## Errata for Gauge Fields, Knots and Gravity

#### (by John Baez and Javier P. Muniain, published by World Scientific, Singapore, 1994)

I begin by listing known errors that appear only in the original version. A new version is now available from World Scientific, in which most of these should be fixed. Near the end of this page you can see errors that remain in the second version.
• p. 3 line 11 - "case" should be "cast".
• p. 9 - There should not be a minus sign in the third displayed equation on this page: duality corresponds to the transformation E → i E.
• p. 9 - In Exercise 1, we should have ik × E = ω E.
• p. 10 - We should have curl E + dB/dt = -jm.
• p. 27 - "Often is is" should read "Often it is".
• p. 32 - Not a typo but an annoying change in notation: throughout most of this section we are studying a map φ: M -> N, but in the sentence beginning "Using our new jargon..." we momentarily switch to a map φ: N -> M.
• p. 33 - In the first equation on this page, the tangent vector lies in Tφ(p)N, not "Tφ(p)M".
• p. 33 - In Exercise 18, we must assume that φ is a diffeomorphism. Also, on line 13 it should read "to obtain a vector field φ*v on N satisfying...".
• p. 33 - In Exercise 19 the minus signs are mixed up. We should have:

φ*x = (cos θ) ∂x + (sin θ)∂y

φ*y = -(sin θ) ∂x + (cos θ) &party

• p. 37 - The roles of ψ and φ should be switched in the formula for [v,w](f)(p). Also, the d's in d2/dtds should be ∂'s, and there should be parentheses around the whole expression being differentiated.
• p. 39 - on the 6th line from the bottom, remove the words "in the" at the end.
• p. 46 - in Fig. 3, v should be -v.
• p. 48 - on the 11th line from the bottom, this line of the equation should read ((φ*v)f)(q). In other words, q instead of p.
• p. 48 - on the 5th line from the bottom, φ* should be φ*
• p. 49 - on the 5th line from the bottom, delete "do".
• p. 53 - on the 6th line from the bottom, the right hand side of the equation should read ∂μ*x'λ). In other words, x' instead of x.
• p. 55 - I misspelled "Grassmann" as "Grassman" in the quotation at the beginning of this section, and also everywhere else: for example, on page 62 and page 155.
• p. 56 - The sentence beginning on the tenth line from the bottom should read "suppose V has a basis dx, dy, dz"
• p. 57 - On line 2, in the equation, wx dy should be wx dx.
• p. 57 - Line 3 should begin with an equals sign.
• p. 58 - On the 7th line from the bottom, "Λ2(V) into ΛV" should be "Λ2V into elements of ΛV" - delete parentheses and insert "into elements of".
• p. 64 - On line 14, it should read "we need the right-hand rule to define the curl".
• p. 67. On line 15, we should have Ωp(N) instead of Ωp(M).
• p. 74 - On line 11, "that stretched" should be "that stretches"
• p. 74 - On the third line from the bottom, g(v,w) + cg(v,w') should be cg(v,w) + g(v,w').
• p. 80 - On the third line from the bottom, g(ei, fj) should be replaced by <ei, fj>. (These should be angle brackets, not less than or greater than signs!)
• p. 120 - On line 9, "an open set U of M" should be "an open set U of M containing p"
• p. 123 - Should be "Hp(M) = Zp(M)/Bp(M)"
• p. 124 - H0(M) = R if and only if M is connected.
• p. 124 - H1(M) = 0 if M is simply connected, but not only if. (In general, it's zero only if the abelianization of the fundamental group of M is a torsion group, which is a weaker condition than M being simply connected.)
• p. 135 - On the 9th line from the bottom, insert the word "field" after "magnetic".
• p. 142 - On line 6, the n-torus Tn should be replaced with S1 × Sn-1. This is the same thing when n = 2, as in Figures 17 and 18.
• p. 142 - Exercise 103 is correct, but what's more relevant is the fact that S1 × Sn-1 has a closed but not exact 1-form on it, which we may call dθ, coming from the closed but not exact 1-form on S1.
• p. 143 - In general, we get a wormhole by removing one point from S1 × Sn-1.
• p. 168 - Should be ρ': G → GL(V') and ρ'(g) T = T ρ(g).
• p. 175 - On line 4, insert "to" after "equivalent".
• p. 175 - On the third line from the bottom, V should be the space of traceless hermitian matrices.
• p. 179 - On line 4, "differ from" should be "differ by"
• p. 181 - On line 3, "a a unitary" should be "a unitary"
• p. 182 - On the 6th line from the bottom, "clue have still not" should be "clue we have still not"
• p. 183 - In Exercise 29, the representation ρ of SL(2,C) on M is given by ρ(g)T = gTg*, not gTg-1.
• p. 186 - On line 9, "that that" should be "that"
• p. 189 - On line 15, γ should be replaced by a gothic lowercase g.
• p. 189 - On the 8th line from the bottom, "is plausible" should be "it plausible".
• p. 195 - On the 8th line from the bottom,"all three J's" should be "all three σ's".
• p. 196 - On the fifth line from the bottom, "algebra" should be "group".
• p. 197 - On line 13, "if this symmetry" should be "if this system".
• p. 199 - On line 4, in the quotation, "Attempt" should be "Attempts".
• p. 201 - On line 10, "we have E is bundle a over M" should be "E is a bundle over M".
• p. 203 - The first line after Fig. 2 should be indented - it's a new paragraph.
• p. 203 - in Fig. 3, the label on the right-hand vertical line should be E'φ(p) , not Eφ(p) .
• p. 204 - On line 6, Tφ(p)M should be Tφ(p)M' .
• p. 205 - On line 8, delete the sentence "A section of E|U is called a section of E over U". Sections don't get defined until page 207, so this should be moved back there.
• p. 206 - On line 4, "each fiber Ep to the fiber {p} × Rn" should be "each fiber Eq to the fiber {q} × Rn". We already have a point p in this discussion; q is any other point in U.
• p. 205 - On line 5, "the the trivialization" should be "the trivialization".
• p. 207 - On the 9th line from the bottom, the V should be an F.
• p. 208 - On the 5th line from the bottom, si should be si.
