3 posts to finish the TSOR adventure!

This blog post will introduce you to basic string theory from my own biased viewpoint. My blog, my rules. I think you concede that!

What is the Universe made of? This really ancient question (both philosophically and scientifically addressed differently from time to time) has not an ultimate answer. Today, we believe there are atoms (elements) that make up the chemistry of life we need. Atoms are not fundamental! Since the 19th century and through the 20th century we discovered lots of particles: electrons, protons, neutrons…An even worst, protons an neutrons are now believed not to be fundamental but made up from quarks. There are 6 quarks (6 flavors or types: up, down, charm, strange, top, bottom). There are 6 leptons (electron, muon, tau, electron neutrino, muon neutrino, tau neutrino). Moreover, there are gauge fields: photon, gluons, W bosons and Z bosons, plus the Higgs bosons found in 2012, 8 years ago. BUT, particles are not fundamental either! They are excitations of quantum fields. Quantum fields are fluid-like stuff permeating the whole Universe. Missing something? Of course: gravity. The Standard Model does NOT contain gravitational fields as gauge fields. Gravity is described not by a Yang-Mills theory but with General Relativity, a minimal theory treating spacetime as the field potential. The metric of spacetime is somehow the field of gravity (more precisely, the metric is the gravitational potential function). Using a torsion-less theory of gravity, you find out that the so called affine connection (the Christoffel symbols) are the equivalent of the classical gravitational field. . However, the nature of the gravitational field is not the affine connection, but the spacetime curvature. The tensor field defined by

(1)

describes the free source theory of relativistic gravity known as general relativity. Gravitons are not included in principle in the theory, gravitational waves are derived in the weak limit of perturbations of flat spacetime . Thus, gravitons are hypothetical transmitters of gravity for quantum gravity. Indeed, if you believe in quantum mechanics, gravitons are inevitable quanta behind the gravitational waves. In summary, we get bosons and fermions. We have reduced the ancient periodic table to a new set of fundamental ingredients. Not counting colors, or helicities or antiparticles, we have 6 quarks, 6 leptons, the photon, the gluon, the W, the Z and the H. Thats all, folks. 17 particles. 17 fields. Why to stop there? 17 are too many. What if some of them are again NOT fundamental and made of other stuff? Are they truly point-like?

Purely point-like particles provide some known issues. The first issue is the divergence of electric (or gravitational or alike) potential energy. Suppose the electron were a uniform sphere with radius (density ). The electric energy of such an sphere is

(2)

If purely point-like, you see there is an infinity electric energy! Giving up the factor arising from spherical symmetry, the classical electron radius is

(3)

Electrons are known to be fundamental much below this scale. In conclusion: a pure point-particle is meaningless at the end, because of the divergence in classical electrodynamics of the electromagnetic energy. But, as you know since you read me here, electromagnetism is not classical at the current state-of-art of the theory. Quantum electrodynamics (QED) is the theory we should use to answer the above questions. Well, even when it is a disaster as a single theory (it produces a so called Landau Pole at very very high energies), we know QED is an approximation to electroweak theory. However, a mystery is the final destiny of the divergence above even in the quantum regime!!!!!!! Yes, for all practical purposes, the QED is an effective theory that, in the end, neglect the infinities due to refined versions of the above ultraviolet (short scale, high energy) catastrophe. The fact that calculations work fine even neglecting those infinities is puzzling. An unsolved theoretical problem to understand why those infinities can be ignored without (excepting the vacuum energy problem) leaving us a completely nonsense mess of theory. The problem of quantum radiative self-corrections to energy-mass of particles is even worse with gravity. Using similar techniques to those used in the Standard Model do NOT work. Then, what is a particle or graviton? Two main alternative paths are possible:

- 1) Save gravity and pointlike particles, but change quantization. This path is the one donde in Loop Quantum Gravity (formerly non-perturbative canonical quantum gravity) or LQQ. Also, this method is followed up by the approach called Asymptotic Safety, and some minor approaches.
- 2) Save usual quantisation, but give-up a purely point-like nature of particles. That’s string theory, or now p-brane theory (even when a first quantized theory of p-branes is not yet available for ).

