Perturbative String Theory Research Articles

Quantisation of type IIB superstring theory and the matrix model

We discuss the path-integral quantisation of perturbative string theory and show equivalence between the Polyakov-type, Schild-type and Nambu-Goto-type formulations of critical type II superstring theory. Remarkably, we also find that the Minkowskian path integral realises causality in the sense that a string does not propagate between points at space-like separation, by giving careful consideration to the measure of the world-sheet metric. We also discuss matrix regularisation of the path integral for type IIB perturbative superstring theory. The obtained matrix models are the Euclidean IKKT matrix model and a modified Minkowskian IKKT model, depending on how the matrix regularisation is applied.

Type II string theory on AdS3×S3×T4 and symmetric orbifolds

We discuss in detail the 1+1-dimensional superconformal field theory dual to type II string theory on AdS3×S3×T4, emphasizing the string theoretic aspects of this duality. For one unit of Neveu-Schwarz (NS-NS) 5-brane flux (Q5=1), this string theory has been suggested to be dual to a grand-canonical ensemble of T4N/SN free symmetric orbifold conformal field theories (CFTs). We show how the string genus expansion emerges to all orders for the free orbifold grand-canonical correlation functions. We also discuss how the strong coupling limit of the NS-NS string theory arises (even at large N) in the free orbifold description, and argue why this limit does not have a weakly coupled RR description. The dual conformal field theory (CFT) includes (for all values of Q5) an extra T4 factor that is decoupled from perturbative string theory. We discuss the exactly marginal deformations that relate the different values of Q5, including the precise JJ¯ deformations mixing this extra T4 with the symmetric orbifold. Published by the American Physical Society 2024

A string theory for two dimensional Yang-Mills theory. Part I

Two dimensional gauge theories with charged matter fields are useful toy models for studying gauge theory dynamics, and in particular for studying the duality of large N gauge theories to perturbative string theories. A useful starting point for such studies is the pure Yang-Mills theory, which is exactly solvable. Its 1/N expansion was interpreted as a string theory by Gross and Taylor 30 years ago, but they did not provide a worldsheet action for this string theory, and such an action is useful for coupling it to matter fields. The chiral sector of the Yang-Mills theory can be written as a sum over holomorphic maps and has useful worldsheet descriptions, but the full theory includes more general extremal-area maps; a formal worldsheet action including all these maps in a “topological rigid string theory” was written by Hořava many years ago, but various subtleties arise when trying to use it for computations. In this paper we suggest a Polyakov-like generalization of Hořava’s worldsheet action which is well-defined, and we show how it reproduces the free limit of the Yang-Mills theory, both by formal arguments and by explicitly computing its partition function in several cases. In the future we plan to generalize this string theory to the finite-coupling gauge theory, and to analyze it with boundaries, corresponding either to Wilson loops or to dynamical matter fields in the fundamental representation.

Emergence of R4-terms in M-theory

It has been recently suggested that the strong Emergence Proposal is realized in M-theory limits by integrating out all light towers of states with a typical mass scale not larger than the species scale, i.e. the eleventh dimensional Planck mass. Within the BPS sector, these are transverse M2- and M5-branes, that can be wrapped and particle-like, carrying Kaluza-Klein momentum along the compact directions. We provide additional evidence for this picture by revisiting and investigating further the computation of R4-interactions in M-theory à la Green-Gutperle-Vanhove. A central aspect is a novel UV-regularization of Schwinger-like integrals, whose actual meaning and power we clarify by first applying it to string perturbation theory. We consider then toroidal compactifications of M-theory and provide evidence that integrating out all light towers of states via Schwinger-like integrals thus regularized yields the complete result for R4-interactions. In particular, this includes terms that are tree-level, one-loop and space-time instanton corrections from the weakly coupled point of view. Finally, we comment on the conceptual difference of our approach to earlier closely related work by Kiritsis-Pioline and Obers-Pioline, leading to a correspondence between two types of constrained Eisenstein series.

String thermodynamics in and out of equilibrium: Boltzmann equations and random walks

We revisit the study of string theory close to the Hagedorn temperature with the aim towards cosmological applications. We consider interactions of open and closed strings in a gas of Dp-branes, and/or one isolated Dp-brane, in an arbitrary number d of flat non-compact dimensions and general compact dimensions. Leading order string perturbation theory is used to obtain the basic interaction rates in a flat background, which are shown to be consistent with the random walk picture of highly excited strings that should apply in more general backgrounds. Using the random walk interpretation we infer the structure of more general semi-inclusive string scattering rates and then write down the corresponding Boltzmann equations describing ensembles of highly excited closed and open strings. We organise the interaction terms in Boltzmann equations so that detailed balance becomes manifest. We obtain the equilibrium solutions and show that they reduce to previously computed solutions for d = 0. We further study the behaviour of non-equilibrium fluctuations and find explicit analytic expressions for the equilibration rates (and for the number of open strings in d = 0). Potential implications for an early universe with strings at high temperatures are outlined.

