BCAP 2013, a conference in philosophy of physics at the University of Bucharest. The talk is about the explanatory power of string dualities.

1. Ioan Muntean
Indiana University-Purdue University, Fort Wayne
&
University of Notre Dame
http://imuntean.net
1

2. Motivations for this talk
I. Assert the role and importance of principles in quantum
gravity (esp. in string theory)
II. Endorse the “model-based” interpretation of string
theory.
III. Discuss the relation between dualities, unification
and ontological fundamentalism
2

3. Some proposals
III. Dualities as relations among models
A duality principle is a general conjecture about
relations among string models
3

4. Prospective results
Duality and unification: Dualities are weaker than unification.
Albeit not unificatory or explanatory in nature, dualities can:
Accommodate
Integrate
Predict
theoretical facts about the dual model.
Duality is not a consequence or a condition of unification
Deflationary result: Perhaps dualities are consequences of other
assumptions, such as supersymmetry
Dualities and structure: Dualities may unveil a deeper structure,
with or without a deeper theory. (Rickles)
Duality and (reductive) fundamentalism in string theory are
inconsistent
4

5. What is a principle in physics?
Several possible answers. Philosophers envisaged principles as:
a) meta-laws (Lange)
b) constraints on physical models of a theory
c) “methodological maxims”
d) as axioms (of an axiomatized system, e.g. classical mechanics. Carnap)
e) as “correspondences” between the abstract mathematical formalism and the
concrete empirical phenomena (Reichenbach)
f) Bold conjectures, but which are not proven to be false
Although all a)-f ) are germane to physics, I take here f ) as the most
representative for quantum gravity.
All in all, principles play a foundational role in the development and
advancement of a theoretical discipline
Q: In what does lie the foundation of a theory T?
A: The principles that ground it.
5

6. Three philosophical stances
How do principles ground a theory?
A Principle monism: a theory T should be grounded in
one principle P
B Principle pluralism: a theory T is grounded in a
number of independent principles {P1….Pn}
C Principle deflationism: grounding a theory one
principle, in certain set of principles, in a completely
different set, or in no principle at all is a matter of
choice.
I will focus on B here.
6

7. A. From the ideal to reality
B. Greene (1999): “Is string theory itself an inevitable consequence of some broader
principle—possibly but not necessarily a symmetry principle—in much the same way
that the equivalence principle inexorably leads to general relativity or that gauge
symmetries lead to the nongravitational forces?”
“String theory is missing a core principle” and string theorists are all “in a position
analogous to an Einstein bereft of the equivalence principle” (Greene, 1999, p. 171).
“there is […] no guarantee that such a fundamental principle exists, but the evolution of
physics during the last hundred years encourages string theorists to have high hopes that
it does.”
L. Smolin (2002, 149) “string theory in its present form most likely has the same
relationship to its ultimate form as Kepler's astronomy had to Newton's physics".
The ideal: one symmetry principle may become The principle of quantum gravity
Some candidates:
“the holographic principle” `t Hooft (1993); Susskind (1995); Susskind & Lindesay (2004).
The duality principle
7

8. Principle monism and unification
Greene (2011, p. 82): “a theory based on vibrating
filaments might not seem to have much in common
with general relativity’s curved spacetime picture of
gravity. Nevertheless, apply string theory’s
mathematics to a situation where gravity matters but
quantum mechanics doesn’t […] and out pop Einstein’s
equations. Vibrating filaments and point particles are
also quite different. But apply string theory’s
mathematics to a situation where quantum mechanics
matters but gravity doesn’t […] and the math of string
theory morphs into the math of quantum field theory”.
8

9. C. Conventionalism and principles
Any theory T can be grounded in a set of principles,
but one can freely choose what set of conjectures
constitutes the principles and what are mere
consequences
Example: the history of General Relativity.
You can reformulate GR with no principle at all.
The independent principles that ground a theory can
change dramatically in time.
Compare and contrast: a system in logic where we can
interchange axioms with theorems and still get the
same expressiveness of the system
9

