Monopole operators, moduli spaces and dualities in 3d CS matter theories Mauricio Romo University of California, Santa Barbara 2011 D. Berenstein and M. R.(to appear in ATMP), arXiv:0909.2856 [hep-th] M. R. JHEP 1109 , 122, arXiv:1011.4733 [hep-th] D. Berenstein and M. R., arXiv:1108.4013 [hep-th]
AdS4 / CFT3 The Setup • M2-brane worldvolume theory AdS 4 × X 7 ↔ 3 d SCFT . • The cone CX 7 over X 7 is contained in M vac , the moduli space of vacua of the corresponding SCFT. • Our purpose is to get X 7 computing M vac . • If X 7 = S 7 / Z k . The theory corresponds to M2 branes probing a C 4 / Z k singularity (ABJM theory). • Type IIA string theory on AdS 4 × X 6 , plus flux. • M vac ∼ C ∗ fiber over a CY 3 .
Outline • BPS states on CS matter theories • Examples • Seiberg-like duality? • Monopole operators ( N = 3 case) • Conclusions
M vac • M vac is characterized by the VEVs �O I � of the different scalar The set of numbers {�O I �} operators (order parameters). labels a vacuum. • There can exist relations between them � a I 1 ··· I n �O I 1 � · · · �O I n � = 0 { I i } • M vac correspond to the variety parameterized by these VEVs modulo relations.
M vac and the chiral ring in 4d SCFTs The operators that compose the coordinate ring of M 4 d are el- ements of the chiral ring. These are holomorphic operators. Their VEVs are classified by the solution of the F-term and D- term equations. For a theory with bifundamental matter fields φ a � ∂W a [ l ] φ [ l ] ) . ∂φ I = 0 W = Tr ( l � � φφ † − φ † φ = ζ FI t ( φ )= i h ( φ )= i We will set ζ FI = 0.
M 4d and quiver representations F-terms can be re-written as a path algebra by associating a nilpotent operator P ( a ) to each vertex. � φ I , P ( a ) � / { ∂W = 0 } A = = C Q/ { ∂W = 0 } Representations are labeled by their dimension vector � d ∈ N Q 0 and the values of the linear maps φ I . In the case of D3-branes (4d SCFTs) D-terms (with ζ FI = 0) will give us a moment map, which will be equivalent to consider GL ( N, C )-classes of A - modules and we have the correspondences ZA ↔ singularity d ↔ branes R � For M2-branes this is more complicated for many reasons. But we still want to do the identification d ↔ branes R � Semiclassical methods will help us to identify the CY 4 singularity.
M 3d with CS gauge fields We will work on theories with at least N = 2 SUSY in 3d ⇔ N = 1 in 4d. So, we can use the usual N = 1 superspace formalism and holomorphy. The vector multiplet will look like √ √ θχ † − i θθσ + 2 θσ µ ¯ V = 2 i ¯ 2 θθ ¯ 2¯ θ ¯ θθχ + θθ ¯ θ ¯ θA µ + i θD. and the supersymmetric CS action � � � S CS ( A ) = k AdA + 2 3 A 3 − χχ + 2 Dσ Tr 4 π canonical kinetic terms will have couplings of the form � φ † Dφ integration of the auxiliary field D will give us the following vac- uum equations...
M 3d with CS gauge fields σ α φ αβ − φ αβ σ β = 0 ∂W = 0 ∂φ αβ � � φφ † − φ † φ = k α σ α t ( φ )= α h ( φ )= α In addition � N G � A ′ ∧ dA D + . . . A D ≡ A i S CS = i =1 decouples from matter.
