2,340 research outputs found
On higher holonomy invariants in higher gauge theory II
This is the second of a series of two technical papers devoted to the
analysis of holonomy invariants in strict higher gauge theory with end
applications in higher Chern--Simons theory. We provide a definition of trace
over a crossed module such to yield surface knot invariants upon application to
2-holonomies. We show further that the properties of the trace are best
described using the theory quandle crossed modules.Comment: Latex, 34 pages, no figure
Reducibility and Gribov Problem in Topological Quantum Field Theory
In spite of its simplicity and beauty, the Mathai-Quillen formulation of
cohomological topological quantum field theory with gauge symmetry suffers two
basic problems: ) the existence of reducible field configurations on which
the action of the gauge group is not free and ) the Gribov ambiguity
associated with gauge fixing, i. e. the lack of global definition on the space
of gauge orbits of gauge fixed functional integrals. In this paper, we show
that such problems are in fact related and we propose a general completely
geometrical recipe for their treatment. The space of field configurations is
augmented in such a way to render the action of the gauge group free and
localization is suitably modified. In this way, the standard Mathai--Quillen
formalism can be rigorously applied. The resulting topological action contains
the ordinary action as a subsector and can be shown to yield a local quantum
field theory, which is argued to be renormalizable as well. The salient feature
of our method is that the Gribov problem is inherent in localization, and thus
can be dealt with in a completely equivariant setting, whereas gauge fixing is
free of Gribov ambiguities. For the stratum of irreducible gauge orbits, the
case of main interest in applications, the Gribov problem is solvable.
Conversely, for the the strata of reducible gauge orbits, the Gribov problem
cannot be solved in general and the obstruction may be described in the
language of sheaf theory. The formalism is applied to the Donaldson--Witten
model.Comment: 37 pages, Plain TeX, no figures, requires AMS font files AMSSYM.DEF
and AMSSYM.TEX, final version to appear in CMP, minor change
Exact renormalization group in Batalin--Vilkovisky theory
In this paper, inspired by the Costello's seminal work, we present a general
formulation of exact renormalization group (RG) within the Batalin-Vilkovisky
(BV) quantization scheme. In the spirit of effective field theory, the BV
bracket and Laplacian structure as well as the BV effective action (EA) depend
on an effective energy scale. The BV EA at a certain scale satisfies the BV
quantum master equation at that scale. The RG flow of the EA is implemented by
BV canonical maps intertwining the BV structures at different scales.
Infinitesimally, this generates the BV exact renormalization group equation
(RGE). We show that BV RG theory can be extended by augmenting the scale
parameter space R to its shifted tangent bundle T[1]R. The extra odd direction
in scale space allows for a BV RG supersymmetry that constrains the structure
of the BV RGE bringing it to Polchinski's form. We investigate the implications
of BV RG supersymmetry in perturbation theory. Finally, we illustrate our
findings by constructing free models of BV RG flow and EA exhibiting RG
supersymmetry in the degree -1 symplectic framework and studying the
perturbation theory thereof. We find in particular that the odd partner of
effective action describes perturbatively the deviation of the interacting RG
flow from its free counterpart.Comment: 52 pages, no figures, introduction thoroughly rewritten, two new
subsections adde
Four dimensional Abelian duality and SL(2,Z) action in three dimensional conformal field theory
Recently, Witten showed that there is a natural action of the group SL(2,Z)
on the space of 3 dimensional conformal field theories with U(1) global
symmetry and a chosen coupling of the symmetry current to a background gauge
field on a 3-fold N. He further argued that, for a class of conformal field
theories, in the nearly Gaussian limit, this SL(2,Z) action may be viewed as a
holographic image of the well-known SL(2,Z) Abelian duality of a pure U(1)
gauge theory on AdS-like 4-folds M bounded by N, as dictated by the AdS/CFT
correspondence. However, he showed that explicitly only for the generator T;
for the generator S, instead, his analysis remained conjectural. In this paper,
we propose a solution of this problem. We derive a general holographic formula
for the nearly Gaussian generating functional of the correlators of the
symmetry current and, using this, we show that Witten's conjecture is indeed
correct when N=S^3. We further identify a class of homology 3-spheres N for
which Witten's conjecture takes a particular simple form.Comment: analysis of sect. 5 generalized; 43 pages, Plain TeX, no figures,
requires AMS font files amssym.def and amssym.te
The Hitchin Model, Poisson-quasi-Nijenhuis Geometry and Symmetry Reduction
We revisit our earlier work on the AKSZ formulation of topological sigma
model on generalized complex manifolds, or Hitchin model. We show that the
target space geometry geometry implied by the BV master equations is
Poisson--quasi--Nijenhuis geometry recently introduced and studied by Sti\'enon
and Xu (in the untwisted case). Poisson--quasi--Nijenhuis geometry is more
general than generalized complex geometry and comprises it as a particular
case. Next, we show how gauging and reduction can be implemented in the Hitchin
model. We find that the geometry resulting form the BV master equation is
closely related to but more general than that recently described by Lin and
Tolman, suggesting a natural framework for the study of reduction of
Poisson--quasi--Nijenhuis manifolds.Comment: 38 pages, no figures, LaTex. One paragraph in sect. 6 and 3
references adde
Competitive Pressure: Competitive Dynamics as Reactions to Multiple Rivals
Competitive dynamics research has focused primarily on interactions between dyads of firms. Drawing on the awareness-motivation-capability framework and strategic group theory we extend this by proposing that firms’ actions are influenced by perceived competitive pressure resulting from actions by several rivals. We predict that firms’ action magnitude is influenced by the total number of rival actions accumulating in the market, and that this effect is moderated by strategic group membership. We test this using data on the German mobile telephony market and find them supported: the magnitude of firm’s actions is influenced by a buildup of actions by multiple rivals, and firms react more strongly to strategically similar rivals
4-d semistrict higher Chern-Simons theory I
We formulate a 4-dimensional higher gauge theoretic Chern-Simons theory. Its
symmetry is encoded in a semistrict Lie 2-algebra equipped with an invariant
non singular bilinear form. We analyze the gauge invariance of the theory and
show that action is invariant under a higher gauge transformation up to a
higher winding number. We find that the theory admits two seemingly
inequivalent canonical quantizations. The first is manifestly topological, it
does not require a choice of any additional structure on the spacial 3-fold.
The second, more akin to that of ordinary Chern-Simons theory, involves fixing
a CR structure on the latter. Correspondingly, we obtain two sets of semistrict
higher WZW Ward identities and we find the explicit expressions of two higher
versions of the WZW action. We speculate that the model could be used to define
2-knot invariants of 4-folds.Comment: 97 pages, LaTex, a few references adde
Competitive Pressure: Competitive Dynamics as Reactions to Multiple Rivals
Competitive dynamics research has focused primarily on interactions between dyads of firms. Drawing on the awareness-motivation-capability framework and strategic group theory we extend this by proposing that firms’ actions are influenced by perceived competitive pressure resulting from actions by several rivals. We predict that firms’ action magnitude is influenced by the total number of rival actions accumulating in the market, and that this effect is moderated by strategic group membership. We test this using data on the German mobile telephony market and find them supported: the magnitude of firm’s actions is influenced by a buildup of actions by multiple rivals, and firms react more strongly to strategically similar rivals.Competitive rivalry; competitive dynamics; strategic groups; mobile telecommunications
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