10. B. Principle pluralism and problems
The most obvious problem for the pluralist is the consistency of
principles.
Other less dramatic is completeness and independence
Typical clash of principles in quantum gravity: principles of relativity
(e.g. equivalence principle) and principles of quantum theory
(superposition)
Enticing results:
Quantum backreaction as a logical result (inconsistency)
The Weinberg-Witten (1980) theorem
Semi-classical models of system are not coherent (Peres&Terno 2001)
Typical solution: create a new theory with new principle(s) that
explain(s) gravitation as a low energy limit of the new theory.
Non-typical solution: postulate dualities that would relate classical and
quantum regimes of different models.
10

11. First claim of this presentation
Dualities do alleviate such a clash of principles
between theories
Dualities are conceptually weaker than unification, but
can replace the unificatory power when unification is
simply a “bridge too far”
11

12. What is a duality?
Dualities can be both underestimated and overestimated.
Trivial dualities are simply notational variants
“Germane” dualities are inter-theoretical relations
Claim: they are very rich philosophically but understudied in
philosophy.
Notable exceptions: E. Castellani (2009), J. Cushing (1990), Muntean
(2013), D. Rickles (2009, 2011, 2013)
There are several dualities in classical EM and in QFT.
A duality relates two theories (or models) such that:
i. One theory is “more classical” than the other (viz. “more quantum”)
ii. One theory is better known than the other
iii. One theory has a better explanation / prediction than the other
iv. One theory is weakly coupled, the other is strongly coupled
12

13. Triviality and non-triviality
Dualities can relate two completely different theories of the same
physical system
Or two completely different systems described by two different theories
The classical and the quantum description of the same system
A classical system to an another, quantum system
A gauge theory to a gravitational theory
A string theory to another string theory
The weak coupling regime of a theory to its strong regime
Note: Mathematical dualities or logical dualities are only remotely
related to ours
Equivalence between the category of sets and the category of complete
atomic Boolean algebras.
Conjunction and disjunction are dual
13

14. A classical duality
A dual theory is obtained by a “duality
transformation”:
ur ur ur ur
E B;BE
It is a rotation duality in the complex vector field E+iB
In QED, this symmetry signals the existence of
magnetic monopoles (g). They attract each other with
a force of greater than the force between two
electrons
Duality explanation: If there are magnetic poles, the
electric charge is quantized, because: eg = 2n;
14
2
2
137
 
 
 

15. Lessons for Maxwell
The “dual invariance” of the classical EM theory.
ur ur ur
If we define E=
then Maxwell equations are:

E
And they are invariant to these transformation
Here the conserved charge is:
ur ur
E = E=
15
 E  iB
0
i

c t

 



 

ur
ur
ur
E
E
ia e


( ) ia q  ig  e q  ig

16. Duality and explanation
From the dual invariance of EM, Dirac (1931) inferred:
1. The quantization of electrical charge from
2. The existence of magnetic monopoles
1 is a strong result. It could not be explained by other means
(except by a 5D compactification mechanism, Klein 1926)
The other option is to take 1 as a “brute fact” of the universe
16
qg  2 mh

17. Bosonization of fermions in 1D
For some range of the coupling constant, bosons are
more useful as fundamental particle than fermions
For some range of constants, fermions are
fundamental.
By turning the couple constants, one becomes more
fundamental than the other.
Does this show a common structure?
17

18. E-M duality and unification
The dual invariance is strongly related to unification
The 4-vector unifies the E and B fields.
There is a deeper structure than what ordinary Maxwell
equations unveil
Maxwell equations are very rich in symmetries and consequently
in dualities (compare to general relativity)
Bianchi identity
Gauge transformation
Conformal symmetries (Baterman Cunningham 1909)
Dual invariance
Other types of symmetries NOT present in EM:
Diffeomorphism
Supersymmetry (SUSY)
18
μν F

19. The E/M duality in particle
parlance
EM duality relates weak and strong coupling of the same theory.
In one regime, α<<1,
the electron charge is “weak” compared to its dynamics, i.e. it does not interact
with its own field.
Electrons “barely” radiate photons. The electric field is weak. They are hard to
excite (can we excite an electron?)
There are magnetic poles, but they have large fields around them, they are
heavy. The poles are hard to separate, spread out, composite, solitonic
excitations. Structurally, they are very rich.
At α>>1, the poles are fundamental and charges are heavy and rich.
Charges are heavier
Monopoles are more elementary
Either the charge is elementary and the poles are composite, OR the poles are
elementary and charges are composite.
At α=1 there is no fundamentality strictly speaking.
All depends on the coupling constant α.
See a philosophical discussion in Castellani 2009, Rickles 2011.
19

