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Prerequisite mathematics for string theory and loop quantum gravity

  1. Feb 17, 2007 #1
    Here is a list of the mathematics one needs to know for string theory (i'm skipping the simple 1st and 2nd year math courses).

    Real analysis
    In real analysis, students learn abstract properties of real functions as mappings, isomorphism, fixed points, and basic topology such as sets, neighborhoods, invariants and homeomorphisms.

    Complex analysis
    Complex analysis is an important foundation for learning string theory. Functions of a complex variable, complex manifolds, holomorphic functions, harmonic forms, Kähler manifolds, Riemann surfaces and Teichmuller spaces are topics one needs to become familiar with in order to study string theory.

    Group theory
    Modern particle physics could not have progressed without an understanding of symmetries and group transformations. Group theory usually begins with the group of permutations on N objects, and other finite groups. Concepts such as representations, irreducibility, classes and characters.

    Differential geometry
    Einstein's General Theory of Relativity turned non-Euclidean geometry from a controversial advance in mathematics into a component of graduate physics education. Differential geometry begins with the study of differentiable manifolds, coordinate systems, vectors and tensors. Students should learn about metrics and covariant derivatives, and how to calculate curvature in coordinate and non-coordinate bases.

    Lie groups
    A Lie group is a group defined as a set of mappings on a differentiable manifold. Lie groups have been especially important in modern physics. The study of Lie groups combines techniques from group theory and basic differential geometry to develop the concepts of Lie derivatives, Killing vectors, Lie algebras and matrix representations.

    Differential forms
    The mathematics of differential forms, developed by Elie Cartan at the beginning of the 20th century, has been powerful technology for understanding Hamiltonian dynamics, relativity and gauge field theory. Students begin with antisymmetric tensors, then develop the concepts of exterior product, exterior derivative, orientability, volume elements, and integrability conditions.

    Homology concerns regions and boundaries of spaces. For example, the boundary of a two-dimensional circular disk is a one-dimensional circle. But a one-dimensional circle has no edges, and hence no boundary. In homology this case is generalized to "The boundary of a boundary is zero." Students learn about simplexes, complexes, chains, and homology groups.

    Cohomology and homology are related, as one might suspect from the names. Cohomology is the study of the relationship between closed and exact differential forms defined on some manifold M. Students explore the generalization of Stokes' theorem, de Rham cohomology, the de Rahm complex, de Rahm's theorem and cohomology groups.

    Lightly speaking, homotopy is the study of the hole in the donut. Homotopy is important in string theory because closed strings can wind around donut holes and get stuck, with physical consequences. Students learn about paths and loops, homotopic maps of loops, contractibility, the fundamental group, higher homotopy groups, and the Bott periodicity theorem.

    Fiber bundles
    Fiber bundles comprise an area of mathematics that studies spaces defined on other spaces through the use of a projection map of some kind. For example, in electromagnetism there is a U(1) vector potential associated with every point of the spacetime manifold. Therefore one could study electromagnetism abstractly as a U(1) fiber bundle over some spacetime manifold M. Concepts developed include tangent bundles, principal bundles, Hopf maps, covariant derivatives, curvature, and the connection to gauge field theories in physics.

    Characteristic classes
    The subject of characteristic classes applies cohomology to fiber bundles to understand the barriers to untwisting a fiber bundle into what is known as a trivial bundle. This is useful because it can reduce complex physical problems to math problems that are already solved. The Chern class is particularly relevant to string theory.

    Index theorems
    In physics we are often interested in knowing about the space of zero eigenvalues of a differential operator. The index of such an operator is related to the dimension of that space of zero eigenvalues. The subject of index theorems and characteristic classes is concerned with

    Supersymmetry and supergravity
    The mathematics behind supersymmetry starts with two concepts: graded Lie algebras, and Grassmann numbers. A graded algebra is one that uses both commutation and anti-commutation relations. Grassmann numbers are anti-commuting numbers, so that x times y = –y times x. The mathematical technology needed to work in supersymmetry includes an understanding of graded Lie algebras, spinors in arbitrary spacetime dimensions, covariant derivatives of spinors, torsion, Killing spinors, and Grassmann multiplication, derivation and integration, and Kähler potentials.