• p. 212 - On line 4 from the bottom, "identity" should be "identify".
• p. 213 - On line 17, αm should be βm.
• p. 213 - On the 5th line from the bottom, π: E→ V should be π: E→ M .
• p. 216 - On the 13th line from the bottom, "states of single particle" should be "states of a single particle".
• p. 217 - On line 11, delete the bar over νμ.
• p. 218 - On line 1, N should be ψ and π should be φ.
• p. 218 - On the 11th from the bottom, N should be ψ and π should be φ.
• p. 235 - On the second line from the bottom, v(t) should be u(t).
• p. 239 - On the 7th line from the bottom, 1p(t) = 0 should be 1p(t) = p.
• p. 245 - On the 6th line from the bottom, "Le" should be "Let".
• p. 249 - On the 2nd line from the bottom, "a E-valued" should be "an E-valued".
• p. 252 - On line 16, the γ should be a gothic letter g just like the g later in this sentence.
• p. 258 - In Exercise 112, it is not true that [A,A] = 0.
• p. 261 - On line 15, "differential forms to be" should be "differential forms be"
• p. 274 - On line 7, delete "we".
• p. 276 - In Exercise 119, the condition p + q = n should be p = q.
• p. 284 - The formula for dD η is true for any End(E)-valued form η, not ω.
• p. 289 - There is a factor of 2 missing from 3 lines of the computation of the derivative of SCS(As).
• p. 296 - The Borromean rings are drawn wrong. We leave it as an exercise to change some of the crossings so that removing any one ring will leave the remaining two unlinked. If you give up, click here.
• p. 309 - The roman numeral I in Fig. 27 should be a roman numeral II.
• p. 328 - The Alexander-Conway polynomial is not multiplicative under disjoint union, and it is not true that one can remove an unknot that is not linked to any other components if one multiplies the polynomial by 1. In fact, one can show using the skein relations that if a link contains an unlinked unknot, the Alexander polynomial of the link is zero. Thus in Figure 45 the number 1 should not be there.
• p. 332 - The formula should read = d2 A2 + dAB + dAB + d2B2
• p. 342 - On the 12th line from the bottom, gauge-invariant should be in italics, not slanted font. (Only a few people will care about this.)
• p. 347 - The Chern-Simons path integral for G = SU(2) does not quite give the Kauffman bracket: our claim that it did appears to be an error found widely in the literature. There is a certain issue of signs. Some aspects of this issue are treated in Steve Sawin's review article on TQFTs. Briefly, the Kaufmann bracket of a link with n components is equal to plus or minus 2n when q = 1, while the Chern-Simons path integral for G = SU(2) should give 2n when q = 1. Why? The former follows from the fact that the "loop value" d is equal to -2 when q = 1, and the fact that the Kauffman bracket does not change when you change overcrossings to undercrossings when q = 1. The latter follows heuristically from the principle of stationary phase: as q → 1, the path integral should be dominated by the value of the integrand at the critical points of the Chern-Simons action, i.e., at flat connections. But for any flat connection on R3, the trace of a Wilson loop labelled with the spin-1/2 representation of SU(2) is 2 (since the spin-1/2 representation is 2-dimensional), so the product of the traces of n such Wilson loops is 2n. Thus it seems that the Chern-Simons path integral gives the Kaufmann bracket times -1 to the winding number of the link when drawn on the page using the blackboard framing.
• p. 366 - On line 17, delete the space after "tensor field".
• p. 366 - On the 10th line from the bottom, the quantities ω1,..., ωs are 1-forms, not vector fields.
• p. 368 - On line 12, (r+s,r'+s') should be (r+r', s+s').
• p. 372 - On the 9th line from the bottom, "for all u,v,w" should be "for all v,w".
• p. 372 - D is torsion free if [v,w] = Dv w - Dw v. The line below should read "[v,w] and the expression Dv w - Dw v are both antisymmetric..."
• p. 382 - On line 6, F = dA + [A,A] should be F = dA + A^A.
• p. 409 - On line 1, "Instead is" should be "Instead it is".
• p. 425 - In the equation in line 10, the subscript i should be a superscript in both places.
• p. 425- On the 9th line from the bottom, in the equation, there should be a dot over the q.
• p. 428 - On line 4, "that many problems and subtleties to arise" should be "that many problems and subtleties arise"
• p. 434 - On the 10th line from the bottom, in the equation, π should be p.
Finally, the notes to Chapter 5 of part 3 refer to a quote by Maxwell which was supposed to appear at the very end of the chapter, but was omitted by a printer's error. The quote goes as follows:
The way in which Faraday made use of his ideas of lines of force in co-ordinating the phenomena of magneto-electric induction shew him to have been in reality a mathematician of a very high order — one from whom the mathematicians of the future may derive valuable and fertile methods. For the advance of the exact sciences depends upon the discovery and development of appropriate and exact ideas, by means of which we may form a mental representation of the facts, sufficiently general, on the one hand, to stand for any particular case, and sufficiently exact, on the other, to warrant the deductions we may draw from them by the application of mathematical reasoning. From the straight line of Euclid to the lines of force of Faraday this has been the character of the ideas by which science has been advanced, and by the free use of dynamical as well as geometrical ideas we may hope for a further advance. The use of mathematical calculations is to compare the results of the application of these ideas with our measurements of the quantities concerned in our experiments.... We are probably ignorant even of the name of the science which will be developed out of the materials we are now collecting.... — James Clerk Maxwell

