Winter 2004
Feb 25: Jacapo Orgera: Dualities in Twistor Space
In a recent paper [hep-th/0312171], it has been noted that some perturbative
amplitudes in 4-dimensional Yang-Mills theories, when Fourier transformed
in a particular way, have some remarkable geometrical properties. These
particular properties are conjectured to hold at any order in the
perturbative expansion if the theory is N=4, the conjecture is justified
using a stringy theoretical construction called the B-Model. We will review
both sides of this beautiful duality.
References:
--E. Witten, "Perturbative Gauge Theory as a String Theory
in twistor space", hep-th/0312171.
Meeting:4:00pm small seminar room. Email: jacorg@physics.ucsb.edu
Feb 19: Nelia Mann: Integrable Spin Chains
I intend to discuss the matter of integrable spin chains. I will
introduce the basic technology of the Bethe ansatz, using the periodic
chain of spin 1/2 particles as my main example. I will explain the
Yang-Baxter equation and the transfer matrix as the origin of an infinite
set of commuting charges. I hope to make this lecture complementary to,
but not dependent on Joe's lectures last summer on continuous
integrability. If there is time at the end, I will then explain how the
Bethe Ansatz has been used recently to study the one-loop anomalous
dimensions of operators in N = 4 Super Yang Mills theory.
References:
---"Quantum Groups in Two-dimensional Physics", by Gomez, Ruiz-Altaba, and
Sierra.
---Joe's lectures last summer
---"The Bethe Ansatz for N = 4 Super Yang Mills" by Minahan and Zarembo,
hep-th/0212208.
Meeting:4:00pm Main Seminar room. Email: nelia@physics.ucsb.edu
Feb 11: Ted Erler: Antifield Formalism and the
Path Integral
Abstract: In the previous lecture, we found that the Lagrangian BRST
construction motivated us to introduce some extra unphysical fields, in
addition to the fields appearing in the classical action: the ghosts,
antifields, and ghost antifields. We found that the space of all trajectories
for these physical and unphysical fields---the "extended covariant phase
space," P_ext---is a supermanifold with a symplectic structure. BRST
transformations on P_ext are generated by the Hamiltonian vector field of
a certain function, called the "master action." BRST invariance implies
that the master action satisfies the so-called "master equation." The
master action, as the name suggests, is the fundamental generalization of
the classical action to the space of fields, ghosts, and thier antifields.
In this lecture, we show how to define a gauge invariant path integral on
the extended covariant phase space. Since the master action has gauge
symmetry, the path integral over the entire P_ext is divergent. However,
if we restrict ourselves to integrate only over a submanifold M of P_ext,
the path integral can converge, provided of course that the master
action on M has no gauge symmetry. Thus the choice of M corresponds to a
gauge fixing procedure. If M is a particular type of submanifold---a
"Lagrangian submanifold"---, the path integral is actually independent of
the choice of M, and therefore the choice of gauge, assuming that the
master action satisfies a quantum mechanical generalization of the master
equation, called the "quantum master equation."
References:
--Ted Erler, "Lecture notes on the classical Antifield formalism"
--Henneaux and Teitelboim, "Quantization of gauge systems"
--J.W. Van Holten, "Aspects of BRST Quantization," hep-th/0201124
--J. Gomis, J. Paris, S. Samuel, "Antibracket, Antifields and Gauge-Theory
Quantizatiion," Physics Reports, Vol. 259, August 1995.
--A. Schwarz, "Geometry of Batalin_Vilkovisky quantization,"
hep-th/9205088
Meeting:4:00pm Small Seminar room. Email: terler@physics.ucsb.edu
Jan 29: Ted Erler: Classical Antifield Formalism
In this lecture I would like to present the concepts of gauge symmetry and
BRST invariance from a classical Lagrangian point of view. I will
explain how in the Lagrangian formalism the fundamental space should be
thought of as the space of all classical trajectories---the so-called
"covariant phase space." Only a submanifold of this space contain
trajectories that satisfy the Euler-Lagrange equations, and this
submanifold is foliated by gauge orbits connecting classical trajectories
related by a gauge transformation. Thus we find a geometric picture quite
analogous to a Hamiltonian description of gauge theories with first class
constraints. Motivated by this analogy, we attempt a description of the
theory's observables in terms of cohomology classes of a Lagrangian BRST
differential, acting on a suitably defined grassmann algebra of functions
on the covariant phase space. In this grassmann algebra of "fields and
antifields" we find a natural symplectic structure which can be used to
define a Poisson-like bracket, the "antibracket." Furthermore, the
Lagrangian BRST differential has a represention in terms of the
antibracket with a certain functional, called the "master action."
Nilpotency of the BRST differential implies that the master action
bracketed with itself must vanish---this is the so-called "master
equation," the central equation of the antifield (or BV) formalism.
References:
--Henneaux and Teitelboim, "Quantization of gauge systems"
--Ted Erler, "Review of BRST formalism, constrained Hamiltonians"
--J.W. Van Holten, "Aspects of BRST Quantization," hep-th/0201124
--J. Gomis, J. Paris, S. Samuel, "Antibracket, Antifields and Gauge-Theory
Quantizatiion," Physics Reports, Vol. 259, August 1995.
Meeting:4:00pm Main Seminar room. Email: terler@physics.ucsb.edu