Results in Localization for Supersymmetric Gauge Theories

UNIVERSITATISACTA UPSALIENSIS

UPPSALA 2020

Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1956

Results in Localization for

Supersymmetric Gauge Theories

ANASTASIOS GORANTIS

ISSN 1651-6214 ISBN 978-91-513-0989-7 urn:nbn:se:uu:diva-416756

Dissertation presented at Uppsala University to be publicly examined in Häggsalen,

Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, Monday, 28 September 2020 at 09:00 for the degree of Doctor of Philosophy. The examination will be conducted in English. Faculty examiner: Professor Kimyeong Lee (School of Physics, Korea Institute for Advanced Study).

Abstract

Gorantis, A. 2020. Results in Localization for Supersymmetric Gauge Theories. Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1956. 136 pp. Uppsala: Acta Universitatis Upsaliensis. ISBN 978-91-513-0989-7.

The strong coupling dynamics of Quantum Field Theories with gauge symmetries constitutes a profound problem in Theoretical Physics. Supersymmetric theories offer rare instances where this elusive problem is tractable and can be a valuable source of information and intuition. One of the most powerful approaches available for this purpose is supersymmetric localization. In this thesis, we employ this technique to study many facets of supersymmetric theories.

After a brief theoretical introduction to the basic concepts that make an appearance in this thesis, we begin with a localization computation for supersymmetric gauges theories on spheres of variable dimension, with eight and four supersymmetries, proving an earlier conjecture. We also analytically continue our results to get a partition function for the four- dimensional N=1 theory, and perform various consistency checks. Then, we consider a special case of these theories, namely two-dimensional Yang-Mills gauge theory, and we study the matrix model that arises from its localization. We analyze the theory in the limit of large gauge group rank using numerical and analytical tools and connect it to previous related results from the literature. Next, we turn our attention to N=2 gauge theories with matter. We construct the theories on a general class of four-dimensional manifolds, ensuring that they are globally well-defined. Then, we reformulate them using twisted variables and perform localization on the resulting cohomological theory. Lastly, we study N=2 theories on another four-dimensional manifold, namely a product of a three-dimensional hyperbolic space and a circle, with the goal of finding infinite-dimensional chiral algebras. To pursue this goal, we once again employ a localization technique and find evidence of this structure in two cases.

Keywords: Supersymmetric localization, Matrix models, Supersymmetric field theories, Cohomological field theories, Chiral algebras

Anastasios Gorantis, Department of Physics and Astronomy, Theoretical Physics, Box 516, Uppsala University, SE-751 20 Uppsala, Sweden.

© Anastasios Gorantis 2020 ISSN 1651-6214

ISBN 978-91-513-0989-7

urn:nbn:se:uu:diva-416756 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-416756)

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IVL \PM [MKWVL ZM^QM_[ QV \PM KI[M WN \PM X]JTQ[PML _WZS[ WZ LM[KZQJM[ QV \PM KI[M WN ]VX]JTQ[PML WVM[ \PM LM^MTWXUMV\[ UILM QV W]Z ZM[MIZKP MNNWZ\[

<PM ZM[\ WN \PQ[ \PM[Q[ Q[ [\Z]K\]ZML I[ NWTTW_[ 1V +PIX\MZ  _M JMOQV _Q\P IV QV\ZWL]K\QWV \W \PM \MKPVQY]M WN []XMZ[aUUM\ZQK TWKITQbI\QWV ?M M`XTIQV PW_ \W KWV[\Z]K\ I []XMZ[aUUM\ZQK \PMWZa WV I K]Z^ML UIVQNWTL IVL XZM[MV\

\PM TWKITQbI\QWV IZO]UMV\ <PMV QV +PIX\MZ  _M KWV\QV]M _Q\P IV M`XW[Q

\QWV WN I NM_ JI[QK \WWT[ NWZ \PM [\]La WN UI\ZQ` UWLMT[ _PQKP WN\MV IZQ[M NZWU TWKITQbI\QWV 1V XIZ\QK]TIZ _M KWVKMV\ZI\M WV \PM IVITa[Q[ WN \PM [ILLTMXWQV\

MY]I\QWV WN \PM UI\ZQ` UWLMT[ 1V +PIX\MZ  _M KWVKT]LM W]Z QV\ZWL]K\WZa UI\MZQIT QV 8IZ\ 1 _Q\P I XZM[MV\I\QWV WN \PM UIQV QLMI[ WN \WXWTWOQKIT \_Q[\QVO QV []XMZ[aUUM\ZQK \PMWZQM[ ?M LMÅVM \PM VW\QWV WN I \WXWTWOQKIT Y]IV\]U ÅMTL \PMWZa IVL \PMV XZWKMML \W M`XTIQV \PM \_Q[\QVO XZWKML]ZM NWZ N = 2

\PMWZQM[

?M JMOQV 8IZ\ 11 _PMZM VW^MT LM^MTWXUMV\[ IZM XZM[MV\ML _Q\P I ZM^QM_ WN 8IXMZ 1 QV +PIX\MZ  1V \PQ[ _WZS _M MUXTWa TWKITQbI\QWV \W KITK]TI\M \PM XIZ

\Q\QWV N]VK\QWV NWZ KMZ\IQV OI]OM \PMWZQM[ QV [XPMZM[ WN ^IZQW][ VW\ VMKM[[IZQTa QV\MOMZ LQUMV[QWV[ d XZW^QVO I KWVRMK\]ZM NWZU]TI\ML QV C5QVIPIV E

)N\MZ LMÅVQVO \PM \PMWZa WV \PM [XPMZM[ _Q\P8 []XMZKPIZOM[ WV d ≤ 5 IVL 4 WV d ≤ 3 _M [M\ ]X \PM TWKITQbI\QWV KWUX]\I\QWV <PMV _M MUJIZS WV KWUX]\QVO \PM []XMZLM\MZUQVIV\[ Ja KWUX]\QVO \PM [XMK\ZI WN \PM ZMTM^IV\

WXMZI\WZ[ 4I[\Ta _M XMZNWZU IV IVITa\QK KWV\QV]I\QWV WN \PM \PMWZa _Q\P 4 []XMZKPIZOM[ NWZ \PM KI[M WN \PM NW]Z[XPMZM IVL KWVL]K\ KPMKS[ NWZ \PM ZM[]T\

<PMV _M XZWKMML _Q\P +PIX\MZ  _PMZM _M XZM[MV\ ]VX]JTQ[PML _WZS WV

\_WLQUMV[QWVIT UI`QUITTa []XMZ[aUUM\ZQK AIVO·5QTT[ <PQ[ MNNWZ\ TM^MZIOM[

\PM ZM[]T\[ WN 8IXMZ 1 \W QV^M[\QOI\M \PM UI\ZQ` UWLMT IZQ[QVO NZWU \PM TWKITQbI

\QWV WN \PM INWZMUMV\QWVML \PMWZa ?M [\]La \PM \PMWZa ][QVO IVITa\QKIT IVL V]UMZQKIT IXXZWIKPM[ IVL ÅVL M^QLMVKM WN I XPI[M \ZIV[Q\QWV ?M KWVKT]LM



\PM KPIX\MZ _Q\P I LQ[K][[QWV WN W]Z XZMTQUQVIZa ZM[]T\[ IVL [SM\KP \PM VM`\

[\MX[ NWZ \PQ[ XZWRMK\

6M`\ QV +PIX\MZ  _M ZM^QM_ 8IXMZ 11 _PMZM _M \ISM WV I [TQOP\Ta LQNNMZ

MV\ ZM[MIZKP LQZMK\QWV IVL M`MK]\M I KWPWUWTWOQKIT \_Q[\ WN I KTI[[ WNN = 2 OI]OM \PMWZQM[ _Q\P UI\\MZ M`\MVLQVO \PM _WZS WN C.M[\]KKQI M\ IT E )N\MZ LMÅVQVO \PM \PMWZa WV I OMVMZIT KTI[[ WN NW]ZUIVQNWTL[ _M QV\ZWL]KM I VW^MT XZWRMK\WZ IK\QVO WV [XQVWZ[ QV \PM [XQZQ\ WN \PM ¹ÆQXXQVO XZWRMK\WZ[º WN C.M[\]K

KQI M\ IT E IVL XMZNWZU \PM \_Q[\QVO 0I^QVO \ZIV[TI\ML \PM \PMWZa QV \PM KWPWUWTWOQKIT TIVO]IOM _M XZWKMML \W TWKITQbM \PM ZM[]T\QVO \PMWZa ][QVO IV QVLM` KWUX]\I\QWV

