arXiv subjects

Raphael Zentner

Publications and source records attributed to Raphael Zentner.

24 records · Page 2Linked to original sources

arXiv2012

The quantum sl(n) graph invariant and a moduli space

We associate a moduli problem to a colored trivalent graph; such graphs, when planar, appear in the state-sum description of the quantum sl(N) knot polynomial due to Murakami, Ohtsuki, and Yamada. We discuss how the resulting moduli space can be thought of a representation variety. We show that the Euler characteristic of the moduli space is equal to the quantum sl(N) polynomial of the graph evaluated at unity. Possible extensions of the result are also indicated.

math.GT↗

arXiv2009

On higher rank instantons & the monopole cobordism program

Witten's conjecture suggests that the polynomial invariants of Donaldson are expressible in terms of the Seiberg-Witten invariants if the underlying four-manifold is of simple type. A higher rank version of the Donaldson invariants was introduced by Kronheimer. Before even having been defined, the physicists Mariño and Moore had already suggested that there should be a generalisation of Witten's conjecture to this type of invariants. We study a generalisation of the classical cobordism program to the higher rank situation and obtain vanishing results which gives evidence that the generalisation of Witten's conjecture should hold.

math.DG↗

arXiv2009

On Casson-type instanton moduli spaces over negative definite four-manifolds

Recently Andrei Teleman considered instanton moduli spaces over negative definite four-manifolds $X$ with $b_2(X) \geq 1$. If $b_2(X)$ is divisible by four and $b_1(X) =1$ a gauge-theoretic invariant can be defined; it is a count of flat connections modulo the gauge group. Our first result shows that if such a moduli space is non-empty and the manifold admits a connected sum decomposition $X \cong X_1 # X_2$ then both $b_2(X_1)$ and $b_2(X_2)$ are divisible by four; this rules out a previously natural appearing source of 4-manifolds with non-empty moduli space. We give in some detail a construction of negative definite 4-manifolds which we expect will eventually provide examples of manifolds with non-empty moduli space.

math.GT↗

arXiv2008

What to expect from $U(n)$ Seiberg-Witten monopoles for $n > 1$

We study generalisations to the structure groups U(n) of the familiar (abelian) Seiberg-Witten monopole equations on a four-manifold $X$ and their moduli spaces. For $n=1$ one obtains the classical monopole equations. For $n > 1$ our results indicate that there should not be any non-trivial gauge-theoretical invariants which are obtained by the scheme `evaluation of cohomology classes on the fundamental cycle of the moduli space'. For, if $b_2^+$ is positive the moduli space should be `cobordant' to the empty space because we can deform the equations so as the moduli space of the deformed equations is generically empty. Furthermore, on Kähler surfaces with $b_2^+ > 1$, the moduli spaces become empty as soon as we perturb with a non-vanishing holomorphic 2-form.

math.DG↗

arXiv2008

$PU(N)$ monopoles, higher rank instantons, and the monopole invariants

A famous conjecture in gauge theory mathematics, attributed to Witten, suggests that the polynomial invariants of Donaldson are expressible in terms of the Seiberg-Witten invariants if the underlying four-manifold is of simple type. Mathematicians have sought a proof of the conjecture by means of a `cobordism program' involving $PU(2)$ monopoles. A higher rank version of the Donaldson invariants was recently introduced by Kronheimer. Before being defined, the physicists Mariño and Moore had already suggested that there should be a generalisation of Witten's conjecture to this type of invariants. We adopt a generalisation of the cobordism program to the higher rank situation by studying $PU(N)$ monopoles. We analyse the differences to the $PU(2)$ situation, yielding evidence that a generalisation of Witten's conjecture should hold.

math.DG↗

arXiv2006

A note on Logarithmic Transformations on the Hopf surface

In this note we study logarithmic transformations in the sense of differential topology on two fibers of the Hopf surface. It is known that such transformations are susceptible to yield exotic smooth structures on four-manifolds. We will show here that this is not the case for the Hopf surface, all integer homology Hopf surfaces we obtain are diffeomorphic to the standard Hopf surface.

math.GT↗