Special Geometry and Born-Infeld Attractors

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1 JOINT ERC WORKSHOP ON SUPERFIELDS, SELFCOMPLETION AND STRINGS & GRAVITY October 22-24, Ludwig-Maximilians-University, Munich Special Geometry and Born-Infeld Attractors Sergio Ferrara (CERN - Geneva) 24th October, 2014

2 Summary of the talk Nilpotent Superfields in Superspace Applications to Rigid and Local Supersymmetry Partial Supersymmetry Breaking in Rigid N=2 Theories Emergence of Volkov-Akulov and Born-Infeld Actions Symplectic Structure and Black-Hole Attractors: analogies and differences New U(1) n Born-Infeld Actions and Theory of Invariant Polynomials

3 Some of the material of this presentation originates from some recent work with Antoniadis, Dudas, Sagnotti; Kallosh, Linde and some work in progress with Porrati, Sagnotti The latter introduces a generalization of the Born-Infeld Action for an arbitrary U(1) n N=2 supergravity with N=2 self-interacting vector multiplets

4 NILPOTENCY CONSTRAINTS IN SPONTANEOUSLY BROKEN N=1 RIGID SUPERSYMMETRY (Casalbuoni, De Curtis, Dominici, Ferruglio, Gatto; Komargodski, Seiberg; Rocek; Lindstrom, Rocek) X chiral superfield ( DȧX = 0) nilpotency: X 2 = 0 solution: X = GG F G + i p 2q G + q 2 F G (G Weyl fermion) Lagrangian: Re X X D + fx F Equivalent to Volkov-Akulov goldstino action More general constraints (Komargodski,Seiberg) independent fields X 2 = 0, XY = 0 y X = G, y Y XW a = 0 independent fields y X = G, F µn, l = f (y X, F µn )

5 NILPOTENT CONSTRAINTS IN LOCAL SUPERSYMMETRY (SUPERHIGGS EFFECT IN SUPERGRAVITY) Pure supergravity coupled to the Goldstino W(X) = fx+ m theory of a massive gravitino coupled to gravity and cosmological constant: L = f 2 3m 2 (Deser, Zumino; Rocek; Antoniadis, Dudas, Sagnotti, S.F.) Volkov-Akulov-Starobinsky supergravity (a linear (inflaton) multiplet T, and a non-linear Goldstino multiplet S, with S 2 =0) V = M2 p 23 2 f M 2 1 e e 2p 2 f 3 a 2 Starobinsky inflaton potential axion field

6 Recently, nilpotent super fields have been used in more general theories of inflation (Kallosh, Linde, S.F.; Kallosh, Linde) For a class of models with SUPERPOTENTIAL W = Sf(T) (Kallosh, Linde, Roest; Kallosh, Linde, Rube) the resulting potential is a no-scale type, and has a universal form V eff = e K(T) K S S f (T) 2 > 0 goldstino y S with T q=0 = j + ia, inflaton: j or a depending which one is lighter during inflation

7 GENERALITIES ON PARTIAL SUPERSYMMETRY BREAKING GLOBAL SUPERSYMMETRY (Witten; Hughes, Polchinski; Cecotti, Girardello, Porrati, Maiani, S.F.; Girardello, Porrati, S.F.; Antoniadis, Partouche, Taylor) a = 1...N dc a fermion variations; J a µa(x) susy Noether current Current algebra relation (Polchinski) Z d 3 y { J 0ȧb (y), Jµa(x)} a = 2saȧT n µn (x)db a + s µaȧcb a Relation to the scalar potential: dc a d c b = V db a + Ca b In N=2 (APT, FGP) C a b = sxa be xyz Q y ^ Q z = 2s xa b(~ E ^ ~ M)x ; Q x = M x E x

8 In Supergravity (partial SuperHiggs) (CGP, FM, FGP) there is an extra term in the potential which allows supersymmetric anti-de-sitter vacua dc a d c b = V d a b + 3Mac M bc M ab = M ba gravitino mass term dy a µ = D µ e a + M ab g µ ē b

9 N=2 RIGID (SPECIAL) GEOMETRY R i k l = C ikp C l p g p p V = X A, U X A īv = 0, D j i V = C jik g k k k V, i V = 0 C ikp = i k p U = U ikp U = X 2 + X3 M Xn+2 M n for M large: U X 2 X3 M M U ABC = d ABC

