The birth of string theory

BCFW construction of the Veneziano Amplitude

In this note we demonstrate how one can compute the Veneziano amplitude for bosonic string theory using the BCFW method. We use an educated ansatz for the cubic amplitude of two tachyons and an arbitrary level string state.

Strings from Massive Higher Spins: The Asymptotic Uniqueness of the Veneziano Amplitude

We consider weakly-coupled theories of massive higher-spin particles. This class of models includes, for instance, tree-level String Theory and Large-N Yang-Mills theory. The S-matrix in such theories is a meromorphic function obeying unitarity and crossing symmetry. We discuss the (unphysical) regime $s,t \gg 1$, in which we expect the amplitude to be universal and exponentially large. We develop methods to study this regime and show that the amplitude necessarily coincides with the Veneziano amplitude there. In particular, this implies that the leading Regge trajectory, $j(t)$, is asymptotic…

New strings for old Veneziano amplitudes II. Group-theoretic treatment

In this part of our four parts work (e.g see Part I, hep-th/0410242) we use the theory of polynomial invariants of finite pseudo-reflection groups in order to reconstruct both the Veneziano and Veneziano-like (tachyon-free) amplitudes and the generating function reproducing these amplitudes. We demonstrate that such generating function can be recovered with help of the finite dimensional exactly solvable N=2 supersymmetric quantum mechanical model known earlier from works by Witten, Stone and others. Using the Lefschetz isomorphisms theorem we replace traditional supersymmetric calculations by…

Veneziano Variations: How Unique are String Amplitudes?

String theory offers an elegant and concrete realization of how to consistently couple states of arbitrarily high spin. But how unique is this construction? In this paper we derive a novel, multi-parameter family of four-point scattering amplitudes exhibiting i) polynomially bounded high-energy behavior and ii) exchange of an infinite tower of high-spin modes, albeit with a finite number of states at each resonance. These amplitudes take an infinite-product form and, depending on parameters, exhibit mass spectra that are either unbounded or bounded, thus corresponding to generalizations of the…

广义 Veneziano 振幅的世界面

Worldsheet for Generalized Veneziano Amplitudes

该研究针对广义 Veneziano 振幅提出基于手征复合线性伸缩子的世界面作用量，计算了更高点扩展和闭弦类似物，展现出部分交叉对称性。

New Strings for Old Veneziano Amplitudes I.Analytical Treatment

The bosonic string theory evolved as an attempt to find physical/quantum mechanical model capable of reproducing Euler's beta function (Veneziano amplitude) and its multidimensional analogue. The multidimensional analogue of beta function was studied mathematically for some time from different angles by mathematicians such as Selberg, Weil and Deligne among many others. The results of their studies apparently were not taken into account in physics literature on string theory. In recent publication hep-th/0212189 [IJMPA 19 (2004) 1655] an attempt was made to restore the missing links. The resul…

Veneziano振幅的复鞍点与时间延迟

Complex saddles and time-delay of Veneziano amplitude

针对弦散射振幅形式积分在物理运动学区域发散的问题，研究采用新积分围道计算Veneziano振幅的时间延迟，经thimble分析发现无穷多复鞍点参与积分，各对应不同洛伦兹时间进入散射区的弦路径。

New strings for old Veneziano amplitudes IV.Connections with spin chains and other stochastic systems

In a series of published papers we reanalyzed treatments of the Veneziano amplitudes and the models associated with them. In this work we demonstrate that the already obtained partition function for these amplitudes can be exactly mapped into that for the Polychronakos-Frahm spin chain which, in turn, is obtainable from the Richardson-Gaudin XXX model. Reshetikhin and Varchenko demonstrated that such a model is recoverable from their WKB-type analysis of solutions of the Knizhnik-Zamolodchikov (K-Z) equations. The linear independence of solutions of these equations is controlled by determinant…

New Strings for Old Veneziano Amplitudes III. Symplectic Treatment

A d-dimensional rational polytope P is a polytope whose vertices are located at the nodes of d-dimensional Z-lattice. Consider a number of points inside the inflated polytope (with coefficient of inflation k, k=1,2, 3...). The Ehrhart polynomial of P counts the number of such lattice points (nodes) inside the inflated P and (may be) at its faces (including vertices). In Part I (hep-th/0410242) of our four parts work we noticed that the Veneziano amplitude is just the Laplace transform of the generating function (considered as a partition function in the sence of statistical mechanics) for the …