    Cohomology is a powerful mathematical technology for classifying differential forms. In the 1960s, work by Sir Michael Atiyah, Isadore Singer, Alexandre Grothendieck, and Friedrich Hirzebruch generalized coholomogy from differential forms to vector bundles, a subject that is now known as K-theory.
    Witten has argued that K-theory is relevant to string theory for classifying D-brane charges. D-brane objects in string theory carry a type of charge called Ramond-Ramond charge. Ramond-Ramond fields are differential forms, and their charges should be classifed by ordinary cohomology. But gauge fields propagate on D-branes, and gauge fields give rise to vector bundles. This suggests that D-brane charge classification requires a generalization of cohomology to vector bundles -- hence K-theory.

    Noncommutative geometry (NCG for short)
    Geometry was originally developed to describe physical space that we can see and measure. After modern mathematics was freed from Euclid's Fifth Axiom by Gauss and Bolyai, Riemann added to modern geometry the abstract notion of a manifold M with points that are labeled by local coordinates that are real numbers, with some metric tensor that determines an extremal length between two points on the manifold.
    Much of the progress in 20th century physics was in applying this modern notion of geometry to spacetime, or to quantum gauge field theory.
    In the quest to develop a notion of quantum geometry, as far back as 1947, people were trying to quantize spacetime so that the coordinates would not be ordinary real numbers, but somehow elevated to quantum operators obeying some nontrivial quantum commutation relations. Hence the term "noncommutative geometry," or NCG for short.

    I was wondering if this list is the same for loop quantum gravity. If not, what can be removed and what should be added?
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  3. Feb 18, 2007 #2


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    Hi Tom, pretty much everything you mentioned comes up in research by one LQG person or another, so if you wanted to prepare yourself to understand EVERY research paper coming out of the LQG community you would have to learn pretty much the whole curriculum.

    As you probably know LQG (as people normally speak of it) is a mix of non-string background-independent QG approaches. It is not codified except as "what Loop people do" as seen in the main international conferences like Loops '05 at the AEI in Germany. And presumably Loops '07 to be held in Morelia this summer.
    Also there's an overview book supposed to come out this year from Cambridge U. Press called "Approaches to Quantum Gravity, Towards a New Understanding of Space, Time, and Matter" edited by Dan Oriti. This will give an up to date picture of the various approaches---with chapters written by around 20 people IIRC. When the book comes out you can scan the chapters and see what all mathematics Loop people, or non-string QG people, use.

    I've annotated your list of mathematical subjects ...
    Real analysis [pervades LQG]

    Complex analysis [pervasive]

    Group theory [pervasive]

    Differential geometry [basic to most LQG]

    Lie groups [absolutely basic]

    Differential forms [absolutely basic]

    Fiber bundles [basic to correct understanding of much LQG]

    Characteristic class/Index [nice to know but not pervasive! Chern-Simon index is essential to ONE QG approach primarily due to Lee Smolin and not AFAIK most other approaches, but see work by Stephon Alexander and that young guy Andy Randono]

    Homotopy [an interesting extension of homotopy is essential to some recent work e.g. by John Baez students and co-workers, also in a general way homotopy bears on LQG because of connection between quantum states of gravity and knots]

    Noncommutative geometry [interaction between NCG and LQG communities]

    A lot of the work within the LQG these days comes under the heading of spinfoams (it would not be recognizable as vintage 1990s LQG which outsiders seem to imagine when they say LQG) and DSR (officially this means deformed special relativity).

    DSR can be understood in terms of quantum groups, hopf algebras and non-commutative geometry. So we see papers be co-authored by prominent LQG people (e.g. Laurent Freidel) on the one hand and by NCG people (Shahn Majid, Jerzy K-G) on the other.
    Another sign of the crossover is that the most recent paper by Alain Connes (NCG) came out with the same results the same day as a paper by John Barrett, whose main line of research is spin foam QG.

    I guess you could say that LQG people and NCG people are playing in each others back yards.

