Phases of five-dimensional theories, monopole walls, and melting crystals

Abstract

(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)

Moduli spaces of doubly periodic monopoles, also called monopole walls or monowalls, are hyperkähler; thus, when four-dimensional, they are self-dual gravitational instantons. We find all monowalls with lowest number of moduli. Their moduli spaces can be identified, on the one hand, with Coulomb branches of five-dimensional supersymmetric quantum field theories on ... ^sup 3^ × T ^sup 2^ and, on the other hand, with moduli spaces of local Calabi-Yau metrics on the canonical bundle of a del Pezzo surface. We explore the asymptotic metric of these moduli spaces and compare our results with Seiberg's low energy description of the five-dimensional quantum theories. We also give a natural description of the phase structure of general monowall moduli spaces in terms of triangulations of Newton polygons, secondary polyhedra, and associahedral projections of secondary fans.

Published for SISSA by Springer

Received: March 27, 2014 Accepted: May 18, 2014 Published: June 4, 2014

Phases of ve-dimensional theories, monopole walls, and melting crystals

Sergey A. CherkisDepartment of Mathematics, University of Arizona, 617 N Santa Rita Ave, Tucson 85716, U.S.A.

E-mail: mailto:[email protected]

Web End [email protected]

Abstract: Moduli spaces of doubly periodic monopoles, also called monopole walls or monowalls, are hyperkahler; thus, when four-dimensional, they are self-dual gravitational instantons. We nd all monowalls with lowest number of moduli. Their moduli spaces can be identied, on the one hand, with Coulomb branches of ve-dimensional supersymmetric quantum eld theories on R3 [notdef] T 2 and, on the other hand, with moduli spaces of local

Calabi-Yau metrics on the canonical bundle of a del Pezzo surface. We explore the asymptotic metric of these moduli spaces and compare our results with Seibergs low energy description of the ve-dimensional quantum theories. We also give a natural description of the phase structure of general monowall moduli spaces in terms of triangulations of Newton polygons, secondary polyhedra, and associahedral projections of secondary fans.

Keywords: Solitons Monopoles and Instantons, Duality in Gauge Field Theories, String Duality

ArXiv ePrint: 1402.7117In memory of Andrei Zelevinsky

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP06(2014)027

Web End =10.1007/JHEP06(2014)027

JHEP06(2014)027

Contents

1 Introduction 1

2 Monowalls and their moduli spaces 32.1 Physical setup 32.2 Mathematical problem 42.3 Dirac monowall 52.4 Moduli problem 62.5 Spectral description and moduli space isometry 72.6 Monowalls fusion 8

3 Monopole walls with four moduli 93.1 Monopole walls with no moduli 103.2 Monopole walls with four moduli 103.2.1 Maximal side of integer length four 113.2.2 Maximal side of integer length three 113.2.3 Maximal side of integer length two 113.2.4 Maximal side of integer length one 123.2.5 Complete list 143.3 Additional equivalence 14

4 Relation to gauge theories and Calabi-Yau moduli spaces 174.1 Five-dimensional theories 174.2 Calabi-Yau moduli spaces 184.3 String theory dualities and the two spectral curves 20

5 Phase space of a monowall 245.1 Litvinov-Maslovs dequantization or tropical geometry 245.2 Amoebas, melting crystals, and the Kahler potential 255.3 The low-dimensional test 275.4 Secondary fan and the phase space 295.5 Comparison 32

1 Introduction

Most known self-dual gravitational instantons admit realizations as moduli spaces. Moreover, usually a gravitational instanton can be viewed as a moduli space in more then one way. Such realizations are very useful in studying their geometry and topology. In particular, they can be represented as moduli spaces of solutions of the self-duality equation

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Type of the

Moduli Space

Transform Nahm Equations (Bows) ALG Periodic Monopoles Hitchin System (Slings)

ALH Doubly-periodic Monopoles Doubly-periodic Monopoles

Table 1. Self-dual Gravitational Instantons as Moduli Spaces.

for Yang-Mills elds or its dimensional reductions. This is a particularly convenient point of view, since antihermitian connections on a hyperkahler space (in particular on the Euclidean space with appropriate boundary conditions) form an innite-dimensional a ne hyperkahler space. This innite-dimensional space of connections carries the triholomorphic action of the group of gauge transformations and the self-dual Yang-Mills equations are the vanishing moment map conditions for this group action. As a result, the moduli space of self-dual connections, up to gauge equivalence, is an innite hyperkahler quotient and thus, itself carries a hyperkahler metric. Whenever it is of real dimension four, its Riemann tensor is self-dual and it is a self-dual gravitational instanton.

The type of self-dual Yang-Mills solutions to consider is dictated by the desired asymptotic behavior of the moduli space. This correspondence is presented in table 1. The four types of the moduli spaces here are distinguished by their volume growth. We distinguish these spaces by how fast the volume of a ball of geodesic radius R centered at some xed point p grows with R. A noncompact self-dual gravitational instanton space is of 1) ALE,2) ALF, 3) ALG, or 4) ALH type if the volume growth is, respectively, 1) quartic, 2) cubic,3) lower than cubic and no less than quadratic, and 4) lower than quadratic.

ALE spaces, such as Eguchi-Hanson space, are moduli spaces of four-dimensional ob

jects: instantons or of zero-dimensional objects: quivers. ALF spaces are moduli spaces of three-dimensional monopoles or of a system of ODEs called the Nahm equations [14]. ALG spaces are moduli spaces of periodic monopoles or of two-dimensional Hitchin systems [5, 6]. ALH spaces, in this view, appear as moduli spaces of doubly periodic monopoles. Thereby, in the pursuit of gravitational instantons we are led to doubly periodic monopoles, also called monopole walls, or monowalls for short. If in all previous cases (as indicated in table 1) the Nahm transform produces a simpler, lower-dimensional object, in the case of ALH space the Nahm transform [7, 8], when applied to a doubly periodic monopole, produces another doubly periodic monopole. Thus we are destined to face the monowall.

In a more extended view, not captured by table 1 above, some ALG spaces appear as moduli spaces of (Zn equivariant) doubly periodic instantons, or, equivalently, under the Nahm transform, of (Zn equavariant) Hitchin systems on a two-torus [9]. The possible values of n in Zn are 2, 3, 4, and 6 and the corresponding instanton gauge groups are

SU(4), SU(3), SU(4), and SU(6). If ! = exp(2i/n) and (z, v) are linear coordinates on R2 [notdef] T 2 [similarequal] C [notdef] (C/(Z + Z)) , then the instanton equivariance condition is A(z, v) =

U1A(!z, !v)U, with U given in terms of the j [notdef] j shift matrices Sj by respectively U =

2

Self-dual Yang-Mills Solution Dual Equivalent Description

ALE Instantons ADHM Equations (Quivers)

ALF Monopoles

ADHM-Nahm

!

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14, 13, 12 [notdef] S2, and 11 [notdef] S2 [notdef] S3. On the Hitchin system side, on the other hand, the

SU(n) Hitchin data ([notdef] = A d + A

d

) satisfy

(! ) = !1Sn

( )S1 and A (! ) = !1SA ( )S1. The intersection diagram of the compact two-cycles of one of these ALG spaces is respectively D4, E6, E7, and E8 a ne Dynkin diagram.

At least one case of an ALH space,1 a hyperkahler deformation of (T 3[notdef]R)/Z2, appears

as a moduli space of triply periodic U(2) monopole with two positive and two negative Dirac singularities. The orbifold limit is reached when one positive singularity is placed atop of a negative one, while the other pair of positive and negative singularities is placed on top of each other at the diametrically opposite point in T 3.

Monowalls were explored analytically and in terms of D-brane congurations in [10] and numerically in [11]. More recently, the asymptotic metric on the moduli space of certain monowalls was computed in [12].

In [13] we associated to each monopole wall a decorated Newton polygon and found that dimension of the monopole wall moduli space is four times the number of internal points of its Newton polygon. We also found that there is a GL(2, Z) action on monopole walls and their Newton polygons that is isometric on their moduli spaces. In the study of gravitational instantons one is interested in four-dimensional moduli spaces. Thus, after reviewing the monowall problem in section 2, we identify all monowalls with no moduli and all monowalls with four moduli in section 3. We nd that all Newton polygons corresponding to monowalls with four moduli are reexive. Furthermore, we nd that some of these moduli spaces are isometric, ending with eight distinct moduli spaces. After discussing their signicance in eld theory and string theory in section 4, we conclude by establishing the phase structure of these moduli spaces in section 5.

2 Monowalls and their moduli spaces

2.1 Physical setup

We consider a monowall, also called a monopole wall, as dened in [13]. In physical terms considering the Yang-Mills-Higgs theory with the action

S[A, ] = 1

g2

ZR[notdef]M3

3

,

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12F F + D D [parenrightbigg]

d4Vol, (2.1)

and the space M3, we ask what magnetically charged static solutions it can have. In particular, if the gauge group outside of a certain ball is broken by the Higgs eld and the space M3 = R3 a typical magnetic eld decays as 1/r2, inversely proportional to the square of the distance from the magetic source a monopole. If the space is M3 = S1 [notdef] R2, as considered in [5], then the magnetic eld decays as 1/r, inversely proportional to the distance from the magnetic source a periodic monopole.

In this paper we consider the case M = T 2 [notdef] R, and the magnetic eld produced by a

compact source is directed along the R component and is constant. In the Maxwell-Higgs case the gauge group is U(1) and this magnetic eld has integer ux through the torus T 2.

1Explored in collaboration with Marcos Jardim.

In the Yang-Mills-Higgs case, however, there is a number of uxes of the magnetic eld. Moreover, each of them does not have to be integer, it can be rational. If the gauge group U(n) is completely broken by the Higgs eld outside of some ball, then one expects one set of magnetic uxes to the right of the ball, and possibly di erent set of magnetic uxes to the left of the ball. This is the monopole wall, or the monowall. In other words, monowall is a domain wall separating two regions with Higgs broken gauge group and di erent constant magnetic eld. Due to its non-abelian nature, such a monowall can be entirely smooth in the interior. Moreover, it can have internal excitations, which we call moduli. The main goal of [13] was to count the number of such excitations, given the xed magnetic uxes, also called monowall charges, to the left and to the right of the monowall. Below we identify all monowalls which a rigid, i.e. which have no internal excitations at all. Then we proceed to classify monowalls with minimal nonzero number of such excitations, which is four.

