Sumários

Lecture 24 (final lecture)

25 maio 2016, 18:00 • Roger Francis Picken

Brief review of the chiral/Abelian anomaly, and the gauge/non-Abelian anomaly. 


Remark that the gauge anomaly is bad for the consistency of the theory, and needs to be avoided or cancelled through a judicious choice of group G and space-time topology/dimension (cancellation of gauge and gravitational anomalies). The chiral anomaly is needed, since it plays a role in an observed physical process: the decay of the pi^0 meson into two photons. The triangular Feynman diagram that is relevant for this process, and the corresponding integral which is non-zero due to the anomaly (Bertlmann Fig 4.1 and eq. 4.79). Remark that evaluating this diagram involves regularizing a divergent 4d integral (the internal 4-momentum), and performing algebraic calculations with gamma matrices and group theory calculations in SU(3) (all of which is typical for perturbation theory). Brief remark about global anomalies, which are like gauge anomalies but for non-infinitesimal ("large") gauge transformations, such as can occur for M= S^4, G= SU(2). Remark about dealing with divergent power series in the coupling constant, such as those that appear in perturbation theory, also in the presence of instantons: through Borel summation, or the more recent approach of resurgence and trans-series. Various topics that might be of interest to follow up: the Higgs boson; the millennium problem concerning the existence of quantum YM theory on R^4; the rigorous construction of YM measure in dimension 2. 

Final remarks about topological QFT (TQFT), which may be in the restricted or the broad sense. In the restricted sense, the path integrals only take into account the topology of the space-time manifold. Example of a TQFT for 2D triangulated or cellular surfaces obtained from colouring the edges or 1-cells with elements of a finite group G, subject to a flatness condition. Its topological invariance is immediate using Pachner moves, and the invariant may be seen as counting flat G-connections modulo gauge transformations. In the broad sense, a TQFT is a QFT that is invariant under changes in the metric. This may be established using supersymmetry arguments, e.g. Witten's article entitled TQFT, which gives a construction of Donaldson invariants, or by starting with a topological action like the Chern-Simons action (Witten's article on QFT and the Jones polynomial). Sketch of the Kontsevich integral (a universal Vassiliev invariant of knots, constructed in terms of a time-ordered/path-ordered integral encoding generalized winding numbers), which can be derived from a QFT perspective.

Some refs (with grateful thanks for some suggestions by J. Mourão) I am happy to provide additional references on any of the topics mentioned.

Anomalies
 
Borel summation/resurgence: 

Higgs boson:

Millennium problem:

Quantum YM theory in 2D:

Lecture 23

23 maio 2016, 12:30 • Roger Francis Picken

At the end of the previous lecture, using the BRST operator s, it was shown that the form of the gauge anomaly (not the overall scale), could be deduced from general assumptions, in particular that the effective action in the quantum theory, being s-exact, has to be s-closed, which extends to the integrand up to an exact form. This is the Wess-Zumino consistency condition, and it can be solved (up to a multiplicative constant) by expanding the Chern-Simons 5-form on a principal bundle P over a 4D manifold M into horizontal and vertical components. This is the "consistent" form of the the gauge anomaly - see below.

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 fermion action coupled to a YM gauge field, and the corresponding (classically) conserved current. The "axial" (or "chiral" or Abelian") anomaly in this conservation law, approached via the Fujikawa method: the breakdown in the symmetry is caused by the non-invariance of the fermion measure in the path integral. The anomalous term is proportional to the trace of gamma_5, which in turn is related to the index of the Dirac operator acting on positive chirality spinors, thus giving an expression for it by applying the index theorem. The analogous discussion of the "non-Abelian" or "gauge" anomaly for a positive chirality fermion coupled to a YM field. Here the eigenvalues are those of an operator which combines the operator in the action with a Dirac operator acting on negative chirality fermions. Using zeta function regularization techniques, the consistent form of the gauge anomaly can be obtained, including the multiplicative constant that was missing in the Wess-Zumino approach. Remark about how using different regularization operators gives rise to other forms of the anomaly (in particular, the debate about the "consistent" anomaly versus the "covariant" anomaly).

Ref: Nakahara

Lecture 22

18 maio 2016, 18:00 • Roger Francis Picken

Main points from the previous lecture: the BRST invariance of the action that emerges from the Faddeev-Popov procedure, and properties of the BRST operator s, i.e. s^2=0, sd+ds=0, and remark that its behaviour resembles the vertical exterior derivative on a principal G-bundle. The Koszul complex and the vertical complex which implement cohomologically the two steps involved in symplectic reduction.