20. E-M duality in QFT
The EM duality as explained before does not survive quantization,
but new dualities arise
Montonen&Olive (1977): At different coupling constants, electric
and magnetic charges exchange roles.
20
Magnetic charge =
topological charge
 Noetherian charge
The magnetic monopoles =
solitons
Magnetic monopoles =
elementary
gauge fields = elementary
particles
Gauge fields=solitons
Weak Strong
Strong Weak

21. Perturbative string theory
The perturbative formulation of string “theory”
contains the highest number of idealizations:
strings are weakly coupled,
the number of strings is relatively small and
strings vibrate against a flat, fixed background
spacetime (background dependence).
The interaction term is a perturbation of the non-interactive
dynamics
All are problematic idealizations, but the third is the
most outrageous and infamous.
21

22. Background dependence as the
major drawback of string theory
“in general, string theory, and other background-dependent
approaches, are […] examples of how not to go about
constructing a theory of quantum gravity” (Rickles French 2006)
background independence and structuralism are “well-matched
bedfellows”
Can we interpret string theory as a structural metaphysics even if
it is not (yet) background independent?
Alternatives:
talk about the promises of a background independent string theory
The concept of background independence needs more
philosophical work.
Dualities may play the central role of smooting out some of these
consequences of the idealizations
22

23. Whacked by the GR community
“What is very frustrating is that […] string theory does
not seem to fully incorporate the basic lesson of GR,
which is that space and time are dynamical rather than
fixed, and relational rather than absolute […] all that
happens is that some strings move against this fixed
background and interact with one another.” (Smolin
2001, 159)
Penrose, Stachel, Woit and others would agree.
But QFT is doing the same (basically)
23

24. Non-perturbative string theory
A more realistic model of strings would assume:
strings interact
string that can split and join (create/annihilate strings)
They have enough energy to interact with spacetime
itself.
In the strong coupling regime, the backreaction with
spacetime is assumed and the background is not
anymore fixed.
24

25. A naïve solution to background
independence
Curved spacetime is reducible to a collection of
gravitons
Given the Witten-Weinberg (no-go result), gravitons
cannot be reduced to any known bosons,
gravitation is simply not “yet another quantum field
theory”
Therefore you need something like strings.
As gravitons are states of strings, the covariant part of
the M space in string theory is in itself dynamical.
25

26. Four interpretations
[1] A collection of theories, most likely all being aspects of a more
fundamental theory (“M-theory"), which is ultimately the theory of
everything (TOE) of our reality. Other theories in physics can be
ultimately reduced to the TOE.
[2] A collection of mathematical models of strings and branes vibrating
in various types of spaces, having different symmetries and properties.
These string models represent aspects of known interactions in
physics: gravitation, gauge theories, black holes, etc.
[3] A collection of conjectures about the relations among the string
models. Some of these string models may (or may not) represent real
interactions in the world;
[4] A collection of conjectures about the relations between string
models (as in [2]) and other theories in physics: gauge theories,
gravitation, black hole thermodynamics, information theory, etc.
I have some reasons to adopt [3] here and to keep an eye on [4]
26

27. String theories
Type
Spacetime
dimensio
ns
SUSY
generators
chiral open strings
heterotic
compactific
ation
gauge group tachyon
Bosonic
(closed)
26 N = 0 no no no none yes
Bosonic
(open)
26 N = 0 no yes no U(1) yes
I 10 N = (1,0) yes yes no SO(32) no
IIA 10 N = (1,1) no no no U(1) no
IIB 10 N = (2,0) yes no no none no
HO 10 N = (1,0) yes no yes SO(32) no
HE 10 N = (1,0) yes no yes E8 × E8 no
M-theory ? 11 N = 1 no no no none no
27

