22
votes

Accepted

### About Simon Donaldson's book on four dimensional manifold

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 ...

- 16.1k

19
votes

### What is an "Instanton" in classical gauge theory? (to a mathematician)

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 ...

- 17.8k

14
votes

Accepted

### Does $(S^1\times S^3)\#(S^1\times S^3)\#(S^2\times S^2)$ admit a symplectic form?

No, $M$ is not symplectic. Consider a double cover $\tilde{M}$ of $M$ along one of the $S^1$ components. Then it is not hard to prove that $\tilde{M}$ is diffeomorphic with $(S^1\times S^3)\#2(S^1\...

- 3,331

11
votes

### About Simon Donaldson's book on four dimensional manifold

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 ...

- 714

11
votes

Accepted

### What is an "Instanton" in classical gauge theory? (to a mathematician)

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 ...

- 16.1k

9
votes

Accepted

### Uhlenbeck's theorem novelty

Denote by $A$ the connection and by $F_A$ its curvature. Then
$$dA=F_A-A\wedge A. $$
If $A$ is in Coulomb gauge we have an additional equation
$$d^*A=0. $$
The advantage is that the operator $d\...

- 32.4k

9
votes

Accepted

### A fibration of classifying spaces

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 ...

- 29.2k

9
votes

### Homotopy Type of the Based Mapping Space $Map_*^{(k,l)}(\mathbb{C}P^2,BU(2))$

[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}...

- 49.8k

9
votes

Accepted

### Gottsche Nakajima Yoshioka define a weird slant product

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 $p_1 = \sum x_j \otimes m_j$ ...

- 17.8k

8
votes

Accepted

### Flat connections on 3-manifold with boundary

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 ...

- 2,904

8
votes

Accepted

### Symmetries of non-Riemannian curvature tensor

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 $R{^c}_{dab}=-R{^c}_{dba}$ (...

- 99.4k

8
votes

### Homology Sphere Embedding into $\mathbb R^4$

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 ...

- 9,759

8
votes

Accepted

### Is the space of connections modulo gauge equivalence paracompact?

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 ...

- 5,172

8
votes

### Langlands dual group in math vs. Goddard-Nyuts-Olive dual group in physics

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 ...

- 11.5k

7
votes

### Literature for gauge field theory on the lattice in geometrical formulation

Rudolph&Schmidt just released their second volume Differential Geometry and Mathematical Physics II: Fiber bundles, Topology and Gauge fields, and it covers exactly this stuff.
These two volumes ...

Community wiki

7
votes

### Flat connections on 3-manifold with boundary

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 ...

- 8,094

7
votes

### Gauge group of tangent bundle and diffeomorphism group

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 $...

- 24.5k

7
votes

Accepted

### $spin_{\mathbb{C}}$ Connection and Charge Parity

I think this is how a physicst would treat spin_c structures: Suppose $(M,g)\cong(\mathbb{R}^n,g_{ij}\mathrm dx^i\mathrm dx^j)$ is a coordinate chart. We can then define $n$ complex $2^{\lceil\frac{n}{...

- 4,444

6
votes

Accepted

### Geometric Construct for Integrating Symmetric Tensors?

Here is one way to construct all "local" gauge-invariant quantities out of a symmetric tensor $A_{ij}$. It would be up to you to decide how it meshes with the intuition you gained from your ...

- 17.8k

6
votes

### Homology Sphere Embedding into $\mathbb R^4$

L Z Gao and S T Yau showed in Invent Math 85 (1986) 637-652 that any compact 3 manifold admits a metric of negative Ricci curvature .J Lohkamp proved that any manifold of dimension 3 or higher admits ...

- 4,031

6
votes

Accepted

### Monopole Floer Homology vs. Heegaard-Floer theory

Ozsvath and Szabo constructed HF as a topological interpretation of SWF, and they noticed many links between the two. Roughly speaking, their Euler characteristics are the same, and the analog of the ...

- 16.1k

6
votes

### About Simon Donaldson's book on four dimensional manifold

A quite "low-brow" book you might find enjoyable reading is The Wild World of Four manifolds, which surveyed the whole theory beginning from handle body decomposition. It has a long list of reference ...

- 5,941

6
votes

### What is an "Instanton" in classical gauge theory? (to a mathematician)

Generally speaking, you could say they are a special type of solution to the field equations of gauge theories. More specifically, an instanton is a classical solution in a classical Euclidean field ...

- 4,397

6
votes

### 1-dimensional pure gauge theory

Just to add something about Gauss' law to the excellent previous answer: The Hamiltonian of a pure gauge theory, as initially derived from the Lagrangean, typically has the structure $H=E^2 +B^2 - A_0 ...

- 4,386

6
votes

Accepted

### Different definitions of "charged spinors": "bundle splicing" vs. "twisted spinor bundles"

I'll assume that the vector space "$V$" occuring in constructions (1) and (2) doesn't have to be the same. In that case I'll rename vector space in construction (2) to "$W$."
Then ...

- 1,679

6
votes

Accepted

### In search of a combinatorial proof on particular set of partitions

Proposition. A partition $\lambda \in \mathcal{syP}_0$ iff it is empty, or both of the following hold:
$\lambda'_1 - \lambda_1 = 1$,
the partition $\mu$ obtained by removing first row and column of $\...

- 5,161

5
votes

### A fibration of classifying spaces

By functoriality there is a map $BG\to B(G/N)$. Let $X$ to be its homotopy fiber. Then you can get a fiber sequence
$\Omega X \to G \to G/N \to X \to BG \to B(G/N)$
using the fact that $\Omega BG = ...

- 15.8k

5
votes

Accepted

### What is the space for the coefficients of the connection 1-form of a connection in a vector bundle?

This answer just extends the remark by Liviu Nicolaescu above. For a general connection on a vector bundle with structure group $G$, you cannot say anything about the connection coefficients. (As an ...

- 1,564

5
votes

### Curvature as infinitesimal holonomy

I have no idea about the sidequestion. For the main question, there is a general answer applicable for any parallel transport, not only for holonomy.
Let $\gamma_{s}$ a family of smooth paths such ...

5
votes

### What is the BRST-anti-BRST formalism?

Jim. I think the following references might be helpful:
Grégoire, Philippe; Henneaux, Marc. Hamiltonian BRST--anti-BRST theory. Communications in Mathematical Physics 157 (1993), no. 2, 279--303. ...

- 17.8k

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