David Berenstein, UCSB DPF 2015 Meeting Based on D.B., Eric Dziankowski, R

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Spinning the fuzzy sphere David Berenstein, UCSB DPF 2015 meeting Based on D.B., Eric Dziankowski, R. Lashof Regas arXiv:1506.01722 This talk is not about this fuzzy sphere. Outline

• Where the problem comes from and why it is interesting.

• Ansatz for spinning fuzzy sphere and its consistency

• Phase diagrams Holography

Some large N ﬁeld theories are equivalent to theories of quantum gravity in higher dimensions.

Many solutions of such ﬁeld theories have geometric interpretations, even at ﬁnite N

Part of the goal is to ﬁnd more examples of such solutions. SO(3) sector of BMN matrix model

1 1 3 H = Tr(P 2 + P 2 + P 2)+ Tr (Xj + i✏ XmXn)2 2 1 2 3 2 0 jmn 1 j=1 X @ A Has SO(3) symmetry with generator

J = L = Tr(XP YP ) Z Y X

And a U(N) gauge invariance. Arises from

• BMN matrix model (B,Maldacena, Nastase 2002)

• Mass deformed N=4 SYM (Vafa, Witten; Polchinski, Strassler, …)

• Equivariant Sphere reductions of Yang Mills on sphere (Kim, Klose, Plefka, 2003) Model is chaotic

Yang Mills (on a box, with translationally invariant conﬁgurations) is chaotic.

Basenyan, Matinyan, Savvidy, Shepelyanskii, Chirikov (early 80’)

Chaos in BFSS matrix theory: Aref’eva, Medvedev, Rytchkov, Volovich (1998)

Chaos in BFSS/BMN: Asplund, D.B., Trancanelli, Dzienkowski (2011,2012), Asano, Kawai, Yoshida (2015) The solutions with H =0aregivenbyfuzzyspheres.Thesearesolutionsoftheequations

[Xi,Xj]=i✏ijkXk (6)

The solutions to these equations are characterized by direct sums of the adjoint matrices for irreducible representations of su(2). For these solutions we have P1,P2,P3 =0andthusare classically gauge invariant according to (2). These solutions carry no angular momentum as L~ =0identically. The Hamiltonian also satisﬁesHas a BPSa BPS inequality inequality bound, where H J (details can be | | found in [23]). Solutions that saturate the bound will be called extremal or BPS. This follows from writing the Hamiltonian as a sum of squares in a slightly di↵erent way 1 1 H =Tr P 2 + (P (X + i✏ [X3,X1]))2 2 3 2 1 ± 2 231 ✓ 1 1 + (P (X + i✏ [X2,X3]))2 + (X + i✏ [X1,X2])2 J (7) 2 2 ⌥ 1 123 2 3 312 ± ◆ The cross terms between X2,P1 and X1,P2 in the squares generate a copy of J that needs 3 1 to be subtracted. The cross terms withHP1 andJ [X ,X ] lead to something that does not automatically cancel for generic matrices, but | after| a bit of reshu✏ing can be shown to be proportional to Can try to look for solutions3 1 that1 saturate2 2 equality: usually these Xrotate([X ,P “rigidly”.]+[X ,P ]) (8) and we recognize the Gauss’ law constraint starting to arise. After imposing the full Gauss’ law constraint, we get Tr(X3[X3,P3]) that does vanish identically. The BPS bound is not directly related to supersymmetry. Instead it is derived from the conformal group in four dimensions, where one can show based on unitarity arguments that

E J from requiring that K = P †, where P, K are the generators of translations and special conformal transformations on S3 R. This bound asserts that the dimension of ⇥ operators is greater than the spin and is usually saturated for free theories, or nearly free theories at leading order in perturbation theory. This should be lifted in general theories (see [24] for a recent discussion). However, the bound does show up in studying supersymmetric BPS states [25, 26] for the full BMN matrix model. This bound descends to any classical solution of Yang-Mills theory which is a conformal ﬁeld theory at the classical level. Now we move on to construct solutions to (4) and (5) with non-zero J. We deﬁne the matrices

+ 1 1 X = (X + iY ),X = (X iY )(9) 2 2

4 When P=0

The minima are angular momentum matrices that satisfy

X` + i✏`mnXmXn =0

These are called fuzzy spheres. Method of solution: ansatz The solutions with H =0aregivenbyfuzzyspheres.Thesearesolutionsoftheequations

[Xi,Xj]=i✏ijkXk (6)

