Well, all previous theories inward physics receive got used the classical manifold geometry (whose Definition volition live sketched momentarily) as i of the basic prerequisites that the theories had to receive got in addition to elaborate upon. This made the classical manifold geometry in addition to its calculations straight relevant for all these theories in addition to the rules of the geometry were so stiff dogmas.
In other words, the theories followed the template:
Dear theory, listen, hither you lot receive got a classical manifold alongside some shape.And the theories just couldn't practise anything else. They were dependent on the geometry of a classical manifold. If in that location were no manifold, in that location was no physical theory. And if 2 manifolds were geometrically different, the physical theories on them had to live distinguishable, too.
What tin you lot gain alongside this pre-existing shape?
Before the regain of special relativity, physics was also dividing spacetime to the absolute fourth dimension in addition to the infinite that exists alongside it. That meant that the "spacetime" as nosotros understood it today had to live basically factorized to \(\RR \times M^3\) where \(\RR\) was the existent axis representing fourth dimension in addition to \(M^3\) was a purely spatial manifold (OK, some time-dependent fibration alongside a dissimilar \(M^3(t)\) at each 2nd fourth dimension was sometimes allowed, too). At most, you lot could receive got picked time-dependent coordinates on that \(M^3\) inward guild to celebrate the Galilean relativity.
But Newton's laws of motion said\[
m\frac{d^2 x^i}{dt^2} = F^i (x^j, v^j).
\] The 2nd derivative of the seat was equal to the forcefulness – which was expressed as a role of the positions in addition to velocities of the dot masses. You may imagine how the metric tensor could come inward if you lot generalized the equations to the motion on a curved 3-dimensional manifold.
At whatsoever rate, in that location was no way to avoid the appearance of the coordinates \(x^i\) that parameterized the classical manifold – the necessary arena for the laws of physics to live formulated at all.
When mechanics was largely superseded past times plain theory – imagine Maxwell's equations – in addition to then the coordinates \(x^i\) in addition to \(t\) to a greater extent than or less naturally merged to the spacetime coordinates \(x^\mu\) in addition to the most interesting resulting theories flora it tardily to live Lorentz-covariant. And these coordinates nevertheless had to appear inward the equations encoding the laws of Nature. Why? The fields such as \(\vec B(x,y,z,t)\) had to live functions of the spacetime coordinates – variables that seat points on a classical manifold.
To brand the story short, classical plain theory was developed to include some really interesting cases – alongside Yang-Mills symmetries in addition to fifty-fifty the diffeomorphism symmetries of full general relativity. The latter allowed the background spacetime manifold to live curved in addition to almost inevitably postulated that all smooth+nice plenty coordinate systems are as good. Finally, quantum plain theory (possibly on curved spaces) added hats on pinnacle of the fields. The fields became operators (or operator distributions). But they nevertheless depended on the spacetime coordinates.
If you lot desire a simple example, a Klein-Gordon equation on a curved spacetime background says\[
g^{\mu\nu} \nabla_\mu \nabla_\nu \Phi(x^\lambda) = 0
\] or something else on the right mitt side. Equations similar that utilisation a spacetime manifold alongside the coordinates \(x^\lambda\) in addition to the metric tensor \(g_{\mu\nu}(x^\lambda)\) on them as edifice blocks that you lot but need.
I should lastly tell you lot what I hateful past times the geometry of classical manifolds that is going to live generalized past times string theory. Well, it's a collection of patches such that each of them is diffeomorphic to an opened upward laid inward \(\RR^n\). The patches may overlap in addition to weather guarantee that inside the intersections, you lot larn the equivalent results if you lot utilisation i patch or another. From the viewpoint of topology as a branch of abstract mathematics, all manifolds may also live understood as "sets of points" alongside some topology that tells you lot which subsets of the manifolds are "open sets". Because of the phrase "sets of points", all the previous theories implicitly assumed that "points are allowed to alive on that manifold", too.
Does string theory require the classical background geometry as well?