    To get an idea of LQG and NCG convergence at leading edge go to arxiv and look up abstracts recent papers by Laurent Freidel, John Barrett, Jerzy Kowalski-Glikman, Shahn Majid, Philippe Roche.

    I don't understand this well enough to describe, it is just something I see going on without having a clear explanation.

    I leave the following for someone else to discuss.
    Index theorems

    The list didn't include Categorics. Probably to correct for that absence Tom, you should have a look at John Baez paper
    Quantum Quandaries
    it is at arxiv, but if you go looking for it at Baez website you will find his QG seminar notes and a whole bunch of stuff on QG-and-Categories

    The question that occurs to me, Tom, is about LEARNING STRATEGY.
    It seems to me that these are all good things to learn, as a math major and math graduate student. I myself once gave a seminar talk on K theory (he says modestly:smile: back in grad school days)
    But it seems to me misguided to treat all this stuff as PREREQUISITES and assume that the various fields within QG are going to hold still.

    By the time you learn all the prerequisites, QG might be unrecognizably different.

    So yeah go ahead and learn differential geometry and fiber bundles and lie groups because they are basic to so much.

    But also right now you might look at some QG literature.

    Like Derek Wise (a John Baez student) has a nice paper with a lot of intuitive content
    "M-M gravity and Cartan geometry"
    Ordinary diff. geom. is constructed using flat tangent spaces. One has tangent vectors and a vectorspace is essentially a flat object. That's how a conventional manifold---the prevailing model of the continuum---is defined. Derek and a dozen other QG people are currently moving into new territory where you have a different idea of the continuum: Cartan's 1923 invention of a manifold with curved local "tangent space-like" approximations.
    Cartan geometry (I don't mean differential forms which Cartan invented, but his actual curved tangent geometry) was neglected by mathematicians and physicists for around 70 years until the 1990s. Derek Wise gives some of the history in his paper.

    Today the LQG people are doing different QG from what they were 5 years ago. In five years they will be doing still more different stuff. It is a rapidly changing field with very inventive people. So I would suggest not treating it as a stationary target. Don't wait for prerequisites. Get a taste of the leading edge research right now.
    Last edited: Feb 18, 2007
  4. Feb 18, 2007 #3
    Thanks alot! I've learned about half of the above list of prerequisite mathematics so far. So I'll just continue to try to finish learning the rest of the mathematics. By then I'll have an adequate mathematical toolbox to study whatever has become of loop quantum gravity and/or string theory and thus be able to study them without reservation.
  5. Feb 19, 2007 #4


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    I would like to note that standards of mathematical rigor are much higher in LQG than in string theory. Unlike LQG, string theory contains a lot of hand waving and intuition. Thus, if one actually wants to learn string theory, it is not the best strategy not to start with it before learning all mathematics that is related to some aspects of string theory. In fact, the book of Zwiebach "A first course in string theory" shows that the main ideas of string theory can be understood without the most advanced mathematics.
  6. Feb 19, 2007 #5


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    I often don't agree with you, Harvey. But I would agree completely here: if a young person's goal is merely to learn string ideas (assuming there is no coherent theory but an intricate thicket of interwoven ideas) then I would advise them to begin at once

    with textbooks like Zwiebach and also this new one by Becker Becker Schwarz, for which I've seen favorable reviews.

    I only answered Tom with respect to non-string QG because I interpreted his question narrowly and tried to respond in a focused way to what he asked.

    But to comment more broadly, his learning strategy seems to me to use string ideas (which currently attract him) as MOTIVATIONAL BAIT to draw himself into generally useful 20th century mathematics.

    String think may serve here as a kind of MENTAL CHEESE. In truth it may only be a piece of Velveeta (semi-edible putty with a nearly cheese-like appearance). But if it lures young people into a world of excellent mathematics then it performs a valuable function.

    On the other hand, as you point out, some people short-circuit this process and swipe the bait without learning the Lie Groups.
    Those doing this may be too smart for their own good! since then all one ends up with is String.

    My hunch is that this observation does not really address Tom. He has decided to learn the math and, by his account, has already made considerable progress with it.

