Resurgence and Topological Strings

The mathematical idea of resurgence allows one to obtain nonperturbative information from the large-order behavior of perturbative expansions. This idea can be very fruitful in physics applications, in particular if one does not have access to such nonperturbative information from first principles. An important example is topological string theory, which is a priori only defined as an asymptotic perturbative expansion in the coupling constant g_s. We show how the idea of resurgence can be combined with the holomorphic anomaly equation to extend the perturbative definition of the topological string and obtain, in a model-independent way, a large amount of information about its nonperturbative structure.Comment: 11 pages, 7 figures. Pedestrian introduction to 1308.1695 and 1407.4821, based on my talk at String Math 2014. Submitted for the proceedings of that conferenc

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: Nonperturbative Closed Strings in Local CP2

The holomorphic anomaly equations describe B-model closed topological strings in Calabi-Yau geometries. Having been used to construct perturbative expansions, it was recently shown that they can also be extended past perturbation theory by making use of resurgent transseries. These yield formal nonperturbative solutions, showing integrability of the holomorphic anomaly equations at the nonperturbative level. This paper takes such constructions one step further by working out in great detail the specific example of topological strings in the mirror of the local CP2 toric Calabi-Yau background, and by addressing the associated (resurgent) large-order analysis of both perturbative and multi-instanton sectors. In particular, analyzing the asymptotic growth of the perturbative free energies, one finds contributions from three different instanton actions related by Z_3 symmetry, alongside another action related to the Kahler parameter. Resurgent transseries methods then compute, from the extended holomorphic anomaly equations, higher instanton sectors and it is shown that these precisely control the asymptotic behavior of the perturbative free energies, as dictated by resurgence. The asymptotic large-order growth of the one-instanton sector unveils the presence of resonance, i.e., each instanton action is necessarily joined by its symmetric contribution. The structure of different resurgence relations is extensively checked at the numerical level, both in the holomorphic limit and in the general nonholomorphic case, always showing excellent agreement with transseries data computed out of the nonperturbative holomorphic anomaly equations. The resurgence relations further imply that the string free energy displays an intricate multi-branched Borel structure, and that resonance must be properly taken into account in order to describe the full transseries solution.Comment: 63 pages, 54 images in 24 figures, jheppub-nosort.sty; v2: corrected figure, minor changes, final version for CM

MHV, CSW and BCFW: field theory structures in string theory amplitudes

Motivated by recent progress in calculating field theory amplitudes, we study applications of the basic ideas in these developments to the calculation of amplitudes in string theory. We consider in particular both non-Abelian and Abelian open superstring disk amplitudes in a flat space background, focusing mainly on the four-dimensional case. The basic field theory ideas under consideration split into three separate categories. In the first, we argue that the calculation of alpha'-corrections to MHV open string disk amplitudes reduces to the determination of certain classes of polynomials. This line of reasoning is then used to determine the alpha'^3-correction to the MHV amplitude for all multiplicities. A second line of attack concerns the existence of an analog of CSW rules derived from the Abelian Dirac-Born-Infeld action in four dimensions. We show explicitly that the CSW-like perturbation series of this action is surprisingly trivial: only helicity conserving amplitudes are non-zero. Last but not least, we initiate the study of BCFW on-shell recursion relations in string theory. These should appear very naturally as the UV properties of the string theory are excellent. We show that all open four-point string amplitudes in a flat background at the disk level obey BCFW recursion relations. Based on the naturalness of the proof and some explicit results for the five-point gluon amplitude, it is expected that this pattern persists for all higher point amplitudes and for the closed string.Comment: v3: corrected erroneous statement about Virasoro-Shapiro amplitude and added referenc

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

Painlev\'e I and exact WKB: Stokes phenomenon for two-parameter transseries

For more than a century, the Painlev\'e I equation has played an important role in both physics and mathematics. Its two-parameter family of solutions was studied in many different ways, yet still leads to new surprises and discoveries. Two popular tools in these studies are the theory of isomonodromic deformation that uses the exact WKB method, and the asymptotic description of transcendents in terms of two-parameter transseries. Combining methods from both schools of thought, and following work by Takei and collaborators, we find complete, two-parameter connection formulae for solutions when they cross arbitrary Stokes lines in the complex plane. These formulae allow us to study Stokes phenomenon for the full two-parameter family of transseries solutions. In particular, we recover the exact expressions for the Stokes data that were recently found by Baldino, Schwick, Schiappa and Vega and compare our connection formulae to theirs. We also explain several ambiguities in relating transseries parameter choices to actual Painlev\'e transcendents, study the monodromy of formal solutions, and provide high-precision numerical tests of our results.Comment: 71 pages, 16 figures, 5 tables and 4 appendices. v2: Minor changes (rewrites, typos, added references

Sample Stability and Protein Composition of Saliva: Implications for Its Use as a Diagnostic Fluid

Saliva is an easy accessible plasma ultra-filtrate. Therefore, saliva can be an attractive alternative to blood for measurement of diagnostic protein markers. Our aim was to determine stability and protein composition of saliva. Protein stability at room temperature was examined by incubating fresh whole saliva with and without inhibitors of proteases and bacterial metabolism followed by Surface Enhanced Laser Desorption/Ionization (SELDI) analyses. Protein composition was determined by sodium dodecyl sulfate polyacrylamide gel electrophoresis (SDS-PAGE) fractionation of saliva proteins followed by digestion of excised bands and identification by liquid chromatography tandem mass spectrometry (LC-MS/MS). Results show that rapid protein degradation occurs within 30 minutes after sample collection. Degradation starts already during collection. Protease inhibitors partly prevented degradation while inhibition of bacterial metabolism did not affect degradation. Three stable degradation products of 2937 Da, 3370 Da and 4132 Da were discovered which can be used as markers to monitor sample quality. Saliva proteome analyses revealed 218 proteins of which 84 can also be found in blood plasma. Based on a comparison with seven other proteomics studies on whole saliva we identified 83 new saliva proteins. We conclude that saliva is a promising diagnostic fluid when precautions are taken towards protein breakdown

Type 0A 2D Black Hole Thermodynamics and the Deformed Matrix Model

Recently, it has been proposed that the deformed matrix model describes a two-dimensional type 0A extremal black hole. In this paper, the thermodynamics of 0A charged non-extremal black holes is investigated. We observe that the free energy of the deformed matrix model to leading order in 1/q can be seen to agree to that of the extremal black hole. We also speculate on how the deformed matrix model is able to describe the thermodynamics of non-extremal black holes.Comment: 12 page