| String Theory and Mathematics : String Theory and Mathematics University of Washington
Science Forum Colloquium
May 20, 2005
Charles Doran
Department of Mathematics
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| String Theory and Mathematics : String Theory and Mathematics Why are mathematicians so excited about string theory?
Physics and mathematics in interaction
Mirror symmetry and mathematics |
| Historical development : Historical development 3000 years of development together
Babylonian description of the metric of flat 2- space for land measurement Babylonian tablet: Plimpton 322 (Columbia University Rare Book and Manuscript Library) |
| Historical development : Historical development 17th century natural philosophers developed mathematical and physical concepts together
Quantitative understanding of laws of nature best expressed in the language of precise abstraction
Physical theories were expressed in terms of mathematical equations |
| Historical development : Historical development Periods of working largely apart on seemingly unrelated problems
More general laws … more abstract language
General relativity
Quantum mechanics
Gauge theory Minkowski space and differential geometry
Hilbert space and operator theory
Lie groups and bundle theory It became reasonable to ask “Why?” |
| Physicist’s perspective : Physicist’s perspective Eugene Wigner, “On the Unreasonable Effectiveness of Mathematics in the Natural Sciences” (Comm. Pure Appl. Math. XIII, 1960)
Posits two reasons:
The laws of invariance: Physical laws are valid at every point of space-time
The empirical law of epistemology: “Article of faith” of the theoretical physicist “that the mathematical formulation of the physicist’s often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena” |
| Physicist’s perspective : Physicist’s perspective Contrasts this with the “nightmare of the theorist”:
“... theories, which we considered to be ‘proved’ by a number of numerical agreements which appears to be large enough for us, are false because they are in conflict with a possible more encompassing theory which is beyond our means of discovery.”
Illustrates it with the problem of quantum gravity:
“The two theories operate with different mathematical concepts … no mathematical formulation exists to which both of these theories are approximations.” |
| Mathematician’s perspective : Mathematician’s perspective Physics as a source of examples and conjectures
Sources of motivation in mathematical research:
Study examples and generalize them
Try to prove mathematical conjectures
Physics as the ur-example:
Classical mechanics: qualitative theory of differential equations, symplectic geometry
Quantum mechanics: “q-analogues” in algebra and geometry, phenomena in operator theory
General relativity: partial differential equations and differential geometry |
| Mathematician’s perspective : Mathematician’s perspective Some famous Conjectures:
Poincaré Conjecture (topology and differential geometry)
Riemann Hypothesis (number theory, probability)
Other $1,000,000 “Millennium Problems”
Best conjectures are those whose statements or proofs relate different types of mathematics
Wigner’s illustration of the theorist’s nightmare, the problem of reconciling quantum mechanics and gravity, becomes a great opportunity! |
| Mathematician’s perspective : Mathematician’s perspective From physics we know:
“The laws of nature are written in the language of mathematics.” -- Wigner (after Galileo)
“The two theories operate with different mathematical concepts …” -- Wigner
“All physicists believe that a union of the two theories is inherently possible and that we shall find it.” – Wigner
Thus any candidate for a consistent theory of quantum gravity must relate in a fundamentally new way the different branches of mathematics used to describe each component theory
In other words: Such a physical theory should suggest important mathematical conjectures! |
| Unification of fundamental forces : Unification of fundamental forces The four fundamental physical forces play key roles in the fundamental physical theories |
| Unification of fundamental forces : Unification of fundamental forces
These theories describe physical interactions mediated by the elementary force carrier particles |
| Unification of fundamental forces : Unification of fundamental forces These theories are valid at widely different energy scales, and unification of general relativity and quantum mechanics may even require a theory of physics at the Planck scale |
| String theory : String theory Candidate theory of quantum gravity
Point particles Strings |
| String theory : String theory Is not a unique consistent theory – there are several variants … … but they all have certain features in common |