Adopting the second approach, if not a point, how do we describe strings or stuff? After all, the simplest object beyond structureless points are strings. We need some kinematical and dynamical basics for doing the math. We should describe strings in a natural invariant way, sticking to special relativity and quantum rules, even trying to extend that to general relativity. It is a surprising historical remark that strings do describe not only strong forces, but also have spin two excitations, aka gravitons! The miracle of string theory, beyond point particle theories, is that it allows us to describe a consistent theory of gauge interactions including the gravitational field. Plus a bonus: string theory (modulo some details concerning the critical dimension and the number of quantum fields) is free of UV divergences in perturbation theory. It is a finite theory of quantum gravity from the beginning. There is no free dimensionless parameters though. There is a link between the string coupling , the string tension and the string length , dependent of the type of string theory, the spacetime dimensions and the nature of the fundamental objects (not only string-like!) in the spectrum of the theory.

String theory fundamental object is a single tiny string, generally speaking in old string theory, but massive states can mismatch that. For instance, if , so you could get objects and string greater than Planck length in non-perturbative fashion with . Or, you can have things below the Planck scale if soft enough. Particles or field are really excitation modes of fundamental strings (in critical string theory). Different modes correspond to different fields or particles.

Classical points are given a a line of world in spacetime, i.e., . Strings should be described by a surface area in spacetime, i.e., . Here, and strings can be open or closed strings. Open strings have and . Closed strings have periodicity in the space-like worldsheet, such as . You see that depends on the target spacetime time, as any string in -spacetime, minkovskian spacetime is the usual selection, consists really of a set of (generally scalar) fields. The next question is: what is the equation of motion of a FREE string? Well, without giving more advanced details, and noting the similarity between strings and waves, recall that free point particles have the equation of motion (in newtonian mechanics, but also single special relativistic case):

(4)

or

(5)

Then, it would be natural for free strings to have the following equation of motion

(6)

You can see that -dimensional free strings are only a set of -dimensional 2d wave equations. Strings carry energy and momentum, but also spin degrees of freeedom. Generally speaking, we have that the string tension and the string length are related via (some people usually prefers as normalized string length . The string also owns a typical energy scale:

(7)

Experimentally, circa 2020, we know that the string mass scale is in a range

You could try to generalize the above for classical p-branes (p is the number of space-like dimensions) in spacetime, and classical (p,q)-branes (p is the number of space-like dimensions, q is the number of time-like dimensions) in spacetime as follows:

(8)

and where

(9)

Thus, general p-branes /(p,q)-branes or extended objects are described by hyperbolic/ultrahyperbolic wave equations in -dimensional target space-time. Only the first quantized version of strings is known at current time.

General solution for 2D dimensional wave equations are available:

(10)

This describes a traveler string wave as the sum of a stringy right-wave/oscillation plus a stringy left-wave/oscillation. Assuming periodic boundary conditions, the most general solution is a Fourier expansion

(11)

Here, are the center of mass and center of momentum of the string. The first quantization of strings is simple and well understood (unlike general p-branes!). Quantization of every wave oscillation in a single string is a mode. Every mode is quantum harmonic oscillator. A string is secretly a field or infinite number of harmonic oscillators.