All of the asymptotic magnetic uxes, as well as other asymptotic parameters dening the problem, we call parameters as opposed to moduli. These are dynamically xed, as it would require innite energy to change any of them. In other words, what would be their e ective kinetic terms are innite. The latter part of the paper is concerned with the dependence of the monowall dynamics on the choice of these asymptotic parameters. In particular, we identify phases in the parameter space which distinguish di erent monowall dynamics.

2.2 Mathematical problem

Mathematically, a monowall is a Hermitian bundle E ! T 2 [notdef] R with a connection (the

gauge eld) A and an endomorphism (the Higgs eld) satisfying the Bogomolny equation

dA + A ^ A = (d + [A, ]), (2.2) and the asymptotic eigenvalues of growing at most linearly along the R component. We denote the linear coordinate along R by z, while the two periodic coordinates on the torus

T 2 are x and y with respective periods S and R, i.e. x x + S and y y + R.

The Bogomolny equation can be viewed as the zero level moment map condition for the hyperkahler reduction of the a ne space of pairs [notdef](A, )[notdef] by the group of gauge

transformations. Thus the space of gauge equivalence classes of its solutions inherits a hyperkahler metric from

| (A, )[notdef]2 = [integraldisplay]T2

[notdef]R

tr ( A ^ A + ^ ) . (2.3)

Note, that this metric is the direct product of the gauge algebra center part and the rest.

As we shall see later, the center, trace u(1), part will have no associated moduli, thus it is only the remaining su(n) part that is of any signicance. In particular, two background solutions that di er only in the trace part will have exactly the same metric in their vicinity. This fact will be signicant for our classication below.

We demand that is smooth everywhere except at a nite number of prescribed points in T 2 [notdef] R, where it has positive or negative Dirac singularities, and that the asymptotic

behavior of the eigenvalues of is at most linear in z. As in [13], in order to introduce and motivate these conditions we rst discuss some abelian solutions.

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2.3 Dirac monowall

Let us consider the rank one case, that is when the monowall elds are abelian. We let = i and A = ia so that the function and the one-form a are real. They satisfy the Bogomolny equation d = da. Geometrically it implies that is harmonic and, via the

Stokes and Chern-Weil theorems, the ux of r through any closed surface is proportional

to 2. The coe cient of proportionality Q+ (and Q) as z ! +1 (and z ! 1) is called

the right (and left) charge of the monowall.The only harmonic function on T 2 [notdef]R that is at most linear at innity is = 2(Qz +

M) with corresponding one-form a = 2 [parenleftBig]

QSR ydx pS dx qRdy[parenrightBig]

. Here the charge Q has to be integer, and M, p, and q are arbitrary real constants. This is the constant energy density solution.

Another way of constructing a monowall solution is by superimposing Dirac monopole solutions arranged along a doubly periodic array. The Dirac solution of eq. (2.2) on R3 is

=

1

2r , a[notdef] =

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1

2

ydx xdy

r(z [notdef] r)

. (2.4)

It satises the Bogomolny equation d = da and is the Greens function satisfying r2 = 2 (z) (y) (z).

Straightforward superposition of Dirac monopoles arranged as a doubly periodic array at the lattice vertices ejk = (jS, kR, 0), with j, k 2 Z with the distance to the ejk vertex

denoted by rjk = [notdef]r ejk[notdef], produces the Higgs eld

1

2r

1

2

1rjk

1

|ejk[notdef]

[parenrightbigg]

= [notdef]z[notdef]

SR + o(z0), (2.5)

with the constant [14]

= 1

R

ln 4R

S

X( j,k)[negationslash]=(0,0)

[parenleftbigg]

4 R

Xm,n K0

2mn S

R

[parenrightbigg]= 1 S

ln 4S

R

4 S

Xm,n K0

2mnR S

.

Such a Higgs eld does not have desired behavior as [notdef]z[notdef] ! 1, namely

SR d

dz = [notdef]

1

2

and is not integer; thus there is no line bundle with a connection a satisfying the Bogomolny equation for this Higgs eld, since it would have to satisfy 1

2

RT 2 da = SR ddz = [notdef]12.

With this in mind, the basic Dirac monowall with Q = 0 and Q+ = 1 and M =

M+ = 0 has the following Higgs eld

= z

SR

1

2r

1

2

1rjk

1

|ejk[notdef]

[parenrightbigg]

+ (2.6)

X( j,k)[negationslash]=(0,0)

[parenleftbigg]

with asymptotic expansions [15, 16]

= z + [notdef]z[notdef]

SR

1

2

Xm,n1pS2m2 + R2n2 e4 2

(mS)2+( nR)2

[notdef]z[notdef]e2i(mSx+ nR y) (2.7)

5

= z

SR +

1 SR ln

[vextendsingle][vextendsingle][vextendsingle]

2 sin

S (xiz)

[vextendsingle][vextendsingle][vextendsingle]

2 R

Xm,n K0

2n R

pz2+(xmS)2

cos

2

R ny

[parenrightbigg](2.8)

= z

SR +

1 SR ln

[vextendsingle][vextendsingle][vextendsingle]

[vextendsingle][vextendsingle][vextendsingle]

2 sin

R(y+iz)

2 S

Xm,n K0

2n S

pz2+(ymR)2

cos

2

S nx

. (2.9)

Series (2.7) converges fast for large values of [notdef]z[notdef], while series (2.8) and (2.9) can be used

for large (z2 + x2)/R2 and (z2 + y2)/S2 respectively. More details of various expansions of this function can be found in [15, 16] and [14].

2.4 Moduli problem

In general we consider rank n solutions of the Bogomolny equation DA = FA on T 2[notdef]R with asymptotic conditions on eigenvalues of the Higgs eld

Eig Val =

2i (Q[notdef],lz + M[notdef],l) + o(z0) [notdef] l = 1, . . . , n

[bracerightbig]

, (2.10)

which split the bundle E[notdef]z ! T 2z over the two-torus at large values of [notdef]z[notdef] into eigen-bundles

of :

E[notdef]z =

f+

j=1 E+j for z ! 1 and E[notdef]z =

f

j=1 Ej for z ! 1. (2.11)

Here f[notdef] are the numbers of distinct pairs (Q[notdef],l, M[notdef],l). We also x the conjugacy classes

of the holonomy of the connection in each E[notdef]j by xing the eigenvalues of the holonomy

around the x-direction to be p[notdef],l and around the y-direction to be q[notdef],l. We also presume

the holonomy conjugacy classes to be generic.

In addition, we choose points r+, and r, at which one of the Higgs eld eigenvalues

has respectively positive and negative Dirac singularity, i.e. one of the eigenvalues of the Higgs eld tends to imaginary positive or imaginary negative innity, so that the Higgs eld is gauge equivalent to:

= i

0

@

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1

2[notdef]rr+, [notdef] 01[notdef](n1)

0(n1)[notdef]1 0(n1)[notdef](n1)

1

A

+ O([notdef]r r+, [notdef]0), (2.12)

= i

0

@

12[notdef]rr, [notdef] 01[notdef](n1)

0(n1)[notdef]1 0(n1)[notdef](n1)

1

A

+ O([notdef]r r, [notdef]0). (2.13)

The complete set of boundary data is thus (Q[notdef],l, M[notdef],l, p[notdef],l, q[notdef],l, r[notdef], ). Among the results of [13] is the statement that the space of solutions for generic boundary data is a smooth hyperkahler manifold of dimension 4 [notdef] IntN, where N is the Newton polygon

(determined purely in terms of the charges Q[notdef],l and the numbers of positive and negative

singularities) and IntN is the number of integer points in the interior of N. Though the Newton polygon N can be constructed directly from the charges [13, section 4.1], one gains more insight by considering how N arises from the spectral curve of the monowall, that we now dene.

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2.5 Spectral description and moduli space isometry

As spelled out in [13], a monowall has two spectral descriptions each corresponding to one of the periodic directions x or y of the torus T 2. A spectral description consists of a spectral curve and a Hermitian holomorphic line bundle over it. Singling out the x-direction for concreteness, the Bogomolny equation (2.2) implies

[Dz iDy, Dx + i ] = 0, (2.14) where Dx, Dy, and Dz are the covariant derivatives Dj = @j + Aj, with j = x, y, orz. As a consequence, the holonomy V (y, z) of Dx + i around the x direction depends holomorphically on z iy (so long as we stay away from the monowall singularities). As a

result, he eigenvalues of V (y, z) are locally meromorphic in s = exp(2(z iy)/R) (away

from the singularities and branch points) with simple poles at s = s+, := exp(2(z+,

iy+, )/R), at the positions of the positive Dirac singularities r+, = (x+, , y+, , z+, ), and simple zeros at s = s, := exp(2(z, iy, )/R), at the positions of the negative Dirac

singularities r, = (x, , y, , z, ),. The spectral curve

x :

(s, t) 2 C [notdef] C [vextendsingle][vextendsingle]

of eigenvalues of the holonomy is an algebraic curve in C [notdef] C with cusps at innity

corresponding either to singularities or to the asymptotic eigenvalues of the Higgs eld at z ! [notdef]1.

As x is a curve of eigenvalues, it carries an associated eigensheaf over itself. So long as the spectral curve x is nondegenerate, this is an eigen line bundle Lx ! x. Since each

ber of this line bundle is a line in the Hermitian ber of E ! T 2 [notdef] R, the line bundle Lx

is also Hermitian.

The pair ( x, Lx ! x) of the spectral curve and the Hermitian line bundle over it is

equivalent to the monowall (A, ). This is a form of the Hitchin-Kobayashi correspondence, still to be proved in this particular setup. It gives a view of the monowall moduli space as a Jacobian bration over the moduli space of curves. Namely, the base is the space of curves in C [notdef] C with xed cusps going to innity; these are determined in terms of the

boundary data (charges, s singularity positions, constant terms in Higgs asymptotics, and asymptotic x-holonomy) of the monowall problem. The ber over a given curve is a set of Hermitian line bundles over it with xed holonomy around the cusps. The holonomy around each cusp is xed by the monowall asymptotic y-holonomy data and x-coordinates of the monowall singularities.

Since the curve x is algebraic, it can be given by a polynomial equation G(s, t) = 0. Marking a lattice point (m, n) for each monomial smtn with nonzero coe cient in G(s, t), the minimal integer convex polygon containing all of these marked points is the Newton polygon Nx. For a given monowall (A, ) its spectral curves x and y generally di er, and so do the line bundles Lx and Ly. However, the Newton polygon of x is the same as

that of y, thus, from now on, we denote Nx by N. N is completely determined by the numbers of positive and negative singularities and by the charges (with their multiplicities) of the monowall.

