1. 3-Sasakian geometry from M2 branes Daniel L. Jafferis Rutgers University Kähler and Sasakian Geometry in Rome 19 June, 2009 Based on work with: A. Tomasiello; X. Yin; D. Gaiotto.; and work in progress

2. Outline Introduction Introduction N=3 Chern-Simons-matter and 3-Sasakian 7- manifolds N=3 Chern-Simons-matter and 3-Sasakian 7- manifolds Quantum corrections to N=3 CSM moduli spaces and duals to AdS4 with D6 branes Quantum corrections to N=3 CSM moduli spaces and duals to AdS4 with D6 branes N=2 speculations N=2 speculations

3. Motivation Extend AdS/CFT correspondence between 4d gauge theory and SE 5-manifolds to relation between 3d Chern-Simons theories and 3- Sasakian and SE 7-manifolds. Extend AdS/CFT correspondence between 4d gauge theory and SE 5-manifolds to relation between 3d Chern-Simons theories and 3- Sasakian and SE 7-manifolds. Requires extra data of a U(1) action that commutes with Reeb vector. What happens when this action has fixed loci? Requires extra data of a U(1) action that commutes with Reeb vector. What happens when this action has fixed loci? New mathematical objects associated to 3-cycles in 8-manifolds with flux? New mathematical objects associated to 3-cycles in 8-manifolds with flux?

4. M2 branes These are 2+1 dimensional objects in 11d M- theory. Their explicit description in physics was mysterious since their discovery over a decade ago. These are 2+1 dimensional objects in 11d M- theory. Their explicit description in physics was mysterious since their discovery over a decade ago. Now we have found that they carry (super)conformally invariant theory of Chern- Simons coupled to charged scalars. Now we have found that they carry (super)conformally invariant theory of Chern- Simons coupled to charged scalars.

5. AdS/CFT Equivalence between 4(3)d theory on the boundary and AdS5(4) x SE5(7) Focus on BPS sector (roughly topological sector) Classical limit: Moduli space of stable representations of a quiver Moduli space of stable objects in the derived category of coherent sheaves on the CY cone

6. Simplest version The moduli space of the abelian quiver, thought of as a quotient of the set of solutions to some equations is the CY cone itself. The moduli space of the abelian quiver, thought of as a quotient of the set of solutions to some equations is the CY cone itself.Example: A1, A2, B1, B2. Equations dW = 0 for W = Tr(A1 B1 A2 B2 – A1 B2 A2 B1). A1, A2, B1, B2. Equations dW = 0 for W = Tr(A1 B1 A2 B2 – A1 B2 A2 B1). Results in C 4 //U(1) acting by 1, 1, -1, -1. Results in C 4 //U(1) acting by 1, 1, -1, -1. This is the cone over T 1,1

7. The correspondence also says that stable representations of the path algebra The correspondence also says that stable representations of the path algebra correspond to stable objects in the derived category of compactly supported sheaves in the conifold. At the level of equations, King’s theorem says that solving μ=θ and quotienting by U(N) is equivalent to imposing an algebraic stability condition and quotienting by GL(N). At the level of equations, King’s theorem says that solving μ=θ and quotienting by U(N) is equivalent to imposing an algebraic stability condition and quotienting by GL(N). A = C [ f 0 ; f 1 ]h A 1 ; A 2 ; B 1 ; B 2 i=h B 1 A i B 2 ¡ B 2 A i B 1 ; A 1 B i A 2 ¡ A 2 B i A 1 i ¹ = A y i A i ¡ B i B y i

8. SE7 with U(1)B action CY4 cone KE6 /U(1)_R CY3 cone //U(1) B /U(1) B M6 SE5 KE4 //U(1) B /U(1) R M2 theory for SE7 is related to D3 theory for SE5 M6 is an S 2 bundle over KE4

9. 3-S with U(1)B action hK2 cone hK1 cone ///U(1) B /U(1) B M6 S3/ΓS3/Γ M2 theory for 3-S is related to hyperKähler quiver for hK1