.QVITTa QV +PIX\MZ  _M XZM[MV\ ]VX]JTQ[PML _WZS WV IVW\PMZ IXXZWIKP \W N = 2 []XMZ[aUUM\ZQK \PMWZQM[ _PMZM _M [MIZKP NWZ KPQZIT ITOMJZI[ IVITWOW][

\W \PW[M WN C*MMU 4MUW[ M\ IT E QV H³× S¹ <PM KPWQKM WN \PQ[ XIZ\QK]

TIZ [XIKM Q[ M`XTIQVML QV \PM KPIX\MZ IVL PI[ \W LW _Q\P Q\[ Q[WUM\Za OZW]X ?M JMOQV Ja UW\Q^I\QVO \PM _WZS IVL ZM^QM_QVO \PM VW\QWV WN I KPQZIT ITOMJZI QV KWVNWZUIT ÅMTL \PMWZa ?M IT[W ZMKITT \PM KWV[\Z]K\QWV WN \PM KPQZIT ITOMJZI QV R<sup>4</sup>QV C*MMU 4MUW[ M\ IT E <PMV _M [\IZ\ \PM M`XW[Q\QWV WN W]Z XZMTQU

QVIZa ZM[]T\[ WV \PM \WXQK ?M LMÅVM I []XMZ[aUUM\ZQK \PMWZa WV H³× S¹ _M M`IUQVM \PM []XMZITOMJZI WN \PM \PMWZa IVL XZWKMML \W LMÅVM \PM \_Q[\ML []

XMZITOMJZI ?M []J[MY]MV\Ta MUXTWa I TWKITQbI\QWV \MKPVQY]M IXXMIZQVO IT[W QV C,ML][PMVSW 8]N]  AIKWJa  E IVL C*WVM\\Q  :I[\MTTQ  E \PI\

ITTW_[ ][ \W ZMKW^MZ KPQZIT ITOMJZI[ ^QI I \MKPVQY]M LQNNMZMV\ NZWU \PM WZQOQVIT _WZS C*MMU 4MUW[ M\ IT E ?M ÅVQ[P \PM KPIX\MZ _Q\P IV M`IUQVI

\QWV WN \PM TQVM WXMZI\WZ[ \PI\ KW]TL JM QVKWZXWZI\ML QV W]Z NZIUM_WZS IVL I LQ[K][[QWV WN \PM XZWRMK\ IVL Q\[ N]\]ZM LQZMK\QWV[



8IZ\ 1"

1V\ZWL]K\WZa UI\MZQIT

 )V JZQMN QV\ZWL]K\QWV \W []XMZ[aUUM\ZQK TWKITQbI\QWV

1V \PQ[ KPIX\MZ _M XZM[MV\ I [PWZ\ OMVMZIT QV\ZWL]K\QWV \W \PM \MKPVQY]M WN []

XMZ[aUUM\ZQK TWKITQbI\QWV _PQKP _M _QTT ][M \PZW]OPW]\ \PM \PM[Q[ ?M JMOQV QV ;MK\QWV  Ja JZQMÆa ZM^QM_QVO \_W KWUUWV IXXZWIKPM[ \W LMÅVM I []XMZ

[aUUM\ZQK ÅMTL \PMWZa WV I K]Z^ML UIVQNWTL <PMV _M XZWKMML QV ;MK\QWV 

_Q\P I XZM[MV\I\QWV WN \PM IZO]UMV\[ JMPQVL []XMZ[aUUM\ZQK TWKITQbI\QWV IVL I LQ[K][[QWV WN \PM UIQV [\MX[ QV I TWKITQbI\QWV KWUX]\I\QWV <PM UW[\ KWU

XTM\M [W]ZKM WN QVNWZUI\QWV WV \PM []JRMK\ Q[ XZWJIJTa \PM ZMKMV\ ZM^QM_ C8M[\]V M\ IT E _PQTM XMLIOWOQKIT QV\ZWL]K\QWV[ \W \PM []JRMK\ KIV JM NW]VL QV \PM TMK\]ZM VW\M[ C*MVQVQ E C+ZMUWVM[Q E IVL C5IZQVW E

 :QOQL []XMZ[aUUM\ZQK ÅMTL \PMWZQM[ QV K]Z^ML [XIKM\QUM

*MNWZM [\IZ\QVO \W LQ[K][[ []XMZ[aUUM\ZQK TWKITQbI\QWV WVM VMML[ \W XW[[M[[ I []XMZ[aUUM\ZQK \PMWZa WV I K]Z^ML UIVQNWTL I[ UW[\ TWKITQbI\QWV KWUX]\I

\QWV[ IZM XMZNWZUML WV KWUXIK\ K]Z^ML [XIKM[ \PM K]Z^I\]ZM IK\[ I[ I K]\WNN NWZ QVNZIZML LQ^MZOMVKM[ <PQ[ KIV JM LWVM Ja \ZIV[NMZZQVO I ÅMTL \PMWZa SVW_V QV ÆI\ [XIKM <W JMOQV \PQ[ XZWKM[[ _M KIV QV[MZ\ \PM K]Z^ML UM\ZQK _PMZM^MZ _M MVKW]V\MZ \PM ÆI\ UM\ZQK QV \PM WZQOQVIT \PMWZa IVL KW^IZQIV\QbM ITT LMZQ^I\Q^M[

.WZ I []XMZ[aUUM\ZQK \PMWZa \PQ[ [PW]TL JM LWVM NWZ JW\P \PM 4IOZIVOQIV IVL

\PM ÅMTL ^IZQI\QWV[ ) N]VLIUMV\IT ZMY]QZMUMV\ NWZ I []XMZ[aUUM\ZQK ÅMTL \PM

WZa Q[ \PI\ \PM ^IZQI\QWV WN Q\[ 4IOZIVOQIV _Q\P ZM[XMK\ \W \PM []XMZKPIZOM[ Q[ I

\W\IT LMZQ^I\Q^M"

δ^(f)L^(f)= ∂μ(. . .)^μ. 

_PMZM \PM []XMZ[KZQX\ (f) LMVW\M[ ÆI\ [XIKM Y]IV\Q\QM[ 0W_M^MZ _PMV WVM XMZNWZU[ \PM[M KPIVOM[ \W \ZIV[NMZ \PM \PMWZa \W I K]Z^ML UIVQNWTL WVM MVL[

]X _Q\P I ÅMTL \PMWZa \PI\ NIQT[ \W JM []XMZ[aUUM\ZQK"

δ^(c)L^(c) = ∇μ(. . .)^μ, 

_PMZM \PM []XMZ[KZQX\ (c) QVLQKI\M[ \PM K]Z^ML [XIKM IVITWO[ WN \PM ÆI\ [XIKM Y]IV\Q\QM[ WJ\IQVML Ja ZMXTIKQVO \PM UM\ZQK IVL KW^IZQIV\QbQVO \PM LMZQ^I\Q^M[

?M _QTT XZM[MV\ \_W IXXZWIKPM[ \W [WT^QVO \PQ[ Q[[]M <PM ÅZ[\ WVM [MMS[

\W ZMUMLa \PM XZWJTMU Ja \ZMI\QVO \PM ÆI\ [XIKM ^IZQI\QWV _Q\P \PM UM\ZQK



IVL LMZQ^I\Q^M ZMXTIKMUMV\[ PMVKMNWZ\P LMVW\ML Ja δ^(c) I[ \PM bMZW\P WZLMZ KWV\ZQJ]\QWV QV IV M`XIV[QWV WN \PM K]Z^I\]ZM [KITM r WN \PM UIVQNWTL"

δ = δ^(c)+

n≥1

1

rⁿδ⁽ⁿ⁾ 

IVL [QUQTIZTa NWZ \PM 4IOZIVOQIV"

L = L^(c)+

n≥1

1

rⁿL⁽ⁿ⁾, 

_PMZML^(c)Q[ \PM ÆI\[XIKM 4IOZIVOQIV _Q\P Q\[ UM\ZQK IVL LMZQ^I\Q^M[ ZMXTIKML

.WTTW_QVO \PQ[ UM\PWL WVM PI[ \W KPMKS \PM KTW[]ZM WN \PM []XMZ[aUUM\Za IT

OMJZI IVL MV[]ZM \PI\ \PM 4IOZIVOQIV Q[ []XMZ[aUUM\ZQK WZLMZ Ja WZLMZ <PM KIZMN]T ZMILMZ _QTT VW\QKM \PI\ \PM ]XXMZ TQUQ\ WN \PM []U[ QV \PM \_W MY]I\QWV[

IJW^M _I[ WUQ\\ML <PM []U PI[ \W \MZUQVI\M I\ [WUM ÅVQ\M WZLMZ JMKI][M r PI[ LQUMV[QWV[m]⁻¹ IVL I\ [WUM XWQV\ WVM Z]V[ W]\ WN ZMTM^IV\ WXMZI\WZ[

