Weyl Subchambers and Hyperplane Arrangements - …math.bu.edu/people/ep/AFRAMATH15/KeynoteAddress.pdfThe University of Massachusetts Boston AFRAMATH15 Keynote Address 4/25/2015 Boston

Societe Scientifique Haıtienne: haitianscientificsociety.orgThe Physicist’s Dream

Genesis of the ProjectKnown Results

Generalization by Esole-Jackson-Noel

Weyl Subchambers and Hyperplane Arrangements

Alfred Gerard NoelJoint work with

Mboyo Esole (Harvard)and Steven Glenn Jackson

The University of Massachusetts Boston

AFRAMATH15 Keynote Address4/25/2015

Boston University Boston, MA, USAOrganizer: Emma Previato (BU)

Esole-Jackson-Noel Weyl Subchambers and Hyperplane Arrangements




Partner in Crime: Mboyo Esole (BPL Harvard)





Partner in Crime: Steven Glenn Jackson( Associate Professor UmassBoston)





Outline

1 The Physicist’s Dream

2 Genesis of the Project

3 Known Results

4 Generalization by Esole-Jackson-Noel





Describe The Laws of Nature Using a Small Numbers of Principles(Preferably Just One) expressible in mathematical language.

Galileo: “ [The universe] cannot be read until we have learnt the languageand become familiar with the characters in which it is written. It is writtenin mathematical language, and the letters are triangles, circles and othergeometrical figures, without which means it is humanly impossible tocomprehend a single word.” Opere Il Saggiatore p. 171.

Wigner: “ A possible explanation of the physicist’s use of mathematics toformulate his laws of nature is that he is a somewhat irresponsibleperson... ...However, it is important to point out that the mathematicalformulation of the physicist’s often crude experience leads in an uncannynumber of cases to an amazingly accurate description of a large class ofphenomena... this shows that it [Mathematics] is, in a very real sense, thecorrect language.” [The Unreasonable Effectiveness of Mathematics in theNatural Sciences]





Newton’s law of universal gravitation states that any two bodies in theuniverse attract each other with a force that is directly proportional to theproduct of their masses and inversely proportional to the square of thedistance between them:

G is the gravitational constant (6.673× 10−11N(m/kg)2)Esole-Jackson-Noel Weyl Subchambers and Hyperplane Arrangements




Newton’s laws of motion are three physical laws that together laid thefoundation for classical mechanics.

First law: When viewed in an inertial reference frame, an object eitherremains at rest or continues to move at a constant velocity, unless actedupon by an external force.

Second law: The vector sum of the external forces F on an object is equalto the mass m of that object multiplied by the acceleration vector a of theobject:

F = ma.

Third law: When one body exerts a force on a second body, the secondbody simultaneously exerts a force equal in magnitude and opposite indirection on the first body.





Newton’s First and Second Law (1687) in Principia Mathematica





Mass-Energy equivalence in special relativity (Einstein-Poincare) (via thespeed of light)

E = mc2

Quantum analog of Newton’s second law is Schodinger’s Equation whichgives a description of a system evolving with time:

i}∂

∂tΨ = HΨ





Maxwell’s equations describe how electric and magnetic fields aregenerated and altered by each other and by charges and currents.

The theoretical basis of Modern technologies (Computers,Communications, Satellites ... )

Gauss’ law, Gauss’magnetism law, Faraday’s law, Ampere’s law.





The Standard Model (1970’s) of particle physics is a theory concerning theelectromagnetic, weak, and strong nuclear interactions, as well asclassifying all the subatomic particles known.(NOT GRAVITY)

“A major goal of physics is to find the ”common ground” that would uniteall of these theories into one integrated theory of everything, of which allthe other known laws would be special cases, and from which the behaviorof all matter and energy could be derived (at least in principle).”[Wikipedia]





Outline



3 Known Results






June 2014 Mboyo gave a talk at the CAARMS 20 (Conference forAfrican-American Researchers in the Mathematical Sciences) organized byWilliam Massey at Princeton University.http://www.caarms.net/custom.aspx?qs1=rkV9040714125933vbBr





•Mboyo spoke on ”Elliptic Fibrations and F-theory”

•Relating String Theory and the geometry of elliptic fibrations andrepresentation theory of Lie Groups (My area of research)

• Decision to collaborate was made in Princeton. Later on (in August)Steven Jackson joined the project and made a fundamental contribution.





Intriligator-Morrision-Seiberg prepotential

The prepotential has been computed by Intriligator-Morrision-Seiberg for atheory in which hypermultiplets transforming in the representation R of agauge group with Lie algebra g:

F(ϕ) =1

2m0

∑a,b

habϕaϕb+

ccl3!

∑a,b,c

dabcϕaϕbϕc+

1

12

(∑α

|α·ϕ|3+∑w

|w ·ϕ|3),

where α are the simple roots of g, and w are the weights of therepresentation R under which the hypermultiplets transform,





The term dabc is the cubic Casimir invariant of the Lie algebra. Amongsimple Lie algebras, this term is non-vanishing only for SU(N) with N > 2.The cubic absolute values generate singularities corresponding tohyperplanes along which new massless particles are generated.

A well defined phase of the Coulomb branch of the theory should be aregion in which α · ϕ and w · ϕ are taking well defined signs. We can fixthe sign of α · ϕ by requiring ϕ to be in the fundamental Weyl chamber.

Then the condition w · ϕ 6= 0 for all w ∈ R means that we are inside asubchamber defined inside the Weyl chamber.