10. Toric hyperKähler 8-manifold We know the associated M2 field theory in the case all p = 1 this last summer. We know the associated M2 field theory in the case all p = 1 this last summer. New development in March: p=0 as well. New development in March: p=0 as well. d s 2 = U ij d ~ x i ¢ d ~ x j + U ij ( d ' i + A i )( d ' j + A j ) A i = d ~ x j ¢ ~ ! ji = d x j a ! a ji @ x j a ! b k i ¡ @ x k b ! a ji = ² a b c @ x j c U k i [Bielawski, Dancer; Gauntlett, Gibbons, Papadopulos, Townsend] U = 1 + P i µ p 2 i h i p i q i h i p i q i h i q 2 i h i ¶ ; h i = 1 2 j p i ~ x 1 q i ~ x 2 j

11. Hyperkahler singularity In particular, the pair of U(1) isometries of the T^2 fiber are compatible with the hyperkahler structure, and one obtains the hypertoric manifold where N is the kernel of the map In particular, the pair of U(1) isometries of the T^2 fiber are compatible with the hyperkahler structure, and one obtains the hypertoric manifold where N is the kernel of the map In the case of D6 branes in CP^3, this reduces to In the case of D6 branes in CP^3, this reduces to ¯ : U ( 1 ) n + 1 ! U ( 1 ) 2 ; ¯ = µ 11 ::: 10 p 1 p 2 ::: p n m ¶ H n + 1 === N, H 3 === U ( 1 ) ; ( m, m, k ) [Bielawski Dancer]

12. D-term equation for M2 These are now cubic equations, and the analog of King’s theorem is not known. Here a,b are Lie algebra indices, and i,j are representation indices. These are now cubic equations, and the analog of King’s theorem is not known. Here a,b are Lie algebra indices, and i,j are representation indices. One branch of solutions has all One branch of solutions has all Here k is diagonal on each U(N) factor, indexed by m. Here k is diagonal on each U(N) factor, indexed by m. ¹ m = ³k m ( k ¡ 1 ) a b ¹ a ( T R ) ij b q j = 0

13. D6 branes in AdS_4 We now know a large class of quiver CSM theories describing a stack of M2 branes at a hypertoric singularity. In the ‘t Hooft limit, the dual geometry is a warped product We now know a large class of quiver CSM theories describing a stack of M2 branes at a hypertoric singularity. In the ‘t Hooft limit, the dual geometry is a warped product Introducing D6 branes wrapping an internal 3- cycle ( in the case) adds fundamental hypermultiplets to the quiver. Introducing D6 branes wrapping an internal 3- cycle ( in the case) adds fundamental hypermultiplets to the quiver. Interestingly, conformality is preserved. Interestingly, conformality is preserved. N = 3 A d S 4 £ w M 6 RP 3 CP 3

14. Dual CFT for. One of the most symmetric 3-Sasakians, its cone is, and it can be written as a quotient One of the most symmetric 3-Sasakians, its cone is, and it can be written as a quotient Classical moduli space is, but quantum corrected to Classical moduli space is, but quantum corrected to Attempts in the 90’s where close: Attempts in the 90’s where close: [ Billo` Fabbri Fre` Merlatti Zaffaroni ] T ¤ CP 2 k -k C 4 = Z k T ¤ CP 2 = H 3 === U ( 1 ) ; ( 1 ; 1 ; 1 ) SU ( 3 ) U ( 1 ) A d S 4 £ N 010

15. Quantum correction The cone is modified to The cone is modified to where U(1) acts as U(1) B on M and with the natural charge m action on C 2 Applied to C 4 this results in the cone over N 010 Applied to C 4 this results in the cone over N 010 ~ M m = ( M £ H )=== U ( 1 )

16. ADHM quiver for D2 in D6 In addition to the branch of moduli space where M2 branes probe the geometry including the lift of the D6 branes (which is always via quantum correction), the D2 branes may dissolve into the D6 branes (M2 branes fractionate). In addition to the branch of moduli space where M2 branes probe the geometry including the lift of the D6 branes (which is always via quantum correction), the D2 branes may dissolve into the D6 branes (M2 branes fractionate). Fundamentals now get a VEV, quiver is exactly ADHM quiver. Fundamentals now get a VEV, quiver is exactly ADHM quiver.

