Flux vacua in String Theory, generalized calibrations and supersymmetry breaking. Luca Martucci
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1 Flux vacua in String Theory, generalized calibrations and supersymmetry breaking Luca Martucci Workshop on complex geometry and supersymmetry, IPMU January
2 Plan of this talk Generalized calibrations and fluxes Generalized calibrations and SUSY vacua Calibrations, SUSY-breaking and 4D potential
3 Generalized calibrations and fluxes
4 Ordinary calibrations Harvey & Lawson `82 An ordinary calibration is a differential -form such that 1) Algebraic condition: 2) Differential condition: for any -dim submanifold Calibrated submanifolds:
5 Ordinary calibrations Harvey & Lawson `82 Main property: calibrated cycles are volume minimizing
6 Ordinary calibrations Harvey & Lawson `82 Main property: calibrated cycles are volume minimizing
7 Ordinary calibrations Harvey & Lawson `82 Main property: calibrated cycles are volume minimizing algebraic condition
8 Ordinary calibrations Harvey & Lawson `82 Main property: calibrated cycles are volume minimizing algebraic condition differential condition
9 Ordinary calibrations Harvey & Lawson `82 Main property: calibrated cycles are volume minimizing algebraic condition differential condition calibration condition
10 Supersymmetric branes and calibrations (no fluxes)
11 Supersymmetric branes and calibrations (no fluxes) In absence of fluxes, a static -brane wrapping has energy
12 Supersymmetric branes and calibrations (no fluxes) In absence of fluxes, a static -brane wrapping has energy We expect BPS bound for supersymmetric branes if
13 Supersymmetric branes and calibrations (no fluxes) In absence of fluxes, a static -brane wrapping has energy We expect BPS bound for supersymmetric branes if Explicitly realized by calibrations bulk SUSY -calibration brane SUSY
14 How to include (type II) String Theory fluxes?
15 How to include (type II) String Theory fluxes? Type II bulk fields NS sector: RR sector:,, (locally ) (locally ) IIA IIB
16 How to include (type II) String Theory fluxes? Type II bulk fields NS sector: RR sector:,, (locally ) (locally ) IIA D-brane world-volume field: flux IIB such that (locally )
17 How to include (type II) String Theory fluxes? Type II bulk fields NS sector: RR sector:,, (locally ) (locally ) IIA D-brane world-volume field: flux IIB such that D-brane energy density: (locally )
18 How to include (type II) String Theory fluxes? Type II bulk fields NS sector: RR sector:,, (locally ) (locally ) IIA D-brane world-volume field: flux IIB such that D-brane energy density: (locally )
19 How to include (type II) String Theory fluxes? Type II bulk fields NS sector: RR sector:,, (locally ) (locally ) IIA D-brane world-volume field: flux IIB such that D-brane energy density: (locally )
20 Generalized calibrations Koerber `05; L.M. & Smyth `05
21 Generalized calibrations Koerber `05; L.M. & Smyth `05 A generalized calibration is a background polyform
22 Generalized calibrations Koerber `05; L.M. & Smyth `05 A generalized calibration is a background polyform such that: 1) 2) Algebraic: Differential:
23 Generalized calibrations Koerber `05; L.M. & Smyth `05 A generalized calibration is a background polyform such that: 1) 2) Algebraic: Differential: `ordinary generalized calibrations Gutowski, Papadopoulos & Townsend `99
24 Generalized calibrations Koerber `05; L.M. & Smyth `05 A generalized calibration is a background polyform such that: 1) 2) Algebraic: Differential: generalized geometry `ordinary generalized calibrations Gutowski, Papadopoulos & Townsend `99
25 Generalized calibrations Koerber `05; L.M. & Smyth `05 A generalized calibration is a background polyform such that: 1) 2) Algebraic: Differential: generalized geometry `ordinary generalized calibrations Gutowski, Papadopoulos & Townsend `99 Calibrated (or BPS) D-branes:
26 The stability argument Main property: calibrated D-branes are stable!
27 The stability argument Main property: calibrated D-branes are stable!
28 The stability argument Main property: calibrated D-branes are stable! algebraic condition
29 The stability argument Main property: calibrated D-branes are stable! algebraic condition differential condition
30 The stability argument Main property: calibrated D-branes are stable! algebraic condition differential condition D-brane calibration condition
31 Generalized calibrations and type II SUSY vacua
32 N=1 vacua and pure spinors General 4+6 backgrounds: plus,,, w (or )
33 N=1 vacua and pure spinors Ordinary Killing spinors:, w (or )
34 N=1 vacua and pure spinors Ordinary Killing spinors: S0(6,6) pure spinors, w (or )
35 N=1 vacua and pure spinors Ordinary Killing spinors: S0(6,6) pure spinors, in IIA w in IIB (or )
36 N=1 vacua and pure spinors Ordinary Killing spinors: S0(6,6) pure spinors w, and contain complete information about: (or ) NS sector:, and ( excluded) Internal spinors: and
37 N=1 vacua and calibrations L.M. & Smyth `05 SUSY conditions Graña, Minasian, Petrini & Tomasiello `05, w
38 N=1 vacua and calibrations L.M. & Smyth `05 SUSY conditions Graña, Minasian, Petrini & Tomasiello `05, w
39 N=1 vacua and calibrations L.M. & Smyth `05 SUSY conditions Graña, Minasian, Petrini & Tomasiello `05, w defines integrable generalized CY structure!
40 N=1 vacua and calibrations L.M. & Smyth `05 SUSY conditions Graña, Minasian, Petrini & Tomasiello `05, w defines integrable generalized CY structure! They have a natural interpretation in terms of D-brane generalized calibrations!
41 N=1 vacua and calibrations L.M. & Smyth `05 is a generalized calibration for space-filling D-branes space-filling
42 N=1 vacua and calibrations L.M. & Smyth `05 is a generalized calibration for space-filling D-branes space-filling For SUSY (BPS) space-filling D-branes, internal satisfies:
43 N=1 vacua and calibrations L.M. & Smyth `05 is a generalized calibration for space-filling D-branes space-filling For SUSY (BPS) space-filling D-branes, internal satisfies: is generalized complex submanifold w.r.t. GCS defined by satisfies the `speciality condition (F-flatness) (D-flatness)
44 N=1 vacua and calibrations L.M. & Smyth `05 BPS lower bound for D-brane energy no open-string tachyons!
45 N=1 vacua and calibrations L.M. & Smyth `05 BPS lower bound for D-brane energy no open-string tachyons! For space-filling D-branes, it originates in bulk SUSY condition Gauge BPSness
46 N=1 vacua and calibrations L.M. & Smyth `05 is a generalized calibration for D-strings string BPSness strings
47 N=1 vacua and calibrations L.M. & Smyth `05 is a generalized calibration for D-strings string BPSness strings is a generalized calibration for domain walls DW BPSness domain walls
48 Summarizing In N=1 compactifications (to flat ) we have Graña, Minasian, Petrini L.M. & Smyth`05 Koerber & L.M. `07 & Tomasiello `05 Equation D-brane BPSness 4D SUGRA int. gauge BPSness string BPSness DW BPSness chiral fields: (N=2 vect. mult.) (N=2 hypermult.)