NZWU \PM \PMWZa NWZ J]QTLQVOL⁽ⁿ⁾

7N KW]Z[M \PQ[ XZWKML]ZM Q[ VW\ O]IZIV\MML \W []KKMML 1N WVM M`PI][\[ ITT \PM ZMTM^IV\ WXMZI\WZ[ WN \PM \PMWZa IVL \PM []XMZ[aUUM\Za ITOMJZI LWM[ VW\ KTW[M WZ \PM 4IOZIVOQIV Q[ VW\ []XMZ[aUUM\ZQK \PMV \PM UIVQNWTL LWM[ VW\ ILUQ\ I []XMZ[aUUM\ZQK ÅMTL \PMWZa _Q\P \PQ[ ÅMTL KWV\MV\

<PQ[ IXXZWIKP _PQTM [\ZIQOP\NWZ_IZL PI[ [WUM LQ[IL^IV\IOM[ .QZ[\ WN ITT

Q\ PI[ \W JM ZMXMI\ML NWZ MIKP UIVQNWTL .]Z\PMZUWZM QV ITT SVW_V M`IUXTM[

_PMZM \PQ[ UM\PWL _I[ []KKM[[N]T Q\ []NÅKML \W ][M ÅZ[\ IVL [MKWVL WZLMZ KWZ

ZMK\QWV[ QV \PM M`XIV[QWV[ NWZ \PM ^IZQI\QWV[  IVL 4IOZIVOQIV  ZM[XMK

\Q^MTa <PM XMZ\]ZJI\Q^M IXXZWIKP LM[KZQJML IJW^M LWM[ VW\ [MMU \W JM IJTM \W XZW^QLM I R][\QÅKI\QWV NWZ \PQ[ WJ[MZ^I\QWV

;WUM WN \PM [PWZ\KWUQVO[ WN \PQ[ UM\PWL OM\ ILLZM[[ML QV IVW\PMZ IXXZWIKP

X]\ NWZ_IZL QV C.M[\]KKQI  ;MQJMZO E <PM I]\PWZ[ WN \PQ[ XIXMZ []OOM[\

\PI\ \W J]QTL I []XMZ[aUUM\ZQK \PMWZa WV I K]Z^ML [XIKM\QUM WVM [PW]TL KW]XTM

\PM WZQOQVIT \PMWZa \W [WUM []XMZOZI^Q\a \PMWZa IVL \ISM \PM [WKITTML ¹ZQOQL TQUQ\º <PQ[ KWZZM[XWVL[ \W"

Œ \ISQVO \PM TQUQ\ WN ^IVQ[PQVO 6M_\WV¼[ KWV[\IV\ G_N

ΠNZMMbQVO \PM UM\ZQK \W \PI\ WN \PM LM[QZML [XIKM IVL

Œ Å`QVO \PM I]`QTQIZa ÅMTL[ \W IXXZWXZQI\M JIKSOZW]VL ^IT]M[ LMXMVLQVO WV \PM V]UJMZ WN []XMZKPIZOM[ \PI\ _M _IV\ \W XZM[MZ^M

<PM KWVLQ\QWV _M VMML \W QUXW[M Q[ \PI\ ITT NMZUQWVQK ÅMTL[ WN \PM OZI^Q\a U]T\Q

XTM\ I[ _MTT I[ \PMQZ ^IZQI\QWV[ [PW]TL JM MY]IT \W bMZW <PQ[ KWVLQ\QWV OQ^M[ ZQ[M

\W \PM OMVMZITQbML 3QTTQVO [XQVWZ MY]I\QWV[ _PQKP IZM I [M\ WN ÅZ[\ WZLMZ XIZ\QIT LQN

NMZMV\QIT MY]I\QWV[ \PI\ KWV\IQV \PM JW[WVQK ÅMTL[ WN \PM OZI^Q\a U]T\QXTM\ <PM []XMZ[aUUM\ZQK \PMWZa KIV JM X]\ WV I [XMKQÅK K]Z^ML UIVQNWTL WVTa QN _M KIV ÅVL VWV^IVQ[PQVO [XQVWZ[ \PI\ [I\Q[Na \PM OMVMZITQbML 3QTTQVO [XQVWZ MY]I\QWV[

<PM [WT]\QWV[ \W \PM OMVMZITQbML 3QTTQVO [XQVWZ MY]I\QWV[ [XIV I ^MK\WZ [XIKM WN LQUMV[QWV MY]IT \W \PM V]UJMZ WN []XMZKPIZOM[ \PI\ \PM JIKSOZW]VL ILUQ\[



1N _M [WT^M \PM OMVMZITQbML 3QTTQVO [XQVWZ MY]I\QWV[ IVL LMZQ^M \PM JIKS

OZW]VL OZI^Q\a U]T\QXTM\ ÅMTL[ \PI\ [I\Q[Na \PMU _M KIV OM\ \PM LM[QZML []XMZ

[aUUM\ZQK \PMWZa Ja \PM ZQOQL TQUQ\ WN \PM KWUJQVML []XMZOZI^Q\a IVL WZQOQVIT

\PMWZa <PM ZM[]T\QVO \PMWZa _QTT KWV\IQV \PM JIKSOZW]VL ^IT]M[ WN \PM []XMZ

OZI^Q\a ÅMTL[ IVL \PM []XMZ[aUUM\Za ^IZQI\QWV WN Q\[ 4IOZIVOQIV _QTT [I\Q[Na"

δL = ∇μ(. . .)^μ. 

1\[ []XMZ[aUUM\Za \ZIV[NWZUI\QWV[ NWZU I []JITOMJZI WN \PM TWKIT []XMZ[aU

UM\Za ITOMJZI IVL \P][ \PM []XMZ[aUUM\Za ITOMJZI _QTT KTW[M _Q\PW]\ IVa IL

LQ\QWVIT MNNWZ\

6W\M \PI\ [WUM WN \PM NWZU]TI[ LMZQ^ML ][QVO \PQ[ UM\PWL KIV JM ZM][ML NWZ W\PMZ JIKSOZW]VL[ \PI\ [I\Q[Na \PM [IUM 3QTTQVO [XQVWZ MY]I\QWV[ QV KWV\ZI[\ \W

\PM ÅZ[\ \MKPVQY]M _PMZM WVM VMML[ \W ZMXMI\ \PM IVITa[Q[ NWZ M^MZa VM_ KI[M

=[QVO \PM ZQOQL []XMZOZI^Q\a IXXZWIKP WN C.M[\]KKQI  ;MQJMZO E _M KIV IT[W [WT^M \PM ZQLLTM WN _Pa \PM M`XIV[QWV[  IVL  \MZUQVI\M I\ ÅZ[\

IVL [MKWVL WZLMZ QV1/r ZM[XMK\Q^MTa <PM M`XIV[QWV KIV JM \PW]OP\ WN I[ IV M`XIV[QWV QV \PM OZI^Q\a U]T\QXTM\ I]`QTQIZa ÅMTL[ <PM[M ÅMTL[ MV\MZ \PM []XMZ

[aUUM\Za \ZIV[NWZUI\QWV[ WVTa TQVMIZTa IVL \PM 4IOZIVOQIV WVTa Y]ILZI\QKITTa

PMVKM \PM \Z]VKI\QWV WN \PM M`XIV[QWV[  IVL 

 <PM []XMZ[aUUM\ZQK TWKITQbI\QWV IZO]UMV\

 1V\ZWL]K\WZa ZMUIZS[

5IVa XZWJTMU[ QV Y]IV\]U ÅMTL \PMWZa IUW]V\ \W KWUX]\QVO M`XMK\I\QWV

^IT]M[ WN WVM WZ UWZM WXMZI\WZ[ <PQ[ Q[ WN\MV [\I\ML QV \MZU[ WN IV QVÅVQ\M

LQUMV[QWVIT XI\P QV\MOZIT"

O₁. . .On = 1 Z

ˆ

[Dφ] O₁. . .One^−S[{φ}], 

_PMZM Oi IZM WXMZI\WZ[ QV \PM \PMWZa φ LMVW\M KWTTMK\Q^MTa \PM ÅMTL[ QV \PM

\PMWZa[Dφ] Q[ [WUM QV\MOZI\QWV UMI[]ZM IVL Z Q[ \PM XIZ\Q\QWV N]VK\QWV WN \PM

\PMWZa"

Z= ˆ

[Dφ] e^−S[{φ}]. 

