The computation of observables in general interacting theories, be them quantum mechanical, field, gauge or string theories, is a non-trivial problem which in many cases can only be addressed by ...resorting to perturbative methods. In most physically interesting problems these perturbative expansions result in asymptotic series with zero radius of convergence. These asymptotic series then require the use of resurgence and transseries in order for the associated observables to become nonperturbatively well-defined. Resurgence encodes the complete large-order asymptotic behaviour of the coefficients from a perturbative expansion, generically in terms of (multi) instanton sectors and for each problem in terms of its Stokes constants. Some observables arise from linear problems, and have a finite number of instanton sectors and associated Stokes constants; some other observables arise from nonlinear problems, and have an infinite number of instanton sectors and Stokes constants. By means of two very explicit examples, and with emphasis on a pedagogical style of presentation, this work aims at serving as a primer on the aforementioned resurgent, large-order asymptotics of general perturbative expansions. This includes discussions of transseries, Stokes phenomena, generalized steepest-descent methods, Borel transforms, nonlinear resonance, and alien calculus. Furthermore, resurgent properties of transseries – usually described mathematically via alien calculus – are recast in equivalent physical languages: either a “statistical mechanical” language, as motions in chains and lattices; or a “conformal field theoretical” language, with underlying Virasoro-like algebraic structures.
Abstract
We study generalized (doubled) structures in 2
D
-dimensional Born geometries in which T-duality symmetry is manifestly realized. We show that spacetime structures of Kähler, hyperkähler, ...bi-hermitian and bi-hypercomplex manifolds are implemented in Born geometries as generalized (doubled) structures. We find that the Born structures and the generalized Kähler (hyperkähler) structures appear as subalgebras of bi-quaternions ℂ × ℍ and split-tetra-quaternions ℍ × Spℍ. We investigate the nature of T-duality for the worldsheet instantons in Born sigma models. This manuscript is based on the original paper 1.
Abstract
We investigate the T-duality relations between hyperkähler and bi-hypercomplex structures using the doubled formalism. In generalized geometry, both the hyperkähler and bi-hypercomplex ...structures are embedded in generalized hyperkähler structures that satisfy the split-bi-quaternion algebra. We write down the analogue of the Buscher rule, which is the T-duality transformation of the hyperkähler and bi-hypercomplex structures. As a practical example, we construct the bi-hypercomplex structure of the
5
2
2
-brane, known as a T-fold, from the hyperkähler structure of the Taub-NUT space using the T-duality transformation. The bi-hypercomplex structures of the T-fold have non-trivial monodromies. This results in the fact that the worldsheet instantons on the T-fold are multi-valued. We comment on the resolution of this issue using the Born sigma model.
We define the analogue of instanton sheaves on the blow-up Pn˜ of the n-dimensional projective space at a point. We choose an appropriate polarisation on Pn˜ and construct rank 2 examples of locally ...free and non locally free (but torsion free) type. In general, the defined instantons also turn out to be the cohomology of monads, although non-linear ones. Moreover, in the five dimensional case, we show that there are continuous families of them that fill, at least, a smooth component in the moduli of semi-stable sheaves.
The question of the dependence of the topological charge q of a gauged Skyrmion, on the gauge field, is studied quantitatively. Two examples, both gauged with SO(2) are studied and contrasted: i) The ...O(3) model in 2+1 dimensions, and ii) The O(4) model in 3+1 dimensions. In case i), where the (usual) Chern–Simons (CS) term is present, the value of q changes sign, going through zero. This evolution is tracked by a parameter characterising the solutions in the given theory. In case ii), in which dimensions no CS density is available, the evolution of q is not observed.
SU(2) 2 -invariant G 2 -instantons Lotay, Jason D; Oliveira, Goncalo
Mathematische annalen,
2018, Letnik:
371, Številka:
1
Journal Article
Recenzirano
We initiate the systematic study of
-instantons with SU(2)
-symmetry. As well as developing foundational theory, we give existence, non-existence and classification results for these instantons. We ...particularly focus on
with its two explicitly known distinct holonomy
metrics, which have different volume growths at infinity, exhibiting the different behaviour of instantons in these settings. We also give an explicit example of sequences of
-instantons where "bubbling" and "removable singularity" phenomena occur in the limit.
A
bstract
We consider general 5d SU(
N
) quiver gauge theories whose nodes form an ADE Dynkin diagram of type
G
. Each node has SU(
N
i
) gauge group of general rank, Chern-Simons level
κ
i
and ...additional
w
i
fundamentals. When the total flavor number at each node is less than or equal to 2
N
i
− 2|
κ
i
|, we give general rules under which the symmetries associated to instanton currents are enhanced to
G
×
G
or a subgroup of it in the UV 5d superconformal theory. When the total flavor number violates that condition at some of the nodes, further enhancement of flavor symmetries occurs. In particular we find a large class of gauge theories interpreted as
S
1
compactification of 6d superconformal theories which are waiting for string/F-theory realization. We also consider hypermultiplets in (anti-)symmetric representation.
A
bstract
We consider the SU(
N
) Yang-Mills theory, whose topological sectors are restricted to the instanton number with integer multiples of
p
. We can formulate such a quantum field theory ...maintaining locality and unitarity, and the model contains both 2
π
-periodic scalar and 3-form gauge fields. This can be interpreted as coupling a topological theory to Yang-Mills theory, so the local dynamics becomes identical with that of pure Yang-Mills theory. The theory has not only ℤ
N
1-form symmetry but also ℤ
p
3-form symmetry, and we study the global nature of this theory from the recent ’t Hooft anomaly matching. The computation of ’t Hooft anomaly incorporates an intriguing higher-group structure. We also carefully examine that how such kinematical constraint is realized in the dynamics by using the large-
N
and also the reliable semiclassics on ℝ
3
×
S
1
, and we find that the topological susceptibility plays a role of the order parameter for the ℤ
p
3-form symmetry. Introducing a fermion in the fundamental or adjoint representation, we find that the chiral symmetry becomes larger than the usual case by ℤ
p
, and it leads to the extra
p
vacua by discrete chiral symmetry breaking. No dynamical domain wall can interpolate those extra vacua since such objects must be charged under the 3-form symmetry in order to match the ’t Hooft anomaly.
General instanton counting and 5d SCFT Hwang, Chiung; Kim, Joonho; Kim, Seok ...
The journal of high energy physics,
07/2015, Letnik:
2015, Številka:
7
Journal Article
Recenzirano
Odprti dostop
A
bstract
Instanton partition functions of 5d
N
=
1
gauge theories are Witten indices for the ADHM gauged quantum mechanics with (0, 4) SUSY. We derive the integral contour prescriptions for these ...indices using the Jeffrey-Kirwan method, for gauge theories with hypermultiplets in various representations. The results can be used to study various 4d/5d/6d QFTs. In this paper, we study 5d SCFTs which are at the UV fixed points of 5d SYM theories. In particular, we focus on the Sp(
N
) theories with
N
f
≤ 7 fundamental and 1 antisymmetric hypermultiplets, living on the D4-D8-O8 systems. Their superconformal indices calculated from instantons all show
E
N
f
+
1
symmetry enhancements. We also discuss some aspects of the 6d SCFTs living on the M5-M9 system. It is crucial to understand the UV incompleteness of the 5d SYM, coming from small instantons in our problem. We explain in our examples how to fix them. As an aside, we derive the index for general gauged quantum mechanics with (0
,
2) SUSY.
PDF