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det (V (z, y) t) = 0

[bracerightbig]

(2.15)

It is more elegant to take a toric view of the spectral curve x not as a curve in C [notdef]C ,

but as a curve x in its toric compactication given by the toric diagram N. Then the cusps are the intersections of x with the innity divisor and the positions of these points of intersection (together with the holonomy of Lx around them) are the asymptotic data of

the monowall.

An important observation for us is that it is the curve and line bundle that are important, and not any special coordinates s and t that C [notdef] C inherited from the

monowall formulation. As argued in [13], the natural GL(2, Z) action on s and t by

a b c d

[parenrightbig]

: (s, t) [mapsto]!(satb, sctd) is an isometry of the monowall moduli spaces. The monowall

changes drastically under such a transformation: its rank, charges, even the numbers of positive and negative singularities change. The moduli spaces, nevertheless, remain isometric.

In terms of the spectral curve x, its intersection with innity divisor of the toric compactication is determined by the coe cients of the monomials in G(s, t) that correspond to the perimeter points of the Newton polygon N. Thus we can vary at will the coe cients in G(s, t) corresponding to the internal points of N while respecting the asymptotic conditions. In other words, the coe cients of G(s, t) at the internal points of N are complex moduli. They coordinatize the base of the monowall moduli space viewed as the Jacobian bration.

As for the dimension of the ber, Hermitian line bundles on a punctured Riemann surface of genus g with xed holonomy around its punctures are parameterized by a points in a 2g-torus T 2g, its Jacobian. The coordinates can be viewed as holonomies of the corresponding at connection on Lx around the 2g generators of the fundamental group

1( x). Thanks to the theorem of Khovanskii [17], the genus of x equals to the number of integer internal points of N. Thus the monowall moduli space is bered by 2g-dimensional real tori over a 2g-real-dimensional base, with g = Int N being the number of integer internal points of N.

2.6 Monowalls fusion

Spectral description and Newton polygons in particular provide a good language for describing monowall fusion or concatenation. A natural question to ask is the following. Given a monowall A and another monowall B when can we arrange them back to back. To begin with let us place A far to the left and B far to the right on T 2[notdef]R. If this can be done,

then we view the result as another monowall C and view this as a fusion or concatenation

A + B ! C.

We would like to know the requirements on A and B for this process to be possible. We would also like to know the properties of the resulting monowall C.

With A far to the left and B far to the right, in the intermediate region eigenvales of are linear. Since away from the monowalls nonabelian cores the eigenvalues with di erent charges are distinct, in the intermediate region between A and B, while still su ciently far from both A and B, all eigenvalues (of di ering charges) associated with monowall A are diverging from each other as z increases, while those associated with monowall B are converging.

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Thus there are two possibilities: no eigenvalue of is associated with both A and B monowall or there is only one value of charge for which QA+ = QB =

and the corresponding eigenvalues of associated with both A and B. The former possibility gives a monowall C in the direct sum of vector bundles of A and B, EA [notdef] EB ! T 2 [notdef] R and the monowall C conguration (A, ) is block-diagonal. In this case there is no interaction whatsoever between A and B. It is a trivial case. The latter case has in the intermediate, between-the-walls, region all eigenvalues of A with a given charge equal to the eigenvalues of B with that same charge, thus they are identied. In terms of the Newton polygons it implies that they have antiparallel edges. For polygons with a common edge we obtain an

associative operation (NA, e)+(NB, e) = NC. It is dened if NA has an edge e = r

[parenrightBigg]

!, with > 0. We orient the edges on a Newton polygon

clockwise, as in [13, section 4.1], and r is the integer length of the edge which equals to the multiplicity of the corresponding charge Q = / . The Newton polygon NC is obtained by joining NA and NB along these two edges and taking the minimal convex polygon with integer vertices containing NA and NB. This process is Viros patchworking [5759] (see [51] for an illustration of its power).

Next, we turn to monowall ssion by identifying elementary monowalls and those with minimal number of independent constituents.

3 Monopole walls with four moduli

We would like to identify all monopole walls with four moduli, moreover, we would like to know which of them have isometric moduli spaces. To begin with we identify all elementary monopole walls, these have no moduli at all. Next, we identify all monopole walls with four moduli up to the action of GL(2, Z) group. This group, acting on the Newton polygon lattice, is generated by

1. reection of one of the axes,

2. T transformation of the lattice (e1, e2) ! (e1, e1 + e2), and

3. S transformation (e1, e2) ! (e2, e1),

which in terms of the monopole wall correspond respectively to

1. the reection of the noncompact and one of the periodic coordinates,

2. adding a charge one constant energy solution in the center of the gauge group, and

3. the Nahm transformation.

Our rst goal in this section is to classify GL(2, Z) inequivalent convex integer polygons with only one internal point. Considering how natural this question is, the answer is probably known since antiquity. However, not nding a good reference, though there must

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and NB has an edge e = r

be many, we proceed obtaining this classication in section 3.2. The answer is sixteen reexive polygons (table 2).

Once all GL(2, Z) inequivalent monopole walls are identied we nd pairs of these which are related by adding some abelian monopole wall in the gauge group center, and thus with isometric moduli spaces. The nal list of monopole walls with nonisometric moduli spaces, table 3, is twice shorter.

3.1 Monopole walls with no moduli

The simplest monopole wall is the direct sum of a number of constant energy density solutions of the same charge. It is translationally invariant in all directions and has no moduli. Its Newton polygon is in fact not a polygon, but a single interval. What are the other mono-pole walls without moduli. Khovanskii proved in [18] that, up to the GL(2, Z) equivalence, the only polygons with no internal points that are not degenerate (i.e. not an interval) are

1. a triangle with sides of integer length two. A representative of this class is a triangle with vertices (0, 0), (2, 0), and (0, 2). It corresponds to a U(2) monopole wall with two negative singularities,

2. a trapezium (or a triangle if k = 0) of integer height 1 and bases of integer lengths k and m with k m. A representative of this class has vertices (0, 0), (m, 0), (1, k),

and (1, 0). This corresponds to a U(1) monopole wall with k positive and m negative singularities.

The spectral curves of the corresponding monopole walls are 1) t2 + (C11s + C10)t +

C02s2 + C01s + C00 = 0 and 2) t = Pm(s)/Qk(s).

In some sense these can be viewed as elementary walls out of which other walls are composed.

3.2 Monopole walls with four moduli

The smallest nonzero number of moduli that a monopole wall can have is four.2 These are particularly interesting since they deliver moduli spaces that are self-dual gravitational instantons. The search for these is among the main motivations for this study.

Since each monopole wall with given singularity structure and given boundary conditions determines a Newton polygon and its number of moduli is four times the number of internal internal points of its Newton polygon, we would like to list all Newton polygons with single integral point. To begin with, integer translations of this polygon do not change the spectral curve and are therefore immaterial. Thus, for now, we choose our polynomial to have the origin as the end of one of its edges with longest integer length.3 Then use GL(2, Z) transformation to have this edge stretch along the positive horizontal axis, and to place the Newton polygon in the upper half-plane. If this edge had integer length l, then, after this transformation, it has end points (0, 0) and (l, 0).

2Note that we have the regular linear growth asymptotic conditions on R [notdef] T 2. In a theory with a

boundary one might expect lower number of moduli.

3Integer length of an edge is one short of the number of integer points on that edge.

10

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Figure 1. The only Newton polygon with a single internal point and a side of integer length four.

If the intersection of the Newton polygon N with the horizontal line h passing through the point (0, 1) is empty then the whole Newton polygon is an interval [(0, 0), (l, 0)]. In that case the monopole wall is GL(2, Z) equivalent to the constant energy solution and N has no internal points and no moduli. Similarly, if this intersection N \ h has no internal

integer points, then the Newton polygon has no internal points at all. As we are looking for a Newton polygon with a single integer internal point, we conclude that the N \ h has

exactly one integer internal point. Now we use an SL(2, Z) transformation of the form 1 q

0 1

to put the internal point at (1, 1). Since N is by denition convex, contains (1, 1) as its only internal point and [(0, 0), (l, 0)] as its side, we conclude that l 4. This conclusion follows

from the fact that N should be contained within the triangle bounded from above by the line containing [(l, 0), (2, 1)] (otherwise, (2,1) is another internal integer point), bounded from the left by the vertical axis (otherwise, (0,1) is another integer internal point), from below by the horizontal axis, and has only integer vertices. There are no such convex integer polygons with (1, 1) as their only internal point for l > 4.

3.2.1 Maximal side of integer length four

For l = 4 there is exactly one such polygon with sides of integer lengths 4, 2, 2 as in gure 1.

3.2.2 Maximal side of integer length three

If l = 3 then the Newton polygon is contained within the triangle ((0, 0), (3, 0), (0, 3)). There are six such Newton polygons that satisfy our conditions. They are of integer sides lengths (3, 3, 3), (3, 2, 1, 2), (3, 2, 1, 1), (3, 1, 1, 2), (3, 1, 2), and (3, 2, 1), presented in gures 2a, 2b, 2c, 2d, 2e, and 2f respectively. The Newton polygon (3, 1, 2) is related to (3, 2, 1) by the transformation 1 1

0 1

[parenrightbig]

followed by a right shift by three units; the Newton polygon (3, 1, 1, 2) is related to (3, 2, 1, 1) by the same transformation. The others are clearly GL(2, Z) inequivalent, as they have either di erent number of sides or their side integer lengths spectra di er.

Thus there are four distinct cases of maximal side of length three.

3.2.3 Maximal side of integer length two

For l = 2 N lies in the strip between the vertical axis, the vertical line passing through the point (2, 0) and the horizontal line passing through (0, 2). There are six Newton polygons

11

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(a) (3,3,3) (b) (3,2,1,2) (c) (3,2,1,1) (d) (3,1,1,2) (e) (3,1,2) (f) (3,2,1)

Figure 2. Newton Polygons with single internal integer point and the longest side of length three.

(a) (2,2,2,2) (b) (2,1,1,1,2) (c) (2,1,1,2) (d) (2,1,1,1,1) (e) (2,1,1,1) (f) (2,1,1)

Figure 3. Newton Polygons with single internal integer point and the longest side of length two.

with one integer internal point (up to GL(2, Z) equivalence). The master polygon of gure 3a is a square with sides of integer length 2. All other cases result from truncating it, so that the result is convex and still contains the internal point (1, 1). Their integer side length spectra are (2, 1, 1, 1, 2), (2, 1, 1, 2), (2, 1, 1, 1, 1), (2, 1, 1, 1), and (2, 1, 1). Since these spectra are all distinct, we have six distinct cases with l = 2.