<PM XI\P QV\MOZIT Q[ IV WJRMK\ \PI\ Q[ VW\WZQW][Ta LQNÅK]T\ \W NWZUITQbM UI\PM

UI\QKITTa IVL IT[W ^MZa LQNÅK]T\ \W KWUX]\M M[XMKQITTa QV QV\MZIK\QVO ÅMTL \PMW

ZQM[ 7^MZ \PM aMIZ[ ^IZQW][ \MKPVQY]M[ PI^M JMMV LM^MTWXML QV WZLMZ \W MVIJTM IVL NIKQTQ\I\M XI\P QV\MOZIT KWUX]\I\QWV[ <PM[M \MKPVQY]M[ ZIVOM NZWU XMZ\]Z

JI\Q^MLQIOZIUUI\QK \W M`IK\ \PM TI\\MZ JMQVO \PM NWK][ WN \PQ[ \PM[Q[

7VM WN \PM UW[\ XW_MZN]T M`IK\ KWUX]\I\QWVIT \MKPVQY]M[ NWZ XI\P QV\M

OZIT[ Q[ \PI\ WN []XMZ[aUUM\ZQK TWKITQbI\QWV <PQ[ UM\PWL Q[ I \aXM WN IV MY]Q^

IZQIV\ TWKITQbI\QWV \PM QV\MZM[\ML ZMILMZ KIV KWV[]T\ \PM UI\PMUI\QKIT TQ\MZ

I\]ZM WV \PM []JRMK\ NWZ QV[\IVKM C*MZTQVM  >MZOVM ! E C,]Q[\MZUII\ 



0MKSUIV ! E IVL C)\QaIP  *W\\ ! E <PM ÅZ[\ MIZTa [QOV[ WN \PM ][M

N]TVM[[ WN TWKITQbI\QWV QV \PM <PMWZM\QKIT 8Pa[QK[ TQ\MZI\]ZM IXXMIZML QV ! 

_PMV -L_IZL ?Q\\MV ÅZ[\ ][ML Q\ QV PQ[ XIXMZ WV []XMZ[aUUM\ZQK Y]IV\]U UMKPIVQK[ C?Q\\MV ! E ?Q\\MV KWV\QV]ML ]\QTQbQVO [QUQTIZ KWVKMX\[ QV PQ[

XIXMZ[ WV \PM \_WLQUMV[QWVIT \WXWTWOQKIT [QOUI UWLMT C?Q\\MV ! JE NW]Z

LQUMV[QWVIT \WXWTWOQKIT OI]OM \PMWZQM[ C?Q\\MV ! IE IVL TI\MZ UQZZWZ [aU

UM\Za C?Q\\MV !!E IVL \_WLQUMV[QWVIT OI]OM \PMWZQM[ C?Q\\MV !!E )V

W\PMZ QUXWZ\IV\ UQTM[\WVM QV \PM M^WT]\QWV WN \PM []JRMK\ Q[ \PM ][M WN \PQ[

\MKPVQY]M Ja 6QSQ\I 6MSZI[W^ QV \PM KWUX]\I\QWV WN \PM QV[\IV\WV XIZ\Q\QWV N]VK\QWV C6 ) 6MSZI[W^ E J]QTLQVO WV XZQWZ _WZS QV C4W[M^ 5WWZM

6MSZI[W^  ;PI\I[P^QTQ !!E C4W[M^ 6MSZI[W^  ;PI\I[P^QTQ !! E C4W[

[M^ 6MSZI[W^  ;PI\I[P^QTQ !!!E IVL C5WWZM 6MSZI[W^  ;PI\I[P^QTQ

JE .QVITTa \PM XIXMZ \PI\ WXMVML \PM ÆWWLOI\M[ NWZ \PM ÅMTL _I[ C8M[

\]V E _PMZM >I[QTa 8M[\]V ][ML \PM \MKPVQY]M \W KWUX]\M \PM XIZ\Q\QWV N]VK\QWV IVL \PM KQZK]TIZ ?QT[WV TWWX M`XMK\I\QWV ^IT]M NWZN = 2 []XMZ AIVO·

5QTT[ WV S⁴IVL [PW_ML \PI\ \PM \MKPVQY]M KIV JM ][ML JMaWVL \WXWTWOQKIT ÅMTL

\PMWZQM[

 0W_ \W XMZNWZU I TWKITQbI\QWV KWUX]\I\QWV

4M\ ][ VW_ \]ZV \W M`XTIQVQVO PW_ WVM KIV ][M \PM \MKPVQY]M +WV[QLMZ I Y]IV

\]U ÅMTL \PMWZa \PI\ XW[[M[[M[ I []XMZKPIZOM Q WZ UWZM OMVMZITTa I /ZI[[

UIVVWLL KPIZOM _PW[M [Y]IZM [I\Q[ÅM["

Q²= B,  

_PMZM B Q[ [WUM JW[WVQK KPIZOM 1\ KIV JM I KWUJQVI\QWV WN OI]OM OTWJIT IVL [XIKM\QUM [aUUM\ZQM[ ?M _QTT ZM[\ZQK\ W]Z I\\MV\QWV \W \PM [M\ WN OI]OM QV^IZQIV\ WXMZI\WZ[ \PI\ [I\Q[Na \PM NWTTW_QVO XZWXMZ\a"

QO*8;= 0. !

<PM[M WXMZI\WZ[O*8;IZM KITTML *8; WXMZI\WZ[ 1\ Q[ \PM M`XMK\I\QWV ^IT]M[ WN M`

IK\Ta \PM[M WXMZI\WZ[ \PI\ WVM KIV KWUX]\M ][QVO []XMZ[aUUM\ZQK TWKITQbI\QWV"

O*8; = 1 Z

ˆ

[Dφ] O*8;e^−S[{φ}]. 

6W\M \PI\ NZWU VW_ WV _M _QTT WUQ\ \PM VWZUITQbI\QWV WN \PM XI\P QV\MOZIT Ja

\PM XIZ\Q\QWV N]VK\QWV \W I^WQL ^Q[]IT KT]\\MZ <PM IZO]UMV\ JMTW_ Q[ QLMV\QKIT _PMV WVM ZM[\WZM[ \PM NIK\WZ1/Z \W \PM M`XZM[[QWV[

6W_ TM\ ][ LMNWZU \PM IK\QWV QV \PM XI\P QV\MOZIT Ja I QM`IK\ \MZU"

O*8;t= ˆ

[Dφ] O*8;e−S[{φ}]−tQV. 



<PM M`XZM[[QWV QV Q[ KWUUWVTa ZMNMZZML \W I[ \PM TWKITQbQVO IK\QWV <PMV WVM KIV [PW_ \PI\O*8;tQV NIK\ LWM[ VW\ LMXMVL WV t"

d

dtO*8;t= − ˆ

[Dφ] O*8;(QV )e−S[{φ}]−tQV

= − ˆ

[Dφ] Q

O*8;V e−S[{φ}]−tQV

= 0. 

)\ \PQ[ XWQV\ _M PI^M I[[]UML \PI\ \PM [aUUM\Za Q Q[ VW\ IVWUITW][ IVL PMVKM \PM QV\MOZI\QWV UMI[]ZM Dφ Q[ QV^IZQIV\ ]VLMZ Q ?M PI^M IT[W I[[]UML

\PMZM IZM VW JW]VLIZa \MZU[ )LLQ\QWVITTa WVM KIV VW_ ]VLMZ[\IVL _Pa _M NWK][ML W]Z MNNWZ\[ WV *8; WXMZI\WZ[" QO = 0 Q[ KZ]KQIT NWZ W]Z IZO]UMV\

;W [QVKMO*8;tQ[ tQVLMXMVLMV\ _M KIV M^IT]I\M \PM XI\P QV\MOZIT NWZ I

^IT]M WN t WN W]Z KPWQKM ?M ][]ITTa KPWW[M \W \ISM \PM TQUQ\ t → +∞ IVL LMUIVL \PI\ \PM JW[WVQK XIZ\ WN QV Q[ OZMI\MZ WZ MY]IT \W bMZW <PQ[ UMIV[

\PI\ UIQV KWV\ZQJ]\QWV NWZ \PM QV\MOZIT _QTT KWUM NZWU \PM [ILLTM XWQV\[ WN QV  KWUUWVTa ZMNMZZML \W I[ \PM TWKITQbI\QWV TWK][

<PM KIVWVQKIT KPWQKM NWZ \PM TWKITQbQVO IK\QWV Q["

LTWK = Q

{λ}

(Qλ)^†λ+ λ^†(Qλ^†)^†

, 

_PMZM _M []U W^MZ \PM NMZUQWVQK ÅMTL KWV\MV\ WN \PM \PMWZa <PM JW[WVQK XIZ\

WN \PM TWKITQbQVO IK\QWV Q[ \PMV



{λ}

|Qλ|²+ |Qλ^†|²



IVL Q\ Q[ XW[Q\Q^M [MUQLMÅVQ\M ?Q\P \PQ[ KPWQKM WN TWKITQbQVO IK\QWV \PM TWKIT

QbI\QWV TWK][ KWV[Q[\[ WN \PM NWTTW_QVO KWVÅO]ZI\QWV["

λ= 0, λ^†= 0, Qλ= 0, Qλ^†= 0. 

