Partition functions of N = 4 Yang-Mills and applications

Transcription

1 Partition functions of N = 4 Yang-Mills and applications Jan Manschot Universität Bonn & MPIM ISM, Puri December 20, 2012

2 Outline 1. Partition functions of topologically twisted N = 4 U(r) Yang-Mills theory on a four-manifold Motivation: S-duality in gauge theory Harder-Narasimhan recursion Explicit expressions and modularity 2. Application A 5d/2d/4d correspondence Based on: w. Haghighat & Vandoren

3 4-dim. Yang-Mills theory Connection and field strength: - Field strength: F = da + A A Ω 2 (S, g) - Action: S[A] = 1 iθ g 2 S Tr F F + Tr F F 8π 2 Electric and magnetic charge (g = u(1)): Q e = 1 F, Q m = 1 F 4π S 2 4π S 2 Solitonic monopoles (g = su(2)): F, φ F u(1) with monopole charge Mass: M a Q e + τ Q m, t Hooft (1974), Polyakov (1976) with τ = θ 2π + 4πi g 2

4 S-duality Electric-magnetic duality: Montonen, Olive (1977) (F, F ) ( F, F ) S : τ 1/τ a τa Translations: 1 8π 2 Tr F F Z T : τ τ + 1 is a symmetry of the theory S + T generate SL 2 (Z) S-duality group

5 Is the path integral invariant under SL 2 (Z)? Topologically twisted N = 4 Yang-Mills on S: Vafa, Witten (1994) h r (τ) = DADψDX e S[A,ψ,X ] S-duality h r (τ) transforms as a modular form: ( ) aτ + b h r (cτ + d) χ(s)/2 h r (τ), cτ + d ( a b c d ) SL 2 (Z)

6 Path integral generating function Path integral localizes on BPS-solutions: - minimize S[A], e.g. instantons: F inst = F inst - instanton number: κ(f ) = 1 8π 2 S Tr F F Z - e S[A] = q κ with q = e 2πiτ h r (τ) = κ Ω(κ) qκ + R(τ, τ) - Ω(κ): Euler number of inst. moduli space BPS-invariant - R(τ, τ): arises due to reducible ( connections: ) A1 0 e.g. r = 2 : A = 0 A 1 Verification of EM-duality for U(r = 1, 2) YM and complex surfaces Vafa, Witten (1994)

7 From YM theory to vector bundles Computation of h r (τ) for rational surfaces is possible using methods for complex surfaces Gottsche, Nakajima, Yoshioka,... Donaldson-Uhlenbeck-Yau theorem: { stable vector bundles ASD connections reducible connections Characterized by their Chern classes ch i H 2i (S, Z): ch 0 = r, ch 1 = i 2π Tr F, ch 2 = 1 8π 2 Tr F F

8 Gieseker stability Choose polarization (Kähler modulus) J C(S), and define the polynomial: p J (F ) := c 1(F ) J + 1 ( c1 (F ) 2 ) K S c 1 (F ) c 2 (F ) r(f ) r(f ) 2 Definition: A bundle F is Gieseker stable if for every subbundle F F, p J (F, n) p J (F, n) ( semi-stable p J (F, n) p J (F, n) ) Agrees with stability based on central charge in large volume limit: lim J Z(F, B + ij) Douglas (2000), Bridgeland (2002)

9 BPS-invariants Definition: Ω(Γ, w; J) := w dim C M J (Γ) w w 1 p(m J (Γ), w), where p(x, s) = 2 dim C (X ) i=0 b i s i with b i = dim H 2 (X, Z) and M(Γ) is the moduli space of semi-stable vector bundles.

10 Generating function Generating function: h r,c1 (z, τ; S, J) := c 2 Ω(Γ, w; J) q r rχ(s) 24 - discriminant: = 1 r (c 2 r 1 2r c1 2) - modular parameter: τ aτ+b cτ+d, q = e2πiτ - elliptic parameter: z z w = e2πiz Also define: cτ+d, f r,c1 (z, τ; S, J) := h r,c 1 (z, τ; S, J) h 1,0 (z, τ; S, J) r Isomorphism of M J (Γ): c 1 (E) c 1 (E) + k, k H 2 (S, rz), invariant

11 Rational surfaces Blow-up Σ1 : two points of view: Σl Blow-up: φ : Σ1 P2 Bott & Tu Rational ruled surface π : Σ1 P1 - H2 (Σ1, Z): generators C, f such that φ (C + f ) = H - Intersection numbers: C 2 = 1, f 2 = 0, C f = 1 - Ample cone: C (Σ1 ) = {m(c + f ) + nf ; m, n > 0}