10 N=2 rigid supersymmetric theory with Fayet-Iliopoulos terms (n vector multiplets, no hypermultiplets) in N=1 notations Kahler potential: K = i(x a Ū A X A U A ), U A = U A X A ( U : N=2 prepotential) Fayet-Iliopoulos terms: triplet of (real) symplectic (Sp(2n)) constant vectors ~Q =(~ M A, ~ m A EA )=(Q c, Q 3 )= 1 + im2 A, e 1A + ie 2A m3 A, e3 A (A = 1,..., n)

11 Superpotential: W =(U A m A X A e A ) Q c =(m A, e A ) D term N=1 F-I magnetic and electric charges: Q 3 =(e A, x A ) Due to the underlying symplectic structure of N=2 rigid special geometry, we can rewrite all expressions by using the symplectic metric W = and symplectic sections V =(X A, U A ) W =(V, Q) =V W Q, K = i(v, V) V F =(Im U 1 AB ) W X A W X B = Q T c M Q c + i ( Q c, Q c )

12 V D = Q T 3 M Q 3 so that V = V F + V D = Q c T (M + iw) Q c + Q3 T M Q 3 The matrix M is a real (positive definite) symmetric and symplectic 2n 2n matrix M T = M MWM = W M > 0 It is related to the F 2, F F terms in the Lagrangian L = g AB (X)F A µnf B µn + q AB (X) F A µn F B µn M = g + qg 1 q qg 1 g 1 q g 1

13 N=1 supersymmetric vacua require (Porrati, Sagnotti, S.F.) W since X A = 0, V D = 0 M > 0 The V D = 0 condition requires Q 3 = 0 The first equation implies (M + iw) Q c = 0 which requires i Q c W Q c < 0 Since Q c M Q c is positive definite, and at the attractor point we have Q c M crit Q c = i Q c WQ c Q c So it is crucial that is complex. To simplify, we will later consider m A real and e A complex, so that the previous condition is m1 A e 2A < 0

14 The theory here considered is the generalization to n vector multiplets of the theory considered by Antoniadis, Partouche, Taylor (1995) Later, this theory (n=1) was shown to reproduce in some limit (Rocek, Tseytlin, 1998) the supersymmetric Born-Infeld action (Cecotti, S.F., 1986). The latter was shown (Bagger, Galperin, 1996) to be the Goldstone action for N=2 partially broken to N=1 where the gauging l = W a is the Goldstone fermion q=0 of the second broken supersymmetry.

15 EXTREMAL BLACK HOLE ANALOGIES In the case of asymptotically flat black holes, the so-called Black-Hole Potential for an extremal (single-center) black-hole solution is (Kallosh, S.F., 1996) V BH = 1 2 QT M Q Q =(m A, e A ) is the asymptotic black-hole charge The theory of BH attractors (Kallosh, Strominger, S.F., 1995) tells that V BH crit = 1 2 Qc M(X crit ) Q = 1 p S(Q) = 1 4p A H where S(Q) is the BH entropy (Bekenstein-Hawking area formula)

16 In the N=2 case (Ceresole, D Auria, S.F.; Gibbons, Kallosh, S.F.) V BH = g i D i Z D Z + Z 2 so that at the BPS attractor point ( D i Z = 0) V BH = Z crit 2 = 1 p S(Q) In analogy to the N=2 partial breaking of the rigid case, the BPS black hole breaks N=2 down to N=1, and the central charge is the quantity replacing the super potential W. The entropy and BI action are both expressed through W.

17 In our problem, the attractors occur at V crit = 0, (because of unbroken space time supersymmetry) and this is only possible because the Fayet-Iliopoulos charge is an SU(2) triplet (charged in the F-term direction) and thus allows the attractor equation (M crit + iw) Q c = 0 being satisfied. We call these vacua Born-Infeld attractors, for reasons will become soon evident

18 The superspace action of the theory in question is L = Im U AB Wa A Wb B eab + W(X) F + X A Ū A X A U A D The Euler-Lagrange equations for X A are U ABC W B W C + U AB (m B D 2 X B ) e A + D 2 Ū A = 0 These are the complete X A equations for the theory in question. The first thing to note is that our attractor gives a mass to the N=1 chiral multiplet X A, but not to the N=1 vector multiplets. So N=2 is broken. W a