17. Fantasy The quiver theory describing D3 branes at a singular Calabi-Yau 3-fold can be determined by resolving the singularity, thus blowing up the fractional branes into wrapped D5 and D7 branes. Mathematically, the arrows in the quiver are the Ext groups between a primitive objects in the derived category of coherent sheaves. The quiver theory describing D3 branes at a singular Calabi-Yau 3-fold can be determined by resolving the singularity, thus blowing up the fractional branes into wrapped D5 and D7 branes. Mathematically, the arrows in the quiver are the Ext groups between a primitive objects in the derived category of coherent sheaves. It would be extremely interesting to find the analog for M2 brane theories. It would be extremely interesting to find the analog for M2 brane theories.

18. The physically natural picture would be the resolution of fractional M2 branes into wrapped M5 branes. There are two differences, however: The physically natural picture would be the resolution of fractional M2 branes into wrapped M5 branes. There are two differences, however: The fractional branes are typically pure torsion There are no supersymmetric 3-cycles in hyperkahler, CY4, Spin(7) 8-manifolds. The resolution is probably that M-theory 4-form flux plays a crucial role. Conjecture that there are supersymmetric resolutions of CY4 singularities with flux. Fuzzy M5 branes would naturally have s-rule. The resolution is probably that M-theory 4-form flux plays a crucial role. Conjecture that there are supersymmetric resolutions of CY4 singularities with flux. Fuzzy M5 branes would naturally have s-rule.

19. Quantum correction to hypermultiplet moduli space In Yang-Mills theories with eight supercharges, the moduli space of hypermultiplets is normally not corrected, since one can promote the coupling to a vector superfield, which decouples from the hypers. In Yang-Mills theories with eight supercharges, the moduli space of hypermultiplets is normally not corrected, since one can promote the coupling to a vector superfield, which decouples from the hypers. In CSM theories, no such argument exists for the CS level. However, corrections to the metric must respect the hyperkahler structure. In CSM theories, no such argument exists for the CS level. However, corrections to the metric must respect the hyperkahler structure. We will find a correction of this type. We will find a correction of this type.

20. Branches of CSM moduli space In Chern-Simons theories, there is no Coulomb branch, as the vector multiplets are all effectively massive. In Chern-Simons theories, there is no Coulomb branch, as the vector multiplets are all effectively massive. N=3 supersymmetry protects the dimensions of ordinary chiral operators formed from the matter fields; the quantum correction depends on the existence of monopole operators. N=3 supersymmetry protects the dimensions of ordinary chiral operators formed from the matter fields; the quantum correction depends on the existence of monopole operators. Rich structure of branches distinguished by the spectrum of allowed monopoles. Rich structure of branches distinguished by the spectrum of allowed monopoles.

21. Chern-Simons-matter theory We first consider the case with N=2 susy. It consists of a vector multiplet in the adjoint of the gauge group, and chiral multiplets in representations We first consider the case with N=2 susy. It consists of a vector multiplet in the adjoint of the gauge group, and chiral multiplets in representations The kinetic term for the chiral multiplets includes couplings The kinetic term for the chiral multiplets includes couplings There is the usual D term There is the usual D term R i ¡ ¹ Á i ¾ 2 Á i ¡ ¹ Ã i ¾ Ã i ¹ Á i D Á i S N = 2 CS = k 4 ¼ R ( A ^ d A + 2 3 A 3 ¡ ¹ ÂÂ + 2 D ¾ )

