Lecture 26

7 junho 2018, 14:00 • Roger Francis Picken

Brief reminder about the Faddeev-Popov construction of the action, its BRST invariance, the chiral and gauge symmetries of a massless fermion action, and the corresponding conserved currents.

The conservation of these currents may be spoilt by an anomaly. One approach (Fujikawa) to this phenomenon is to argue that the effective action is not invariant due to the non-invariance of the spinor measure. The general form of the gauge anomaly can be deduced from the Wess-Zumino consistency condition, which uses the nilpotency of the BRST operator. Solving this condition via the descent equations can be interpreted geometrically using the Ehresmann connection and the Chern-Simons form on the total space of the principal G-bundle. Discussion of the Fujikawa method for the chiral anomaly, which involves the trace of gamma_5, and applying the index theorem. Remarks about the need to cancel the "bad" gauge anomaly by means of gravitational anomalies, and how the chiral anomaly is "good" since it corresponds to a physical process described by a triangle Feynman diagram.

Discussion about topics for the seminar and essay part of the assessment (see below for a list of suggestions with references). Artistic landscape of gauge theory by John Huerta.


1. Anomalies
R. Bertlmann, Anomalies in Quantum Field Theory, 1996 Oxford University Press password protected zip file  

4. Quantum YM theory in 2D
5. TQFT
D. Dugger, Quantum Theory for Topologists
E. Witten, Topological quantum field theory, Comm. Math. Phys.Volume 117, Number 3 (1988), 353-386.
E. Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. Volume 121, Number 3 (1989), 351-399.

6. Instanton solutions and the ADHM construction
7. Monopoles in Yang-Mills-Higgs (two topics, but related)
M. Atiyah, Geometry of Yang-Mills fields, 1995   password protected zip file
M. Atiyah and N. Hitchin The Geometry and Dynamics of Magnetic Monopoles, 1998
P. Goddard and D. Olive, Magnetic monopoles in gauge field theories, Rep. Prog. Phys., Vol. 41, 1978. pdf N. Manton and P. Sutcliffe, Topological Solitons,  2004   password protected zip file
C. Nash and S. Sen, Topology and Geometry for Physicists, 1983   password protected zip file
R. Ward and R. Wells, Twistor Geometry and Field Theory, 1990 password protected zip file

8. Characteristic classes and/or Chern-Weil theory
9. Index theorem via susy models
L. Alvarez-Gaumé, Supersymmetry and the Atiyah-Singer Index Theorem, Commun. Math. Phys. 90, 161-173 (1983)  password protected zip file

10. The representation theory of the standard model
J.C. Baez and J. Huerta - The Algebra of Grand Unified Theories

11. The representation theory of the Poincare group
S. Sternberg - Group Theory and Physics

12. Gauge theory in the topology of 4-manifolds
A. Scorpan - The Wild World of 4-Manifolds  password protected zip file

With grateful thanks for some suggestions of references by J. Mourão.

Lecture 25

5 junho 2018, 11:00 • Roger Francis Picken

Outline of the Faddeev-Popov (FP) procedure for the Yang-Mills action (plus matter fields). Gauge-fixing conditions. Adding a gauge-fixing term (for the Lorenz gauge) to the action, and considering its second order variation under an infinitesimal gauge transformation, the corresponding path integral formally yields the determinant of a second order differential operator in the denominator. This leads to the introduction of a third term, the FP term, with fermionic fields ("ghosts" and "anti-ghosts"). The corresponding path integral formally gives the previous determinant in the numerator. Thus the path integral for the combined action terms over all fields equals the path integral over the moduli space (which was the aim), and is more convenient, since it is susceptible to perturbation theory / Feynman diagram methods of calculation. A final manipulation improves the gauge-fixing action by introducing a (bosonic) auxiliary field. The observation of Becchi, Rouet, Stora (BRS) and Tyutin (T), was that the resulting action admits a supersymmetry s, which in addition satisfies s^2=0, providing the link with cohomology already mentioned.

Real and complex Clifford algebras. Some examples and remarks about their structure as matrix algebras over R, C, H, mentioning Bott periodicity (mod 8 for real and mod 2 for complex Clifford algebras). Spinors take values in Clifford modules via these isomorphisms. Gamma matrices and the Dirac equation. For more details, see Coquereaux.

Introduction of the "gamma_5" element of the Clifford algebra, and the notion of chiral, or Weyl, fermions, with eigenvalue 1 or -1 for gamma_5. The chiral symmetry of a massless fermion action coupled to a YM gauge field, and the corresponding conserved current. Likewise there is a conserved current associated to the gauge symmetry for a positive chirality fermion coupled to a YM field. Both these symmetries are potentially subject to an anomaly in the quantum theory (to be discussed in the final lecture).