12 Outline of computations n J ε,1 n W 1 W 2 W 3 O m O m 1. Determine invariants for suitable polarization 2. Wall-crossing 3. Blow-up/down

13 Harder-Narasimhan recursion I Crucial technique for computation of BPS invariants: Vector bundles on Riemann surfaces: Harder, Narasimhan (1975), Atiyah, Bott (1982) r-dim. A C 1 Quivers: Reineke (2003) d1 γ 13 d3 App. to BH s Denef (2002), M. Pioline, Sen ( ),... γ 12 γ 23 d2

14 Harder-Narasimhan recursion II Computation of Ω(Γ, w) proceeds in 2 steps: 1. Determine the cohomology of the space of all bundles BG (Γ): h r (z; C) = w r 2 (1 g) (1 + w 2r 1 ) 2g 1 w 2r Harder, Narasimhan (1975), Atiyah, Bott (1982) 2. Subtract the unstable bundles r 1 l=1 (1 + w 2l 1 ) 2g (1 w 2l ) 2

15 Harder-Narasimhan recursion III Definition: A Harder-Narasimhan filtration of a vector bundle F is a filtration 0 F 1 F 2 F l = F such that the quotients E i = F i /F i 1 are semi-stable and satisfy d i /r i > d i+1 /r i+1 for all i. If l 2 then F is unstable. Enumerate vector bundles with a given HN filtration: w P l i<j (r i d j r j d i ) Ω(Γ i, w) Leads to a recursion in Γ for gcd(r, d) = 1, e.g. for Γ = (2, 1): i=1 Ω(Γ, w) = h 2 (z; C) + w 2 g = 2 = Recursion solved by Zagier (1996), Reineke (2003) 1 w 4 h 1(z; C) 2 1 w w 1 w 5 + 4w 4 + 7w w w

16 Suitable polarization c 1 f 0 mod r: Ω(Γ, w; J ε,1 ) = 0 c 1 f = 0 mod r: Conjecture: M. (2011) The contribution to h r,c1 (z, τ; Σ l, J) with c 1 f = 0 mod r from the set of vector bundles whose restriction to the fibre is semi-stable is given by: H r (z, τ; f ) := i ( 1) r 1 η(τ) 2r 3 θ 1 (2z, τ) 2 θ 1 (4z, τ) 2... θ 1 ((2r 2)z, τ) 2 θ 1 (2rz, τ), with η(τ) = q 1 24 n=1 (1 qn ), θ 1 (z, τ) = r Z+ 1 ( 1) r q r2 2 w r Proven for r = 1 and 2 Gottsche (1990) & Yoshioka (1995) 2

17 Explicit expressions for Σ 1 Rank 1: Gottsche (1990) i h 1,c1 (z, τ; S) = θ 1 (z, τ) η(τ) b 2 (S) 1 Rank 2: w 2α f 2,βC αf (z, τ; Σ 1, J m,n) = δ β,0 1 w δ α,0 + iη(τ)! 3 θ 1 (4z, τ) Yoshioka (1995/6), Gottsche, Zagier (1996) + X (a,b)= (α,β) mod 2 b 0 1 < w < q ( sgn(b) sgn(bn am) ) w b+2a q 1 4 b ab n Indefinite theta function: - + D ab = O + - J m + -

18 Rank 3: Σ 1 Explicit expression: f 3, C f (z, τ; Σ 1, J m,n) = 1 P 4 (a,b)=( 2,2) mod 3 (c,d)=(a,b) mod 2, d 0 ( sgn(b) sgn(bn am)) ( sgn(d) sgn(d a c b ) ) (w b+2a w b 2a )w d+2c q 1 12 b ab+ 1 4 d cd signature of lattice is (2,2) - quadratic condition on the lattice points

19 Modularity Rank 1: h 1,c1 (z, τ; S) has expected transformation properties Rank 2: h 2,c1 (z, τ; Σ 1, J m,n ) is not modular However, replace: sgn x 1 O -1 2 τ 2 x 2 e π u2 du 0 1 O -1 f 2,c1 transforms as a modular form Zwegers (2000), Ramanujan ( 1915) - Approaches sgn(x) for τ 2 - Main physical importance: modularity & continuous in J Related to continuity of hypermultiplet metric, Gaiotto, Moore, Neitzke (2008), Alexandrov, JM, Pioline (2012) - Modular properties of f 3,c1 (z, τ; P 2 ) under investigation Bringmann, JM, Zwegers - How to derive from path integral?