Every excitation mode , represents a harmonic oscillator. States in vacuum are labeled by center of mass momentum . Excitations of L/R type gives a frequency . And finally, the quantum string state is a ket:

(12)

Remark: consider equal number of left/right moving stringy quanta. Then, there is a Tower of String EXcitations characterized by oscillation number . The first levels of this quantization provides:

(13)

For bosonic strings, this represents a spectrum spin-dependent

(14)

with , . The tachyon mode is erased with supersymmetry (SUSY) in superstring theory (in 10d spacetime!). sets the string scale, and represents massless tensorial states (gravitons!). In the higher energy regime, we expect resonances depending of mass and spin, roughly , with . Just a further comment: closed strings give sense with the above argument to graviton-like excitations, where do photon-like degrees of freedom come from? From open strings! Copy-cat the same program of classical solutions and quantisation with the right suitable boundary conditions. The result is that free endpoints can move freely along an object called a -brane, or (p+1)-dimensional hypersurface of spacetime (cf. Polchinski 1996). With boundary conditions assumed, the massless gauge U(1)-like fields will be

(15)

String excitations along 1 Dp-brane: gauge fields , with N-coincident Dp-branes is promoted to gauge symmetry gauge bosons! Moreover, Dp-branes at intersection provides matter fields (chiral fermions) in bifundamental reprentations . Thus, we have a stringy/Dp-brane machine to generate gauge field theory and derive the Standard Model. The problem is: there are too many ways to do it! Even if with string theory, gauge theory implies gravity, or gravity and gauge theory are included in the same set-up, we do not know how to generate uniquely the Standard Model, and the vacuum we call our Universe… However:

- Strings interact by joining and splitting. Open string endpoints can join to form a stable closed string. (The converse is not always true).
- Behaviour consistent with universality of gravity: photons provide gravity, and somehow, gravity is the square of a gauge theory. That is the motto gravity= popular theses days.
- In string theory, gauge interactions and gravity are not independent. They are linked by the internal consistency of the theory. String theory is the only known theory with this property. Even more, consistency implies critical dimensions: 26d the original bosonic string theory, 10d the superstring, and 11d M-theory (12d F-theory, 13d S-theory,\ldots).
- UV finiteness and the end of divergences. The general picture is that string theory has an intrinsic UV regulator (the string length). High energy scattering probes that lenght and non-local behaviour is obtained. Point-like interaction vertices are smothened/erased. Quantitatively precise, loop diagrams in perturbative string theory can be checked to be FINITE. No more UV divergences.
- Strings are special? Can a particle have even higher-dimensional substructure? Model particle as a membrane (Dirac pioneered this with the electron-membrane model): 2 spatial dimensions .Tubes of length and radius have spatial . Quantum fluctuations of p-branes: Long, thin tubes can form without energy cost and that is an issue. Membranes automatically describe multi-particle states. No first quantisation of higher-branes à la strings possible. Quantum membranes have continuous spectrum no one know how to discretize like strings.

However, string theory puzzles further. It implies:

- Internal consistency conditions make further predictions: spacetime is not 4-dimensional, but 10-dimensional (26d in bosonic string theory with no supersymmetry).
- In 10 dimensions there is only one unique type of string theory. It has many equivalent formulations which are dual to each other.
- Witten 1995 showed that there is a single theory, dubbed M-theory, with six duality-related limites: 11d SUGRA, Heterotic SO(32), Heterotic , Type IIA, Type IIB, and Type I. These 6 theories are related with a web of T-dualities and S-dualities, and a general U-duality group much more general.
- The 10-dim. theory/11d maximal SUGRA/11d M-theory is supersymmetric, and every boson has a fermionic superpartner. This does NOT imply that supersymmetry must be found at LHC. SUSY energy scale can be in any point between tested energies and Planck energy.
- Superstring theory is well-defined and unique (up to dualities) in 10d/11d and lower/higher dimensions(higher dimensions are usually neglected due to higher spins or extra time-like dimensions).
- The low energy regime os superstring predicts indeed Einstein general relativity plus stringy corrections with several gauge fields.
- Within the full 10d bulk a graviton propagates, and along lower dimensional D-branes a gauge boson propagates.
- Within the high energy regime , characteristic tower of massive string excitations provide measurable (in principle) as resonances (Kaluza-Klein states)! Energy dependence of interactions differs from field theory.
- The scattering amplitudes are ultra-violet finite without the need for renormalisation. It is believed that string theory interactions represent the fundamental (as opposed to effective) theory, but heavy -brane states also arised in the second string revolution.

