n=2 supersymmetric gauge theory and mock theta...

N = 2 supersymmetric gauge theory and Mock theta functions

N = 2 supersymmetric gauge theory and Mocktheta functions

Andreas MalmendierGTP Seminar

(joint work with Ken Ono)

November 7, 2008


q-series in differential topology

“Theorem” (M-Ono)The following q-series is a topological invariant

M(q8) := q−1∞∑

n=0

(−1)n+1q8(n+1)2∏n

k=1(1− q16k−8)∏n+1k=1(1 + q16k−8)2

= −q7 + 2q15 − 3q23 + · · · .

Remark

This series appears in Ramanujan’s “Lost Notebook”.


Overview:

1 SU(2) and SO(3) gauge theory in mathematics/physics

2 The Coulomb branch and the u-plane integral

3 SO(3)-Donaldson invariants for CP2 and Mock thetafunctions


Setup

(X , g): four-dim., compact, smooth, simply connectedmanifold w/ Riemannian metric g ,Hodge star ∗ : Λp(X ) → Λ4−p(X ) with ∗2 = (−1)p;

Simplest example of a manifold which is not of simple type:X = CP2 w/ the Fubini-Study metric g ,g is Kahler, has positive scalar curvature,self-dual Kahler form ω ∈ H2(X ) = H2+(X ),H = PD(ω) ∈ H2(X );

ω determines homology orientation:any element Σ ∈ H2(X ) is given by S =

∫Σ ω ∈ Z;


SU(2) gauge theory

P → X is SU(2)-principal bundles, classified by second Chernclass c2(P)[X ] = k;

A ∈ A connection on P, A∗ the subset of irreducibleconnections, gauge group G;

c2(P)[X ] = 18π2

∫X tr(FA ∧ FA) = 1

8π2

(||F−A ||2 − ||F

+A ||2

);

anti-selfdual instanton = connection A ∈ A∗ w/ ∗FA = −FA;

moduli space of solutions Mk ⊂ A∗/G;Mk is a smooth and oriented manifold of dimension2dk = 8k − 3(1 + b+

2 ) = 8k − 6;


Universal bundle construction

the universal bundle is the SU(2)-principal bundle:

L = (P × A∗)/G → X × A∗/G;

L has a natural connection D with curvatureF ∈ Λ2(X ×M)⊗ su(2);

p1(L) = − 1

8π2tr(F ∧ F) =

∑i

βi ⊗ γi ,βi ∈ H∗(X , Q)γi ∈ H∗(M, Q)

,

µ(α) = p1(L)/α =

(∫α

βi

)γi , α ∈ H∗(X ) ;

for X = CP2, two interesting classes:p ∈ H0(X ), H ∈ H2(X ): µ(p) ∈ H4(M), µ(H) ∈ H2(M);


Donaldson invariants

Donaldson invariants are the linear function:

Φk : Sym∗

(H0(X ; Z)⊕ H2(X ; Z)

)→ Q

Φk(pm,Σn) =

∫Mk

µ(p)mµ(Σ)n ;

invariants can be assembled in a generating function

X = CP2 : Z =∞∑

k=1

∞∑m,n=0

pm

m!

Sn

n!Φk

(pm, Hn

);


Results for X = CP2

Ellingsrud and Gottsche computed (hard work) the Donaldsoninvariants for CP2 of degree smaller or equal to 50:

Theorem (Ellingsrud and Gottsche, 1998)

(k, dk) Z(p,S) =

(1, 1) −32S

(2, 5) S5 − p S3 − 138 p2 S

(3, 9) 3 S9 + 154 p S7 − 11

16p2 S5 − 14164 p3 S3 − 879

256p4 S

(4, 13) 54 S13 + 24 pS11 + 1598 p2S9 + 51

16p3S7 − 459128p4S5

−1515256 p5S3 − 36675

4096 p6S...

...


SO(3) gauge theory

V → X is SO(3)-principal bundles, classified by firstPontryagin class p1(V )[X ] = −l and w2(V );

p1(V )[X ] = − 18π2

∫X tr(FA ∧ FA),

moduli space of asd instantons Ml ⊂ A∗/G;

Ml is a smooth and oriented manifold of dimension2dl = 2l − 3(1 + b+

2 );

if w2(V ) = 0 then V = Ad(P) = P ×SU(2) so(3)and p1(AdP) = −4c2(P) (from trace identity);

for moduli spaces which do not arise from SU(2)-bundles takep1(V )[X ] ≡ w2(V )2[X ] mod 4,for X = CP2: w2

2 (V )[X ] ≡ 1 mod 4 ⇒ l = 3, 7, 11, . . .