3.2.4 Maximal side of integer length one

For l = 1 all sides of the Newton polygon are of integer length 1, i.e. all integer points on the perimeter of the Newton polygon are vertices. By our construction, one of the sides is ((0, 0), (1, 0)). Let us focus on the other side originating at (0, 0). Let us denote the coordinates of its other end by (p, q). Using 1 q

0 1

[parenrightbig]

transformation of GL(2, Z) we can make sure that p 0 and q > 0. Then, by convexity of N, the triangle = ((0, 0), (p, q), (1, 0))

is contained within the Newton polygon and thus should have zero or one internal point. Picks formula relates the number of internal points I to the area A and integer perimeter length P of an integer polygon:

I = A

followed by the upward shift by one unit. Now,

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1

2P + 1. (3.1)

For the triangle we have I 1, A = 12q, and P = 3. Thus Picks formula implies q 3.

This leaves three possibilities (p, q) = (0, 1), (1, 2), or (2, 3), since we allow only (1, 1) as the internal point of the Newton polygon.

There are nine such polygons: one hexagon, two pentagon, ve quadrilaterals, and one triangle as in gure 4. The second pentagon (gure 4c) is related to the rst pentagon (gure 4b) by the transformation 1 0

1 1

[parenrightbig]

(a) Hexagon (b) First Pentagon (c) Second Pentagon

(d) Parallelogram or Rhombus (e) Symmetric Quadrilateral (f) Curious Quadrilateral

(g) Ambitious Quadrilateral (h) Obnoxious Quadrilateral (i) Triangle

Figure 4. Newton Polygons with single internal integer point and all sides of length one.

turning to the quadrilaterals, we look at their GL(2, Z) invariant quantities. The areas spanned by pairs of their adjacent edges are (1, 1, 1, 1), (12, 1, 32, 1), (1, 12, 1, 32), (32, 1, 12, 1), and (1, 32, 1, 12) respectively for the parallelogram, symmetric, curious, ambitious, and obnoxious quadrilaterals of gure 4. This indicates that parallelogram is distinct, while the other four might be GL(2, Z) equivalent. Indeed, the symmetric quadrilateral is transformed

into the curious quadrilateral by a shift down by one followed by 1 1

1 0

[parenrightbig]

13

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transforma-

tion,

into the ambitious quadrilateral by a shift by (2, 2) followed by

mation, and

into the obnoxious quadrilateral by a shift by (1, 0) followed by

Thus for l = 1 case we have ve distinct Newton polygons: a hexagon, a pentagon, a parallelogram, a symmetric quadrilateral, and a triangle.

3.2.5 Complete list

The three master polygons are those in gures 2a, 3a, and 1. All the others can be obtained by deleting some of the perimeter points of these three. We organize them in table 2 according to their integer perimeter length. We also put the triangle of gure 4i in a more elegant form applying the 2 1

1 0

[parenrightbig]

transformation.

There are sixteen cases in total. This is exactly the celebrated list of reexive polygons that are signicant in numerous elds, see e.g. [19] for some insightful relations. Each has its own GL(2, Z) class of monowalls with the same moduli space.

There might be some other equivalence that would establish that some of these spaces are isometric. Clearly, any isometric spaces would have equal number of deformation parameters, so such an isometry would only relate spaces on the same line of gure 2. As demonstrated in [13], number of the moduli space deformations is the number of relevant parameters in the monowall problem and it equals to #Perim(N) 3, integer perimeter

length of N minus three. In the next section we formulate relation between monopole walls with di erent, non GL(2, Z) related, Newton polygons that induces isometry of their moduli spaces.

3.3 Additional equivalence

Given any abelian monopole wall (a, ) we consider a map acting on all monopole walls:

(A, ) [mapsto]!(A + aI, + I), (3.2) where I is the identity matrix. The abelian spectral curve (a,)x is given by some rational function of s, so that (a,)x : t = P (s)/Q(s). Then, if the original spectral curve of (A, ) was given by G(A, )(s, t) = 0, the spectral curve (A+a, +)x of (A + aI, + I) is given by G(A+a, +)(s, tQ(s)/P (s)) = 0. What is equally important is that the trace part of any monopole wall is completely determined by the boundary and the singularity data and is independent of its moduli. As it lies in the center of the algebra, it plays no role in the moduli space metric computation, thus the transform of eq. (3.2), while changing the spectral curve, acts isometrically on the moduli spaces.

The spectral curve of gure 3a, for example, has the form L(s) + M(s)t + R(s)t2 = 0,

with L(s), M(s), and R(s) some quadratic polynomials. Adding a U(1) monowall with the spectral curve t = R(s) amounts to the substitution t ! t/R(s). Adding any U(1) solution

does not alter the moduli space metric, while the resulting spectral curve is now given by L(s)R(s) + M(s)t + t2 = 0 with its Newton polygon of gure 1. Thus the spectral curve

14

0 1

1 2

[parenrightbig]

transfor-

1 1

2 1

[parenrightbig]

.

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p = 9

#

p = 8

[arrowsoutheast] [arrowsouthwest] [arrowsoutheast] [arrowsouthwest]

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p = 7

[arrowsouthwest] [arrowsoutheast] [arrowsouthwest][arrowsoutheast] [arrowsouthwest] [arrowsoutheast]

p = 6

[arrowsoutheast] [arrowsoutheast] [arrowsouthwest] [arrowsouthwest]

p = 5

[arrowsouthwest] [arrowsoutheast] [arrowsouthwest] [arrowsoutheast]

p = 4

#

p = 3

Table 2. Relations between moduli spaces.

L(s) + M(s)t + R(s)t2 = 0 is mapped to L(s)R(s) + M(s)t + t2 = 0 via the substitution t ! t/R(s). This is the total transform, it maps a monopole wall with n+ positive and n

negative singularities to a monopole wall with (n+ + n) only negative singularities.

If r is one of the roots of R(s) then adding a single Dirac monowall with negative singularity at s = r amounts to making a substitution t ! t/(s r). It puts the spectral

curve in the form L(s)(s r) + M(s)t + R(s)srt2 = 0 with the Newton polygon of gure 2b.

We call this a partial transform.

We conclude that all moduli spaces of monopole walls corresponding to the Newton polygons of line p = 8 of table 2 are isometric.

Now we apply the same argument to each line of that table.

For p = 7 line the argument is exactly the same with L(s) and M(s) quadratic and R(s) linear, so all of these spaces are isometric to each other.

15

(a) E6 (b) E0 = 1

(c) E5 = Spin(10) (d) E1 = SU(2) (e)

(f) E4 = SU(5) (g) E2 = SU(3) [notdef] U(1)

(h) E3 = SU(3) [notdef] SU(2)

Table 3. All monopole walls with four moduli and distinct moduli spaces.

For p = 6 the spectral curve for the Newton polygon (2, 1, 1, 2) (second in p = 6 line of table 2) has L(s) quadratic and M(s) and R(s) linear; the above transformation maps it to the spectral curve of the Newton polygon (3, 1, 2) (the fourth one on p = 6 line of table 2).

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eE1 = U(1)

The spectral curve of the hexagon, on the other hand, has the form L1(s) + M1(s)t +

sN1(s)t2 = 0 and, via the transformation t ! t/N1(s), it is mapped to a curve with the

Newton polygon (2, 1, 1, 1, 1) (the third one on line p = 6 of table 2).

It remains to relate the two pairs. We choose to consider (2, 1, 1, 1, 1) and (2, 1, 1, 2). To begin with, we interchange the two axes by applying ( 0 110 ) . The spectral curves of the resulting Newton polygons are respectively L1(s) + M2(s)t + R1(s)t2 = 0 and P2(s) +

Q2(s)t + t2 = 0 and they are related by the total transform.

Thus all moduli spaces corresponding to perimeter six, p = 6, Newton polygons are also isometric to each other.

For p = 5 we already established the equivalence of the two pentagons 4b and 4c. The spectral curve of the latter has the form L1(s) + M1(s)t + sN1(s)t2 = 0 and by the total transform t ! t/N1(s) is mapped to the spectral curve L1(s)N1(s) + M1(s)t + st2 = 0 with

the Newton polygon (2, 1, 1, 1) of gure 3e. Thus all p = 5 Newton polygons have isometric moduli spaces.

So far all Newton polygons with a given integer perimeter length had isometric moduli spaces. In other words, each line of table 2, besides p = 4 line, corresponds to one distinct family of moduli spaces. For p = 4 the situation is di erent. Of the three Newton polygons on p = 4 line of table 2 the rst and last are equivalent via the total transform. The middle one the symmetric quadrilateral is distinct.

We end up with a list of only eight monopole wall spaces of dimension four. These appear in table 3.

This list relates to the classication of ve-dimensional superconformal eld theories with one-dimensional Coulomb branch of vacua of [20, 22, 23]. Table 3 lists the global symmetry groups of the corresponding theories. This relation is not coincidental, as we explain in the following section.

4 Relation to gauge theories and Calabi-Yau moduli spaces

4.1 Five-dimensional theories

In [20] Seiberg identied superconformal ve-dimensional eld theories with En, n 8, global symmetries. Heterotic string theory view of these theories appeared in [21]. For low values of n these global symmetry groups are: E1 = SU(1), E2 = SU(1) [notdef] U(1), E3 =

SU(3) [notdef] SU(2), E4 = SU(5), and E5 = Spin(10); while E6, E7, and E8 are the exceptional

ones. In [22, 23] two more theories were added to this list with

eE1 = U(1) and E0 = 1.

Relevant deformations of ENf+1 theory are interpreted as the supersymmetric SU(2) gauge theories with Nf quarks4 with masses mi, i = 1, . . . , Nf.

Even though these ve-dimensional theories are non-renormalizable and should rather be viewed as an intermediate e ective description of any of the string realizations mentioned

4As table 3 and the geometric engineering [22] indicate, the number of quarks relates to the integer perimeter length p of the Newton polygon via Nf = p 4.

17

JHEP06(2014)027

below, they give a good description of the moduli space of vacua. Namely, according to [20], they have one-dimensional Coulomb branch with the metric

0

@t0 + 16

Nf

Xi=1

| mi[notdef] + [notdef] + mi[notdef]

[parenrightbig]

1

Ad2, (4.1)

while their Higgs branches are isometric to the moduli space of En, SO(2k), or SU(m) instantons (with the gauge groups determined by the remaining global symmetry at the point where the Higgs branch is intersecting the Coulomb branch). A general classication of such theories with higher-dimensional Coulomb branches appeared in [24].