The solutions to these equations are characterized by direct sums of the adjoint matrices for irreducible representations of su(2). For these solutions we have P1,P2,P3 =0andthusare classically gauge invariant according to (2). These solutions carry no angular momentum as L~ =0identically. The Hamiltonian also satisﬁes a BPS inequality bound, where H J (details can be | | found in [23]). Solutions that saturate the bound will be called extremal or BPS. This follows from writing the Hamiltonian as a sum of squares in a slightly di↵erent way 1 1 H =Tr P 2 + (P (X + i✏ [X3,X1]))2 2 3 2 1 ± 2 231 ✓ 1 1 + (P (X + i✏ [X2,X3]))2 + (X + i✏ [X1,X2])2 J (7) 2 2 ⌥ 1 123 2 3 312 ± ◆ The cross terms between X2,P1 and X1,P2 in the squares generate a copy of J that needs 3 1 to be subtracted. The cross terms with P1 and [X ,X ] lead to something that does not automatically cancel for generic matrices, but after a bit of reshu✏ing can be shown to be proportional to X3([X1,P1]+[X2,P2]) (8) and we recognize the Gauss’ law constraint starting to arise. After imposing the full Gauss’ law constraint, we get Tr(X3[X3,P3]) that does vanish identically. The BPS bound is not directly related to supersymmetry. Instead it is derived from the conformal group in four dimensions, where one can show based on unitarity arguments that

E J from requiring that K = P †, where P, K are the generators of translations and special conformal transformations on S3 R. This bound asserts that the dimension of ⇥ operators is greater than the spin and is usually saturated for free theories, or nearly free theories at leading order in perturbation theory. This should be lifted in general theories (see [24] for a recent discussion). However, the bound does show up in studying supersymmetric BPS states [25, 26] for the full BMN matrix model. This bound descends to any classical solution of Yang-Mills theory which is a conformal ﬁeld theory at the classical level. Now we move on to constructeasier solutionsusing complex to (4) matrices and (5) with non-zero J. We deﬁne the matrices

+ 1 1 X = (X + iY ),X = (X iY )(9) 2 2

4 + + X = X + X,Y= i(X X)(10) Consider the ansatz We will make an ansatz for a solution more general than the fuzzy spheres

0 a1 exp(i!1t)0 ...

0 00a2 exp(i!2t) ... 1 + . . . . X (t)=B ...... C (11) B C B C B 0 ... 0 aN 1 exp(i!N 1t)C B C B C Ba exp(i! t)0 ... 0 C B N N C @ + A with ai constants, and X(t)=(X (t))† (this is the transpose complex conjugate). At the same time we also take Z time independent Z(t) = diag(z1,...,zN )(12)

independent of time and real. We will explain the origin of this ansatz in the following This rotates rigidly because there is still gauge invariance section. Notice that all fuzzy sphere ground states are solutions of this kind already, with that can let us shift the w at the expense of introducing a non-trivial ! =0,andsomeofthea = 0. Gauge transformations that commute with Z allow us to i i A_0. vary the relative phases of the ai. Therefore we can assume that they are real, and a common phase can be translated away by choosing the starting time appropriately. This ansatz is di↵erent to those that have been studied before [23] (previous works also look for solutions of the BMN and BFSS matrix model where more than three matrices are oscillating [28, 29]). Other time dependent solutions can be found in [30, 31, 33], but again, these involve more matrices being turned on. In particular, we allow for multiple frequencies to arise in the ansatz, rather than just one. This does not a↵ect fact that the system is rigidly rotating. One can use a gauge transformation to make the frequencies the same, but one pays the

price that A0, the connection in the time direction, becomes non-trivial. This is actually very useful for the BPS states, where A Z [25]. 0 / The method of solving the equations is then to ﬁrst solve for the conjugate momenta by

using equation (4). Then, we solve the Gauss’ law constraint (2) to relate the !i to each other. These can then be substituted into the angular momentum equation (3), so that we

can express the !i in terms of the total angular momentum Lz = J and the ai. One can then show that the system of equations of motion (5) reduces consistently to an algebraic

set of real equations for the ai,zi and that it has as many unknowns as there are variables. One therefore expects to generally ﬁnd a discrete (possibly empty) set of solutions to the equations. We will show eventually that this set of solutions is non-empty for all J.

