Integrability in Supersymmetric Gauge Theories and Topological Strings Andrei Marshakov Lebedev Institute & ITEP, Moscow Galileo Galilei Institute, Florence, May 2009
Old story (1995): N = 2 supersymmetric Yang-Mills theory = Yang-Mills-Higgs system plus fermions: • Higgs field falls into condensate � Φ � ∈ h , and breaks the gauge group up to maximal torus (in general position); • supersymmetry ensures (partial) cancelation of perturba- tive corrections, and existence of light BPS states, with masses ∼ | q · a + g · a D | , ( q, g ) - set of electric and magnetic charges.
One may speak on moduli space of the theory: u ∼ � TrΦ 2 � , or generally the set coefficients of P ( z ) = � det( z − Φ) � (1) Classical moduli space: singular point at the origin u = 0, where the gauge group restores, and nothing interesting ... but this is in domain of strong coupling, where quasiclassics does not work.
Quantum moduli space u D D a + a =0 a =0 Gauge group never restores, but there are singularities where BPS states become massless: e.g. the monopole at a D = 0 and dyon at a + a D = 0.
Seiberg-Witten theory: N = 2 supersymmetric Yang-Mills the- ory ( U ( N c ) gauge group) L 0 = 1 µν + | D µ Φ | 2 + [Φ , ¯ Φ] 2 + . . . F 2 � � Tr (2) g 2 0 so that [Φ , ¯ Φ] = 0 ⇒ Φ = diag( a 1 , . . . , a N c ), and D µ Φ ⇒ [ A µ , Φ] ij = A ij µ ( a i − a j ), so that only A ii µ ≡ A i µ remain massless. SW theory gives a set of effective couplings T ij ( a ) in the low- energy N = 2 SUSY Abelian U (1) rank gauge theory. L eff = Im T ij ( a ) F i µν F j µν + . . . (3) �� Λ � 2 N c � weak coupling log a i − a j with T ij → + O . Λ a
N = 2 kinematics encodes nontrivial information in holomor- � d 4 θ F ( Φ )). ∂ 2 F phic prepotential T ij = ∂a i ∂a j (effective action is Im The prepotential itself is determined by: Σ of genus=rank, with a meromorphic differential dS SW such that δdS SW ≃ holomorphic (4) or by an integrable system . Period variables { a i = � A i dS SW } and F are introduced by dS SW = ∂ F � a D i = (5) ∂a i B i ∂ 2 F consistent by symmetricity of ∂a i ∂a j = T ij ( a ) period matrix of Σ (integrability from Riemann bilinear identities).
Famous example of Σ: let P N c ( z ) = � det( z − Φ) � , then N c w + Λ 2 N c � ( z − v i ) = P N c ( z ) = w (6) i =1 dS SW ≃ zdw w Integrable system is N c -periodic Toda chain. Simplest possible(?) example N c = 2, z → momentum, log w → coordinate, the curve Σ and dS SW turn into the Hamiltonian and Jacobi form of physical pendulum or the 1d “sine-Gordon” (Λ → 0: Liouville) system w + Λ 4 w = z 2 − u
In fact the simplest possible example is N c = 1 ( U (1) N = 2 supersymmetric gauge theory?) w + 1 � � Λ = z − v (7) w � z dw 2 a 2 t 1 + e t 1 , with Λ 2 = e t 1 , a = giving rise to F = 1 w = v . Indeed, the Toda “chain” (dispersionless limit): ∂ 2 F = exp ∂ 2 F ∂t 2 ∂a 2 1 Stringy solution F = 1 2 a 2 t 1 + e t 1 : a system of particles a D = ∂ F ∂a = at 1 with constant velocity = number = a .
Topological A-string on P 1 with quantum cohomology OPE: ̟ · ̟ ≃ e t 1 1 , primary operators t 1 ↔ ̟ , a ↔ 1 : F ∼ � exp ( a 1 + t 1 ̟ ) � is a truncated generation function. Toda hierarchy - the descendants: t k +1 ↔ σ k ( ̟ ), T n ↔ σ n ( 1 ), ( a ≡ − T 0 ) then F = a 2 t 1 + e t 1 ⇒ F ( t , a ) ⇒ F ( t , T ) (8) 2 being still a solution to the Toda equation ∂ 2 F = exp ∂ 2 F ∂t 2 ∂a 2 1
Solution is found via dual “Landau-Ginzburg” B-model (the N c = 1 SW curve) w + 1 � � (9) z = v + Λ w by construction of a function with asymptotics, t k z k − 2 T n z n (log z − c n )+ � � S ( z ) = z →∞ k> 0 n> 0 (10) +2 a log z − ∂ F 1 ∂ F � ∂a − 2 kz k ∂t k k> 0 1 ( c k = � k i ), whose “tail” defines the gradients of prepoten- i =1 tial (analogs of the dual periods), e.g. ∂ F B zdw � ∂a ∼ w ∼ [ S ] 0
B 1 B B 3 2 A A 1 A 2 3 Smooth Riemann surface (of genus 3) with fixed A - and B -cycles. w= 8 z= 8 B w=0 A z= 8 � w + 1 � Cylinder z = v + Λ w with degenerate B - cycle.
What is the sense of this oversimplified example? Topological A-string: the prepotential counts asymptotics of the Hurwirz numbers, number of ramified covers by string world-sheets of the (target!) P 1 . Gauge-string duality: sum over partitions ≡ summing instan- tons in 4D N = 2 SUSY gauge theory (Nekrasov partition function). U (1) gauge theory: non-commutative instantons, Toda hier- archy - the deformation of the UV prepotential t k 2 τ Φ 2 → F UV = F UV, 0 = 1 k + 1Φ k � k> 0 with τ = t 1 ∼ log Λ.