When one of the ve space-time directions is compact, the Coulomb branch doubles its dimension. The additional dimensions correspond to the eigenvalues of the vacuum expectation value of the holomony around the compact direction. Such gauge theories on R1,3[notdef]S1 were solved in [25] with the full quantum-corrected metric on the Coulomb branch given in terms of special geometry. One loop asymptotic analysis of ve-dimensional gauge theory with two periodic directions was carried out in [26] in complete agreement with eq. (4.1).

There is a relation, discovered by Seiberg and Witten [27], between quantum vacua of super-Yang-Mills with eight real supercharges in three dimensions and classical monopoles on R3. For super-QCD with n quarks [3], the appropriate monopoles are those with n Dirac singularities. In both cases, realizing the relevant gauge theory via the Chalmers-Hanany-

Witten brane conguration [28, 29] makes the relation to the dynamics of monopoles transparent. Along the similar lines [30], super-QCD with eight supercharges on R3[notdef]S1 is related

to periodic monopoles with Dirac singularities. Thus, from this Chalmers-Hanany-Witten point of view, it is not surprising that the doubly periodic monopoles with singularities relate to super-QCD on R3 [notdef] T 2. For more detailed reasoning with concrete brane cong

urations and the string theory duality chain see [13] or the diagrams of section 4.3 below.

4.2 Calabi-Yau moduli spaces

The ve-dimensional theories of [20] can be realized in string theory either by considering a D4-brane probe in type I[prime] string theory on R/Z2 with Nf D8-branes [21] or via geometric engineering of [31] by compactifying M theory on a Calabi-Yau manifold with a smooth four-cycle S (of complex dimension two) shrinking to a point [20, 22]. Equally relevant to our monowall picture is the fact that the same theories appear as e ective theories on M theory ve-brane wrapped on a curve in C [notdef] C (the same curve as the monowall spectral curve)

and as theories on (p, q)-networks of ve-branes [32, 33] (see [34] for a recent discussion).

In the geometric engineering picture [31], the shrinking surface inside a Calabi-Yau space has to be a del Pezzo surface and it is Gorenstein. The local Calabi-Yau geometry is that of the total space of the canonical bundle of S. All Gorenstein toric del Pezzo surfaces are in one-to-one correspondence with reexive convex polygons. (In fact this holds in general dimension [35].) Thus, it is not surprising that our intermediate result above in gure 2, is similar to the del Pezzo tree [36, gure 1]. All En theories are geometrically

18

JHEP06(2014)027

engineered by compactifying M theory on a local Calabi-Yau that is a canonical bundle to a del Pezzo surface. Whenever this del Pezzo is toric, it corresponds to a monopole wall, with the toric diagram of the former being the Newton polygon of the latter.

This correspondence extends to the level of the moduli spaces. The map between the moduli and parameters of the gauge theory and M theory and string theory descriptions is explored in detail in [22, 26]. Viewing the del Pezzo surface S as a bration over a projective line P1B, the generic ber is P1f (see [22]) and each singular ber is a pair of

P1s intersecting at a point. Monowall parameters correspond to 1) the size of the base P1B (which corresponds to the coupling of the ve-dimensional quantum theory) and 2) the di erence of the sizes of the two P1s in each special ber (which correspond to the masses of the ve-dimensional theory matter multiplets). The monowall modulus corresponds to the size of the generic ber P1f. So far, considering the Calabi-Yau geometries (with xed parameters) the Kahler moduli space is one-dimensional. The corresponding monowall has four moduli. What are the remaining three periodic moduli?

Since we are interested in a ve-dimensional theory on R1,2 [notdef]T 2, the relevant M theory

compactication to engineer it is on the direct product of a Calabi-Yau space with a two-torus. For the two-torus being a direct product of two circles, T 2 = S1S [notdef]S1R, the remaining three moduli are

RP1f [notdef]S1S C3, RP1f [notdef]S1R C3, and RS[notdef]T2 C6. We can view any one of the rst two, together with the ber size modulus to give the complexication of the Calabi-Yau Kahler structure moduli space. The remaining two are coordinates on a torus bration over it. Selecting either

RP1f [notdef]S1S C3 or RP1f [notdef]S1R C3 to complexify the size of P1 gives the choice of two complex structures on the same moduli space. These are the same two complex structures on the moduli space that emerge from the spectral curve of a monowall in the x- or y-direction. Since the moduli space is hyperkahler, the moduli come in multiples of four, the deformation parameters, however, come in triplets (M, p, q). Let us identify these in terms of the Calabi-Yau geometry. M corresponds to the di erence in sizes of the two P1s, say

P1A and P1B of each singular ber. While p and q correspond to the di erence in C3 uxes:

R(P1AP1B)[notdef]S1S C3 and R(P1AP1B)[notdef]S1R C3.

Compactifying M theory on one of the two circles, we are left with a type IIA theory on the same Calabi-Yau space product with the remaining circle. The size of P1f is the real modulus that is complexied by adding as imaginary component

RP1f BNS2. The torus ber

RS CRR4.

In the mirror [37], type IIA version, the relevant Calabi-Yau space W is written directly in terms of the spectral curve of the monowall. Type IIA theory is compactied on W [notdef]S1.

If the monowall spectral curve is x : [notdef](s, t) 2 C [notdef]C [notdef] G(s, t) = 0[notdef], then the mirror Calabi-

Yau space is W : [notdef](s, t, u, v) 2 C [notdef] C [notdef] C [notdef] C [notdef] uv = G(s, t)[notdef]. In our case x is genus one

with a number of punctures. Choose generators of 1( x) so that some correspond to the

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RP1f [notdef]S1 CRR3 and RS[notdef]S1 CRR5.

Equivalently, [26], the same moduli space can be realized as the moduli space of type IIB string theory on the same Calabi-Yau space with the noncompact modulus being the size of the ber P1f and three periodic moduli

RPf CRR2, RP1f BNS2, and

coordinates are

punctures while and are the remaining two generators, corresponding to the genus. The compact Lagrangian three-cycles of W , and , correspond to the cycles and , as in [31]. Now, it is the complex structure moduli space of W that is relevant; it is the space of curves x and it forms the base of our monowall moduli space. While the ber is the intermediate Jacobian of W with coordinates

R CRR3 and R CRR3.

4.3 String theory dualities and the two spectral curves

There are three distinct limits to consider, each has its interpretation as a monopole, quantum eld theory, and a spectral curve degeneration. Each poses new interesting problems for monowall and eld theory interpretation. String theory dualities relate these tree points of view via the following diagram:

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M5-brane wrapped on

x spectral curve

O 5

a

O.

M 0 1 2 3 4

O 6 7 8 9 10

O

2 M5 x x x x x xM5 x x x x x x S =S

(p,q)-network or 5D QFT on R3 [notdef] T 2

.

[d37]

O 6 7 8 92 NS5 x x x x x xk D4 x x x x x

[d79]

ST

[d15]

O 6 7 8 92 NS5 x x x x x xk D5 x x x x x x

[d111]

O 5

O 5

b

O.

IIB 0 1 2 3 4

T [d47][d47]

f

O.

IIA 0 1 2 3 4

[d79]

S

[d15]

(q,p)-network or EM dual 5D QFT on R3 [notdef] T 2

c

O.

IIB 0 1 2 3 4

Monowall

O 5

O 5

O 6 7 8 92 D5 x x x x x xk NS5 x x x x x x

[d79]

ST

[d15]

[d73]

O 6 7 8 92 D5 x x x x x xk D3 x x x x

[d79]

T

[d15]

S =S

g

O.

IIB 0 1 2 3 4

Nahm Transformed Monowall

D4 wrapped on y

O 6 7 8 92 D3 x x x xk D5 x x x x x x

[d79]

T [d15]

O 5

O 5

T

d

O.

IIB 0 1 2 3 4

h

O.

IIA 0 1 2 3 4

O 6 7 8 92 D4 x x x x xk D4 x x x x x

S =S

[d79]

[d22]

[d9]

M5-brane wrapped on

y spectral curve

D4 wrapped on x

O 5

O 5

e

O.

IIA 0 1 2 3 4

O 6 7 8 92 D4 x x x x xk D4 x x x x x

i

O.

M 0 1 2 3 4

O 6 7 8 9 10

O

2 M5 x x x x x x M5 x x x x x x

O, and g

O above relevant gauge theory emerges in the world-volume of the D5-branes [40, 41]. Before we proceed with the discussion of various limits, let us recall the standard string theory duality relations and their M theory origin [4249]. They are important in understanding all scales, couplings, and sizes involved:

Given M theory with Planck scale lpl = L compactied on a two torus S1R [notdef] S1r, there

are two ways of describing it as a type IIA string theory. Each requires selecting

20

In string theory brane congurations b

O, c

O, d

an M theory circle. Identifying the rst circle S1R as the M theory circle, S1M = S1R, for example, acquire an equivalent description as the type IIA string theory on the remaining S1r[prime], [42, 43]. We denote this relation by

M theory Type IIA String Theory

lpl = L, S1R [notdef] S1r

S1M =S1R [d47][d47]

S1r[prime] g=(RL)3/2, l = L

3/2

R1/2

32 is the string theory coupling, l = L3/2/R1/2 is the string scale, while the string theory circle radius is r[prime] = r.

Under the T duality, the type IIA string theory on S1r, with string coupling g = , and string scale l = a is equivalent to the type IIB string theory on S1r[prime] with r[prime] = a2/r,

same string scale l = a, and string coupling g = a/r :

S1r

g= ,l=a

JHEP06(2014)027

Here g = R

[d111] T [d47][d47]

S1a2/r

g= a

r ,l=a .

S duality [d111][d111] S [d47][d47] inverts the string coupling and changes the string scale a to apg, [44]. In string frame it implies:

S1r

g= ,l=a

S1r

g= 1 ,l=ap .

Symbol [notdef] in S1r [notdef] S1r[prime] [notdef] S1r[prime] [notdef] S1r signies the interchange of the two S1

factors.

Schematically we assemble these facts in the relation

2

6

6

6

6

6

6

6

6

6

6

4

M theory R

r

L

[d111] S [d47][d47]

3

7

7

7

7

7

7

7

7

7

7

5

S1M =S1R

3/2

!

2

6

6

6

6

6

6

6

6

6

6

6

4

IIA gA = R

L

lA = L

32

R

!

