Table Of ContentSeiberg-WiNen Theory
and Integrable Systems
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Seiberg-Witfen Theory
and Integroble Systems
Andrei Marshahov
P N Lebedev Physics Institute and
Institute of Theoretical and Experimental Physics, Russia
World Scientific
Singapore • New Jersey • London • Hong Kong
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World Scientific Publishing Co. Pte. Ltd.
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SEIBERG-WnTEN THEORY AND INTEGRABLE SYSTEMS
Copyright © 1999 by World Scientific Publishing Co. Pte. Ltd.
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ISBN 981-02-3636-0
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1
Contents
1 Introduction 4
2 SUSY Yang-Mills theories 9
2.1 Gauge theories: perturbation theory and instantons 9
2.2 SUSY Yang-Mills theories as vacua of string theory 13
2.3 Seiberg-Witten effective theory 18
3 Integrable systems 26
3.1 Integrable systems: main definitions 26
3.2 KP and Toda lattice hierarchies 28
3.3 Finite-gap solutions: the Krichever construction 32
3.4 Hitchin systems 38
3.5 Deformation of the finite-gap solutions and
tau-function of Whitham hierarchy 43
4 Integrable equations in 2D topological string theories 52
4.1 Topological theories: prepotential and WDVV equations . .. 53
4.2 Topological gravity and topological string theory 64
4.3 2D quantum gravity 67
5 The Seiberg-Witten anzatz 71
5.1 The Seiberg-Witten map 71
5.2 SU(2) pure gauge theory 74
5.3 The periodic Toda chain and pure gluodynamics . . . . . . .. 76
2
5.4 Integrable deformations of the Toda chain: coupling to the ad
joint and fundamental matter 84
5.5 5D SUSY gauge theories and relativistic integrable systems . . 94
5.6 XYZ spin chain and the Sklyanin algebra . 99
6 Generating differential and Whitham hierarchy 105
6.1 Symplectic structure of the finite-gap systems 105
6.2 Geometric properties of generating differential Ill
6.3 SW theory and solutions to Whitham equations . . . . . . . . . 115
6.4 Whitham hierarchy and renormalization group equations . . . 125
7 Prepotential of the Seiberg-Witten Theory 144
7.1 The associativity equations 145
7.2 The proof of the associativity equations 148
7.3 Algebraic construction of generic associativity equations . . .. 152
7.4 Perturbative examples 161
8 Seiberg-Witten theory from strings 182
8.1 Strings, M-theory and compactifications 182
8.2 D-branes 183
8.3 Hyperelliptic curves as part of space-time geometry 187
8.4 Type HB and type IIA integrable systems 190
8.5 Compactification to 3+1 (compact) dimensions . . . . . . . .. 195
8.6 N=l degenerate spectral curve and solitons in
periodic Toda chain 198
3
9 Conclusion 204
A Riemann surfaces and Theta-functions 206
B KP hierarchy and theory of free fermions 216
C Residue formula for the N=2 Calogero-Moser system 222
D Algebra of differentials for the Calogero-Moser system 226
E Explicit derivation in elliptic case 233
4
1 Introduction
Quantum field theory (QFT), or, more strictly, quantum theory of gauge fields
has already demonstrated that it is an adequate language for the physics of ele
mentary particles and their interactions and it certainly has much wider scope
of applications (see for example [1]). Most of the well-known results of QFT
were obtained, however, in the regime of weak coupling - where the constant
of interaction is small and in the zeroth approximation the theory can be for
mulated in terms of free fields or infinite systems of harmonic oscillators. The
expansion of perturbation theory in coupling constant allows to compute with
high accuracy all physical effects in Abelian gauge theory - quantum electro
dynamics as well as many effects in the theory of weak and strong interactions,
governed already by non Abelian gauge fields (see for example [2]).
Nevertheless, QFT as perturbation theory is far from being complete phys
ical theory. First of all, one of the most intriguing problems of the theory of
strong interaction - quantum chromodynamics (QCD) - the confinement of
quarks in mesons and hadrons occur in the regime when coupling constant
is big and interaction is strong, i.e. in the regime which is far from naive
perturbation theory. Second, even from pure internal or theoretical point of
view it is clear that perturbation theory is not complete and, even in the weak
coupling phase, there may be contributions non-analytic in coupling constant.
Such sort of remarks immediately lead to a conclusion that one should try to
search for the consistent formulation of QFT beyond the perturbation theory.
There are many different approaches and investigations in this direction.
Neither of them can be distinguished at the moment and considered as defi
nitely successful. Moreover, there is no consistent theory for the QFT in the
strong coupling regime. It is even more exciting on such background that it
is possible to achieve sufficient progress in understanding how some quantum
5
theories of gauge fields behaves in the strong coupled phase. Of course that
was done only for restricted set of theories and, moreover, only for certain
phenomena in these theories, containing, however, a very important ingredi
ent of effective actions for the light (or massless) sector - the sector which is
observable from macroscopic point of view. Another important point that this
effective action (or the set of coupling constants) can be determined exactly,
i.e. the coupling constants are found as exact (and known) functions of the
parameters of the theory - the ultraviolet couplings and vacuum expectation
values of fields.
The approach to nonperturbative quantum field theory to be discussed be
low is inspired by nonperturbative string theory or M-theory * [5]. The basic
concept is that the physically interesting quantum field theories could be con
sidered as various vacua of M-theory and the stringy symmetries or dualities
may relate spectra and correlation functions in one QFT with those in another
QFT; then string duality allows in principle to establish the correspondence
between the perturbative regime in one model with the nonperturbative in
another one.
This general idea till the moment was put to more solid ground only for the
case of complex backgrounds in string theory (see for example [3] for details)
which are equivalent to supersymmetric (SUSY) quantum field theories [4].
In such theories physical data of the model (masses, couplings, etc) can be
considered as functions on moduli spaces of complex manifolds and the duality
symmetry can be regarded as action of a modular group. Despite some progress
1In the literature this theory is called in many different ways: Theory of Everything
(TOE), M-theory etc. We will not go into the terminological details, preserving the "old"
name of String theory (sometimes with the capital letter) for the TOE which partially - in
some region of coupling constants - can be formulated in the language of one-dimensional
objects (strings) and summation over two-dimensional geometries instead of paths (see [1, 3]
and references therein).