| String theory : String theory Hidden extra dimensions “curled up” or compactified |
| String theory : String theory Requires that 6 dimensions be compactified – i.e., these are 6 + 3 + 1 = 10 dimensional theories
Shape of these compactified dimensions is what mathematicians call a Calabi-Yau manifold |
| String theory : String theory Stringy dynamics on Calabi-Yau manifold M
Topology and geometry of M particle spectrum that can arise
Topology:
Hodge numbers
(h1,1 , h2,1)
Geometry: Notion of distance given by metric |
| String theory : String theory Special metric on Calabi-Yau manifolds was what physicists needed
Conjectured by Eugenio Calabi in 1954, proved by Shing-Tung Yau in 1976
Yau’s proof entirely non-constructive -- but OK since physicists just needed existence and uniqueness, not explicit description
Next question became:
How many Calabi-Yau manifolds are there? |
| How many Calabi-Yau manifolds? : How many Calabi-Yau manifolds? Mid 1980’s Handful of examples
Mid 1990’s Hundreds of thousands
Plot the Hodge numbers and a pattern emerges … Horizontal axis: 2(h1,1 – h2,1)
Vertical axis: h1,1 + h2,1
Points represent Calabi-Yau manifolds … a Mirror Symmetry |
| Mirror symmetry : Mirror symmetry
Mirror pairs of Calabi-Yau manifolds:
Calabi-Yau manifold M Calabi-Yau manifold W
h1,1(M) = h2,1(W)
h2,1(M) = h1,1(W)
Shocking prediction to mathematicians
First example of a string duality
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| String dualities : String dualities String dualities relate different string theories:
Mirror Symmetry = Type IIA/IIB string duality
Non-uniqueness of string theory becomes a useful feature as string dualities themselves yield new mathematical conjectures!
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| String dualities : String dualities Are “extra-mathematical” in origin
Mathematical descriptions of two string theories can be very different
Identifying these sometimes results in …
Easy problem = Hard problem
(known how to compute) (math not yet developed)
… which can make for some very precise mathematical predictions coming from physics
The more precise these predictions, the better for mathematicians seeking to refine the conjectures they inspire |
| Mathematical mirror symmetry : Mathematical mirror symmetry Combinatorics and the topological mirror symmetry conjecture
Most constructions of Calabi-Yau manifolds are based on the combinatorics of reflexive polytopes
Polar duality for reflexive polytopes implies topological properties of mirror symmetry
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| Mathematical mirror symmetry : Mathematical mirror symmetry Victor Batyrev established the Hodge number identities relating Calabi-Yau manifold M and W constructed from such polar pairs
Another pair (in 3D) |
| Mathematical mirror symmetry : Mathematical mirror symmetry Much more than just topological consequences
Correlation functions in IIA-model and IIB-model topological quantum field theories yield:
IIB-model IIA-model
calculation problem
on W on M
Precise enumerative predictions: Numbers of rational curves via the theory of Gromov-Witten invariants
Inspired a revolution in enumerative geometry over the last decade |
| Mathematical mirror symmetry : Mathematical mirror symmetry Motivating conjectures to emerge from mirror symmetry?
1. Strominger-Yau-Zaslow Conjecture
Motivated by T-duality and D-brane physics
Mirror symmetry as a sort of Fourier transform on the fibers of a torus-fibered Calabi-Yau manifold
Advantage of being constructive
Works well in low dimensions, difficulties with singularities in dimension 6
Has inspired much research in differential geometry
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| Mathematical mirror symmetry : Mathematical mirror symmetry Homological Mirror Symmetry Conjecture
Due to Maxim Kontsevich
Expresses duality as most general mathematical duality – an equivalence of categories
Category:
Objects (mathematical structures)
Arrows (ways to transform between structures)
Bounded derived category of coherent sheaves on M
Category associated to the mirror manifold W has yet to be constructed
Easy problem Hard problem paradigm taken to another level
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| Unreasonable effectiveness : Unreasonable effectiveness … of String Theory in Mathematics!
String theory has produced
“derivations” of mathematical theories like toric geometry and K-theory, and
a host of string-motivated conjectures in virtually every field of mathematics
Whatever its eventual status as a physical theory of quantum gravity, the inevitability of string theory as a mathematical theory of the highest order is hard to dispute. |