Lecture 24

29 maio 2018, 11:00 • Roger Francis Picken

Reminder about supersymmetric quantum mechanics and the Witten index from the last lecture. Remark that this index can be understood as the Brouwer degree of W=h', the function determining the potential of this class of models. Brief reminder of bosonic and fermionic Gaussian integrals, and generalization to complex fermionic variables. Using lecture notes by Bastianelli, some path integral representations of the transition function, the partition function and the supertrace (Witten index), with a Wick rotation to put the action in Gaussian form. Observation about rescaling W, suggesting that the path integral contributions should be concentrated around constant solutions at critical points of h. Expanding to 2nd order gives a ratio of determinants yielding the sign of W'=h'' at the critical point. Thus the path integral approach gives the Brouwer degree of h' via local contributions (and illustrates the cancellation effect of fermionic degrees of freedom in the action, with no need for regularization in this case).

As a stepping stone towards the final topic of BRST cohomology in gauge theory and anomalies, a brief discussion of the Atiyah-Singer index theorem in a more general setting than susy qm, namely on a compact manifold M, endowed with vector bundles E and F and an elliptic operator D from the sections of E to the sections of F. The adjoint of D is defined with respect to inner products on the respective spaces of sections. Then the index theorem relates the anaytical index of D to to a topological index coming from M, E and F, which may be given in terms of characteristic classes like the Chern classes seen earlier. As an example, one can consider Hodge theory, with d acting on p-forms, its adjoint delta, and the Laplacian (d+delta)^2. Using the fundamental theorem of Hodge theory one shows that every de Rham cohomology class has a unique harmonic representative, and then the index of D = d + delta mapping from even to odd degree forms is the Euler number of M. Remark about physicists' proofs of various cases of the index theorem by setting up a susy model whose canonical quantization gives the analytical index and whose path integral quantization reproduces the characteristic classes of the topological index.

Bastianelli lecture notes

Lecture 23

24 maio 2018, 14:00 • Roger Francis Picken

Reminder about how the BRST complex involves an odd operator Q and the exterior algebra of the Lie algebra of the symmetry group which we are quotienting out. Brief discussion about the tempting route of symplectic reduction and geometric quantization, and why it is not necessarily the best approach for gauge theory: the warning example of how flat G-connections on the torus modulo gauge transformations can lead to a non-manifold quotient that is not even Hausdorff. Brief description of Berezin integration and the Pfaffian of an antisymmetric matrix as an approach to compensating unwanted gauge degrees of freedom in the path-integral. Observation that this approach to gauge theory leads to a BRST action where Q is not merely odd, but a supersymmetry of the action.

Start of a digression on fermions and supersymmetry. The bosonic and fermionic harmonic 1D harmonic oscillator using Fock space. Combining them into a supersymmetric model and generalizing the potential leads to a class of supersymmetric models known as Witten's susy quantum mechanics. The Witten index and the arguments showing that it is 0 when restricted to states of energy greater than 0. Representation of the bosonic operators by their action on square integrable functions of q (the position variable), and of the fermionic operators as 2 by 2 matrices. Then Witten's susy QM models are parametrized by a function W(q). The relation between the Witten index and the behaviour of W(q) at plus or minus infinity can be viewed as a simple type of index theorem, related to the Morse theory of the potential.

J. Nelson, R. Picken, Parametrization of the moduli space of flat SL(2,R) connections on the torus, Lett. Math. Phys. 59:215-226 (2002) arxiv version

E. Witten, Constraints on supersymmetry breaking

Lecture 22: BV-BRST 5

22 maio 2018, 11:00 • John Huerta

We rehashed what I started last time about the geometry of BV-BRST. Remember, the idea of BRST is to replace the quoutient A/G of the space of gauge fields modulo gauge transformations with a more well-behavior object.

In this lecture, unlike in field theory, we'll assume A is a finite-dimensional vector space - this isn't so bad, because in gauge theory, once we pick a connection to perturb around, the space of connections really is vector spae, albeit infinite-dimensional. We will take A to be a representation of our group of gauge transformations, G, a finite-dimensional connected Lie group.

Now, we reason as follows: we replace spaces with their algebras of functions. The algebra of functions on O(A/G) on A/G is the same as the  space of G-invariant functions on A, O(A)^G. Since G is connected, this is the same as the space of g-invariant functions, O(A)^g, where g is the Lie algebra of G. And finally, as you will show in your homework, this last space is the 0th cohomology of the Chevalley-Eilenberg complex for g with coefficients in O(A):

O(A)^g = H^0(CE(g, O(A)))

The idea of resolving things in homological algebra is to find an complex whose H^0 is the object we're really interested in. So let's get to know the Chevalley-Eilenberg complex. In degree p, it is the dual of the pth exterior power of g tensored with O(A):

CE^p(g, O(A)) = Lambda^p(g*) (x) O(A).

This is the same thing as as the space linear maps from Lambda^p(g) to O(A):

CE^p(g, O(A)) = Hom(Lambda^p(g), O(A)).

On this complex, we defined a differential, Q: CE^p --> CE^{p+1}. It has a long formula you can look up on Wikipedia, linked to below.

But I want to think geometrically. So note that elements of g[1], which physicists call ghosts, are of odd degree. So, like any good fermions, they better anticommute with each other. So, with this informal idea, it makes sense to define the algebra of functions on g[1] to be the exterior algebra:

 O(g[1]) = Lambda(g*) = Lambda^0(g*) (+) Lambda^1(g*) (+) ...