On time-dependent backgrounds in 1 + 1 dimensional string theory

In perturbative string theory, one is generally interested in asymptotic observables, such as the S-matrix in flat spacetime, and boundary correlation functions in anti-de Sitter spacetime. However, there are backgrounds in which such observables do not exist. We study examples of such backgrounds in 1 + 1 dimensional string theory. In these examples, the Liouville wall accelerates and can become spacelike in the past and/or future. When that happens, the corresponding null infinity, at which the standard scattering states are defined, is shielded by the Liouville wall. We compute scattering and particle production amplitudes in these backgrounds in the region in parameter space where the wall remains timelike, and discuss the continuation of this picture to the spacelike regime. We also discuss the physics from the point of view of the dynamics of free fermions in backgrounds with a time-dependent Fermi surface.

BPS fivebrane stars. Part II. Fluctuations

We investigate quantum fluctuations of metric components in coherent 1/2-BPS bound states of n1 fundamental strings and n5 NS5-branes. The leading order contribution in an expansion in 1/(n1n5) is calculated via a combination of analytical and numerical methods. We find that the fluctuations are small away from a tiny distance from the source, comparable to the 6d Planck scale. Comparing this result with an analysis in the literature of fluctuations in the maximally mixed state, we conclude that the large fluctuations previously found for the latter are statistical rather than quantum in nature, and that perturbative string theory provides an accurate description of these backgrounds.

BPS fivebrane stars. Part I. Expectation values of observables

We study ensembles of 1/2-BPS bound states of fundamental strings and NS-fivebranes (NS5-F1 states) in the AdS decoupling limit. We revisit a solution corresponding to an ensemble average of these bound states, and find that the appropriate duality frame for describing the near-source structure is the T-dual NS5-P frame, where the bound state is a collection of momentum waves on the fivebranes. We find that the fivebranes are generically well-separated; this property results in the applicability of perturbative string theory. The geometry sourced by the typical microstate is not close to that of an extremal non-rotating black hole; instead the fivebranes occupy a ball whose radius is parametrically much larger than the “stretched horizon” scale of the corresponding black hole. These microstates are thus better characterized as BPS fivebrane stars than as small black holes.When members of the ensemble spin with two fixed angular potentials about two orthogonal planes, we find that the spherical ball of the non-rotating ensemble average geometry deforms into an ellipsoid. This contrasts with ring structures obtained when fixing the angular momenta instead of the angular potentials; we trace this difference of ensembles to large fluctuations of the angular momentum in the ensemble of fixed angular potential.

New lower bounds on scattering amplitudes: non-locality constraints

Under reasonable working assumptions including the polynomial boundedness, one proves the well-known Cerulus-Martin lower bound on how fast an elastic scattering amplitude can decrease in the hard-scattering regime. In this paper we consider two non-trivial extensions of the previous bound. (i) We generalize the assumption of polynomial boundedness by allowing amplitudes to exponentially grow for some complex momenta and prove a more general lower bound in the hard-scattering regime. (ii) We prove a new lower bound on elastic scattering amplitudes in the Regge regime, in both cases of polynomial and exponential boundedness. A bound on the Regge trajectory for negative momentum transfer squared is also derived. We discuss the relevance of our results for understanding gravitational scattering at the non-perturbative level and for constraining ultraviolet completions. In particular, we use the new bounds as probes of non-locality in black-hole formation, perturbative string theory, classicalization, Galileons, and infinite-derivative field theories, where both the polynomial boundedness and the Cerulus-Martin bound are violated.

Fayet–Iliopoulos D-term in non-supersymmetric heterotic string orbifolds

The Fayet–Iliopoulos D-term is a common feature in N=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {N}}=1$$\\end{document} string vacua that contain an anomalous U(1) gauge symmetry, and arises from a one-loop diagram in string perturbation theory. The same diagram is generated in string vacua in which supersymmetry is broken directly at the string scale, either via spontaneous Scherk–Schwarz breaking, in which case the gravitino mass is determined by the radius of the circle used in the Scherk–Schwarz mechanism, or via explicit supersymmetry breaking by the GSO projections. We analyse the resulting would-be Fayet–Illiopoulos D-term in the non-supersymmetric string vacua and its contribution to the vacuum energy. A numerical estimate in an explicit tachyon-free string-derived model suggests that the would-be D-term contribution may uplift the vacuum energy to a positive value.

