The Hilbert Series of SQCD Matti J arvinen University of Crete 2 - PowerPoint PPT Presentation

The Hilbert Series of SQCD Matti J¨ arvinen University of Crete 2 March 2012 1/26

Motivation: Hilbert series vs. Brane decay 1. Hilbert series of N = 1 supersymmetric QCD [Chen,Mekareeya arXiv:1104.2045] N c N c 1 d τ a 1 � � � g ( t , ˜ 2 π | ∆( z ) | 2 t ) = � N f (1 − ˜ N c ! 1 − tz − 1 tz a ) N f � a =1 a =1 a 2. (A contribution to the) emission amplitude for a closed string from a decaying brane (half S -brane) N N Z ( w ) = 1 d τ a � � 2 π | ∆( z ) | 2 � | 1 − wz a | 2 i ω N ! a =1 a =1 ◮ Same integrals (for t = ˜ t )! ◮ Our work: known results from brane decay applied to the Hilbert series (+some new results) [Jokela,MJ,Keski-Vakkuri, arXiv:1112.5454] 2/26

Outline ◮ Introduction to Hilbert series (mostly stolen from Amihay Hanany’s talks) ◮ Defining SQCD Hilbert series ◮ Hilbert series and Schur polynomials ◮ Hilbert series of SQCD in the Veneziano limit: the log-gas approach 3/26

Introduction to Hilbert Series Main idea: ◮ Hilbert series = generating function for numbers of gauge invariant BPS operators in N = 1 supersymmetric gauge theory ◮ For a theory having n U (1) symmetries � c k 1 ,..., k n t k 1 1 · · · t k n H ( t 1 , . . . , t n ) = n k 1 ,..., k n ◮ c k 1 ,..., k n : number of operators having charges k 1 , . . . , k n under the symmetries ◮ Variables t i termed “chemical potentials” or “fugacities” ◮ Admits a generalization to non-Abelian symmetries (definition however complicated. . . ) ◮ Usually one restricts to one Abelian fugacity t → operators counted by their dimension 4/26

Introduction to Hilbert Series – Example Example: SQCD with SU (2) gauge group and N f = 1 1 1 − t 2 = 1 + t 2 + t 4 + t 6 + · · · H ( t ) = g ( t ) = ◮ One single-trace operator Q 1 Q 2 ◮ All operators of various degrees 1 , Q 1 Q 2 , ( Q 1 Q 2 ) 2 , ( Q 1 Q 2 ) 3 , . . . ◮ “Freely generated” moduli space, dimension one 5/26

Introduction to Hilbert Series – Generic Features Q ( t ) P ( t ) H ( t ) = (1 − t ) k = (1 − t ) dim ( M ) ◮ Q ( t ), P ( t ) polynomials ◮ k is dimension of “embedding space” (For SQCD, mesonic+baryonic operators) ◮ dim ( M ), dimension of (classical) moduli space, equals the degree of the pole at t = 1 ◮ P ( t = 1) is “degree of M ” (AdS/CFT → volume of the dual Sasaki-Einstein manifold) 6/26

Introduction to Hilbert Series – More Examples More SQCD examples ( N f , N c ) [Gray,He,Hanany,Mekareeya,Jejjala, arXiv:0803.4257] Palindromic property of P ( t ) ⇒ moduli space is a Calabi-Yau 7/26

Introduction to Hilbert Series – Moduli Spaces Q ( t ) H ( t ) = (1 − t ) k Three families of moduli spaces: 1. Freely generated ( N c > N f ): Q ( t ) = 1 2. Complete intersection ( N c = N f ): Q ( t ) = 1 − t d 3. The rest ( N c < N f ) 8/26

Introduction to Hilbert Series – Plethystics Plethystic exponential PE and logarithm PL = PE − 1 � ∞ � f ( t k ) � PE [ f ( t )] ≡ exp k k =1 Plethystic logarithm of H : ◮ Generators of the moduli space – first positive terms ◮ Relations of the moduli space – first negative terms For SQCD, taking the plethystic logarithm of H ( N f , N c ) ( t ): PL [ H (1 , 2) ( t )] = t 2 PL [ H (2 , 2) ( t )] = 6 t 2 − t 4 PL [ H (2 , 3) ( t )] = 4 t 2 PL [ H (3 , 2) ( t )] = 15 t 2 − 15 t 4 + 35 t 6 − · · · PL [ H (3 , 3) ( t )] = 9 t 2 + 2 t 3 − t 6 9/26

