27 research outputs found
Remarks on Chern-Simons invariants
The perturbative Chern-Simons theory is studied in a finite-dimensional
version or assuming that the propagator satisfies certain properties (as is the
case, e.g., with the propagator defined by Axelrod and Singer). It turns out
that the effective BV action is a function on cohomology (with shifted degrees)
that solves the quantum master equation and is defined modulo certain canonical
transformations that can be characterized completely. Out of it one obtains
invariants.Comment: 33 pages; minor corrections, new appendices with technical details,
new references, new example; to appear in Commun. Math. Phys
On Pure Spinor Superfield Formalism
We show that a certain superfield formalism can be used to find an off-shell
supersymmetric description for some supersymmetric field theories where
conventional superfield formalism does not work. This "new" formalism contains
even auxiliary variables in addition to conventional odd super-coordinates. The
idea of this construction is similar to the pure spinor formalism developed by
N.Berkovits. It is demonstrated that using this formalism it is possible to
prove that the certain Chern-Simons-like (Witten's OSFT-like) theory can be
considered as an off-shell version for some on-shell supersymmetric field
theories. We use the simplest non-trivial model found in [2] to illustrate the
power of this pure spinor superfield formalism. Then we redo all the
calculations for the case of 10-dimensional Super-Yang-Mills theory. The
construction of off-shell description for this theory is more subtle in
comparison with the model of [2] and requires additional Z_2 projection. We
discover experimentally (through a direct explicit calculation) a non-trivial
Z_2 duality at the level of Feynman diagrams. The nature of this duality
requires a better investigation
Classical BV theories on manifolds with boundary
In this paper we extend the classical BV framework to gauge theories on
spacetime manifolds with boundary. In particular, we connect the BV
construction in the bulk with the BFV construction on the boundary and we
develop its extension to strata of higher codimension in the case of manifolds
with corners. We present several examples including electrodynamics, Yang-Mills
theory and topological field theories coming from the AKSZ construction, in
particular, the Chern-Simons theory, the theory, and the Poisson sigma
model. This paper is the first step towards developing the perturbative
quantization of such theories on manifolds with boundary in a way consistent
with gluing.Comment: The second version has many typos corrected, references added. Some
typos are probably still there, in particular, signs in examples. In the
third version more typoes are corrected and the exposition is slightly
change
The Poisson sigma model on closed surfaces
Using methods of formal geometry, the Poisson sigma model on a closed surface
is studied in perturbation theory. The effective action, as a function on
vacua, is shown to have no quantum corrections if the surface is a torus or if
the Poisson structure is regular and unimodular (e.g., symplectic). In the case
of a Kahler structure or of a trivial Poisson structure, the partition function
on the torus is shown to be the Euler characteristic of the target; some
evidence is given for this to happen more generally. The methods of formal
geometry introduced in this paper might be applicable to other sigma models, at
least of the AKSZ type.Comment: 32 pages; references adde
The Complexity of Drawing a Graph in a Polygonal Region
We prove that the following problem is complete for the existential theory of
the reals: Given a planar graph and a polygonal region, with some vertices of
the graph assigned to points on the boundary of the region, place the remaining
vertices to create a planar straight-line drawing of the graph inside the
region. This strengthens an NP-hardness result by Patrignani on extending
partial planar graph drawings. Our result is one of the first showing that a
problem of drawing planar graphs with straight-line edges is hard for the
existential theory of the reals. The complexity of the problem is open in the
case of a simply connected region.
We also show that, even for integer input coordinates, it is possible that
drawing a graph in a polygonal region requires some vertices to be placed at
irrational coordinates. By contrast, the coordinates are known to be bounded in
the special case of a convex region, or for drawing a path in any polygonal
region.Comment: Appears in the Proceedings of the 26th International Symposium on
Graph Drawing and Network Visualization (GD 2018
Introduction to Integral Discriminants
The simplest partition function, associated with homogeneous symmetric forms
S of degree r in n variables, is integral discriminant J_{n|r}(S) = \int
e^{-S(x_1 ... x_n)} dx_1 ... dx_n. Actually, S-dependence remains the same if
e^{-S} in the integrand is substituted by arbitrary function f(S), i.e.
integral discriminant is a characteristic of the form S itself, and not of the
averaging procedure. The aim of the present paper is to calculate J_{n|r} in a
number of non-Gaussian cases. Using Ward identities -- linear differential
equations, satisfied by integral discriminants -- we calculate J_{2|3},
J_{2|4}, J_{2|5} and J_{3|3}. In all these examples, integral discriminant
appears to be a generalized hypergeometric function. It depends on several
SL(n) invariants of S, with essential singularities controlled by the ordinary
algebraic discriminant of S.Comment: 36 pages, 19 figure
Towards Holography in the BV-BFV Setting
We show how the BV-BFV formalism provides natural solutions to descent equations and discuss how it relates to the emergence of holographic counterparts of given gauge theories. Furthermore, by means of an AKSZ-type construction we reproduce the Chern–Simons to Wess–Zumino–Witten correspondence from infinitesimal local data and show an analogous correspondence for BF theory. We discuss how holographic correspondences relate to choices of polarisation relevant for quantisation, proposing a semi-classical interpretation of the quantum holographic principle