5 shape of the configuration and are a small perturbation of a single fuzzy sphere configuration. After all, this is how the giant torus configurations in supergravity are constructed [18]. It turns out that the answer is no. The way to see this is the following. Assume that Z is a Hermitian matrix with eigenvalues that are not degenerate. A configuration will be rotationally invariant around the Z-axis if anaiverotationofthematricesintoeachothercanbeundonewithagaugetransformation. This is the same way that the method of images works for D-branes on orbifolds when considering a discrete subset of the rotation group [34] (this is also the mechanism for rotational invariance of monopole solutions in nonabelian gauge theories [35, 36]). That is, if for any angle ✓ we canSymmetry find a unitary matrix U(✓) such that

1 U(✓)ZU (✓)=Z

1 U(✓)XU (✓)=X cos(✓) Y sin(✓)(19) 1 U(✓)YU (✓)=Y cos(✓)+X sin(✓)

Because Z hasSet non-degenerate of unitaries (gauge eigenvalues, transformations) the first equation that tells rotate us that theU must be diagonal in the same basis that Z is (theyconfiguration commute withinto eachitself. other). We gauge transform to such abasiswithoutlossofgenerality.Theangle✓ will be identified with the time evolution Ansatz we have is the only one that preserves a itself later on. The uniform rotation motion of the configuration requires that Z is time ZN independent, so that Z˙ =0. Let 1 ,..., N denote the eigenvectors of Z. It is also convenient to rewrite the last two | i | i equations of (19) in terms of the general matrix X+ = X + iY and it’s adjoint. The matrix X+ is unconstrained. Then we have that

+ 1 + U(✓)X U (✓)=exp(i✓)X (20)

Using a general expression for X+ m = X+ n ,andU diag(exp(i✓ )) we ﬁnd that X+ | i mn | i ' i mn transforms under conjugation by U as

X+ exp(i(✓ ✓ ))X+ (21) mn ! m n mn which can only be equal to exp(i✓)X if ✓ ✓ = ✓ mod (2⇡)orifX+ =0.Picking✓ mn m n mn arbitrarily small, we can only satisfy ✓ ✓ = ✓ for some of the components, and because m n ✓ is very small, we can arrange the kets n such that the ✓ are strictly decreasing as we | i n + increase n. This shows that U(1) invariance requires Xmn to be upper triangular with zeros

8 Gauss law constraint states that

a 2! = a 2! | i| i | j| j An direct computation shows that

J =4N! a 2 i| i| + and similarly, the terms with two copies of X or X cancel even before taking the trace. We ﬁnd that

+ + 2 2 J =2Tr(X X⌦ + X⌦X )=2Tr(diag(!i ai + !i 1 ai 1 )) (35) | | | | Now, notice that because of Gauss’ law constraint, all the ! a 2 are equal to each other. i| i| Thus, even before taking the trace, the matrix version of J is proportional to the identity. We can interpret this as having uniform density of angular momentum per unit D0-brane of the corresponding fuzzy membrane. We use this to ﬁnd that