Outline



3 Known Results






The incidence geometry of this type of arrangement is known forN = 3,N = 4,N = 5by Mboyo Esole, Shu-Heng Shao, Shing-Tung Yau (All from Harvard)

• Singularities and Gauge Theory Phases (preprint)

• Singularities and Gauge Theory Phases II (preprint)

Available on the math archives http://arxiv.org/





N=3

The Weyl chamber is spanned by the two vectors µ1 and µ2, and isdivided by the interior wall Ww2 into two subchambers C±. The interiorwall Ww2 is the Higgs branch root where matter fields become massless.The two boundary walls are the lines generated by µ1 and µ2 where theW -bosons become massless.





N=4

The three interior walls are W+, W 0, W− where some matter fieldsbecome massless. The Coulomb branch is partitioned into foursubchambers C±± by the three interior walls, which further intersect at theline L lying at the bottom of the Weyl chamber. The three boundary wallsare spanned by any pair of the three µi ’s.





N=5

There are nine interior walls intersecting at the following nine planes:

P1 = w53 ∩ w10

4 ∩ w106 : φ1 − φ4 = φ2 − φ3 = 0,

P2 = w52 ∩ w5

3 ∩ w105 : φ2 = −φ1 + 2φ3 + φ4 = 0,

P3 = w53 ∩ w5

4 ∩ w108 : φ3 = −φ1 − 2φ2 + φ4 = 0,

P4 = w103 ∩ w10

6 : φ1 = φ2 − φ3 + 2φ4 = 0,

P5 = w104 ∩ w10

7 : φ1 = φ2 − φ3 − 3φ4 = 0,

P6 = w106 ∩ w10

8 : φ2 = −2φ1 − φ3 + 2φ4 = 0,

P7 = w105 ∩ w10

6 : φ3 = −2φ1 + φ2 + 2φ4 = 0,

P8 = w103 ∩ w10

4 : φ4 = 3φ1 + φ2 − φ3 = 0,

P9 = w106 ∩ w10

7 : φ4 = −2φ1 + φ2 − φ3 = 0.

(1)





N=5

The nine planes further intersect along the following four lines :

L1 = P1 ∩ P2 ∩ P3 ∩ P6 ∩ P7 : φ2 = φ3 = φ1 − φ4 = 0,

L2 = P1 ∩ P4 ∩ P5 ∩ P8 ∩ P9 : φ1 = φ4 = φ2 − φ3 = 0,

L3 = P4 ∩ P6 : φ1 = φ2 = −φ3 + 2φ4 = 0,

L4 = P7 ∩ P9 : φ3 = φ4 = −2φ1 + φ2 = 0.

(2)

Finally, the four lines intersect at the origin of the Weyl chamber :

O = L1 ∩ L2 ∩ L3 ∩ L4. (3)





N=5

Figure: Each circle represents a subchamber and each edge corresponds to acommon interior wall between two adjacent subchambers. Physically, the interiorwalls labeled by a weight wi (the lines in the figure) correspond to the Higgsbranch roots where the matter fields with weight wi become massless.





Weierstrass model g R Polynomial Pg,R(t)

Is2 A1 2 1 + t

Is3 A2 3 1 + t + 2t2

Is4 A3 4⊕ 6 1 + t + 3t2 + 4t3

Is5 A4 5⊕ 10 1 + 4t + 9t2 + 9t3 + 12t4

GOAL: Find

Pg,R(t) = n0 + n1t + n2t2 + · · ·+ nr t

r .

We always have n0 = 1 since all the interior walls intersect at a uniquepoint: the origin of the Weyl chamber. The integer nr is the number ofsubchambers partitioned by the interior walls from the representation Rand nr−1 is the number of interior walls.





Outline



3 Known Results






Observations:

• Previous methods involve solving equations do not scale well when N islarge.

• Geometrically nr counts the number of subchambers (faces of dimensionr) while ni for i < r count the number of flats (intersections ofhyperplanes) of dimension i .

Strategy:• Embed SU(N) in GL(N), the set of invertible N × N-matrices ( TheSimplicial geometry is simpler )

• Find generating polynomials for computing the number of k-faces andk-flats.

• Restrict these polynomials to SU(N) to find the desired generatingfunction.





x2 = 0

x1 = 0

x1 + x2 = 0

x1 − x2 = 0

W

C4

C3

C2C1

Figure: For GL(2), we have exactly 4 subchambers, 5 half-lines and the origin.There are one 2-flat (W ) , three 1-flats (the interior walls), and one 0-flat.





The key object is the following poset that encodes the incidence geometry:

(−−) (+−) (++)

(0−) (+0)

Figure: E2.

The extreme rays of the polygonal cone are the vertices of the poset andthe number of k-faces in the arrangement corresponds to the number ofchains of length k in the poset and vice versa. We see that there are 4chains of length two, 5 chains of length one , and of course one chain oflength zero, the origin.





Theorem (E-J-N)

Face count polynomial for GLn:

P0 = 1,

Pn(x) = (1 + x)Pn−1(x) +n∑

i=1

xPn−i (x).

P1(x) = (1 + x)P0(x) + xP0(x) = 1 + 2xP2(x) = (1 + x)(1 + 2x) + x(2 + 2x) = 1 + 5x + 4x2





With some effort (Mboyo) one obtains:

Pn(x) =Sn(x)

2√

x(1 + x)

where

Sn(x) =(√

x(1 + x)− x)(

1 + x −√

x(1 + x))n

+(x +

√x(1 + x)

)(1 + x +

√x(1 + x)

)nThe Flat count is more subtle. We have obtained similar generatingfunctions for SU(N) also.





In Memory ofMy Mother

Suze Auguste10/21/1934 - 3/10/2015


Documents

Weyl Subchambers and Hyperplane Arrangements - …math.bu.edu/people/ep/AFRAMATH15/KeynoteAddress.pdfThe University of Massachusetts Boston AFRAMATH15 Keynote Address 4/25/2015 Boston