Yes in addition to no. Influenza A virus subtype H5N1 item simple subset of descriptions of string theory, or some situations inward it, depend on the pre-existing spacetime arena inward the same way as mechanics or plain theory. In other words, this way of doing string theory nevertheless obeys the template
Dear strings, hither you lot receive got a background spacetime geometry.You saw that inward the quote above, Mother Nature was straight talking to strings alongside the pretense of superiority (indicating that She didn't consider strings to live to a greater extent than than just the piddling light-green men or dot masses) in addition to as if they were completely well-defined in addition to the exclusively ones that mattered. That agency that we're exclusively talking close perturbative string theory where all quantities are computed as might serial inward the string coupling constant, \(g_s\). In perturbative string theory defined similar that in addition to exclusively inward it, strings are well-defined in addition to stand upward for existing in addition to the exclusively degrees of freedom. That way of thinking close string theory is exclusively safe for \(g_s\ll 1\).
Show us, beloved strings, how tin you lot trip the calorie-free fantastic in addition to split upward in addition to bring together on that geometry.
OK, how practise strings obey the guild of Mother Nature? They may obey it if the theory is defined as the so-called non-linear sigma model, kickoff written past times Gell-Mann in addition to Lévy inward 1960, total 8 years earlier the nativity of string theory. The model is named "sigma" because the plain was labeled \(\Sigma\) past times the authors. And it's nonlinear because the equations of motion aren't just linear inward \(\Sigma\) – equivalently, because the Lagrangian\[
\mathcal{L}={1\over 2}g_{ab}(\Sigma) (\partial^\mu \Sigma^a) (\partial_\mu \Sigma^b) - V(\Sigma).
\] which has to live integrated over the 2D basis sheet, \(S=\int d^2\sigma {\mathcal L}\) to larn the action, isn't purely bilinear or quadratic inward these fields \(\Sigma\). Note that the bilinear Klein-Gordon Lagrangian is multiplied past times additional functions of \(\Sigma^a\) – imagine that the latter are Taylor-expanded inward \(\Sigma^a\). What does this Lagrangian describe? It describes a plain theory living on an auxiliary infinite parameterized past times the coordinates \(\sigma^\mu\) – banking company annotation that I had to pick some alphabetic lineament for the coordinates in addition to \(x^\mu\) is "morally taken" because it looks similar a goodness synonym for \(\Sigma^a\) (you should also persistently detect the evolving conventions for the indices: at this point, I am using \(\mu,\nu\) indices for the basis canvas coordinates in addition to \(a,b\) for the spacetime ones) – in addition to at each dot of the so-called basis canvas or basis volume indicated past times these coordinates \(\sigma^\mu\), in that location are several fields \(\Sigma^a(\sigma^\mu)\) that send similar some "generalized Klein-Gordon fields" from the basis volume dot of view.
I promise that you lot won't larn confused past times the basis volume coordinates \(\sigma^\mu\) in addition to the fields \(\Sigma^a\). The model is named later the fields \(\Sigma^a\). Fine. But if nosotros receive got a greater set out of fields \(\Sigma^a\) on the dot given past times \(\sigma^\mu\), nosotros may sympathise a item selection of the fields \(\Sigma^a(\sigma^\mu)\), it describes an embedding \[
\text{world sheet} \to \text{spacetime}
\] where the basis canvas has coordinates \(\sigma^\mu\) in addition to the spacetime has coordinates \(\Sigma^a\). When you lot receive got such embeddings, i.e. a shape of the string (or membrane or brane) embedded inward the spacetime, you lot may written report how it vibrates if it has some tension that tries to shrink it. And the Lagrangian higher upward is morally the same as the Lagrangian for a truthful pianoforte string or a security band that oscillates inward the pre-determined classical manifold whose geometry is encoded inward the functions (the metric tensor)\[
g_{ab}(\Sigma^c(\sigma^\mu)).
\] So far null besides master copy has taken place. The motion of strings – the instance of strings is equivalent to a two-dimensional infinite ("world sheet") parameterized past times \(\sigma^\mu\), i.e. past times the province of affairs inward which the index \(\mu\) has 2 possible values – depends on the pre-existing spacetime manifold just similar the mechanics of dot masses did, or just similar the classical plain theory or quantum plain theory did. It's just dissimilar objects that are "ordered to alive in addition to vibrate" on the given manifold.