The combination of the two complexes into a double complex to give a cohomological framework for the symplectic reduction of M to M^~, in terms of smooth functions on M. The BRST cohomology is that of a complex with differential D equal to the sum of the horizontal and vertical differentials of the double complex. Reformulation of the double complex to exhibit its (graded) super Poisson algebra structure. Derivations of a super Poisson algebra, and examples of inner derivations with respect to an element of the algebra of degree k. The horizontal and vertical differentials of the double complex are inner derivations, and the element corresponding to their sum D is an element of degree 1 (which appears in the Batalin-Vilkoviskii approach to BRST). In the cointext of symplectic reduction via a Poisson action of a Lie group G on a symplectic manifold, if one is prepared to work with smooth functions of M_0, the BRST cohomology is equal to the (Chevalley-Eilenberg) Lie algebra cohomology with coefficients in the module of these functions. Some comments on the significance of Lie algebra cohomology of degree 1 and 2 with coefficients in the trivial module.

As an introduction to anomalies, and an application of these algebraic methods, a discussion of the Wess-Zumino consistency condition for the gauge anomaly (non-abelian anomaly) for YM theory in space-time dimension 4. A solution is given by means of the descent equations, which come from separating into horizontal and vertical components the statement that on the principal bundle P, the exterior derivative of the Chern-Simons 5-form is proportional to the cube of the curvature 2-form.

Ref (for the cohomology part): Figueroa-O'Farrill

Lecture 21

16 maio 2016, 12:30 • Roger Francis Picken

Reminder about the main points of the previous lecture: the symplectic reduction procedure for a Poisson action of a Lie group G on a symplectic manifold M; the constraint hypersurface M_0 (the inverse image of a regular value under the moment map); the reduced symplectic manifold M^~= M_0/G; the special case when M is a cotangent bundle; the analogous situation for gauge theory; the warning example showing that it is better to quantize, then constrain, rather than constrain, then quantize.

Outline of the Faddeev-Popov (FP) procedure for the Yang-Mills action (plus matter fields). Gauge-fixing conditions, and remark about complete / incomplete conditions, as well as the possible absence of a single condition (Gribov ambiguity). 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, Stota (BRS) and Tyutin (T), was that the resulting action admits an infinitesimal (super)symmetry s, which in addition satisfies s^2=0, suggesting a link with cohomology. Remark about a possible geometric interpretation of the BRST transformation, s acting on the ghost, based on its similarity with the Maurer-Cartan equation for the left-invariant 1-form on G: as a vertical differential in a principal fibre bundle. Remark that BRST approaches in classical and quantum systems have been widely studied, not just in gauge theory.

Following Figueroa-O'Farrill, the description of a cohomological approach to the two steps in symplectic reduction. For the restriction from M to M_0, the appropriate framework is the Koszul complex, obtained from a regular sequence of elements of a ring R (which can be the constraint functions in the ring of smooth functions on M). The 0'th cohomology of this complex (which has trivial cohomology in other degrees) is then isomorphic to the ring of smooth functions on M_0. Remark that for general sequences of elements, there is a similar approach via the so-called Koszul-Tate complex. The second step of reduction is the quotient of M_0 by G, and can be described in cohomological terms by introducing a vertical differential on a complex of vertical forms on M_0. In the next lecture these will be combined to give the BRST complex in terms of functions on M, before starting the discussion of anomalies in gauge theory.

Lecture 20

11 maio 2016, 18:00 • Roger Francis Picken

Start of a new topic: BRST symmetry. Gauge theory is characterized by gauge symmetry, which motivates the study of symmetry in classical mechanics, closely tied to the notion of constraints - since symmetries imply conserved quantities, the system is effectively constrained to stay on level sets of these functions.
Reminder of some concepts of symplectic manifolds. The definition of the symplectic complement of a subspace of a symplectic vector space; the definition of isotropic, co-isotropic, Lagrangian and symplectic subspaces, and the corresponding types of submanifold of a symplectic manifold. Brief digression about the induced Poisson bracket on a symplectic submanifold (Dirac bracket). Symplectic actions of a Lie group G. Hamiltonian actions, with an example of an action that is only locally Hamiltonian. The definition of a moment map and its dual version. The definition of a Poisson action. Example of the Poisson action of the Poincaré group on the cotangent bundle of R^3. Marsden and Weinstein's theorem on symplectic reduction, and a simple example of a reduced symplectic manifold, which is compact, the 2-sphere. Brief mention of a quantization method that is well-adapted to such reduced manifolds: geometric quantization.
Discussion about how this route is not appropriate for gauge theory, despite the existence of a symmetry group of gauge transformations acting on a configuration space of connections on a principal G-bundle. This would lead to the cotangent space of the quotient as the reduced phase space for gauge theory. Warning example of how such a quotient is not well-behaved, from a study of the moduli space of flat G=SL(2,R) connections on the torus. Via the holonomy of these connections, they can be identified with a finite-dimensional manifold, whose elements are pairs of commuting elements of G. The group of gauge transformations is isomorphic to G, acting by simultaneous conjugation on the pairs. The resulting quotient is not a manifold, and the natural choice of topology (quotient topology) is not Hausdorff. This motivates the need for other methods "with reduction in mind". The BRST approach (to be introduced in the next lecture) combines ideas of supersymmetry and cohomology.

Refs:
(Symplectic reduction) J. Figueroa-O'Farrill, PhD thesis 1989, Section II.2
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