The chiral ring P µ | 0 � = Q | 0 � = Q | 0 � = M µν | 0 � = 0 , the expectation value w.r.t. | 0 � of a general superfield O ( θ, ¯ θ, x ) satisfies ∂ µ �O� = ∂ θ �O� = ∂ ¯ θ �O� = 0 , Moreover O ( θ, ¯ θ, x ) = { D, G ( θ, ¯ θ, x ) } ⇒ �O� = 0 Therefore we only have to worry about equivalence classes of chiral operators D O = 0 . For O chiral we have the nice properties ∂ x 1 �O ( x 1 ) O ( x 2 ) � = ∂ x 2 �O ( x 1 ) O ( x 2 ) � = 0 , �O ( x 1 ) O ( x 2 ) � = �O ( x 1 ) ��O ( x 2 ) � , Definition . The chiral ring is the subset of chiral operators ( D O = 0) θ, x ) = 0 � � � R = � O = � θ, x ) �� O| D α O ( θ, ¯ D, G ( θ, ¯ ,
The chiral ring For SCFTs additional constrains can be imposed over the oper- ators on R due to the large amount of (super-)symmetry. This boils down to consider operators whose lowest component φ is a superprimary in the chiral ring (i.e. its equivalence class can be represented by a superprimary). More importantly this casts φ as a BPS state satisfying ∆ φ ∼ R φ , with ∆ φ the scaling dimension of φ and R φ its R-charge. In particular, for d = 3 ∆ φ = R φ . So, the moduli space of these theories can be written as � � M ∼ � φ �|O = φ + ¯ θψ + . . . , O ∈ R = ,
M vac in SCFTs When working in the cylinder R × S d , then we can identify ∆ with the Hamil- tonian of the system. BPS equations can be solved classically. The classical Hamiltonian and R-charge in terms of the momenta are given by � � � a + 1 a ) − 1 Π φ a Π ¯ a ∇ φ a ∇ ¯ φ ¯ H = ( K ,a ¯ a + K ,a ¯ 4 K + V D + V F φ ¯ S 2 � � a � Π φ a γ a φ a − Π ¯ a ¯ φ ¯ Q R = i a γ ¯ φ ¯ S 2 The classical BPS eqs. H − Q R = 0 reduce to a sum of squares that have to vanish separatedly φ a ˙ iγ a φ a , = ∇ φ a = 0 V D = V F = 0
M vac in SCFTs Additionally we have the constraint coming from the A 0 e.o.m � � � � � − k i F ( i ) Π φ a φ a + i Π φ a φ a + i a − i φ ¯ φ ¯ a ¯ a ¯ a = S 2 − i Π ¯ Π ¯ φ ¯ φ ¯ π t ( a )= i h ( a )= i t (¯ a )= i h (¯ a )= i The pullback of ω , the symplectic form of the φ a phase space, to the manifold of BPS solutions can be written as a dφ a ∧ d ¯ ω = iK ,a ¯ φ ¯ a = − 2 dφ a ∧ d Π φ a this shows that we can holomorphically quantize the φ a ’s. Wave functions will take the form � φ m a a a the A 0 equations can be written as a constraint on the exponents (for the U (1) l case) � � − k i F ( i ) = − i m a + i m a π t ( a )= i h ( a )= i
M vac in SCFTs Summarizing ∂W = 0 ∂φ a � � k i F ( i ) ψ = φ a ∂ φ a − φ a ∂ φ a ψ t ( a )= i h ( a )= i F ( i ) ∈ Z � φ m a ψ = a a
Example 1: ABJM Stack of N M2-branes probing a C 4 / Z k singularity ( N = 6). G = U ( N ) × U ( N ) O. Aharony, O. Bergman, D. L. Jafferis and J. Maldacena, JHEP 0810 , (2008) � � � � A µ ∂ ν A λ + 2 i A λ − 2 i d 3 x 2 Kε µνλ Tr 3 A µ A ν A λ − ˆ A µ ∂ ν ˆ A µ ˆ ˆ A ν ˆ S ABJM = A λ 3 � 4 KDσ + 4 K ˆ − D ˆ σ � � V W e V � Z e − V Z e ˆ V − ¯ W e − ˆ d 3 xd 4 θTr − ¯ + � � � 1 d 3 xd 2 θTr ε AC ε BD Z A W B Z C W D + 4 K � � W D � 1 W B ¯ Z A ¯ d 3 xd 2 θTr ε AC ε BD ¯ ¯ Z C ¯ + 4 K Field U ( N ) U ( N ) Z , ¯ W � � W , ¯ Z � �
Example 1: ABJM Point-like brane G = U (1) k × U (1) − k � Z 1 W 1 Z 2 W 2 − Z 1 W 2 Z 2 W 1 � W c = Tr , Classical Moduli equations A c = � Z i , W j , P a � / { dW c = 0 } � � � � 0 0 0 z A W A = Z A = w A 0 0 0 Wave functions ( z 1 ) i 1 ( z 2 ) i 2 ( w 1 ) j 1 ( w 2 ) j 2 i 1 + i 2 − j 1 − j 2 ∈ k Z The variables ( z, w ) describe the coordinate ring of C 4 / Z k . This is the moduli space of one M2-brane in the bulk .
