Research Interest of Yanwen Shang

      Publication 1: Resonance in asymmetric warped geometry (URL)
      Publication 2: Quantum Cosmology of Classically Constrained Gravity (URL)
      Publication 3: Pressure as a Source of Gravity (URL)

    URL of Research: http://individual.utoronto.ca/ywshang/doc/statement.pdf

Statement of Interest:


  I am currently pursuing several research directions.

N=(2,2) supersymmetric gauge theories 
in 2-d prove to be very fruitful.  Besides its applications in string
theory and the fundamental understanding of the quantum gauge theories
it's also of great mathematical interests as a powerful tool 
in providing physical intuitions in the studies of mirror symmetry.  
The case with Abelian gauge group was well studied (Witten).  
The extension to non-Abelian theories, however, was only studied 
recently (Tong and Hori) for $U(k)/SU(k)$ gauge groups.  There, Witten 
index was computed and some IR dynamics were discussed.  In 
particular a simple criterion for the theory to become singular at IR 
was given.  Many features discovered there were novel, only seen in 
the non-Abelian case.  I have recently further extended the studies to 
other gauge groups.  Many conclusions seem to alter dramatically.  I 
counted the vacua in the moduli space away from the strong 
coupling regions including the one-loop quantum corrections in the 
theories endowed with a twisted mass .  In the case when twisted mass
vanishes, a criterion for the IR singularities were also derived for the gauge
groups of $SO(k)$ and $Sp(k)$.  It turns out to be simple but very different
from that in the case of $SU(k)$.  A major challenge in these researches is 
the analysis near the strong coupling regions in the moduli space, where
a geometrical description of the theory becomes important.  For a full
computation of Witten index, such an investigation is essential. We are
currently working on it and it appears that many interesting and 
surprising features are to be discovered.

I'm also  working on the gauge anomaly in the theories with chiral 
fermions on the lattice.  It's well-known fact that any theory on a
 finite lattice naively possesses no chiral/gauge anomalies due to a 
peculiar doubling of the fermion spectrum.  Despite this, it was 
demonstrated (Ginsparg, Wilson, Neuberger, Luscher and et. al.) 
that anomalies could still be understood on the lattice by introducing 
an explicit chiral symmetry breaking operator that vanishes in the 
continuous limit, and with a slightly modified chiral rotation on the 
lattice.  The construction in vector-like theories became trivial but 
the attempts to carry out similar analysis in chiral theories remained 
difficult.  Although it's been proven possible in the case of $U(1)$ gauge
group, no explicit constructions are available.  Recently I noticed one of 
the key issues that brought in the troubles and proposed a further 
modified chiral rotation.  Such a modification completely clarifies the
situation in the case when gauge field is kept fixed as background.  
To include the dynamical gauge field continues to be problematic and 
we are currently working on this subject in several different ways.  One 
of the possibilities involves integrating out heavy field so that an explicit
fermion measure can be dynamically generated in the low energy.

Another somewhat related problem is the consistency of anomalous 
gauge theory in $2$-d.  The widely accepted lore is that these theories 
are consistent as long as the photon is endowed with a mass, which is 
subject to a lower bound by unitarity.   On the other hand, some 
Hamiltonian analysis seems to suggest that in a certain gauge choice, 
the theory can also be consistent without making the photon massive if
Lorenz symmetry is sacrificed.  We'd like to understand this in the path-
integral language and examine whether any higher dimensional analogies
exist.  Furthermore we'd love to find their holographic descriptions, if exist,
that aid our understanding in these lower dimensional gauge theories.