J =4N! a 2 (36) i| i| so that we can substitute J ! = (37) i 4N a 2 | i| Knowing the a ,thez and J, we can evaluate the energy by substituting the above results | i| i in the Hamiltonian (1). The result for the kinetic energy is 1 1 E (J, a )= Tr(P 2 + P 2 )= Tr(P + iP )(P iP )(38) kin i X Y X + Y X Y | and| similarly,2 the terms with two2 copies of X or X cancel even before taking the trace. We find that4 + = Tr(⌦X X⌦)(39) 2 + + 2 2 NJ =2Tr(X X⌦ + X⌦NX )=2Tr(diag(!i ai + !i 1 ai 1 )) (35) 2 | | | | 2 2 1 J + =2 ai !i = 2 (40) and similarly, the terms with two copies of X or XNow, cancel notice even that| because before| of taking Gauss’8N 2 the law trace. constraint,a 2 all the !i ai are equal to each other. i=1 i=1 | i| | | We find that Thus, evenX before taking the trace, theX matrix version of J is proportional to the identity. What is important to realizeWe can is interpret that this as looks having veryuniform similar density of to angular an angular momentum momentum per unit D0-brane cen- + + 2 2 J =2Tr(X X⌦ + X⌦X )=2Tr(diag(of the! correspondingi ai + !i 1 fuzzyai 1 membrane.)) We (35) use this to find that trifugal potential, where each| particle| | (associated | to the radial variable ai)hasthesame 2 2 J =4N!i ai (36) Now, notice that becauseangular of Gauss’ momentum law constraint,J/N alland the the!i a samei are equalmass to (in each this other. case| the| mass would be interpreted | | Thus, even before taking the trace, the matrix versionso that of J weis can proportional substitute to the identity. as 4). To specify the full problem, we need to evaluateJ the potential energy as well. A ! = (37) We can interpret this as having uniform density of angular momentum per unit D0-branei 4N a 2 straightforward, though rather tedious procedure shows that| i| of the corresponding fuzzy membrane. We use this toKnowing find that the a ,thez and J, we can evaluate the energy by substituting the above results | i| i in theN Hamiltonian1 (1). The result for the kinetic energy is J =4N! a 2 2 2 2(36) 2 2 V ( ai ,zii)=i [zi +2ai 1 12 ai ] +2(1+1 zi+1 zi) ai (41) | | | | 2 | | | 2| 2 | | i=1 Ekin(J, ai )= Tr(PX + PY )= Tr(PX + iPY )(PX iPY )(38) X | | 2 2 so that we can substitute 4 + One can show that solutions are= equivalentTr(⌦X X⌦ )(39)to having critical points where the i are definedJ modulo N. We denote2 the full energy of a configuration (J, ai ,zi) ! = N (37) N i 2 of the energy function 1 J 2 | | 4N ai =2 a 2!2 = (40) by | | | i| i 8N 2 a 2 Knowing the a ,thez and J, we can evaluate the energy by substituting the abovei=1 results i=1 | i| | i| i X X What(J, is importanta ,z)= to realizeE ( isJ, thata this)+ looksV ( verya ,z similar)(42) to an angular momentum cen- in the Hamiltonian (1). The result for the kinetic energyE is| i| i kin | i| | i| i trifugal potential, where each particle (associated to the radial variable ai)hasthesame 1 1 E (J, aWhen)= Tr( weP 2 consider+ P 2 )= theTr(angular energyP + momentumiP function)(P J/NiP and)(38)(J, theai same,zi)andrecallthat mass (in this case the massPZ =0inouransatz, would be interpreted kin | i| 2 X Y 2 X Y X EY | | as 4). To specify the full problem, we need to evaluate the potential energy as well. A it is straightforward4 + to notice that the equations of motion of the Z variables are exactly the = Tr(⌦X X⌦)(39) 2 straightforward, though rather tedious procedure shows that N N 1 J 2 N 2 2 12 1 2 2 2 2 2 =2 ai !i = 2 2 V ( ai ,zi)= [zi +2ai 1(40) 2 ai ] +2(1+zi+1 zi) ai (41) | | 8N ai | | 2 | | | | | | i=1 i=1 i=1 X X | | X where the i are defined modulo N. We denote the full energy of a configuration (J, a ,z) What is important to realize is that this looks very similar to an angular momentum cen- | i| i by trifugal potential, where each particle (associated to the radialSolve variable a nastyai )hasthesamealgebra problem (J, ai ,zi)=Ekin(J, ai )+V ( ai ,zi)(42) angular momentum J/N and the same mass (in this case the mass wouldE be| | interpreted | | | | When we consider the energy function (J, a ,z)andrecallthatP =0inouransatz, as 4). To specify the full problem, we need to evaluate the potential energy as well.E A| i| i Z it is straightforward to notice that the equations of motion of the Z variables are exactly the straightforward, though rather tedious procedure shows that 12 N 1 2 2 2 2 2 V ( ai ,zi)= [zi +2ai 1 2 ai ] +2(1+zi+1 zi) ai (41) | | 2 | | | | | | i=1 X where the i are defined modulo N. We denote the full energy of a configuration (J, a ,z) | i| i by (J, a ,z)=E (J, a )+V ( a ,z)(42) E | i| i kin | i| | i| i When we consider the energy function (J, a ,z)andrecallthatP =0inouransatz, E | i| i Z it is straightforward to notice that the equations of motion of the Z variables are exactly the

12 The fact that they are critical points of a function that depends on J implies that one can use Morse theory to analyze the problem: solutions come in continuous families parametrized by J and they can disappear only at multi-critical values (special values of J)

The z appear as a quadratic form, so they can be eliminated as functions of the a. at r =1,andalwaysE 0, then the energy must be proportional to r2(1 r)2. This has amaximumatr =1/2, which is the sphere at half radius. This solution also matches the solution at Q =0thatwealreadyhadwhenwesetP =1/4. There is similarly a reflected solution with Q<0, where we take the other square root branch cut of equation (52). The new solution of the fuzzy sphere at half radius migrates to higher angular momentum as we increase P from 1/16 to 1/4andisasaddlepoint.ThereforeithasMorseindexone.This solution and the one that is reflected by taking Q Q can cancel the two minima from ! the BPS solution as they meet at J =1withthetrivialsolutionthathasZ =0throughout. The full set of solutions is depicted in figure 1. There we can see that for low angular momenta there are five critical points. Three are minima and two are saddles with Morse index one. The saddles and two of the minima are reflected into each other by the Z2 symmetry, and one minimum if at the fixed point. The two saddles and two of the minima annihilate each other when J = 1. Because the minima that annihilate with the saddles are BPS, the saddles need to approach the extremal limit and thus must touch the fixed point set, because one can not descend further from the saddle to the fixed point otherwise.

a1 | |0.6 0.31

0.5 0.33

0.305 0.4 2x2 matrices solvable analytically.

