61 research outputs found
Topological Symmetry Breaking on Einstein Manifolds
It is known that if gauge conditions have Gribov zero modes, then topological
symmetry is broken. In this paper we apply it to topological gravity in
dimension . Our choice of the gauge condition for conformal
invariance is , where is the Ricci scalar curvature. We find
when , topological symmetry is not broken, but when
and solutions of the Einstein equations exist then topological symmetry is
broken. This conditions connect to the Yamabe conjecture. Namely negative
constant scalar curvature exist on manifolds of any topology, but existence of
nonnegative constant scalar curvature is restricted by topology. This fact is
easily seen in this theory. Topological symmetry breaking means that BRS
symmetry breaking in cohomological field theory. But it is found that another
BRS symmetry can be defined and physical states are redefined. The divergence
due to the Gribov zero modes is regularized, and the theory after topological
symmetry breaking become semiclassical Einstein gravitational theory under a
special definition of observables.Comment: 16 pages, Late
Dimensional Reduction of Seiberg-Witten Monopole Equations, N=2 Noncommutative Supersymmetric Field Theories and Young Diagrams
We investigate the Seiberg-Witten monopole equations on noncommutative(N.C.)
R^4 at the large N.C. parameter limit, in terms of the equivariant cohomology.
In other words, N}=2 supersymmetric U(1) gauge theories with hypermultiplet on
N.C. R}^4 are studied. It is known that after topological twisting partition
functions of N}>1 supersymmetric theories on N.C. R^2D are invariant under
N.C.parameter shift, then the partition functions can be calculated by its
dimensional reduction. At the large N.C. parameter limit, the Seiberg-Witten
monopole equations are reduced to ADHM equations with the Dirac equation
reduced to 0 dimension. The equations are equivalent to the dimensional
reduction of non-Abelian U(N) Seiberg-Witten monopole equations in N -> \infty.
The solutions of the equations are also interpreted as a configuration of brane
anti-brane system. The theory has global symmetries under torus actions
originated in space rotations and gauge symmetries. We investigate the
Seiberg-Witten monopole equations reduced to 0 dimension and the fixed point
equations of the torus actions. We show that the Dirac equation reduced to 0
dimension is trivial when the fixed point equations and the ADHM equations are
satisfied. For finite N, it is known that the fixed points of the ADHM data are
isolated and are classified by the Young diagrams. We give a new proof of this
statement by solving the ADHM equations and the fixed point equations
concretely and by giving graphical interpretations of the field components and
these equations.Comment: v2: 28+1 pages, 13 figures, appendix for convention added, references
added, some descriptions improved, v3: eq.(113),(117) corrected, v4: 29+1
pages, minor correction
Partition functions of Supersymmetric Gauge Theories in Noncommutative R^{2D} and their Unified Perspective
We investigate cohomological gauge theories in noncommutative R^{2D}. We show
that vacuum expectation values of the theories do not depend on noncommutative
parameters, and the large noncommutative parameter limit is equivalent to the
dimensional reduction. As a result of these facts, we show that a partition
function of a cohomological theory defined in noncommutative R^{2D} and a
partition function of a cohomological field theory in R^{2D+2} are equivalent
if they are connected through dimensional reduction. Therefore, we find several
partition functions of supersymmetric gauge theories in various dimensions are
equivalent. Using this technique, we determine the partition function of the
N=4 U(1) gauge theory in noncommutative R^4, where its action does not include
a topological term. The result is common among (8-dim, N=2), (6-dim, N=2),
(2-dim, N=8) and the IKKT matrix model given by their dimensional reduction to
0-dim.Comment: 45 pages, no figures, Appendices B and C are added, changes in the
text, references are adde
Integrability of Matrix Model as -body Harmonic Oscillator System
We study a Hermitian matrix model with a kinetic term given by , where is a positive definite Hermitian matrix, similar as in the
Kontsevich Matrix model, but with its potential replaced by .
We show that its partition function solves an integrable Schr\"odinger-type
equation for a non-interacting -body Harmonic oscillator system.Comment: 16 pages, 3 figure
Exact Solutions v.s. Perturbative Calculations of Finite - Hybrid-Matrix-Model
There is a matrix model corresponding to a scalar field theory called
Grosse-Wulkenhaar model, which is renormalizable by adding a harmonic
oscillator potential to scalar theory on Moyal spaces. There are
more unknowns in matrix model than in matrix model, for
example, in terms of integrability. We then construct a one-matrix model
(- Hybrid-Matrix-Model) with multiple potentials, which is
a combination of a -point interaction and a -point interaction, where the
-point interaction of is multiplied by some positive definite
diagonal matrix . This model is solvable due to the effect of this . In
particular, the connected -point function
of - Hybrid-Matrix-Model is studied in detail. This
-point function can be interpreted
geometrically and corresponds to the sum over all Feynman diagrams (ribbon
graphs) drawn on Riemann surfaces with boundaries (punctures). Each
represents external lines coming from
the -th boundary (puncture) in each Feynman diagram. First, we construct
Feynman rules for - Hybrid-Matrix-Model and calculate
perturbative expansions of some multipoint functions in ordinary methods.
Second, we calculate the path integral of the partition function
and use the result to compute exact solutions for -point
function with -boundary, -point function with
-boundary, -point function with -boundaries, and -point
function with -boundaries. They include
contributions from Feynman diagrams corresponding to nonplanar Feynman diagrams
or higher genus surfaces.Comment: 28 pages. Typos were corrected, three references were added, and an
annotation was added in Section
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