Reducible connections

V is a reducible: V = L⊕ ε w/ FA = FA ⊗ iσ3,ε: trivial oriented real line bundle,L: line bundle w/ c1(L) = 1

2πFA:

p1(V )[X ] = − 1

8π2

∫X

tr(FA ∧ FA) =1

4π2

∫X

FA ∧ FA = c21 (L)

for stable classes: w2(V ) = w2(L) ≡ c1(L) mod 2⇒ p1(V )[X ] ≡ w2

2 (V )[X ] mod 4;

for X = CP2:

w2(V ) ≡ 0 mod 2 : c1(L) = 2n ωw2(V ) ≡ ω mod 2 : c1(L) = (2n + 1)ω


Results for X = CP2

Theorem (Kotschick and Lisca, 1995)

(l , dl) Zw2=1(p,S) =

(3, 0) −1

(7, 4) −3 S4 − 5 p S2 − 19 p2

(11, 8) −232 S8 − 152 p S6 − 136 p2 S4 − 184 p3 S2 − 680 p4

(15, 12) −69525 S12 − 26907 p S10 − 12853 p2 S8 − 7803 p3 S6

−6357 p4 S4 − 8155 p5 S2 − 29557 p6

......

Theorem (Zagier and Gottsche, 1998)

Explicit formula for all coefficients in terms of Jacobi ϑ-functions.


Physics interpretation: Witten, 1988

∃ twisted N = 2 supersymmetric topological SU(2) orSO(3)-Yang-Mills theory on M;

bosonic fields = differential forms, values in Ad(P) = P ×G g;A connection on V → X , Φ ∈ Γ(X ,Ad(P));

twisted = supersymmetry charge Q w/ Q2 = 0 is a scalar,fermionic BRST-operator;Q: exterior derivative on X × A∗/G ⊃ X ×M;

action: S = p1(V ) + {Q, . . . };observables = Q-cohomology classes of H∗(M)(hence topological invariants)

expectation values = cup-product to the top-degree evaluatedon fundamental class;


Physics interpretation: Seiberg, Witten, 1994

moduli space of vacua of TQFT

= Coulomb branch + Seiberg-Witten branchZ(p,S) = Zu(p,S) + ZSW (p,S)

Coulomb branch, SW branch: moduli spaces of simplerphysical theories;

Seiberg-Witten branch: moduli space of solutions to(mathematical) SW-equations / gauge transformations;

for CP2 or CP2 with a small number of points blown up:positive scalar curvature + maximum’s principle⇒ SW-invariants vanish, ZSW (p,S) = 0;


Low energy effective field theory

vev 〈trΦ2〉 = 2u breaks SU(2) or SO(3) → U(1);

Coulomb branch = moduli space of vacua of a N = 2supersymmetric U(1)-gauge theory;

classical bosonic fields: connection A on a line bundle L → Xand a complex scalar field ϕ;

discrete modulus: 〈c1(L) ∪ ω, [X ]〉 = 2k + w2(V ),continuous modulus: a the minimum of the scalar field,complex gauge coupling: τ ∈ H/Γ0(4),Γ0(4) duality group, duality transformation, e.g., τ 7→ τ + 4;

bosonic action:

Sbos =

∫X

i Reτ16π

FA ∧ FA︸︷︷︸=πi Reτ

4c21 (L)

+i w2(X )∧FA+Imτ

16πFA∧∗FA+Imτ 〈dϕ, dϕ〉


Semi-classical approximation

path integral for a supersymmetric action S can be definedwith mathematical rigor by the stationary phase approx.;

quadratic approximation S (2) around a critical point isHessian; S (2) determines a free field theory in the collectedvariations of the Bose fields Φ and Fermi fields Ψ (=coordinates of the normal bundle at the critical points):

S (2) =

∫X

volM

( ⟨Φ,∆(k,a)Φ

⟩+

⟨Ψ, i /D(k,a)Ψ

⟩ ),

where ∆ is a family of second-order, elliptic operators and Da family of skew-symmetric first-order operator;


Semi-classical path integral

functional integration over the fluctuations isinfinite-dimensional Gaussian integral, define∫ [

