23 research outputs found
Exact results for N = 2 supersymmetric gauge theories on compact toric manifolds and equivariant Donaldson invariants
We provide a contour integral formula for the exact partition function of N = 2 supersymmetric U(N) gauge theories on compact toric four-manifolds by means of supersymmetric localisation. We perform the explicit evaluation of the contour integral for U(2) N = 2 17 theory on CP2 for all instanton numbers. In the zero mass case, corresponding to the N = 4 supersymmetric gauge theory, we obtain the generating function of the Euler characteristics of instanton moduli spaces in terms of mock-modular forms. In the decoupling limit of infinite mass we find that the generating function of local and surface observables computes equivariant Donaldson invariants, thus proving in this case a longstanding conjecture by N. Nekrasov. In the case of vanishing first Chern class the resulting equivariant Donaldson polynomials are new. \ua9 2016, The Author(s)
Instantons and 2d Superconformal field theory
A recently proposed correspondence between 4-dimensional N=2 SUSY SU(k) gauge
theories on R^4/Z_m and SU(k) Toda-like theories with Z_m parafermionic
symmetry is used to construct four-point N=1 super Liouville conformal block,
which corresponds to the particular case k=m=2.
The construction is based on the conjectural relation between moduli spaces
of SU(2) instantons on R^4/Z_2 and algebras like \hat{gl}(2)_2\times NSR. This
conjecture is confirmed by checking the coincidence of number of fixed points
on such instanton moduli space with given instanton number N and dimension of
subspace degree N in the representation of such algebra.Comment: 13 pages, exposition improved, references adde
Matrix Model Conjecture for Exact BS Periods and Nekrasov Functions
We give a concise summary of the impressive recent development unifying a
number of different fundamental subjects. The quiver Nekrasov functions
(generalized hypergeometric series) form a full basis for all conformal blocks
of the Virasoro algebra and are sufficient to provide the same for some
(special) conformal blocks of W-algebras. They can be described in terms of
Seiberg-Witten theory, with the SW differential given by the 1-point resolvent
in the DV phase of the quiver (discrete or conformal) matrix model
(\beta-ensemble), dS = ydz + O(\epsilon^2) = \sum_p \epsilon^{2p}
\rho_\beta^{(p|1)}(z), where \epsilon and \beta are related to the LNS
parameters \epsilon_1 and \epsilon_2. This provides explicit formulas for
conformal blocks in terms of analytically continued contour integrals and
resolves the old puzzle of the free-field description of generic conformal
blocks through the Dotsenko-Fateev integrals. Most important, this completes
the GKMMM description of SW theory in terms of integrability theory with the
help of exact BS integrals, and provides an extended manifestation of the basic
principle which states that the effective actions are the tau-functions of
integrable hierarchies.Comment: 14 page
A & B model approaches to surface operators and Toda theories
It has recently been argued by Alday et al that the inclusion of surface
operators in 4d N=2 SU(2) quiver gauge theories should correspond to insertions
of certain degenerate operators in the dual Liouville theory. So far only the
insertion of a single surface operator has been treated (in a semi-classical
limit). In this paper we study and generalise this proposal. Our approach
relies on the use of topological string theory techniques. On the B-model side
we show that the effects of multiple surface operator insertions in 4d N=2
gauge theories can be calculated using the B-model topological recursion
method, valid beyond the semi-classical limit. On the mirror A-model side we
find by explicit computations that the 5d lift of the SU(N) gauge theory
partition function in the presence of (one or many) surface operators is equal
to an A-model topological string partition function with the insertion of (one
or many) toric branes. This is in agreement with an earlier proposal by Gukov.
Our A-model results were motivated by and agree with what one obtains by
combining the AGT conjecture with the dual interpretation in terms of
degenerate operators. The topological string theory approach also opens up new
possibilities in the study of 2d Toda field theories.Comment: 43 pages. v2: Added references, including a reference to unpublished
work by S.Gukov; minor changes and clarifications
Vertices, Vortices & Interacting Surface Operators
We show that the vortex moduli space in non-abelian supersymmetric N=(2,2)
gauge theories on the two dimensional plane with adjoint and anti-fundamental
matter can be described as an holomorphic submanifold of the instanton moduli
space in four dimensions. The vortex partition functions for these theories are
computed via equivariant localization. We show that these coincide with the
field theory limit of the topological vertex on the strip with boundary
conditions corresponding to column diagrams. Moreover, we resum the field
theory limit of the vertex partition functions in terms of generalized
hypergeometric functions formulating their AGT dual description as interacting
surface operators of simple type. Analogously we resum the topological open
string amplitudes in terms of q-deformed generalized hypergeometric functions
proving that they satisfy appropriate finite difference equations.Comment: 22 pages, 4 figures; v.2 refs. and comments added; v.3 further
comments and typo
Seizure prediction : ready for a new era
Acknowledgements: The authors acknowledge colleagues in the international seizure prediction group for valuable discussions. L.K. acknowledges funding support from the National Health and Medical Research Council (APP1130468) and the James S. McDonnell Foundation (220020419) and acknowledges the contribution of Dean R. Freestone at the University of Melbourne, Australia, to the creation of Fig. 3.Peer reviewedPostprin