2

6

6

6

6

6

6

6

6

6

6

4

IIBgB = gA lArA

lB = lA

rB = (l

A)2 rA

3

7

7

7

7

7

7

7

7

7

7

5

S

!

2

6

6

6

6

6

6

6

6

6

6

4

IIB B = 1

gB

TrA

(4.2)

~lB = lB

pgB

12

rA = r

3

7

7

7

7

7

7

7

7

7

7

7

5

~rB = rB

3

7

7

7

7

7

7

7

7

7

7

5

Another important fact to note is that the Yang-Mills coupling on a Dp-brane is determined in terms of the string scale l and the string coupling g by the relation g2Y M =

lp3g, [43]. In our case, the most relevant is the D5-brane with g2Y M = l2g.

Consider M theory on a tree-torus T 3 = S1A [notdef] S1B [notdef] S1C (that is a product of three

circles of respective radii A, B, and C) with an M ve-brane wrapped on [notdef] S1B [notdef] R1,2

21

T 3 [notdef] R23,6 [notdef] R3 [notdef] R1,2, where S1A [notdef] S1C [notdef] R3 [notdef] R6 [equalorsimilar] C [notdef] C . This allows for three

type IIA six type IIB descriptions. The sequence of dualities relating these (and most of the equivalent brane congurations on page 20) is captured by the following diagram

M theory on a

OS1A [notdef] S1B [notdef] S1C, lpl = L

JHEP06(2014)027

S1

M =S

1

A

S1

M =S

1

C

S1

M =S

1

B

[d121]

[d15]

[d37]

IIA on T 2at string coupling g and string scale l

S1

B [notdef]S

1

C

S1

A[notdef]S

1

C

S1

A[notdef]S

1

B

g=( AL )3/2,l=

L3/2 A1/2

[d79]

e

O

g=( BL )3/2,l=

L3/2 B1/2

g=( CL )3/2,l=

L3/2 C1/2

[d99]

[d60]

[d98]

[d59]

[d79]

TC

TA

TC

TA

TB

TB

[d15]

d

O[d124][d124]

c

O

[d35]

b

O[d123][d123]

g

O

[d34]

f

O

[d15]

S1

C

S1L3

AB [notdef]

S1

C

S1

B [notdef]S

S1L3

AC [notdef]

S1

B

S1

A[notdef]S

S1

A[notdef]S

IIB on T 2at string coupling g and string scale l

S1L3 AB [notdef]

g= A

B ,l=

L3/2 A1/2

[d91]

g= B

A ,l=

L3/2 B1/2

AC g= A

C ,l=

L3/2 A1/2

g= C

A ,l=

L3/2 C1/2

BC g= B

C ,l=

L3/2 B1/2

[d91]

BC g= C

B ,l=

L3/2 C1/2

[d67]

[d67]

S

S

We mark the corresponding brane congurations of the duality table of page 20 by respective circled letters. For example, M theory ve-brane wrapped on the spectral curve x marked by a

O resides in M theory on top of this diagram. The monowall conguration on the D5-brane world-volume of g

O is in the second from the right type IIB theory at the bottom row. The ve-dimensional theory of R3[notdef]T 2 in the world-volume of the (p, q)-ve-brane

network c

O is in third from the left the type IIB theory in the bottom row of the diagram.

The complete picture of the action of T and S duality on M theory on a three torus is represented by the following diamond diagram.

22

M theory on T 3

IIA on T 2 at string coupling g and string scale l

IIB on T 2 at string coupling g and string scale l

IIB on T 2 at string coupling g and string scale l

IIA on T 2 at string coupling g and string scale l

M theory on T 3

[notdef][notdef]S

S1

C

[notdef][notdef]S

1

C [notdef]S

g= A

B ,l=

L3/2A1/2 [notdef]

[d87]

AB g= A

B ,l=

L3/2 A1/2

TB [d60][d60]

[d98]

TB

[d124]

[d34]

1

B [notdef]S

1

C

[notdef][notdef]S

[notdef][notdef]S

S1L3

AB

g=( AL )3/2,l=

L3/2 A1/2

g= L3/2A1/2

BC ,l=

L3/2 A1/2

[d72]

[d98] TC c

S

[d60]

[d86]

[d34]

O

TC

[d124]

[notdef][notdef]S

1

B [notdef]S

S1

B

Dual 5D QFT

AC g= A

C ,l=

L3/2 A1/2 [notdef]

[notdef][notdef]S

g= A

C ,l=

L3/2 A1/2

[d63]

1

A

[d9]

S1

M =S

1

A

S1L3

AB [notdef][notdef][notdef]

S1

C

2

3

pABC , S1L3

BC

[notdef] S1L3

AC

[notdef] S1L3

AB

i

O

Nahm Dual Monopole

O [notdef]

S1

C [notdef][notdef][notdef]S

d

g= B

A ,l=

L3/2 B1/2

AB g= B

A ,l=

L3/2 B1/2

lpl = L

lpl = L,

S1A

[notdef] S1B

[notdef] S1C

a

O

S1

M =S

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TA [d60][d60]

[d98]

TA

e

O

[d124]

[d34]

S1

M =S

1

B

[d47]

S1

A[notdef][notdef][notdef]S

1

C

S

S1

M =S

1

B

Spectral Curves

g=( BL )3/2,l=

L3/2 B1/2

h

O

S1L3

BC [notdef][notdef][notdef]

S1L3

AB

g= L3/2B1/2

AC ,l=

L3/2 B1/2

[d111]

[d98] TC

[d60]

[d34]

[d124]

TC

g

O

S1

A[notdef][notdef][notdef]S

S1

A

BC g= B

C ,l=

L3/2B1/2 [notdef]

S1L3

BC [notdef][notdef][notdef]

Monopole

g= B

C ,l=

L3/2 B1/2

[d74]

S1

M =S

1

C

[d127]

S1

M =S

1

C

S1L3

AC [notdef]

S1

B [notdef][notdef]

g= C

A ,l=

[d60]

5D QFT

L3/2C1/2 [notdef]

S1

B [notdef]S

L3/2 C1/2

TA b

O

[d98]

f

O

[d22]

TA

[d124]

[d34]

[d8]

S

S1

A[notdef]S

1

B [notdef][notdef]

AC [notdef][notdef] g= L3/2C1/2

AB ,l=

L3/2 C1/2

S1L3

BC [notdef]

S1L3

g=( CL )3/2,l=

L3/2 C1/2

[d98]

[d60]

TB

[d34]

[d22]

TB

[d124]

S1

A[notdef]S

L3/2 C1/2 [notdef]

S1L3

BC [notdef]

S1

A[notdef][notdef]

g= C

B ,l=

L3/2 C1/2

The full U-duality group [60] acting on M theory on a d-dimensional torus T d is Ed(d) (see [61] for references and the broad picture). In our case of M theory on T 3 the U-duality group is E3(3) = SL(3, Z)[notdef]SL(2, Z). The rst component, SL(3, Z), is the modular group of

the torus T 3 contains elements interchanging the rows of the above diagram. S element of the second, SL(2, Z), factor acts on the diagram as a reection with respect to the vertical axis.

Note, that the complete description of the original monowall or the full ve-dimensional theory with two nite periodic directions involves two spectral curves x and y in C [notdef]C .

These appear as M ve-brane curves in, respectively, congurations a

O and i

O.

Now we have a few interesting limits to mention:

23

Tropical limit: tracing the above diamond diagram, a monowall is realized on the brane conguration g

O in the world-volume of the D5-brane. The Yang-Mills coupling gY M

on this brane satises g2Y M = l2g = L3/C. We are holding it xed. Under T duality on the second circle it becomes a D4-brane of conguration e

O wrapped on a curve x

a

O this is the M ve-brane with world-volume R1,20,1,2 [notdef] x [notdef] S1B R1,20,1,2 [notdef] S1A [notdef] S1B [notdef] S1C [notdef] R3 [notdef] R6 [notdef] R37,8,9. The tropical limit is A =

C = h ! 0. To hold the Yang-Mills coupling of the monowall xed we need L3 h. Tracing the diamond diagram, the monowall g

O has periods A = h and 1/B and in this limit one of the circles degenerates to zero size and the monowall becomes the Hitchin system on a cylinder. The ve-dimensional theory b

O, on the other hand, is now on R1,20,1,2 [notdef] S11/h [notdef] S1B, and in this limit becomes the four-dimensional Seiberg-

Witten theory with one periodic direction and nite coupling g2Y M = L3/A.

Note, that in this limit the spectral curve y, on which the M ve-brane of i

O is

wrapped, remains regular y S11/B [notdef] S11/B [notdef] R3 [notdef] R6 [similarequal] C [notdef] C , while the middle circle S1B[prime] = S1L3/(AC) = S11/h opens up becoming R and the Planck scale remains nite lPl B

13 . This is the only remaining Seiberg-Witten curve of the theory in this limit, as the other curve x turns into a tropical curve in this limit.

5D eld theory limit: if we are interested in exploring the ve-dimensional quantum theory of b

O without any periodic directions, we are to send B ! 1 and

L3

AC

This can be achieved within the above tropical limit by sending B ! 1. Since S1B was the spectator circle, i.e. it did not involve the spectral curve, the curve x still tends to the tropical limit. What is new, is that the other spectral curve y involves S1A[prime] [notdef]S1C[prime] = S11/B [notdef]S11/B becomes tropical as well. This is the reason why our analysis

of monowalls in the tropical limit reproduces the Seiberg and Morrison results of [23] and the asymptotic metric of [20].

3D monopole limit: the decompactication limit of the monowall torus of g

O is A ! 1

and L3

BC

! 1, while holding the Yang-Mills coupling on the D5-brane xed: g2Y M = L3/C = Const. This implies L3 C while A ! 1 and B ! 0.

The ve-dimensional space of the eld theory b

O degenerates to a three-dimensional space, since both circles in S1L3/(AC) [notdef] S1B shrink to zero size. The limiting theory

is the N = 4 three-dimensional super-QCD of [27]. The monowall in this limit be

comes a monopole in R3 and the moduli space becomes ALF in complete agreement with [2730].

5 Phase space of a monowall

5.1 Litvinov-Maslovs dequantization or tropical geometry

Following [50], consider the positive real line R+ with the standard operations + : (u, v) [mapsto]!

u + v and [notdef] : (u, v) [mapsto]!u [notdef] v. The exponential map e : R ! R+ induces operations and [circledot]

on the full real line R, namely, letting u = ea and v = eb, we have

+ : (ea, eb) [mapsto]!ea + eb = ea b, (5.1)

24

S1A[notdef]S1C[notdef]R3[notdef]R6 [similarequal] C [notdef]C . In M theory

JHEP06(2014)027

! 1.