However, strings are to a greater extent than creative as good as to a greater extent than self-confident when they larn the liberty to vibrate on a pre-existing background. All other objects that are given the Lebensraum to alive volition but alive – barely hold upward – in addition to die along the geometry of the spacetime manifold. However, vibrating strings come upward at diverse release energy levels in addition to some of the massless or low-lying ones may live identified alongside the gravitons. If you lot create a coherent terra firma of strings inward the graviton terra firma of vibration, the resultant of these strings (through the splitting in addition to joining interactions of strings) on all other strings volition live exactly equivalent to a modification of the geometry of the background spacetime manifold.
While the plain \(g_{ab}(\Sigma(\sigma^\mu))\) seems similar an unchangeable classical parameter describing the spacetime geometry for some seemingly "totally generic" objects – strings – moving inside it, you lot volition regain out that the strings inward a item internal vibration terra firma (in the graviton state) create fields living on the spacetime \(\Sigma^a\) in addition to these fields \(h_{ab}(\Sigma^c)\) may live naturally added to the background \(g_{ab}\) to larn the full, dynamical metric on the spacetime manifold.
Just to live sure, the total geometry is also dynamical – governed past times some partial differential equations – inward the full general theory of relativity so the "amount of flexibility" of the spacetime is comparable inward full general relativity in addition to inward string theory. However, a deviation is that inward full general relativity, you lot ended upward alongside a dynamical metric tensor because you lot postulated it. You haven't derived Einstein's gravity from anything else: you lot just assumed it all along. On the other hand, inward string theory, you lot exclusively assumed some seemingly dissimilar objects – strings – in addition to Einstein's gravity was implied past times that assumption.
Influenza A virus subtype H5N1 cool thing is that Yang-Mills in addition to Proca fields, Dirac in addition to Weyl fields, Kaluza-Klein fields, in addition to other fields inward particle physics arise as well, from the same supposition that strings exist – they arise from other vibration states of the strings – which agency that string theory reduces the set out of independent assumptions inward physics. Also, all the approximate in addition to Yukawa in addition to Higgs self-coupling in addition to other interactions all arise from the same unproblematic splitting-and-joining interactions of strings. In some counting, perturbative string theory reduces all assumptions of physics into one: the basis is made of vibrating strings alongside some tension. And all the interactions are allowed as shortly as you lot allow the basis sheets to receive got whatsoever topology it wants – which is a rattling natural liberty or "right" inward a theory of gravity (and the theory on the basis canvas is a theory of gravity, too). (More technically, the exclusively axiom you lot demand is described past times two-dimensional conformal in addition to modular-invariant theories on the basis sheet.)
This reduction makes string theory much to a greater extent than constrained, unique, in addition to predictive than all previous theories inward physics. I've discussed it inward older spider web log posts.
But I haven't spent much fourth dimension alongside the unproblematic yet profound observation that the classical geometry (with manifolds' patches in addition to points inward them) is no longer cardinal in addition to necessary inside string theory.
The perturbative stringy generalization of the geometry has many aspects. First, a "not so new" novelty. Einstein's equations say that the Einstein tensor\[
R_{ab} - \frac 12 R g_{ab} = \dots
\] is equal to the stress-energy tensor multiplied past times a constant, in addition to so on. In principle, in that location could live additional damage on the left mitt side that expect similar \(R_{ac} R_b{}^c\) or something similar that. They could live polynomial inward the Riemann curved shape tensor – in addition to its covariant derivatives of diverse sorts. Such extra damage don't spoil whatsoever symmetries of full general relativity. The resulting theory is nevertheless invariant nether all coordinate transformations.
Well, perturbative string theory allows you lot to derive Einstein's equations in addition to it indeed produces infinitely many higher-derivative corrections similar that. Influenza A virus subtype H5N1 item string theory produces item corrections. Relatively to the existing terms, the novel damage are multiplied past times diverse powers of the dimensionless tensor\[
\ell_{\rm string}^2 R_{abcd}
\] alongside diverse contractions of the indices. Some \(\ell_{\rm string}^2 \nabla_a\dots \nabla_b \dots\) may live present, too. The constant \(\ell_{\rm string} = \alpha'\) is the so-called (squared) string length (the inverse string tension \(\alpha' = 1/2\pi T\)). And this "small" parameter guarantees that whenever the curved shape radii are much longer than this length \(\ell_{\rm string}\), the novel damage are negligible.