Partition function in deformed gauge theory (at T n = δ n, 1 ) m 2 tk 1 k +1 ch k +1 ( a, k , � ) ∼ � k � k> 0 � 2 Z ( a, t ; � ) = ( − � 2 ) | k | e (11) k � 1 � ∼ exp � 2 F ( a, t ) + . . . is some over set of partitions k = k 1 ≥ k 2 ≥ . . . with the Plancherel measure 1 ≤ i<j ≤ ℓ k ( k i − k j + j − i ) � k i − k j + j − i � m k = = (12) � ℓ k j − i i =1 ( ℓ k + k i − i )! i<j and particular (Chern) polynomials ch 2 ( a, k ) = a 2 + 2 � 2 | k | ch 0 ( a, k ) = 1 , ch 1 ( a, k ) = a, ch 3 ( a, k ) = a 3 + 6 � 2 a | k | + 3 � 3 � k i ( k i + 1 − 2 i ) (13) i . . .
or � ∞ ∞ u l e u ( a + � (1 � � u 2 − e − � u 2 − i + k i )) = � � e l ! ch l ( a, k , � ) (14) 2 i =1 l =0 coming from the Chern classes of the universal bundle over the instanton moduli space. The T -dependence Z ( a, t ) → Z ( a, t , T ) is restored from the Virasoro constraints L n ( t , T ; ∂ t , ∂ T ; ∂ 2 (15) t ) Z ( a, t , T ; � ) = 0 , n ≥ − 1
Non Abelian theory: U ( N c ) gauge group, nontrivial SW theory. Partition function more complicated, but quasiclassics always given by solution to the same functional problem: � � dxf ′′ ( x ) F UV ( x ) − 1 xf ′′ ( x ) f ′′ (˜ F = x dxd ˜ x ) F ( x − ˜ x )+ 2 x> ˜ N c � � � dx xf ′′ ( x ) a D a i − 1 � + i 2 i =1 (16) k> 0 t kx k +1 with F UV ( x ) = � k +1 , and log x − 3 � � log m 2 k → F ( x ) ∝ x 2 2 when integrated with (double derivative of the) shape function
f(x) a Shape function for partitions (Young diagrams) f ( x ) = | x − a | + ∆ f ( x ) f ext a Extremal shape for large partition
a a 1 2 Non-Abelian theory: extremal shape for N c = 2
δ F From the functional one gets for S ( z ) = d δf ′′ ( z ) dz � t k z k − dxf ′′ ( x )( z − x ) (log( z − x ) − 1) − a D � S ( z ) = k> 0 (17) with vanishing real part Re S ( x ) = 1 (18) 2 ( S ( x + i 0) + S ( x − i 0)) = 0 on the cut, where ∆ f ( x ) � = 0. On the double cover N c y 2 = ( z − x + i )( z − x − � i ) (19) i =1 � Φ( x ) = dS � S is odd under y ↔ − y , then f ′ ( x ) ∼ jump , and dx s ( z ) dz d Φ = ± (20) �� N c i =1 ( z − x + i )( z − x − i )
If all t k = 0, for k > 1, t 1 = log Λ N c , T n = δ n, 1 : ∓ 2 N c log z ± 2 N c log Λ + O ( z − 1 ) Φ = (21) P → P ± and there exists a meromorphic function w = Λ N c exp ( − Φ), satisfying N c w + Λ 2 N c � (22) = P N c ( z ) = ( z − v i ) w i =1 which restores the SW curve.
To restore the dependence on descendants σ n ( 1 ) quasiclassi- cally (influenced by Saito formula) ∂ F � = ( − ) n n ! ( S n ) 0 � (23) � ∂T n � t where d n S n (24) n ≥ 0 dz n = S, or S n is the n -th primitive (odd under w ↔ 1 w ). For higher t k � = 0, exp ( − Φ) has an essential singularity and cannot be described algebraically. Implicitly it is fixed by � � f ′′ ( x ) dx = − 2 πi, d Φ = − iπ A j I j (25) � res P ± d Φ = ∓ 2 N c , d Φ = 0 B j
d ≥ 0 q d F d , Instanton expansion in 4d gauge theory F = � q ∼ Λ 2 N c , log Λ ∼ t 1 . Topological string expansion: � is background parameter (IR cutoff) in 4d gauge theory. Topological string condensate: � σ 1 ( 1 ) � � = 0, T n = δ n, 1 is the simplest possible background, while a ∼ T 0 is the gauge theory condensate itself.
In the pertirbative limit Λ → 0 cuts shrink to the points z = a j , j = 1 , . . . , N c : the curve is N c � w pert = P N c ( z ) = ( z − v i ) (26) i =1 k> 0 t k z k ; T ( x ) ≡ � n> 0 T n x n ) endowed with ( t ( z ) ≡ � N c σ ( z ; v j ) + t ′ ( z ) � S ( z ) = − 2 j =1 (27) T ( k ) ( x ) ( z − x ) k (log( z − x ) − c k ) � σ ( z ; x ) = k ! k> 0
Logic: • restrict to the N -th class of backgrounds, with only T 1 , . . . , T N � = 0; • the “minimal” theory was with T n = δ n, 1 and F = F ( a, t ); T 1 = 1 corresponds to the condensate � σ 1 ( ̟ ) � � = 0; • N + 1-th derivative of S becomes single-valued.
Perturbative solution: i = S ( v i ) = ∂ F pert a D (28) ∂a i gives rise to N c � F pert ( a 1 , . . . , a N c ; t , T ) = F UV ( a j ; t , T )+ j =1 (29) � + F ( a i , a j ; T ) i � = j a j = T ( v j ) , j = 1 , . . . , N c
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