22. We integrate out D,, and ¾Â Note that this action has classically marginal couplings. It is has been argued that it does not renormalize, up to shift of k, and so is a CFT. [Gaiotto Yin]

23. N=3 CS-matter To obtain a more supersymmetric theory, begin with N=4 YM-matter. Then add the CS term, breaking to N=3. To obtain a more supersymmetric theory, begin with N=4 YM-matter. Then add the CS term, breaking to N=3. Thus we add a chiral multiplet,,with no kintetic term in the adjoint, and the matter chiral multiplets, must come in pairs. Thus we add a chiral multiplet,,with no kintetic term in the adjoint, and the matter chiral multiplets, must come in pairs. There is a superpotential, from the CS term. © i ; ~ © i ' W = ¡ k 8 ¼ T r ( ' 2 )

24. Integrating out one obtains the same action as before, but with a superpotential: Integrating out one obtains the same action as before, but with a superpotential: These N=3 theories are completely rigid, and hence superconformal. It is impossible to have more supersymmetry in a YM-CS-matter theory, but we shall see that for particular choices of gauge groups and matter representations, the pure CSM can have enhanced supersymmetry. These N=3 theories are completely rigid, and hence superconformal. It is impossible to have more supersymmetry in a YM-CS-matter theory, but we shall see that for particular choices of gauge groups and matter representations, the pure CSM can have enhanced supersymmetry. [Schwarz; Gaiotto Yin] [Schwarz; Gaiotto Yin] ' W = 4 ¼ k ( ~ © i T a R i © i )( ~ © j T a R j © j )

25. Simple example Consider a U(1) x U(1) CSM theory, with a BF- like Chern-Simons coupling Consider a U(1) x U(1) CSM theory, with a BF- like Chern-Simons coupling Take a pair of matter hypers, in the fundamental of the first and second U(1). Take a pair of matter hypers, in the fundamental of the first and second U(1). In this theory the supersymmetry is enhanced to N=4; one can check that the boson-fermion coupling is invariant under a separate SU(2) acting of each fundamental hyper. In this theory the supersymmetry is enhanced to N=4; one can check that the boson-fermion coupling is invariant under a separate SU(2) acting of each fundamental hyper. 2 k 4 ¼ R a ^ db ( X ; ~ X ) ; ( Y ; ~ Y )

26. Classical moduli space The superpotential is dictated by N=3 supersymmetry to be The superpotential is dictated by N=3 supersymmetry to be Thus there are two branches, and, on which the respective U(1) is unbroken. Thus there are two branches, and, on which the respective U(1) is unbroken. Naively, one would quotient by the nontrivially acting U(1), but would leave 3d, so can’t be. Naively, one would quotient by the nontrivially acting U(1), but would leave 3d, so can’t be. is only invariant under a Z_k. is only invariant under a Z_k. W = 4 ¼ k ( X ~ X )( Y ~ Y ) M a M b 2 k 4 ¼ R a ^ db M a = M b = C 2 = Z k

27. Extra massless fields at origin The two branches intersect at the origin, where there are extra massless fields. In particular, on which is parameterized by the X fields have a mass The two branches intersect at the origin, where there are extra massless fields. In particular, on which is parameterized by the X fields have a mass We will see that integrating out these fields changes the singularity at the origin. We will see that integrating out these fields changes the singularity at the origin. 4 ¼ k ¹ Y ( a Y b ) = 4 ¼ k µ 1 2 (j Y j 2 ¡ j ~ Y j 2 ) ; Y ~ Y ; ¹ Y ¹ ~ Y ¶ ´ ~ m X M a ( Y ; ~ Y )