3. Issues with contemporary string theory

Our world/Universe is apparently (3+1, minkovskian metric). We need to compactify extra unobserved string theory dimensions, with or without brane world metrics, to derive our Universe (SM plus gravity in the form of General Relativity). The problem is, as I told you before, there is no a unique way to do it. Thus, the model building to get the Universe as a single solution is doomed in current string theory! The set of every possible string theory compatification providing a Universe like ours has a name: the string landscape.

Superstring theory is well-defined only if spacetime is 10 d/11 d as M-theory. It is thus an example of a theory of extra dimensions. You can build up string theories having point particles 0-branes, strings 1-branes, 2-branes (M-theory, 11d SUGRA), and so on. Extra dimensions are compact and very small. For instance, pick a 5d world with coordinates . The extra invisible dimension is folded/wrapped a tiny circle with radius . If this radius becomes very tiny, the world will appear to be 4d, but, with enough energy, you could reach that dimension. To arrive at 4 large extra dimensions we need to compactify 6 dimensions (or 7 in M-theory). The simplest solution (of course, not the only solution), every dimension is a circle , i.e. internal space is a six-dimensional torus :

(16)

More general 6-dimensional/7-dimensional spaces allowed (Calabi-Yau manifolds were popular in the past). Every consistent compactification yields a solution to string equation of motions with specific physics in 4D. This gives you a landscape of theories. Configuration of multiple branes are related to gauge groups. The intersection pattern is related to charged matter and specifics of geometry is related to interactions (computable!).The field of model building or String phenomenology is to explore interplay of string geometry and physics in 4 dimensions.

The landscape of string vacua biggest puzzle: every consistent compactification is a solution to string equations of motion. Every 4d solution is called a 4d string vacuum. In 10d: All interactions uniquely determined. In 4d: Plethora of consistent solutions exists – the landscape of string vacua Existence of many solutions is typical in physics: Einstein gravity is one theory with many solutions! Pressing question: Consequences for physics in 4D physics? Solution to fine-tuning problems (Higgs, Cosmological Constant)? Harder in the Landscape!

Even worse is the more recent ideas of Swampland versus Landscape: Which EFT(effective field theories) can be coupled to a fundamental theory of QG? There is also a Swampland of inconsistent EFTs related to the Landscape of consistent quantum gravitational theories. Swampland conjectures of general scope, but not sharply proven. The Weak Gravity Conjecture is also analyzed from the viewpoint of the string theory landscape these times.

Is string theory as a framework for QG allows to test explicit conjectures? More ideas (for quantitative check of swampland conjectures and sharper formulation):

- Study manifestations of swampland conjectures in string geometry.
- String Geometry Geometry of compactification space involves Physics in 4d (or higher). Holographic principle is tested as well.
- Strings as extended objects probe geometry differently than points. it opens door for fascinating interplay between mathematics and physics: new physics ways to think about geometry by translating into physics. For instance: classification of singularities in geometry, singularities occur when submanifolds shrink to zero size and branes can wrap these vanishing cycles and give rise to massless particles in effective theory.
- String theories and the Landscape/Swampland give interpretation for classification of singularities in mathematics and guidelines for new situations unknown to mathematicians.
- String theory is a maximally economic quantum theory of gravity, gauge interactions and matter.
- Assumption of stringlike nature of particles leads to calculable theory without UV divergences.
- Challenge for String Phenomenology: understanding the vacuum of this theory
- String Theory as modern mathematical physics: deep interplay with sophisticated mathematics(e.g.: Mirror symmetry, D-brane categories,. . . ).
- String Theory as a tool: Holographic principle like the AdS/CFT, or Kerr dS/dS correspondences. String Theory is a framework for modern physics.