AdS Veneziano振幅的单值性

Single-valuedness of the AdS Veneziano amplitude

本文研究了IIB型弦理论中胶子散射的AdS Veneziano振幅，提出曲率修正源于软引力子插入并表现为实轴上的单值函数，据此验证了第一曲率修正并计算了第二曲率修正。

The Veneziano Limit of N=2 Superconformal QCD: Towards the String Dual of N=2 SU(N_c) SYM with N_f =2 N_c

We attack the long-standing problem of finding the AdS dual of N = 2 superconformal QCD, the N=2 super Yang Mills theory with gauge group SU(N_c) and N_f = 2 N_c fundamental hyper multiplets. The theory admits a Veneziano expansion of large N_c and large N_f, with N_f/N_c and lambda = g^2 N_c kept fixed. The topological structure of large N diagrams motivates a general conjecture: the flavor-singlet sector of a gauge theory in the Veneziano limit is dual to a closed string theory; single closed string states correspond to "generalized single-trace" operators, where adjoint letters and flavor-c…

Geometric transitions, Chern-Simons gauge theory and Veneziano type amplitudes

We consider the geometric transition and compute the all-genus topological string amplitudes expressed in terms of Hopf link invariants and topological vertices of Chern-Simons gauge theory. We introduce an operator technique of 2-dimensional CFT which greatly simplifies the computations. We in particular show that in the case of local Calabi-Yau manifolds described by toric geometry basic amplitudes are written as vacuum expectation values of a product vertex operators and thus appear quite similar to the Veneziano amplitudes of the old dual resonance models. Topological string amplitudes can…

The Conformal Equivalence of the half string 3-vertex and the Sciuto,Caneschi-schwimmer-veneziano 3-vertex

In this paper we construct the operator connecting the three vertex in half string field theory and the dual model vertex of Sciuto, Caneshi Schwimmer and Veneziano (SCSV). This construction results in an explicit conformal transformation linking the two interactions in the matter sector at all levels. Thus establishing the equivalence in the matter sector between the two theories at least for N = 3.

The off-shell Veneziano amplitude in Schnabl gauge

We give a careful definition of the open string propagator in Schnabl gauge and present its worldsheet interpretation. The propagator requires two Schwinger parameters and contains the BRST operator. It builds surfaces by gluing strips of variable width to the left and to the right of off-shell states with contracted or expanded local frames. We evaluate explicitly the four-point amplitude of off-shell tachyons. The computation involves a subtle boundary term, crucial to enforce the correct exchange symmetries. Interestingly, the familiar on-shell physics emerges even though string diagrams pr…

From Veneziano to Riemann: A String Theory Statement of the Riemann Hypothesis

We discuss a precise relation between the Veneziano amplitude of string theory, rewritten in terms of ratios of the Riemann zeta function, and two elementary criteria for the Riemann hypothesis formulated in terms of integrals of the logarithm and the argument of the zeta function. We also discuss how the integral criterion based on the argument of the Riemann zeta function relates to the Li criterion for the Riemann hypothesis. We provide a new generalization of this integral criterion. Finally, we comment on the physical interpretation of our recasting of the Riemann hypothesis in terms of t…

Accumulation-Point Amplitudes in String Theory

We point out some common qualitative features of the Coon amplitude$\unicode{x2014}$a family of deformations of the Veneziano amplitude with logarithmic Regge trajectories$\unicode{x2014}$and the open string scattering amplitude for strings ending on a D-brane in AdS. Both reduce to the Veneziano amplitude at relatively low energies. Both systems have an accumulation point in their spectrum, with an infinite number of states below a certain energy. The approach to this point is very similar. Both have the same high-energy fixed-angle behavior. Nevertheless, we find some differences in the spec…

AdS上弦振幅的单值关系

Monodromy Relations for String Amplitudes on AdS

本文针对IIB型弦论$AdS_5 \ imes S^3$背景下的四胶子散射$AdS$ Veneziano振幅，提出一组色序振幅间的单值关系，其源于开弦散射的涌现世界面描述，平坦空间下退化为标准单值关系，小曲率展开逐阶成立，且符合现有文献结果。

任意维度下的 Veneziano 振幅与一个 Virasoro-Shapiro 分振幅

The Veneziano Amplitude in any Dimension and a Virasoro-Shapiro Partial Amplitude

本文将 Veneziano 分振幅推广至任意维度世界面，并在 d=2 时由共形交比分解 Virasoro-Shapiro 振幅，同时提出残差自旋交替的自举原理以寻找无快子超对称形式。

Large-N QCD and the Veneziano Amplitude

We consider four scalar mesons scattering in large-Nc QCD. Using the worldline formalism we show that the scattering amplitude can be written as a formal sum over Wilson loops. The AdS/CFT correspondence maps this sum into a sum over string worldsheets in a confining background. We then argue that for well separated mesons the sum is dominated by flat space configurations. Under additional assumptions about the dual string path integral we obtain the Veneziano amplitude.