Because derivatives of arbitrarily high orders practise appear, the resulting theory looks "slightly nonlocal" as a plain theory. However, because you lot may seek out that all these damage resultant from the local propagation in addition to splitting/joining of strings at item spacetime points (processes which don't depend on the demeanour of strings at whatsoever other points), the resulting theory really is just local or causal inward some really physical sense, fifty-fifty though the complicated construction alongside the higher derivatives naively indicates something else.
Also, string theory contains damage that are "even smaller" than whatsoever \((R_{abcd})^n\) for \(n\in \ZZ\) inward the \(\ell_{\rm string}\to 0\) limit, the basis canvas instantons. The two-dimensional basis sheets may larn wrapped on some locally minimum 2-cycles inward the spacetime manifold in addition to create corrections that scale similar \(\exp(- Influenza A virus subtype H5N1 / \ell_{\rm string}^2)\). Note that when \(\ell_{string}\) is a rattling pocket-sized set out inward the SI units, the exponential is really, really tiny. It may live \(0.000\dots\) followed past times \(10^{70}\) zeroes in addition to and then a nonzero digit, if the proper expanse \(A\) is a squared meter in addition to the string length is comparable to the Planck length.
Fine. String theory generalizes the geometry of full general relativity because the simplest Einstein's equations are no longer accurate – it produces corrections at every guild in addition to also corrections beyond all orders. But this is non the most profound "generalization of geometry" that string theory brings us, I think. Another, perhaps deeper dot is that
Perturbative string theories may live defined fifty-fifty without whatsoever spacetime manifold in addition to its metric tensor.
How is it possible? I receive got already mentioned that just similar dot masses or piddling light-green men, strings are capable of hearing Mother Nature's guild "this spacetime manifold is your Lebensraum, alive as you lot can". But strings are to a greater extent than clever so they non exclusively seat the spacetime manifold as something that is pretty hospitable to life but they naturally start to curved shape in addition to improve this environment. They're capable of changing its geometry in addition to independently determine the weather that the curved shape of the environs should obey.
But strings tin practise something else, too. They may supersede the axioms for a manifold alongside completely dissimilar ones – novel axioms that are effectively equivalent for the non-linear sigma models but that also acknowledge other, non-geometric solutions. What are the former axioms in addition to the novel axioms?
By the former axioms of the geometry, I meant the Definition of a manifold as something that includes the atlas i.e. collection of overlapping patches in addition to each patch is equivalent to an opened upward laid inward \(\RR^n\), a laid whose elements may live called "points inward the continuum". So fifty-fifty though strings are one-dimensional, they nevertheless piece of work on something (the manifold) that as allows zero-dimensional dot masses to alive in that location as well.
However, i may written report the properties of the nonlinear sigma models in addition to he finds out that
the non-linear sigma models on a Ricci-flat (plus stringy correction) target spacetime are two-dimensional, modular-invariant, conformal plain theories.Within the shape of non-linear sigma models, the Ricci flatness of the target spacetime manifold is basically equivalent to the conformal invariance (basically just the scale invariance nether the scalings of the basis canvas coordinates). So the Einsteinian "Ricci flatness" (or equivalent "Einstein flatness" if you lot wish) is equivalent to the "conformal symmetry of the basis canvas theory". The beta-functions receive got to vanish inward a scale invariant theory – in addition to the beta-functions of the non-linear sigma-model for the "metric tensor components" coupling constants are all the corresponding Ricci tensors.
However, when you lot replaced the old-fashioned, field-theory-style "Ricci flatness" past times a novel axiom, the "conformal symmetry of the basis sheet", you lot may completely forget close the former Einsteinian axiom – the Ricci flatness – in addition to exclusively piece of work alongside the novel i – the conformal symmetry on the basis sheet. And you lot volition regain out novel solutions – theories on the basis canvas that don't expect similar non-linear sigma models at all. In other words, they don't seem to live derived from any old-fashioned manifold alongside its atlas in addition to its metric tensor.