String theories have extra field-theoretic degrees of freedom. Consider firstly the next four dimensional action

(17)

where is a four-form with solutions to the EOM(equations of motion)

(18)

It is easily probed that it gives a contribution to the cosmological constant/vacuum energy

(19)

This gives rise to the moduli (flux compatifications) problem in string theory. In string theory, is quantized, but you are provided many of such four-form (and even other grade forms!) contributions:

If you have MANY and is arbitrary, can be tuned to a very somall value under VERY special conditions, but not all clear! You can try to get how many values you need of these vacua, to get , and see how many string theory solutions give you the SM plus Gravity Universe we live in! Terrible result: usual string theory gives you possible Universes in 10d/11d, or even worst, using F-theory technology, you guess an upper bound about (Vafa)…

4.1. The string spectrum and M-atrix models

As strings need extra dimensions, they also have different quantum numbers in addition to common particle quantum numbers. The dimensional compactification provides the level of Kaluza-Klein resonance. BUT, the winding number around the extra dimension/s is also a purely stringy quantum number. For KK-modes:

(20)

and for a the winding w-modes

(21)

so finally, including the R/L excitation modes and the continous part of the quantized string, we get with units:

(22)

Note the symmetry under and , known as -duality. In non-pertubative settings, we also get a symmetry between the strong and weak coupling (S-duality). It anticipated the Dp-brane revolution with monopoles in a famous Montonen-Olive conjecture.

Even when a general formulation of what M-theory is, one proposal was made called M-atrix theory. This M-atrix theory reveals that M-theory is a emergent model from the dynamics of a matrix model of D0-branes. Pick up a very large set of matrices . These matrices (one for each space dimension, ) represent the position of N-pointlike D0-branes. The energy is a hamiltonian object, formally

(23)

At low energies, these matrices all commute, their eigenvalues behave like normal spatial coordinates. Thus, ordinary spacetime is emergent from the M-atrix. But, in the regime where quantum fluctuations become large or strong, the full M-atrix structure, non-commutative and (sometimes non-associative in strings!) highly non-linear must be considered. M-theory is a highly non-local theory seen in this way. If it can be simulated with quantum computing is something to be tested in the future! Maybe, even M-theory tools will be more powerful than usual quantum computational tools.

5. Epilogue: a Multiverse of Madness nightmare?

Even when string theory or SUSY are very powerful, the duality revolution has touched two angular pieces they have left unanswered:

- The selection of our vacuum or Universe. There are too many possible solutions, and that leaves us with the option of the Multiverse or that our vacuum could be not stable but metastable.
- What is the fundamental theory/degrees of freeedom of superstring/M-theory. Duality maps change and challenge what is the fundamental entity in a dual theory. Holographic maps included, you can have a field theory with no gravity and change into a higher-dimensional gravitational theory and vice versa. You can calculate with magnetic branes instead electric branes. What is string theory? After all, we have no first quantized theory of membranes like those in M-theory yet.

Parallel to all this, Nima-Arkami Hamed discovered the amplituhedron: a new tool to simplify Feynman diagram computations based on higher-dimensional entities of polytopal class. They are intrinsically non-local. He has envisioned a future in which locality, SR, GR, and QFT are derived from a new set of structures. Long ago, when the 26d four-point function for the scattering of four tachyons is the Shapiro-Virasoro amplitude was derived to be:

people wondered what was behind that. It showed to be the string…What is the p-brane generalization of this amplitude, if it exists?

In the end, the problem with String Theory is that we are not sure of what the symmetry of the whole theory is. We lack an invariant notion/relativity+equivalence principle for string/p-branes! There are only a few ideas circulating about what these new relativity principle/new equivalence principle are! But that is the subject of my final TSOR blog post, after a twistor interlude!

See you in my next blog post!

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