A two-dimensional string cosmology

We study two-dimensional string theory on a time-dependent background, whose worldsheet description consists of Liouville theory at central charge c = 1 and Liouville theory at central charge c = 25, together with the conformal ghosts. We compute the tree-level three-point and four-point components of the cosmological wavefunction in string perturbation theory. The latter is evaluated numerically by decomposing the Liouville four-point correlation functions into Virasoro conformal blocks and three-point function coefficients and integrating over the moduli space of the four-punctured sphere string diagram. This computation numerically confirms a surprisingly simple conjectural result for the four-point wavefunction component whose physical interpretation remains to be clarified.

Keldysh rotation in the large- N expansion and string theory out of equilibrium

We extend our study of the large-$N$ expansion of general nonequilibrium many-body systems with matrix degrees of freedom $M$, and its dual description as a sum over surface topologies in a dual string theory, to the Keldysh-rotated version of the Schwinger-Keldysh formalism. The Keldysh rotation trades the original fields ${M}_{\ifmmode\pm\else\textpm\fi{}}$---defined as the values of $M$ on the forward and backward segments of the closed time contour---for their linear combinations ${M}_{\mathrm{cl}}$ and ${M}_{\mathrm{qu}}$, known as the ``classical'' and ``quantum'' fields. First we develop a novel ``signpost'' notation for nonequilibrium Feynman diagrams in the Keldysh-rotated form, which simplifies the analysis considerably. Before the Keldysh rotation, each world-sheet surface $\mathrm{\ensuremath{\Sigma}}$ in the dual string theory expansion was found to exhibit a triple decomposition into the parts ${\mathrm{\ensuremath{\Sigma}}}^{\ifmmode\pm\else\textpm\fi{}}$ corresponding to the forward and backward segments of the closed time contour, and ${\mathrm{\ensuremath{\Sigma}}}^{\ensuremath{\wedge}}$ which corresponds to the instant in time where the two segments meet. After the Keldysh rotation, we find that the world-sheet surface $\mathrm{\ensuremath{\Sigma}}$ of the dual string theory undergoes a very different natural decomposition: $\mathrm{\ensuremath{\Sigma}}$ consists of a ``classical'' part ${\mathrm{\ensuremath{\Sigma}}}^{\mathrm{cl}}$ and a ``quantum embellishment'' part ${\mathrm{\ensuremath{\Sigma}}}^{\mathrm{qu}}$. We show that both parts of $\mathrm{\ensuremath{\Sigma}}$ carry their own independent genus expansion. The nonequilibrium sum over world-sheet topologies is naturally refined into a sum over the double decomposition of each $\mathrm{\ensuremath{\Sigma}}$ into its classical and quantum part. We apply this picture to the classical limits of the quantum nonequilibrium system (with or without interactions with a thermal bath), and find that in these limits, the dual string perturbation theory expansion reduces to its appropriately defined classical limit.

Laplace-eigenvalue equations for length three modular iterated integrals

A space of modular iterated integrals sits inside the space of real analytic modular forms. We present a theorem for producing length three modular iterated integrals which are not simply combinations of real analytic Eisenstein series; each function has an associated Laplace-eigenvalue equation. This can be viewed as an extension of the length two case recently given by F. Brown, a review of which is included in this paper. We discuss how modular iterated integrals could help understand the modular graph functions which arise in string perturbation theory.

Explicit microstates at the Schwarzschild horizon

The claim that the microstates of Schwarzschild black holes in perturbative string theory amount to the modes of long folded strings in the vicinity of its horizon is supported by more evidence.

Topological modular forms and the absence of a heterotic global anomaly

Abstract Spacetime theories obtained from perturbative string theory constructions are automatically free of perturbative anomalies, but it is not settled whether they are always free of global anomalies. Here we discuss a possible $\mathbb{Z}_{24}$-valued pure gravitational anomaly of heterotic compactifications down to two spacetime dimensions, and point out that it can be shown to vanish using the theory of topological modular forms, assuming the validity of the Stolz–Teichner conjecture.