When you lot run across such a deep statement, you lot should ever know some simple plenty yet nontrivial examples. OK, what are the theories on the basis canvas in addition to their fields that may live used instead of the fields \(\Sigma^a\) labeling a simple embedding of the basis canvas to a target spacetime?
Fermionization, bosonization
One really simple instance are basis canvas theories alongside fermionic fields only. Imagine that instead of (or on pinnacle of) the bosonic fields \(\Sigma^a(\sigma^\mu)\), you lot receive got many gratis fermionic fields\[
\psi^\alpha(\sigma^\mu)
\] on the basis sheet. We are formally adding new, fermionic coordinates onto the basis sheet. If point-like particles were propagating on this spacetime, the improver of the fermionic coordinates would live a rather trivial thing. Note that if you lot receive got \(N\) fermionic i.e. Grassmann variables, every role of these coordinates \(\psi^\alpha\) may live Taylor-expanded in addition to because \((\theta^a)^2=0\) already vanishes, the exponent of each \(\psi^\alpha\) may exclusively live null or one. So this exclusively allows you lot \(2^N\) nonzero damage inward the Taylor expansion: each possible fermionic coordinate is exclusively allowed to appear null times or in i trial inward the product. For point-like particles, whatsoever finite set out of fermionic spacetime coordinates is just a bookkeeping device to merge \(2^N\) fields that depend on the bosonic coordinates into i plain of all the "supercoordinates", i.e. into i superfield.
Again, for point-like particles, superspaces are non really needed or they don't create anything fundamentally new. When you lot written report theories inward which the superspaces are helpful, peculiarly supersymmetric theories, you lot may ever piece of work inward "components" instead, too.
The province of affairs is dissimilar inward string theory. If one-dimensional strings probe the target spacetime, i fermionic spacetime coordinate multiplies the total set out of degrees of liberty non past times 2 but past times infinitely many. And inward fact, when it comes to the counting, 2 gratis fermionic fields on the basis canvas are equivalent to i bosonic one! ;-)
How is it possible that for strings, fermionic spacetime coordinates multiply the set out of degrees of liberty past times an infinite factor, just similar the bosonic ones? It's non difficult to reckon why. The fermions \(\psi^\alpha\) on the basis canvas may live Fourier-expanded inward \(\sigma^1\), the coordinate along the string, in addition to nosotros larn the modes \(\psi^\alpha_n\) where \(n\in \ZZ\). The \(n=0\) fashion is equivalent to the fermionic spacetime coordinate seen past times a point-like particle theory but on pinnacle of it, you lot receive got infinitely many other novel coordinates for nonzero values of \(n\). It agency that the improver of \(\psi^\alpha(\sigma^1)\) multiplies the set out of damage inward the "Taylor expansion" non just past times \(2\) but past times \(2^\infty\): the exponent is infinite because the Fourier index \(n\) tin receive got infinitely many values.
If you lot written report string theory or conformal plain theory seriously, you lot volition receive got to reckon at i dot why 2 fermions are equivalent to i boson. Locally, if you lot receive got i bosonic plain \(x(\sigma,\tau)\) and/or a theory alongside 2 fermionic fields \(\psi,\bar\psi\), the maps betwixt these 2 equivalent descriptions of the same theory are basically\[
\partial_z x = \psi\bar\psi, \quad \psi = :\exp(ix):, \quad \bar\psi = :\exp(-ix):
\] It seems totally counterintuitive – why a fermionic bilinear would live indistinguishable from the derivative of a bosonic field, in addition to why the fermion itself would live indistinguishable from the exponential of a boson – but it's true. You may build the Hilbert spaces (they're gratis theories inward the simplest case) in addition to count the degeneracy at each release energy level. Two fermions but destination upward existence equivalent to i boson.
This equivalence is just the "simplest, moral template" or a "local sketch" of a full-blown equivalence. For well-defined string theories, you lot receive got to live careful close all allowed boundary weather for the fermions in addition to bosons in addition to the corresponding projections on states. And in that location be diverse possible projections for a grouping of (an fifty-fifty number) of fermions that are equivalent to some theories alongside bosons (their set out is 1/2 of those of the fermions) in addition to the corresponding projections.