0.3 0.31

0.2 0.315 0.305 0.325 0.305 0.32 0.31

0.1

0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 a | 2| FIG. 1. Parametric plot of the solutions for a1,a2 derived either from equation (52), from P =1/4 or from the solution Q =0witha1 = a2. Superposed we ﬁnd level sets of the energy function at J =0.3, with the energy levels shown, and we see that the curves pass through the critical points of the energy function.

17 0.435

0.430

0.425

0.420

0.415 0.390 0.395 0.400 0.405 0.410 0.415 0.420

FIG. 4. Energy contours in a , a = a at J =2.1. The saddles are hard to see. The energy | 1| | 2| | 3| contours have energies starting at E =2.1 and spaced at E =2 10 7 showing ten contours. ai ⇥ “Phase diagram” for 3x3 | | matrices 0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0 0.5 1.0 1.5 2.0

FIG. 5. Full Phase diagram of solutions as a function of J on the abscissa,J and the ai of a single | | solution are plotted on the ordinate. Each color is a different solution this should persist in the analytic continuation to the right. This indicates that there should be a saddle of index 0 joining BC. This process is depicted in figure 4 Once we include the new saddles after this point, we can complete the phase diagram in figure 5. What we see from the figure is that first, there is a maximal angular momentum after which there is only one solution, and this solution preserves the maximal dihedral symmetry. Second, the trajectory of the maximal sphere goes through a phase transition at a finite J before reaching this maximal angular momentum. We have checked that after the first transition

23 FIG. 6. On the left, partial phase diagram of 4 4 solutions as a function of J on the abscissa, and ⇥ FIG. 6. Onthe theSomea left,of aa partial single values solution phase for diagram plotted 4x4 on of the 4 ordinate4 solutionsSome in the as same z a values function color. Notice for of J 4x4 theon similarity the abscissa, of the and | i| ⇥ the a of atransitions single solution to the ones plotted for 3 on3 the matrices ordinate in figure in the 3. On same the right, color. we Notice evaluate the also similarity the zi values of the | i| ⇥ for the given solution of the a . The figure is not symmetric under z z which indicates that transitions to theThere ones for is 3always3 matrices ia critical in figure value 3. On after the right,which we ionly evaluate i one also the zi values ⇥ | | ! many of the phases are not ‘parity invariant’ with respect to the dihedral discrete symmetry group. for the given solution of the a . The figuresolution is not remains symmetric under z z which indicates that | i| i ! i many of thephase. phases Because are not there ‘parity are many invariant’ more with solutions respect at J to=0ateach the dihedralN (the discrete number symmetry is bounded group. below at least by twice the partitions of N), the full phase diagram is more complicated as phase. Becausewe increase thereN are, and many it is clear more from solutions the Morse at analysisJ =0ateach that weN are(the missing number quite a is number bounded below at leastof saddles. by twice Although the partitions two new solutions of N), always the full exists phase for any diagramN, the is maximal more complicated fuzzy sphere as we increaseandN the, and sphere it is at clear half from size, one the can Morse expect analysis in general that that we there are missing are quite quite a number a number of intrinsically new (indecomposable) solutions that appear at any N. of saddles. Although two new solutions always exists for any N, the maximal fuzzy sphere For illustration purposes, we also show some of the solutions for 10 10 matrices in figure and the sphere at half size, one can expect in general that there are⇥ quite a number of

intrinsically new (indecomposable)4 solutions that appear at any N. For illustration purposes, we also show some of the solutions for 10 10 matrices in ﬁgure 2 ⇥

4 10 20 30 40 50 60

-2 2

-4

10 20 30 40 50 60

FIG. 7. We evaluate zi(J) for three di↵erent solutions of 10 10 matrices, the maximal fuzzy -2 ⇥ sphere, a near maximal fuzzy sphere bordered by zero, and a solution for a fuzzy sphere of spin 7/2 1/2. -4

25 FIG. 7. We evaluate z (J) for three di↵erent solutions of 10 10 matrices, the maximal fuzzy i ⇥ sphere, a near maximal fuzzy sphere bordered by zero, and a solution for a fuzzy sphere of spin 7/2 1/2.