弦论从无限宽度神经网络中实现

String Theory from Infinite Width Neural Networks

本研究通过无限宽度神经网络实现了弦论，并新计算了基础性幅作为神经网络相关函数。

"New" Veneziano amplitudes from "old" Fermat (hyper) surfaces

The history of discovery of bosonic string theory is well documented. This theory evolved as an attempt to find a multidimensional analogue of Euler's beta function. Such an analogue had in fact been known in mathematics literature at least in 1922 and was studied subsequently by mathematicians such as Selberg, Weil and Deligne among others. The mathematical interpretation of this multidimensional beta function is markedly different from that described in physics literature. This paper aims to bridge the gap between the existing treatments. Preserving all results of conformal field theories in…

Reminiscence on the Birth of String Theory

These are my personal impressions of the environment in which string theory was born, and what the important developments affecting my work were during the hadronic string era, 1968-1974. I discuss my motivations and concerns at the time, particularly in my work on loop amplitudes and on closed strings.

Threshold Effects Beyond the Standard Model

In this contribution to the Festschrift celebrating Gabriele Veneziano on his 65th birthday, I discuss the threshold effects of extra dimensions and their applications to physics beyond the standard model, focusing on superstring theory.

Birth of String Theory

This is a brief summary of an introductory lecture for students and scholars in general given by the author at Nambu Memorial Symposium which was held at Osaka City University on September 29, 2015. We review the invention of string theory by Professor Yoichiro Nambu following the discovery of the Veneziano amplitude. We also discuss Professor Nambu's proposal on string theory in the Schild gauge in 1976 which is related to the matrix model of Yang-Mills type.

弦谱与散射的引导原理

Bootstrap Principle for the Spectrum and Scattering of Strings

本文证明弦理论的 Veneziano 振幅是特定引导问题的唯一解，由高能衰减和自旋抵消假设导出，并推广至三参数族。

Personnal recollections about the birth of string theory

Invited contribution to the collective book "The Birth of String Theory"

Non-Perturbative Yang-Mills from Supersymmetry and Strings, Or, in the Jungles of Strong Coupling

I summarize some recent developments in the issue of planar equivalence between supersymmetric Yang-Mills theory and its orbifold/orientifold daughters. This talk is based on works carried out in collaboration with Adi Armoni, Sasha Gorsky and Gabriele Veneziano.

Birth of Closed Strings and Death of Open Strings during Tachyon Condensation

The tremendous progress achieved through the study of black holes and branes suggests that their time dependent generalizations called Spacelike branes (S-branes) may prove similarly useful. An example of an established approach to S-branes is to include a string boundary interaction and we first summarize evidence for the death of open string degrees of freedom for the homogeneous rolling tachyon on a decaying brane. Then, we review how to extract the flat S-brane worldvolumes describing the homogeneous rolling tachyon and how large deformations correspond to creation of lower dimensional str…

On the unitarity and low energy expansion of the Coon amplitude

The Coon amplitude is a deformation of the Veneziano amplitude with logarithmic Regge trajectories and an accumulation point in the spectrum, which interpolates between string theory and field theory. With string theory, it is the only other solution to duality constraints explicitly known and it constitutes an important data point in the modern S-matrix bootstrap. Yet, its basics properties are essentially unknown. In this paper we fill this gap and derive the conditions of positivity and the low energy expansion of the amplitude. On the positivity side, we discover that the amplitude switche…

新生弦是无张力的

New-born strings are tensionless

本文首次提出无张力弦的物理起源，通过在有限寿命设定下构建弦动力学，揭示无张力弦仅在诞生时刻出现，其全球超局部卡罗尔结构特征被发现。

Birth of the Universe as anti-tunnelling from the string perturbative vacuum

The decay of the string perturbative vacuum, if triggered by a suitable, duality-breaking dilaton potential, can efficiently proceed via the parametric amplification of the Wheeler-De Witt wave function in superspace, and can appropriately describe the birth of our Universe as a quantum process of pair production from the vacuum.

