Arithmetic and physics in discrete algebraic geometry · Arithmetic and physics in discrete algebraic geometry Tam as Hausel ... March 2012 Fejes-T oth Lecture University of Calgary

Arithmetic and physics indiscrete algebraic geometry

Tamas Hausel

Royal Society URF at University of Oxfordhttp://www.maths.ox.ac.uk/∼hausel/talks.html

March 2012Fejes-Toth Lecture

University of Calgary

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Number theory - Geometry - Physics

many historical examples

for example in the work of Minkowski (1864-1909)

study of lattices in Rn ⇔ algebraic number theory inMinkowski’s ”Geometrie der Zahlen” (1896)

(Minkowski 1907) introduces the Minkowski spacetime R3,1

”The views of space and time which I wish to lay before youhave sprung from the soil of experimental physics, and thereinlies their strength. They are radical. Henceforth space byitself, and time by itself, are doomed to fade away into mereshadows, and only a kind of union of the two will preserve anindependent reality.”

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Frobenius on group characters

Frobenius, F.G.: Uber Gruppencharaktere (1896):

” I shall develop the concept [of character for arbitrary finitegroups] here in the belief that through its introduction, grouptheory will be substantially enriched. ”

After proving the orthogonality relations (k = 2; g = 0 below)Frobenius’ first application was the g = 0 case of

Theorem (Frobenius 1896, Hurwitz 1902, Freed-Quinn 1993,...)

Let C1, . . . , Ck ⊂ G be conjugacy classes in a finite group G then

#{ci ∈ Ci , aj , bj ∈ G|c1c2 · · · ck [a1, b1] . . . [ag , bg ] = 1} =

=∑

χ∈Irr(G)|G|2g−1

χ(1)2g−2

∏ki=1

χ(Ci )|Ci |χ(1)

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Arithmetic harmonic analysis for GLn(Fq)

G = GLn(Fq)

character table of GLn(Fq) was calculated by(Jordan, Schur, 1907) for n = 2(Steinberg, 1951) for n = 3, 4(Green, 1955) for all n

(Hausel–Letellier–Villegas, 2008) calculated explicitly(using Macdonald polynomials)

∑χ∈Irr(GLn(Fq))

|G|2g−1

χ(1)2g−2

k∏i=1

χ(Ci )|Ci |χ(1)

=

#{ci ∈ Ci , aj , bj ∈ G|c1c2 · · · ck [a1, b1] . . . [ag , bg ] = 1},

where Ci ⊂ GLn(Fq) are generic semisimple conjugacy classes

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Character table of GL2(Fq)

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Example GL2(Fq)

G = GL2(Fq), k = 1 C1 = {−1} ⊂ GL2(Fq)

1

|PGL2(Fq)|(q − 1)2g#{aj , bj ∈ G|[a1, b1] . . . [ag , bg ] = −1} =

1

|PGL2(Fq)|(q − 1)2g

∑χ∈Irr(GL2(Fq))

|G|2g−1

χ(1)2g−2

χ(−1)

χ(1)=

(q2−1)2g−2+q2g−2(q2−1)2g−2−1

2q2g−2(q−1)2g−2−1

2q2g−2(q+1)2g−2.

e.g. g = 0 gives 0 when g = 1 it gives 1

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Example SLn(Fq)

character table for SL2(Fq) by (Jordan 1907), (Schur 1907). . . for SLn(Fq) (Bonnafe 2006), (Shoji 2006)

for G = SL2(Fq), k = 1, C1 = {−1} ⊂ SL2(Fq)

1

|PGL2(Fq)|#{aj , bj ∈ G|[a1, b1] . . . [ag , bg ] = −1} =

1

|PGL2(Fq)|∑

χ∈Irr(SL2(Fq))

|G|2g−1

χ(1)2g−2

χ(−1)

χ(1)=

(q2−1)2g−2+q2g−2(q2−1)2g−2−1

2q2g−2(q−1)2g−2−1

2q2g−2(q+1)2g−2+

+(22g − 1)q2g−2(

(q−1)2g−2−(q+1)2g−2

2

).

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Character table of SL2(Fq)

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Character varieties for GLn and SLn

fix integers n > 1 and d such that (n, d) = 1 andζn primitive nth root of unity; coefficients in C or Fq or Z[ζn]

the GLn-character variety:

M := {(Ai ,Bi )i=1..g ∈ GL2gn | [A1,B1] . . . [Ag ,Bg ] = ζd

n In}//PGLn

non-singular, affine

the SLn-character variety:

M := {(Ai ,Bi )i=1..g ∈ SL2gn | [A1,B1] . . . [Ag ,Bg ] = ζd

n In}//PGLn

non-singular, affine

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Character variety for PGLn

(GL1)2g acts on M

Γ ∼= (Zn)2g ⊂ (GL1)2g acts on M

the PGLn-character variety: M := M/Γ ∼=M/(GL1)2g is anaffine orbifold

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Character variety over a finite field

Frobenius’ character formula

for GLn(Fq) ; #M(Fq) =∑

χ∈Irr(GLn(Fq))

|GLn(Fq)|2g−2

χ(1)2g−2

χ(ξdn In)

χ(1)

for SLn(Fq) ; #M(Fq) =∑

χ∈Irr(SLn(Fq))

|SLn(Fq)|2g−2

χ(1)2g−2

χ(ξdn In)