25 7. This shows the procedure for building up solutions by bordering by zero or adding up previous solutions in more detail. It is clear that the full phase diagram is quite complicated, some of the solutions ending in ﬁrst order phase transitions, and some others ending in what appear to be multi-critical points. Here we basically show that the problem is amenable to computer calculations. We should point out that for larger N we do not have a good strategy to search for the intrinsically new solutions yet. The graphical method that worked for N =3isunsuitedforhigherdimensions,aswecannotvisualizethedata.

VIII. LARGE N

7. This shows the procedure for building up solutions by bordering by zero or adding up An interestingprevious way to solutions proceed in more todetail. large It is clearN thatis the the full following. phase diagram Since is quite complicated, the energy function in some of the solutions ending in first order phase transitions, and some others ending in what equation (41) is ofappear nearest to be multi-critical neighbor points. type, Here we we can basically think show that of it the as problem a discretized is amenable version of an energy which is anto computer integral calculations. of an energy We should density. point out that Indeed, for larger thisN we is do what not have the a good interpretation of strategy to search for the intrinsically new solutions yet. The graphical method that worked the solutions as offor matrixN =3isunsuitedforhigherdimensions,aswecannotvisualizethedata. mechanics as discretized membranes indicates we should be doing 2 2 [44, 45]. With that in mind, we want to replace zi z˜(✓)andthesamefor ai a˜ (✓), VIII. LARGE N ! | | ! | | where ✓ is a periodic coordinate with period one, rather than a discrete set with N elements. We basically take iAn interestingN✓. Then way to expressions proceed to large N ofis nearest the following. neighbor Since the energy di↵erences function in get replaced by equation' (41) is of nearest neighbor type, we can think of it as a discretized version of an 1 derivatives zi+1 energyzi whichN is@ an✓z integral˜. We of also an energy need density. to remember Indeed, this is what that the the interpretation maximal of fuzzy spheres the solutions! as of matrix mechanics as discretized membranes indicates we should be doing are of size N when we have N D0-branes, and we want to rescale this out of the energy. [44, 45]. With that in mind, we want to replace z z˜(✓)andthesamefor a 2 a˜ 2(✓), i ! | i| ! | | where ✓ is a periodic coordinate with2 period one,2 rather2 than a discrete set with N elements. Therefore, we write zi Nz(✓), and ai N a (✓)and i = N d✓. When we do this, We basically' take i N✓. Then| expressions| ' of| nearest| neighbor di↵erences get replaced by ' we find that 1 derivatives z z N @ z˜. We also need to remember thatP the maximal fuzzyR spheres Can alsoi+1 take i ! large ✓N (potential is nearest neighbor finite are of size N when we have N D0-branes, and we want to rescale this out of the energy. differences),21 2 then2 2 Therefore, we write zi Nz3(✓), and ai N a (✓)and 2 = N d✓. When we do2 this,2 Vpot = V ( ai ,zi) 'N d✓| | ' z | | 2@✓ a i +2(1+@✓z) a (68) we find that| | ! 2 | | | | Z  P R 1 2 V = V ( a ,z) N 3 d✓ z 2@ a 2 +2(1+@ z)2 a 2 (68) 3 and similarly, the terms thatpot contain| i| i ! angular2 momentum, ✓| | when✓ | we| rescale J N |˜,give Z  ! and similarly, the terms that contain angular momentum, when we rescale J N 3|˜,give us ! us 2 3 2 |˜ Ekin = N 3 d✓ |˜ (69) Ekin = N d✓ 2 (69) 88a(a✓)(2✓) ZZ | | | | The point is that in this rescaling, we get a common factor of N 3 in front of the energy that The point is that in this rescaling, weConstrained get a common to factor of N 3 in front of the energy that can be dropped if we want to, to obtain a classical membrane energy which is independent can be dropped ifof weN. The want value to, of N tocan obtain be changed a e classical↵ectively by membrane changing the periodicity energy of ✓ whichwithout is independent

changing the energy density further. Wed✓ use=1 the relation Ntot = N d✓ to convert the of N. The value of N can be changed e↵ectively by changing the periodicity of ✓ without changing period of ✓ into a di↵erentZ value of N, and we can always changeR scales by taking changing the energy density further. We use26 the relation Ntot = N d✓ to convert the changing period of ✓ into a di↵erent value of N, and we can always changeR scales by taking

26 Can do a variational principle and interpret in terms of a non-trivial dynamics on a curved space with a background magnetic ﬁeld and a non-trivial effective potential |a|2 1.5

Need to shoot for periodic orbits of the right period

1.0

0.5

z -1.5 -1.0 -0.5 0.5 1.0 1.5

FIG. 8. Parametric plots of (z(✓), a 2(✓)) for various initial conditions at fixed value of| ˜2 =0.1. | | We show a BPS trajectory in purple, and examples of scanning over parameters to find periodic trajectories in blue and red. The fixed point is marked. The green solution which was found by scanning winds twice around the fixed point and it is not BPS.

any period. Other periodic orbits need to be found by trial and error, and once a suciently approximate solution of the periodicity condition is found it is possible to zoom into it.