Then the algebra of functions on A (+) g[1] is the Chevalley-Eilenberg complex:

O(A (+) g[1]) = O(A) (x) O(g[1]) = O(A) (x) Lambda(g*) = CE^0(g,O(A)) (+) CE^1(g,O(A)) (+) ...

This is indeed an algebra, because O(A) is an algebra and Lambda(g*) is an algebra. The differential Q acts on this algebra:

Q: O(A + g[1]) --> O(A + g[1])

Prop Q is a square-zero derivation of degree 1. That is, Q^2 = 0 (square-zero), Q(ab) = (Qa) b + (-1)^|a| a (Qb)  (derivation property) and Q sends elements in degree p to degree p+1.

That last property, of being a derivation, should be familiar from ordinary differential geometry. If M if a manifold, a derivation of O(M) = C^oo(M) is a vector field.

So, let us think of Q as a vector field on A (+) g[1].

In summary, BRST says we should replace A/G with the space A (+) g[1], equipped with a vector field Q, which is of degree 1 and squares to 0: Q^2 = [Q,Q] = 0.

Now let us combine this with BV. In previous lectures, we have seen that the BV complex lives on the complex of polyvector fields, Lambda TM. As above, I want to think of this complex as being the algebra of functions on some space. I claim this space deserves to be called the shifted cotangent bundle, T*[-1]M.

Let us see why in the case where M is a vector space, V. Then the familiar cotangent bundle T*V is just the sum of V with its dual:

T*V = V (+) V*.

The shifted cotangent bundle just shifts that dual up in degree:

T*[-1]V = V (+) V*[-1]

Using our earlier definition that tha algebra of functions on a space of odd degree should be the exterior algebra, we see that:

O(T*[-1]V) = O(V) (x) O(V*[-1]) = O(V) (x) Lambda(V**) = O(V) (x) Lambda(V).

We have used that V** = V for finite-dimensional vector spaces. We think of Lambda(V) as being polyvectors in V, with constant coefficients. Tensoring with O(V) introduces nonconstant coefficients, so this really is the complex of polyvector fields on V.

In summary, the BV formalism says that we should replace the space of our fields M with shifted cotangent bundle T*[-1]M.

Now, let us do BV-BRST. First, we replace A/G with A (+) g[1] as BRST tells us to do, and then we replace this with the shifted cotangent bundle:

T*[-1]( A (+) g[1] ) = A (+) g[1] (+) (A (+) g[1])*[-1] = A (+) g[1] (+) (A* (+) g*[-1])[-1] = A (+) g[1] (+) A*[-1] (+) g*[-2]

This is the space of the BV-BRST formalism. Let me denote it by E:

E = g[1] (+) A (+) A*[-1] (+) g*[-2]

and let us introduce each of the degrees in turn:

• in degree -1, the Lie algebra g[1] is called the space of ghosts.
  
• in degree 0, A, is called the space of fields.
  
• in degree 1, the dual of A, A*[-1], is called the space of anitfields.
  
• Finally, in degree 2, the dual of the Lie algebra, g*[-2], is called the space of antighosts.

All of these fields will ultimately show up in our BV action, which is the sum of two terms:

S_BV = S_cl + S_Gauge

where S_cl is the classical action, but S_Gauge is new! S_Gauge comes from the vector field Q that we introduce on the BRST space, A (+) g[1]. Q lifts to a vector field on the cotangent bundle the shifted cotangent bundle, E = T*[-1](A + g[1]), in the natural way. Like an ordinary cotangent bundle, E comes with a symplectic structure, but now shifted into degree -1. We can use this to define a function, S_Gauge, as the Hamiltonian that generates the vector field Q:

dS_Gauge = <Q, ->

where <-,-> is the shifted symplectic structure Q. We can compute S_Gauge takes a very nice form:

Prop S_Gauge = 1/2 <c(A), A*> + <[c,c], c*>

where c is a ghost, A a field, A* an antifield, and c* an antighost. Finally, it is for the BV action that we wish to define the path integral. This takes even more work. In particular, we have to make sure that the BV action satisfies the quantum master equation. I gave one version in lecture, but here is another: Let d be the operator

d = {S_BV, -} + hbar Delta

This is the same as the quantum BV operator we had before, except S_BV is now the BV action. Then the quantum master equation says:

d^2 = 0.

In general, S_BV need not satisfy the quantum master equation, and finding an action which does can be nontrivial. The quantum master equation is deep way to encode all of the following facts at once:

• The Lie bracket of infinitesimal gauge symmetries satisfies the Jacobi identity.
  
• The infinitesimal gauge symmetries preserve the classical action.
  
• The action of the Lie algebra on the fields preserves the "Lebesgue measure" on this space.
• The action of the Lie algebra on itself preserves the "Lebesgue measure" on the space of gauge transformations.

The last two items become ill-defined in infinite-dimensions, and renormalization is required to make sense of them. This can, however, but done rigorously.

Wikipedia - Lie algebra cohomology