28. Mukhi-Papageorgakis effect Forget about the mutliplet for a moment. Going onto the moduli space by turning on gives a mass to the broken gauge field, b. Forget about the mutliplet for a moment. Going onto the moduli space by turning on gives a mass to the broken gauge field, b. Integrating out b gives Yang-Mills kinetics to the unbroken gauge field, a! It can then be dualized to a scalar, which transforms under the U(1) in the same way as the phase of the hyper Y, but with charge k. The Z_k arises by gauge fixing. Integrating out b gives Yang-Mills kinetics to the unbroken gauge field, a! It can then be dualized to a scalar, which transforms under the U(1) in the same way as the phase of the hyper Y, but with charge k. The Z_k arises by gauge fixing. ( X ; ~ X )( Y ; ~ Y ) ' Y

29. Correction to the hyperkahler metric As familiar from the Coulomb branch of N=4 2+1 gauge theories, integrating out a charged massive hypermultiplet at 1 loop gives rise to a term As familiar from the Coulomb branch of N=4 2+1 gauge theories, integrating out a charged massive hypermultiplet at 1 loop gives rise to a term Note that this already introduces a Yang-Mills term for the gauge field a. Note that this already introduces a Yang-Mills term for the gauge field a. Z 1 8 ¼ j ~ m j ( @ ¹ ~ m ¢ @ ¹ ~ m ¡ j d a j 2 ) + ² ¹º½ ² ij k @ i ( 1 8 ¼ j ~ m j ) a ¹ @ º m j @ ½ m k = Z 1 8 ¼ j ~ m j ( @ ¹ ~ m ¢ @ ¹ ~ m ¡ j d a j 2 ) + ( ¤ d a ) ¹ ! i @ ¹ m i

30. Before integrating out the broken gauge field b, we dualize a, treating F_a as the fundamental variable. Before integrating out the broken gauge field b, we dualize a, treating F_a as the fundamental variable. Integrating out F_a leads to Integrating out F_a leads to The U(1)_b acts on the space of. The metric is nontrivial due to the quantum correction as seen. The U(1)_b acts on the space of. The metric is nontrivial due to the quantum correction as seen. Z j D ¹ Y j 2 + 1 8 ¼ j ~ m j ( @ ¹ ~ m ¢ @ ¹ ~ m ¡ j ~ F a j 2 ) + Z ~ F a ^ ( d ' Y + ! i d m i + k 2 ¼ b ) Z j D ¹ Y j 2 + 1 8 ¼ j ~ m j @ ¹ ~ m ¢ @ ¹ ~ m + 2 ¼ j ~ m j( @ ¹ ' Y + ! i @ ¹ m i + k 2 ¼ b ¹ ) 2 Y ! e i ¤ Y ; b ¹ ! b ¹ + @ ¹ ¤ ; ' Y ! ' Y ¡ k 2 ¼ ¤ Y ; ~ Y, an d ' Y. M a = C 2 = Z k + 1

31. Monopoles in the chiral ring There are monopole operators in YM-CS-matter theories, which we follow to the IR CSM. There are monopole operators in YM-CS-matter theories, which we follow to the IR CSM. In radial quantization, it is a classical background with magnetic flux, and constant scalar,. Of course, in the CSM limit, In radial quantization, it is a classical background with magnetic flux, and constant scalar,. Of course, in the CSM limit, It is crucial that Y is not charged under a. It is crucial that Y is not charged under a. Call this monopole operator T. Call this monopole operator T. R S 2 F a = 2 ¼n ¾ = n = 2 ¾ a = 1 = k (j Y j 2 ¡ j ~ Y j 2 ) [Borokhov Kapustin Wu]

32. CS induced charge of T The Chern-Simons term induces a charge for the operator T we have just defined. Writing in the monopole background, it is a particle of charge n k under U(1)_b The Chern-Simons term induces a charge for the operator T we have just defined. Writing in the monopole background, it is a particle of charge n k under U(1)_b Equivalently, in radial quantization, the Gauss’ law constraint is modified, and some matter field zero modes must be turned on. Equivalently, in radial quantization, the Gauss’ law constraint is modified, and some matter field zero modes must be turned on. 2 k 4 ¼ R a ^ db = k n R ra d i us b