<PM VM`\ [\MX Q[ \W M`XIVL W]Z ÅMTL[ IZW]VL \PM TWKITQbI\QWV TWK]["

φ= φ₀+√1

tφ^, 

_Q\P φ^ LMVW\QVO \PM Æ]K\]I\QWV[ 1V \PM TQUQ\ t → +∞ \PM WVTa \MZU[ \PI\

[]Z^Q^M NZWU \PM IZO]UMV\ WN \PM M`XWVMV\QIT QV \PM XI\P QV\MOZIT  IZM

\PW[M \PI\ IZM VW\ []XXZM[[ML Ja XW_MZ[ WN1/√

t ;W _M MVL ]X _Q\P IV M`IK\

[MUQKTI[[QKIT WVMTWWX KWUX]\I\QWV _Q\P ZM[XMK\ \W \PM XIZIUM\MZ t"

O*8; = ˆ

{φ0}[Dφ₀]O*8;

{φ0}e^−S[φ0]Z_TWWX[{φ₀}]

= ˆ

{φ0}[Dφ0]O*8;

{φ0}e^−S[φ0] 1

;,M\

δ²STWK[{φ0}]

δφ²₀

, 

!

_PMZM _M QV\MOZI\M W^MZ \PM TWKITQbI\QWV TWK][ IVL ;,M\ LMVW\M[ \PM []XMZ

LM\MZUQVIV\ WN \PM Æ]K\]I\QWV[ ?M PI^M _Q\VM[[ML I []J[\IV\QIT ZML]K\QWV QV

\PM QV\MOZI\QWV [XIKM" NZWU \PM WZQOQVIT QVÅVQ\MLQUMV[QWVIT [XIKM WN ÅMTL[ \W

\PM *8; TWK][ <PM [QbM WN \PQ[ [XIKM Q[ LM\MZUQVML Ja \PM [XIKM\QUM LMXMV

LMVKM WN \PM ÅMTL[ \PI\ KWUXZQ[M Q\ <PQ[ KIV JM I Y]IV\]U ÅMTL \PMWZa WN TW_MZ LQUMV[QWVITQ\a WZ QV \PM UW[\ LZIUI\QK KI[M[ I bMZWLQUMV[QWVIT UI\ZQ` UWLMT

KWZZM[XWVLQVO \W WZLQVIZa ÅVQ\MLQUMV[QWVIT QV\MOZIT[ 1V \PM TI\\MZ KI[M WVM Q[ IJTM \W ][M \PM \WWT[ _M _QTT QV\ZWL]KM QV +PIX\MZ  \W [\]La \PM \PMWZa

<PM TI[\ [\MX \W IZZQ^M \W IV M`XTQKQ\ M`XZM[[QWV QV \PM M^IT]I\QWV WN \PM XI\P QV\MOZIT Q[ \PM KWUX]\I\QWV WN \PM []XMZLM\MZUQVIV\ QV  <PMZM M`Q[\ \_W JI[QK IXXZWIKPM[ \W IKKWUXTQ[P \PQ[ <PM ÅZ[\ _PQKP _M MUXTWa QV 8IXMZ 1 Q[ \W KWUX]\M \PM [XMK\ZI WN \PM WXMZI\WZ[ \PI\ IXXMIZ QV \PM Æ]K\]I\QWV M`XZM[[QWV[

NWZ \PM [XIKM I\ PIVL <PM [MKWVL M`XTWQ\ML QV 8IXMZ 11 MV\IQT[ QV^WSQVO \PM )\QaIP·;QVOMZ QVLM` \PMWZMU I[[]UQVO \PM IK\QWV WN IV MTTQX\QK WXMZI\WZ WZ I OMVMZITQbI\QWV WN \PM \PMWZMU NWZ I \ZIV[^MZ[ITTa MTTQX\QK WVM *W\P IXXZWIKPM[

PI^M \PMQZ UMZQ\[ IVL \PMQZ _MISVM[[M[ ?PMV ][QVO \PM ÅZ[\ IXXZWIKP _M WN\MV WJ[MZ^M UIVa KIVKMTTI\QWV[ 1V \PI\ [MV[M \PM [MKWVL IXXZWIKP Q[ UWZM MNÅKQMV\ [QVKM _M KWVKMV\ZI\M [WTMTa \W \PM UWLM[ \PI\ IZM VW\ OWQVO \W KIVKMT QV[\MIL WN \PM N]TT [XMK\ZI 7V \PM W\PMZ PIVL \PM QVLM` UM\PWL Q[ LQUMV[QWV

LMXMVLMV\^IVL Q[ VW\ _Q\PW]\ []J\TM\QM[

.QVITTa I[ \PM ZMILMZ UQOP\ VW\QKM QV \PM XZWKML]ZM XZM[MV\ML IJW^M \PMZM PI^M JMMV I NM_ XWQV\[ _PMZM _M PI^M I NZMMLWU WN KPWQKM 5WZM [XMKQÅKITTa

_M KIV KPWW[M IVa WN \PM []XMZKPIZOM[ WN \PM \PMWZa I[[]UQVO WN KW]Z[M \PI\

\PM \PMWZa PI[ U]T\QXTM \W TWKITQbM IVL _M KIV KPWW[M LQNNMZMV\ TWKITQbQVO IK

\QWV[ QV[\MIL WN  <PQ[ NZMMLWU Q[ [WUM\QUM[ ZMNMZZML \W I[ ][QVO LQNNMZMV\

TWKITQbI\QWV [KPMUM[ +ITK]TI\QWV[ XMZNWZUML _Q\P LQNNMZMV\ [KPMUM[ UQOP\ aQMTL ZM[]T\[ \PI\ IXXMIZ \W LQ[IOZMM _Q\P MIKP W\PMZ 0W_M^MZ I UWZM KIZMN]T [\]La _QTT ZM^MIT \PI\ \PM LQNNMZMV\ ZM[]T\[ IOZMM _Q\P MIKP W\PMZ ]X \W XI\P QV\MOZIT[ WN QM`IK\ M`XZM[[QWV[ _PQKP [PW]TL JM MY]IT \W bMZW L]M \W I []XMZ[aUUM\ZQK IVITWO WN \PM ;\WSM[¼ \PMWZMU

,QUMV[QWVQVLMXMVLMVKM Q[ I [QVM Y]I VWV NWZ \PM KWUX]\I\QWV QV 8IXMZ 1



 ) JZQMN QV\ZWL]K\QWV \W UI\ZQ` UWLMT[

1V \PQ[ KPIX\MZ _M IZM OWQVO \W XZW^QLM I [PWZ\ QV\ZWL]K\QWV \W \PM IVITa[Q[ WN UI\ZQ` UWLMT[ ?PQTM UI\ZQ` UWLMT[ IZM I ^MZa QV\MZM[\QVO \WXQK IVL PI^M V]

UMZW][ IXXTQKI\QWV[ _M IZM OWQVO \W NWK][ WV \PM [\]La WN UI\ZQ` UWLMT[ NWZ

\PMQZ IXXTQKI\QWV QV \PM []XMZ[aUUM\ZQK TWKITQbI\QWV TQ\MZI\]ZM 5I\ZQ` UWLMT[

IZQ[M QV KMZ\IQV KI[M[ _PMZM \PM [XIKM\QUM LMXMVLMVKM WN \PM TWKITQbI\QWV TWK][

ÅMTL KWVÅO]ZI\QWV[ Q[ \ZQ^QIT <PM QV\MZM[\ML ZMILMZ KIV ÅVL I UWZM LM\IQTML XMLIOWOQKIT QV\ZWL]K\QWV QV C5IZQVW E ?M _QTT JMOQV QV ;MK\QWV  Ja M`XTIQVQVO \PM JI[QK [M\]X WN I UI\ZQ` UWLMT IVL \PMV _M _QTT XZWKMML QV ;MK

\QWV  \W M`IUQVQVO Q\[ [ILLTMXWQV\ MY]I\QWV ][QVO \PM ZM[WT^MV\ \MKPVQY]M

 *I[QK [M\]X

5I\ZQ` UWLMT[ XZM[MV\ IV WXXWZ\]VQ\a \W [\]La OI]OM \PMWZQM[ QV I [QUXTMZ [M\

\QVO I[ \PMa IZM bMZWLQUMV[QWVIT ÅMTL \PMWZQM[ QM \PMa PI^M VW [XIKM\QUM LMXMVLMVKM <PMQZ XIZ\Q\QWV N]VK\QWV PI[ \PM NWTTW_QVO NWZU"

Z = ˆ

[DM] e^−S[M], 

_PMZM M Q[ IV N× N 0MZUQ\QIV UI\ZQ` [DM] Q[ IV IXXZWXZQI\M QV\MOZI\QWV UMI[]ZM IVL S[M] Q[ \PM IK\QWV -`XMK\I\QWV ^IT]M[ WN WXMZI\WZ[ O(M) KIV JM KWUX]\ML QV \PM ][]IT _Ia"

O(M) = 1 Z

ˆ

[DM] e^−S[M]O(M). 