28. Five string models
There are some relevant string models in D=10
1. I: SUSY, open and closed strings, group symmetry SO(32)
2. IIA: SUSY, open and closed strings, non-chiral fermions. D-branes are the
boundaries of open strings
3. IIB: SUSY, open and closed strings, chiral fermions. D-branes are the
boundaries of open strings
4. Type II:
5. HO: Heterotic model: SUSY, closed strings only, right moving strings and
left moving strings differ, symmetry group SO(32)
6. HE: Heterotic model: SUSY, as above, but symmetry group is E8xE8
Dualities:
1/4: I and HO
2/5: IIA and HE
Witten’s conjecture (1995): all string models are related to each other by
dualities.
28

29. Interpretation of string models
1 Empiricist: If this string model were true, what
physics would look like?
2. What are the realist commitments of these models?
29

30. Dualities
Dualities are not:
Approximations of a theory by another theory
Correspondences (as in “theory correspondence”)
Notation variants of the same theory
It is not (always) a symptom of a gauge freedom
But
There is representational ambiguity in dualities
“Dualities can highlight which features of our ontological
picture are not fundamental”. (Rickles 2011) and
They may or may not point towards a deeper structure
Can be related to the symmetries of a theory
Hence their importance to underdetermination and scientific
realism
30

31. A definition of dualities (Vafa)
A theory is characterized by a moduli space M of the
coupling constants
In this space we have several regions conventionally
designated as weak coupling and strong coupling.
A physical system can be represented in various places of
the moduli space with various observables:
Q[M,O ] 
Two physical systems Q and Q’ are dual if:
M M ';O O '    
i.e. there is an isomorphism between their moduli spaces
and another between their observables
31

32. Duality symmetries
In string theory, the moduli space is very rich. Each model is
characterized by a “moduli space” formed by the constants of the
theory.
The string coupling constant gs
The topology of the manifold
Other fields in the background
Everything (?)
The flow of theories is important but more complicated than in QFT
In this moduli space a duality symmetry can relate:
The weak coupling region of T1 with the strong coupling region of T2
The weak coupling region of T to the weak coupling region of the same
T
Interchange elementary quanta with solitons (collective excitations)
Exchange what is fundamental with what is composite
32

33. Some known dualities
33

34. S-duality, informally
gs is the coupling constant in string theory. But in string theory
gs is a field, has its equation, changes from place to place
A string can split into two strings. The probability if this process
is ≈ gs.
The D0-brane and “strings” can interchange the role of
fundamental entity.
When gs is small f-strings are fundamental, light, hard to split.
D0-branes or D-1 are complicated and heavy. Their excitations
are the particles (photons, gravitons)
When D0 branes are light, f-string are heavier
When gs=1 , f-strings and D-branes look the same.
And topology is part of the moduli space. We can change
dimensionality or topology as we walk in the moduli space.
See (Sen 2002) for details
34

35. T-dualities
T1 and T2 can be different (different symmetries,
gauge invariants, topologies), or can be the same
theory (self-duality)!
Weak Weak
T1 T2
35

36. A standard S-duality map
This is not a “self-duality”.
T1 and T2 are structurally different (different
symmetries, gauge invariants, topologies)
They can be string models or other models in QCD
etc.
Weak Weak
T1 T2
Strong
Strong
36

37. Rickles on dualities and unification
Rickles 2011: “the dualities point to the fact that the
five consistent superstring theories, that were believed
to be distinct entities, are better understood as
different perturbative expansions of some single,
deeper theory.” (aka M-theory)
For Rickles and some string enthusiasts, the five points
in moduli space are representations of a single M-theory.
This is the received view in the community.
But….
37

38. Dualities without unification
He admits that “the duality relationships between the
points in moduli space will hold independently of the
existence of an M-theory” Rickles 2012
“in order to achieve a computable scheme for the whole of
the moduli space (including regions away from the
distinguished 'perturbation-friendly' points) such an
underlying theory is required” Rickles 2011, my emphasis
My argument is that we do not need M-theory to see the
deep structure of the S-dualities
One on my assumptions is to analyze these as models, not
as theories
Conceptually I separate the discussion on dualities from
the discussion on unification
38

39. Non-perturbative string theory and
S-duality
If we knew how to relate the weak coupling to the
strong coupling we would relate the non-perturbative
physics to the perturbative physics.
In the ADS/CFT duality this is the path to a
background independent theory.
39