DΦ DΨ]

e−S(2)to be

pfaff /Dk,a√det ∆k,a

;

to integrate this section over the moduli space, check:1) line bundle is flat = vanishing of the local anomaly;2) no monodromy = vanishing of the global anomaly;3) line bundle has canonical trivialization = ratio ofdeterminants is function on the moduli space;


Semi-classical path integral

integrate over continuous moduli, sum over the discretemoduli to obtain the semi-classical approximation of thepartition function:

Zu =∑k

∫d2a e−S(0)(k,a)

∫ [DΦ DΨ

]e−S(2)

=∑k

∫d2a e−S(0)(k,a) pfaff /Dk,a√

det ∆k,a

physical considerations guarantee that the semi-classicalapproximation is exact;


Semi-classical generating function

we are not only interested in the partition function, but thegenerating function with the inclusion of observables:

Donaldson theory → Low Energy Effective Field Theoryµ(p) 7→ 2u = 〈trΦ2〉µ(Σ) 7→ T (u)

path integral:

Zu(p,S) =∑2 H2(M;Z)+w2(V )

∫da ∧ d a e2 p u+S2 bT (u) e−S(0) pfaff /Dk, a√

det ∆k, a


Coulomb branch

Coulomb branch: rational Weierstrass elliptic surfaces,holomorphic fibration π : Z → CP1 where [u : 1] ∈ CP1:

Eu : y2 = 4 x3 − g2(u) x − g3(u)

Discriminant: ∆ = g32 − 27g2

3 (smoothness cond. 6= 0);

Kodaira (1950) classified singular fibers: they only depend onthe vanishing order of g2, g3,∆;

u = ±1 node I1 0 0 1monopole/dyon becomes massless

u = ∞ 9 lines meeting in D8 I ∗4 2 3 10weak coupling limit


The u-plane

analytical marking du ∧ dxy of elliptic surface:

period integrals:∫A-cycle du ∧ dx

y = ωωωdu;

period integrals:∫A-cycle λSW = a, da

du = ωωω;

elliptic fiber Eu is C/〈ωωω, τ ωωω〉;effective gauge coupling depends holomorphically on scalarcomponent of N = 2 vector multiplet, τ = τ(a);

Oguiso, Shidoa, 1991 classified the Mordell-Weil groups of allrational elliptic surfaces; rational elliptic surface is # 64 anduniversal curve for modular group Γ0(4);


The u-plane

J−−−→

Figure: The mapping from H/Γ0(4) (with the six copies of thefundamental domain) to the u-plane (with the points u = 1 (�) andu = −1 (©) removed)


Photon partition function

For X = CP2 we have

w2(X ) ≡ ω , w2(V ) ≡ w2 ω ,

c1(L) =1

2πFA = (2k + w2) ω ,

S (0) =

∫X

i Reτ16π

FA ∧ FA + i w2(X ) ∧ FA +Imτ

16πFA ∧ ∗FA ,

and ∑k∈Z

e−S(0)(k,a) pfaff /Dk,a√det ∆k,a

= C∑k∈Z

e−S(0)(k,a)

[∫H

(F+

A +S

Imτ ωωωω

)]


Photon partition function

we obtain∑k∈Z

e−iπτ(2k+w2)2+iπ(k+

w22 )

[(k +

w2

2

)+

S

Imτ ωωω

]

=

{iη3(τ) if w2 = 1

SImτ ωωω ϑ4(τ) if w2 = 0

.


The u-plane integral

Theorem (Moore and Witten, 1997)

For X = CP2, it follows

Zu(p,S) =

∫ reg

H/Γ0(4)

dτ ∧ d τ

Imτ32−w2

e2 p u+S2 bT (u)

du

dτ

∆18

ωωω32−w2

{η3(τ) if w2 = 1

S ϑ4(τ) if w2 = 0.

with

ϑ4(τ) =∑∞

n=−∞(−1)n qn2

2 , η(τ) = q124

∏n≥1 (1− qn) .


Remarks about the u-plane integral

integrand is modular invariant under Γ0(4);

integrand has singularities at nodes at cusps u = ±1,∞,regularization procedure must be applied,the cusps contributions are the only contributions to Zu:

Zu(p,S) = Zu(p,S) |u=−1 +Zu(p,S) |u=1 +Zu(p,S) |u=∞

Renormalization group action:xΛ = Λ2 x , yΛ = Λ3 y , g2,Λ = Λ4 g2, g3,Λ = Λ6 g3,uΛ = Λ2 u, aΛ = Λ a,

T =(

∂uΛ∂Λ

)Λ = 1aΛ = const


Results for X = CP2

Proposition (M-Ono)

For X = CP2 and w2 = 1, it follows:

Zu(p,S) |u=±1 = 0

Zu(p,S) |u=∞ = Z(p,S)


Proof

method: integration by parts using nonholomorphic modular formof weight (1

2 , 0) for Γ0(4):

near cusp u = ∞, Imτ →∞ of type I ∗4 , τ = x + iy :

Zu ∼∫

dτ ∧ d τ∂

∂τ

(. . .

