57 research outputs found
An A_r threesome: Matrix models, 2d CFTs and 4d N=2 gauge theories
We explore the connections between three classes of theories: A_r quiver
matrix models, d=2 conformal A_r Toda field theories and d=4 N=2 supersymmetric
conformal A_r quiver gauge theories. In particular, we analyse the quiver
matrix models recently introduced by Dijkgraaf and Vafa and make detailed
comparisons with the corresponding quantities in the Toda field theories and
the N=2 quiver gauge theories. We also make a speculative proposal for how the
matrix models should be modified in order for them to reproduce the instanton
partition functions in quiver gauge theories in five dimensions.Comment: 43 pages; v2: minor typos corrected, reference added. v3: Added
clarifying discussions in sections 4.1 and 5.1; typos correcte
The Dynamics of Knotted Strings Attached to D-Branes
We extend the general solution to the Cauchy problem for the relativistic
closed string (Phys. Lett. B404 (1997) 57-65, hep-th/9704084) to the case of
open strings attached to Dp-branes, including the cases where the initial data
has a knotlike topology. We use this extended solution to derive intrinsic
dynamical properties of open and closed relativistic strings attached to
Dp-branes. We also study the singularity structure and the oscillating periods
of this extended solution.Comment: 9 pages, 4 figures, Plain Te
Borel and Stokes Nonperturbative Phenomena in Topological String Theory and c=1 Matrix Models
We address the nonperturbative structure of topological strings and c=1
matrix models, focusing on understanding the nature of instanton effects
alongside with exploring their relation to the large-order behavior of the 1/N
expansion. We consider the Gaussian, Penner and Chern-Simons matrix models,
together with their holographic duals, the c=1 minimal string at self-dual
radius and topological string theory on the resolved conifold. We employ Borel
analysis to obtain the exact all-loop multi-instanton corrections to the free
energies of the aforementioned models, and show that the leading poles in the
Borel plane control the large-order behavior of perturbation theory. We
understand the nonperturbative effects in terms of the Schwinger effect and
provide a semiclassical picture in terms of eigenvalue tunneling between
critical points of the multi-sheeted matrix model effective potentials. In
particular, we relate instantons to Stokes phenomena via a hyperasymptotic
analysis, providing a smoothing of the nonperturbative ambiguity. Our
predictions for the multi-instanton expansions are confirmed within the
trans-series set-up, which in the double-scaling limit describes
nonperturbative corrections to the Toda equation. Finally, we provide a
spacetime realization of our nonperturbative corrections in terms of toric
D-brane instantons which, in the double-scaling limit, precisely match
D-instanton contributions to c=1 minimal strings.Comment: 71 pages, 14 figures, JHEP3.cls; v2: added refs, minor change
Nonperturbative Ambiguities and the Reality of Resurgent Transseries
In a wide range of quantum theoretical settings -- from quantum mechanics to
quantum field theory, from gauge theory to string theory -- singularities in
the complex Borel plane, usually associated to instantons or renormalons,
render perturbation theory ill-defined as they give rise to nonperturbative
ambiguities. These ambiguities are associated to choices of an integration
contour in the resummation of perturbation theory, along (singular) Stokes
directions in the complex Borel plane (rendering perturbative expansions
non-Borel summable along any Stokes line). More recently, it has been shown
that the proper framework to address these issues is that of resurgent analysis
and transseries. In this context, the cancelation of all nonperturbative
ambiguities is shown to be a consequence of choosing the transseries median
resummation as the appropriate family of unambiguous real solutions along the
coupling-constant real axis. While the median resummation is easily implemented
for one-parameter transseries, once one considers more general multi-parameter
transseries the procedure becomes highly dependent upon properly understanding
Stokes transitions in the complex Borel plane. In particular, all Stokes
coefficients must now be known in order to explicitly implement multi-parameter
median resummations. In the cases where quantum-theoretical physical
observables are described by resurgent functions and transseries, the methods
described herein show how one may cancel nonperturbative ambiguities, and
define these observables nonperturbatively starting out from perturbation
theory. Along the way, structural results concerning resurgent transseries are
also obtained.Comment: 62 pages, 4 figures; v2: corrected typos, added small discussion on
topological sectors, two new figure
The Resurgence of Instantons in String Theory
Nonperturbative effects in string theory are usually associated to D-branes.
In many cases it can be explicitly shown that D-brane instantons control the
large-order behavior of string perturbation theory, leading to the well-known
(2g)! growth of the genus expansion. This paper presents a detailed treatment
of nonperturbative solutions in string theory, and their relation to the
large-order behavior of perturbation theory, making use of transseries and
resurgent analysis. These are powerful techniques addressing general
nonperturbative contributions within non-linear systems, which are developed at
length herein as they apply to string theory. The cases of topological strings,
the Painleve I equation describing 2d quantum gravity, and the quartic matrix
model, are explicitly addressed. These results generalize to minimal strings
and general matrix models. It is shown that, in order to completely understand
string theory at a fully nonperturbative level, new sectors are required beyond
the standard D-brane sector.Comment: 108 pages; v2,v3: references added; v4: improved pedagogical content,
final version for CNTP; v5: typos correcte
Resurgent Transseries and the Holomorphic Anomaly
The gauge theoretic large N expansion yields an asymptotic series which
requires a nonperturbative completion in order to be well defined. Recently,
within the context of random matrix models, it was shown how to build resurgent
transseries solutions encoding the full nonperturbative information beyond the
't Hooft genus expansion. On the other hand, via large N duality, random matrix
models may be holographically described by B-model closed topological strings
in local Calabi-Yau geometries. This raises the question of constructing the
corresponding holographically dual resurgent transseries, tantamount to
nonperturbative topological string theory. This paper addresses this point by
showing how to construct resurgent transseries solutions to the holomorphic
anomaly equations. These solutions are built upon (generalized) multi-instanton
sectors, where the instanton actions are holomorphic. The asymptotic expansions
around the multi-instanton sectors have both holomorphic and anti-holomorphic
dependence, may allow for resonance, and their structure is completely fixed by
the holomorphic anomaly equations in terms of specific polynomials multiplied
by exponential factors and up to the holomorphic ambiguities -- which
generalizes the known perturbative structure to the full transseries. In
particular, the anti-holomorphic dependence has a somewhat universal character.
Furthermore, in the nonperturbative sectors, holomorphic ambiguities may be
fixed at conifold points. This construction shows the nonperturbative
integrability of the holomorphic anomaly equations, and sets the ground to
start addressing large-order analysis and resurgent nonperturbative completions
within closed topological string theory.Comment: 59 pages, jheppub-nosort.sty; v2: small additions, minor changes,
refs updated; v3: more minor corrections, final version for AH
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