Whole classes of theories, including totally realistic string theories, be that exclusively incorporate fermionic basis canvas fields. In particular, the "free fermionic heterotic models'" exclusively bosonic fields on the basis canvas are those for the four spacetime coordinates nosotros know. All the other degrees of liberty – equivalent to the supersymmetric \(D=10\) heterotic spacetime – are carried past times fermionic fields on the basis sheet.
Ising, minimal, Gepner models
I don't desire to live terribly technical in addition to this is covered inward many squeamish textbooks, e.g. Joe Polchinski's textbook of string theory. But gratis fermions, spell simple, are non the exclusively cool in addition to of import models on the basis canvas that die beyond a non-linear sigma models. An of import extra shape are the "minimal models". They are generalizations of the Ising model – in addition to inward this stringy discussion, past times the Ising model nosotros normally hateful the long-distance, scale-invariant bound of it. The simplest Ising model is basically equivalent to a fermion as well. But its generalizations alongside many states (like the Potts model etc.) – are dissimilar theories.
All the operator-product expansions in addition to the spectrum of operators may live determined completely past times the bootstrap, past times the consistency weather in addition to axioms of conformal plain theories, in addition to they may live proven to be in addition to live unique (having no continuous deformations etc.). So these minimal models – generalization of the Ising model on the basis canvas – incorporate no fields that could live identified as \(\Sigma^a\), the coordinates on a curved classical background manifold i.e. alongside the fields of a non-linear sigma models. But according to the stringy axioms of working, all these theories piece of work great.
Various types of the minimal models on the basis canvas may live combined inward diverse ways to obtain modular-invariant theories in addition to they're exactly as consistent as a theory e.g. alongside the compactified half dozen dimensions spanning a Calabi-Yau manifold. Except that you lot cannot reckon whatsoever manifold inward these combinations of the minimal models – which are normally called the Gepner models later the physicist who played alongside them for the kickoff time.
Funnily enough, you lot may regain out for many such constructions that fifty-fifty though in that location is no visible, self-evident classical manifold underlying the constructions, some of these simple, geometry-independent constructions are really equivalent to a non-linear sigma model i.e. a classical geometry – except that it's i that you lot normally wouldn't guess without lots of experience. So some Gepner models (combinations of several minimal models) are equivalent to non-linear models on Calabi-Yau manifolds of item topologies. The sizes in addition to curved shape radii of these manifolds that are equivalent to the Gepner models are of guild \(\ell_{\rm string}\). That also agency that all the stringy corrections to Einstein's equations etc. are of guild 100% – i.e. as of import as Einstein's master copy terms.
T-duality, mirror symmetry, in addition to many-to-one equivalences of the geometries
When dot masses or piddling light-green men "live" on a predetermined spacetime manifold, you lot may live pretty certain that they may probe or stair out its geometry in addition to they volition live able to distinguish each 2 manifolds from each other. Two manifolds are the same if in that location is a diffeomorphism betwixt them – which maps a dot of the kickoff manifold to the corresponding dot of the 2nd manifold in addition to vice versa, inward such a way that the distances betwixt all the points destination upward existence the same inward both languages.
Two plain theories or configurations on the same classical manifold may live dissimilar – because the physical theory or configuration adds extra information on pinnacle of the classical geometry – but 2 plain theories or arrangements on 2 dissimilar geometries are unavoidably different, too, because the spacetime geometry is classically equivalent to the "constitutional rules" that no other rules or phenomena may ever overrule.
However, string theory goes beyond this "predetermined" dominion to create upward one's heed close the equivalence of 2 manifolds. In my constitutional analogy, strings are capable of sensibly editing their constitution as good to bargain alongside curved in addition to other spaces. In the footnote, I am grateful to Barack Obama for discussions. ;-) Perturbative string theory – fifty-fifty if nosotros verbalize close the non-linear sigma models – on 2 manifolds that are clearly different according to the former rules of equivalence of manifolds (sketched inward the previous paragraphs) may nevertheless live exactly in addition to perfectly physically equivalent.