33. Anomalous dimension The dimension of the monopole operator will be the sum of the two contributions The dimension of the monopole operator will be the sum of the two contributions and the dimension of the scalar fields used in the dressing. and the dimension of the scalar fields used in the dressing. This was calculated in Borokhov-Kapustin-Wu by quantizing the matter fields in the monopole background with constant This was calculated in Borokhov-Kapustin-Wu by quantizing the matter fields in the monopole background with constant Q 0 = 1 2 ( X i 2 h yper ¡ X i 2 vec t or )j q i j ¾ = n = 2

34. The result was that the spectrum of fermions from the hypermultiplets became asymmetric, The result was that the spectrum of fermions from the hypermultiplets became asymmetric, The spectrum of scalars was found to be symmetric. Thus only the fermions contributed to the vacuum energy, which is exactly the anomalous dimension of the operator. The spectrum of scalars was found to be symmetric. Thus only the fermions contributed to the vacuum energy, which is exactly the anomalous dimension of the operator. We will include the CS terms simply noting this operator is charged under the gauge group. We will include the CS terms simply noting this operator is charged under the gauge group. This is sensible since the matter fields needed to “dress” the operator are neutral under the magnetic U(1). Needed for it to be in chiral ring This is sensible since the matter fields needed to “dress” the operator are neutral under the magnetic U(1). Needed for it to be in chiral ring E + p = j n j= 2 + p ; E ¡ p = ¡ j n j= 2 ¡ p ; E 0 = + j n j= 2

35. Our example We have monopoles, the first two on the branch, and the latter pair on We have monopoles, the first two on the branch, and the latter pair on Each has one hypermultiplet charged under the associated U(1), so it gets a dimension ½ Each has one hypermultiplet charged under the associated U(1), so it gets a dimension ½ The CS induced charge of T is (0,k) under the U(1) x U(1) gauge group. The CS induced charge of T is (0,k) under the U(1) x U(1) gauge group. The chiral operators on are The chiral operators on are exactly as expected for exactly as expected for T a ; ~ T a ; T b ; ~ T b M a M b Y ~ Y, T ~ Y k, ~ TY k M a M a = C 2 = Z k + 1

36. D6 branes in AdS_4 x CP^3 We consider introducing D6 branes wrapping the AdS. This should be similar to adding D7 branes in AdS5. We consider introducing D6 branes wrapping the AdS. This should be similar to adding D7 branes in AdS5. They wrap an cycle in the internal manifold. Thus there is a Z_2 Wilson line, distinguishing two types of D6 branes. They wrap an cycle in the internal manifold. Thus there is a Z_2 Wilson line, distinguishing two types of D6 branes. One can also add D6 branes to more general N=3 AdS4 backgrounds, where they wrap One can also add D6 branes to more general N=3 AdS4 backgrounds, where they wrap RP 3 S 3 = Z n [DJ Tomasiello] CSM quiver with n nodes

37. IIB engineering Consider N D3 branes wrapping a circle and intersecting an NS5 and (1,k) 5 brane. This engineers the ABJM theory. Consider N D3 branes wrapping a circle and intersecting an NS5 and (1,k) 5 brane. This engineers the ABJM theory. Add some D5 branes, some intersecting each half of the stack of D3. Breaks supersymmetry to N=3, and adds fundamentals to the quiver. Add some D5 branes, some intersecting each half of the stack of D3. Breaks supersymmetry to N=3, and adds fundamentals to the quiver. NS5 (1,k) 5 D5

38. M-theory lift T-dualize the circle: NS5 branes turn into Taub- NUT, D5 charge become D6, D3 becomes D2. T-dualize the circle: NS5 branes turn into Taub- NUT, D5 charge become D6, D3 becomes D2. Near the D2 horizon, lift to M-theory: Near the D2 horizon, lift to M-theory: Gibbons-Gauntless-Papadopolus-Townsend showed this is purely geometry Gibbons-Gauntless-Papadopolus-Townsend showed this is purely geometry U = U 1 + 1 2 j ~ x 1 j µ 10 00 ¶ + 1 2 j ~ x 1 + k ~ x 2 j µ 1 k kk 2 ¶ + m 2 j ~ x 2 j µ 00 01 ¶