)= 2 i

∫ 4

0dx

(. . .

);

integration involves η(τ)3 as divergence ⇒ mock theta Q(τ);

prove exponential convergence after∫

dx , then compute

Zu = limy→∞

2 i

∫ 4

0dx

(. . .

)=

∑m,n≥0

DDDm,2npmS2n

m!(2n)!

Gymnastics with heat operators and differential operators


Mock theta function

For η3(τ) (modular form of weight 32) and q = e2πiτ , τ = x + iy ,

we look for solutions of

∂

∂τ

[Q+(τ) + Q−(τ, y)

]=

1√

yη3(τ) .

Q+(τ): mock modular form of weight 12 ,

holomorphic but not quite modular;

Q−(τ): correction term, non-holomorphic;has only neqative Fourier modes ∼ q−n;each Fourier coefficient has exponential in y →∞;

Q+(τ) + Q−(τ) is modular form for Γ(2) ∩ Γ0(4);


Mock theta function

Q+(τ) = 1

q18

∑α≥0 Hα q

α2 ;

Q−(τ) = 1

q18

∑α≥0 H−α(y) q−α w/ limy→∞H−α(y)q−α = 0;

u-plane integral:

Zu ∼ limy→∞

∫ 4

0dx

∑β≥0

Cβ qβ4

[Q+(τ) + Q−(τ, y)

]since

∫ 40 dx q

α4 = 4 δα,0 it follows:

Zu ∼ limy→∞

Coeffq0

∑β≥0

Cβ qβ4

[Q+(τ) + Q−(τ, y)

]= 4Coeffq0

∑β≥0

Cβ qβ4 Q+(τ)


Coefficients Dm,2n

A gory and lengthy calculation gives:

DDDm,2n =n∑

k=0

k∑j=0

(−1)k+j+1

2n−2j−1 3n−j

(2n)!

(n − k)! j! (k − j)!

Γ(

12

)Γ

(j + 1

2

)× Coefq0

[ϑ9

4(τ)[ϑ4

2(τ) + ϑ43(τ)

]m+n−k

[ϑ2(τ) ϑ3(τ)]2m+2n+3E k−j

2

(q

d

dq

)j

Q+

].


What does this have to do with M(q)?

Q+(q) + Q−(q) compatible with cusp width at singularpoints:

u Esing Q+

∞ I ∗4 Q+(τ) = q−18

(1 + 28 q

12 + 39 q + 196 q

32 + 161 q2 + · · ·

)±1 I1 Q+(τS) = 1√

−i τsQ+

(− 1

τs

)= q−

18

(52 + 111

2 q + 4132 q2 + · · ·

)Q+(q8) is given by Ramanujan’s mock theta function M(q):

4M(q8)+28η(16τ)8

η(8τ)7+

3η(8τ)5

2η(16τ)4+

48η(32τ)8

η(8τ)3η(16τ)4− η(8τ)5

2η(16τ)4


The end game!

We can now show that

ZZZ (p,S) = ZZZu(p,S) |u=∞ .

For every m, n, just show that zero is the constant of

n∑k=0

k∑j=0

(−1)j(2n)!

(n − k)! j! (k − j)!

ϑ84

[ϑ4

2 + ϑ43

]m

[ϑ2 ϑ3]2m+2n+3

E k−j2

×[(−1)n+1

2k−3 3k

(n − k)!

(2n − 2k)!

[ϑ4

2 + ϑ43

]jF2(n−k)

− (−1)k+1

2n−2j−1 3n−j

Γ(

12

)Γ

(j + 1

2

)ϑ4(τ)[ϑ4

2(τ) + ϑ43

]n−k(

qd

dq

)j

Q

].


Freeman Dyson, 1987

“My dream is that I will live to see the day when our youngphysicists, struggling to bring the predictions of superstring theoryinto correspondence with the facts of nature, will be led to enlargetheir analytic machinery to include mock theta-functions. . . ”

⇑q-series

n=2 supersymmetric gauge theory and mock theta...

Documents