20 Geometric Engineering I Limit of string theory on non-compact CY: O( K Σ1 ) Σ 1 N = 2, SU(2) gauge theory in 4 dimensions Katz, Klemm, Vafa (1997) Field theory particles: Particles D-brane W-boson D2 on f Monopole D4 on Σ 1 (Flavor charge D2 on blow-up C i ) Dyon spectra N = 4 Yang-Mills partition functions Charges: # D4 s: P = r # D2 s: Q = c 1 + rk S /2 # D0 s: Q 0 = r + 1 2r Q2 + rχ(s)/24

21 Geometric Engineering II Field theory particles are the light D-branes. Magnitude of periods: large volume limit field theory limit: Limit of Kähler classes: e 2πi(B C +ij C ) = lim ε 0 ε 4 e 4c 0 ; ( e 2πi(B f +ij f ) 1 = lim 1 + 2π2 ε 2 ) u ε 0 4 M0 2 Diaconescu et al. (2012)

22 Geometric Engineering III No walls: Arguments of periods For r 2, D0-brane charge > 0 for Ω(Γ; J) 0 Dyon spectrum: Ω((1, 1 2 n ef, 0); J) = 1 (n m, n e ) = (±1, 2n), n Z Dimofte, Gukov, Soibelman (2009); Diaconescu et al. (2012)

23 Geometric Engineering IV In agreement with: - Vacuum moduli space Seiberg, Witten (1994/5), Bilal, Ferrari (1996) - Computation of index of Dirac operator on monopole moduli space: O(r) SO(2N f ) Sethi, Stern, Zaslow (1995), Gauntlett, Harvey (1996) M r = R 3 S1 f M r Z r. Ground states of quantum mechanics on M r : - 4r boson-fermion pairs (X m, ψ m ) - 2rN f fermions χ A

24 A 5d/2d/4d correspondence I Five dimensional gauge theory on R 3 S 1 t S 1 M : BPS objects: - Magnetic string: T 2 = r - Electric charge: Z e = φ - Instanton: Z I = 1 + κ 2g5 2 2 φ ( φ 2g κ 4 φ2 ) - Momentum around S 1 M : Q 0 - Hypermultiplet mass: m i, i = 1,..., N f with κ = 2(8 N f ) Seiberg (1996), Morrison, Seiberg (1996) Engineered by M-theory on O(K Σ1 ) Σ 1 : { J = f 4g 2 5 φ K BNf N f i,j m i C 1 D n,ij d j k = 1 2 n ef n I K BNf 1 2 Nf i n f,i d j

25 A 5d/2d/4d correspondence II Worldvolume theory on magnetic string (0, 4) σ-model with target space M: - tension of the string: T = T radius of S 1 2 : φ - momentum p: n e - winding w: n I - oscillator level: Q 0 CFT elliptic genus: [ ] Z CFT,r (τ, y) = Tr 1 R 2 F 2 ( 1) F q L 0 c L 24 q L 0 c R 24 e 2πiy i J0 i 2 - L 0 c L 24 = p φ T +w 2 φ Q 2 T 0 - BPS condition L 0 c R 24 = p φ 2 2 T +w φ 2 T - spectral flow Z CFT,r (τ, y) = µ hcft r,µ (τ) Θ r,µ (τ, y)

26 A 5d/2d/4d correspondence III Proposal: Z CFT,r (τ, y) = Z YM,r (τ, y; J) Haghighat, JM, Vandoren (2012) Prediction of elliptic genera of hyper-kähler monopole moduli spaces!

27 A 5d/2d/4d correspondence IV Proven for SU(2) gauge theory with N f flavors and r = 1 (0,4) σ-model: - Condition on gauge theory charges: ( 1) ne = ( 1) H lift to orbifold CFT - Θ DNf flavor symmetry SO(2N f ) N = 4 Yang-Mills: - H 2 (S, Z) : ( ) ( 8 n Q Dn ) - gluing vectors g i

28 A 5d/2d/4d correspondence V For r 2 various additional intriguing aspects arise: - Dependence on J renormalization group flow? - What does the (0, 4) elliptic genus compute in terms of the target space?

29 Conclusions Partition functions of N = 4, U(r) Yang-Mills theory on rational surfaces can be determined explicitly Applications: Geometric engineering: - dyon spectra - a 5d/2d/4d correspondence Elliptic Calabi-Yau manifolds w. Klemm & Wotschke D3-instantons w. Alexandrov & Pioline Localization Quivers...