    I would still advise reading Derek Wise paper because it underscores the importance of the de Sitter group (not the ANTI de Sitter!) and both things referred to in Derek's title: M-M gravity and Cartan geometry.
    Both these things have barely been touched yet and have territory-size potential.

    Also I don't know what you would say to this, Harvey, but I would advise to expose the 14-year old brain to a few pages of
    Connes latest NCG paper, the one that actually GETS the standard model.

    Gravity and the standard model with neutrino mixing
    Ali H. Chamseddine, Alain Connes, Matilde Marcolli
    71 pages, 7 figures

    "We present an effective unified theory based on noncommutative geometry for the standard model with neutrino mixing, minimally coupled to gravity. The unification is based on the symplectic unitary group in Hilbert space and on the spectral action. It yields all the detailed structure of the standard model with several predictions at unification scale. Besides the familiar predictions for the gauge couplings as for GUT theories, it predicts the Higgs scattering parameter and the sum of the squares of Yukawa couplings. From these relations one can extract predictions at low energy, giving in particular a Higgs mass around 170 GeV and a top mass compatible with present experimental value. The geometric picture that emerges is that space-time is the product of an ordinary spin manifold (for which the theory would deliver Einstein gravity) by a finite noncommutative geometry F. The discrete space F is of KO-dimension 6 modulo 8 and of metric dimension 0, and accounts for all the intricacies of the standard model with its spontaneous symmetry breaking Higgs sector."

    By contrast with the handwaving you mentioned, IMHO there is no vague suggestion in Connes work. He is thoroughly rigorous, even checking steps in algebra on a computer to avoid logical error. It is still an effective theory but it seems to work, to be unique/coherent and make predictions. Apparently one can say precisely what the theory is. So it may not be right, it may not be fundamental, but I think it is real cheese.

    Please let me know if and how your opinion differs.
    Last edited: Feb 19, 2007
  7. Feb 19, 2007 #6
  8. Feb 20, 2007 #7


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    Marcus, I also want to point the following.
    Although LQG is formulated very rigorously, even more rigorously than ordinary quantum field theory, it does not mean that this theory cannot be understood by intuitive theorethical physicists who are not mathematical physicists. For that kind of physicists I would recommend the book of Rovelli (as well as other reviews written by him).

    And of course, as always in physics, insisting on high level of mathematical rigor may often prevent discovery of interesting physical results. Hand waving and intuition is also an important part of physics, which makes it different from mathematics.
  9. Feb 20, 2007 #8
    hope not to crash on this thread but i have a question on LQG.
    i wanted to know which universities outside of the US offer research in LQG, or any non-string theory?
  10. Feb 20, 2007 #9


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    this list is off-the-cuff from memory, vague in places and may have mistakes and major omissions---you are in Israel IIRC so maybe you can say what Universities have QG research there.

    Waterloo ( Perimeter group)
    U. West Ontario (Dan Christensen's group)
    U. B. C. (occasional QG papers, no regular group)

    Nottingham (Barrett, Krasnov group)
    London (Dowker, Isham group, Majid)
    Cambridge (Williams, some postdocs but no other faculty)
    some place in Scotland Aberdeen IIRC (papers but no regular group)

    Marseille (Rovelli, Perez group)
    Montpellier Campus B (Roche, Buffenoir, maybe Noui)
    Lyon (Livine)
    (wild card Alain Connes)

    Utrecht (Ambjorn, Loll, large group)

    Mainz (Reuter)
    Aachen (the venerable Kastrup who advised both Bojowald and Thiemann)
    AEI-Golm (Thiemann, AEI connects to nearby institutions in Berlin/Potsdam area)

    (strong Jerzy Lewandowski group, never sure if it is U. Warsaw or Wroclaw, maybe both)

    Morelia campus of the National University (Corichi and several others)

    various institutions (Ganashyam Date and several others)

    Montevideo (Gambini, small group)

    various institutions
    Last edited: Feb 20, 2007
  11. Feb 20, 2007 #10
    from what iv'e seen in courses which are offered at the hebrew university and tel aviv university, there isn't even a mention of one graduate course in LQG, but there are courses in string theory and ofcourse qft.

    btw, are you sure about cambridge?
    i had a discussion in the academic forum here in pf, and someone apparently a graduate student from there, stated that they solely are working on string theory and not on LQG.
  12. Feb 20, 2007 #11


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    It seems that the person you say is "apparently a graduate student from [Cambridge]" didn't know everything that is going on at Cambridge.