Example 2: Non-toric quiver W ∼ Tr ( ABC ) B∂ B − C∂ C k A 0 0 0 0 0 0 C∂ C − A∂ A 0 k B 0 0 0 0 0 F ψ = ψ A 1 ∂ A 1 − B 1 ∂ B 1 A 1 ∂ A 2 − B 2 ∂ B 1 0 0 k C 0 0 0 A 2 ∂ A 1 − B 1 ∂ B 2 A 2 ∂ A 2 − B 2 ∂ B 2 0 0 0 k C 0 0 F ∈ Z M = B 2 k C C 2 k C + k A ˜ M = B 2 k C C 2 k C + k A M ˜ M ∼ ( ACB ) 2 k C
Seiberg-like duality? Studied for the case of a single gauge group U ( N c ) k → U ( N f − N c + | k | ) − k What about quivers? [ B ] → [ B ] [ � [ B i ] → B i ] ≡ [ B i ] + n i [ B ] If ([ B ] , [ B i ]) have ranks ( N, N i ), then � N → N i n i − N. i so → − k k k i → k i + n i k. equivalently we can have → − k k → k i + ˜ k i n i k.
Seiberg-like duality? Consider a N = 3 theory with superpotential and field content given by � n � n � � ϕ i ( B i A i − A i − 1 B i − 1 ) − 1 k i ϕ 2 W = Tr i 2 i =1 i =1 � ( w − ϕr i ) uv = i n − i n � � j r i = k j + nk j j =1 j =1
Seiberg-like duality? Fractional brane branches are allowed iff r i = r l for i � = l , this is equivalent to l � k j = 0 j = i i.e. some consecutive subset of the k i ’s add up to zero. This also is equivalent to have a singularity at a point distinct than u = v = w = ϕ = 0. Therefore fractional branes have dimension vector j ∈ [ i, l ] d j = 1 if d j = 0 otherwise
Seiberg-like duality? In both cases the bulk moduli space is the same but fractional brane branches describe different singularities
Seiberg-like duality? M = a k 1 1 a k 2 + k 1 M = b k 1 1 b k 2 + k 1 M ∼ z 2 k 1 + k 2 � ⇒ M � 2 2 M ′ = a k M ′ = b k ⇒ M ′ � M ′ ∼ z k �
Seiberg-like duality? 1 b k 2 + k 1 1 a k 2 + k 1 M = a k M = b k M ∼ z k + k 1 + k 2 � ⇒ M � 2 2 M ′ = a k 1 M ′ = b k 1 ⇒ M ′ � M ′ ∼ z k 1 �
Monopole operators In CS theories with matter, monopole operators (operators with vortex charge) can be BPS. Bare monopoles are not gauge in- variant, but they can be paired with matter fields to form an operator in R � X m i O H ( X ) = T H i i Bare monopoles T H are characterized by an element of h ⊆ g � H = n a h a n a ∈ Z a classically � � 1 F ∼ ∗ d H | x |
Recommend
More recommend