40. Dual nature of duality and
conclusions
We use the weak coupling sector as a calculation device,
but we trust the strong coupling sector in respect of its
ontology and its structure
If duality is isomorphic there is no remainder in the dual
strong sector that cannot be explained away from the weak
coupling sector
If we take dualities seriously we may need to investigate a
new type of explanation based on dualities.
We may want to take a throughout look at string models
related to dualities (and not to “string theories”)
40

41. AdS/CFT duality
We start from a IIB theory (D=10, SUSY with N=4, open
and closed strings)
(Maldacena 1998) but esp (Klebanov 2002)
This theory has a 16 supersymmetric group, the smallest
algebra in D=10.
We take Nc parallel D3-branes close one to the other.
If Ncgs ≪ 1, the low coupling regime, closed strings live in
empty space and the open strings end on the D-branes and
describe excitations of the D-branes.
Open and closed strings are decoupled from each
other.
41

42. Strong coupling and the YM
When Ncgs ≫ 1 the gravitational effect of the D-branes
on the spacetime metric is important, leading to a
curved geometry and to a “black brane”.
But near the horizon the strings are redshifted and
have low energy.
Gauge theory exists and the physics on the D-branes is
nothing else than a gauge theory
The physics is now described by a Yang-Mills CFT with
gYM=4πgs
42

43. Maldacena’s duality conjecture
The 4D, N=4 SUSY SU(Nc) gauge theory (Yang Mills)
is dual to a IIB string theory with the AdS/CFT
boundary.
43

44. Polchinski on branes
“We start with strings in a flat background and discover that a
massless closed string state corresponds to fluctuations of the
geometry. Here we found first a flat hyperplane, and then
discovered that a certain open string state corresponds to
fluctuations of its shape. We should not be surprised that the
hyperplane has become dynamical.”
“Thus the hyperplane is indeed a dynamical object, a Dirichlet
membrane, or D-brane for short. The p-dimensional D-brane,
from dualizing 25-p dimensions, is a Dp-brane. In this
terminology, the original U(n) open string theory contains n
D25-branes.”
A D25-brane fills space, so the string endpoint can be anywhere:
it just corresponds to an ordinary Chan-Paton factor.
44

45. Current research (post 2003)
People try to falsify the AdS/CFT. No success yet
More opportunistically, others try to understand black
holes through the duality as hot gases of fermions
Others try to look for more realistic dualities:
dS/CFT
Dualities with no SUSY, but with holographic principle
(T. Banks).
Understand hadrons with the IIB theory (“the
boomerang kid”)
45

46. Explanation/unification/prediction
in AdS/CFT?
Explanation? I do not see it
Unification? Not directly, without the M-theory.
Prediction? Perhaps, if we relate it to the Higgs
mechanism
Computational advantages? Yes, definitely!
Is the holographic principle doing any
explanatory/unification/predictive work?
46

47. Emergence in string theory
Duality, when correctly interpreted, may illuminate the classical-quantum relation
People believe spacetime emerges in AdS/CFT (Seiberg, Koch Murugan)
In S-duality the non-perturbative aspects emerge from the perturbative aspects but we
compute in the perturbative sector.
Some believe the curved spacetime of general relativity is a holographic emergent
construct from a quantum gauge field without gravity rather than a fundamental feature
of reality.
Strong emergence
Any spacetime emerges from any gauge theory
Weak emergence:
A special type of spacetime (the IIB gravitation) emerges from a special type of gauge
theory (SUSY YM in D=4)
Witness we assume here that:
“If a theory does not have general covariance, then the theory lacks an underlying
spacetime”
47

48. Deflationism: ex SUSY quodlibet
sequitur
SUSY can be a great candidate for the deeper structure
underlying S-dualities and the AdS/CFT duality.
SUGR is also present in the string sector
This is a an argument from symmetry
There are dS/CFT dualities but no non-SUSY dualities.
SUSY is part of the CFT Yang-Mills story because it is
the limit of a IIB theory.
I am not pluralist about SUSY, but I am
pluralist/functionalist about spacetime (and
background independence)
48

49. But symmetry is not enough in
AdS/CFT
In AdS/CFT we do not have a symmetry in the moduli
space.
It is not like a gauge orbifold in the moduli space
There is something else than symmetry here.
49

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