<PM QV\MOZI\QWV UMI[]ZM KWUUWVTa MUXTWaML Q["

[DM] =

N i=1

dM_ii

i<j

d(:MMij)

i<j

d(1UMij), 

_PQTM \PM IK\QWV Q[ OMVMZITTa WN \PM NWZU"

S[M] = 1

2g²<Z M²+ 1

g²V(M), 

_Q\P I XW\MV\QIT V(M)"

V(M) =

k≥3

g_k

k<Z M^k. 



<PM KI[M _PMZM V(M) = 0 Q[ KITTML \PM /I][[QIV UI\ZQ` UWLMT

<PM XZM[MVKM WN \PM \ZIKM QV \PM IK\QWV MVLW_[ ][ _Q\P I OI]OM [aUUM\Za"

M → UMU⁻¹, 

_PMZM U Q[ IVa N×N ]VQ\IZa UI\ZQ` <PQ[ UMIV[ \PI\ _M KIV \ISM IL^IV\IOM WN \PQ[ [aUUM\Za \W ZML]KM \PM N²LMOZMM[ WN NZMMLWU ][QVO I .ILLMM^·8WXW^

XZWKML]ZM 1V M[[MVKM _M IZM [MMSQVO \W LQIOWVITQbM M ^QI I ]VQ\IZa \ZIV[NWZ

UI\QWV  QV\W"

Λ = LQIO(λ₁, . . . , λ_N), 

ZML]KQVO W]Z UWLMT \W N LMOZMM[ WN NZMMLWU ?M JMOQV Ja QV[MZ\QVO I ]VQ\"

ˆ

DΛDUδ^(N2)

U M U⁻¹− Λ

Δ²(Λ) = 1  

QV \PM XI\P QV\MOZIT NWZ \PM XIZ\Q\QWV N]VK\QWV _PQKP JMKWUM["

Z = ˆ N

i=1

dλ_iΔ²(Λ)e^{−iS({λi})}, !

_PMZM _M PI^M WUQ\\ML IV W^MZITT KWV[\IV\ NIK\WZ MY]IT \W \PM ^WT]UM WN \PM ]VQ\IZa OZW]X I[ Q\ [QUXTQÅM[ QV \PM KWUX]\I\QWV WN M`XMK\I\QWV ^IT]M[ 6M`\

_M VMML \W XZW^QLM IV M`XTQKQ\ M`XZM[[QWV NWZΔ²(Λ) ][QVO Q\[ LMÅVQVO MY]I

\QWV   .WZ I Å`ML M  \PM UIQV KWV\ZQJ]\QWV QV \PM QV\MOZIT KWUM[ NZWU

\MZU[ \PI\ ^MZa KTW[M \W \PM ]VQ\IZa UI\ZQ` U \PI\ LQIOWVITQbM[ Q\ <PQ[ XZWUX\[

][ \W XMZNWZU \PM NWTTW_QVO KPIVOM[ WN ^IZQIJTM["

U = (1 + R)U^ 

_Q\P IV QVÅVQ\M[QUIT IV\QPMZUQ\QIV R <PMV \PM LMT\I N]VK\QWV JMKWUM["

δ^(N2)(UMU⁻¹− Λ) = δ^(N2)

(1 + R)U^M U^−1(1 − R) − Λ

= δ^(N2)(RΛ^− Λ^R+ Λ^− Λ), 

_PMZMΛ^ = U^M U^−1 <PM UI\ZQKM[Λ IVL Λ^ IZM LQIOWVIT IVL [W Q[ \PMQZ LQNNMZMVKM 7V \PM W\PMZ PIVL \PM KWUU]\I\WZ[R, Λ^] PI[ [WTMTa WNNLQIOWVIT KWUXWVMV\[ <P][ _M KIV [XTQ\ \PM LMT\I N]VK\QWV QV\W \_W"

δ^(N2)(RΛ^− Λ^R+ Λ^− Λ)) = δ^(N2−N))([R, Λ^])δ^(N)(Λ^− Λ). 

<PMV Ja _ZQ\QVO \PM KWUU]\I\WZ I["

[R, Λ^] = Rij(λ^_j− λ^_i), 

_M KIV ZM_ZQ\M \PM KWVLQ\QWV   I[ NWTTW_["

1 =ˆ 

i<j

d(:M Rij) d(1U Rij) Δ²(Λ^) δ^(N2−N)

R_ij(λ^_j − λ^_i)





IVL ÅVITTa OM\ IV M`XZM[[QWV NWZΔ({λi})"

Δ({λi}) =

i<j

(λj− λi), 

_PQKP _M ZMKWOVQbM I[ \PM >IVLMZUWVLM LM\MZUQVIV\

;W IN\MZ \PM L][\ PI[ [M\\TML _M IZM TMN\ _Q\P \PM NWTTW_QVO XIZ\Q\QWV N]VK\QWV"

Z =ˆ 

i

dλ_i

i<j

(λj− λi)²e^−S({λi})=ˆ 

i

dλ_ie^{−SMNN({λi})}, 

_PMZM _M PI^M QV\ZWL]KML \PM MNNMK\Q^M IK\QWV"

S_MNN({λi}) = 1 2g²



i

λ²_i + 1

g²V({λi}) − 2

i<j

TWO(λi− λj). 

?PQTM UI\ZQ` UWLMT[ IZM \PM [QUXTM[\ OI]OM \PMWZQM[ _M KIV [\]La \PM XZWJ

TMU[ XW[ML Ja \PMQZ IVITa[Q[ IZM [\QTT Y]Q\M LQNÅK]T\ \W \IKSTM QV \PMQZ N]TT OMVMZ

ITQ\a ;W _M _QTT \ISM I PQV\ NZWU \PM O(N²) [KITQVO JMPI^QWZ WN \PM MNNMK\Q^M IK\QWV  IVL \Za \W IVITabM \PM \PMWZa QV \PM ¼\ 0WWN\ TQUQ\ _PMZM N → +∞

IVL λ= g²N = Å`ML C¼\ 0WWN\ !E 1V \PM ¼\ 0WWN\ TQUQ\ \PM XI\P QV\MOZIT Q[ LWUQVI\ML Ja \PM [ILLTM XWQV\[ WN \PM MNNMK\Q^M IK\QWV 0MVKM \PM MY]I\QWV _M VMML \W [\]La Q["

∂S_MNN

∂λ_i = 0,  

WZ 1

g²λ_i= 2

j=i

1

λ_i− λj − 1 g²

∂V

∂λ_i. !

6W\M \PI\ \PM [ILLTMXWQV\ MY]I\QWV _PQTM IK\]ITTa LM[KZQJQVO \PM JMPI^QWZ WN \PM MQOMV^IT]M[ WN \PM UI\ZQ` UWLMT KIV IT[W JM QV\MZXZM\ML QV \PM NWTTW_QVO _Ia 7VM KIV \PQVS WN MIKP MQOMV^IT]M λ_iI[ I XIZ\QKTM ]VLMZ \PM QVÆ]MVKM WN I XW\MV\QIT V IVL I TWOIZQ\PUQK QV\MZIK\QWV JM\_MMV \PM MQOMV^IT]M[ ,MXMVLQVO WV \PM QV\MZXTIa IVL \PM ZMTI\Q^M UIOVQ\]LM WN \PM[M \_W NIK\WZ[ _M KIV MVL ]X _Q\P LQNNMZMV\ KWVÅO]ZI\QWV[ NWZ \PM MQOMV^IT]M[XIZ\QKTM[ 1V XIZ\QK]TIZ

I[ _M _QTT LQ[K][[ TI\MZ QV \PM KI[M _PMZM \PM XW\MV\QIT V PI[ UWZM \PIV WVM UQVQU]U \PM MQOMV^IT]M[ KIV [XTQ\ QV\W OZW]X[

 1V^M[\QOI\QVO \PM [ILLTMXWQV\ MY]I\QWV

<PM [\IZ\QVO XWQV\ NWZ \PM [\]La WN \PM [ILLTMXWQV\ MY]I\QWV Q[ \PM QV\ZWL]K\QWV WN \PM MQOMV^IT]M LMV[Q\a LQ[\ZQJ]\QWV ρ(z)"

ρ(z) := 1 N

N i=1

δ(z − zi) 



_PMZM z_i IZM \PM MQOMV^IT]M[ \PI\ [I\Q[Na \PM [ILLTMXWQV\ MY]I\QWV^ 1V \PM ¼\

0WWN\ TQUQ\ _M KIV ZMXTIKM \PM []U _Q\P IV QV\MOZIT W^MZ I KWV\W]Z C"

1 N

N i=1

→ ˆ

C

dz. 