19 Indeed, our action is invariant under a second supersymmetry h a, which acts on the N=1 chiral super fields (X A, Wa A ) (Bagger, Galperin n=1) dx A = h a W A a, dw A a = h a (m A D 2 X A ) i aȧ X A hȧ and because of the m A parameter, the second supersymmetry is spontaneously broken. Note that the m A, e 2A parameters are those which allow the equations W Z A = U AB m B e A = 0 to have solutions

20 Expanding the fields around their classical value cancel the linear terms in the action and we obtain (Porrati, Sagnotti, S.F.) dl dx A = 0 ) d ABC h i W B W C + X B (m C D 2 X C ) + D 2 Ū A = 0 The BI approximation corresponds to have D 2 Ū A = 0, (which we solve letting U A = 0 as operator condition) The N=2 Born-Infeld generalized lagrangian turns out to be (Porrati, Sagnotti, S.F.) L N=2 BI = Im W(X) q 2 = Re FA e 2A + Im F A e 1A with the chiral superfields X A solutions of the above constraints

21 The super field BI constraint allows to write the n chiral fermions in terms of the n gauginos, so l g combination of the n (dressed) gauginos is a linear In Black-Hole physics, the superpotential is the N=2 central charge, and S entropy = p Z 2 crit

22 SOLUTION OF THE BORN-INFELD ATTRACTOR EQUATIONS For n=1, the above equation is the BI constraint X = m W 2 D 2 X (electric magnetic self-dual BG inherited from the linear theory) which also implies the nilpotency (Komargodsky, Seiberg; Casalbuoni et al) Type of constraints XW a = 0, X 2 = 0 This type of constraints have been recently used in inflationary supergravity dynamics to simplify and to provide a more general supergravity breaking sector (Kallosh, Linde, S.F.; Kallosh, Linde; Antoniadis, Dudas, Sagnotti, S.F.)

23 The Born-Infeld action comes by solving the of the chiral superfield equation q 2 component L h i X A ) d ABC G+G B + C + F B (m C F C ) q 2 = 0 G± A = Fµn A ± i and we have set 2 F µn A D A = 0 by taking the real and imaginary parts of this equation we have d ABC H B + mb 2 m C 2 H C d ABC Im F B m C = = d ABC d ABC G B G C G B G C + Im F B ImF C and we have set Re F A = ma 2 H A

24 CLASSIFICATION OF d ABC. THEORY OF INVARIANT POLYNOMIALS (Mumford, Gelfand, Dieudonné, et al) U(X) = 1 3! d ABCX A X B X C n=2 case: d ABC (4 entries) spin 3/2 representation of SL(2) It has a unique (quartic) invariant which is also the discriminant of the cubic (Cayley hyperdeterminant) (Duff, q-bit entanglement in quantum information theory) I 4 = 27 d d d2 221 d d 222d 111 d 211 d 221 4d 111 d d 222 d 3 211

25 CLASSIFICATION OF d ABC. THEORY OF INVARIANT POLYNOMIALS (Mumford, Gelfand, Dieudonné, et al) I 4 = 27 d d d2 221 d d 222d 111 d 211 d 221 4d 111 d d 222 d Four orbits I 4 > 0, I 4 < 0, I 4 = 0, I 4 = 0 Two of them ( I 4 < 0, I 4 = 0 ) give trivial models. The other two ( I ) give non-trivial U(1) BI theories Extremal black-hole analogy: q the model in question corresponds to the T 3 model, I 4 is the BH Berenstein-Hawking entropy and the four orbits correspond to large BPS and non-bps as well as small BH

26 An example: Explicit solution of the I 4 >0 theory U = 1 3! X3 1 2 XY2 (d 111 = 1, d 122 = 1) Im F X =(m 2 X + m 2 Y ) 1 (m X R X + m Y R Y ) Im F Y =(m 2 X + m 2 Y ) 1 ( m Y R X + m X R Y ) H X = 1 p 2 q S 2 X + S2 Y S X 1/2 H Y = 1 p 2 qs 2 X + S2 Y + S X R X = R Y = 2G X G Y G X G X + G Y G Y S X = T m2 X X 4 + m2 Y 4 1/2 S Y = T Y + m X m Y 2 H 2 X + H 2 Y = S X, 2H X H Y = S Y (H A equation) T X = G X G X + Im F X Im F X + G Y G Y Im F Y Im F Y T Y = 2(G X G Y + Im F X Im F Y ) Note: for G X,Y µn = 0! F X = F Y = 0