49 Pre-GG Example: type IIB warped CY Graña & Polchinski `00; Giddings, Kachru & Polchinski`01 The internal metric is CY, up to warping:
50 Pre-GG Example: type IIB warped CY Graña & Polchinski `00; Giddings, Kachru & Polchinski`01 The internal metric is CY, up to warping:
51 Pre-GG Example: type IIB warped CY Graña & Polchinski `00; Giddings, Kachru & Polchinski`01 The internal metric is CY, up to warping: In this case:,
52 SUSY wcy and calibrations
53 SUSY wcy and calibrations Gauge BPSness: calibrated D3 & D7 branes
54 SUSY wcy and calibrations Gauge BPSness: calibrated D3 & D7 branes String BPSness: calibrated D3 & D7 branes
55 SUSY wcy and calibrations Gauge BPSness: calibrated D3 & D7 branes String BPSness: DW BPSness: calibrated D3 & D7 branes calibrated D5 & D7 branes
56 Calibrations, SUSY-breaking and 4D potential Lüst, Marchesano, L.M. & Tsimpis`07
57 SUSY breaking? Pre-GG prototypical example: IIB wcy Graña & Polchinski `00; Giddings, Kachru & Polchinski`01
58 SUSY breaking? Pre-GG prototypical example: IIB wcy Graña & Polchinski `00; Giddings, Kachru & Polchinski`01 N=1 and N=0 shear the same geometry with 03/O7 and D3/D7
59 SUSY breaking? Pre-GG prototypical example: IIB wcy Graña & Polchinski `00; Giddings, Kachru & Polchinski`01 N=1 and N=0 shear the same geometry with 03/O7 and D3/D7 flux induced SUSY-breaking:
60 SUSY breaking? Pre-GG prototypical example: IIB wcy Graña & Polchinski `00; Giddings, Kachru & Polchinski`01 N=1 and N=0 shear the same geometry with 03/O7 and D3/D7 flux induced SUSY-breaking: Many `phenomenological models are based on these classical vacua e.g. Kachru, Kallosh,, Linde & Trivedi `03; + MacAllister & Maldacena `03 Balasubramanian,, Berglund,, Conlon & Quevedo `05
61 SUSY wcy and calibrations In this case:, Gauge BPSness: calibrated D3 & D7 branes String BPSness: calibrated D3 & D7 branes DW (non)bpsness: ( ) calibrated D5 & D7 branes
62 Generalized SUSY-breaking
63 Generalized SUSY-breaking We are led to consider backgrounds that fulfill gauge and string BPSness, but not DW BPSness
64 Generalized SUSY-breaking We are led to consider backgrounds that fulfill gauge and string BPSness, but not DW BPSness BPS bounds for,, space-filling strings domain walls
65 Generalized SUSY-breaking We are led to consider backgrounds that fulfill gauge and string BPSness, but not DW BPSness BPS bounds for,, space-filling strings domain walls DWSB backgrounds
66 Generalized SUSY-breaking Equation D-brane BPSness 4D SUGRA int. gauge BPSness string BPSness DW (non)bpsness DWSB backgrounds
67 Generalized SUSY-breaking In wcy, the DWSB has rather specific form crucial to solve 10D e.o.m.
68 Generalized SUSY-breaking In wcy, the DWSB has rather specific form crucial to solve 10D e.o.m. Can we analogously restrict the generalized DWSB??
69 1-parameter DWSB
70 1-parameter DWSB Take generalized fibration (Dirac structure)
71 1-parameter DWSB Take generalized fibration (Dirac structure) spanned by mobile D-branes
72 1-parameter DWSB Take generalized fibration (Dirac structure) spanned by mobile D-branes Consider DWSB of the form
73 1-parameter DWSB Take generalized fibration (Dirac structure) spanned by mobile D-branes Consider DWSB of the form SUSY-breaking parameter
74 1-parameter DWSB Take generalized fibration (Dirac structure) spanned by mobile D-branes Consider DWSB of the form SUSY-breaking parameter In wcy case:
75 1-parameter DWSB Take generalized fibration (Dirac structure) spanned by mobile D-branes Consider DWSB of the form SUSY-breaking parameter In wcy case: (and thus )
76 1-parameter DWSB Take generalized fibration (Dirac structure) spanned by mobile D-branes Consider DWSB of the form SUSY-breaking parameter In wcy case: (and thus ) spanned by mobile D3
77 1-parameter DWSB Take generalized fibration (Dirac structure) spanned by mobile D-branes Consider DWSB of the form SUSY-breaking parameter In wcy case: (and thus ) spanned by mobile D3
78 1-parameter DWSB Take generalized fibration (Dirac structure) spanned by mobile D-branes Consider DWSB of the form SUSY-breaking parameter In wcy case: spanned by mobile D3 (and thus ),
79 1-parameter DWSB Take generalized fibration (Dirac structure) Consider DWSB of the form SUSY-breaking parameter How to check that 10D e.o.m. are satisfied? Which conditions must fullfill?