The SQCD Hilbert Series – U(N) For U ( N c ) gauge group (simpler), refined Hilbert series � 2 π N c N c N f 1 d τ a 1 � 2 π | ∆( z ) | 2 � � g N f , U ( N c ) ( t i , ˜ t i ) = (1 − t i z − 1 N c ! a )(1 − ˜ t i z a ) 0 a =1 a =1 i =1 t i and z a = e i τ a are the flavor, “antiflavor” and color ◮ t i , ˜ fugacities, respectively ◮ ∆( z ) is the Vandermonde determinant Understanding the expression: ◮ � N c � N f 1 generates all operators involving Q a a =1 1 − t i z − 1 i =1 i a ◮ � N c � N f 1 t i z a generates all operators involving ˜ Q a a =1 i =1 1 − ˜ i � 2 π 2 π | ∆( z ) | 2 picks up gauge invariant terms � N c 1 d τ a ◮ a =1 N c ! 0 10/26

The SQCD Hilbert Series – SU(N) For SU ( N c ), add a constraint for the phases τ a � 2 π N c ∞ � � 1 d τ a � � � 2 π | ∆( z ) | 2 g N f , SU ( N c ) ( t i , ˜ t i ) = δ τ a − 2 π k N c ! 0 a =1 a k = −∞ N f N c 1 � � × (1 − t i z − 1 a )(1 − ˜ t i z a ) a =1 i =1 Some notation (important to recall!): ◮ For U ( N f ) L fugacities (separation into SU ( N f ) L × U (1) Q ) � � x 1 , x 2 1 ( t 1 , t 2 , . . . , t N f ) ≡ , . . . , t ≡ (˜ x 1 , ˜ x 2 , . . . , ˜ x N f ) t x 1 x N f − 1 ◮ For U ( N f ) R fugacities � 1 � , y 1 (˜ t 1 , ˜ t 2 , . . . , ˜ ˜ y N f )˜ t N f ) ≡ t ≡ (˜ , . . . , y N f − 1 y 1 , ˜ y 2 , . . . , ˜ t y 1 y 2 11/26

The SQCD Hilbert Series – Unrefining (Unrefined, standard) Hilbert series: set all x i = 1 = y j (or t = t 1 = · · · t n ), e.g. � 2 π N c N c t )= 1 d τ a � � a ) − N f (1 − g N f , U ( N c ) ( t , ˜ 2 π | ∆( z ) | 2 (1 − tz − 1 ˜ tz a ) − N f N c ! 0 a =1 a =1 Often in addition set t = ˜ t (the most interesting case) ⇒ real integrand 12/26

The SQCD Hilbert Series – As Matrix Integral � 2 π N c N c t )= 1 d τ a � � a ) − N f (1 − 2 π | ∆( z ) | 2 (1 − tz − 1 tz a ) − N f g N f , U ( N c ) ( t , ˜ ˜ N c ! 0 a =1 a =1 � d µ U ( N c ) det( 1 − tU † ) − N f det( 1 − ˜ tU ) − N f = � � det( 1 − tU † ) − N f det( 1 − ˜ tU ) − N f = CUE ◮ An expectation value in the circular unitary ensemble ◮ d µ U ( N c ) is the Haar measure Integrals can be evaluated ⇒ Toeplitz determinant t )=det T [ f ] ≡ det(ˆ g N f , U ( N c ) ( t , ˜ f i − j ) i , j =1 ,..., N c with � ˜ t n � − N f � n ≥ 0 ˆ t ) = ( − 1) n f n ( t , ˜ 1; t ˜ 2 F 1 ( N f + | n | , N f , | n | + t ) × t − n | n | n < 0 [Chen,Mekareeya; Jokela,MJ,Keski-Vakkuri] However result cumbersome for N c � 3 13/26