[notdef] : (ea, eb) [mapsto]!eaeb = ea[circledot]b. (5.2)

ah and v = e

eah + ebh[parenrightBig]

, (5.3)

a [circledot]h b = a + b. (5.4)

Let us denote the real line R with these operations by Sh, then (Sh, h, [circledot]h) is a semiring.

For any positive values of h the semiring (Sh, h, [circledot]h) is isomorphic to (R+, +, [notdef]) with

conventional addition and multiplication (under the exponential isomorphism given above sending a 2 Sh to exp(a/h)). What makes the operations h and [circledot]h interesting, however,

is that they have a good limit as h vanishes, namely

a 0 b = lim

h!0

eah + ebh[parenrightBig]= max(a, b), (5.5)

a [circledot]0 b = a + b. (5.6)

The semiring (S0, 0, [circledot]0) di ers from (R+, +, [notdef]), in particular, it is idempotent since

a 0 a = a.

Litvinov and Maslov formulated a broad correspondence principle in which (Sh, h, [circledot]h) with nonzero h is viewed as a quantum object, while (S0, 0, [circledot]0) is its classical limit; thus

the name dequantization. According to this principle various statements over (Sh, h, [circledot]h) have corresponding statements over (S0, 0, [circledot]0).

For some given Newton polygon N, let F (u, v) =

P(m,n)2N Fm,numvn be its corresponding family of real polynomials and let us look at the family of curves F (u, v) = 0 in R2+ with the coe cients Fm,n parameterizing this family and explore its dequantization. To begin with, we let Fm,n = exp(fm,n/h) and F (u, v) = exp(f(a, b)/h), then in S0 this algebraic relation becomes f(a, b) = max(m,n)2N [notdef]ma + nb + fm,n[notdef] . An ana

logue of a real algebraic variety given by F (u, v) = 0 is the set where the function f(a, b) = max(m,n) (ma + nb + fm,n) is not di erentiable,5 it is called the tropical curve.

In fact, a more elegant form of dequantization by using hyperelds (see [52] and [53]).

The function f(a, b) = max(m,n) (ma + nb + fm,n) is linear on every connected component of the complement of the tropical curve. A geometric way of viewing this is by considering an arrangement of planes c = ma + nb + fm,n in three-space R3 with coordinates (a, b, c). Given such a plane arrangement there is a hull or crystal which is formed by the closure of the set of all points above all of these planes C = [notdef](a, b, c)[notdef]c ma+nb+fm,n[notdef].

Our spectral curves are complex, so a dequantization of C is in order. Hyperelds again provide appropriate language, however, for our limited purposes here, the dequantization of the real line as described above will be su cient.

5.2 Amoebas, melting crystals, and the Kahler potential

To see clearly what is happening in the limit h ! 0 it is useful to introduce an amoeba [54]

associated to a curve = [notdef](s, t)[notdef]s 2 C , t 2 C ,

P(m,n)2N Gm,nsmtn = 0[notdef]. If we view 5See [51] for the exact description and the relation to Ragsdale conjecture and Hilberts 16th problem.

25

So a b = ln ea + eb

the following operations on R :

a h b = h ln

h ln

[parenrightbig]

and a[circledot]b = a+b. More generally, letting u = e

bh induces

JHEP06(2014)027

C [notdef] C as a direct product of two cylinders C [notdef] C = (R [notdef] S1) [notdef] (R [notdef] S1) = R2 [notdef] T 2,

then we can project the curve C [notdef] C onto the rst factor in R2 [notdef] T 2 viaLog : C [notdef] C ! R2 (5.7)

(s, t) [mapsto]!(log [notdef]s[notdef], log [notdef]t[notdef]). (5.8) The amoeba A of the curve is the image of this curve under this map: A = Log( ).

A slightly di erent approach to amoebas and tropical varieties is via the Ronkin function [55], dened by averaging log [notdef]G(s, t)[notdef] of the polynomial G(s, t) =

P(m,n)2N Gm,nsmtn

along the torus factor:

dt

t . (5.9)

In the vicinity of any point (a, b) such that the torus T 2(a,b) does not intersect the curve this function is linear in a and b, so one can dene the amoeba A as the set of points where

the Ronkin function is not linear. Moreover, each connected component of the complement of the amoeba is a convex region of the (a, b)-plane R2. It corresponds to one of the integer points (m, n) of the Newton polygon of G(s, t), and the Ronkin function is linear on this connected component with the slope equal to (m, n).

Now let us keep track of the sizes of the two cylinders in the above picture. C [notdef] C = R [notdef] S1x [notdef] R [notdef] S1y, where the two circles S1x and S1y are parameterized respectively by

coordinates x and y each with the same period 2h. Then

s = e(a+ix)/h and t = e(b+iy)/h. (5.10)

For the curve G(s, t) = 0 dene a rescaled Ronkin function

r(a, b) := hR(a, b) = h [integraldisplay]

0

Consider the region H above the graph of the function r(a, b)

H = [notdef](a, b, c)[notdef]c r(a, b)[notdef] . (5.12)

In the limit h = 0 this region is exactly the crystal C of the plane arrangement

c = ma + nb + gm,n

[vextendsingle][vextendsingle]

[bracerightbig]

consisting of the closure of all points above all of these planes. For nonzero h the region H is that same crystal C with melted corners a melted crystal.

We shall distinguish the planes corresponding to perimeter points of the Newton polygon and those corresponding to the internal points. For each perimeter point (m, n) 2 @N

26

JHEP06(2014)027

R(a, b) =

dy

h . (5.11)

In these (a, b) coordinates the region of nonlinearity of r(a, b) shrinks as h is sent to zero, and the tropical variety dened above can be viewed as a classical limit of the amoeba. It is called the skeleton of the amoeba A. Moreover, in the limit limh!0 r(a, b) =

max(m,n)2N(ma + nb + gm,n), where [notdef]Gm,n[notdef] = exp

gm,n

h

[integraldisplay][notdef]x[notdef]=a,[notdef]y[notdef]=b

2h

log [notdef]G(s, t)[notdef]

ds s

[integraldisplay]

2h

0 log [notdef]G(e(a+ix)/h, e(b+iy)/h)[notdef]

dx h

[parenrightbig]

(m, n) 2 N

.

,

with the G(s, t) coe cient Gm,n we call the plane c = ma + nb + log [notdef]Gm,n[notdef] the (m, n)-

perimeter plane or just the perimeter plane. For an internal point (m, n) 2 Int N we call

the plane c = ma + nb + log [notdef]Gm,n[notdef] an internal plane. As dened above for G(s, t), the

crystal C is the domain above all perimeter and internal planes:

C = [notdef](a, b, c)[notdef]c ma + nb + log [notdef]Gm,n[notdef] , (m, n) 2 N[notdef] . (5.13)

The complete crystal Cper is the domain above all of the perimeter planes only:

Cper = [notdef](a, b, c)[notdef]c ma + nb + log [notdef]Gm,n[notdef] , (m, n) 2 @N[notdef] . (5.14)

The melted crystal H is the domain above the Ronkin function:

H = [notdef](a, b, c)[notdef]c R(a, b)[notdef]. (5.15) These denitions and the convexity of the Ronkin function imply that H C Cper. Let us call the volume of Cper \ H the region above the perimeter planes and under the

Ronkin function the melted volume and denote it by Vmelt, and the volume of Cper \ C the regularized volume and denote it by Vreg. Since H C, Vmelt > Vreg while both are

functions of the coe cients Gm,n of the polynomial G(s, t).

We can now formulate the conjecture that the leading part of the Kahler potential on the moduli space of a monopole wall with the Newton polygon N is given by the melted volume Vmelt. The perimeter coe cients are the parameters, and they are held xed. The internal coe cients Gm,n constitute half of the moduli space coordinates and Vmelt is a function of their absolute values. This structure of the asymptotic Kahler potential implies existence of asymptotic isometries of the moduli space metric.

5.3 The low-dimensional test

Let us illustrate the conjecture in the case of a four-dimensional monowall moduli space. In this case the Newton polygon N has only one internal point; by shifts let us arrange this internal point to be at the origin, so that its coe cient, the modulus, is G0,0. The corresponding plane L0,0 is horizontal, positioned at the hight := log [notdef]G0,0[notdef], which is one of the mono-

wall moduli. Let us denote the remaining three periodic moduli by 1 = Arg G0,0, 2, and .

As is the case for other monopole moduli spaces with only four moduli, asymptotically the moduli space metric approaches one with a triholomorphic isometry. Any such asymptotic metric has the form

Ud2 + (d + !)2

U , (5.16)

where U is a harmonic function of , 1, 2 and ! is a one-form on the same space. For monopoles on R3 the function U behaves as 1/r; for periodic monopoles it behaves as log r.

In our case of doubly periodic monopoles, with only one noncompact coordinate, U has to be linear and thus has the form U = c0 + c1. The one form ! is dual to U, i.e. dU = d!,

which in our case implies e.g. ! = c1 1d 2. Since the -circle is to form a bration over the two-torus of 1 and 2, the constant c1 has to be integer.

27

JHEP06(2014)027

According to the conjecture

. (5.19)

In these expressions we are using the conventions of [54] with the area of a basic simplex [(0, 0), (1, 0), (1, 0)] being equal to 1 instead of 1/2. Thus AreaGKZ is twice the conventional area Area : AreaGKZ = 2Area.

Let us add to this two more geometric facts: Picks formula

1

2AreaGKZ(N_) = Area(N_) = Int(N_) +

and the perimeter relation for reexive polygons6

Perim(N) + Perim(N_) = 12. (5.21)

Thus, in our case with a single internal point, we nd that AreaGKZ(N_) = Perim(N_) = 12 Perim(N), and the asymptotic metric of eq. (5.16) has

U = d2

d2 Vmelt = 2AreaGKZ(N_) + O(0) = (24 2Perim(N)) + O(0). (5.22)

Now we have all of the needed geometric ingredients in order to compare to the gauge theory computations of [20] and [26], we recall the relation between the Newton polygon

6See [19] for three intriguing explanations of this fact.

28

U = d2

d2 Vmelt, (5.17) up to terms exponentially small in . Now let us focus on the leading behavior of Vmelt for

large :1. In the limit of a large internal coe cient the Ronkin function is well approximated by the maximum of the tropical planes; thus in this limit Vmelt [similarequal] Vreg.

