M theory black holes and 3d gauge theories Alberto Zaffaroni Universit` a di Milano-Bicocca StringGeo, Mainz, September 2015 [work in collaboration with F. Benini, K. Hristov] F. Benini-AZ; arXiv 1504.03698 F. Benini-K.Hristov-AZ; arXiv 1510.xxxxx [Thanks to A. Tomasiello for many related discussions] Alberto Zaffaroni (Milano-Bicocca) M theory black holes and 3d gauge theories Sestri 1 / 26
Introduction Introduction In this talk I consider BPS black holes in AdS 4 . ◮ One of the success of string theory is the microscopic counting of asymptotically flat black holes made with D-branes [Vafa-Strominger’96] ◮ No similar result for AdS black holes But AdS should be simpler and related to holography: counting of states in the dual CFT. People failed for AdS 5 black holes (states in N=4 SYM). Alberto Zaffaroni (Milano-Bicocca) M theory black holes and 3d gauge theories Sestri 2 / 26
Introduction Introduction There are many 1/4 BPS asymptotically AdS 4 static black holes ◮ solutions asymptotic to magnetic AdS 4 and with horizon AdS 2 × S 2 � ◮ Characterized by a collection of magnetic charges S 2 F ◮ preserving supersymmetry via a twist ( ∇ µ − iA µ ) ǫ = ∂ µ ǫ = ⇒ ǫ = cost Various solutions with regular horizons, some embeddable in AdS 4 × S 7 . [Cacciatori, Klemm; Gnecchi, Dall’agata; Hristov, Vandoren] ; Alberto Zaffaroni (Milano-Bicocca) M theory black holes and 3d gauge theories Sestri 3 / 26
Introduction Introduction Examples in a N = 2 gauged supergravity with 3 vector multiplets the STU model. � � 2 d t 2 − e −K ( X ) d r 2 c d s 2 = e K ( X ) � 2 − e −K ( X ) r 2 d s 2 gr + � S 2 2 gr c gr + 2 gr Truncation of M theory on AdS 4 × S 7 ◮ four abelian vectors U (1) 4 ⊂ SO (8) that come from the reduction on S 7 . ◮ One is the graviphoton, three enter in vector multiplets. Alberto Zaffaroni (Milano-Bicocca) M theory black holes and 3d gauge theories Sestri 4 / 26
Introduction Introduction Examples in a N = 2 gauged supergravity with 3 vector multiplets the STU model. � � 2 d t 2 − e −K ( X ) d r 2 c d s 2 = e K ( X ) � 2 − e −K ( X ) r 2 d s 2 gr + � S 2 2 gr c gr + 2 gr √ F = − 2 i X 0 X 1 X 2 X 3 √ � ¯ � e −K ( X ) = i X Λ F Λ − X Λ ¯ F Λ = 16 X 0 X 1 X 2 X 3 X i = 1 4 − β i X 0 = 1 4 + β 1 + β 2 + β 3 r , r with arbitrary parameters β 1 , β 2 , β 3 . Alberto Zaffaroni (Milano-Bicocca) M theory black holes and 3d gauge theories Sestri 4 / 26
Introduction Introduction Examples in a N = 2 gauged supergravity with 3 vector multiplets the STU model. � � 2 d t 2 − e −K ( X ) d r 2 c d s 2 = e K ( X ) � 2 − e −K ( X ) r 2 d s 2 gr + � S 2 2 gr c gr + 2 gr The parameters are related to the magnetic charges supporting the black hole � � n i = 1 S 2 F ( i ) , n 1 , n 2 , n 3 , n 4 , n i = 2 2 π by n 1 = 8( − β 2 1 + β 2 2 + β 2 3 + β 2 β 3 ) , n 2 = 8( − β 2 2 + β 2 1 + β 2 3 + β 1 β 3 ) , n 3 = 8( − β 2 3 + β 2 1 + β 2 2 + β 1 β 2 ) . Alberto Zaffaroni (Milano-Bicocca) M theory black holes and 3d gauge theories Sestri 4 / 26
Introduction Introduction Examples in a N = 2 gauged supergravity with 3 vector multiplets the STU model. � � 2 d t 2 − e −K ( X ) d r 2 c d s 2 = e K ( X ) � 2 − e −K ( X ) r 2 d s 2 gr + � S 2 2 gr c gr + 2 gr The horizon is AdS 2 × S 2 and the entropy is � S = 8 r 2 X 0 ( r h ) X 1 ( r h ) X 2 ( r h ) X 3 ( r h ) h for example, for n 1 = n 2 = n 3 � 1 + ( − 1 + 2 n 1 ) 3 / 2 √ − 1 + 6 n 1 − 6 n 2 − 1 + 6 n 1 Alberto Zaffaroni (Milano-Bicocca) M theory black holes and 3d gauge theories Sestri 4 / 26
Introduction Introduction General vacua of a bulk effective action L = − 1 2 R + F µν F µν + V ... with a metric d +1 = dr 2 ds 2 r 2 + ( r 2 ds 2 M d + O ( r )) A = A M d + O (1 / r ) and a gauge fields profile, correspond to CFTs on a d-manifold M d and a non trivial background field for the symmetry L CFT + J µ A µ Alberto Zaffaroni (Milano-Bicocca) M theory black holes and 3d gauge theories Sestri 5 / 26
Introduction Introduction In the case of the AdS 4 black holes ◮ the boundary is S 2 × R (or S 2 × S 1 after Wick rotation) ◮ bulk gauge fields induce magnetic backgrounds for R and global symmetries in the CFT ◮ bulk supersymmetry induce boundary susy (twist) ( ∇ µ − iA µ ) ǫ = ∂ µ ǫ = 0 Alberto Zaffaroni (Milano-Bicocca) M theory black holes and 3d gauge theories Sestri 6 / 26