    Remember the question you asked was about non-string QG, which includes a wide range of things which people normally mean when they use the blanket term "LQG"

    The March April QGQG school features Dr. Williams from Cambridge (DAMPT) among other people

    in other words Williams is on the short list of internationally recognized non-string QG authorities.

    Until recently Oriti was there (currently he is at Utrecht) and he is the editor of the forthcoming book Approaches to Quantum Gravity: Towards a New Understanding of Space Time and Matter. Williams was Oriti's PhD thesis advisor at Cambridge.

    Williams field is simplicial gravity (Regge). Oriti's thesis was in spinfoam gravity and subsequent papers were in spinfoam and group field theory (another nonstring QG approach related to spinfoam). There are other people in the Cambridge QG group---Ryan comes to mind. But Williams and Oriti are the most visible internationally known folks.

    From your question about research opportunities, I assumed you were interested in WHAT ARE UNIVERSITIES WHERE YOU CAN GET INTO NON-STRING QG RESEARCH. That is, where do they have groups, where can you find a mentor, where are people writing papers? Maybe you are asking a different question? Like: Where is a university where there is a course being taught called "Introduction to Background Independent Quantum Gravity"

    That's a reasonable question but it does not correspond to where the research opportunities are----the good places to get into QG research.

    People who go into QG research have usually already gotten to the point in graduate or postdoctorate study where they dont need a course called "Introduction to QG"---they need an experienced one-on-one coach who can advise on reading and research problems, and a seminar room with a few other people like them to talk with.

    there may actually be one or more QG groups in Israel, that you and I dont know about---but you wouldn't necessarily find them by reading University course catalogs.

    You can't tell what research opportunities a University offers by looking in the General Catalog of courses. What matters more is who is there, what kind of PhD theses their graduate students write, nd the quality of the papers posted on arxiv co-authored by young people at that University.
    Last edited: Feb 20, 2007
  13. Feb 20, 2007 #12


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    There's more if you look around. Thing is as Marcus said, many individuals, few groups.

    Edinburgh has some people on 3D Quantum Geometry, Claus Kiefer in Cologne published a book called Quantum Gravity, Christian Fleischhack is leading an Emmy Noether group in Leipzig.

    It's a couple of years old but Thiemann's book on LQG (on the arxiv) contains a list of research destinations as well.
  14. Feb 20, 2007 #13
    my assumption was, that if research is being conducted in a spscific field in a particular university, then they also offer graduate courses and seminars in this field, am i wrong in assuming that?
    so iv'e looked in courses catalogs which offer also seminars for masters and phd students, and so far i havent found something which isnt related to string or qft.
    i know that the best method is looking for the people themselves, but i dont think LQG research is being conducted in israel.
    anyway, i will keep trying, perhaps i missed or havent looked thouroughly.
  15. Feb 20, 2007 #14
    thanks, f-h, i'll look into it.
  16. Feb 20, 2007 #15


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    Correction, Fleischhack is actually in Hamburg now.
  17. Feb 20, 2007 #16


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    As far as I know you're right! I watch the literature and I don't recall any non-string QG papers coming from Israel at least in the past 2 years.
    (memory far from perfect, but I'll offer that for what it's worth.)

    Maybe a more efficient search proceedure for you would be to write email to Danny Terno at Perimeter.
    He recently came from Israel to Perimeter, like IIRC in 2005.

    He would know who is working on what, in Israel.
    His field is Quantum Information Theory, but he is easily able to cross over into quantum theory of spacetime and has done some of that since 2005.

