93 research outputs found
On the space of elliptic genera
Invariance under modular transformations and spectral flow restrict the
possible spectra of superconformal field theories (SCFT). This paper presents a
technique to calculate the number of constraints on the polar spectra of
N=(2,2) and N=(4,0) SCFT's by analyzing the elliptic genus. The polar spectrum
corresponds to the principal part of a Laurent expansion derived from the
elliptic genus. From the point of view of the AdS_3/CFT_2 correspondence, these
are the states which lie below the cosmic censorship bound in classical
gravity. The dimension of the space of elliptic genera is obtained as the
number of coefficients of the principal part minus the number of constraints.
As an additional illustration of the technique, the constraints on the spectrum
of N=4 topologically twisted Yang-Mills on CP^2 are discussed.Comment: 31 pages, published versio
BPS invariants of semi-stable sheaves on rational surfaces
BPS invariants are computed, capturing topological invariants of moduli
spaces of semi-stable sheaves on rational surfaces. For a suitable stability
condition, it is proposed that the generating function of BPS invariants of a
Hirzebruch surface takes the form of a product formula. BPS invariants for
other stability conditions and other rational surfaces are obtained using
Harder-Narasimhan filtrations and the blow-up formula. Explicit expressions are
given for rank <4 sheaves on a Hirzebruch surface or the projective plane. The
applied techniques can be applied iteratively to compute invariants for higher
rank.Comment: 26 pages, version submitted to journa
The Betti numbers of the moduli space of stable sheaves of rank 3 on P2
This article computes the generating functions of the Betti numbers of the
moduli space of stable sheaves of rank 3 on the projective plane P2 and its
blow-up. Wall-crossing is used to obtain the Betti numbers on the blow-up.
These can be derived equivalently using flow trees, which appear in the physics
of BPS-states. The Betti numbers for P2 follow from those for the blow-up by
the blow-up formula. The generating functions are expressed in terms of modular
functions and indefinite theta functions.Comment: 15 pages, final versio
Donaldson-Witten theory and indefinite theta functions
We consider partition functions with insertions of surface operators of
topologically twisted N=2, SU(2) supersymmetric Yang-Mills theory, or
Donaldson-Witten theory for short, on a four-manifold. If the metric of the
compact four-manifold has positive scalar curvature, Moore and Witten have
shown that the partition function is completely determined by the integral over
the Coulomb branch parameter , while more generally the Coulomb branch
integral captures the wall-crossing behavior of both Donaldson polynomials and
Seiberg-Witten invariants. We show that after addition of a Q-exact surface
operator to the Moore-Witten integrand, the integrand can be written as a total
derivative to the anti-holomorphic coordinate using Zwegers'
indefinite theta functions. In this way, we reproduce G\"ottsche's expressions
for Donaldson invariants of rational surfaces in terms of indefinite theta
functions for any choice of metric.Comment: 23 pages + appendices, comments welcome. v2: published versio
Identities for generalized Appell functions and the blow-up formula
In this paper, we prove identities for a class of generalized Appell
functions which are based on the root lattice. The
identities are reminiscent of periodicity relations for the classical Appell
function, and are proven using only analytic properties of the functions.
Moreover they are a consequence of the blow-up formula for generating functions
of invariants of moduli spaces of semi-stable sheaves of rank 3 on rational
surfaces. Our proof confirms that in the latter context, different routes to
compute the generating function (using the blow-up formula and wall-crossing)
do arrive at identical -series. The proof also gives a clear procedure for
how to prove analogous identities for generalized Appell functions appearing in
generating functions for sheaves with rank
The Coulomb Branch Formula for Quiver Moduli Spaces
In recent series of works, by translating properties of multi-centered
supersymmetric black holes into the language of quiver representations, we
proposed a formula that expresses the Hodge numbers of the moduli space of
semi-stable representations of quivers with generic superpotential in terms of
a set of invariants associated to `single-centered' or `pure-Higgs' states. The
distinguishing feature of these invariants is that they are independent of the
choice of stability condition. Furthermore they are uniquely determined by the
-genus of the moduli space. Here, we provide a self-contained summary
of the Coulomb branch formula, spelling out mathematical details but leaving
out proofs and physical motivations.Comment: 24 pages. v2: final version; minor changes, including a new diagra
A 5d/2d/4d correspondence
We propose a correspondence between two-dimensional (0,4) sigma models with
target space the moduli spaces of r monopoles, and four-dimensional N=4, U(r)
Yang-Mills theory on del Pezzo surfaces. In particular, the two- and
four-dimensional BPS partition functions are argued to be equal. The
correspondence relies on insights from five-dimensional supersymmetric gauge
theory and its geometric engineering in M-theory, hence the name "5d/2d/4d
correspondence". We provide various tests of our proposal. The most stringent
ones are for r=1, for which we prove the equality of partition functions.Comment: 39 pages, final versio
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