Colored BPS Pyramid Partition Functions, Quivers and Cluster. and Cluster Transformations

Transcription

1 Colored BPS Pyramid Partition Functions, Quivers and Cluster Transformations Richard Eager IPMU Based on: arxiv:. Tuesday, January 4, 0

2 Introduction to AdS/CFT Quivers, Brane Tilings, and Geometry Pyramids from Quivers Pyramid Partitions Recursive Calcula Joint work with Figure: Mauricio Romo and Sebastion Franco

3 Quiver Gauge Theories Figure: N D-branes The world-volume theory of a stack of coincident D-branes placed at a Calabi-Yau singularity takes the form of a quiver gauge theory. A quiver Q = (V, A, h, t) is a collection of Vertices V Arrows A Maps h, t : A V

4 Toric Quivers (a) (b) Figure: a) Periodic quiver and b) brane tiling for the SPP. A brane tiling is a bipartite graph G = (G ± 0, G ) embedded into the two-torus where the faces form a tiling of the torus. A periodic quiver Q = (Q 0, Q, Q, h, t) is the dual graph of the brane tiling. Vertices Q 0 of the periodic quiver are dual to faces of the brane tiling.

5 Superpotential W CQ/[CQ, CQ] is the sum over all plaquettes with the sign of each term given by the corresponding orientation, W = w P w P, (.) P Q + P Q The superpotential algebra A = CQ/( W ) is obtained from identifying elements in the path algebra using the relations given by the partial derivatives of the superpotential.

6 Infinite Pyramids q p q p q Figure: Flavored quiver for SPP with n = top stones. q i s are indicated in red and p j s in blue. Framing with q i : Infinite Pyramids Introduce flavors q i transforming in the fundamental representation of gauge groups α i, (i =,..., n). Introduce flavors p j transforming in the antifundamental representation of gauge groups β j, (j =,..., n ). Add to the superpotential couplings for the flavors.

7 Every stone corresponds to an open oriented path ending with a q i. Paths containing a p j are eliminated by setting p j = 0, which implies that F qi = 0 and we are left with the F-term equations of the p j as relations. There are n top stones and (n ) relations, resulting in an infinite pyramid. W rels =p O q + (p O + p Õ )q (p n O n + p n Õ n )q n + p n Õ n q n Figure: Infinite pyramid for the SPP.

8 q p q p q Figure: Finite pyramid for SPP Framing with p j : Finite Pyramids Introduce flavors q i transforming in the fundamental representation of gauge groups α i, (i =,..., n). Introduce flavors p j transforming in the antifundamental representation of gauge groups β j, (j =,..., n ). Add to the superpotential couplings for the flavors.

9 Pyramid Partitions as Posets up q p q p q Figure: SPP Finite Pyramid Figure: Hasse diagram

10 up The pyramid partitions are in one-to-one correspondence with the ideals of. Introduce variable y i for each gauge group in the quiver i.e. for each type or color of stone. To every ideal Ω we assign the weight i Q 0 y n i i, where n i is the number of stones of type i in Ω. The colored partition function associated to a pyramid is defined as Z = y n i i. Ω i

11 Colored Pyramid Partition Function up Figure: Hasse diagram for stones in the poset Z = + y }{{} +y y + y y + y y + y y y + y y y + y y y top stones + y y y 4 + y y y 5. }{{} all stones

12 Conifold with Flavors Figure: Framed Quiver Figure: Pyramid for Conifold Z conifold = ( + q aq b )( + q aq b ) ( + q a )

13 Duality Cascades () p j (n ) () q i (n ) (n ) () q i q () n () p (n ) j () (n ) p j () q i n (a) (b) (c) Figure: a) The original SPP theory with framing flavors. b) The quiver after dualizing node. c) The quiver after further dualizing node. We have included a superindex to indicate the step in the dualization sequence at which flavors are generated. The final theory is identical to the original one (including superpotential coupling) after a rotation and a reduction n n.

14 Cluster Algebras A labeled seed is a triple (Z, x, B) consisting of Z = (Z,..., Z n ) a cluster, x = (x,... x n ) an n-tuple of coefficients, B = (b ij ) an n n integer matrix that is skew-symmetrizable. Quiver gauge theories are generalized cluster algebras where B is the anti-symmetrized adjacency matrix of the quiver. x are prefactors Cluster Variables Z = Pyramid Partition Functions

15 Seiberg Duality = Cluster Mutation x j = { x k if j = k, x j arr(k j) x k if j k, (5.) and Z k = arr(k j) Z j + x k arr(j k) Z j Z k. (5.)

16 Y p,q Examples For the Y p,q geometries he recursive equation takes the form Z n Z n N = Z n a Z n N+a + x n Z n c Z n N+c (5.) where N = a + b is the number of gauge groups in the corresponding quiver and x n = a+b i= y g n i i g = ( q a )( q b ), (5.4) where g = n g nq n. Setting y i =, the dp partition functions count the number of pyramid partitions and reduce to the Somos-4 sequence: Somos-4 sequence:,, 7,, 59, 4, 59, 809,... (5.5)

17 Example Y, (dp ) c 0 b 00 a c bd c d bc 0 b 0 a 0 a a c bd c d bc d c bd c 0 a a 0 a b 00 a a d bc d c bd c d bc d 0 b 0 a 0 a a d bc d c bd 0 b 00 a d Figure: A finite pyramid for dp

18 Z = + q Z = q Z + Z Z 0 Z 4 = q Z + Z Z Z 5 = q 5 Z + Z 4 Z Z 6 = q6 Z 4 + Z 5Z Z

19 Figure: A finite pyramid for dp Z dp 6 (q) = + q + q + q + 4q 4 + 5q 5 + 6q 6 + 6q 7 + 5q 8 + 5q 9 + 5q 0 + 4q + q + q + q 4 + q 5

20 Thank You!