2. The cross section Cper \ L0,0, which is the horizontal face of the crystal C, at high

modulus approaches a log [notdef]G0,0[notdef] multiple of the polar polygon

N_ = [notdef]v[notdef](v, w) > 1, 8w 2 N[notdef]. (5.18) To be exact, at some su ciently large value of = log [notdef]G0,0[notdef] the cross section Cper \ L0,0 is

a polygon bounded by lines parallel to the lines in the denition (5.18) of the polar polygon N_. As we increase and, accordingly, move up the plane L0,0, these lines are moved away from the center at constant rates. Thus, the larger the value of the less signicant the initial positions of the lines become. Asymptotically, the cross section approaches N_.

3. One other way of seeing the previous statement is by rst considering the situation when all the perimeter planes are passing through the origin. In this case Cper is a cone, and the volume of that cone below L0,0 is exactly Vreg = 13AreaGKZ(N_) (log [notdef]C0,0[notdef])3 . Now,

moving one of the perimeter planes up or down by a nite distance changes Vreg by amount

at most quadratic in , therefore

Vreg = 13AreaGKZ(N_) (log [notdef]G0,0[notdef])3 + O (log [notdef]G0,0[notdef])2 [parenrightbig]

JHEP06(2014)027

1

2Perim(N_) 1 (5.20)

integer perimeter length Perim(N) and the number Nf of quarks in the corresponding SU(2) gauge theory: Nf = Perim(N) 4. This gives

U = (16 2Nf) + const, (5.23)

in perfect agreement with Seibergs one loop gauge theory result of eq. (4.1).

5.4 Secondary fan and the phase space

For a given monowall, its moduli space depends on the boundary data consisting of the asymptotic behavior at z ! [notdef]1 and the positions of the positive and negative singularities.

We call these parameters. Parameters are distinct from moduli: if moduli parameterize L2 deformations and give coordinates on the moduli space, the parameters determine the shape of the moduli space and correspond to perturbations of the solution that are not L2. For generic values of parameters the moduli space is smooth, however, as we change these parameters the space can degenerate and undergo some drastic changes. We would like to describe the structure of the space of parameters. In particular, we would like to understand the phase structure, i.e. to describe the walls on which degenerations can occur and the connected domains within which the moduli spaces are di eomorphic (though not necessarily isometric). Here we restrict our discussions to the tropical limit, relying heavily on the beautiful combinatorics and geometry of [54].

Near the tropical limit the amoeba is very close to its skeleton. A generic skeleton, in turn, is dual to some regular triangulation of the Newton polygon. A triangulation is regular if it can be constructed in the following architectural manner. Use the Newton polygon as a oor plan and erect columns of various hights at each integer point of the Newton polygon. Next, through a canvas over these columns and stretch it down by its perimeter. As a result you have built a tent over the Newton polygon; the roof of this tent is piecewise linear (with some columns supporting it and, perhaps, some not even reaching the roof) with linear planar pieces joining at edges. Projection of these edges gives a regular subdivision of the Newton polygon. Generically, this subdivision is a triangulation (unless the tops of four of more columns lie in a common plane). Such triangulations are called regular triangulations. Every regular subdivision is dual to a skeleton of some amoeba, and vice versa. Few examples are given in gure 5.

As we vary the moduli (the internal column heights), holding the parameters (the perimeter column heights) xed, we can go through a series of triangulations, each describing a region of the moduli space.

For example, the Newton polygon of the E2 monowall admits ten regular triangulations listed in gure 6. Triangulations listed in the rst line are using only the perimeter points as vertices. Such triangulations are special; we call them associahedral triangulations.

What is the relation between the (tropical) spectral polynomial and the corresponding triangulation? The answer to this question is given in [54] in terms of the secondary polyhedron (N) that is constructed as follows. Given a Newton polygon, consider all of its regular triangulations. Each triangulation T r determines a vector [vector]vTr in a d-dimensional space, where d is the number of integer points of the Newton polygon N. Namely, if

29

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Figure 5. Some regular triangulations and Corresponding Amoebas.

(a) (5,1,4,4,1,0) (b) (3,4,2,5,1,0) (c) (1,5,2,4,3,0) (d) (1,3,4,2,5,0) (e) (3,1,5,2,4,0)

(f) (1,3,2,2,3,4) (g) (4,1,3,3,1,3) (h) (3,1,3,2,2,4) (i) (2,2,2,2,2,5) (j) (3,2,2,3,1,4)

Figure 6. All regular triangulations of E2 with their ve-vector.

e1, . . . , ed are the integer points of N, then the i-th component vi of the vector vTr equals to the area covered by all of the triangles of T r for which ei is a vertex. (For simplicity, to keep all vector coordinates integer, the area of a basic simplex is taken to be 1 instead of 12.)

vi =

For example, for the vertex numbering of gure 7, the six-vectors of the triangulation are given under each triangulation in gure 6.

Clearly, the vectors are constrained, namely

1. the sum of vector coordinates is thrice the area of N :

triangle area contributes tree times.

2. the sum of vector components weighted by the corresponding integer point

is 23Area(N) times of the center of mass of N.

30

X 2T rvi2Vert

AreaGKZ( ) (5.24)

Pdi=1 vi = 3Area N, since every

Pdi=1 eivi

3

6 4

2

1

Figure 7. Newton polygon of E2 monowall with labelled vertices.

(a) Vertex Rising (b) Flop

Figure 8. Two possible changes relating neighbouring regular triangulations.

Thus all [vector]vTr are lying in a d 3 hyperspace. Moreover, they are vertices of a convex

polyhedron (N) called the secondary polyhedron of N. The edges of this polyhedron connect triangulations related by two kinds of possible transitions of gure 8.

For example, the secondary polyhedron of E2 monowall is shown in gure 9. The biggest face of the secondary polyhedron (N) is formed by the vertices corresponding to the triangulations that do not use any of the internal points of N - the associahedral triangulations [56]. In other words this face is the secondary polyhedron of the perimeter of N, (Perim(N)). It is called the associahedron of Perim(N), thus we call this the associahedral face.

Now, consider the normal fan Fan(N) of (N). It is called the secondary fan. Its rays are outward normals to the faces of (N) and it divides the space of all coe cients [notdef]Gm,n[notdef]

into cones, each cone of maximal dimension is labelled by the corresponding triangulation. Two cones share a face is the corresponding vertices in (N) are connected by its edge. All coe cient values forming a vector in a given cone Tr have the tropical curve dual to the cones corresponding triangulation T r of N.

What is important to us is that we have two kids of coe cients: 1. the perimeter coe cients, that are the parameters, and 2. the internal points coe cients, that are the moduli. Starting with some point we keep the parameters xed, while varying the moduli. Thus, as we change the moduli, we shall be crossing various cones. Our goal is to identify the phases, i.e. to divide the space of parameters into domains, such that within each domain the sequence of cones we cross as we vary the moduli is the same. Any two points in parameter space belong to the same phase, only if every point on the interval connecting them has the same sequence of cones crossed when we vary the moduli.

From the geometric picture above it is clear that we should project the secondary fan Fan(N) onto the associahedral plane along the moduli subspace. The resulting projection of Fan(N) is a new fan Ph(N). We call it the phase fan of N. Each point in a given cone of Ph(N) corresponds to the same sequence of triangulations given by the cones of Fan(N) above it. For the E2 example this projection is given by the vitruvian diagram in gure 10.

31

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Figure 9. The secondary polyhedron (of the monowall with Newton polygon of gure 7) together with the rays of its normal fan. The vertices correspond to the regular triangulations of gure 6.

5.5 Comparison

Let us now turn to the Seiberg and Morrison results of [23] comparing the phase structure of the ve-dimensional supersymmetric En eld theories with extreme transitions in Calabi-

Yau manifolds. Our phase diagram of gure 10 reproduces the E2 eld theory diagram [23, gure 1].7 The upward ray is indeed governed by the E1 Newton sub-polygon of E2, while the direct downward ray by that of E0 sub-polygon. The slanted downward rays correspond to[notdef]1. It is straightforward to identify D1 diagram in gure 11a with SU(2) theory at nite coupling with a massless fundamental multiplet, by considering its dual

7For the exact comparison, we recall that, instead of (t0, m) coordinates, [23, gure 1] is drawn in (m0, m) coordinates, with m0 = t0 2[notdef]m[notdef].

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Figure 10. This is the phase diagram Ph(N) of the E2 monopole wall. The sequence of triangulations within each cone is to be read outwards, so that it corresponds to increasing the modulus.

as a (p,q)-brane network. Sending the mass to innity amounts to moving the horizontal ray in the network downwards, removing the left upper point of D1 and leaving the D0 diagram of gure 11b. Similarly, removing that ray upwards, removing the left lower point of D1 and leaves the ~D0 diagram of gure 11c. The asymptotic rays of the (p,q)-network correspond to the same theory as D0 but at a di erent Chern-Simons level. Thus Ph(N)

33

(a) D1 (b) D0 (c)

Figure 11. D-type diagrams in the E2 monowall phase space.

describing the phase structure of a monowall at the same time describes the phase structure of the ve-dimensional supersymmetric eld theories.

What is the relation to the del Pezzo diagram [23, gure 2]? The phase structure Ph(N) emerged by projecting the secondary fan Fan(N) of gure 9 onto the associahedral plane. Some of the cones of the secondary fan, namely those that are downwards directed in gure 9, correspond to the vertices of the associahedral face. Call these the associahedral cones of Fan(N). The del Pezzo diagram [23, gure 2] is the subfan of Fan(N) consisting of all of its non-associahedral cones. In terms of the secondary polygon, diagram [23, gure 2] and its cross-section [23, gure 3] are dual to the graph formed by the top (i.e. non-associahedral) skeleton of the secondary polytope.

To summarize, for a monowall with the Newton polygon N, the phase structure of its moduli space (in tropical limit) is given by the projection of Fan(N) on the associahedral subspace along the moduli subspace (such as e.g. [23, gure 1]), i.e. along the directions corresponding to the internal points of N. In order to obtain the local Calabi-Yau extremal transitions diagram (such as [23, gure 2]), restrict Fan(N) to a subfan consisting of cones that are not associahedral.

Acknowledgments

This work was supported in part by a grant from the Simons Foundation (#245643 to SCh). The author is grateful to Oleg Viro for illuminating discussions, to John Schwarz for comments on the manuscript, to the Simons Center for Geometry and Physics and to the Caltech Particle Theory group for hospitality.

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/

Web End =CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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