Even though strings are normally "as sensitive to the scaling of distances" as point-like particles, string theory on a spacetime alongside a circular dimension of radius \(R\) is exactly equivalent to the theory on a manifold whose circular dimension has the radius of \(\ell_{\rm string}^2 / R\). That's known as the T-duality. The equivalence holds because the momentum modes in addition to the winding modes larn perfectly interchanged. You may apply this T-duality on 3-dimensional toroidal fibers of a to a greater extent than complex manifold, a Calabi-Yau three-fold, in addition to you lot volition larn the mirror symmetry – the equivalence of string theory on 2 target spacetimes whose geometry is completely (even topologically) completely dissimilar according to the old-fashioned rules of the geometry of manifolds!
You should appreciate how deep this resultant is in addition to what it means. It agency that spell string theory often agrees alongside the old-fashioned rules what it agency for 2 spacetime manifolds to live physically equivalent, it stands above the most straightforward, old-fashioned factoids. String theory thinks dissimilar in addition to inward some cases, it may say that 2 geometries are completely physically equivalent fifty-fifty though all other objects, point-like particles, in addition to piddling light-green men could recollect that those are completely different.
To larn a precise physical understanding inward all physically measurable quantities inward a grand theory that contains everything you lot demand inward a theory of everything is highly nontrivial but string theory ofttimes does it. These equivalences – T-duality, mirror symmetry, in addition to perhaps other dualities – also present that the selection of the consistent theories of quantum gravity is extremely limited in addition to constrained. The set out of vacua of a consistent theory of quantum gravity is so depression that string theory prefers to sell each solution "several times". It may sell it as a Calabi-Yau manifold, or as its mirror. There is exclusively one physical theory, non 2 inequivalent ones, but this i theory is capable of adopt "at to the lowest degree 2 jobs". The same theory may live presented as the consistent quantum theory of gravity associated alongside 2 (or more) distinct spacetime manifolds etc.
As I said, many of the non-linear sigma models (i.e. old-fashioned geometries, as probed past times vibrating strings) are just equivalent to seemingly non-geometric theories on the basis canvas such as the Gepner models. Within string theory, the old-fashioned rules dictating "which 2 manifolds are physically the same" no longer hold. You must utilisation string theory's own, to a greater extent than advanced rules – in addition to they exclusively trim back to the old-fashioned rules if all the feature distances such as radii of compact dimensions in addition to curved shape radii are much longer than \(\ell_{\rm string}\) at all times. In other words, string theory tells you lot that the former axioms in addition to rules may exclusively live trusted inward the long-distance bound in addition to it explicitly tells you lot what the rules receive got to live replaced alongside when some radii or curved shape radii are brusk enough.
In the text above, I discussed both non-geometric vacua similar the Gepner models (collections of minimal models) as good as the geometric, non-linear sigma models. One may combine the phenomena inward diverse ways. For example, unorientable theories may live defined on a Möbius strip – which is a strip alongside a \(\ZZ_2\) twist applied earlier the reconnection. However, aside from the left-right \(\ZZ_2\) map that defines the Möbius strip, the stringy geometry allows you lot other approximate transformations, i.e. the \(\ZZ_2\) T-duality transformation. You die around a Möbius-like strip in addition to the basis becomes T-dual to what it was when you lot larn back: the momentum in addition to winding modes larn reverted if you lot walk around your house! ;-) Compactifications of this variety comprehend a large shape of clearly non-geometric vacua of string theory in addition to as far as I know, the truthful experts inward this sub-industry nevertheless disagree whether all non-geometric vacua of perturbative string theory may live constructed inward this way, as "locally geometric ones" connected past times some T-dualities.
D-brane generalized geometry
So far inward this text, I discussed the "generalizations of geometry" that are encoded inward perturbative string theory i.e. inward a modular-invariant, two-dimensional, conformal plain theory on a basis sheet. But when \(g_s\ll 1\) no longer holds, nonperturbative string phenomena start to live important. Lots of facts are known close non-perturbative physics of string theory – for example, the \(g_{\rm string}\to \infty\) limits of many string vacua are exactly known (and they are normally other, weakly coupled vacua of string theory or M-theory). And nosotros fifty-fifty receive got total nonperturbatively exact definitions of string theory for whatsoever finite \(g_{\rm string}\) such as matrix string theory.