Particularly simple examples of this search can be performed if the solutions are reﬂection symmetric under z z. Then the solution can be characterized by the value of a 2(✓ ), ! | | 0 when z(✓ )=0,andtheconditionforsymmetryforces@ a2 (✓ )=0.Wecanthenmove 0 ✓| | 0 the velocity of @✓z(✓0)=⇠ as a scanning parameter. Some examples of this procedure are depicted in ﬁgure 8.

When the orbit returns to z(✓1)=0atsomelatertime(notnecessarilytheﬁrsttime around), we can the compute  = @ a 2(✓ )fromsolvingtheequationsofmotion,andas ✓| | 1 we scan over ⇠ we look for sign changes in . We can then zoom in for the parameter ⇠ that solves the periodicity condition. The point of intersection can be the same one as before or it can be di↵erent. If it is di↵erent we double the time of this half orbit, and it becomes periodic.

It is also interesting to study the ﬁrst order di↵erential equations that arise for states

29 that saturate the BPS inequality at ﬁnite J. These equations are given by

|˜ that saturate the BPS inequality at finite@ z =J.1+ These equations are given by (78) For BPS✓ states, it is4 simpler:a 2 | | 2 1 |˜ @✓ @a z== z 1+ (79) (78) | |✓ 2 4 a 2 | | and are obtained from requiring the vanishing2 of1 the squares in equation (7) after substituting @✓ a = z (79) our ansatz. These can in turn be derived| | from2 a variational principle for an auxiliary andLagrangian are obtained of fromthe form requiringEquivalent the to a vanishingdynamics with of the the Following squares Lagrangian in equation (7) after substituting |˜ L =˙q2 q + log q (80) our ansatz. These can in turn be derived from4 a variational principle for an auxiliary 2 Where Lagrangianwhere q = ofa the, and formz plays the role of the canonical conjugate of q (namely z =2˙q = pq). | | q = a2 2 | ||˜|˜ The e↵ective one dimensional potentialL =˙q(q)=qq+ 4 loglog qq is bounded from below and goes (80) U 4 to infinity at q 0andalsoatq , so it produces automatically closed periodic orbits where q = a 2, and! z plays the role!1 of the canonical conjugate of q (namely z =2˙q = p ). without| self| intersections. There is also only one minimum at q =˜|/4. q The e↵ective one dimensional potential (q)=q |˜ log q is bounded from below and goes These trajectories are interpreted geometricallyU as multiply4 wrapped tori that wrap the to infinitysame torus, at q where0andalsoat the wrappingq number, so is the it produces number of automatically times we have to closed go around periodic the orbits ! !1 withoutorbit self so that intersections. the period matches There is the also number only of one D0-branes. minimum at q =˜|/4. TheseThe trajectories period of the are orbit interpreted can be calculated geometrically using standard as multiply techniques wrapped as follows: tori that wrap the dq dq same torus, where the wrapping number⇥ = is the number of times we have to go around(81) the q˙ / z orbit so that the period matches the numberI ofI D0-branes. where the integral is over a periodic orbit, which is characterized by the parametric equation The period of the orbit can be calculated using standard techniques as follows:

2 |˜ 2 |˜ =˙q + q log(qdq)=z + qdq log(q)(82) W ⇥ =4 4 (81) q˙ / z where is a constant of integration (theI energy associatedI to the lagrangian function L). W whereThis the solves integral for z isas over a function a periodic of q orbit,readily. which Indeed, is characterized the parameter by thethis parametric way defines equation a W curve with a di↵erential, and then ⇥ is|˜ the period integral over|˜ the di↵erential. We have =˙q2 + q log(q)=z2 + q log(q)(82) to be careful with thisW interpretation: the4 curve is real analytic 4 and not a complex curve. More general solutions for the BPS states in the continuum limit were found in [26], and where is a constant of integration (the energy associated to the lagrangian function L). theyW similarly show up with various logarithms 3. This solves for z as a function of q readily. Indeed, the parameter this way defines a Because (q)hasanon-trivialthirdderivativeabouttheminimum,itispossibletoshowW curve with a diU↵erential, and then ⇥ is the period integral over the di↵erential. We have that the period for orbits very near the fixed point have a decreasing period as we go away to be careful with this interpretation: the curve is real analytic and not a complex curve. 3 The variable W in their work is related to q in ours, with q W 2 More general solutions for the BPS states in the continuum' limit were found in [26], and 3 they similarly show up with various logarithms30 . Because (q)hasanon-trivialthirdderivativeabouttheminimum,itispossibletoshow U that the period for orbits very near the fixed point have a decreasing period as we go away