?M _QTT IT[W LMUIVL \PI\ \PM LQ[\ZQJ]\QWV N]VK\QWV Q[ VWZUITQbML"

ˆ

C

dz ρ(z) = 1. 

<PM QV\MOZI\QWV KWV\W]Z C Q[ \ISMV W^MZ \PM QV\MZ^IT[ WN ÅVQ\M []XXWZ\ WN ρ(z)

1N _M WVTa PI^M WVM []KP QV\MZ^IT _M IZM LQ[K][[QVO I WVMK]\ [WT]\QWV# W\PMZ_Q[M _M IZM LQ[K][[QVO I U]T\QK]\ [WT]\QWV

*a MUXTWaQVO MY]I\QWV  _M ZM_ZQ\M \PM [ILLTMXWQV\ MY]I\QWV I[ NWT

TW_["

1 λz+ 1

λ

∂V

∂z = 2 dz^ ρ(z^)

z− z^. 

?M _QTT VW_ XZWKMML \W [MIZKP NWZ I [WT]\QWV WN \PM [ILLTMXWQV\ MY]I\QWV ÅZ[\

Ja I[[]UQVO \PI\ ITT \PM MQOMV^IT]M[ z_iTQM WV I [QVOTM QV\MZ^IT IVL \PMV OMVMZ

ITQbM \W \PM KI[M _PMZM \PM MQOMV^IT]M[ TQM WV U]T\QXTM QV\MZ^IT[

 7VMK]\ [WT]\QWV

?M JMOQV Ja QV\ZWL]KQVO \PM ZM[WT^MV\"

ω(z) := 1 N



i

1

z_i− z, 

WZ QV \PM KWV\QV]W][ TQUQ\"

ω(z) = ˆ

R

dz^ ρ(z^)

z^− z. 

<PQ[ Q[ I N]VK\QWV _PQKP Q[ IVITa\QK M^MZa_PMZM QV \PM KWUXTM` XTIVM M`KMX\

NWZ \PM TWKI\QWV WN \PM MQOMV^IT]M[ WV IV QV\MZ^IT C ?M KIV IT[W M`\ZIK\ \PM I[aUX\W\QK JMPI^QWZ WN \PM ZM[WT^MV\ ][QVO \PM VWZUITQbI\QWV KWVLQ\QWV WN \PM MQOMV^IT]M LMV[Q\a LQ[\ZQJ]\QWV ρ(z) "

ω(z) −→

z→+∞−1

z. 

?M ZMVIUM W]Z MQOMV^IT]M[ \W ziI^WQL KWVN][QWV _Q\P \PM ¼\ 0WWN\ KW]XTQVO QV \PM KWV\QV]W][

TQUQ\



4M\¼[ VW_ [MM PW_ \W \PM ZM[WT^MV\ MVIJTM[ ][ \W _ZQ\M LW_V I [WT]\QWV NWZ

\PM [ILLTMXWQV\ MY]I\QWV  1N _M M^IT]I\M \PM ZM[WT^MV\ I\ z± i0 _M OM\"

ω(z ± i0) = ˆ

dz^ ρ(z^) z^− z ∓ i0

= dz^ ρ(z^) z^− z ± πi

ˆ

dzρ(z^)δ(z^− z)

= dz^ ρ(z^)

z^− z ± πiρ(z). 

<PMV Ja ][QVO \PM [ILLTMXWQV\ MY]I\QWV "

ω(z + i0) + ω(z − i0) = −1 λz− 1

λV^(z),  I

ω(z + i0) − ω(z − i0) = 2πiρ(z).  J

?M VW_ PI^M I :QMUIVV·0QTJMZ\ XZWJTMU I\ W]Z PIVL[ ;QVKM _M IZM TWWSQVO NWZ I WVMK]\ [WT]\QWV \PM [QVO]TIZ XIZ\ WN \PM ZM[WT^MV\ ωs(z)[PW]TL PI^M \PM NWTTW_QVO NWZU"

ω_s(z) ∼

(z − a1)(z − a2), !

_PMZM a_1,2IZM \PM MVLXWQV\[ WN \PM QV\MZ^ITC <PMV Ja ZMXTIKQVO \PQ[ M`XZM[

[QWV QV  I _M OM\"

ω(z) =

(z − a1)(z − a2) 2πλ

ˆ _a2

a1

dxV^(x) + x x− z

1

(a2− x)(x − a1). 

?M KIV \PMV Å` \PM MVLXWQV\[ a_1,2Ja QV[XMK\QVO \PM I[aUX\W\QK JMPI^QWZ WN \PM ZM[WT^MV\"

ω(z) = z

1 −^a1+a_z ² 2πλ

ˆ _a2

a1

dx V^(x) + x (a₂− x)(x − a₁)



−1 z − x

z²



, 

IVL [W NWZ MIKP XW_MZ WN z _M OM\"

1 2πλ

ˆ _a2

a1

dx V^(x) + x

(a₂− x)(x − a₁) = 0, 

IVL 1

2πλ ˆ _a2

a1

dx x(V^(x) + x)

(a₂− x)(x − a₁) = 1. 

 5]T\QK]\ [WT]\QWV

<PM LQ[K][[QWV WN \PM XZMKMLQVO []J[MK\QWV KIV JM OMVMZITQbML NWZ \PM KI[M WN I [WT]\QWV _Q\P U]T\QXTM K]\[ 5WZM XZMKQ[MTa \PM UW[\ OMVMZIT [WT]\QWV PI[ n K]\[



_PMZM n Q[ [UITTMZ WZ MY]IT \W \PM V]UJMZ WN UQVQUI WN \PM XW\MV\QIT V  1V \PQ[

KI[M \PM MQOMV^IT]M[ IZM LQ[\ZQJ]\ML QV n LQ[RWQV\ QV\MZ^IT[Ci i∈ {1, . . . , n}

?M VW_ ZMY]QZM \PI\ \PM ZM[WT^MV\ ω PI[2n JZIVKP XWQV\[ _PQKP Å`M[ Q\[ NWZU QV\W"

ω(z) := 1 2πλ



²ⁿ

k=1

(z − ak) ˆ

dx(V^(x) + x) x− z

_2n 1

k=1(x − ak)

, 

_PMZM a_kIZM \PM MVLXWQV\[ WN \PM QV\MZ^IT[CiIVL \PM QV\MOZI\QWV [PW]TL JM XMZ

NWZUML WV ITTCi¼[ <PM MVLXWQV\[ IZM []JRMK\ \W \PM NWTTW_QVO n−1 KWVLQ\QWV["

1 2πλ

ˆ

dx x^m(V^(x) + x)

_2n

k=1(x − ak) = δmn, 

_PMZM m= 1, . . . , n <PQ[ UMIV[ \PI\ _M IZM TMN\ _Q\P 2n − (n − 1) = n + 1 WN

\PM QV\MZ^IT MVLXWQV\[ ZMUIQVQVO \W JM Å`ML <PM XW[Q\QWV WN \PM[M XWQV\[ _QTT PI^M \W JM Å`ML ][QVO [WUM ILLQ\QWVIT KWVLQ\QWV []KP I[ X]\\QVO \PM K]\[ I\

MY]QXW\MV\QIT TQVM[ 1\ Q[ IT[W KWUUWV XZIK\QKM \W QV\ZWL]KM \PM ÅTTQVO NZIK\QWV[

f_i"

f_i = ˆ

Ci

dλ ρ(λ), 

[]KP \PI\"

n i=1

f_i = 1. 