But it's nevertheless fair to say that the "exact rules" of the generalized geometry – as probed non just past times strings but all the objects inward the theory nosotros nevertheless telephone weep upward string theory – are much less understood than the rules inward perturbative string theory where the conformal fields theories on the basis canvas "know close everything" inward the game. Perturbative string theory – i.e. conformal theories on the basis canvas – are so good understood that fifty-fifty rigorous mathematicians basically receive got an axiomatic framework to written report it. Non-perturbative string theory remains so incompletely understood that fifty-fifty physicists – who tin hold upward alongside much less rigor – don't know of whatsoever universally applicable laid of axioms (yet).
Inside in addition to fifty-fifty exterior perturbative string theory, nosotros know lots of things that may live considered "stringy generalizations of geometries". For example, the interactions of strings on a background alongside a nonzero B-field may live seen to live equivalent to field theories (with all the excited string fields) on a noncommutative geometry. These are plain theories alongside fields that are multiplied non inward the pointwise way – but using the star-product or, equivalently, as operators that depend on \(\hat x,\hat p\) noncommuting coordinates on a stage space. And these theories alongside the star-product are also equivalent to the "matrix-like generalizations" of the spacetime coordinates that are seen past times \(N\) D-branes.
Coincident D-branes generalize the geometry inward yet some other way. If you lot receive got \(N\) ordinary, old-fashioned particles living on a manifold, their coordinates are given past times \(x^a_i\) where \(a=1,2,\dots D\) [thanks, Bill] in addition to \(i=1,2,\dots N\). However, if you lot supersede these particles past times the stringy Dirichlet particles i.e. D0-branes, the degrees of liberty larn organized to \(x^a_{ij}\), matrix elements labeled past times 2 in addition to non just i indices \(i,j=1,2,\dots N\). It's similar to the way how quantum mechanics generalizes probability distributions: it replaces them alongside the density matrix that has the former probabilities on the diagonal but may also receive got anything (Hermitian) exterior the diagonal.
The spacetime geometry as probed past times D-branes brings you lot novel equivalences betwixt geometries – in addition to non-geometries – that should live added to the equivalences seen past times the perturbative strings.
To summarize, perturbative strings, D-branes, in addition to other objects inward string theory tin reckon the geometry of spacetime manifolds that you lot "force upon them" inward a similar way as on whatsoever objects propagating on the geometry. But they send inward a much to a greater extent than mature way on this geometry than whatsoever other objects – from "non-string theories" – you lot could recollect of. They are capable of living "without the crutches of an explicit geometry" in addition to they are also able to reckon that 2 or to a greater extent than geometries (or non-geometries) are really physically equivalent fifty-fifty if all the other probes (point masses in addition to piddling light-green men) would recollect that they are non equivalent.
Much is known close the novel rules alongside which string/M-theory modifies or replaces the old-fashioned rules of geometry. Much remains unknown. By the stringy or quantum generalized geometry, nosotros may hateful whatsoever shape of rules or phenomena that trim back to those of the classical geometries when the radii in addition to curved shape radii are kept much longer than the string length at all times but that also say something coherent when this status is relaxed. With this definition, the seemingly mathematical rules of "stringy or quantum generalized geometry" larn pretty much a synonym for the physical laws of string/M-theory itself. Physical phenomena larn primary.
You may reckon that because of its might to create the old-fashioned rules of geometry inward some limits but also imply something else, string theory in addition to string theorists are no longer reduced to uncritical users of the products of mathematics in addition to mathematicians. They larn active builders or discoverers of novel rules of mathematics in addition to novel structures inward mathematics – rules that mathematicians were ofttimes ignorant close because they were exclusively probing the geometry past times clumsy probes (and clumsy ways of thinking which emulated the clumsy probes inward people's hands). String theory in addition to string theorists tin practise better, live to a greater extent than accurate, map the construction of all consistent theories to a greater extent than accurately, in addition to larn rid of axioms that are sometimes non quite truthful in addition to shouldn't live believed uncritically.
The spacetime in addition to all rules governing it are emergent inward string theory. String theory may modify, generalize, in addition to fifty-fifty alive without these structures in addition to assumptions – yet rest as well-defined in addition to fifty-fifty to a greater extent than consistent at the same moment.