From Quarks to Strings

In this article, prepared for the book "The birth of string theory", I recall the sequence of ideas which led to non-critical strings and gauge/strings duality. I also comment on some promising future directions.

Birth of the Universe as quantum scattering in string cosmology

In a Wheeler-de Witt approach to quantum string cosmology, the present state of the Universe arises from the scattering and reflection of the wave function representing the initial string vacuum in superspace. This scenario is described and compared with the more conventional quantum cosmology picture, in which the birth of the Universe is represented as a process of tunnelling "from nothing" in superspace.

Evolution of FLRW spacetime after the birth of a cosmic string

We consider the evolution of an initially FLRW universe after the formation of a long, straight, cosmic string with arbitrary tension and mass per unit length. The birth of the string sources scalar and tensor-type perturbations in the background metric and both density and velocity perturbations in the background fluid, which compensate for the string mass and maintain energy conservation. The former generate the deficit angle within the light cone of the string and a gravitational shock front at the cosmological horizon, whereas the latter are confined within the sound cone. We study the pro…

The beginning of string theory: a historical sketch

In this note we follow the historical development of the ideas that led to the formulation of String Theory. We start from the inspired guess of Veneziano and its extension to the scattering of $N$ scalar particles, then we describe how the study of its factorization properties allowed to identify the physical spectrum making the string worldsheet manifest and finally we discuss how the critical values of the intercept of the Regge trajectory and of the critical dimension were fixed to 1 and 26.

Memoirs of an Early String Theorist

I worked on String Theory over a period of five years during the First String Era, the most intellectually satisfying years of my scientific life. One of the early prospectors in the String Theory Mine, I was fortunate enough to contribute to the birth of this subject, which retains after these many years, its magical hold on our imaginations and expectations.

Birth of the Universe in string cosmology

The decay of the string perturbative vacuum into our present cosmological state is associated to the transition from a phase of growing curvature and growing dilaton, to a phase of decreasing curvature and frozen dilaton. The possible approaches to a classical and quantum description of such a transition are introduced and briefly discussed.

Polynomial Gauge Invariants of a Bosonic String

An open bosonic string is considered with the aim to construct a general gauge invariant, being a polynomial of Fubini-Veneziano (FV) fields. The FV fields are transformed as 1-forms on $S^1$, that allows to formulate the problem in geometric terms. We introduce a most general anzats for these invariants and explicitly resolve the invariance conditions in the framework of the anzats. The invariants are interpreted as integrals of n-form over a gauge invariant domains in an n-dimensional torus, where the invariance of these domains is considered with respect to the action of the diagonal of the…

Birth of the Universe from the Landscape of String Theory

We show that a unique, most probable and stable solution for the wavefunction of the universe, with a very small cosmological constant $Λ_1 \simeq (\fracπ{l_p N})^2$, can be predicted from the supersymmetric minisuperspace with $N$ vacua, of the landscape of string theory without reffering to the antropic principle. Due to the nearest neighbor tunneling in moduli space lattice, the $N$-fold degeneracy of vacua is lifted and a discrete spectrum of bound state levels over the whole minisuperspace emerges. $SUSY$ is spontaneously broken by these bound states, with discrete nonzero energy levels $…

O(d,d) duality transformations in F(R) theories of gravity

The argument of Hodge duality symmetry is introduced starting from the electromagnetic field. Introducing bosonic string theory, O(d,d) duality symmetry can be implemented when there exist d-symmetries, which allows one to write Hodge-dual fields. A tree-level effective gravitational action of bosonic string theory coupled with the dilaton field is considered. This theory inherits the Busher's duality of its parent string theory. The dilaton field can be recast into the Weyl's mode of the metric tensor in the Jordan frame. This maps the effective one-loop bosonic string theory of gravity into …

Aspects of dual models many years ago

Invited contribution to the collection of articles: `The Birth of String Theory', edited by Andrea Cappelli, Elena Castellani, Filippo Colomo and Paolo Di Vecchia.

论暴胀前非高斯性

On Pre-Inflationary non Gaussianities

本文研究弦理论高尺度超对称破缺场景中暴胀前曲率扰动三点振幅，发现与攀爬阶段转折点相关的第二类贡献在e-fold数62-66内可能可观测。