<PM ÅTTQVO NZIK\QWV[ QVLQKI\M \PM V]UJMZ WN MQOMV^IT]M[ KWV\IQVML QV MIKP K]\

Ci



 ) JZQMN QV\ZWL]K\QWV \W \WXWTWOQKIT \_Q[\QVO

1V \PQ[ KPIX\MZ _M _QTT UISM I JZQMN QV\ZWL]K\QWV \W \PM \WXQK WN \WXWTWOQKIT Y]IV\]U ÅMTL \PMWZQM[ <9.<[ IVL \PMQZ KWV[\Z]K\QWV NZWUN = 2 []XMZ[aU

UM\ZQK \PMWZQM[ ^QI \PM XZWKML]ZM WN \_Q[\QVO <PM [\]La WN <9.<[ PI[ JMMV WV

\PM UW[\ QUXWZ\IV\ LM^MTWXUMV\[ WN \PM TI[\ LMKILM[ QV UI\PMUI\QKIT XPa[QK[

[QVKM Q\ PI[ ZM^MITML LMMX KWVVMK\QWV[ JM\_MMV Y]IV\]U ÅMTL \PMWZa OMWUM\Za IVL \WXWTWOa IVL PIL [QOVQÅKIV\ QV[QOP\[ \W WNNMZ \W JW\P []JRMK\[

?M JMOQV \PQ[ KPIX\MZ Ja QV\ZWL]KQVO \PM KWVKMX\ WN I \WXWTWOQKIT Y]IV\]U ÅMTL \PMWZa QV ;MK\QWV  ;QVKM W]Z X]ZXW[M Q[ \W WNNMZ IV QV\ZWL]K\QWV \W _WZS KWVL]KML QV 8IXMZ 11 _M _QTT TQUQ\ W]Z XZM[MV\I\QWV \W \PM KI[M NW]Z

LQUMV[QWVIT \PMWZQM[ <PMV QV ;MK\QWV  _M \]ZV \W \PM \_Q[\QVO XZWKML]ZM NWZ N = 2 []XMZ[aUUM\ZQK \PMWZQM[ ) OZMI\ LMIT WN ILLQ\QWVIT QVNWZUI\QWV IVL ZMNMZMVKM[ KIV JM NW]VL QV \PM JWWS C4IJI[\QLI  5IZQVW E

 <WXWTWOQKIT Y]IV\]U ÅMTL \PMWZQM[

+WV[QLMZ I :QMUIVVQIV UIVQNWTL M _Q\P I UM\ZQK gμν 1N _M KWV[\Z]K\ I Y]IV\]U ÅMTL \PMWZa WV \PM UIVQNWTL \PMV \PM \PMWZa _QTT OMVMZITTa LMXMVL WV \PM [XIKM\QUM UM\ZQK g_μν 0W_M^MZ \PMZM M`Q[\ KI[M[ WN \PMWZQM[ \PI\ XW[[M[[

I [MK\WZ _Q\P WJ[MZ^IJTM[ \PI\ IZM QVLMXMVLMV\ WN \PM UM\ZQK"

δ

δg_μνO = 0. 

<PM[M \PMWZQM[ IZM KITTML \WXWTWOQKIT Y]IV\]U ÅMTL \PMWZQM[ <9.<[ ?M KIV LQ[\QV

O]Q[P \_W \aXM[ WN <9.<¼["

Œ ;KP_IZb \aXM <9.<[" \PM[M IZM ÅMTL \PMWZQM[ _PMZM ITT \PM QVOZMLQMV\[ IZM M`XTQKQ\Ta QVLMXMVLMV\ WN \PM [XIKM\QUM UM\ZQK ) KMTMJZI\ML M`IUXTM

\PI\ JMTWVO[ QV \PQ[ KI\MOWZa Q[ +PMZV·;QUWV[ \PMWZa ÅZ[\ XZM[MV\ML Ja

?Q\\MV QV C?Q\\MV ! IE

Œ ?Q\\MV \aXM WZ KWPWUWTWOQKIT <9.<[" \PM[M IZM ÅMTL \PMWZQM[ _PMZM \PM IK

\QWV IVL [WUM WXMZI\WZ[ [MMU \W LMXMVL WV \PM UM\ZQK 0W_M^MZ \PMa XW[[M[[ I [aUUM\Za WN \PM IK\QWV δ"

δS[{φi}] = 0. 

<PQ[ [aUUM\Za MV[]ZM[ \PI\ \PM M`XMK\I\QWV ^IT]M[ WN I [M\ WN WXMZI\WZ[ QV

\PM \PMWZa LW VW\ LMXMVL WV \PM UM\ZQK IVL PMVKM IZM \WXWTWOQKIT <PQ[



KIV JM [MMV I[ NWTTW_[ <PM MVMZOaUWUMV\]U \MV[WZ Q[ LMÅVML I[ \PM

^IZQI\QWV WN \PM IK\QWV _Q\P ZM[XMK\ \W \PM UM\ZQK"

T_μν = δS[{φi}]

δg^μν . 

1N TμνKIV JM ZMVLMZML QV I δM`IK\ NWZU"

T_μν = −iδRμν, 

_PMZM R_μνQ[ [WUM IZJQ\ZIZa \MV[WZ \PMV \PM M`XMK\I\QWV ^IT]M WN IV WX

MZI\WZO \PI\ Q[ QV^IZQIV\ ]VLMZ \PM IK\QWV WN \PM [aUUM\Za JMKWUM["

δO

δg^μν = O Tμν = −iO δRμν = ±iδ(O Rμν) = 0, 

_PMZM _M PI^M I[[]UML \PI\ \PM [aUUM\Za PI[ VW IVWUITa IVL \PI\ \PMZM IZM VW JW]VLIZa \MZU KWV\ZQJ]\QWV[ _PMV XMZNWZUQVO \PM ÅMTL[XIKM QV

\MOZI\QWV Ja XIZ\[ ;]KP I [aUUM\Za Q[ [WUM\QUM[ ZMNMZZML \W I[ \WXWTWOQKIT

1V \PM NWTTW_QVO _M _QTT KWVKMV\ZI\M WV ?Q\\MV \aXM <9.<[ L]M \W \PMQZ ZMTM^IVKM NWZ 8IXMZ 11

.WZ I ?Q\\MV \aXM \PMWZa \PM WJ[MZ^IJTM[ _M IZM QV\MZM[\ML QV IZM \PW[M _PQKP IZM QV^IZQIV\ ]VLMZ \PM [aUUM\Za δ [W \PI\ \PM IZO]UMV\ WN MY]I\QWV 

PWTL[ J]\ KIVVW\ JM _ZQ\\MV \PMU[MT^M[ I[ \PM δ^IZQI\QWV WN IVW\PMZ WXMZI\WZ

QV _PQKP KI[M \PMQZ M`XMK\I\QWV ^IT]M _W]TL JM QLMV\QKITTa bMZW :M[\I\ML QV I UWZM NWZUIT UIVVMZ _M KWVKMV\ZI\M WV WXMZI\WZ[ \PI\ JMTWVO \W \PM KWPW

UWTWOa WN \PM [aUUM\Za WXMZI\WZ δ" 3MZ δ/1U δ PMVKM \PM IT\MZVI\Q^M VIUM KWPWUWTWOQKIT <9.< NWZ \PQ[ \aXM WN \PMWZa 1VLMML \PM [aUUM\Za WXMZI\WZ δ Q[

/ZI[[UIVVQIV QV ITT KI[M[ NW]VL [W NIZ J]\ Q\ Q[ VW\ IT_Ia[ VQTXW\MV\ δ² = 0

1V NIK\ QV UIVa \PMWZQM[ δ [Y]IZM[ \W I KWUJQVI\QWV WN OI]OM IVL OTWJIT [aU

UM\ZQM["

δ² = B, 

TMILQVO ][ \W [\]La WXMZI\WZ[ \PI\ IZM QV^IZQIV\ ]VLMZ \PM IK\QWV WN B MY]Q^IZQIV\

KWPWUWTWOa

 _{N = 2} \WXWTWOQKIT \_Q[\QVO

1V \PQ[ [MK\QWV _M _QTT LM[KZQJM [PWZ\Ta IV QUXWZ\IV\ \MKPVQY]M NWZ WJ\IQVQVO I KWPWUWTWOQKIT <9.< NZWU I NW]ZLQUMV[QWVITN = 2 []XMZ[aUUM\ZQK ÅMTL

\PMWZa QV\ZWL]KML Ja ?Q\\MV QV C?Q\\MV ! IE ZMNMZZML \W I[ \_Q[\QVO

<PM \PMWZa XW[[M[[M[ IV SU(2)+× SU(2)−ZW\I\QWV [aUUM\Za OZW]X IVL IV SU(2)I×U(1) QV\MZVIT [aUUM\Za OZW]X <PM TMN\ IVL ZQOP\PIVLML []XMZ

[aUUM\Za OMVMZI\WZ[ IZM Qαi IVL ¯Q_˙αiZM[XMK\Q^MTa <PMa [I\Q[Na \PM NWTTW_QVO