3 The variable W in their work is related to q in ours, with q W 2 '

30 that saturate the BPS inequality at ﬁnite J. These equations are given by

|˜ @ z = 1+ (78) ✓ 4 a 2 | | 1 @ a 2 = z (79) ✓| | 2 and are obtained from requiring the vanishing of the squares in equation (7) after substituting our ansatz. These can in turn be derived from a variational principle for an auxiliary Lagrangian of the form |˜ L =˙q2 q + log q (80) 4 where q = a 2, and z plays the role of the canonical conjugate of q (namely z =2˙q = p ). | | q The e↵ective one dimensional potential (q)=q |˜ log q is bounded from below and goes U 4 to inﬁnity at q 0andalsoatq , so it produces automatically closed periodic orbits ! !1 without self intersections. There is also only one minimum at q =˜|/4. These trajectories are interpreted geometrically as multiply wrapped tori that wrap the same torus, where the wrapping number is the number of times we have to go around the orbit so that the period matches the number of D0-branes. The period of the orbit can be calculated using standard techniques as follows:

dq dq ⇥ = (81) One gets a constantq˙ / ofz motion I I where the integral is over a periodic orbit, which is characterized by the parametric equation

|˜ |˜ =˙q2 + q log(q)=z2 + q log(q)(82) W 4 4 where is a constant of integration (the energy associated to the lagrangian function L). W Shape of the 2-dimensional D-brane in 3-dimensions. This solves forIt’sz rotationallyas a function invariant of q readily. 2-torus Indeed, (possibly the parameter multiply wound).this way deﬁnes a W curve with a di↵erential, and then ⇥ is the period integral over the di↵erential. We have to be careful with this interpretation: the curve is real analytic and not a complex curve. More general solutions for the BPS states in the continuum limit were found in [26], and they similarly show up with various logarithms 3. Because (q)hasanon-trivialthirdderivativeabouttheminimum,itispossibletoshow U that the period for orbits very near the ﬁxed point have a decreasing period as we go away

3 The variable W in their work is related to q in ours, with q W 2 '

30 We go from sphere to torus at ﬁnite J.

Torus is similar to giant torus of Nishioka+Takayanagi (2008).

It arises from condensation of strings that stretch between north and south pole. For some 4x4 matrices, ﬁnding effective geometry we get a very non- round torus.

FIG. 11. Transition from a very distorted sphere to a very distorted tours for 4 4 matrices. The ⇥ values are at J 4.24 and J 4.31 for the maximal sphere. Very deformed' ' geometry due to finiteness of N. sphere transitioning to a torus near the phase transition for the maximal sphere, but before it. From these examples it should be clear that as far as the bosonic degrees of freedom are concerned, the change of topology from a sphere to a torus is smooth. Moreover, it does not occur immediately as in the large N limit. The fermions do detect the topology change. The tori that are obtained this way for the maximal sphere for such small values of N are very distorted. This should improve as we increase N. One could also analyze the geometry and topology in terms of the ideas of fuzzy Riemann surfaces found in [49, 50]. For the analysis of topology, one uses properties of the eigenvalues of Z interpreted as a Morse function. This only works for very large N. As far as the geometry is concerned, we find that the finite matrices for some suciently classical states (at suciently large N again) would give rise to fuzzy approximations to (82) for some + values of W , where q (X X)isinterpretedasamatrixandisanormalorderedformof ' the product. These solutions are not algebro-geometric in nature because of the logarithm. In this case q and Z commute and can be thought of as classical variables on the torus that can be constrained by an equation. Deviations from satisfying the precise equation W = z2 + q 1/4˜| log(q)forfixedW should then be interpreted as “quantum corrections” in 1/N . These are not quantum in the sense of due to a non-trivial vale of the Planck constant ~, but should be thought of instead of as quantum corrections to geometry due

34 Conclusions

• A method for ﬁnding interesting solutions of BMN matrix model is proposed.

• It looks at critical points of an energy function, for which a lot to mathematics can be used to analyze solutions and follow them numerically.

• These can also be interpreted geometrically in higher dimensions and show very deformed tori, which can be interpreted asa quantum corrected geometry for ﬁnite N.