrac{1}{2}Rg)$ is bad, a kind of coupled backwards/forwards heat equation, i.e. no short time existence. 2) The linearization of $Rc$ is elliptic however (after fixing the ...


9

Here is one problem. The instantons or the monopoles are critical points of certain energy functionals and thus they satisfy Euler-Lagrange equations. These are second-order p.d.e.'s. However the Yang-Mills equations or the Seiberg-Witten equations are first-order p.d.e.-s. The reason for this fortunate accident is that the instantons and the monopoles ...


9

First, it follows from the work of Kirby and Siebenmann that in dimensions $\le 6$ PL and DIFF categories are equivalent. In particular, if you are working in dimension 4 (where gauge-theoretic invariants are mostly used) then the answer to your your question is negative. Starting in dimension 7, there are smooth manifolds which are PL-equivalent but not ...


9

I'm afraid that this answer will be somewhat physics-y. Apologies if this is deemed inappropriate for MO. First of all, I think that it is slightly misleading to say that one has "fractional charge in $SU(N)$ gauge theory." A careful reading of the lectures linked in the question reveals that one actually has a Yang-Mills-Higgs theory with group $G$ and ...


9

[This version has been updated in response to comments from the OP] Recall that $B$ gives an endofunctor of the category of abelian topological groups. We can apply $B$ to the obvious map $\mathbb{Z}\to\mathbb{Z}/2$ to get a homomorphism $S^1=B\mathbb{Z}\to B\mathbb{Z}/2$, then compose with $\det\colon U(2)\to S^1$ to get a homomorphism $U(2)\to B\mathbb{Z}...


9

The slant product is defined in many basic algebraic topology books, eg Hatcher's. If you are willing to work over a field, then you'd take the Kunneth decomposition of $p_1 = \sum x_j \otimes m_j$ with $x_j \in H^*(X)$ and $m_j \in H^*(M(P))$. Then $p_1/\beta = \sum x_j(\beta) \otimes m_j \in H^*(M(P))$, where the evaluation $x_j(\beta)$ is declared to be $...


9

A linguistic remark: "Instantons" are the same mathematically to "solitons", particle-like solutions of classical field theories (explaining the suffix "on"). Unlike solitons, instantons are structures in time (explaining the prefix "instant"). A mathematical remark (using Donaldson's book on Yang-Mills Floer homology, appendix C of section 2.8), which ...


8

This is an edited extract from a book in preparation (Bruner, Catanzaro, May) tentatively titled Characteristic Classes and is therefore overlong for an answer. This is similar to Denis Nardin's answer, but more bundle theoretic; he did refer to an old Memoir of mine, so I thought I'd give an answer. Let $N$ be a closed normal subgroup of a topological ...


8

Part of the confusion is that your positional notational convention is not the standard one. In most books, what you are writing as $R_{ab}{^c}_d$ would be written as $R{^c}_{dab}=-R{^c}_{dba}$ (though the letters used would probably be different). That said, the condition $R{^c}_{cab}=0$ is the condition that the connection (locally) admit a parallel ...


8

Another good reference is Chris Herald's paper. Legendrian cobordism and Chern-Simons theory on 3-manifolds with boundary. Comm. Anal. Geom. 2 (1994), no. 3, 337–413. It is an easy exercise with Poincare duality to see that the image of the map in cohomology (so at the level of "Zaraski" tangent spaces) $$ H^1(Y;ad_\rho) \to H^1(\Sigma;ad_\rho) $$ is half ...


8

There are certainly closed hyperbolic 3-manifolds that embed in $\mathbb R^4$, that also have arbitrarily big volume. A construction goes as follows: construct a Mazur manifold with one 1-handle and one 2-handle. More specifically, by attaching a 1-handle to $D^4$ you get $D^3 \times S^1$. Then you attach a 2-handle along a knot $K \subset \partial (D^3 \...


8

The groups $G^{\vee}$ and $G'$ are the same. It is clear from the description in the paper you refer to. Their definitions are the same as well, just swapping roots and coroots. You can do it in compact Lie groups or reductive algebraic groups. There is further definition of $G^{\vee}$ from geometric Satake. Its physical significance has been looked into ...


7

this strand model ("strand" = your "rope") of gauge interactions is the brain child of Christoph Schiller, who has written a 400+ page textbook on his theory. Chapter 9 systematically goes through the various groups, U(1), SU(2), SU(3) --- I guess this is the source you want to study if you wish to pursue this idea. For a discussion of the physics behind ...


7

Picture changing operators in supergeometry and superstring theory, Alexander Belopolsky (1997).


7

For the case of compact $G$, see Proposition 3.27 in: Freed, Daniel S. Classical Chern-Simons theory. I. Adv. Math. 113 (1995), no. 2, 237–303. For the case of complex reductive $G$, see Theorem 61 in: Sikora, Adam S. Character varieties. Trans. Amer. Math. Soc. 364 (2012), no. 10, 5173–5208.


7

Yes, the space of gauge orbits of connections is paracompact (even when you the use Fréchet topology). First, the space of all connections is paracompact since it is an affine space modelled on a nuclear Fréchet space (and/or it is metrisable). Narasimhan & Ramadas (Geometry of SU(2) Gauge Fields) showed that the action of the group of gauge ...


7

The tangent bundle is a natural bundle of first order: $\phi\mapsto T\phi$ is an injective group homomorphism $\operatorname{Diff}(M) \to \operatorname{Gau}(TM)$. You can view $TM$ as a vector bundle $E\to M$ together with a 1-form $\theta\in \Omega^1(M,E)$ with values in $E$ which is a linear isomorphism fiberwise (called a soldering form). For a connection ...


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