80 10D e.o.m. from 4D potential General configurations of the form plus,,, D-branes & orientifolds
81 10D e.o.m. from 4D potential General configurations of the form plus,,, D-branes & orientifolds The full set of 10D e.o.m. can be obtained from
82 10D e.o.m. from 4D potential General configurations of the form plus,,, D-branes & orientifolds The full set of 10D e.o.m. can be obtained from We need to express the potential in terms of and
83 Potential and pure spinors We found, schematically (see also Cassani `08) (...)
84 Potential and pure spinors We found, schematically (see also Cassani `08) > - 0 > - 0 > - 0 > - 0 < - 0 (...) < - 0
85 Potential and pure spinors We found, schematically (see also Cassani `08) > - 0 (D-brane BPS bound) > - 0 (gauge BPSness) 2 ~ > - 0 (string BPSness) 2 ~ > - 0 < - 0 (DW BPSness) 2 (DW BPSness) 2 ~ ~ (...) < - 0 (string + DW BPSness) 2 ~
86 Potential for 1-param. DWSB > - 0 > - 0 > - 0 > - 0
87 Potential for 1-param. DWSB calibrated D-branes > - 0 > - 0 > - 0 > - 0
88 Potential for 1-param. DWSB > - 0 gauge BPSness ( ) > - 0 > - 0 > - 0
89 Potential for 1-param. DWSB > - 0 > - 0 string BPSness ( ) > - 0 > - 0
90 Potential for 1-param. DWSB > - 0 > - 0 > - 0 > - 0 calibrated generalized fibration ( )
91 Potential for 1-param. DWSB > - 0 > - 0 > - 0 > - 0 calibrated generalized fibration -dependence disappears: no-scale structure ( )
92 An example
93 Example: SU(3)-structure with D5/O5 [cfr. Camara & Graña `07]
94 Example: SU(3)-structure with D5/O5 In this case:, [cfr. Camara & Graña `07]
95 Example: SU(3)-structure with D5/O5 In this case:, [cfr. Camara & Graña `07] Gauge + string BPSness: non-kähler ISD
96 Example: SU(3)-structure with D5/O5 In this case:, [cfr. Camara & Graña `07] Gauge + string BPSness: non-kähler ISD DWSB:
97 Example: SU(3)-structure with D5/O5 In this case:, [cfr. Camara & Graña `07] Gauge + string BPSness: non-kähler ISD DWSB: Notice that: non-integrable complex structure if
98 Explicit example on twisted torus
99 Explicit example on twisted torus
100 Explicit example on twisted torus rd Twisted periodicity in 3 direction:
101 Explicit example on twisted torus rd Twisted periodicity in 3 direction: Metric and fields,
102 Explicit example on twisted torus rd Twisted periodicity in 3 direction: Metric and fields SU(3)-structure,
103 Explicit example on twisted torus rd Twisted periodicity in 3 direction: Metric and fields SU(3)-structure, DWSB twist-induced DWSB
104 Conclusion
105 Conclusion Generic type II flux compactifications are naturally described by generalized calibrations
106 Conclusion Generic type II flux compactifications are naturally described by generalized calibrations SUSY integrable GC and calibration structures emergent N=1 4D effective structures
107 Conclusion Generic type II flux compactifications are naturally described by generalized calibrations SUSY SUSY integrable GC and calibration structures emergent N=1 4D effective structures integrable GC but still calibration structures less clear N=1 4D structures
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