39. Hyperkahler singularity In particular, the pair of U(1) isometries of the T^2 fiber are compatible with the hyperkahler structure, and one obtains the hypertoric manifold where N is the kernel of the map In particular, the pair of U(1) isometries of the T^2 fiber are compatible with the hyperkahler structure, and one obtains the hypertoric manifold where N is the kernel of the map In the case of D6 branes in CP^3, this reduces to In the case of D6 branes in CP^3, this reduces to ¯ : U ( 1 ) n + 1 ! U ( 1 ) 2 ; ¯ = µ 11 ::: 10 p 1 p 2 ::: p n m ¶ H n + 1 === N, H 3 === U ( 1 ) ; ( m, m, k ) [Bielawski Dancer]

40. Quantum corrected geometric branch There are ordinary chiral operators of the form There are ordinary chiral operators of the form On the moduli space of diagonal matrices, the diagonal U(1)^N is unbroken, and there are monopoles operators with such magnetic fluxes. On the moduli space of diagonal matrices, the diagonal U(1)^N is unbroken, and there are monopoles operators with such magnetic fluxes. They have CS induced charge k, and anomalous dimension m/2. They have CS induced charge k, and anomalous dimension m/2. For m=1, k=1, at dimension 1, one has 8 gauge invariant operators as expected for For m=1, k=1, at dimension 1, one has 8 gauge invariant operators as expected for T r ( A i B j A k B l ) T r ( A i B j ) ; TB i ; ~ TA i H 3 ==== U ( 1 ) 111

41. Completely Higgsed branch If the number of fundamentals is at least twice the rank of the gauge groups, there is a branch in which the entire gauge symmetry is Higgsed. If the number of fundamentals is at least twice the rank of the gauge groups, there is a branch in which the entire gauge symmetry is Higgsed. This branch must have all moments set to zero, resulting exactly the ordinary Kahler quotient for the ADHM quiver of N instantons of rank m on C^2/Z_n. This branch must have all moments set to zero, resulting exactly the ordinary Kahler quotient for the ADHM quiver of N instantons of rank m on C^2/Z_n. FI parameters resolve the singularity, each node is a fractional brane that blows up into a D4. FI parameters resolve the singularity, each node is a fractional brane that blows up into a D4. k i = Z S 2 ( i ) F 2

42. Supergravity limit The volume of these 3-Sasakians is known: The volume of these 3-Sasakians is known: This implies that the radius of curvature in M- theory is given by This implies that the radius of curvature in M- theory is given by It is a warped compactification, but using the inverse of the lightest D0 mass, a typical value of It is a warped compactification, but using the inverse of the lightest D0 mass, a typical value of V o l ( M 7 ) = V o l ( S 7 ) m + 2 k 2 ( m + k ) 2 R M 7 ` p = ³ 2 6 ¼ 2 N ( m + k ) 2 m + 2 k ´ 1 6 R 2 s t r » R 3 M 7 m + k = 8 ¼ N 1 = 2 p m + 2 k

43. Stuffing fundamentals with dof It is simplest to determine the number of degrees of freedom at high temperature from the M-theory supergravity limit. It is dominated by the large AdS4 black hole, and the internal manifold only enters via the four dimension Planck scale. It is simplest to determine the number of degrees of freedom at high temperature from the M-theory supergravity limit. It is dominated by the large AdS4 black hole, and the internal manifold only enters via the four dimension Planck scale. Note the enhancement of N m! Note the enhancement of N m! ¯ F = ¡ 2 7 = 2 3 ¡ 2 ¼ 2 N 3 = 2 ( m + k ) p 2 p m + 2 k V 2 T 2 » N 2 p ¸ + 3 m N 4 p ¸ + :::