χ(1)

for PGLn(Fq) ; #M(Fq) =#M(Fq)(q−1)2g

in all these cases the count is a polynomial in q

(Katz 2008) ; if for a variety #(X (Fq)) ∈ Z[q] is apolynomial, then

#(X (Fq)) = E (X ; q) =∑

dim(Wi/Wi−1(Hkc (X/C; Q)))(−1)k q

i2

is the Serre polynomial

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Topological Mirror Test

X non-singular algebraic variety/C, Γ finite group acting on X

define stringy Serre-polynomial of the orbifold X/Γ by

E Bst (X/Γ; q) =

∑[γ]∈[Γ]

E (X γ/C (γ), LBγ ; q)(q)F (γ)

motivating: (Kontsevich 1995) for Y → X/Γ crepant ;

Est(X/Γ; q) = E (Y ; q)

recall that the PGLn-character variety M = M/Γ is anorbifold with Γ ∼= (Zn)2g

(Hausel–Thaddeus 2001, Hausel–Villegas 2004) charactervarieties for Langlands dual groups are ”mirror symmetric”

Conjecture ( Hausel–Villegas 2004, Topological Mirror Test)

E (M; q) = E Bst (M; q).

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Example

when n = 2

E (M; q)− E (M; q) = (22g − 1)q2g−2

((q − 1)2g−2 − (q + 1)2g−2

2

)

Mγ can be identified with (C×)2g−2 ;

E (Mγ/Γ, LB,γ ; q) =(q − 1)2g−2 − (q + 1)2g−2

2

implies Topological Mirror Test when n = 2

similar argument settles n = p

ongoing work (Hausel-Mereb-Villegas 2012) handles all n; character formulae reminiscent of the fundamental lemmafor SLn in the Langlands program

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Mirror symmetry for Langlands dual Hitchin systems

by the non-Abelian Hodge theorem M diff' MDol with moduliHiggs bundles ; Hitchin map χ :MDol → A

M

χ AAA

AAAA

A M

χ~~}}}}

}}}

A; SYZ construction for mirror symmetric Calabi-Yau’s; M and M could be considered mirror symmetric!; can be deduced from (Kapustin-Witten 2006) S-duality

Topological Mirror Test is the agreement of Hodge numbers; relative of Ngo’s geometric fundamental lemma

for n = 2 we proved Topological Mirror Test from certainpatterns in Irr(SL2(Fq)) vs. Irr(GL2(Fq))due to (Schur 1907) and (Jordan 1907)

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Jordan’s character table of PGL2(Fq)

JORDAN: Group-Characters of Various Types of Linear Groups. 403

We define x = a + U as the invariant of the substitution (a,' ).

The substitution R = (0' p where p is a primitive root of the GF[2f'],

generates a cyclic group of order s- 1. RI is conjugate to R-4 and is always

distinct from it. Hence the powers of R represent s 2 classes, each contain-

ing s (e + 1) substitutions.

Let a be a primitive root of as+1 = 1. The substitution S= (O' a) is of

period s + 1. Sb is conjugate to 8-b and is distinct fromi it; thus the powers of S represent a classes, each containing s (s-1) substitutions.

The substitution T= (O' 1) of period two and invariant zero is one of ss-i

conjugate substitutions. We denote this class by (0) and the identity by (X). The total number of classes of conjugate substitutions is s + 1. Below is given the table of group-characters.

N 1 1 2n-1_ 1 2n-1

& 1 2n 2n + 1 2n_ 1

Xo 1 0 1 -1

.t(Ra) 1 ra + -a 0

W(Sb) 1 -1 0 -tb t

where r and t are the roots (except unity) of the respective equations r8-l 1, =+1- 1. As before e f.

II.

The Binary Linear Fractional Group F1 in the GF[pn], p>2, of all Determinants not Zero.

The order of F1 is h-= _- 1). The substitutions will be supposed written in the normal form, i. e., of determinant unity or a particular not-square in the GF[pn].

We shall denote the determinant aS-l3y of the substitution V= (a' )

by r, where X=1, or v a particular not-square; and we shall call i i (a + ^) the invariant of V.

JORDAN: Group- Characters of Various Types of Linear Groups. 405

primitive root of the GF [p2], and consequently S is of period s + 1. With

the exception of S8+21 which is conjugate only to itself, S is conjugate to 8-

and is distinct from it. We have therefore 8+1 classes represented by the powers

of S, each containing s (s - 1) substitutions, except SY , the class represented by which contains Is (s - 1) substitutions.

The classes represented by the powers of R (S) are characterized by the property that x 2 -t is a square (not-square) in the GF [p4], where t = v or 1 according as the index is odd or even.

The substitution

MT (0 1) y a mark t O of the GF[p_],

of invariant i 1 and determiniant unity, is one of S2 - 1 conjugate substitutions forming a class (y).

The total number of classes of conjugate substitutions is s + 2. Below is given the table of group-characters.

N | 1 1 1 S-3 s-i 2 2

^ 81 1 s s s+1 s-1

4 01 1 0 0 1 -1

X (R2a) 1 1 1 1 r2a + r-2a 0

% (S 1 1 -1 -1 0 tlb t-2b

z (3R2a+1) 1 -1 1 -1 r2a+l + r-(2a+l) 0

X; (SZb+l) | - 1 -1 1 0 -t2b+1 t-(2b+l)

where r and t are the roots (except 4i 1) of r8- -- 1 and t8+" = 1 respectively. As before e = f.

MICHIGAN COLLEGE OF MINES, HoueHTON, MICH.

52

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Schur’s character table of SL2(Fq)

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