Summing over geometries in string theory

We examine the question how string theory achieves a sum over bulk geometries with fixed asymptotic boundary conditions. We discuss this problem with the help of the tensionless string on {mathrm{mathcal{M}}}_3times {mathrm{S}}^3times {mathbbm{T}}^4 (with one unit of NS-NS flux) that was recently understood to be dual to the symmetric orbifold SymN ( {mathbbm{T}}^4 ). We strengthen the analysis of [1] and show that the perturbative string partition function around a fixed bulk background already includes a sum over semi-classical geometries and large stringy corrections can be interpreted as various semi-classical geometries. We argue in particular that the string partition function on a Euclidean wormhole geometry factorizes completely into factors associated to the two boundaries of spacetime. Central to this is the remarkable property of the moduli space integral of string theory to localize on covering spaces of the conformal boundary of ℳ3. We also emphasize the fact that string perturbation theory computes the grand canonical partition function of the family of theories ⊕N SymN ( {mathbbm{T}}^4 ). The boundary partition function is naturally expressed as a sum over winding worldsheets, each of which we interpret as a ‘stringy geometry’. We argue that the semi-classical bulk geometry can be understood as a condensate of such stringy geometries. We also briefly discuss the effect of ensemble averaging over the Narain moduli space of {mathbbm{T}}^4 and of deforming away from the orbifold by the marginal deformation.

Loop amplitudes monodromy relations and color-kinematics duality

Color-kinematics duality is a remarkable conjectured property of gauge theory which, together with double copy, is at the heart of a wealth of new developments in scattering amplitudes. So far, its validity has been verified in most cases only empirically, with limited ab initio understanding beyond tree-level. In this paper we provide initial steps in a first-principle understanding of color-kinematics duality and double-copy at loop level, through a detailed analysis of the field-theory limit of the monodromy relations of string theory at one loop. In this limit, we dissect the type of Feynman graphs generated and the relations they obey. We find that graphs with contact-terms are unavoidable and are generated in the field theory limit of “bulk” contours which do not have a standard physical interpretation in string perturbation theory. We show how they are related to ambiguities in the definition of the loop momentum and that their role is precisely to cancel those ambiguities.

Single-Valued Integration and Superstring Amplitudes in Genus Zero

We study open and closed string amplitudes at tree-level in string perturbation theory using the methods of single-valued integration which were developed in the prequel to this paper (Brown and Dupont in Single-valued integration and double copy, 2020). Using dihedral coordinates on the moduli spaces of curves of genus zero with marked points, we define a canonical regularisation of both open and closed string perturbation amplitudes at tree level, and deduce that they admit a Laurent expansion in Mandelstam variables whose coefficients are multiple zeta values (resp. single-valued multiple zeta values). Furthermore, we prove the existence of a motivic Laurent expansion whose image under the period map is the open string expansion, and whose image under the single-valued period map is the closed string expansion. This proves the recent conjecture of Stieberger that closed string amplitudes are the single-valued projections of (motivic lifts of) open string amplitudes. Finally, applying a variant of the single-valued formalism for cohomology with coefficients yields the KLT formula expressing closed string amplitudes as quadratic expressions in open string amplitudes.

Dynamical Majorana neutrino masses and axions I

We discuss dynamical mass generation for fermions and (string-inspired) pseudoscalar fields (axion-like particles (ALP)), in the context of effective theories containing Yukawa type interactions between the fermions and ALPs. We discuss both Hermitian and non-Hermitian Yukawa interactions, which are motivated in the context of some scenarios for radiative (anomalous) Majorana sterile neutrino masses in some effective field theories. The latter contain shift-symmetry breaking Yukawa interactions between sterile neutrinos and ALPs. Our models serve as prototypes for discussing dynamical mass generation in string-inspired theories, where ALP potentials are absent in any order of string perturbation theory, and could only be generated by stringy non perturbative effects, such as instantons. The latter are also thought of as being responsible for generating our Yukawa interactions, which are thus characterised by very small couplings. We show that, for a Hermitian Yukawa interaction, there is no (pseudo)scalar dynamical mass generation, but there is fermion dynamical mass generation, provided one adds a bare (pseudo)scalar mass. The situation is opposite for an anti-Hermitian Yukawa model: there is (pseudo)scalar dynamical mass generation, but no fermion dynamical mass generation. In the presence of additional attractive four-fermion interactions, dynamical fermion mass generation can occur in these models, under appropriate conditions and range of their couplings.

Correlation functions of symmetric orbifold from AdS3 string theory

The paper examines correspondence among correlation functions of symmetric orbifold and string theory on AdS3 described by sl(2) Wess-Zumino-Novikov-Witten (WZNW) model. We start by writing down n-point function of twist operators in the symmetric orbifold in terms of the data of effective Riemann surface. It is then shown that the correlation function can be reproduced from the sl(2) WZNW model. The computation is based on the claim that string worldsheet is given by the same Riemann surface and the reduction method from sl(2) WZNW model to Liouville field theory. We first consider the genus zero surface and then generalize the analysis to the case of generic genus. The radius of AdS3 is related to the level k of the WZNW model. For k = 3, our result should be an important ingredient for deriving AdS3/CFT2 correspondence with tensionless superstrings to all orders in string perturbation theory. For generic k, relations involving specific forms of correlation functions for strings on AdS3× X were obtained.