11 research outputs found
Topological twists of massive SQCD, Part II
This is the second and final part of ‘Topological twists of massive SQCD’. Part I is available at Lett. Math. Phys. 114 (2024) 3, 62. In this second part, we evaluate the contribution of the Coulomb branch to topological path integrals for N=2 supersymmetric QCD with Nf ≤ 3 massive hypermultiplets on compact four-manifolds. Our analysis includes the decoupling of hypermultiplets, the massless limit and the merging of mutually non-local singularities at the Argyres–Douglas points. We give explicit mass expansions for the four-manifolds ℙ2 and K3. For ℙ2, we find that the correlation functions are polynomial as function of the masses, while infinite series and (potential) singularities occur for K3. The mass dependence corresponds mathematically to the integration of the equivariant Chern class of the matter bundle over the moduli space of Q-fixed equations. We demonstrate that the physical partition functions agree with mathematical results on Segre numbers of instanton moduli spaces
Topological twists of massive SQCD, Part II
This is the second and final part of ``Topological twists of massive SQCD''.
Part I is available at arXiv:2206.08943. In this second part, we evaluate the
contribution of the Coulomb branch to topological path integrals for
supersymmetric QCD with massive hypermultiplets on
compact four-manifolds. Our analysis includes the decoupling of
hypermultiplets, the massless limit and the merging of mutually non-local
singularities at the Argyres-Douglas points. We give explicit mass expansions
for the four-manifolds and . For , we find
that the correlation functions are polynomial as function of the masses, while
infinite series and (potential) singularities occur for . The mass
dependence corresponds mathematically to the integration of the equivariant
Chern class of the matter bundle over the moduli space of -fixed equations.
We demonstrate that the physical partition functions agree with mathematical
results on Segre numbers of instanton moduli spaces.Comment: 80 pages + appendices, 13 figures. Second part of a series of two
papers, first part is available at arXiv:2206.0894
Taming Binarized Neural Networks and Mixed-Integer Programs
There has been a great deal of recent interest in binarized neural networks,
especially because of their explainability. At the same time, automatic
differentiation algorithms such as backpropagation fail for binarized neural
networks, which limits their applicability. By reformulating the problem of
training binarized neural networks as a subadditive dual of a mixed-integer
program, we show that binarized neural networks admit a tame representation.
This, in turn, makes it possible to use the framework of Bolte et al. for
implicit differentiation, which offers the possibility for practical
implementation of backpropagation in the context of binarized neural networks.
This approach could also be used for a broader class of mixed-integer
programs, beyond the training of binarized neural networks, as encountered in
symbolic approaches to AI and beyond.Comment: 9 pages, 4 figure
Elliptic Loci of SU(3) Vacua
The space of vacua of many four-dimensional, supersymmetric
gauge theories can famously be identified with a family of complex curves. For
gauge group , this gives a fully explicit description of the low-energy
effective theory in terms of an elliptic curve and associated modular
fundamental domain. The two-dimensional space of vacua for gauge group
parametrizes an intricate family of genus two curves. We analyze this family
using the so-called Rosenhain form for these curves. We demonstrate that two
natural one-dimensional subloci of the space of vacua,
and , each parametrize a family of elliptic curves. For these
elliptic loci, we describe the order parameters and fundamental domains
explicitly. The locus contains the points where mutually local
dyons become massless, and is a fundamental domain for a classical congruence
subgroup. Moreover, the locus contains the superconformal
Argyres-Douglas points, and is a fundamental domain for a Fricke group.Comment: 39 pages + Appendices, 5 figures, v2: minor changes and extended
discussion on automorphism
Hybrid Methods in Polynomial Optimisation
The Moment/Sum-of-squares hierarchy provides a way to compute the global
minimizers of polynomial optimization problems (POP), at the cost of solving a
sequence of increasingly large semidefinite programs (SDPs). We consider
large-scale POPs, for which interior-point methods are no longer able to solve
the resulting SDPs. We propose an algorithm that combines a first-order
Burer-Monteiro-type method for solving the SDP relaxation, and a second-order
method on a non-convex problem obtained from the POP. The switch from the first
to the second-order method is based on a quantitative criterion, whose
satisfaction ensures that Newton's method converges quadratically from its
first iteration. This criterion leverages the point-estimation theory of Smale
and the active-set identification. We illustrate the methodology to obtain
global minimizers of large-scale optimal power flow problems
Decay channels for double extremal black holes in four dimensions
We explore decay channels for charged black holes with vanishing temperature in supersymmetric compactifications of string theory. If not protected by supersymmetry, such extremal black holes are expected to decay as a consequence of the weak gravity conjecture. We concentrate on double extremal, non-supersymmetric black holes for which the values of the scalar fields are constant throughout space-time, and explore decay channels for which decay into BPS and anti-BPS constituents is energetically favorable. We demonstrate the existence of decay channels at tree level for large families of double extremal black holes. For specific charges, we also find stable non-supersymmetric black holes, suggesting recombination of (anti)-supersymmetric constituents to a non-supersymmetric object
Four flavors, triality, and bimodular forms
We consider supersymmetric gauge theory with
massive hypermultiplets. The duality group of this theory contains
transformations acting on the UV-coupling as well as on the
running coupling . We establish that subgroups of the duality group act
separately on and , while a larger group acts
simultaneously on and . For special choices of the
masses, we find that the duality groups can be identified with congruence
subgroups of . We demonstrate that in such cases, the
order parameters are instances of bimodular forms with arguments and
. Since the UV duality group of the theory contains the
triality group of outer automorphisms of the flavour symmetry ,
the duality action gives rise to an orbit of mass configurations. Consequently,
the corresponding order parameters combine to vector-valued bimodular forms
with acting simultaneously on the two couplings.Comment: 29 pages + Appendices, 7 figures, v2: minor changes, updated Appendix
Cutting and gluing with running couplings in 𝒩=2 QCD
We consider the order parameter u=\left as function
of the running coupling constant of asymptotically free
QCD with gauge group and massive
hypermultiplets. If the domain for is restricted to an appropriate
fundamental domain , the function is one-to-one. We
demonstrate that these domains consist of six or less images of an keyhole fundamental domain, with appropriate identifications
of the boundaries. For special choices of the masses, does not give rise to
branch points and cuts, such that is a modular function for a congruence
subgroup of and the fundamental domain is
. For generic masses, however, branch points and
cuts are present, and subsets of are being cut and glued
upon varying the mass. We study this mechanism for various phenomena, such as
decoupling of hypermultiplets, merging of local singularities, as well as
merging of non-local singularities which give rise to superconformal
Argyres-Douglas theories.Comment: 56 pages + Appendices, 22 figure
Topological twists of massive SQCD, Part I
We consider topological twists of four-dimensional
supersymmetric QCD with gauge group SU(2) and fundamental
hypermultiplets. The twists are labelled by a choice of background fluxes for
the flavour group, which provides an infinite family of topological partition
functions. In this Part I, we demonstrate that in the presence of such fluxes
the theories can be formulated for arbitrary gauge bundles on a compact
four-manifold. Moreover, we consider arbitrary masses for the hypermultiplets,
which introduce new intricacies for the evaluation of the low-energy path
integral on the Coulomb branch. We develop techniques for the evaluation of
these path integrals. In the forthcoming Part II, we will deal with the
explicit evaluation.Comment: 41 pages + appendices, 8 figures. First part of a series of two
papers, second part available at arXiv:2312.1161