Hilbert Series & Schur Polynomials Schur polynomials: symmetric polynomials of n variables � z λ n − i +1 + i − 1 � det j i , j =1 ,..., n s λ ( z 1 , z 2 , . . . , z n ) = � � z i − 1 det j i , j =1 ,..., n z λ n z λ n z λ n � · · · � 1 2 n � � z λ n − 1 +1 z λ n − 1 +1 z λ n − 1 +1 � � · · · 1 � n � 1 2 = � � . . . . ∆( z ) � � . . � � � � z λ 1 + n − 1 z λ 1 + n − 1 z λ 1 + n − 1 · · · � � 1 2 n ◮ λ = ( λ 1 , . . . , λ n ) partition of | λ | = � i λ i or a Young diagram ◮ Example: λ = (2 , 1 , 1) = ( z 1 , z 2 , z 3 , z 4 ) = z 2 1 z 2 z 3 + z 1 z 2 2 z 3 + z 1 z 2 z 2 3 + z 2 s 1 z 2 z 4 + z 1 z 2 2 z 4 + z 2 1 z 3 z 4 +3 z 1 z 2 z 3 z 4 + z 2 2 z 3 z 4 + z 1 z 2 3 z 4 + z 2 z 2 3 z 4 + z 1 z 2 z 2 4 + z 1 z 3 z 2 4 + z 2 z 3 z 2 4 14/26

Hilbert Series & Schur Polynomials – Properties The Schur polynomials have special properties ◮ Orthogonality ( z i = e i τ i ) � 2 π n 1 d τ i � 2 π | ∆( z ) | 2 s λ ( z 1 , . . . , z n ) s κ (¯ z 1 , . . . , ¯ z n ) = δ λ,κ n ! 0 i =1 ◮ The Cauchy identity n m 1 � � � = s λ ( z ) s λ ( w ) 1 − z i w j i =1 j =1 λ x 1 , x 2 1 � � ◮ s λ (˜ x 1 , ˜ x 2 , . . . , ˜ x N f ) = s λ x 1 , . . . , are characters of x Nf − 1 SU ( N f ) 15/26

Hilbert Series & Schur Polynomials – Results Using the properties one easily proves an earlier conjecture for the refined Hilbert series of U ( N c ) SQCD: [Constable, Larsen, hep-th/0305177] � t ) | λ | s λ (˜ g N f , U ( N c ) ( t , ˜ ( t ˜ t , x , y ) = x ) s λ (˜ y ) λ : ℓ ( λ ) ≤ min( N f , N c ) ◮ ℓ ( λ ) is the width of the Young diagram λ For SU ( N c ) some extra algebra is required, giving ∞ � g N f , SU ( N c ) ( t , ˜ t , x , y ) = I k with k = −∞ �  t | λ | s λ (˜ t | κ | s κ (˜ x )˜ y ) δ λ,κ + k ; k ≥ 0 ,    λ,κ I k = � t | λ | s λ (˜ t | κ | s κ (˜ x )˜ y ) δ λ + | k | ,κ ; k < 0 .    λ,κ 16/26

Hilbert Series & Schur Polynomials – U(N) Results In some cases one can use Cauchy identity to sum the series: ◮ For N f ≤ N c : (freely generated) N f N f 1 � � g N f , U ( N c ) ( t , ˜ t , x , y ) = 1 − t ˜ t ˜ x i ˜ y j i =1 j =1 ◮ For N f = N c + 1: (complete intersection) t ) N c +1 � N c +1 1 � � g N c +1 , U ( N c ) ( t , ˜ 1 − ( t ˜ t , x , y ) = 1 − t ˜ t ˜ x i ˜ y j i , j =1 ◮ N f = N c + 2 also calculable, a big mess ◮ These are new results! 17/26

Hilbert Series & Schur Polynomials – SU Results ◮ For N f < N c : (freely generated) N f N f 1 � � g N f , SU ( N c ) ( t , ˜ t , x , y ) = 1 − t ˜ t ˜ x i ˜ y j i =1 j =1 ◮ For N f = N c : (complete intersection) N c N c t ) N c 1 − ( t ˜ 1 � � g N c , SU ( N c ) ( t , ˜ t , x , y ) = (1 − t N c ) (1 − ˜ 1 − t ˜ t N c ) t ˜ x i ˜ y j i =1 j =1 ◮ N f = N c + 1 also calculable, a big mess 18/26