Introduction Introduction AdS black holes are dual to a topologically twisted CFT on S 2 × S 1 with background magnetic fluxes for the global symmetries Alberto Zaffaroni (Milano-Bicocca) M theory black holes and 3d gauge theories Sestri 7 / 26
The twisted index The background Consider an N = 2 gauge theory on S 2 × S 1 ds 2 = R 2 � d θ 2 + sin 2 θ d ϕ 2 � + β 2 dt 2 with a background for the R-symmetry proportional to the spin connection: A R = − 1 2 cos θ d ϕ = − 1 2 ω 12 so that the Killing spinor equation D µ ǫ = ∂ µ ǫ + 1 4 ω ab µ γ ab ǫ − iA R ⇒ µ ǫ = 0 = ǫ = const Alberto Zaffaroni (Milano-Bicocca) M theory black holes and 3d gauge theories Sestri 8 / 26
The twisted index The background This is just a topological twist. [Witten ’88] The result becomes interesting when supersymmetric backgrounds for the flavor symmetry multiplets ( A F µ , σ F , D F ) are turned on: � u F = A F t + i σ F , q F = S 2 F F = iD F and the path integral, which can be exactly computed by localization, becomes a function of a set of magnetic charges q F and chemical potentials u F . [Benini-AZ; arXiv 1504.03698] Alberto Zaffaroni (Milano-Bicocca) M theory black holes and 3d gauge theories Sestri 9 / 26
The twisted index A topologically twisted index The path integral can be re-interpreted as a twisted index: a trace over the Hilbert space H of states on a sphere in the presence of a magnetic background for the R and the global symmetries, � ( − 1) F e iJ F A F e − β H � Tr H Q 2 = H − σ F J F holomorphic in u F where J F is the generator of the global symmetry. Alberto Zaffaroni (Milano-Bicocca) M theory black holes and 3d gauge theories Sestri 10 / 26
The twisted index Localization Exact quantities in supersymmetric theories with a charge Q 2 = 0 can be obtained by a saddle point approximation � � S | class × det fermions e − S = e − S + t { Q , V } = t ≫ 1 e − ¯ Z = det bosons Very old idea that has become very concrete recently, with the computation of partition functions on spheres and other manifolds supporting supersymmetry. Alberto Zaffaroni (Milano-Bicocca) M theory black holes and 3d gauge theories Sestri 11 / 26
The twisted index The partition function The path integral for an N = 2 gauge theory on S 2 × S 1 with gauge group G localizes on a set of BPS configurations specified by data in the vector multiplets V = ( A µ , σ, λ, λ † , D ) � ◮ A magnetic flux on S 2 , m = 1 S 2 F in the co-root lattice 2 π ◮ A Wilson line A t along S 1 ◮ The vacuum expectation value σ of the real scalar Up to gauge transformations, the BPS manifold is � � ( u = A t + i σ, m ) ∈ M BPS = H × h × Γ h / W Alberto Zaffaroni (Milano-Bicocca) M theory black holes and 3d gauge theories Sestri 12 / 26
The twisted index The partition function The path integral reduces to a the saddle point around the BPS configurations � � u Z cl +1-loop ( u , ¯ dud ¯ u , m ) m ∈ Γ h ◮ The integrand has various singularities where chiral fields become massless ◮ There are fermionic zero modes The two things nicely combine and the path integral reduces to an r -dimensional contour integral of a meromorphic form � � 1 Z int ( u , m ) | W | C m ∈ Γ h Alberto Zaffaroni (Milano-Bicocca) M theory black holes and 3d gauge theories Sestri 13 / 26
The twisted index The partition function ◮ In each sector with gauge flux m we have a a meromorphic form Z int ( u , m ) = Z class Z 1-loop Z CS class = x k m x = e iu � x ρ/ 2 � ρ ( m ) − q +1 � Z chiral 1-loop = q = R charge 1 − x ρ ρ ∈ R � Z gauge (1 − x α ) ( i du ) r 1-loop = α ∈ G ◮ Supersymmetric localization selects a particular contour of integration C and picks some of the residues of the form Z int ( u , m ). [Jeffrey-Kirwan residue - similar to Benini,Eager,Hori,Tachikawa ’13; Hori,Kim,Yi ’14] Alberto Zaffaroni (Milano-Bicocca) M theory black holes and 3d gauge theories Sestri 14 / 26
The twisted index A Simple Example: SQED The theory has gauge group U (1) and two chiral Q and ˜ Q � � x � m + n � x − 1 � − m + n 1 1 1 � 2 y 2 y dx 2 2 Z = 1 − xy 1 − x − 1 y 2 π i x m ∈ Z U (1) g U (1) A U (1) R 1 1 1 Q ˜ Q − 1 1 1 Consistent with duality with three chirals with superpotential XYZ � � 2 n − 1 � � − n +1 � � − n +1 y − 1 y − 1 y 2 2 Z = 1 − y 2 1 − y − 1 1 − y − 1 Alberto Zaffaroni (Milano-Bicocca) M theory black holes and 3d gauge theories Sestri 15 / 26
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