3D Partitions and Geometry

Amer Iqbal

1) Motivation: Understanding Moduli Spaces

2) Toric geometry:

3) 1D-2D-3D partitions

4) Geometry and partitions

To Study Moduli Space of Curves living in a Complex Manifold. A Moduli Space is the space of solutions of a given problem.

df (x) Example: = f (x) dx f (x) = Ae x If f (x) is taken to be a real valued function then A ∈ (− ∞ ,+ ∞ ). In this case the moduli space of solutions is IR.

Example: What is the moduli space of quadratic polynomials with real coefficients and real zeros. c P(x) = x2 + bx + c − + 2 − b b 4c b x± = 2

x± are real if b2 - 4c ≥ 0.

P(x) = (x − A)(x − B) A, B ∈ IR. Since interchanging A,B makes no difference the moduli space of quaratic polynomials with real roots is (IR × IR)/ ~

Where ~ is the Z2 action exchanging A and B. The moduli Spaces are the same i.e. there exists a map from one to the other such that the boundaries are mapped to each other. For this reason we are interested in invariants associated with the moduli spaces.

= 1 M S χ = ( M ) 0 χ (K) = ∑ (− 1)i dim H i (K) i = 0...dim K

= CP1 M 1 = CP1 × CP1 / ~= CP2 M 2 χ ( ) = 2 M 1 χ = ( M 2) 3

Toric geometry

θ X = | X | ei | X |2 ≥ 0

C 2 → (X ,Y ) X = | X | eiθ ,Y = |Y | eiϕ

| Y |2 ≥ 0

| X |2 ≥ 0

C 3,(X ,Y, Z) X = | X | eiθ ,Y = |Y | eiϕ , Z = | Z | eiψ

| Z |2 ≥ 0

| X |2 ≥ 0 |Y |2 ≥ 0

More complicated geometries CP 2 ,[X ,Y, Z] = [λ X , λ Y, λ Z] 2 Also CP 2 = S 3 / S1 | B | S 3 = {| A |2 + | B |2 = 1, A, B ∈ C} | A |2

2 | B |2 | B |

| A |2 | A |2

P1 × P1 h ,h ∈ H (P1 × P1, Z) 1 2 2 + 2h1 2h2

− × O 1 ( 2) C P

K

The of the Euler characteristic of the moduli space of curves.

2 g − 2 3 ∞ ∑  ∫ c g − 1 1 g M Z = ∏ = e g = −  − n n q e n= 0 (1 q )

This is also the partition function of the U(1) non-commuative gauge theory on C^3.

x3 x2 x2 x 1 x1 1 x1

∞ 1 Z = ∑ q# of boxes = ∑ (# of configurations with n boxes) qn = 1D − configurations n= 0 1 q

= 1 + q + q2 + q3 + q4 + q5 + q6 + q7 + .....

y 1 x

y 2 y y 1 x 1 x x2

y 2 y y 1 1 x 1 x x2

y3 y 2 y 2 x y yx yx2 yx3 1 x x2 x3 ∞ ∞ 1 Z (q) = ∑ q#of boxes = ∑ P qk = ∏ 2D k − n configurations k = 0 n= 1 (1 q ) = 1+ q + 2q2 + 3q3+ 5q4 + 7q5 + 11q6 + 15q7 + ... = Pk #of configurations with k boxes = # of Young diagrams with k boxes

{1}

{1, x} {1, y} {1, z}

2 {1, x, z} {1, x, x } {1, x, y}

2 {1, z, z 2} {1, y, z} {1, y, y }

3D partitions

5 4 2 1 1 3 2 2 2 A plane partition or 3D partition of 22.

∞ ∞ 1 Z (q) = ∑ q#of boxes = ∑ M (k)qk = ∏ 3D − n n configurations k = 0 n= 1 (1 q )

= 1+ q + 3q2 + 6q3 + 13q4 + 24q5 + 48q6 + 86q7 + ......

M (k) = # of configurations with k boxes = # of 3D partitions with k boxes.

Product structure of the generating function of 4D partitions is not known!

Percy Alexander MacMahon

3 ∈ iθ (z1, z2 , z3 ) C = i zi | zi | e = 2 ≥ pi | zi | , pi 0. = ∧ = ∧ θ k i ∑ dzi d zi ∑ dpi d i i = 1,2,3 i = 1,2,3

C[x, y, z] = C 3

{1}= C[x, y, z]/{x, y, z}

{1, x}= C[x, y, z]/{1, x2 , y, z}

2 2 {1, x, z} = C[x, y, z]/{1, y, x , z }

For a geometry corresponding to C[x,y,z]/{monomials} ∞ 1 Z = ∏ = " ∑ qnumber of monomial" 3D − n n n= 0 (1 q ) monomials = ∑ qVolume geometries

Cut the partition by X=Y Plane.

5 4 2 1 1 3 2 2 2 A plane partition or 3D partition of 22.

= #of boxes on the right #boxes on the left = Z3D (q1,q2 ) ∑ q1 q2 configurations ∞ ∞ 1 = k1 / 2 k2 / 2 = ∑ M (k ,k )q q ∏ − 1 2 1 2 k − 1 2 k2 1 2 = = − 1 k1 ,k2 0 k1 ,k2 1 (1 q1 q2 )

= + 1/ 2 1/ 2 + 3/ 2 1/ 2 + + 1/ 2 3/ 2 + 5/ 2 1/ 2 + 2 + 3/ 2 3/ 2 + 2 + 1/ 2 5/ 2 + 1 (q1 q2 ) (q1 q2 q1q2 q1 q2 ) (q1 q2 q1 q2 2q1 q2 q1q2 q1 q2 ) ......

1 2 1 1 3 2 1 1 1 1

1 2 1 1 1 1 1 1 1 1

The idea is to interpret Z_3D as sum over geometries i.e., as a path integral. = iS (k ) = Z(M ) ∫ Dk e Z3D (q) 1 1 S(k) = ∫ k ∧ k ∧ k = Vol(M )  M  k ∈ H 1,1(M , ℜ ) M = C 3

 = string coupling constant M = 3 dimensional complex manifold = with c1(M ) 0 (i.e., M has a Ricci flat metric).

∧ = + k k0  k

∧ ∧ ∧ i 1 ∧ ∧ vol(M ) ∫ k k k 2 3! Z(M ) = e 6 ∑ q M ∧ k ∧ ∧ ∧ = q = ei k0 k 0 k 0 ∧ ∈ Ζ ∀ ∈ ∫ k , C H 2 (M , Z). C

Blowup is a local construction and locally can be described as follows: suppose that the ideal I is generated by f1,f2,..fk,then blowup of M is the closure of the graph in M × Pk − 1 ∧ = ∈ c M {(z,( f1, f2 ,.., fk )) | z M \ Z} = Z set of common zeroes of f1,f2 ,..,fk . Example: Consider the case I={x,y,z} corresponding to a single box. ∧ C 3 = {(w,(x : y : z) | w ∈ C 3 \ (0,0,0)} ⊂ C 3 × P2 ∧ C 3 is C 3 away from the origin. And the origin is replaced by a P2. The size of this P2 is  2.