    But maybe nobody is doing non-string QG in Israel and the sensible thing is simply to stop looking :smile:

    Before 2004 I think Terno was probably at Technion and his teacher/advisor was probably Asher Peres.
    Danny Terno's email address at Perimeter would be listed in the more recent papers from this list:
    1. gr-qc/0611135 [abs, ps, pdf, other] :
    Title: Quantum causal histories in the light of quantum information
    Authors: Etera R. Livine, Daniel R. Terno
    Comments: 9 pages, 8 eps figures

    2. gr-qc/0603008 [abs, ps, pdf, other] :
    Title: Reconstructing Quantum Geometry from Quantum Information: Area Renormalisation, Coarse-Graining and Entanglement on Spin Networks
    Authors: Etera R. Livine, Daniel R. Terno
    Comments: 27 pages, 12 figures, RevTex4

    3. quant-ph/0512167 [abs, ps, pdf, other] :
    Title: Physical accessibility of non-completely positive maps
    Authors: Hilary Carteret, Daniel R. Terno, Karol Zyczkowski
    Comments: 4 pages, 1 eps figure Substantially revised and modified version. The definition of accesibility is now unambiguous and physically well-motivated, and the main focus of the applications is on quantum gates and process tomography

    4. gr-qc/0512072 [abs, ps, pdf, other] :
    Title: Quantum information in loop quantum gravity
    Authors: Daniel R. Terno
    Comments: RevTex, 5 pages. Proceedings of QG'05, Cala Gonone, 2005 Relations of the coarse-graining and partial tracing are clarified, and the references are updated
    Journal-ref: J.Phys.Conf.Ser. 33 (2006) 469-474

    5. gr-qc/0508085 [abs, ps, pdf, other] :
    Title: Quantum Black Holes: Entropy and Entanglement on the Horizon
    Authors: Etera R. Livine, Daniel R. Terno
    Comments: Revtex4, 25 pages, 4 figures
    Journal-ref: Nucl.Phys. B741 (2006) 131-161

    6. quant-ph/0508049 [abs, ps, pdf, other] :
    Title: Introduction to relativistic quantum information
    Authors: Daniel R. Terno
    Comments: Lecture notes for NATO ASI Quantum Computation and Quantum Information, Chania, Crete 2005
    Journal-ref: Quantum Information Processing: From Theory to Experiment, edited D.G. Angelakis et al., (IOP Press, 2006) p. 61

    7. gr-qc/0505068 [abs, ps, pdf, other] :
    Title: From qubits to black holes: entropy, entanglement and all that
    Authors: Daniel R. Terno
    Comments: Honorable Mention in the 2005 Gravity Research Foundation Essay Competition
    Journal-ref: Int.J.Mod.Phys. D14 (2005) 2307-2314

    8. quant-ph/0502043 [abs, ps, pdf, other] :
    Title: Entanglement of zero angular momentum mixtures and black hole entropy
    Authors: Etera R. Livine, Daniel R. Terno
    Comments: 5 pages, revtex4 Final version. A discussion of local orthogonality and entanglement is added
    Journal-ref: Phys. Rev. A 72, 022307 (2005)

    9. hep-th/0403142 [abs, ps, pdf, other] :
    Title: Entropy, holography and the second law
    Authors: Daniel R. Terno
    Comments: 4 pages, RevTex 4, 1 eps figure
    Journal-ref: Phys.Rev.Lett. 93 (2004) 051303
    Last edited: Feb 20, 2007
  18. Feb 21, 2007 #17
    Wow, that was a lot of good stuff, thanks for the contributions.

    Do they do string research in singapore? More specifically NUS (National University of Singapore).

    I am there currently and Brett McInnes and Edward Teo seem to do some kind of research in that advanced physics field.

  19. Feb 21, 2007 #18

    Gib Z

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    Well if thats where you are, why don't you ask them?..I dont know..
  20. Feb 21, 2007 #19
    what does "hand waving" mean? is it like looking at a part of a solution that requires mathematical rigour to justify but then you just say "Blah, that's pretty obvious, let's move on..." the purpose of this is to avoid being distracted by mathematical rigour so as to be more focused on the physical interpretation of the problem?
  21. Feb 21, 2007 #20


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    One typical example of "hand waving" in string theory is the way how string theorists "derive" various effective field theories from strings. But if you haven't seen such derivations, it is hard to explain it here.
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