Here's a list of known errors that appear in both the first and second version:

• p. 47 - ω is in T*qN, not TqN.
• p. 91 - In Exercise 68, the signature should be (n-s,s) instead of (s,n-s). In other words, there should be s minus signs in the metric, not s plus signs.
• p. 116 - On line 10, f ∘ φα should be f ∘ φα-1.
• p. 181 - In the far right side of the third displayed equation, Vj(g)Vj(g') should be Vj(h)Vj(h').
• p. 262 - On the 7th line from the bottom, ★ d ★ F should be ★ dD ★ F.
• p. 263 - Apparently

J=D_{k}(F_{ij}) \otimes *(dx^k\wedge *(dx^i \wedge dx^j))

should be

J=D_{k}(F_{ij}) \otimes *(dx^k\wedge *(dx^i \wedge dx^j)) + F_{ij} \otimes *(d*(dx^i \wedge dx^j)).

I need to look at the book to see what's going on, but the main result is still true: you can gauge transform a solution of the Yang-Mills equations and get a new solution.

• pp. 408–411 - John Engle wrote:
Your convention for the Riemann tensor as compared with other authors:

R_(Baez)^a_{bcd} = R_(Penrose)_{bcd}^a
= -R_(Wald)_{bcd}^a
= -R_(Ashtekar)_{bcd}^a

The major difference in your notation is that the "a" index above is in the
first "column", whereas all other authors put it in the last column.

I think this difference in index ordering on the Riemann tensor causes some
confusion when you introduce the "imitation Riemann tensor" in the chapter
on the Palatini action.

On page 408, you have:

R^c_{ab}^d = F_{ab}^{IJ} (e_I^c) (e_J^d)

(where I have omitted the tilde over the R, and replaced the Greek indices
with lower case Latin indices due to the typographical limits of email.)
I'm pretty sure the "c" and "d" on the right hand side should be reversed.
I know this is just a definition and so cannot be right or wrong per se,
but

(1) if the imitation Riemann is truly to be analogous to the Riemann,
the "I" index above on F should really correspond to the "d" index on R,
and the "J" index on F should really correspond to the "c" index on R
when using the tetrad to "transform" internal indices to space-time indices.

(2) In any case, I concluded that it was impossible to do exercise
35 without modifying the definition of the imitation Riemann in the manner
I described.

While I'm at it, I think there's a missing factor of 1/2 in the equation
atop p. 411.  More precisely, I believe there should be a factor of 1/2
in front of the second term on the second line.  I think you forgot it
when expanding out the antisymmetrization brackets.


John Huerta wrote:

I think I may have spotted another erratum!

This has to do with what Jonathan Engle was writing to you about:

On page 408, you have:

R^c_{ab}^d = F_{ab}^{IJ} (e_I^c) (e_J^d)

(where I have omitted the tilde over the R, and replaced the Greek indices
with lower case Latin indices due to the typographical limits of email.)
I'm pretty sure the "c" and "d" on the right hand side should be reversed.
I know this is just a definition and so cannot be right or wrong per se.

This has consequences, not just for Exercise 35, but also for the main text.
Specifically, transforming δS from the form with the coframe field
and internal curvature, shown in the last line of page 409, to the form
with the imitation Einstein tensor shown in the first equation of page 410,
one gets the wrong sign (at least for the first term, but it must work the
same for the second), unless you make the correction Jonathan Engle suggests!

• p. 426 - It should say dH = pd\dot{q} + \dot{q}dp - dL and not dH = \dot{p}d\dot{q} + \dot{q}dp - dL.